Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2013 Article ID 239384 10 pageshttpdxdoiorg1011552013239384

Research ArticleGaussian Estimation of One-Factor Mean Reversion Processes

Freddy H Mariacuten Saacutenchez and J Sebastian Palacio

Basic Science Department Eafit University Carrera 49 No 7 Sur 50 Medellin Colombia

Correspondence should be addressed to Freddy H Marın Sanchez fmarinsaeafiteduco

Received 25 April 2013 Accepted 28 August 2013

Academic Editor Aera Thavaneswaran

Copyright copy 2013 F H Marın Sanchez and J S Palacio This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

We propose a new alternative method to estimate the parameters in one-factor mean reversion processes based on the maximumlikelihood technique This approach makes use of Euler-Maruyama scheme to approximate the continuous-time model and builda new process discretized The closed formulas for the estimators are obtained Using simulated data series we compare the resultsobtained with the results published by other authors

1 Introduction

Different applications of stochastic differential equations haveattracted the attention of many researchers in relation to thetheoretical advances needed for modeling and simulationof the dynamic behavior of the main variables affectingthe evolution of some phenomena Applications in fields ofeconomics finance physics insurance and others have beendrivers of these developments Particularly through diffusionprocesses have been carried out modeling interest ratesand commodity prices where they used in their modelingprocesses mean reversion Some trends of these studies arebased on some a priori knowledge to determine the structureof the model to be used while other alternatives are partof a general structure and statistical procedures throughthe particular structure are determined to represent thephenomenon Whichever approach is used in general inthe specification of the model it is necessary to estimateparameters to achieve a proper fit of the model

Although some parameter estimationmethods have beenextensively studied in diffusion processes this has a seriousproblem which is that the specification of the models is donein continuous time but the data that is available for adjustingthe parameters are discretely sampled in time A commonsolution to this problem is to discretize the continuous-timemodel based on Euler-Maruyama scheme and perform pro-cedures on the theoretical formulation resulting in discrete

time From this scheme alternatives can be proposed forparameter estimation approach among which stands out themaximum likelihood estimation

Following the classification proposed by Yu and Phillips[1] different parameter estimation procedures can be clas-sified globally into three categories according to how toproceed from the model formulation discretization of con-tinuous process and estimation of the parameters with theresulting discretized process estimation of the transitionprobability function for the continuous-time model andindependent estimations of the trend and volatility compo-nents by kernel methods

Thus among the methods that are based on the dis-cretization of the continuous-time model is Nowman [2]who considers a stochastic differential equation with twounknown variables in the drift term and two in the volatilityelasticity allowing the variance Similarly Yu and Phillips[1] present an improvement to the method proposed byNowman [2] which is based on a nonuniform sampling ofthe observations and the estimation process makes use ofthe equivalence of the methods of maximum likelihood andleast squares under the conditions laid down in the modelAlternatively Yu [3] proposes a new scheme that is basedon the idea of transforming a stochastic differential equation(EDE) in its discrete-time equivalent an AR(1) to determinethe formulas that approximate the bias error present in theestimation of one of the parameters

2 Journal of Probability and Statistics

Some of the works that could be classified as methodsin which it seeks to find the transition probability functionand use Martingale properties to approximate the likeli-hood function are Durham and Gallant [4] Valdivieso etal [5] and Elerian et al [6] Durham and Gallant [4]propose a scheme based on simulation Establishing differentmodel modifications Pedersen [7] improves computationalperformance and characteristics of the model are studiedusing simulated synthetic series Koulis and Thavaneswaran[8] study combined martingale estimation functions fordiscretely observed interest rates modelsThey derive closed-form expressions for the first four conditional moments ofthe CIR model via Itorsquos formula and for extended versionsof the CIR model using the Milsteinrsquos approximation andconstruct the optimal estimating functions They also showthat combined estimating functions have more informationwhen the conditional mean and variance of the observedprocess depend on the parameter of interest Valdivieso et al[5] proposed a method for estimating the transfer functionthrough the inversion of the characteristic function of theprocess and the use of Fourier transform This methodologyis developed for some particular cases of D-OU processes(Ornstein-Uhlenbeck processes with broken car distributionD) Elerian et al [6] used simulation-based techniquesfor estimating the transition probability depending on thesample to incorporate auxiliary dummy observations Theparameters are obtained by simulation and optimizationusing numerical algorithms

Methods based on kernel techniques approximate densityfunctions of the components and spreading tendency of theprocess The works of Jiang and Knight [9] and Florens-Zmirou [10] are within this framework

In this paper we propose an alternative scheme forparameter estimation in one-factor mean reversion modelsusing the Euler-Maruyama discretization Using the prop-erties of discretized error normality the maximum likeli-hood technique to obtain closed formulas for the estimatorsapplies We present some performance measures of themaximum likelihood estimators obtained with the proposedmethod in simulated processes and we compare them withthe results published by Nowman [2] Yu and Phillips [1] andDurham and Gallant [4]

This paper is organized as follows In Section 2 a briefdescription of mean reversion processes of one factor withconstant parameters Some Gaussian estimation techniquesare presented in Section 3 Section 4 develops the proposedmethod and Section 5 presents some experimental numeri-cal results with different sampling rates

2 Mean Reversion Processes

Mean reversion processes have been considered as a naturalchoice to describe the dynamic behavior of interest rates inVasicek [11] Brennan and Schwartz [12] Cox et al [13] andChan et al [14] daily prices of some commodities such asoil and natural gas in Lari-Lavassani et al [15] have a fun-damental role in modeling spot prices when they exhibitedlong-term trends such as electrical energy (Pilipovic [16])

Mean reversion processes of one factor with constantparameters can be written as

119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

120574

119905119889119861119905 119905 isin [0 119879] (1)

with initial condition 1198830= 119909 where 120572 gt 0 120590 gt 0 120583 isin R

and 120574 isin [0 32] are constants and 119861119905119905ge0

is a unidimensionalstandard Brownianmotion defined on a complete probabilityspace (ΩFP)

The 120572 parameter is called reversion rate 120583 is the meanreversion level or trend of long-run equilibrium 120590 is theparameter associated with the volatility and 120574 determines thesensitivity of the variance to the level of119883

119905

The model (1) is known as CKLS proposed by Chan et al[14] and is a generalization of the models of interest rate inthe short term with constant parameters in which 120574 = 0 12and 1 or 120574 = 32 with 120572 = 0

In general mean reversion processes are stationary Forexample note that (1) can be written in integral form as

119883119905minus 1198830= 120572int

119905

0

(120583 minus 119883119904) 119889119904 + 120590int

119905

0

119883120574

119904119889119861119904 (2)

Taking the expected value 119864[119883119905] minus 119864[119883

0] = 120572 int

119905

0(120583 minus

119864[119883119904])119889119904 and thus we obtain the differential equation

(119905) = 120572 (120583 minus 119898 (119905)) 119898 (119905) = 119864 [119883119905] (3)

The solution of this equation is 119898(119905) = 120583 + (1198830minus 120583)119890minus120572119905

so lim119905rarrinfin

119864[119883119905] = lim

119905rarrinfin119898(119905) = 120583

It can be concluded then that119898(119905) = 120583 is the equilibriumlevel for the linear differential equation In other words theexpected value of the process119898(119905) returns in the long term to120583 In a finite-time horizon 120583 plays the role of attractor in thesense that when 119883

119905gt 120583 the trend term 120572(120583 minus 119883

119905) lt 0 and

therefore 119883119905decreases and when 119883

119905lt 120583 a similar argument

establishes that119883119905grows

The stationarity condition is the key to perform theGaussian estimation of parameters 120572 120590 and 120583 assuming thatthe elasticity (120574 = 0 12 1 32) is given a priori

Although the general model (1) derives various interestrate models studied in the literature our interest is focusedprimarily on those models that are strictly stationary

For example when 120574 = 0 it is said that the process hasadditive noise so that we get the linear stochastic differentialequation

119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119889119861

119905 1198830= 119909 119905 isin [0 119879] (4)

This model is also known as Ornstein-Uhlenbeck meanreversion process type used by Vasicek [11] in modelingthe equilibrium price of the coupon This Gaussian processhas been widely used in the valuation of options on bondsfutures options on futures and other financial derivativeinstruments while in Lari-Lavassani et al [15] it has beenused to study the dynamic behavior of the prices of oil andnatural gas

Although the model of Vasicek [11] is analyticallytractable it has some drawbacks Since the interest rate in theshort term is normally distributed for each 119905 isin [0 119879] exists

Journal of Probability and Statistics 3

Table 1 Summary of the models

Model 120572 120583 120590 120574

Vasicek 0Brennan-Schwartz 1CIR SR 12

CIR VR 0 32

CKLS [0 32]

a probability that 119883119905is negative and this is not reasonable

from an economic standpoint Another drawback of thismodel is that conditional volatility assumes changes in theinterest rate that are constants independent of the level of119883

119905

If 120574 = 1 it is said that the process has proportional noiseand gives the nonhomogeneous linear stochastic differentialequation given by119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

119905119889119861119905 1198830= 119909 119905 isin [0 119879] (5)

This model is used by Brennan and Schwartz [12] toobtain a numerical model of convertible bond prices Thisprocess is also used by Courtadon [17] in developing a modelfor option pricing on discount bonds

When 120574 = 12 we obtain the nonlinear stochastic differ-ential equation

119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

12

119905119889119861119905

2120572120583 ge 1205902 1198830= 119909 119905 isin [0 119879]

(6)

This process is naturally linked to the 120594 centered-squaredistribution and is the (square root) version of the model ofBrennan and Schwartz [12] proposed byCox et al [18] whichproduces types of instantaneous short-term interest strictlypositive

This model has also been used extensively in the develop-ment of pricingmodels for derivative instruments sensitive tointerest rates Examples include pricingmodels formortgage-backed in Dunn and McConnell [19] models of options fordiscount bonds in Cox et al [18] future and option pricingmodels for futures in Ramaswamy and Sundaresan [20] theswap valuationmodels in Sundaresan [21] and option pricingmodels for the convenience yield in Longstaff [22]

Finally for the special case in which 120574 = 32 and 120572 = 0equation (1) takes the form

119889119883119905= 12059011988332

119905119889119861119905 1198830= 119909 119905 isin [0 119879] (7)

This model is introduced by Cox et al [13] in their study ofvariable-rate (VR) securities A similar model is also usedby Constantinides and Ingersoll [23] for value bonds in thepresence of taxes

Table 1 shows the summary of the models associatedwith (1) that are of interest in this work (Vasicek Brennan-Schwartz CIR-SRCIR-VR and CKLS)

3 Gaussian Estimation of Diffusion Models

Many theoretical models in economics and finance are for-mulated in continuous time as either a diffusion or diffu-sion systems For this reason identification estimation and

therefore the investigation of the asymptotic properties ofsamples of the estimators of these processes turn out to bea difficult task These problems arise mainly due to the lackof availability of continuous sampling observations In factthe estimation of diffusion models has been considered inthe literature for many years and inmost scientific journals itonly refers to the estimate considering continuous samplingobservations although the data that these models describedcan only be sampled at discrete points in time (See eg Jiangand Knight [9] and references therein)

There are various techniques and methods for estimationof models of mean reversion of a single factor with differentstructural forms in the trend and distribution functionsmostsimulation techniques-based on statistical and quite rigorousarguments The work of Jiang and Knight [9] proposesa parametric identification and estimation procedure fordiffusion processes based on discrete sampling observationsThe nonparametric kernel estimator for diffusion functionsdeveloped in this work deals with general diffusion processesand avoids any functional form specification for either thetrend function or distribution function They show thatunder certain regularity conditions the nonparametric esti-mator function is consistent and timely dissemination alsoasymptotically follows a mixed normal distribution In Jiangand Knight [24] there are some numerical experimentalresults obtained by a Monte Carlo study using the results ofJiang and Knight [9] which include parameter estimationfor Ornstein-Uhlenbeck process In this sense the lack ofanalytical solutions has hampered empirical evidence andvalidation of alternative hypotheses about these continuous-time models

Although the maximum likelihood method is a widelyused parametric method the analytical expressions for thelikelihood functions exist only for processes of mean rever-sion with additive noise In fact in most of the work on esti-mation of diffusion models processes becomemore complexprocesses of Ornstein-Uhlenbeck type For example Gietand Lubrano [25] consider a diffusion process in which theelasticity constant of the nonlinear term of diffusion can befreely chosen achieving after reducing it a transformationto an Ornstein-Uhlenbeck process The major drawback isthat there is no degree of freedom in the form of the trendfunction which is completely determined by the shape of thedistribution function and the choice of the underlying linearmodel Valdivieso et al [5] proposed a methodology to esti-mate maximum likelihood parameters of a one-dimensionalstationary process of Ornstein-Uhlenbeck type which isdriven by general Levy process and which is constructedthrough a distribution D self-decomposed This approachis based on the inversion of the characteristic function andusing the discrete fractional fast Fourier transform Themaximization of the likelihood function is performed usingnumerical methods

The framework proposed in our work has some featuresin common with Nowman [2] Yu and Phillips [1] andElerian et al [6] The similarity of our approach lies in theuse of Euler numerical scheme using discretely sampleddata and maximum likelihood estimation but the maindifference is that in our proposal does not need any numerical

4 Journal of Probability and Statistics

optimization because it is possible to obtain a closed formulafor the estimators

For example in Elerian et al [6] developed a techniquebased on Bayesian estimation of nonlinear EDEs approxi-mating theMarkov transition densities of broadcasts includ-ing Ornstein-Uhlenbeck processes and elasticity models forinterest rates when observations are discretely sampled Theframe of the estimate is based on the introduction of aux-iliary data to complete the diffusion latent missing betweeneach pair of measurements Combine methods of MonteCarlo Markov Chain algorithm based on the Metropolis-Hastings in conjunctionwith the numerical scheme of Euler-Maruyama to sample the posterior distribution of latent dataand model parameters

Another alternative for Gaussian estimation was pro-posed by Nowman [2] In this paper an estimate using theexact discretization of the model is proposed

119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

120574

(119904Δ)119889119861119905 119904Δ le 119905 lt (119904 + 1) Δ

(8)

assuming that volatility stays constant on the intervals [119904Δ(119904 + 1)Δ) 119904 = 0 1

In this way we obtain an equation of the form

119883119905= 119890minus120572Δ

119883(119905minusΔ)

+ 120583 (1 minus 119890minus120572Δ

) + 120578119905 (9)

where the conditional distribution 120578119905| 119865119861119905minus1

asymp 119873(0 (1205902

2120572)(1 minus 119890minus2120572Δ)119883119905minus12120574)

However despite the obtainable approximate conditionaldistribution this approach does not correspond to the Eulerdiscretization applied to (1) This estimation technique re-quires the discrete model corresponding to (2) which is givenby

119883119905= 119890minus120572119883(119905minus1)

+ 120583 (1 minus 119890minus120572) + 120578119905 (119905 = 0 1 119879) (10)

where 120578119905 (119905 = 0 1 119879) satisfies the conditions 119864[120578

119904120578119905] = 0

(119904 = 119905) and

119864 [1205782

119905] = int119905

119905minus1

119890minus2(119905minus120591)120572

1205902119883(119905minus1)

2120574

119889120591

=1205902

2120572(1 minus 119890

minus2120572) 119883(119905minus1)

2120574

= 1198982

119905119905

(11)

This states that the logarithmof theGaussian likelihood func-tion may be written as

119871 (120579) =

119879

sum119905=1

[2 log119898119905119905+

119883119905minus 119890minus120572119883

(119905minus1)+ 120583 (1 minus 119890minus120572)

119898119905119905

2

]

=

119879

sum119905=1

[2 log119898119905119905+ 1205762

119905]

(12)

where 120576 = [1205761 120576

119879] is a vector whose elements can be

calculatedThe optimization of 119871(120579) is carried out using the methods

developed by Bergstrom [26ndash30] and Nowman [2]

Yu and Phillips [1] propose a Gaussian estimation usingrandom time changes which is based on the idea that anycontinuous time martingale can be written as a Brownianmotion after a suitable time change It considers the explicitsolution (1) to obtain the expression

119883(119905+ℎ)

=120583 (1minus119890minus120572ℎ

)+119890minus120572ℎ

119883(119905)+intℎ

0

120590119890minus120572(ℎminus120591)

119883(119905+120591)

120574

119889119861120591

for ℎ gt 0(13)

Let 119872(ℎ) = 120590 intℎ

0119890minus120572(ℎminus120591)119883

(119905+120591)120574119889119861120591 119872(ℎ) is a contin-

uous martingale (In Yu and Phillips [1] 119872(ℎ) was assumedas a continuous martingale but this assumption is generallynot warranted and [119872]

ℎdoes not induce a Brownianmotion

as Yu and Phillips showed in [31] In addition they establishthat a simple decomposition splits the error process into atrend component and a continuousmartingale process) withquadratic variation

[119872]ℎ= 1205902intℎ

0

119890minus2(ℎminus120591)120572

119883(119905+120591)

2120574

119889120591 (14)

They introduce a sequence of positive numbers ℎ119895which

deliver the required time changes For any fixed constant 119886 gt0 let

ℎ119895+1

= inf 119904 | [119872119895]119904ge 119886

= inf 119904 | 1205902 int119904

0

119890minus2(119904minus120591)120572

119883(119905119895+120591)

2120574

119889120591 ge 119886

(15)

Then a succession of points 119905119895 is constructed using the

iterations 119905119895+1

= 119905119895+ ℎ119895+1

with 1199051= 0 to obtain

119883(119905119895+1)

= 120583 (1 minus 119890minus120572ℎ119895+1) + 119890

minus120572ℎ119895+1119883(119905119895)

+119872ℎ119895+1

(16)

where119872ℎ119895+1

= 119861(119886) asymp 119873(0 119886)Note that according to (16) (12) takes the form

119871 (120579) =1

119873sum119895

[[

[

2 log 119886

+

119883(119905119895+1)

minus120583 (1minus119890minus120572ℎ119895+1)minus119890minus120572ℎ119895+1119883(119905119895)

119886

2

]]

]

(17)

Since the data are observed in discrete time the timechange formula cannot be applied directly and thereforemustbe approximated by

ℎ119895+1

= Δmin119904 |119904

sum119894=1

1205902119890minus2120572(119904minus119894)Δ

119883(119905119895+119894Δ)

2120574

ge 119886 (18)

This should be selected so that 119886 is the volatility of theerror term unconditional in the equation

119883(119905+Δ)

= 120583 (1 minus 119890minus120572Δ

) + 119890minus120572Δ

119883(119905)+ 120576 (19)

Journal of Probability and Statistics 5

The implementation of this method is based on theestimate given by Nowman [2] to find the parameters esti-mates 120590 and 120574 Furthermore the estimation of the remainingparameters 120583 and 120572 is carried out using the numericaloptimization algorithm by Powell [32]

4 Gaussian Estimation ofOne-Factor Mean Reversion Processes withConstant Parameters

The development of this section will focus on the descriptionof the proposed technique for the Gaussian estimation ofmean reversion models of a single factor with linear andnonlinear functions of diffusion constant parameters andlinear trend functions with constant parameters

As the density function for most diffusion processes isnot known analytically the proposed idea is not to find orapproximate the transition density function or the marginaldensity function of the process Otherwise we use thenumerical approximation of Euler on the EDE to build a newdiscrete process Huzurbazar [33] and use the fact that inthis approximation the normalized residuals resulting fromBrownian process are independent standard normal randomvariables As pointed out by Pedersen [7] Kohatsu-Higa andOgawa [34] and Bally and Talay [35] the density functionof the discrete process obtained with Euler scheme convergesto the density function of the continuous process and asdiscussed by Zellner [36] these residues can be interpretedas a vector of unknown parameters with the advantage ofknowing their distribution So we build a new variable forthe models defined in (1) so that it can adapt to the observeddata and later can be evaluated by examining the normalizedresiduals that will be useful in the estimation process Byknowing the density function of this new process we canfind discretized likelihood function and use the conventionalmethod of maximum likelihood estimation to find a closedformula for the parameters

Consider the stochastic differential equation (1) with theinitial condition119883

0= 119909 The terms 120572(120583 minus 119883

119905) and 120590119883120574

119905being

the trend and diffusion functions respectively that dependon 119883119905and a vector of unknown parameters 120579 = [120572 120583 120590 120574]

where 119861119905119905ge0

is a unidimensional standard Brownianmotiondefined on a complete probability space (ΩFP) The trendfunction describes the movement of the process due tochanges in time while the diffusion function measures themagnitude of the randomfluctuations around the trend func-tion Suppose that the conditions under which we guaranteethe existence and uniqueness of the solution of Mao [37] aresatisfied Assume further the process observations in discretetime119883

119905= 119883(120591

119905) at times 120591

0 1205911 120591

119879whereΔ

119905= 120591119905minus120591119905minus1

ge

0 119905 = 1 2 119879 The objective is to estimate the unknownparameter vector 120579 given the historical information of thesample119883 = (119883

0 1198831 119883

119879)1015840 with equally spaced data Note

that the explicit solution of (1) is given by

119883119905= 120583 + (119883

0minus 120583) 119890

minus120572119905+ int119905

0

120590119890minus120572(119905minus119904)

119883120574

119904119889119861119904 (20)

In the context of the likelihood the estimation of 120579 isbased on the likelihood function

log 119871 (1198831 1198832 119883

119879| 1198830 120579) =

119879

sum119905=1

[log119892 (119883119905+1

| 119883119905 120579)]

(21)

where 119892(119883119905+1

| 119883119905 120579) are the Markov transition densities

or 119871(120579 | 1198831 1198832 119883

119879) = 119891(119883

1 1198832 119883

119879 120579) and 119891(119883

1

1198832 119883

119879 120579) is the joint density function

Although the differential equation (1) can be solved ana-lytically on [0 119879] the functions 119892(119883

119905+1| 119883119905 120579) and 119891(119883

1

1198832 119883

119879 120579) are not available in closed form and therefore

the estimate can not be accomplished except for the casewhere 120574 = 0

Note that the exact discretized solution of (1) takes theform

119883(120591119905) = 120583 + (119883 (120591

119905minus1) minus 120583) 119890

minus120572(120591119905minus120591119905minus1)

+ int120591

120591119905minus1

120590119890minus120572(120591119905minus119904)119883

120574

(119904)119889119861119904

(22)

which can be approximated by

119883(120591119905) = 120583 + (119883 (120591

119905minus1) minus 120583) 119890

minus120572Δ 119905

+ 120590119890minus120572Δ 119905119883

120574(120591119905minus1) Δ119861119905

(23)

where Δ119861119905= 119861120591119905minus 119861120591119905minus1

Thus

119883(120591119905) = 119883 (120591

119905minus1) + 120572 (120583 minus 119883 (120591

119905minus1)) Δ119905+ 119883120574(120591119905minus1) 120576120591119905

(24)

where 120576120591119905sim 119873(0 1205902Δ

119905) corresponding to the numerical ap-

proximation of EulerNow define the new variable

119884119894=119883119894minus 119883119894minus1

minus 120572 (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

= 120576119894 Δ constant (25)

Note that 119884119894 119894 = 1 2 3 119879 are independent random

variables distributed normal119873(0 1205902Δ)In terms of the density function can be written

119891 (1198841 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(1198831minus1198830minus120572 (120583minus119883

0) Δ

119883120574

0

)

2

]

6 Journal of Probability and Statistics

119891 (1198842 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(1198832minus1198831minus120572 (120583minus119883

1) Δ

119883120574

1

)

2

]

119891 (119884119879 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(119883119879minus119883119879minus1

minus120572 (120583minus119883119879minus1

) Δ

119883120574

119879minus1

)

2

]

(26)

Since each 119884119894is independent can be expressed the joint

density function as their marginal densities product that is

119891 (1198841 1198842 119884

119879) = 119891 (119884

1 120579) 119891 (119884

2 120579) sdot sdot sdot 119891 (119884

119879 120579)

(27)

Consequently the likelihood function is given by

119871 (120579 | 119884119879) = (

1

21205871205902Δ)1198792

times exp[ minus1

21205902Δ

119879

sum119894=1

(119883119894minus119883119894minus1minus120572 (120583minus119883

119894minus1) Δ

119883120574

119894minus1

)

2

]

(28)

Taking the natural logarithm on both sides it follows that

log 119871 = minus119879

2log (21205871205902Δ)

minus1

21205902Δ[

119879

sum119894=1

(119883119894minus 119883119894minus1

minus 120572 (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

]

(29)

Now consider the problem of maximizing the likelihoodfunction given by

120597 log (119871)120597120579

= 0 ie 120579 = Arg120579

max (log (119871)) (30)

The estimation process leads to the estimated parameters120572 and 120583 are the solutions of which the next equation systemwhere it is assumed that the value of 120574 is known Consider

119860 minus 119864 (1 minus 120572Δ) minus 120572120583Δ119861 = 0

119862 minus 119861 (1 minus 120572Δ) minus 120572120583Δ119863 = 0(31)

where

119860 =

119879

sum119894=1

119883119894119883119894minus1

1198832120574

119894minus1

119861 =

119879

sum119894=1

119883119894minus1

1198832120574

119894minus1

119862 =

119879

sum119894=1

119883119894

1198832120574

119894minus1

119863 =

119879

sum119894=1

1

1198832120574

119894minus1

119864 =

119879

sum119894=1

(119883119894minus1

119883120574

119894minus1

)

2

(32)

Thus the estimated parameters are given by

=119864119863 minus 119861

2 minus 119860119863 + 119861119862

(119864119863 minus 1198612) Δ

120583 =119860 minus 119864 (1 minus Δ)

119861Δ

(33)

Additionally the parameter 120590 is determined by an equa-tion independent of the system of (31) and is completelydetermined by the estimated parameters

= radic1

119879Δ

119879

sum119894=1

(119883119894minus 119883119894minus1

minus (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

(34)

This estimation scheme has several important qualitiesFirst you come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simple and does not requiresophisticated routines to perform calculations very extensive

5 Simulations

This sectionwill present comparisons of the results with thosepublished by other authors using the techniques describedabove Comparisons are made with the estimate based onsimulated data sets to analyze statistical properties of theproposed estimators

Yu andPhillips [1] present the results of a simulation studywith the Monte Carlo technique in comparing their methodwith Nowmanrsquos [2] To make this comparison are suggestedthree scenarios to resolve data monthly weekly and daily Inall cases the basic pattern of square root is defined as

119889119883119905= (120582 + 120573119883

119905) 119889119905 + 120590119883

120574

119905119889119861119905

with 120582 gt 0 120573 lt 0 120590 gt 0 120574 = 05(35)

Note that with appropriate substitution of this modelparameters are equivalent to those described in (1) (Theequivalence of models (1) and (35) is satisfied by having120582 = 120572120583 and 120573 = minus120572 Originally in Yu and Phillips [1] theparameter 120582 in (35) is defined as 120572 but was changed to avoidconfusion with the model (1))

The specific parameters used in each stage are shown inTable 2

To perform the first comparison parameters are set asspecified in Table 2 and obtained a series simulatedThen youset the value to 120574 = 05 and proceed to the estimation by themethod of Section 4 to obtain 120573 and This procedure wasrepeated 1000 times (as inYu andPhillips [1]) Some statisticalestimators are presented in Tables 3 4 and 5

Tables 3 4 and 5 show that the method proposed byNowman [2] gives good estimates for the parameters 120590 and120574 as manifested by Yu and Phillips [1] it becomes clearthat a starting point for the method is defined by the latterFurthermore through the resulting estimate of the parameter120590 our method becomes much more accurate than Nowman[2] introducing a bias less than 02 for the monthly dataseries and a small value for cases of weekly and daily series

Journal of Probability and Statistics 7

Table 2 Monte Carlo study parameters

Monthly Weekly DailySize 500 1000 2000120582 072 3 6120573 minus012 minus05 minus1120590 06 035 025120574 05 05 05

In the case of 120582 and 120573 are parameters observed inefficientperformance (as discussed Yu and Phillips [1 page 220]) inthe method of Nowman 120582 yielding bias of 87 47 and64 for data monthly weekly and daily respectively Alsofor 120573 the bias present in the estimate is 94 46 and54 for the respective monthly weekly and daily series Themethod proposed byYu andPhillips has a better performancethan Nowman measured from the bias in the parametersreducing it by 15 8 and 16 for the case of 120582 and 5 6and 6 in the case of 120573 for the respective monthly weeklyand daily series Our method has a performance similar tothat of Yu and Phillips for daily data series reducing the biasregarding Nowman method in 15 for 120582 and 4 for 120573 Forweekly data series for reducing 120582 bias regarding the methodof Nowman is 7 whereas for 120573 is 6 as reduction Yu andPhillips method

An important point to note in the results obtained byour method is the condition that the parameter 120574 is knowna priori This is necessary in order to find closed formulas forthe estimators

In the approach to the model proposed by Nowman andYu-Phillips 119889119883

119905= (120582 + 120573119883

119905)119889119905 + 120590119883

120574

119905119889119861119905is implied to have

particular difficulty in estimating the parameter of speed ofreversion To make this condition clear Table 6 shows theresults obtained by the method of Durham and Gallant [4]and our method to estimate the parameters of the model119889119883119905= 120572(120583 + 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905with 120572 gt 0 120583 gt 0 120590 gt 0

120574 = 05 The results are the average of 512 estimations ofmonthly series with 1000 observations

The results shown in Table 6 show that the errors inestimation of 120583 and 120590 are small (less than 1) for bothmethods On the other hand the estimated speed of reversionhas an error of about 10 resulting in Durham and Gallantwhile ourmethod has an error close to 7 From this analysisit is clear that the parameter 120572 has a higher error in estimatingthan the other parameter estimates

In this regard the estimation error of 120572 in Yu [3] providesan analysis of the bias in the parameter estimation inherentreversion total observation time which is sampled in thecontinuous process for the special case where the long-termaverage 120583 is zero Reference is made to the fact that the bias inparameter depends asymptotically on total observation timewhich of the process and not the number of observationsmade in this period

Figures 1 2 and 3 show the results of the estimationof the parameters for different observation horizons withdifferent sampling frequencies (The results are obtained asthe average of the individual estimates on 1000 simulated

0 10 20 30 40 50 60 70 80 9014

145

15

155

16

165

17

175

18

185

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120572 = 15

Figure 1 120572 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

0 10 20 30 40 50 60 70 80 900497

04975

0498

04985

0499

04995

05

05005

0501

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120583 = 05

Figure 2 120583 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

series in 4 sampling scenarios (a) bimonthly series with 500observations (b) monthly series with 1000 observations (c)weekly series with 4333 observations and (d) daily series with21000 observations The reference model for the simulationswas 119889119883

119905= 120572(120583 minus 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905 with 120572 = 15 120583 = 05

120574 = 05 and 120590 = 04)Figure 1 shows the evolution of the estimate of 120572 for

observations from 10 to 83 years Behavior of the estimatesin the four scenarios can be established empirically by thedependence of the bias on the estimate and the horizonof observation as they have more years of observation

8 Journal of Probability and Statistics

Table 3 Comparison of monthly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 13440 minus02332 12330 minus02275 06173 04919 12797 minus02200 05989Bias 06240 minus01132 05130 minus01075 00173 minus00081 05597 minus01000 minus00011Var 05897 01713 05732 01589 00099 00071 05406 00149 00004MSE 09791 01841 08363 01705 00102 00071 08533 00249 00004The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 500observations each The real values of the parameters are 120582 = 072 120573 = minus012 120590 = 06 and 120574 = 05

Table 4 Comparison of weekly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 44090 minus07320 41650 minus07011 03762 04925 42038 minus07022 03499Bias 14090 minus02320 11650 minus02011 00262 minus00075 12038 minus02022 minus00001Var 40663 01109 38608 01030 00172 00357 35427 00992 00001MSE 60855 01647 52180 01435 00179 00358 49883 01400 00001The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 3 120573 = minus05 120590 = 035 and 120574 = 05

Table 5 Comparison of daily data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 98250 minus15360 88440 minus14750 02521 04970 89542 minus14933 02500Bias 38250 minus05360 28440 minus04750 00021 minus00030 29542 minus04933 00000Var 191959 05219 178220 04798 00244 00746 155818 04334 00000MSE 338266 08092 259100 07054 00244 00746 242938 06763 00000The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 60 120573 = minus10 120590 = 025 and 120574 = 05

Table 6 Comparison of Durham and Gallant and proposed methods

Parameter(lowast) Real Bias(a) Bias(b) Real Bias(a) Bias(b)

120572 05 0048900 0037700 05 0059730 0021400120583 006 0000610 0000127 006 minus0000310 minus0000021120590 015 0000200 0000032 022 0000120 minus0000046120572 05 0043780 0032400 04 0045890 0032200120583 006 0000050 0000016 006 0000770 0000046120590 003 0000010 minus0000003 015 0000080 minus0000025The data are obtained with 512 simulations monthly series (119889119905 = 112) with 1000 observations each and the parameters are listed as real(a)Results of the method proposed by Durham and Gallant(b)Results of the method proposed in this paper(lowast)In Durham and Gallant [4] 120583 = 120579

1 120572 = 120579

2 and 120590 = 120579

3

bias decreases with no evidence of a significant differencebetween the estimates different frequencies

From Figure 2 it is observed that after few observationsvirtually independent of the year of observation estimationsfor 120583 are obtained with a bias of less than 1 showingthe efficiency of the estimation of the long-term averagereversion processes average In the case of estimation of 120590Figure 3 shows that the bias in the parameter estimates is

more dependent on the number of observations which is thetotal time of observation of the processThis is reflected in thefact that the estimates are in series with a higher sampling ratereaches a lower bias for the same observation horizon thanlower sample rate and therefore have fewer observations Onthe other hand as for the case of the estimate of 120583 the biaspresent in the estimates of 120590 is practically negligible (less than1) in most scenarios

Journal of Probability and Statistics 9

0 10 20 30 40 50 60 70 80 900386

0388

039

0392

0394

0396

0398

04

0402

0404

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120590 = 04

Figure 3 120590 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

6 Conclusions and Comments

This paper presents a new alternative for Gaussian estimationmodels of mean reversion of one factor under the Eulernumerical approximation that includes among others somemodels of interest rates in the short term

Monte Carlo simulations show a good approximation ofthe diffusion term parameter but less approximation in atrend term parameterThe estimation results have almost thesame order as that in Nowmanrsquos [2] and in some cases betterthan that in Durham and Gallantrsquos [4] It should be notedthat the proposed methodology outperforms the estimate of120590 compared to the other alternatives presented These resultsbecome more relevant since in practice the estimation ofvolatility is extremely useful in the evaluation of financialderivatives in which the trend term loses significance whenit is absorbed by the processes of change measurement andvaluation risk neutral

The estimate based on this new method though doesnot eliminate the bias in estimation of the reversion rateparameter for a small sample is in some cases comparablewith the results in Yu and Phillips [1] if samples have longertime horizons

This estimation scheme has several important qualitiesFirst we come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simpler and does not requiresophisticated routines to perform extensive calculations

References

[1] J Yu and P C B Phillips ldquoA Gaussian approach for continuoustime models of the short-term interest raterdquo The EconometricsJournal vol 4 no 2 pp 210ndash224 2001

[2] K Nowman ldquoGaussian estimation of single-factor continuoustime models of the term structure of interest ratesrdquoThe Journalof Finance vol 52 pp 1695ndash1703 1997

[3] J Yu ldquoBias in the estimation of themean reversion parameter incontinuous time modelsrdquo Journal of Econometrics vol 169 no1 pp 114ndash122 2012

[4] G B Durham and A R Gallant ldquoNumerical techniques formaximum likelihood estimation of continuous-time diffusionprocessesrdquo Journal of Business amp Economic Statistics vol 20 no3 pp 297ndash338 2002

[5] L ValdiviesoW Schoutens and F Tuerlinckx ldquoMaximum like-lihood estimation in processes of Ornstein-Uhlenbeck typerdquoStatistical Inference for Stochastic Processes vol 12 no 1 pp 1ndash192009

[6] O Elerian S Chib and N Shephard ldquoLikelihood inference fordiscretely observed nonlinear diffusionsrdquo Econometrica vol 69no 4 pp 959ndash993 2001

[7] A R Pedersen ldquoConsistency and asymptotic normality ofan approximate maximum likelihood estimator for discretelyobserved diffusion processesrdquo Bernoulli vol 1 no 3 pp 257ndash279 1995

[8] T Koulis and A Thavaneswaran ldquoInference for interest ratemodels using Milsteinrsquos approximationrdquo The Journal of Math-ematical Finance vol 3 pp 110ndash118 2013

[9] G J Jiang and J L Knight ldquoA nonparametric approach to theestimation of diffusion processes with an application to a short-term interest rate modelrdquo Econometric Theory vol 13 no 5 pp615ndash645 1997

[10] D Florens-Zmirou ldquoOn estimating the diffusion coefficientfrom discrete observationsrdquo Journal of Applied Probability vol30 no 4 pp 790ndash804 1993

[11] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 pp 177ndash186 1977

[12] M Brennan and E Schwartz ldquoAnalyzing convertible bondsrdquoJournal of Financial and Quantitative Analysis vol 15 pp 907ndash929 1980

[13] J Cox J Ingersoll and S Ross ldquoAn analysis of variable rate loancontractsrdquoThe Journal of Finance vol 35 pp 389ndash403 1980

[14] K Chan F Karolyi F Longstaff and A Sanders ldquoAn empiricalcomparison of alternative models of short term interest ratesrdquoThe Journal of Finance vol 47 pp 1209ndash1227 1992

[15] A Lari-Lavassani A Sadeghi and A Ware ldquoMean revertingmodels for energy option pricingrdquo Electronic Publications of theInternational Energy Credit Association 2001 httpwwwiecanet

[16] D Pilipovic Energy Risk Valuing andManaging Energy Deriva-tives McGraw-Hill New York NY USA 2007

[17] G Courtadon ldquoThe Pricing of options on default-free bondsrdquoJournal of Financial and Quantitative Analysis vol 17 pp 75ndash100 1982

[18] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[19] K Dunn and J McConnell ldquoValuation of GNMA Mortgage-backed securitiesrdquoThe Journal of Finance vol 36 no 3 pp 599ndash616 1981

[20] K Ramaswamy and S Sundaresan ldquoThe valuation of floating-rate instrumentsrdquo Journal of Financial Economics vol 17 pp251ndash272 1986

[21] S Sundaresan Valuation of Swaps Working Paper ColumbiaUniversity 1989

10 Journal of Probability and Statistics

[22] F Longstaff ldquoThe valuation of options on yieldsrdquo Journal ofFinancial Economics vol 26 no 1 pp 97ndash122 1990

[23] G Constantinides and J Ingersoll ldquoOptimal bond trading withpersonal taxesrdquo Journal of Financial Economics vol 13 pp 299ndash335 1984

[24] G Jiang and J Knight ldquoFinite sample comparison of alternativeestimators of Ito diffusion processes a Monte Carlo studyrdquoTheJournal of Computational Finance vol 2 no 3 1999

[25] L Giet andM LubranoBayesian Inference in Reducible Stochas-tic Differential Equations Document de Travail No 2004ndash57GREQAM University drsquoAix-Marseille 2004

[26] A R Bergstrom ldquoGaussian estimation of structural parametersin higher order continuous time dynamic modelsrdquo Economet-rica vol 51 no 1 pp 117ndash152 1983

[27] A Bergstrom ldquoContinuous time stochastic models and issuesof aggregation over timerdquo in Handbook of Econometrics ZGriliches and M D Intriligator Eds vol 2 Elsevier Amster-dam The Netherlands 1984

[28] A Bergstrom ldquoThe estimation of parameters in nonstationaryhigher-order continuous-time dynamic modelsrdquo EconometricTheory vol 1 pp 369ndash385 1985

[29] A Bergstrom ldquoThe estimation of open higher-order continuoustime dynamic models with mixed stock and flow datardquo Econo-metric Theory vol 2 pp 350ndash373 1986

[30] A Bergstrom Continuous Time Econometric Modelling OxfordUniversity Press Oxford UK 1990

[31] P C B Phillips and J Yu ldquoCorrigendum to lsquoA Gaussianapproach for continuous time models of short-term interestratesrsquordquoThe Econometrics Journal vol 14 no 1 pp 126ndash129 2011

[32] M J D Powell ldquoAn efficient method for finding the minimumof a function of several variables without calculating deriva-tivesrdquoThe Computer Journal vol 7 pp 155ndash162 1964

[33] V S Huzurbazar ldquoThe likelihood equation consistency and themaxima of the likelihood functionrdquo Annals of Eugenics vol 14pp 185ndash200 1948

[34] A Kohatsu-Higa and S Ogawa ldquoWeak rate of convergence foran Euler scheme of nonlinear SDErsquosrdquoMonte Carlo Methods andApplications vol 3 no 4 pp 327ndash345 1997

[35] V Bally andD Talay ldquoThe law of the Euler scheme for stochasticdifferential equations error analysis with Malliavin calculusrdquoMathematics and Computers in Simulation vol 38 no 1ndash3 pp35ndash41 1995

[36] A Zellner ldquoBayesian analysis of regression error termsrdquo Journalof the American Statistical Association vol 70 no 349 pp 138ndash144 1975

[37] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing England UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Probability and Statistics

Some of the works that could be classified as methodsin which it seeks to find the transition probability functionand use Martingale properties to approximate the likeli-hood function are Durham and Gallant [4] Valdivieso etal [5] and Elerian et al [6] Durham and Gallant [4]propose a scheme based on simulation Establishing differentmodel modifications Pedersen [7] improves computationalperformance and characteristics of the model are studiedusing simulated synthetic series Koulis and Thavaneswaran[8] study combined martingale estimation functions fordiscretely observed interest rates modelsThey derive closed-form expressions for the first four conditional moments ofthe CIR model via Itorsquos formula and for extended versionsof the CIR model using the Milsteinrsquos approximation andconstruct the optimal estimating functions They also showthat combined estimating functions have more informationwhen the conditional mean and variance of the observedprocess depend on the parameter of interest Valdivieso et al[5] proposed a method for estimating the transfer functionthrough the inversion of the characteristic function of theprocess and the use of Fourier transform This methodologyis developed for some particular cases of D-OU processes(Ornstein-Uhlenbeck processes with broken car distributionD) Elerian et al [6] used simulation-based techniquesfor estimating the transition probability depending on thesample to incorporate auxiliary dummy observations Theparameters are obtained by simulation and optimizationusing numerical algorithms

Methods based on kernel techniques approximate densityfunctions of the components and spreading tendency of theprocess The works of Jiang and Knight [9] and Florens-Zmirou [10] are within this framework

In this paper we propose an alternative scheme forparameter estimation in one-factor mean reversion modelsusing the Euler-Maruyama discretization Using the prop-erties of discretized error normality the maximum likeli-hood technique to obtain closed formulas for the estimatorsapplies We present some performance measures of themaximum likelihood estimators obtained with the proposedmethod in simulated processes and we compare them withthe results published by Nowman [2] Yu and Phillips [1] andDurham and Gallant [4]

This paper is organized as follows In Section 2 a briefdescription of mean reversion processes of one factor withconstant parameters Some Gaussian estimation techniquesare presented in Section 3 Section 4 develops the proposedmethod and Section 5 presents some experimental numeri-cal results with different sampling rates

2 Mean Reversion Processes

Mean reversion processes have been considered as a naturalchoice to describe the dynamic behavior of interest rates inVasicek [11] Brennan and Schwartz [12] Cox et al [13] andChan et al [14] daily prices of some commodities such asoil and natural gas in Lari-Lavassani et al [15] have a fun-damental role in modeling spot prices when they exhibitedlong-term trends such as electrical energy (Pilipovic [16])

Mean reversion processes of one factor with constantparameters can be written as

119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

120574

119905119889119861119905 119905 isin [0 119879] (1)

with initial condition 1198830= 119909 where 120572 gt 0 120590 gt 0 120583 isin R

and 120574 isin [0 32] are constants and 119861119905119905ge0

is a unidimensionalstandard Brownianmotion defined on a complete probabilityspace (ΩFP)

The 120572 parameter is called reversion rate 120583 is the meanreversion level or trend of long-run equilibrium 120590 is theparameter associated with the volatility and 120574 determines thesensitivity of the variance to the level of119883

119905

The model (1) is known as CKLS proposed by Chan et al[14] and is a generalization of the models of interest rate inthe short term with constant parameters in which 120574 = 0 12and 1 or 120574 = 32 with 120572 = 0

In general mean reversion processes are stationary Forexample note that (1) can be written in integral form as

119883119905minus 1198830= 120572int

119905

0

(120583 minus 119883119904) 119889119904 + 120590int

119905

0

119883120574

119904119889119861119904 (2)

Taking the expected value 119864[119883119905] minus 119864[119883

0] = 120572 int

119905

0(120583 minus

119864[119883119904])119889119904 and thus we obtain the differential equation

(119905) = 120572 (120583 minus 119898 (119905)) 119898 (119905) = 119864 [119883119905] (3)

The solution of this equation is 119898(119905) = 120583 + (1198830minus 120583)119890minus120572119905

so lim119905rarrinfin

119864[119883119905] = lim

119905rarrinfin119898(119905) = 120583

It can be concluded then that119898(119905) = 120583 is the equilibriumlevel for the linear differential equation In other words theexpected value of the process119898(119905) returns in the long term to120583 In a finite-time horizon 120583 plays the role of attractor in thesense that when 119883

119905gt 120583 the trend term 120572(120583 minus 119883

119905) lt 0 and

therefore 119883119905decreases and when 119883

119905lt 120583 a similar argument

establishes that119883119905grows

The stationarity condition is the key to perform theGaussian estimation of parameters 120572 120590 and 120583 assuming thatthe elasticity (120574 = 0 12 1 32) is given a priori

Although the general model (1) derives various interestrate models studied in the literature our interest is focusedprimarily on those models that are strictly stationary

For example when 120574 = 0 it is said that the process hasadditive noise so that we get the linear stochastic differentialequation

119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119889119861

119905 1198830= 119909 119905 isin [0 119879] (4)

This model is also known as Ornstein-Uhlenbeck meanreversion process type used by Vasicek [11] in modelingthe equilibrium price of the coupon This Gaussian processhas been widely used in the valuation of options on bondsfutures options on futures and other financial derivativeinstruments while in Lari-Lavassani et al [15] it has beenused to study the dynamic behavior of the prices of oil andnatural gas

Although the model of Vasicek [11] is analyticallytractable it has some drawbacks Since the interest rate in theshort term is normally distributed for each 119905 isin [0 119879] exists

Journal of Probability and Statistics 3

Table 1 Summary of the models

Model 120572 120583 120590 120574

Vasicek 0Brennan-Schwartz 1CIR SR 12

CIR VR 0 32

CKLS [0 32]

a probability that 119883119905is negative and this is not reasonable

from an economic standpoint Another drawback of thismodel is that conditional volatility assumes changes in theinterest rate that are constants independent of the level of119883

119905

If 120574 = 1 it is said that the process has proportional noiseand gives the nonhomogeneous linear stochastic differentialequation given by119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

119905119889119861119905 1198830= 119909 119905 isin [0 119879] (5)

This model is used by Brennan and Schwartz [12] toobtain a numerical model of convertible bond prices Thisprocess is also used by Courtadon [17] in developing a modelfor option pricing on discount bonds

When 120574 = 12 we obtain the nonlinear stochastic differ-ential equation

119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

12

119905119889119861119905

2120572120583 ge 1205902 1198830= 119909 119905 isin [0 119879]

(6)

This process is naturally linked to the 120594 centered-squaredistribution and is the (square root) version of the model ofBrennan and Schwartz [12] proposed byCox et al [18] whichproduces types of instantaneous short-term interest strictlypositive

This model has also been used extensively in the develop-ment of pricingmodels for derivative instruments sensitive tointerest rates Examples include pricingmodels formortgage-backed in Dunn and McConnell [19] models of options fordiscount bonds in Cox et al [18] future and option pricingmodels for futures in Ramaswamy and Sundaresan [20] theswap valuationmodels in Sundaresan [21] and option pricingmodels for the convenience yield in Longstaff [22]

Finally for the special case in which 120574 = 32 and 120572 = 0equation (1) takes the form

119889119883119905= 12059011988332

119905119889119861119905 1198830= 119909 119905 isin [0 119879] (7)

This model is introduced by Cox et al [13] in their study ofvariable-rate (VR) securities A similar model is also usedby Constantinides and Ingersoll [23] for value bonds in thepresence of taxes

Table 1 shows the summary of the models associatedwith (1) that are of interest in this work (Vasicek Brennan-Schwartz CIR-SRCIR-VR and CKLS)

3 Gaussian Estimation of Diffusion Models

Many theoretical models in economics and finance are for-mulated in continuous time as either a diffusion or diffu-sion systems For this reason identification estimation and

therefore the investigation of the asymptotic properties ofsamples of the estimators of these processes turn out to bea difficult task These problems arise mainly due to the lackof availability of continuous sampling observations In factthe estimation of diffusion models has been considered inthe literature for many years and inmost scientific journals itonly refers to the estimate considering continuous samplingobservations although the data that these models describedcan only be sampled at discrete points in time (See eg Jiangand Knight [9] and references therein)

There are various techniques and methods for estimationof models of mean reversion of a single factor with differentstructural forms in the trend and distribution functionsmostsimulation techniques-based on statistical and quite rigorousarguments The work of Jiang and Knight [9] proposesa parametric identification and estimation procedure fordiffusion processes based on discrete sampling observationsThe nonparametric kernel estimator for diffusion functionsdeveloped in this work deals with general diffusion processesand avoids any functional form specification for either thetrend function or distribution function They show thatunder certain regularity conditions the nonparametric esti-mator function is consistent and timely dissemination alsoasymptotically follows a mixed normal distribution In Jiangand Knight [24] there are some numerical experimentalresults obtained by a Monte Carlo study using the results ofJiang and Knight [9] which include parameter estimationfor Ornstein-Uhlenbeck process In this sense the lack ofanalytical solutions has hampered empirical evidence andvalidation of alternative hypotheses about these continuous-time models

Although the maximum likelihood method is a widelyused parametric method the analytical expressions for thelikelihood functions exist only for processes of mean rever-sion with additive noise In fact in most of the work on esti-mation of diffusion models processes becomemore complexprocesses of Ornstein-Uhlenbeck type For example Gietand Lubrano [25] consider a diffusion process in which theelasticity constant of the nonlinear term of diffusion can befreely chosen achieving after reducing it a transformationto an Ornstein-Uhlenbeck process The major drawback isthat there is no degree of freedom in the form of the trendfunction which is completely determined by the shape of thedistribution function and the choice of the underlying linearmodel Valdivieso et al [5] proposed a methodology to esti-mate maximum likelihood parameters of a one-dimensionalstationary process of Ornstein-Uhlenbeck type which isdriven by general Levy process and which is constructedthrough a distribution D self-decomposed This approachis based on the inversion of the characteristic function andusing the discrete fractional fast Fourier transform Themaximization of the likelihood function is performed usingnumerical methods

The framework proposed in our work has some featuresin common with Nowman [2] Yu and Phillips [1] andElerian et al [6] The similarity of our approach lies in theuse of Euler numerical scheme using discretely sampleddata and maximum likelihood estimation but the maindifference is that in our proposal does not need any numerical

4 Journal of Probability and Statistics

optimization because it is possible to obtain a closed formulafor the estimators

For example in Elerian et al [6] developed a techniquebased on Bayesian estimation of nonlinear EDEs approxi-mating theMarkov transition densities of broadcasts includ-ing Ornstein-Uhlenbeck processes and elasticity models forinterest rates when observations are discretely sampled Theframe of the estimate is based on the introduction of aux-iliary data to complete the diffusion latent missing betweeneach pair of measurements Combine methods of MonteCarlo Markov Chain algorithm based on the Metropolis-Hastings in conjunctionwith the numerical scheme of Euler-Maruyama to sample the posterior distribution of latent dataand model parameters

Another alternative for Gaussian estimation was pro-posed by Nowman [2] In this paper an estimate using theexact discretization of the model is proposed

119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

120574

(119904Δ)119889119861119905 119904Δ le 119905 lt (119904 + 1) Δ

(8)

assuming that volatility stays constant on the intervals [119904Δ(119904 + 1)Δ) 119904 = 0 1

In this way we obtain an equation of the form

119883119905= 119890minus120572Δ

119883(119905minusΔ)

+ 120583 (1 minus 119890minus120572Δ

) + 120578119905 (9)

where the conditional distribution 120578119905| 119865119861119905minus1

asymp 119873(0 (1205902

2120572)(1 minus 119890minus2120572Δ)119883119905minus12120574)

However despite the obtainable approximate conditionaldistribution this approach does not correspond to the Eulerdiscretization applied to (1) This estimation technique re-quires the discrete model corresponding to (2) which is givenby

119883119905= 119890minus120572119883(119905minus1)

+ 120583 (1 minus 119890minus120572) + 120578119905 (119905 = 0 1 119879) (10)

where 120578119905 (119905 = 0 1 119879) satisfies the conditions 119864[120578

119904120578119905] = 0

(119904 = 119905) and

119864 [1205782

119905] = int119905

119905minus1

119890minus2(119905minus120591)120572

1205902119883(119905minus1)

2120574

119889120591

=1205902

2120572(1 minus 119890

minus2120572) 119883(119905minus1)

2120574

= 1198982

119905119905

(11)

This states that the logarithmof theGaussian likelihood func-tion may be written as

119871 (120579) =

119879

sum119905=1

[2 log119898119905119905+

119883119905minus 119890minus120572119883

(119905minus1)+ 120583 (1 minus 119890minus120572)

119898119905119905

2

]

=

119879

sum119905=1

[2 log119898119905119905+ 1205762

119905]

(12)

where 120576 = [1205761 120576

119879] is a vector whose elements can be

calculatedThe optimization of 119871(120579) is carried out using the methods

developed by Bergstrom [26ndash30] and Nowman [2]

Yu and Phillips [1] propose a Gaussian estimation usingrandom time changes which is based on the idea that anycontinuous time martingale can be written as a Brownianmotion after a suitable time change It considers the explicitsolution (1) to obtain the expression

119883(119905+ℎ)

=120583 (1minus119890minus120572ℎ

)+119890minus120572ℎ

119883(119905)+intℎ

0

120590119890minus120572(ℎminus120591)

119883(119905+120591)

120574

119889119861120591

for ℎ gt 0(13)

Let 119872(ℎ) = 120590 intℎ

0119890minus120572(ℎminus120591)119883

(119905+120591)120574119889119861120591 119872(ℎ) is a contin-

uous martingale (In Yu and Phillips [1] 119872(ℎ) was assumedas a continuous martingale but this assumption is generallynot warranted and [119872]

ℎdoes not induce a Brownianmotion

as Yu and Phillips showed in [31] In addition they establishthat a simple decomposition splits the error process into atrend component and a continuousmartingale process) withquadratic variation

[119872]ℎ= 1205902intℎ

0

119890minus2(ℎminus120591)120572

119883(119905+120591)

2120574

119889120591 (14)

They introduce a sequence of positive numbers ℎ119895which

deliver the required time changes For any fixed constant 119886 gt0 let

ℎ119895+1

= inf 119904 | [119872119895]119904ge 119886

= inf 119904 | 1205902 int119904

0

119890minus2(119904minus120591)120572

119883(119905119895+120591)

2120574

119889120591 ge 119886

(15)

Then a succession of points 119905119895 is constructed using the

iterations 119905119895+1

= 119905119895+ ℎ119895+1

with 1199051= 0 to obtain

119883(119905119895+1)

= 120583 (1 minus 119890minus120572ℎ119895+1) + 119890

minus120572ℎ119895+1119883(119905119895)

+119872ℎ119895+1

(16)

where119872ℎ119895+1

= 119861(119886) asymp 119873(0 119886)Note that according to (16) (12) takes the form

119871 (120579) =1

119873sum119895

[[

[

2 log 119886

+

119883(119905119895+1)

minus120583 (1minus119890minus120572ℎ119895+1)minus119890minus120572ℎ119895+1119883(119905119895)

119886

2

]]

]

(17)

Since the data are observed in discrete time the timechange formula cannot be applied directly and thereforemustbe approximated by

ℎ119895+1

= Δmin119904 |119904

sum119894=1

1205902119890minus2120572(119904minus119894)Δ

119883(119905119895+119894Δ)

2120574

ge 119886 (18)

This should be selected so that 119886 is the volatility of theerror term unconditional in the equation

119883(119905+Δ)

= 120583 (1 minus 119890minus120572Δ

) + 119890minus120572Δ

119883(119905)+ 120576 (19)

Journal of Probability and Statistics 5

The implementation of this method is based on theestimate given by Nowman [2] to find the parameters esti-mates 120590 and 120574 Furthermore the estimation of the remainingparameters 120583 and 120572 is carried out using the numericaloptimization algorithm by Powell [32]

4 Gaussian Estimation ofOne-Factor Mean Reversion Processes withConstant Parameters

The development of this section will focus on the descriptionof the proposed technique for the Gaussian estimation ofmean reversion models of a single factor with linear andnonlinear functions of diffusion constant parameters andlinear trend functions with constant parameters

As the density function for most diffusion processes isnot known analytically the proposed idea is not to find orapproximate the transition density function or the marginaldensity function of the process Otherwise we use thenumerical approximation of Euler on the EDE to build a newdiscrete process Huzurbazar [33] and use the fact that inthis approximation the normalized residuals resulting fromBrownian process are independent standard normal randomvariables As pointed out by Pedersen [7] Kohatsu-Higa andOgawa [34] and Bally and Talay [35] the density functionof the discrete process obtained with Euler scheme convergesto the density function of the continuous process and asdiscussed by Zellner [36] these residues can be interpretedas a vector of unknown parameters with the advantage ofknowing their distribution So we build a new variable forthe models defined in (1) so that it can adapt to the observeddata and later can be evaluated by examining the normalizedresiduals that will be useful in the estimation process Byknowing the density function of this new process we canfind discretized likelihood function and use the conventionalmethod of maximum likelihood estimation to find a closedformula for the parameters

Consider the stochastic differential equation (1) with theinitial condition119883

0= 119909 The terms 120572(120583 minus 119883

119905) and 120590119883120574

119905being

the trend and diffusion functions respectively that dependon 119883119905and a vector of unknown parameters 120579 = [120572 120583 120590 120574]

where 119861119905119905ge0

is a unidimensional standard Brownianmotiondefined on a complete probability space (ΩFP) The trendfunction describes the movement of the process due tochanges in time while the diffusion function measures themagnitude of the randomfluctuations around the trend func-tion Suppose that the conditions under which we guaranteethe existence and uniqueness of the solution of Mao [37] aresatisfied Assume further the process observations in discretetime119883

119905= 119883(120591

119905) at times 120591

0 1205911 120591

119879whereΔ

119905= 120591119905minus120591119905minus1

ge

0 119905 = 1 2 119879 The objective is to estimate the unknownparameter vector 120579 given the historical information of thesample119883 = (119883

0 1198831 119883

119879)1015840 with equally spaced data Note

that the explicit solution of (1) is given by

119883119905= 120583 + (119883

0minus 120583) 119890

minus120572119905+ int119905

0

120590119890minus120572(119905minus119904)

119883120574

119904119889119861119904 (20)

In the context of the likelihood the estimation of 120579 isbased on the likelihood function

log 119871 (1198831 1198832 119883

119879| 1198830 120579) =

119879

sum119905=1

[log119892 (119883119905+1

| 119883119905 120579)]

(21)

where 119892(119883119905+1

| 119883119905 120579) are the Markov transition densities

or 119871(120579 | 1198831 1198832 119883

119879) = 119891(119883

1 1198832 119883

119879 120579) and 119891(119883

1

1198832 119883

119879 120579) is the joint density function

Although the differential equation (1) can be solved ana-lytically on [0 119879] the functions 119892(119883

119905+1| 119883119905 120579) and 119891(119883

1

1198832 119883

119879 120579) are not available in closed form and therefore

the estimate can not be accomplished except for the casewhere 120574 = 0

Note that the exact discretized solution of (1) takes theform

119883(120591119905) = 120583 + (119883 (120591

119905minus1) minus 120583) 119890

minus120572(120591119905minus120591119905minus1)

+ int120591

120591119905minus1

120590119890minus120572(120591119905minus119904)119883

120574

(119904)119889119861119904

(22)

which can be approximated by

119883(120591119905) = 120583 + (119883 (120591

119905minus1) minus 120583) 119890

minus120572Δ 119905

+ 120590119890minus120572Δ 119905119883

120574(120591119905minus1) Δ119861119905

(23)

where Δ119861119905= 119861120591119905minus 119861120591119905minus1

Thus

119883(120591119905) = 119883 (120591

119905minus1) + 120572 (120583 minus 119883 (120591

119905minus1)) Δ119905+ 119883120574(120591119905minus1) 120576120591119905

(24)

where 120576120591119905sim 119873(0 1205902Δ

119905) corresponding to the numerical ap-

proximation of EulerNow define the new variable

119884119894=119883119894minus 119883119894minus1

minus 120572 (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

= 120576119894 Δ constant (25)

Note that 119884119894 119894 = 1 2 3 119879 are independent random

variables distributed normal119873(0 1205902Δ)In terms of the density function can be written

119891 (1198841 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(1198831minus1198830minus120572 (120583minus119883

0) Δ

119883120574

0

)

2

]

6 Journal of Probability and Statistics

119891 (1198842 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(1198832minus1198831minus120572 (120583minus119883

1) Δ

119883120574

1

)

2

]

119891 (119884119879 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(119883119879minus119883119879minus1

minus120572 (120583minus119883119879minus1

) Δ

119883120574

119879minus1

)

2

]

(26)

Since each 119884119894is independent can be expressed the joint

density function as their marginal densities product that is

119891 (1198841 1198842 119884

119879) = 119891 (119884

1 120579) 119891 (119884

2 120579) sdot sdot sdot 119891 (119884

119879 120579)

(27)

Consequently the likelihood function is given by

119871 (120579 | 119884119879) = (

1

21205871205902Δ)1198792

times exp[ minus1

21205902Δ

119879

sum119894=1

(119883119894minus119883119894minus1minus120572 (120583minus119883

119894minus1) Δ

119883120574

119894minus1

)

2

]

(28)

Taking the natural logarithm on both sides it follows that

log 119871 = minus119879

2log (21205871205902Δ)

minus1

21205902Δ[

119879

sum119894=1

(119883119894minus 119883119894minus1

minus 120572 (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

]

(29)

Now consider the problem of maximizing the likelihoodfunction given by

120597 log (119871)120597120579

= 0 ie 120579 = Arg120579

max (log (119871)) (30)

The estimation process leads to the estimated parameters120572 and 120583 are the solutions of which the next equation systemwhere it is assumed that the value of 120574 is known Consider

119860 minus 119864 (1 minus 120572Δ) minus 120572120583Δ119861 = 0

119862 minus 119861 (1 minus 120572Δ) minus 120572120583Δ119863 = 0(31)

where

119860 =

119879

sum119894=1

119883119894119883119894minus1

1198832120574

119894minus1

119861 =

119879

sum119894=1

119883119894minus1

1198832120574

119894minus1

119862 =

119879

sum119894=1

119883119894

1198832120574

119894minus1

119863 =

119879

sum119894=1

1

1198832120574

119894minus1

119864 =

119879

sum119894=1

(119883119894minus1

119883120574

119894minus1

)

2

(32)

Thus the estimated parameters are given by

=119864119863 minus 119861

2 minus 119860119863 + 119861119862

(119864119863 minus 1198612) Δ

120583 =119860 minus 119864 (1 minus Δ)

119861Δ

(33)

Additionally the parameter 120590 is determined by an equa-tion independent of the system of (31) and is completelydetermined by the estimated parameters

= radic1

119879Δ

119879

sum119894=1

(119883119894minus 119883119894minus1

minus (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

(34)

This estimation scheme has several important qualitiesFirst you come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simple and does not requiresophisticated routines to perform calculations very extensive

5 Simulations

This sectionwill present comparisons of the results with thosepublished by other authors using the techniques describedabove Comparisons are made with the estimate based onsimulated data sets to analyze statistical properties of theproposed estimators

Yu andPhillips [1] present the results of a simulation studywith the Monte Carlo technique in comparing their methodwith Nowmanrsquos [2] To make this comparison are suggestedthree scenarios to resolve data monthly weekly and daily Inall cases the basic pattern of square root is defined as

119889119883119905= (120582 + 120573119883

119905) 119889119905 + 120590119883

120574

119905119889119861119905

with 120582 gt 0 120573 lt 0 120590 gt 0 120574 = 05(35)

Note that with appropriate substitution of this modelparameters are equivalent to those described in (1) (Theequivalence of models (1) and (35) is satisfied by having120582 = 120572120583 and 120573 = minus120572 Originally in Yu and Phillips [1] theparameter 120582 in (35) is defined as 120572 but was changed to avoidconfusion with the model (1))

The specific parameters used in each stage are shown inTable 2

To perform the first comparison parameters are set asspecified in Table 2 and obtained a series simulatedThen youset the value to 120574 = 05 and proceed to the estimation by themethod of Section 4 to obtain 120573 and This procedure wasrepeated 1000 times (as inYu andPhillips [1]) Some statisticalestimators are presented in Tables 3 4 and 5

Tables 3 4 and 5 show that the method proposed byNowman [2] gives good estimates for the parameters 120590 and120574 as manifested by Yu and Phillips [1] it becomes clearthat a starting point for the method is defined by the latterFurthermore through the resulting estimate of the parameter120590 our method becomes much more accurate than Nowman[2] introducing a bias less than 02 for the monthly dataseries and a small value for cases of weekly and daily series

Journal of Probability and Statistics 7

Table 2 Monte Carlo study parameters

Monthly Weekly DailySize 500 1000 2000120582 072 3 6120573 minus012 minus05 minus1120590 06 035 025120574 05 05 05

In the case of 120582 and 120573 are parameters observed inefficientperformance (as discussed Yu and Phillips [1 page 220]) inthe method of Nowman 120582 yielding bias of 87 47 and64 for data monthly weekly and daily respectively Alsofor 120573 the bias present in the estimate is 94 46 and54 for the respective monthly weekly and daily series Themethod proposed byYu andPhillips has a better performancethan Nowman measured from the bias in the parametersreducing it by 15 8 and 16 for the case of 120582 and 5 6and 6 in the case of 120573 for the respective monthly weeklyand daily series Our method has a performance similar tothat of Yu and Phillips for daily data series reducing the biasregarding Nowman method in 15 for 120582 and 4 for 120573 Forweekly data series for reducing 120582 bias regarding the methodof Nowman is 7 whereas for 120573 is 6 as reduction Yu andPhillips method

An important point to note in the results obtained byour method is the condition that the parameter 120574 is knowna priori This is necessary in order to find closed formulas forthe estimators

In the approach to the model proposed by Nowman andYu-Phillips 119889119883

119905= (120582 + 120573119883

119905)119889119905 + 120590119883

120574

119905119889119861119905is implied to have

particular difficulty in estimating the parameter of speed ofreversion To make this condition clear Table 6 shows theresults obtained by the method of Durham and Gallant [4]and our method to estimate the parameters of the model119889119883119905= 120572(120583 + 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905with 120572 gt 0 120583 gt 0 120590 gt 0

120574 = 05 The results are the average of 512 estimations ofmonthly series with 1000 observations

The results shown in Table 6 show that the errors inestimation of 120583 and 120590 are small (less than 1) for bothmethods On the other hand the estimated speed of reversionhas an error of about 10 resulting in Durham and Gallantwhile ourmethod has an error close to 7 From this analysisit is clear that the parameter 120572 has a higher error in estimatingthan the other parameter estimates

In this regard the estimation error of 120572 in Yu [3] providesan analysis of the bias in the parameter estimation inherentreversion total observation time which is sampled in thecontinuous process for the special case where the long-termaverage 120583 is zero Reference is made to the fact that the bias inparameter depends asymptotically on total observation timewhich of the process and not the number of observationsmade in this period

Figures 1 2 and 3 show the results of the estimationof the parameters for different observation horizons withdifferent sampling frequencies (The results are obtained asthe average of the individual estimates on 1000 simulated

0 10 20 30 40 50 60 70 80 9014

145

15

155

16

165

17

175

18

185

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120572 = 15

Figure 1 120572 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

0 10 20 30 40 50 60 70 80 900497

04975

0498

04985

0499

04995

05

05005

0501

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120583 = 05

Figure 2 120583 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

series in 4 sampling scenarios (a) bimonthly series with 500observations (b) monthly series with 1000 observations (c)weekly series with 4333 observations and (d) daily series with21000 observations The reference model for the simulationswas 119889119883

119905= 120572(120583 minus 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905 with 120572 = 15 120583 = 05

120574 = 05 and 120590 = 04)Figure 1 shows the evolution of the estimate of 120572 for

observations from 10 to 83 years Behavior of the estimatesin the four scenarios can be established empirically by thedependence of the bias on the estimate and the horizonof observation as they have more years of observation

8 Journal of Probability and Statistics

Table 3 Comparison of monthly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 13440 minus02332 12330 minus02275 06173 04919 12797 minus02200 05989Bias 06240 minus01132 05130 minus01075 00173 minus00081 05597 minus01000 minus00011Var 05897 01713 05732 01589 00099 00071 05406 00149 00004MSE 09791 01841 08363 01705 00102 00071 08533 00249 00004The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 500observations each The real values of the parameters are 120582 = 072 120573 = minus012 120590 = 06 and 120574 = 05

Table 4 Comparison of weekly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 44090 minus07320 41650 minus07011 03762 04925 42038 minus07022 03499Bias 14090 minus02320 11650 minus02011 00262 minus00075 12038 minus02022 minus00001Var 40663 01109 38608 01030 00172 00357 35427 00992 00001MSE 60855 01647 52180 01435 00179 00358 49883 01400 00001The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 3 120573 = minus05 120590 = 035 and 120574 = 05

Table 5 Comparison of daily data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 98250 minus15360 88440 minus14750 02521 04970 89542 minus14933 02500Bias 38250 minus05360 28440 minus04750 00021 minus00030 29542 minus04933 00000Var 191959 05219 178220 04798 00244 00746 155818 04334 00000MSE 338266 08092 259100 07054 00244 00746 242938 06763 00000The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 60 120573 = minus10 120590 = 025 and 120574 = 05

Table 6 Comparison of Durham and Gallant and proposed methods

Parameter(lowast) Real Bias(a) Bias(b) Real Bias(a) Bias(b)

120572 05 0048900 0037700 05 0059730 0021400120583 006 0000610 0000127 006 minus0000310 minus0000021120590 015 0000200 0000032 022 0000120 minus0000046120572 05 0043780 0032400 04 0045890 0032200120583 006 0000050 0000016 006 0000770 0000046120590 003 0000010 minus0000003 015 0000080 minus0000025The data are obtained with 512 simulations monthly series (119889119905 = 112) with 1000 observations each and the parameters are listed as real(a)Results of the method proposed by Durham and Gallant(b)Results of the method proposed in this paper(lowast)In Durham and Gallant [4] 120583 = 120579

1 120572 = 120579

2 and 120590 = 120579

3

bias decreases with no evidence of a significant differencebetween the estimates different frequencies

From Figure 2 it is observed that after few observationsvirtually independent of the year of observation estimationsfor 120583 are obtained with a bias of less than 1 showingthe efficiency of the estimation of the long-term averagereversion processes average In the case of estimation of 120590Figure 3 shows that the bias in the parameter estimates is

more dependent on the number of observations which is thetotal time of observation of the processThis is reflected in thefact that the estimates are in series with a higher sampling ratereaches a lower bias for the same observation horizon thanlower sample rate and therefore have fewer observations Onthe other hand as for the case of the estimate of 120583 the biaspresent in the estimates of 120590 is practically negligible (less than1) in most scenarios

Journal of Probability and Statistics 9

0 10 20 30 40 50 60 70 80 900386

0388

039

0392

0394

0396

0398

04

0402

0404

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120590 = 04

Figure 3 120590 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

6 Conclusions and Comments

This paper presents a new alternative for Gaussian estimationmodels of mean reversion of one factor under the Eulernumerical approximation that includes among others somemodels of interest rates in the short term

Monte Carlo simulations show a good approximation ofthe diffusion term parameter but less approximation in atrend term parameterThe estimation results have almost thesame order as that in Nowmanrsquos [2] and in some cases betterthan that in Durham and Gallantrsquos [4] It should be notedthat the proposed methodology outperforms the estimate of120590 compared to the other alternatives presented These resultsbecome more relevant since in practice the estimation ofvolatility is extremely useful in the evaluation of financialderivatives in which the trend term loses significance whenit is absorbed by the processes of change measurement andvaluation risk neutral

The estimate based on this new method though doesnot eliminate the bias in estimation of the reversion rateparameter for a small sample is in some cases comparablewith the results in Yu and Phillips [1] if samples have longertime horizons

This estimation scheme has several important qualitiesFirst we come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simpler and does not requiresophisticated routines to perform extensive calculations

References

[1] J Yu and P C B Phillips ldquoA Gaussian approach for continuoustime models of the short-term interest raterdquo The EconometricsJournal vol 4 no 2 pp 210ndash224 2001

[2] K Nowman ldquoGaussian estimation of single-factor continuoustime models of the term structure of interest ratesrdquoThe Journalof Finance vol 52 pp 1695ndash1703 1997

[3] J Yu ldquoBias in the estimation of themean reversion parameter incontinuous time modelsrdquo Journal of Econometrics vol 169 no1 pp 114ndash122 2012

[4] G B Durham and A R Gallant ldquoNumerical techniques formaximum likelihood estimation of continuous-time diffusionprocessesrdquo Journal of Business amp Economic Statistics vol 20 no3 pp 297ndash338 2002

[5] L ValdiviesoW Schoutens and F Tuerlinckx ldquoMaximum like-lihood estimation in processes of Ornstein-Uhlenbeck typerdquoStatistical Inference for Stochastic Processes vol 12 no 1 pp 1ndash192009

[6] O Elerian S Chib and N Shephard ldquoLikelihood inference fordiscretely observed nonlinear diffusionsrdquo Econometrica vol 69no 4 pp 959ndash993 2001

[7] A R Pedersen ldquoConsistency and asymptotic normality ofan approximate maximum likelihood estimator for discretelyobserved diffusion processesrdquo Bernoulli vol 1 no 3 pp 257ndash279 1995

[8] T Koulis and A Thavaneswaran ldquoInference for interest ratemodels using Milsteinrsquos approximationrdquo The Journal of Math-ematical Finance vol 3 pp 110ndash118 2013

[9] G J Jiang and J L Knight ldquoA nonparametric approach to theestimation of diffusion processes with an application to a short-term interest rate modelrdquo Econometric Theory vol 13 no 5 pp615ndash645 1997

[10] D Florens-Zmirou ldquoOn estimating the diffusion coefficientfrom discrete observationsrdquo Journal of Applied Probability vol30 no 4 pp 790ndash804 1993

[11] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 pp 177ndash186 1977

[12] M Brennan and E Schwartz ldquoAnalyzing convertible bondsrdquoJournal of Financial and Quantitative Analysis vol 15 pp 907ndash929 1980

[13] J Cox J Ingersoll and S Ross ldquoAn analysis of variable rate loancontractsrdquoThe Journal of Finance vol 35 pp 389ndash403 1980

[14] K Chan F Karolyi F Longstaff and A Sanders ldquoAn empiricalcomparison of alternative models of short term interest ratesrdquoThe Journal of Finance vol 47 pp 1209ndash1227 1992

[15] A Lari-Lavassani A Sadeghi and A Ware ldquoMean revertingmodels for energy option pricingrdquo Electronic Publications of theInternational Energy Credit Association 2001 httpwwwiecanet

[16] D Pilipovic Energy Risk Valuing andManaging Energy Deriva-tives McGraw-Hill New York NY USA 2007

[17] G Courtadon ldquoThe Pricing of options on default-free bondsrdquoJournal of Financial and Quantitative Analysis vol 17 pp 75ndash100 1982

[18] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[19] K Dunn and J McConnell ldquoValuation of GNMA Mortgage-backed securitiesrdquoThe Journal of Finance vol 36 no 3 pp 599ndash616 1981

[20] K Ramaswamy and S Sundaresan ldquoThe valuation of floating-rate instrumentsrdquo Journal of Financial Economics vol 17 pp251ndash272 1986

[21] S Sundaresan Valuation of Swaps Working Paper ColumbiaUniversity 1989

10 Journal of Probability and Statistics

[22] F Longstaff ldquoThe valuation of options on yieldsrdquo Journal ofFinancial Economics vol 26 no 1 pp 97ndash122 1990

[23] G Constantinides and J Ingersoll ldquoOptimal bond trading withpersonal taxesrdquo Journal of Financial Economics vol 13 pp 299ndash335 1984

[24] G Jiang and J Knight ldquoFinite sample comparison of alternativeestimators of Ito diffusion processes a Monte Carlo studyrdquoTheJournal of Computational Finance vol 2 no 3 1999

[25] L Giet andM LubranoBayesian Inference in Reducible Stochas-tic Differential Equations Document de Travail No 2004ndash57GREQAM University drsquoAix-Marseille 2004

[26] A R Bergstrom ldquoGaussian estimation of structural parametersin higher order continuous time dynamic modelsrdquo Economet-rica vol 51 no 1 pp 117ndash152 1983

[27] A Bergstrom ldquoContinuous time stochastic models and issuesof aggregation over timerdquo in Handbook of Econometrics ZGriliches and M D Intriligator Eds vol 2 Elsevier Amster-dam The Netherlands 1984

[28] A Bergstrom ldquoThe estimation of parameters in nonstationaryhigher-order continuous-time dynamic modelsrdquo EconometricTheory vol 1 pp 369ndash385 1985

[29] A Bergstrom ldquoThe estimation of open higher-order continuoustime dynamic models with mixed stock and flow datardquo Econo-metric Theory vol 2 pp 350ndash373 1986

[30] A Bergstrom Continuous Time Econometric Modelling OxfordUniversity Press Oxford UK 1990

[31] P C B Phillips and J Yu ldquoCorrigendum to lsquoA Gaussianapproach for continuous time models of short-term interestratesrsquordquoThe Econometrics Journal vol 14 no 1 pp 126ndash129 2011

[32] M J D Powell ldquoAn efficient method for finding the minimumof a function of several variables without calculating deriva-tivesrdquoThe Computer Journal vol 7 pp 155ndash162 1964

[33] V S Huzurbazar ldquoThe likelihood equation consistency and themaxima of the likelihood functionrdquo Annals of Eugenics vol 14pp 185ndash200 1948

[34] A Kohatsu-Higa and S Ogawa ldquoWeak rate of convergence foran Euler scheme of nonlinear SDErsquosrdquoMonte Carlo Methods andApplications vol 3 no 4 pp 327ndash345 1997

[35] V Bally andD Talay ldquoThe law of the Euler scheme for stochasticdifferential equations error analysis with Malliavin calculusrdquoMathematics and Computers in Simulation vol 38 no 1ndash3 pp35ndash41 1995

[36] A Zellner ldquoBayesian analysis of regression error termsrdquo Journalof the American Statistical Association vol 70 no 349 pp 138ndash144 1975

[37] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing England UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 3

Table 1 Summary of the models

Model 120572 120583 120590 120574

Vasicek 0Brennan-Schwartz 1CIR SR 12

CIR VR 0 32

CKLS [0 32]

a probability that 119883119905is negative and this is not reasonable

from an economic standpoint Another drawback of thismodel is that conditional volatility assumes changes in theinterest rate that are constants independent of the level of119883

119905

If 120574 = 1 it is said that the process has proportional noiseand gives the nonhomogeneous linear stochastic differentialequation given by119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

119905119889119861119905 1198830= 119909 119905 isin [0 119879] (5)

This model is used by Brennan and Schwartz [12] toobtain a numerical model of convertible bond prices Thisprocess is also used by Courtadon [17] in developing a modelfor option pricing on discount bonds

When 120574 = 12 we obtain the nonlinear stochastic differ-ential equation

119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

12

119905119889119861119905

2120572120583 ge 1205902 1198830= 119909 119905 isin [0 119879]

(6)

This process is naturally linked to the 120594 centered-squaredistribution and is the (square root) version of the model ofBrennan and Schwartz [12] proposed byCox et al [18] whichproduces types of instantaneous short-term interest strictlypositive

This model has also been used extensively in the develop-ment of pricingmodels for derivative instruments sensitive tointerest rates Examples include pricingmodels formortgage-backed in Dunn and McConnell [19] models of options fordiscount bonds in Cox et al [18] future and option pricingmodels for futures in Ramaswamy and Sundaresan [20] theswap valuationmodels in Sundaresan [21] and option pricingmodels for the convenience yield in Longstaff [22]

Finally for the special case in which 120574 = 32 and 120572 = 0equation (1) takes the form

119889119883119905= 12059011988332

119905119889119861119905 1198830= 119909 119905 isin [0 119879] (7)

This model is introduced by Cox et al [13] in their study ofvariable-rate (VR) securities A similar model is also usedby Constantinides and Ingersoll [23] for value bonds in thepresence of taxes

Table 1 shows the summary of the models associatedwith (1) that are of interest in this work (Vasicek Brennan-Schwartz CIR-SRCIR-VR and CKLS)

3 Gaussian Estimation of Diffusion Models

Many theoretical models in economics and finance are for-mulated in continuous time as either a diffusion or diffu-sion systems For this reason identification estimation and

therefore the investigation of the asymptotic properties ofsamples of the estimators of these processes turn out to bea difficult task These problems arise mainly due to the lackof availability of continuous sampling observations In factthe estimation of diffusion models has been considered inthe literature for many years and inmost scientific journals itonly refers to the estimate considering continuous samplingobservations although the data that these models describedcan only be sampled at discrete points in time (See eg Jiangand Knight [9] and references therein)

There are various techniques and methods for estimationof models of mean reversion of a single factor with differentstructural forms in the trend and distribution functionsmostsimulation techniques-based on statistical and quite rigorousarguments The work of Jiang and Knight [9] proposesa parametric identification and estimation procedure fordiffusion processes based on discrete sampling observationsThe nonparametric kernel estimator for diffusion functionsdeveloped in this work deals with general diffusion processesand avoids any functional form specification for either thetrend function or distribution function They show thatunder certain regularity conditions the nonparametric esti-mator function is consistent and timely dissemination alsoasymptotically follows a mixed normal distribution In Jiangand Knight [24] there are some numerical experimentalresults obtained by a Monte Carlo study using the results ofJiang and Knight [9] which include parameter estimationfor Ornstein-Uhlenbeck process In this sense the lack ofanalytical solutions has hampered empirical evidence andvalidation of alternative hypotheses about these continuous-time models

Although the maximum likelihood method is a widelyused parametric method the analytical expressions for thelikelihood functions exist only for processes of mean rever-sion with additive noise In fact in most of the work on esti-mation of diffusion models processes becomemore complexprocesses of Ornstein-Uhlenbeck type For example Gietand Lubrano [25] consider a diffusion process in which theelasticity constant of the nonlinear term of diffusion can befreely chosen achieving after reducing it a transformationto an Ornstein-Uhlenbeck process The major drawback isthat there is no degree of freedom in the form of the trendfunction which is completely determined by the shape of thedistribution function and the choice of the underlying linearmodel Valdivieso et al [5] proposed a methodology to esti-mate maximum likelihood parameters of a one-dimensionalstationary process of Ornstein-Uhlenbeck type which isdriven by general Levy process and which is constructedthrough a distribution D self-decomposed This approachis based on the inversion of the characteristic function andusing the discrete fractional fast Fourier transform Themaximization of the likelihood function is performed usingnumerical methods

The framework proposed in our work has some featuresin common with Nowman [2] Yu and Phillips [1] andElerian et al [6] The similarity of our approach lies in theuse of Euler numerical scheme using discretely sampleddata and maximum likelihood estimation but the maindifference is that in our proposal does not need any numerical

4 Journal of Probability and Statistics

optimization because it is possible to obtain a closed formulafor the estimators

For example in Elerian et al [6] developed a techniquebased on Bayesian estimation of nonlinear EDEs approxi-mating theMarkov transition densities of broadcasts includ-ing Ornstein-Uhlenbeck processes and elasticity models forinterest rates when observations are discretely sampled Theframe of the estimate is based on the introduction of aux-iliary data to complete the diffusion latent missing betweeneach pair of measurements Combine methods of MonteCarlo Markov Chain algorithm based on the Metropolis-Hastings in conjunctionwith the numerical scheme of Euler-Maruyama to sample the posterior distribution of latent dataand model parameters

Another alternative for Gaussian estimation was pro-posed by Nowman [2] In this paper an estimate using theexact discretization of the model is proposed

119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

120574

(119904Δ)119889119861119905 119904Δ le 119905 lt (119904 + 1) Δ

(8)

assuming that volatility stays constant on the intervals [119904Δ(119904 + 1)Δ) 119904 = 0 1

In this way we obtain an equation of the form

119883119905= 119890minus120572Δ

119883(119905minusΔ)

+ 120583 (1 minus 119890minus120572Δ

) + 120578119905 (9)

where the conditional distribution 120578119905| 119865119861119905minus1

asymp 119873(0 (1205902

2120572)(1 minus 119890minus2120572Δ)119883119905minus12120574)

However despite the obtainable approximate conditionaldistribution this approach does not correspond to the Eulerdiscretization applied to (1) This estimation technique re-quires the discrete model corresponding to (2) which is givenby

119883119905= 119890minus120572119883(119905minus1)

+ 120583 (1 minus 119890minus120572) + 120578119905 (119905 = 0 1 119879) (10)

where 120578119905 (119905 = 0 1 119879) satisfies the conditions 119864[120578

119904120578119905] = 0

(119904 = 119905) and

119864 [1205782

119905] = int119905

119905minus1

119890minus2(119905minus120591)120572

1205902119883(119905minus1)

2120574

119889120591

=1205902

2120572(1 minus 119890

minus2120572) 119883(119905minus1)

2120574

= 1198982

119905119905

(11)

This states that the logarithmof theGaussian likelihood func-tion may be written as

119871 (120579) =

119879

sum119905=1

[2 log119898119905119905+

119883119905minus 119890minus120572119883

(119905minus1)+ 120583 (1 minus 119890minus120572)

119898119905119905

2

]

=

119879

sum119905=1

[2 log119898119905119905+ 1205762

119905]

(12)

where 120576 = [1205761 120576

119879] is a vector whose elements can be

calculatedThe optimization of 119871(120579) is carried out using the methods

developed by Bergstrom [26ndash30] and Nowman [2]

Yu and Phillips [1] propose a Gaussian estimation usingrandom time changes which is based on the idea that anycontinuous time martingale can be written as a Brownianmotion after a suitable time change It considers the explicitsolution (1) to obtain the expression

119883(119905+ℎ)

=120583 (1minus119890minus120572ℎ

)+119890minus120572ℎ

119883(119905)+intℎ

0

120590119890minus120572(ℎminus120591)

119883(119905+120591)

120574

119889119861120591

for ℎ gt 0(13)

Let 119872(ℎ) = 120590 intℎ

0119890minus120572(ℎminus120591)119883

(119905+120591)120574119889119861120591 119872(ℎ) is a contin-

uous martingale (In Yu and Phillips [1] 119872(ℎ) was assumedas a continuous martingale but this assumption is generallynot warranted and [119872]

ℎdoes not induce a Brownianmotion

as Yu and Phillips showed in [31] In addition they establishthat a simple decomposition splits the error process into atrend component and a continuousmartingale process) withquadratic variation

[119872]ℎ= 1205902intℎ

0

119890minus2(ℎminus120591)120572

119883(119905+120591)

2120574

119889120591 (14)

They introduce a sequence of positive numbers ℎ119895which

deliver the required time changes For any fixed constant 119886 gt0 let

ℎ119895+1

= inf 119904 | [119872119895]119904ge 119886

= inf 119904 | 1205902 int119904

0

119890minus2(119904minus120591)120572

119883(119905119895+120591)

2120574

119889120591 ge 119886

(15)

Then a succession of points 119905119895 is constructed using the

iterations 119905119895+1

= 119905119895+ ℎ119895+1

with 1199051= 0 to obtain

119883(119905119895+1)

= 120583 (1 minus 119890minus120572ℎ119895+1) + 119890

minus120572ℎ119895+1119883(119905119895)

+119872ℎ119895+1

(16)

where119872ℎ119895+1

= 119861(119886) asymp 119873(0 119886)Note that according to (16) (12) takes the form

119871 (120579) =1

119873sum119895

[[

[

2 log 119886

+

119883(119905119895+1)

minus120583 (1minus119890minus120572ℎ119895+1)minus119890minus120572ℎ119895+1119883(119905119895)

119886

2

]]

]

(17)

Since the data are observed in discrete time the timechange formula cannot be applied directly and thereforemustbe approximated by

ℎ119895+1

= Δmin119904 |119904

sum119894=1

1205902119890minus2120572(119904minus119894)Δ

119883(119905119895+119894Δ)

2120574

ge 119886 (18)

This should be selected so that 119886 is the volatility of theerror term unconditional in the equation

119883(119905+Δ)

= 120583 (1 minus 119890minus120572Δ

) + 119890minus120572Δ

119883(119905)+ 120576 (19)

Journal of Probability and Statistics 5

The implementation of this method is based on theestimate given by Nowman [2] to find the parameters esti-mates 120590 and 120574 Furthermore the estimation of the remainingparameters 120583 and 120572 is carried out using the numericaloptimization algorithm by Powell [32]

4 Gaussian Estimation ofOne-Factor Mean Reversion Processes withConstant Parameters

The development of this section will focus on the descriptionof the proposed technique for the Gaussian estimation ofmean reversion models of a single factor with linear andnonlinear functions of diffusion constant parameters andlinear trend functions with constant parameters

As the density function for most diffusion processes isnot known analytically the proposed idea is not to find orapproximate the transition density function or the marginaldensity function of the process Otherwise we use thenumerical approximation of Euler on the EDE to build a newdiscrete process Huzurbazar [33] and use the fact that inthis approximation the normalized residuals resulting fromBrownian process are independent standard normal randomvariables As pointed out by Pedersen [7] Kohatsu-Higa andOgawa [34] and Bally and Talay [35] the density functionof the discrete process obtained with Euler scheme convergesto the density function of the continuous process and asdiscussed by Zellner [36] these residues can be interpretedas a vector of unknown parameters with the advantage ofknowing their distribution So we build a new variable forthe models defined in (1) so that it can adapt to the observeddata and later can be evaluated by examining the normalizedresiduals that will be useful in the estimation process Byknowing the density function of this new process we canfind discretized likelihood function and use the conventionalmethod of maximum likelihood estimation to find a closedformula for the parameters

Consider the stochastic differential equation (1) with theinitial condition119883

0= 119909 The terms 120572(120583 minus 119883

119905) and 120590119883120574

119905being

the trend and diffusion functions respectively that dependon 119883119905and a vector of unknown parameters 120579 = [120572 120583 120590 120574]

where 119861119905119905ge0

is a unidimensional standard Brownianmotiondefined on a complete probability space (ΩFP) The trendfunction describes the movement of the process due tochanges in time while the diffusion function measures themagnitude of the randomfluctuations around the trend func-tion Suppose that the conditions under which we guaranteethe existence and uniqueness of the solution of Mao [37] aresatisfied Assume further the process observations in discretetime119883

119905= 119883(120591

119905) at times 120591

0 1205911 120591

119879whereΔ

119905= 120591119905minus120591119905minus1

ge

0 119905 = 1 2 119879 The objective is to estimate the unknownparameter vector 120579 given the historical information of thesample119883 = (119883

0 1198831 119883

119879)1015840 with equally spaced data Note

that the explicit solution of (1) is given by

119883119905= 120583 + (119883

0minus 120583) 119890

minus120572119905+ int119905

0

120590119890minus120572(119905minus119904)

119883120574

119904119889119861119904 (20)

In the context of the likelihood the estimation of 120579 isbased on the likelihood function

log 119871 (1198831 1198832 119883

119879| 1198830 120579) =

119879

sum119905=1

[log119892 (119883119905+1

| 119883119905 120579)]

(21)

where 119892(119883119905+1

| 119883119905 120579) are the Markov transition densities

or 119871(120579 | 1198831 1198832 119883

119879) = 119891(119883

1 1198832 119883

119879 120579) and 119891(119883

1

1198832 119883

119879 120579) is the joint density function

Although the differential equation (1) can be solved ana-lytically on [0 119879] the functions 119892(119883

119905+1| 119883119905 120579) and 119891(119883

1

1198832 119883

119879 120579) are not available in closed form and therefore

the estimate can not be accomplished except for the casewhere 120574 = 0

Note that the exact discretized solution of (1) takes theform

119883(120591119905) = 120583 + (119883 (120591

119905minus1) minus 120583) 119890

minus120572(120591119905minus120591119905minus1)

+ int120591

120591119905minus1

120590119890minus120572(120591119905minus119904)119883

120574

(119904)119889119861119904

(22)

which can be approximated by

119883(120591119905) = 120583 + (119883 (120591

119905minus1) minus 120583) 119890

minus120572Δ 119905

+ 120590119890minus120572Δ 119905119883

120574(120591119905minus1) Δ119861119905

(23)

where Δ119861119905= 119861120591119905minus 119861120591119905minus1

Thus

119883(120591119905) = 119883 (120591

119905minus1) + 120572 (120583 minus 119883 (120591

119905minus1)) Δ119905+ 119883120574(120591119905minus1) 120576120591119905

(24)

where 120576120591119905sim 119873(0 1205902Δ

119905) corresponding to the numerical ap-

proximation of EulerNow define the new variable

119884119894=119883119894minus 119883119894minus1

minus 120572 (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

= 120576119894 Δ constant (25)

Note that 119884119894 119894 = 1 2 3 119879 are independent random

variables distributed normal119873(0 1205902Δ)In terms of the density function can be written

119891 (1198841 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(1198831minus1198830minus120572 (120583minus119883

0) Δ

119883120574

0

)

2

]

6 Journal of Probability and Statistics

119891 (1198842 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(1198832minus1198831minus120572 (120583minus119883

1) Δ

119883120574

1

)

2

]

119891 (119884119879 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(119883119879minus119883119879minus1

minus120572 (120583minus119883119879minus1

) Δ

119883120574

119879minus1

)

2

]

(26)

Since each 119884119894is independent can be expressed the joint

density function as their marginal densities product that is

119891 (1198841 1198842 119884

119879) = 119891 (119884

1 120579) 119891 (119884

2 120579) sdot sdot sdot 119891 (119884

119879 120579)

(27)

Consequently the likelihood function is given by

119871 (120579 | 119884119879) = (

1

21205871205902Δ)1198792

times exp[ minus1

21205902Δ

119879

sum119894=1

(119883119894minus119883119894minus1minus120572 (120583minus119883

119894minus1) Δ

119883120574

119894minus1

)

2

]

(28)

Taking the natural logarithm on both sides it follows that

log 119871 = minus119879

2log (21205871205902Δ)

minus1

21205902Δ[

119879

sum119894=1

(119883119894minus 119883119894minus1

minus 120572 (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

]

(29)

Now consider the problem of maximizing the likelihoodfunction given by

120597 log (119871)120597120579

= 0 ie 120579 = Arg120579

max (log (119871)) (30)

The estimation process leads to the estimated parameters120572 and 120583 are the solutions of which the next equation systemwhere it is assumed that the value of 120574 is known Consider

119860 minus 119864 (1 minus 120572Δ) minus 120572120583Δ119861 = 0

119862 minus 119861 (1 minus 120572Δ) minus 120572120583Δ119863 = 0(31)

where

119860 =

119879

sum119894=1

119883119894119883119894minus1

1198832120574

119894minus1

119861 =

119879

sum119894=1

119883119894minus1

1198832120574

119894minus1

119862 =

119879

sum119894=1

119883119894

1198832120574

119894minus1

119863 =

119879

sum119894=1

1

1198832120574

119894minus1

119864 =

119879

sum119894=1

(119883119894minus1

119883120574

119894minus1

)

2

(32)

Thus the estimated parameters are given by

=119864119863 minus 119861

2 minus 119860119863 + 119861119862

(119864119863 minus 1198612) Δ

120583 =119860 minus 119864 (1 minus Δ)

119861Δ

(33)

Additionally the parameter 120590 is determined by an equa-tion independent of the system of (31) and is completelydetermined by the estimated parameters

= radic1

119879Δ

119879

sum119894=1

(119883119894minus 119883119894minus1

minus (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

(34)

This estimation scheme has several important qualitiesFirst you come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simple and does not requiresophisticated routines to perform calculations very extensive

5 Simulations

This sectionwill present comparisons of the results with thosepublished by other authors using the techniques describedabove Comparisons are made with the estimate based onsimulated data sets to analyze statistical properties of theproposed estimators

Yu andPhillips [1] present the results of a simulation studywith the Monte Carlo technique in comparing their methodwith Nowmanrsquos [2] To make this comparison are suggestedthree scenarios to resolve data monthly weekly and daily Inall cases the basic pattern of square root is defined as

119889119883119905= (120582 + 120573119883

119905) 119889119905 + 120590119883

120574

119905119889119861119905

with 120582 gt 0 120573 lt 0 120590 gt 0 120574 = 05(35)

Note that with appropriate substitution of this modelparameters are equivalent to those described in (1) (Theequivalence of models (1) and (35) is satisfied by having120582 = 120572120583 and 120573 = minus120572 Originally in Yu and Phillips [1] theparameter 120582 in (35) is defined as 120572 but was changed to avoidconfusion with the model (1))

The specific parameters used in each stage are shown inTable 2

To perform the first comparison parameters are set asspecified in Table 2 and obtained a series simulatedThen youset the value to 120574 = 05 and proceed to the estimation by themethod of Section 4 to obtain 120573 and This procedure wasrepeated 1000 times (as inYu andPhillips [1]) Some statisticalestimators are presented in Tables 3 4 and 5

Tables 3 4 and 5 show that the method proposed byNowman [2] gives good estimates for the parameters 120590 and120574 as manifested by Yu and Phillips [1] it becomes clearthat a starting point for the method is defined by the latterFurthermore through the resulting estimate of the parameter120590 our method becomes much more accurate than Nowman[2] introducing a bias less than 02 for the monthly dataseries and a small value for cases of weekly and daily series

Journal of Probability and Statistics 7

Table 2 Monte Carlo study parameters

Monthly Weekly DailySize 500 1000 2000120582 072 3 6120573 minus012 minus05 minus1120590 06 035 025120574 05 05 05

In the case of 120582 and 120573 are parameters observed inefficientperformance (as discussed Yu and Phillips [1 page 220]) inthe method of Nowman 120582 yielding bias of 87 47 and64 for data monthly weekly and daily respectively Alsofor 120573 the bias present in the estimate is 94 46 and54 for the respective monthly weekly and daily series Themethod proposed byYu andPhillips has a better performancethan Nowman measured from the bias in the parametersreducing it by 15 8 and 16 for the case of 120582 and 5 6and 6 in the case of 120573 for the respective monthly weeklyand daily series Our method has a performance similar tothat of Yu and Phillips for daily data series reducing the biasregarding Nowman method in 15 for 120582 and 4 for 120573 Forweekly data series for reducing 120582 bias regarding the methodof Nowman is 7 whereas for 120573 is 6 as reduction Yu andPhillips method

An important point to note in the results obtained byour method is the condition that the parameter 120574 is knowna priori This is necessary in order to find closed formulas forthe estimators

In the approach to the model proposed by Nowman andYu-Phillips 119889119883

119905= (120582 + 120573119883

119905)119889119905 + 120590119883

120574

119905119889119861119905is implied to have

particular difficulty in estimating the parameter of speed ofreversion To make this condition clear Table 6 shows theresults obtained by the method of Durham and Gallant [4]and our method to estimate the parameters of the model119889119883119905= 120572(120583 + 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905with 120572 gt 0 120583 gt 0 120590 gt 0

120574 = 05 The results are the average of 512 estimations ofmonthly series with 1000 observations

The results shown in Table 6 show that the errors inestimation of 120583 and 120590 are small (less than 1) for bothmethods On the other hand the estimated speed of reversionhas an error of about 10 resulting in Durham and Gallantwhile ourmethod has an error close to 7 From this analysisit is clear that the parameter 120572 has a higher error in estimatingthan the other parameter estimates

In this regard the estimation error of 120572 in Yu [3] providesan analysis of the bias in the parameter estimation inherentreversion total observation time which is sampled in thecontinuous process for the special case where the long-termaverage 120583 is zero Reference is made to the fact that the bias inparameter depends asymptotically on total observation timewhich of the process and not the number of observationsmade in this period

Figures 1 2 and 3 show the results of the estimationof the parameters for different observation horizons withdifferent sampling frequencies (The results are obtained asthe average of the individual estimates on 1000 simulated

0 10 20 30 40 50 60 70 80 9014

145

15

155

16

165

17

175

18

185

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120572 = 15

Figure 1 120572 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

0 10 20 30 40 50 60 70 80 900497

04975

0498

04985

0499

04995

05

05005

0501

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120583 = 05

Figure 2 120583 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

series in 4 sampling scenarios (a) bimonthly series with 500observations (b) monthly series with 1000 observations (c)weekly series with 4333 observations and (d) daily series with21000 observations The reference model for the simulationswas 119889119883

119905= 120572(120583 minus 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905 with 120572 = 15 120583 = 05

120574 = 05 and 120590 = 04)Figure 1 shows the evolution of the estimate of 120572 for

observations from 10 to 83 years Behavior of the estimatesin the four scenarios can be established empirically by thedependence of the bias on the estimate and the horizonof observation as they have more years of observation

8 Journal of Probability and Statistics

Table 3 Comparison of monthly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 13440 minus02332 12330 minus02275 06173 04919 12797 minus02200 05989Bias 06240 minus01132 05130 minus01075 00173 minus00081 05597 minus01000 minus00011Var 05897 01713 05732 01589 00099 00071 05406 00149 00004MSE 09791 01841 08363 01705 00102 00071 08533 00249 00004The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 500observations each The real values of the parameters are 120582 = 072 120573 = minus012 120590 = 06 and 120574 = 05

Table 4 Comparison of weekly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 44090 minus07320 41650 minus07011 03762 04925 42038 minus07022 03499Bias 14090 minus02320 11650 minus02011 00262 minus00075 12038 minus02022 minus00001Var 40663 01109 38608 01030 00172 00357 35427 00992 00001MSE 60855 01647 52180 01435 00179 00358 49883 01400 00001The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 3 120573 = minus05 120590 = 035 and 120574 = 05

Table 5 Comparison of daily data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 98250 minus15360 88440 minus14750 02521 04970 89542 minus14933 02500Bias 38250 minus05360 28440 minus04750 00021 minus00030 29542 minus04933 00000Var 191959 05219 178220 04798 00244 00746 155818 04334 00000MSE 338266 08092 259100 07054 00244 00746 242938 06763 00000The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 60 120573 = minus10 120590 = 025 and 120574 = 05

Table 6 Comparison of Durham and Gallant and proposed methods

Parameter(lowast) Real Bias(a) Bias(b) Real Bias(a) Bias(b)

120572 05 0048900 0037700 05 0059730 0021400120583 006 0000610 0000127 006 minus0000310 minus0000021120590 015 0000200 0000032 022 0000120 minus0000046120572 05 0043780 0032400 04 0045890 0032200120583 006 0000050 0000016 006 0000770 0000046120590 003 0000010 minus0000003 015 0000080 minus0000025The data are obtained with 512 simulations monthly series (119889119905 = 112) with 1000 observations each and the parameters are listed as real(a)Results of the method proposed by Durham and Gallant(b)Results of the method proposed in this paper(lowast)In Durham and Gallant [4] 120583 = 120579

1 120572 = 120579

2 and 120590 = 120579

3

bias decreases with no evidence of a significant differencebetween the estimates different frequencies

From Figure 2 it is observed that after few observationsvirtually independent of the year of observation estimationsfor 120583 are obtained with a bias of less than 1 showingthe efficiency of the estimation of the long-term averagereversion processes average In the case of estimation of 120590Figure 3 shows that the bias in the parameter estimates is

more dependent on the number of observations which is thetotal time of observation of the processThis is reflected in thefact that the estimates are in series with a higher sampling ratereaches a lower bias for the same observation horizon thanlower sample rate and therefore have fewer observations Onthe other hand as for the case of the estimate of 120583 the biaspresent in the estimates of 120590 is practically negligible (less than1) in most scenarios

Journal of Probability and Statistics 9

0 10 20 30 40 50 60 70 80 900386

0388

039

0392

0394

0396

0398

04

0402

0404

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120590 = 04

Figure 3 120590 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

6 Conclusions and Comments

This paper presents a new alternative for Gaussian estimationmodels of mean reversion of one factor under the Eulernumerical approximation that includes among others somemodels of interest rates in the short term

Monte Carlo simulations show a good approximation ofthe diffusion term parameter but less approximation in atrend term parameterThe estimation results have almost thesame order as that in Nowmanrsquos [2] and in some cases betterthan that in Durham and Gallantrsquos [4] It should be notedthat the proposed methodology outperforms the estimate of120590 compared to the other alternatives presented These resultsbecome more relevant since in practice the estimation ofvolatility is extremely useful in the evaluation of financialderivatives in which the trend term loses significance whenit is absorbed by the processes of change measurement andvaluation risk neutral

The estimate based on this new method though doesnot eliminate the bias in estimation of the reversion rateparameter for a small sample is in some cases comparablewith the results in Yu and Phillips [1] if samples have longertime horizons

This estimation scheme has several important qualitiesFirst we come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simpler and does not requiresophisticated routines to perform extensive calculations

References

[1] J Yu and P C B Phillips ldquoA Gaussian approach for continuoustime models of the short-term interest raterdquo The EconometricsJournal vol 4 no 2 pp 210ndash224 2001

[2] K Nowman ldquoGaussian estimation of single-factor continuoustime models of the term structure of interest ratesrdquoThe Journalof Finance vol 52 pp 1695ndash1703 1997

[3] J Yu ldquoBias in the estimation of themean reversion parameter incontinuous time modelsrdquo Journal of Econometrics vol 169 no1 pp 114ndash122 2012

[4] G B Durham and A R Gallant ldquoNumerical techniques formaximum likelihood estimation of continuous-time diffusionprocessesrdquo Journal of Business amp Economic Statistics vol 20 no3 pp 297ndash338 2002

[5] L ValdiviesoW Schoutens and F Tuerlinckx ldquoMaximum like-lihood estimation in processes of Ornstein-Uhlenbeck typerdquoStatistical Inference for Stochastic Processes vol 12 no 1 pp 1ndash192009

[6] O Elerian S Chib and N Shephard ldquoLikelihood inference fordiscretely observed nonlinear diffusionsrdquo Econometrica vol 69no 4 pp 959ndash993 2001

[7] A R Pedersen ldquoConsistency and asymptotic normality ofan approximate maximum likelihood estimator for discretelyobserved diffusion processesrdquo Bernoulli vol 1 no 3 pp 257ndash279 1995

[8] T Koulis and A Thavaneswaran ldquoInference for interest ratemodels using Milsteinrsquos approximationrdquo The Journal of Math-ematical Finance vol 3 pp 110ndash118 2013

[9] G J Jiang and J L Knight ldquoA nonparametric approach to theestimation of diffusion processes with an application to a short-term interest rate modelrdquo Econometric Theory vol 13 no 5 pp615ndash645 1997

[10] D Florens-Zmirou ldquoOn estimating the diffusion coefficientfrom discrete observationsrdquo Journal of Applied Probability vol30 no 4 pp 790ndash804 1993

[11] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 pp 177ndash186 1977

[12] M Brennan and E Schwartz ldquoAnalyzing convertible bondsrdquoJournal of Financial and Quantitative Analysis vol 15 pp 907ndash929 1980

[13] J Cox J Ingersoll and S Ross ldquoAn analysis of variable rate loancontractsrdquoThe Journal of Finance vol 35 pp 389ndash403 1980

[14] K Chan F Karolyi F Longstaff and A Sanders ldquoAn empiricalcomparison of alternative models of short term interest ratesrdquoThe Journal of Finance vol 47 pp 1209ndash1227 1992

[15] A Lari-Lavassani A Sadeghi and A Ware ldquoMean revertingmodels for energy option pricingrdquo Electronic Publications of theInternational Energy Credit Association 2001 httpwwwiecanet

[16] D Pilipovic Energy Risk Valuing andManaging Energy Deriva-tives McGraw-Hill New York NY USA 2007

[17] G Courtadon ldquoThe Pricing of options on default-free bondsrdquoJournal of Financial and Quantitative Analysis vol 17 pp 75ndash100 1982

[18] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[19] K Dunn and J McConnell ldquoValuation of GNMA Mortgage-backed securitiesrdquoThe Journal of Finance vol 36 no 3 pp 599ndash616 1981

[20] K Ramaswamy and S Sundaresan ldquoThe valuation of floating-rate instrumentsrdquo Journal of Financial Economics vol 17 pp251ndash272 1986

[21] S Sundaresan Valuation of Swaps Working Paper ColumbiaUniversity 1989

10 Journal of Probability and Statistics

[22] F Longstaff ldquoThe valuation of options on yieldsrdquo Journal ofFinancial Economics vol 26 no 1 pp 97ndash122 1990

[23] G Constantinides and J Ingersoll ldquoOptimal bond trading withpersonal taxesrdquo Journal of Financial Economics vol 13 pp 299ndash335 1984

[24] G Jiang and J Knight ldquoFinite sample comparison of alternativeestimators of Ito diffusion processes a Monte Carlo studyrdquoTheJournal of Computational Finance vol 2 no 3 1999

[25] L Giet andM LubranoBayesian Inference in Reducible Stochas-tic Differential Equations Document de Travail No 2004ndash57GREQAM University drsquoAix-Marseille 2004

[26] A R Bergstrom ldquoGaussian estimation of structural parametersin higher order continuous time dynamic modelsrdquo Economet-rica vol 51 no 1 pp 117ndash152 1983

[27] A Bergstrom ldquoContinuous time stochastic models and issuesof aggregation over timerdquo in Handbook of Econometrics ZGriliches and M D Intriligator Eds vol 2 Elsevier Amster-dam The Netherlands 1984

[28] A Bergstrom ldquoThe estimation of parameters in nonstationaryhigher-order continuous-time dynamic modelsrdquo EconometricTheory vol 1 pp 369ndash385 1985

[29] A Bergstrom ldquoThe estimation of open higher-order continuoustime dynamic models with mixed stock and flow datardquo Econo-metric Theory vol 2 pp 350ndash373 1986

[30] A Bergstrom Continuous Time Econometric Modelling OxfordUniversity Press Oxford UK 1990

[31] P C B Phillips and J Yu ldquoCorrigendum to lsquoA Gaussianapproach for continuous time models of short-term interestratesrsquordquoThe Econometrics Journal vol 14 no 1 pp 126ndash129 2011

[32] M J D Powell ldquoAn efficient method for finding the minimumof a function of several variables without calculating deriva-tivesrdquoThe Computer Journal vol 7 pp 155ndash162 1964

[33] V S Huzurbazar ldquoThe likelihood equation consistency and themaxima of the likelihood functionrdquo Annals of Eugenics vol 14pp 185ndash200 1948

[34] A Kohatsu-Higa and S Ogawa ldquoWeak rate of convergence foran Euler scheme of nonlinear SDErsquosrdquoMonte Carlo Methods andApplications vol 3 no 4 pp 327ndash345 1997

[35] V Bally andD Talay ldquoThe law of the Euler scheme for stochasticdifferential equations error analysis with Malliavin calculusrdquoMathematics and Computers in Simulation vol 38 no 1ndash3 pp35ndash41 1995

[36] A Zellner ldquoBayesian analysis of regression error termsrdquo Journalof the American Statistical Association vol 70 no 349 pp 138ndash144 1975

[37] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing England UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Probability and Statistics

optimization because it is possible to obtain a closed formulafor the estimators

For example in Elerian et al [6] developed a techniquebased on Bayesian estimation of nonlinear EDEs approxi-mating theMarkov transition densities of broadcasts includ-ing Ornstein-Uhlenbeck processes and elasticity models forinterest rates when observations are discretely sampled Theframe of the estimate is based on the introduction of aux-iliary data to complete the diffusion latent missing betweeneach pair of measurements Combine methods of MonteCarlo Markov Chain algorithm based on the Metropolis-Hastings in conjunctionwith the numerical scheme of Euler-Maruyama to sample the posterior distribution of latent dataand model parameters

Another alternative for Gaussian estimation was pro-posed by Nowman [2] In this paper an estimate using theexact discretization of the model is proposed

119889119883119905= 120572 (120583 minus 119883

119905) 119889119905 + 120590119883

120574

(119904Δ)119889119861119905 119904Δ le 119905 lt (119904 + 1) Δ

(8)

assuming that volatility stays constant on the intervals [119904Δ(119904 + 1)Δ) 119904 = 0 1

In this way we obtain an equation of the form

119883119905= 119890minus120572Δ

119883(119905minusΔ)

+ 120583 (1 minus 119890minus120572Δ

) + 120578119905 (9)

where the conditional distribution 120578119905| 119865119861119905minus1

asymp 119873(0 (1205902

2120572)(1 minus 119890minus2120572Δ)119883119905minus12120574)

However despite the obtainable approximate conditionaldistribution this approach does not correspond to the Eulerdiscretization applied to (1) This estimation technique re-quires the discrete model corresponding to (2) which is givenby

119883119905= 119890minus120572119883(119905minus1)

+ 120583 (1 minus 119890minus120572) + 120578119905 (119905 = 0 1 119879) (10)

where 120578119905 (119905 = 0 1 119879) satisfies the conditions 119864[120578

119904120578119905] = 0

(119904 = 119905) and

119864 [1205782

119905] = int119905

119905minus1

119890minus2(119905minus120591)120572

1205902119883(119905minus1)

2120574

119889120591

=1205902

2120572(1 minus 119890

minus2120572) 119883(119905minus1)

2120574

= 1198982

119905119905

(11)

This states that the logarithmof theGaussian likelihood func-tion may be written as

119871 (120579) =

119879

sum119905=1

[2 log119898119905119905+

119883119905minus 119890minus120572119883

(119905minus1)+ 120583 (1 minus 119890minus120572)

119898119905119905

2

]

=

119879

sum119905=1

[2 log119898119905119905+ 1205762

119905]

(12)

where 120576 = [1205761 120576

119879] is a vector whose elements can be

calculatedThe optimization of 119871(120579) is carried out using the methods

developed by Bergstrom [26ndash30] and Nowman [2]

Yu and Phillips [1] propose a Gaussian estimation usingrandom time changes which is based on the idea that anycontinuous time martingale can be written as a Brownianmotion after a suitable time change It considers the explicitsolution (1) to obtain the expression

119883(119905+ℎ)

=120583 (1minus119890minus120572ℎ

)+119890minus120572ℎ

119883(119905)+intℎ

0

120590119890minus120572(ℎminus120591)

119883(119905+120591)

120574

119889119861120591

for ℎ gt 0(13)

Let 119872(ℎ) = 120590 intℎ

0119890minus120572(ℎminus120591)119883

(119905+120591)120574119889119861120591 119872(ℎ) is a contin-

uous martingale (In Yu and Phillips [1] 119872(ℎ) was assumedas a continuous martingale but this assumption is generallynot warranted and [119872]

ℎdoes not induce a Brownianmotion

as Yu and Phillips showed in [31] In addition they establishthat a simple decomposition splits the error process into atrend component and a continuousmartingale process) withquadratic variation

[119872]ℎ= 1205902intℎ

0

119890minus2(ℎminus120591)120572

119883(119905+120591)

2120574

119889120591 (14)

They introduce a sequence of positive numbers ℎ119895which

deliver the required time changes For any fixed constant 119886 gt0 let

ℎ119895+1

= inf 119904 | [119872119895]119904ge 119886

= inf 119904 | 1205902 int119904

0

119890minus2(119904minus120591)120572

119883(119905119895+120591)

2120574

119889120591 ge 119886

(15)

Then a succession of points 119905119895 is constructed using the

iterations 119905119895+1

= 119905119895+ ℎ119895+1

with 1199051= 0 to obtain

119883(119905119895+1)

= 120583 (1 minus 119890minus120572ℎ119895+1) + 119890

minus120572ℎ119895+1119883(119905119895)

+119872ℎ119895+1

(16)

where119872ℎ119895+1

= 119861(119886) asymp 119873(0 119886)Note that according to (16) (12) takes the form

119871 (120579) =1

119873sum119895

[[

[

2 log 119886

+

119883(119905119895+1)

minus120583 (1minus119890minus120572ℎ119895+1)minus119890minus120572ℎ119895+1119883(119905119895)

119886

2

]]

]

(17)

Since the data are observed in discrete time the timechange formula cannot be applied directly and thereforemustbe approximated by

ℎ119895+1

= Δmin119904 |119904

sum119894=1

1205902119890minus2120572(119904minus119894)Δ

119883(119905119895+119894Δ)

2120574

ge 119886 (18)

This should be selected so that 119886 is the volatility of theerror term unconditional in the equation

119883(119905+Δ)

= 120583 (1 minus 119890minus120572Δ

) + 119890minus120572Δ

119883(119905)+ 120576 (19)

Journal of Probability and Statistics 5

The implementation of this method is based on theestimate given by Nowman [2] to find the parameters esti-mates 120590 and 120574 Furthermore the estimation of the remainingparameters 120583 and 120572 is carried out using the numericaloptimization algorithm by Powell [32]

4 Gaussian Estimation ofOne-Factor Mean Reversion Processes withConstant Parameters

The development of this section will focus on the descriptionof the proposed technique for the Gaussian estimation ofmean reversion models of a single factor with linear andnonlinear functions of diffusion constant parameters andlinear trend functions with constant parameters

As the density function for most diffusion processes isnot known analytically the proposed idea is not to find orapproximate the transition density function or the marginaldensity function of the process Otherwise we use thenumerical approximation of Euler on the EDE to build a newdiscrete process Huzurbazar [33] and use the fact that inthis approximation the normalized residuals resulting fromBrownian process are independent standard normal randomvariables As pointed out by Pedersen [7] Kohatsu-Higa andOgawa [34] and Bally and Talay [35] the density functionof the discrete process obtained with Euler scheme convergesto the density function of the continuous process and asdiscussed by Zellner [36] these residues can be interpretedas a vector of unknown parameters with the advantage ofknowing their distribution So we build a new variable forthe models defined in (1) so that it can adapt to the observeddata and later can be evaluated by examining the normalizedresiduals that will be useful in the estimation process Byknowing the density function of this new process we canfind discretized likelihood function and use the conventionalmethod of maximum likelihood estimation to find a closedformula for the parameters

Consider the stochastic differential equation (1) with theinitial condition119883

0= 119909 The terms 120572(120583 minus 119883

119905) and 120590119883120574

119905being

the trend and diffusion functions respectively that dependon 119883119905and a vector of unknown parameters 120579 = [120572 120583 120590 120574]

where 119861119905119905ge0

is a unidimensional standard Brownianmotiondefined on a complete probability space (ΩFP) The trendfunction describes the movement of the process due tochanges in time while the diffusion function measures themagnitude of the randomfluctuations around the trend func-tion Suppose that the conditions under which we guaranteethe existence and uniqueness of the solution of Mao [37] aresatisfied Assume further the process observations in discretetime119883

119905= 119883(120591

119905) at times 120591

0 1205911 120591

119879whereΔ

119905= 120591119905minus120591119905minus1

ge

0 119905 = 1 2 119879 The objective is to estimate the unknownparameter vector 120579 given the historical information of thesample119883 = (119883

0 1198831 119883

119879)1015840 with equally spaced data Note

that the explicit solution of (1) is given by

119883119905= 120583 + (119883

0minus 120583) 119890

minus120572119905+ int119905

0

120590119890minus120572(119905minus119904)

119883120574

119904119889119861119904 (20)

In the context of the likelihood the estimation of 120579 isbased on the likelihood function

log 119871 (1198831 1198832 119883

119879| 1198830 120579) =

119879

sum119905=1

[log119892 (119883119905+1

| 119883119905 120579)]

(21)

where 119892(119883119905+1

| 119883119905 120579) are the Markov transition densities

or 119871(120579 | 1198831 1198832 119883

119879) = 119891(119883

1 1198832 119883

119879 120579) and 119891(119883

1

1198832 119883

119879 120579) is the joint density function

Although the differential equation (1) can be solved ana-lytically on [0 119879] the functions 119892(119883

119905+1| 119883119905 120579) and 119891(119883

1

1198832 119883

119879 120579) are not available in closed form and therefore

the estimate can not be accomplished except for the casewhere 120574 = 0

Note that the exact discretized solution of (1) takes theform

119883(120591119905) = 120583 + (119883 (120591

119905minus1) minus 120583) 119890

minus120572(120591119905minus120591119905minus1)

+ int120591

120591119905minus1

120590119890minus120572(120591119905minus119904)119883

120574

(119904)119889119861119904

(22)

which can be approximated by

119883(120591119905) = 120583 + (119883 (120591

119905minus1) minus 120583) 119890

minus120572Δ 119905

+ 120590119890minus120572Δ 119905119883

120574(120591119905minus1) Δ119861119905

(23)

where Δ119861119905= 119861120591119905minus 119861120591119905minus1

Thus

119883(120591119905) = 119883 (120591

119905minus1) + 120572 (120583 minus 119883 (120591

119905minus1)) Δ119905+ 119883120574(120591119905minus1) 120576120591119905

(24)

where 120576120591119905sim 119873(0 1205902Δ

119905) corresponding to the numerical ap-

proximation of EulerNow define the new variable

119884119894=119883119894minus 119883119894minus1

minus 120572 (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

= 120576119894 Δ constant (25)

Note that 119884119894 119894 = 1 2 3 119879 are independent random

variables distributed normal119873(0 1205902Δ)In terms of the density function can be written

119891 (1198841 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(1198831minus1198830minus120572 (120583minus119883

0) Δ

119883120574

0

)

2

]

6 Journal of Probability and Statistics

119891 (1198842 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(1198832minus1198831minus120572 (120583minus119883

1) Δ

119883120574

1

)

2

]

119891 (119884119879 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(119883119879minus119883119879minus1

minus120572 (120583minus119883119879minus1

) Δ

119883120574

119879minus1

)

2

]

(26)

Since each 119884119894is independent can be expressed the joint

density function as their marginal densities product that is

119891 (1198841 1198842 119884

119879) = 119891 (119884

1 120579) 119891 (119884

2 120579) sdot sdot sdot 119891 (119884

119879 120579)

(27)

Consequently the likelihood function is given by

119871 (120579 | 119884119879) = (

1

21205871205902Δ)1198792

times exp[ minus1

21205902Δ

119879

sum119894=1

(119883119894minus119883119894minus1minus120572 (120583minus119883

119894minus1) Δ

119883120574

119894minus1

)

2

]

(28)

Taking the natural logarithm on both sides it follows that

log 119871 = minus119879

2log (21205871205902Δ)

minus1

21205902Δ[

119879

sum119894=1

(119883119894minus 119883119894minus1

minus 120572 (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

]

(29)

Now consider the problem of maximizing the likelihoodfunction given by

120597 log (119871)120597120579

= 0 ie 120579 = Arg120579

max (log (119871)) (30)

The estimation process leads to the estimated parameters120572 and 120583 are the solutions of which the next equation systemwhere it is assumed that the value of 120574 is known Consider

119860 minus 119864 (1 minus 120572Δ) minus 120572120583Δ119861 = 0

119862 minus 119861 (1 minus 120572Δ) minus 120572120583Δ119863 = 0(31)

where

119860 =

119879

sum119894=1

119883119894119883119894minus1

1198832120574

119894minus1

119861 =

119879

sum119894=1

119883119894minus1

1198832120574

119894minus1

119862 =

119879

sum119894=1

119883119894

1198832120574

119894minus1

119863 =

119879

sum119894=1

1

1198832120574

119894minus1

119864 =

119879

sum119894=1

(119883119894minus1

119883120574

119894minus1

)

2

(32)

Thus the estimated parameters are given by

=119864119863 minus 119861

2 minus 119860119863 + 119861119862

(119864119863 minus 1198612) Δ

120583 =119860 minus 119864 (1 minus Δ)

119861Δ

(33)

Additionally the parameter 120590 is determined by an equa-tion independent of the system of (31) and is completelydetermined by the estimated parameters

= radic1

119879Δ

119879

sum119894=1

(119883119894minus 119883119894minus1

minus (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

(34)

This estimation scheme has several important qualitiesFirst you come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simple and does not requiresophisticated routines to perform calculations very extensive

5 Simulations

This sectionwill present comparisons of the results with thosepublished by other authors using the techniques describedabove Comparisons are made with the estimate based onsimulated data sets to analyze statistical properties of theproposed estimators

Yu andPhillips [1] present the results of a simulation studywith the Monte Carlo technique in comparing their methodwith Nowmanrsquos [2] To make this comparison are suggestedthree scenarios to resolve data monthly weekly and daily Inall cases the basic pattern of square root is defined as

119889119883119905= (120582 + 120573119883

119905) 119889119905 + 120590119883

120574

119905119889119861119905

with 120582 gt 0 120573 lt 0 120590 gt 0 120574 = 05(35)

Note that with appropriate substitution of this modelparameters are equivalent to those described in (1) (Theequivalence of models (1) and (35) is satisfied by having120582 = 120572120583 and 120573 = minus120572 Originally in Yu and Phillips [1] theparameter 120582 in (35) is defined as 120572 but was changed to avoidconfusion with the model (1))

The specific parameters used in each stage are shown inTable 2

To perform the first comparison parameters are set asspecified in Table 2 and obtained a series simulatedThen youset the value to 120574 = 05 and proceed to the estimation by themethod of Section 4 to obtain 120573 and This procedure wasrepeated 1000 times (as inYu andPhillips [1]) Some statisticalestimators are presented in Tables 3 4 and 5

Tables 3 4 and 5 show that the method proposed byNowman [2] gives good estimates for the parameters 120590 and120574 as manifested by Yu and Phillips [1] it becomes clearthat a starting point for the method is defined by the latterFurthermore through the resulting estimate of the parameter120590 our method becomes much more accurate than Nowman[2] introducing a bias less than 02 for the monthly dataseries and a small value for cases of weekly and daily series

Journal of Probability and Statistics 7

Table 2 Monte Carlo study parameters

Monthly Weekly DailySize 500 1000 2000120582 072 3 6120573 minus012 minus05 minus1120590 06 035 025120574 05 05 05

In the case of 120582 and 120573 are parameters observed inefficientperformance (as discussed Yu and Phillips [1 page 220]) inthe method of Nowman 120582 yielding bias of 87 47 and64 for data monthly weekly and daily respectively Alsofor 120573 the bias present in the estimate is 94 46 and54 for the respective monthly weekly and daily series Themethod proposed byYu andPhillips has a better performancethan Nowman measured from the bias in the parametersreducing it by 15 8 and 16 for the case of 120582 and 5 6and 6 in the case of 120573 for the respective monthly weeklyand daily series Our method has a performance similar tothat of Yu and Phillips for daily data series reducing the biasregarding Nowman method in 15 for 120582 and 4 for 120573 Forweekly data series for reducing 120582 bias regarding the methodof Nowman is 7 whereas for 120573 is 6 as reduction Yu andPhillips method

An important point to note in the results obtained byour method is the condition that the parameter 120574 is knowna priori This is necessary in order to find closed formulas forthe estimators

In the approach to the model proposed by Nowman andYu-Phillips 119889119883

119905= (120582 + 120573119883

119905)119889119905 + 120590119883

120574

119905119889119861119905is implied to have

particular difficulty in estimating the parameter of speed ofreversion To make this condition clear Table 6 shows theresults obtained by the method of Durham and Gallant [4]and our method to estimate the parameters of the model119889119883119905= 120572(120583 + 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905with 120572 gt 0 120583 gt 0 120590 gt 0

120574 = 05 The results are the average of 512 estimations ofmonthly series with 1000 observations

The results shown in Table 6 show that the errors inestimation of 120583 and 120590 are small (less than 1) for bothmethods On the other hand the estimated speed of reversionhas an error of about 10 resulting in Durham and Gallantwhile ourmethod has an error close to 7 From this analysisit is clear that the parameter 120572 has a higher error in estimatingthan the other parameter estimates

In this regard the estimation error of 120572 in Yu [3] providesan analysis of the bias in the parameter estimation inherentreversion total observation time which is sampled in thecontinuous process for the special case where the long-termaverage 120583 is zero Reference is made to the fact that the bias inparameter depends asymptotically on total observation timewhich of the process and not the number of observationsmade in this period

Figures 1 2 and 3 show the results of the estimationof the parameters for different observation horizons withdifferent sampling frequencies (The results are obtained asthe average of the individual estimates on 1000 simulated

0 10 20 30 40 50 60 70 80 9014

145

15

155

16

165

17

175

18

185

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120572 = 15

Figure 1 120572 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

0 10 20 30 40 50 60 70 80 900497

04975

0498

04985

0499

04995

05

05005

0501

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120583 = 05

Figure 2 120583 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

series in 4 sampling scenarios (a) bimonthly series with 500observations (b) monthly series with 1000 observations (c)weekly series with 4333 observations and (d) daily series with21000 observations The reference model for the simulationswas 119889119883

119905= 120572(120583 minus 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905 with 120572 = 15 120583 = 05

120574 = 05 and 120590 = 04)Figure 1 shows the evolution of the estimate of 120572 for

observations from 10 to 83 years Behavior of the estimatesin the four scenarios can be established empirically by thedependence of the bias on the estimate and the horizonof observation as they have more years of observation

8 Journal of Probability and Statistics

Table 3 Comparison of monthly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 13440 minus02332 12330 minus02275 06173 04919 12797 minus02200 05989Bias 06240 minus01132 05130 minus01075 00173 minus00081 05597 minus01000 minus00011Var 05897 01713 05732 01589 00099 00071 05406 00149 00004MSE 09791 01841 08363 01705 00102 00071 08533 00249 00004The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 500observations each The real values of the parameters are 120582 = 072 120573 = minus012 120590 = 06 and 120574 = 05

Table 4 Comparison of weekly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 44090 minus07320 41650 minus07011 03762 04925 42038 minus07022 03499Bias 14090 minus02320 11650 minus02011 00262 minus00075 12038 minus02022 minus00001Var 40663 01109 38608 01030 00172 00357 35427 00992 00001MSE 60855 01647 52180 01435 00179 00358 49883 01400 00001The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 3 120573 = minus05 120590 = 035 and 120574 = 05

Table 5 Comparison of daily data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 98250 minus15360 88440 minus14750 02521 04970 89542 minus14933 02500Bias 38250 minus05360 28440 minus04750 00021 minus00030 29542 minus04933 00000Var 191959 05219 178220 04798 00244 00746 155818 04334 00000MSE 338266 08092 259100 07054 00244 00746 242938 06763 00000The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 60 120573 = minus10 120590 = 025 and 120574 = 05

Table 6 Comparison of Durham and Gallant and proposed methods

Parameter(lowast) Real Bias(a) Bias(b) Real Bias(a) Bias(b)

120572 05 0048900 0037700 05 0059730 0021400120583 006 0000610 0000127 006 minus0000310 minus0000021120590 015 0000200 0000032 022 0000120 minus0000046120572 05 0043780 0032400 04 0045890 0032200120583 006 0000050 0000016 006 0000770 0000046120590 003 0000010 minus0000003 015 0000080 minus0000025The data are obtained with 512 simulations monthly series (119889119905 = 112) with 1000 observations each and the parameters are listed as real(a)Results of the method proposed by Durham and Gallant(b)Results of the method proposed in this paper(lowast)In Durham and Gallant [4] 120583 = 120579

1 120572 = 120579

2 and 120590 = 120579

3

bias decreases with no evidence of a significant differencebetween the estimates different frequencies

From Figure 2 it is observed that after few observationsvirtually independent of the year of observation estimationsfor 120583 are obtained with a bias of less than 1 showingthe efficiency of the estimation of the long-term averagereversion processes average In the case of estimation of 120590Figure 3 shows that the bias in the parameter estimates is

more dependent on the number of observations which is thetotal time of observation of the processThis is reflected in thefact that the estimates are in series with a higher sampling ratereaches a lower bias for the same observation horizon thanlower sample rate and therefore have fewer observations Onthe other hand as for the case of the estimate of 120583 the biaspresent in the estimates of 120590 is practically negligible (less than1) in most scenarios

Journal of Probability and Statistics 9

0 10 20 30 40 50 60 70 80 900386

0388

039

0392

0394

0396

0398

04

0402

0404

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120590 = 04

Figure 3 120590 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

6 Conclusions and Comments

This paper presents a new alternative for Gaussian estimationmodels of mean reversion of one factor under the Eulernumerical approximation that includes among others somemodels of interest rates in the short term

Monte Carlo simulations show a good approximation ofthe diffusion term parameter but less approximation in atrend term parameterThe estimation results have almost thesame order as that in Nowmanrsquos [2] and in some cases betterthan that in Durham and Gallantrsquos [4] It should be notedthat the proposed methodology outperforms the estimate of120590 compared to the other alternatives presented These resultsbecome more relevant since in practice the estimation ofvolatility is extremely useful in the evaluation of financialderivatives in which the trend term loses significance whenit is absorbed by the processes of change measurement andvaluation risk neutral

The estimate based on this new method though doesnot eliminate the bias in estimation of the reversion rateparameter for a small sample is in some cases comparablewith the results in Yu and Phillips [1] if samples have longertime horizons

This estimation scheme has several important qualitiesFirst we come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simpler and does not requiresophisticated routines to perform extensive calculations

References

[1] J Yu and P C B Phillips ldquoA Gaussian approach for continuoustime models of the short-term interest raterdquo The EconometricsJournal vol 4 no 2 pp 210ndash224 2001

[2] K Nowman ldquoGaussian estimation of single-factor continuoustime models of the term structure of interest ratesrdquoThe Journalof Finance vol 52 pp 1695ndash1703 1997

[3] J Yu ldquoBias in the estimation of themean reversion parameter incontinuous time modelsrdquo Journal of Econometrics vol 169 no1 pp 114ndash122 2012

[4] G B Durham and A R Gallant ldquoNumerical techniques formaximum likelihood estimation of continuous-time diffusionprocessesrdquo Journal of Business amp Economic Statistics vol 20 no3 pp 297ndash338 2002

[5] L ValdiviesoW Schoutens and F Tuerlinckx ldquoMaximum like-lihood estimation in processes of Ornstein-Uhlenbeck typerdquoStatistical Inference for Stochastic Processes vol 12 no 1 pp 1ndash192009

[6] O Elerian S Chib and N Shephard ldquoLikelihood inference fordiscretely observed nonlinear diffusionsrdquo Econometrica vol 69no 4 pp 959ndash993 2001

[7] A R Pedersen ldquoConsistency and asymptotic normality ofan approximate maximum likelihood estimator for discretelyobserved diffusion processesrdquo Bernoulli vol 1 no 3 pp 257ndash279 1995

[8] T Koulis and A Thavaneswaran ldquoInference for interest ratemodels using Milsteinrsquos approximationrdquo The Journal of Math-ematical Finance vol 3 pp 110ndash118 2013

[9] G J Jiang and J L Knight ldquoA nonparametric approach to theestimation of diffusion processes with an application to a short-term interest rate modelrdquo Econometric Theory vol 13 no 5 pp615ndash645 1997

[10] D Florens-Zmirou ldquoOn estimating the diffusion coefficientfrom discrete observationsrdquo Journal of Applied Probability vol30 no 4 pp 790ndash804 1993

[11] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 pp 177ndash186 1977

[12] M Brennan and E Schwartz ldquoAnalyzing convertible bondsrdquoJournal of Financial and Quantitative Analysis vol 15 pp 907ndash929 1980

[13] J Cox J Ingersoll and S Ross ldquoAn analysis of variable rate loancontractsrdquoThe Journal of Finance vol 35 pp 389ndash403 1980

[14] K Chan F Karolyi F Longstaff and A Sanders ldquoAn empiricalcomparison of alternative models of short term interest ratesrdquoThe Journal of Finance vol 47 pp 1209ndash1227 1992

[15] A Lari-Lavassani A Sadeghi and A Ware ldquoMean revertingmodels for energy option pricingrdquo Electronic Publications of theInternational Energy Credit Association 2001 httpwwwiecanet

[16] D Pilipovic Energy Risk Valuing andManaging Energy Deriva-tives McGraw-Hill New York NY USA 2007

[17] G Courtadon ldquoThe Pricing of options on default-free bondsrdquoJournal of Financial and Quantitative Analysis vol 17 pp 75ndash100 1982

[18] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[19] K Dunn and J McConnell ldquoValuation of GNMA Mortgage-backed securitiesrdquoThe Journal of Finance vol 36 no 3 pp 599ndash616 1981

[20] K Ramaswamy and S Sundaresan ldquoThe valuation of floating-rate instrumentsrdquo Journal of Financial Economics vol 17 pp251ndash272 1986

[21] S Sundaresan Valuation of Swaps Working Paper ColumbiaUniversity 1989

10 Journal of Probability and Statistics

[22] F Longstaff ldquoThe valuation of options on yieldsrdquo Journal ofFinancial Economics vol 26 no 1 pp 97ndash122 1990

[23] G Constantinides and J Ingersoll ldquoOptimal bond trading withpersonal taxesrdquo Journal of Financial Economics vol 13 pp 299ndash335 1984

[24] G Jiang and J Knight ldquoFinite sample comparison of alternativeestimators of Ito diffusion processes a Monte Carlo studyrdquoTheJournal of Computational Finance vol 2 no 3 1999

[25] L Giet andM LubranoBayesian Inference in Reducible Stochas-tic Differential Equations Document de Travail No 2004ndash57GREQAM University drsquoAix-Marseille 2004

[26] A R Bergstrom ldquoGaussian estimation of structural parametersin higher order continuous time dynamic modelsrdquo Economet-rica vol 51 no 1 pp 117ndash152 1983

[27] A Bergstrom ldquoContinuous time stochastic models and issuesof aggregation over timerdquo in Handbook of Econometrics ZGriliches and M D Intriligator Eds vol 2 Elsevier Amster-dam The Netherlands 1984

[28] A Bergstrom ldquoThe estimation of parameters in nonstationaryhigher-order continuous-time dynamic modelsrdquo EconometricTheory vol 1 pp 369ndash385 1985

[29] A Bergstrom ldquoThe estimation of open higher-order continuoustime dynamic models with mixed stock and flow datardquo Econo-metric Theory vol 2 pp 350ndash373 1986

[30] A Bergstrom Continuous Time Econometric Modelling OxfordUniversity Press Oxford UK 1990

[31] P C B Phillips and J Yu ldquoCorrigendum to lsquoA Gaussianapproach for continuous time models of short-term interestratesrsquordquoThe Econometrics Journal vol 14 no 1 pp 126ndash129 2011

[32] M J D Powell ldquoAn efficient method for finding the minimumof a function of several variables without calculating deriva-tivesrdquoThe Computer Journal vol 7 pp 155ndash162 1964

[33] V S Huzurbazar ldquoThe likelihood equation consistency and themaxima of the likelihood functionrdquo Annals of Eugenics vol 14pp 185ndash200 1948

[34] A Kohatsu-Higa and S Ogawa ldquoWeak rate of convergence foran Euler scheme of nonlinear SDErsquosrdquoMonte Carlo Methods andApplications vol 3 no 4 pp 327ndash345 1997

[35] V Bally andD Talay ldquoThe law of the Euler scheme for stochasticdifferential equations error analysis with Malliavin calculusrdquoMathematics and Computers in Simulation vol 38 no 1ndash3 pp35ndash41 1995

[36] A Zellner ldquoBayesian analysis of regression error termsrdquo Journalof the American Statistical Association vol 70 no 349 pp 138ndash144 1975

[37] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing England UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

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ProbabilityandStatistics

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Volume 2014

Advances in

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Complex AnalysisJournal of

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OptimizationJournal of

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Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

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Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

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Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 5

The implementation of this method is based on theestimate given by Nowman [2] to find the parameters esti-mates 120590 and 120574 Furthermore the estimation of the remainingparameters 120583 and 120572 is carried out using the numericaloptimization algorithm by Powell [32]

4 Gaussian Estimation ofOne-Factor Mean Reversion Processes withConstant Parameters

The development of this section will focus on the descriptionof the proposed technique for the Gaussian estimation ofmean reversion models of a single factor with linear andnonlinear functions of diffusion constant parameters andlinear trend functions with constant parameters

As the density function for most diffusion processes isnot known analytically the proposed idea is not to find orapproximate the transition density function or the marginaldensity function of the process Otherwise we use thenumerical approximation of Euler on the EDE to build a newdiscrete process Huzurbazar [33] and use the fact that inthis approximation the normalized residuals resulting fromBrownian process are independent standard normal randomvariables As pointed out by Pedersen [7] Kohatsu-Higa andOgawa [34] and Bally and Talay [35] the density functionof the discrete process obtained with Euler scheme convergesto the density function of the continuous process and asdiscussed by Zellner [36] these residues can be interpretedas a vector of unknown parameters with the advantage ofknowing their distribution So we build a new variable forthe models defined in (1) so that it can adapt to the observeddata and later can be evaluated by examining the normalizedresiduals that will be useful in the estimation process Byknowing the density function of this new process we canfind discretized likelihood function and use the conventionalmethod of maximum likelihood estimation to find a closedformula for the parameters

Consider the stochastic differential equation (1) with theinitial condition119883

0= 119909 The terms 120572(120583 minus 119883

119905) and 120590119883120574

119905being

the trend and diffusion functions respectively that dependon 119883119905and a vector of unknown parameters 120579 = [120572 120583 120590 120574]

where 119861119905119905ge0

is a unidimensional standard Brownianmotiondefined on a complete probability space (ΩFP) The trendfunction describes the movement of the process due tochanges in time while the diffusion function measures themagnitude of the randomfluctuations around the trend func-tion Suppose that the conditions under which we guaranteethe existence and uniqueness of the solution of Mao [37] aresatisfied Assume further the process observations in discretetime119883

119905= 119883(120591

119905) at times 120591

0 1205911 120591

119879whereΔ

119905= 120591119905minus120591119905minus1

ge

0 119905 = 1 2 119879 The objective is to estimate the unknownparameter vector 120579 given the historical information of thesample119883 = (119883

0 1198831 119883

119879)1015840 with equally spaced data Note

that the explicit solution of (1) is given by

119883119905= 120583 + (119883

0minus 120583) 119890

minus120572119905+ int119905

0

120590119890minus120572(119905minus119904)

119883120574

119904119889119861119904 (20)

In the context of the likelihood the estimation of 120579 isbased on the likelihood function

log 119871 (1198831 1198832 119883

119879| 1198830 120579) =

119879

sum119905=1

[log119892 (119883119905+1

| 119883119905 120579)]

(21)

where 119892(119883119905+1

| 119883119905 120579) are the Markov transition densities

or 119871(120579 | 1198831 1198832 119883

119879) = 119891(119883

1 1198832 119883

119879 120579) and 119891(119883

1

1198832 119883

119879 120579) is the joint density function

Although the differential equation (1) can be solved ana-lytically on [0 119879] the functions 119892(119883

119905+1| 119883119905 120579) and 119891(119883

1

1198832 119883

119879 120579) are not available in closed form and therefore

the estimate can not be accomplished except for the casewhere 120574 = 0

Note that the exact discretized solution of (1) takes theform

119883(120591119905) = 120583 + (119883 (120591

119905minus1) minus 120583) 119890

minus120572(120591119905minus120591119905minus1)

+ int120591

120591119905minus1

120590119890minus120572(120591119905minus119904)119883

120574

(119904)119889119861119904

(22)

which can be approximated by

119883(120591119905) = 120583 + (119883 (120591

119905minus1) minus 120583) 119890

minus120572Δ 119905

+ 120590119890minus120572Δ 119905119883

120574(120591119905minus1) Δ119861119905

(23)

where Δ119861119905= 119861120591119905minus 119861120591119905minus1

Thus

119883(120591119905) = 119883 (120591

119905minus1) + 120572 (120583 minus 119883 (120591

119905minus1)) Δ119905+ 119883120574(120591119905minus1) 120576120591119905

(24)

where 120576120591119905sim 119873(0 1205902Δ

119905) corresponding to the numerical ap-

proximation of EulerNow define the new variable

119884119894=119883119894minus 119883119894minus1

minus 120572 (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

= 120576119894 Δ constant (25)

Note that 119884119894 119894 = 1 2 3 119879 are independent random

variables distributed normal119873(0 1205902Δ)In terms of the density function can be written

119891 (1198841 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(1198831minus1198830minus120572 (120583minus119883

0) Δ

119883120574

0

)

2

]

6 Journal of Probability and Statistics

119891 (1198842 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(1198832minus1198831minus120572 (120583minus119883

1) Δ

119883120574

1

)

2

]

119891 (119884119879 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(119883119879minus119883119879minus1

minus120572 (120583minus119883119879minus1

) Δ

119883120574

119879minus1

)

2

]

(26)

Since each 119884119894is independent can be expressed the joint

density function as their marginal densities product that is

119891 (1198841 1198842 119884

119879) = 119891 (119884

1 120579) 119891 (119884

2 120579) sdot sdot sdot 119891 (119884

119879 120579)

(27)

Consequently the likelihood function is given by

119871 (120579 | 119884119879) = (

1

21205871205902Δ)1198792

times exp[ minus1

21205902Δ

119879

sum119894=1

(119883119894minus119883119894minus1minus120572 (120583minus119883

119894minus1) Δ

119883120574

119894minus1

)

2

]

(28)

Taking the natural logarithm on both sides it follows that

log 119871 = minus119879

2log (21205871205902Δ)

minus1

21205902Δ[

119879

sum119894=1

(119883119894minus 119883119894minus1

minus 120572 (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

]

(29)

Now consider the problem of maximizing the likelihoodfunction given by

120597 log (119871)120597120579

= 0 ie 120579 = Arg120579

max (log (119871)) (30)

The estimation process leads to the estimated parameters120572 and 120583 are the solutions of which the next equation systemwhere it is assumed that the value of 120574 is known Consider

119860 minus 119864 (1 minus 120572Δ) minus 120572120583Δ119861 = 0

119862 minus 119861 (1 minus 120572Δ) minus 120572120583Δ119863 = 0(31)

where

119860 =

119879

sum119894=1

119883119894119883119894minus1

1198832120574

119894minus1

119861 =

119879

sum119894=1

119883119894minus1

1198832120574

119894minus1

119862 =

119879

sum119894=1

119883119894

1198832120574

119894minus1

119863 =

119879

sum119894=1

1

1198832120574

119894minus1

119864 =

119879

sum119894=1

(119883119894minus1

119883120574

119894minus1

)

2

(32)

Thus the estimated parameters are given by

=119864119863 minus 119861

2 minus 119860119863 + 119861119862

(119864119863 minus 1198612) Δ

120583 =119860 minus 119864 (1 minus Δ)

119861Δ

(33)

Additionally the parameter 120590 is determined by an equa-tion independent of the system of (31) and is completelydetermined by the estimated parameters

= radic1

119879Δ

119879

sum119894=1

(119883119894minus 119883119894minus1

minus (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

(34)

This estimation scheme has several important qualitiesFirst you come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simple and does not requiresophisticated routines to perform calculations very extensive

5 Simulations

This sectionwill present comparisons of the results with thosepublished by other authors using the techniques describedabove Comparisons are made with the estimate based onsimulated data sets to analyze statistical properties of theproposed estimators

Yu andPhillips [1] present the results of a simulation studywith the Monte Carlo technique in comparing their methodwith Nowmanrsquos [2] To make this comparison are suggestedthree scenarios to resolve data monthly weekly and daily Inall cases the basic pattern of square root is defined as

119889119883119905= (120582 + 120573119883

119905) 119889119905 + 120590119883

120574

119905119889119861119905

with 120582 gt 0 120573 lt 0 120590 gt 0 120574 = 05(35)

Note that with appropriate substitution of this modelparameters are equivalent to those described in (1) (Theequivalence of models (1) and (35) is satisfied by having120582 = 120572120583 and 120573 = minus120572 Originally in Yu and Phillips [1] theparameter 120582 in (35) is defined as 120572 but was changed to avoidconfusion with the model (1))

The specific parameters used in each stage are shown inTable 2

To perform the first comparison parameters are set asspecified in Table 2 and obtained a series simulatedThen youset the value to 120574 = 05 and proceed to the estimation by themethod of Section 4 to obtain 120573 and This procedure wasrepeated 1000 times (as inYu andPhillips [1]) Some statisticalestimators are presented in Tables 3 4 and 5

Tables 3 4 and 5 show that the method proposed byNowman [2] gives good estimates for the parameters 120590 and120574 as manifested by Yu and Phillips [1] it becomes clearthat a starting point for the method is defined by the latterFurthermore through the resulting estimate of the parameter120590 our method becomes much more accurate than Nowman[2] introducing a bias less than 02 for the monthly dataseries and a small value for cases of weekly and daily series

Journal of Probability and Statistics 7

Table 2 Monte Carlo study parameters

Monthly Weekly DailySize 500 1000 2000120582 072 3 6120573 minus012 minus05 minus1120590 06 035 025120574 05 05 05

In the case of 120582 and 120573 are parameters observed inefficientperformance (as discussed Yu and Phillips [1 page 220]) inthe method of Nowman 120582 yielding bias of 87 47 and64 for data monthly weekly and daily respectively Alsofor 120573 the bias present in the estimate is 94 46 and54 for the respective monthly weekly and daily series Themethod proposed byYu andPhillips has a better performancethan Nowman measured from the bias in the parametersreducing it by 15 8 and 16 for the case of 120582 and 5 6and 6 in the case of 120573 for the respective monthly weeklyand daily series Our method has a performance similar tothat of Yu and Phillips for daily data series reducing the biasregarding Nowman method in 15 for 120582 and 4 for 120573 Forweekly data series for reducing 120582 bias regarding the methodof Nowman is 7 whereas for 120573 is 6 as reduction Yu andPhillips method

An important point to note in the results obtained byour method is the condition that the parameter 120574 is knowna priori This is necessary in order to find closed formulas forthe estimators

In the approach to the model proposed by Nowman andYu-Phillips 119889119883

119905= (120582 + 120573119883

119905)119889119905 + 120590119883

120574

119905119889119861119905is implied to have

particular difficulty in estimating the parameter of speed ofreversion To make this condition clear Table 6 shows theresults obtained by the method of Durham and Gallant [4]and our method to estimate the parameters of the model119889119883119905= 120572(120583 + 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905with 120572 gt 0 120583 gt 0 120590 gt 0

120574 = 05 The results are the average of 512 estimations ofmonthly series with 1000 observations

The results shown in Table 6 show that the errors inestimation of 120583 and 120590 are small (less than 1) for bothmethods On the other hand the estimated speed of reversionhas an error of about 10 resulting in Durham and Gallantwhile ourmethod has an error close to 7 From this analysisit is clear that the parameter 120572 has a higher error in estimatingthan the other parameter estimates

In this regard the estimation error of 120572 in Yu [3] providesan analysis of the bias in the parameter estimation inherentreversion total observation time which is sampled in thecontinuous process for the special case where the long-termaverage 120583 is zero Reference is made to the fact that the bias inparameter depends asymptotically on total observation timewhich of the process and not the number of observationsmade in this period

Figures 1 2 and 3 show the results of the estimationof the parameters for different observation horizons withdifferent sampling frequencies (The results are obtained asthe average of the individual estimates on 1000 simulated

0 10 20 30 40 50 60 70 80 9014

145

15

155

16

165

17

175

18

185

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120572 = 15

Figure 1 120572 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

0 10 20 30 40 50 60 70 80 900497

04975

0498

04985

0499

04995

05

05005

0501

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120583 = 05

Figure 2 120583 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

series in 4 sampling scenarios (a) bimonthly series with 500observations (b) monthly series with 1000 observations (c)weekly series with 4333 observations and (d) daily series with21000 observations The reference model for the simulationswas 119889119883

119905= 120572(120583 minus 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905 with 120572 = 15 120583 = 05

120574 = 05 and 120590 = 04)Figure 1 shows the evolution of the estimate of 120572 for

observations from 10 to 83 years Behavior of the estimatesin the four scenarios can be established empirically by thedependence of the bias on the estimate and the horizonof observation as they have more years of observation

8 Journal of Probability and Statistics

Table 3 Comparison of monthly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 13440 minus02332 12330 minus02275 06173 04919 12797 minus02200 05989Bias 06240 minus01132 05130 minus01075 00173 minus00081 05597 minus01000 minus00011Var 05897 01713 05732 01589 00099 00071 05406 00149 00004MSE 09791 01841 08363 01705 00102 00071 08533 00249 00004The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 500observations each The real values of the parameters are 120582 = 072 120573 = minus012 120590 = 06 and 120574 = 05

Table 4 Comparison of weekly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 44090 minus07320 41650 minus07011 03762 04925 42038 minus07022 03499Bias 14090 minus02320 11650 minus02011 00262 minus00075 12038 minus02022 minus00001Var 40663 01109 38608 01030 00172 00357 35427 00992 00001MSE 60855 01647 52180 01435 00179 00358 49883 01400 00001The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 3 120573 = minus05 120590 = 035 and 120574 = 05

Table 5 Comparison of daily data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 98250 minus15360 88440 minus14750 02521 04970 89542 minus14933 02500Bias 38250 minus05360 28440 minus04750 00021 minus00030 29542 minus04933 00000Var 191959 05219 178220 04798 00244 00746 155818 04334 00000MSE 338266 08092 259100 07054 00244 00746 242938 06763 00000The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 60 120573 = minus10 120590 = 025 and 120574 = 05

Table 6 Comparison of Durham and Gallant and proposed methods

Parameter(lowast) Real Bias(a) Bias(b) Real Bias(a) Bias(b)

120572 05 0048900 0037700 05 0059730 0021400120583 006 0000610 0000127 006 minus0000310 minus0000021120590 015 0000200 0000032 022 0000120 minus0000046120572 05 0043780 0032400 04 0045890 0032200120583 006 0000050 0000016 006 0000770 0000046120590 003 0000010 minus0000003 015 0000080 minus0000025The data are obtained with 512 simulations monthly series (119889119905 = 112) with 1000 observations each and the parameters are listed as real(a)Results of the method proposed by Durham and Gallant(b)Results of the method proposed in this paper(lowast)In Durham and Gallant [4] 120583 = 120579

1 120572 = 120579

2 and 120590 = 120579

3

bias decreases with no evidence of a significant differencebetween the estimates different frequencies

From Figure 2 it is observed that after few observationsvirtually independent of the year of observation estimationsfor 120583 are obtained with a bias of less than 1 showingthe efficiency of the estimation of the long-term averagereversion processes average In the case of estimation of 120590Figure 3 shows that the bias in the parameter estimates is

more dependent on the number of observations which is thetotal time of observation of the processThis is reflected in thefact that the estimates are in series with a higher sampling ratereaches a lower bias for the same observation horizon thanlower sample rate and therefore have fewer observations Onthe other hand as for the case of the estimate of 120583 the biaspresent in the estimates of 120590 is practically negligible (less than1) in most scenarios

Journal of Probability and Statistics 9

0 10 20 30 40 50 60 70 80 900386

0388

039

0392

0394

0396

0398

04

0402

0404

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120590 = 04

Figure 3 120590 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

6 Conclusions and Comments

This paper presents a new alternative for Gaussian estimationmodels of mean reversion of one factor under the Eulernumerical approximation that includes among others somemodels of interest rates in the short term

Monte Carlo simulations show a good approximation ofthe diffusion term parameter but less approximation in atrend term parameterThe estimation results have almost thesame order as that in Nowmanrsquos [2] and in some cases betterthan that in Durham and Gallantrsquos [4] It should be notedthat the proposed methodology outperforms the estimate of120590 compared to the other alternatives presented These resultsbecome more relevant since in practice the estimation ofvolatility is extremely useful in the evaluation of financialderivatives in which the trend term loses significance whenit is absorbed by the processes of change measurement andvaluation risk neutral

The estimate based on this new method though doesnot eliminate the bias in estimation of the reversion rateparameter for a small sample is in some cases comparablewith the results in Yu and Phillips [1] if samples have longertime horizons

This estimation scheme has several important qualitiesFirst we come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simpler and does not requiresophisticated routines to perform extensive calculations

References

[1] J Yu and P C B Phillips ldquoA Gaussian approach for continuoustime models of the short-term interest raterdquo The EconometricsJournal vol 4 no 2 pp 210ndash224 2001

[2] K Nowman ldquoGaussian estimation of single-factor continuoustime models of the term structure of interest ratesrdquoThe Journalof Finance vol 52 pp 1695ndash1703 1997

[3] J Yu ldquoBias in the estimation of themean reversion parameter incontinuous time modelsrdquo Journal of Econometrics vol 169 no1 pp 114ndash122 2012

[4] G B Durham and A R Gallant ldquoNumerical techniques formaximum likelihood estimation of continuous-time diffusionprocessesrdquo Journal of Business amp Economic Statistics vol 20 no3 pp 297ndash338 2002

[5] L ValdiviesoW Schoutens and F Tuerlinckx ldquoMaximum like-lihood estimation in processes of Ornstein-Uhlenbeck typerdquoStatistical Inference for Stochastic Processes vol 12 no 1 pp 1ndash192009

[6] O Elerian S Chib and N Shephard ldquoLikelihood inference fordiscretely observed nonlinear diffusionsrdquo Econometrica vol 69no 4 pp 959ndash993 2001

[7] A R Pedersen ldquoConsistency and asymptotic normality ofan approximate maximum likelihood estimator for discretelyobserved diffusion processesrdquo Bernoulli vol 1 no 3 pp 257ndash279 1995

[8] T Koulis and A Thavaneswaran ldquoInference for interest ratemodels using Milsteinrsquos approximationrdquo The Journal of Math-ematical Finance vol 3 pp 110ndash118 2013

[9] G J Jiang and J L Knight ldquoA nonparametric approach to theestimation of diffusion processes with an application to a short-term interest rate modelrdquo Econometric Theory vol 13 no 5 pp615ndash645 1997

[10] D Florens-Zmirou ldquoOn estimating the diffusion coefficientfrom discrete observationsrdquo Journal of Applied Probability vol30 no 4 pp 790ndash804 1993

[11] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 pp 177ndash186 1977

[12] M Brennan and E Schwartz ldquoAnalyzing convertible bondsrdquoJournal of Financial and Quantitative Analysis vol 15 pp 907ndash929 1980

[13] J Cox J Ingersoll and S Ross ldquoAn analysis of variable rate loancontractsrdquoThe Journal of Finance vol 35 pp 389ndash403 1980

[14] K Chan F Karolyi F Longstaff and A Sanders ldquoAn empiricalcomparison of alternative models of short term interest ratesrdquoThe Journal of Finance vol 47 pp 1209ndash1227 1992

[15] A Lari-Lavassani A Sadeghi and A Ware ldquoMean revertingmodels for energy option pricingrdquo Electronic Publications of theInternational Energy Credit Association 2001 httpwwwiecanet

[16] D Pilipovic Energy Risk Valuing andManaging Energy Deriva-tives McGraw-Hill New York NY USA 2007

[17] G Courtadon ldquoThe Pricing of options on default-free bondsrdquoJournal of Financial and Quantitative Analysis vol 17 pp 75ndash100 1982

[18] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[19] K Dunn and J McConnell ldquoValuation of GNMA Mortgage-backed securitiesrdquoThe Journal of Finance vol 36 no 3 pp 599ndash616 1981

[20] K Ramaswamy and S Sundaresan ldquoThe valuation of floating-rate instrumentsrdquo Journal of Financial Economics vol 17 pp251ndash272 1986

[21] S Sundaresan Valuation of Swaps Working Paper ColumbiaUniversity 1989

10 Journal of Probability and Statistics

[22] F Longstaff ldquoThe valuation of options on yieldsrdquo Journal ofFinancial Economics vol 26 no 1 pp 97ndash122 1990

[23] G Constantinides and J Ingersoll ldquoOptimal bond trading withpersonal taxesrdquo Journal of Financial Economics vol 13 pp 299ndash335 1984

[24] G Jiang and J Knight ldquoFinite sample comparison of alternativeestimators of Ito diffusion processes a Monte Carlo studyrdquoTheJournal of Computational Finance vol 2 no 3 1999

[25] L Giet andM LubranoBayesian Inference in Reducible Stochas-tic Differential Equations Document de Travail No 2004ndash57GREQAM University drsquoAix-Marseille 2004

[26] A R Bergstrom ldquoGaussian estimation of structural parametersin higher order continuous time dynamic modelsrdquo Economet-rica vol 51 no 1 pp 117ndash152 1983

[27] A Bergstrom ldquoContinuous time stochastic models and issuesof aggregation over timerdquo in Handbook of Econometrics ZGriliches and M D Intriligator Eds vol 2 Elsevier Amster-dam The Netherlands 1984

[28] A Bergstrom ldquoThe estimation of parameters in nonstationaryhigher-order continuous-time dynamic modelsrdquo EconometricTheory vol 1 pp 369ndash385 1985

[29] A Bergstrom ldquoThe estimation of open higher-order continuoustime dynamic models with mixed stock and flow datardquo Econo-metric Theory vol 2 pp 350ndash373 1986

[30] A Bergstrom Continuous Time Econometric Modelling OxfordUniversity Press Oxford UK 1990

[31] P C B Phillips and J Yu ldquoCorrigendum to lsquoA Gaussianapproach for continuous time models of short-term interestratesrsquordquoThe Econometrics Journal vol 14 no 1 pp 126ndash129 2011

[32] M J D Powell ldquoAn efficient method for finding the minimumof a function of several variables without calculating deriva-tivesrdquoThe Computer Journal vol 7 pp 155ndash162 1964

[33] V S Huzurbazar ldquoThe likelihood equation consistency and themaxima of the likelihood functionrdquo Annals of Eugenics vol 14pp 185ndash200 1948

[34] A Kohatsu-Higa and S Ogawa ldquoWeak rate of convergence foran Euler scheme of nonlinear SDErsquosrdquoMonte Carlo Methods andApplications vol 3 no 4 pp 327ndash345 1997

[35] V Bally andD Talay ldquoThe law of the Euler scheme for stochasticdifferential equations error analysis with Malliavin calculusrdquoMathematics and Computers in Simulation vol 38 no 1ndash3 pp35ndash41 1995

[36] A Zellner ldquoBayesian analysis of regression error termsrdquo Journalof the American Statistical Association vol 70 no 349 pp 138ndash144 1975

[37] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing England UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Probability and Statistics

119891 (1198842 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(1198832minus1198831minus120572 (120583minus119883

1) Δ

119883120574

1

)

2

]

119891 (119884119879 120579) = (

1

21205871205902Δ)12

timesexp[ minus1

21205902Δ(119883119879minus119883119879minus1

minus120572 (120583minus119883119879minus1

) Δ

119883120574

119879minus1

)

2

]

(26)

Since each 119884119894is independent can be expressed the joint

density function as their marginal densities product that is

119891 (1198841 1198842 119884

119879) = 119891 (119884

1 120579) 119891 (119884

2 120579) sdot sdot sdot 119891 (119884

119879 120579)

(27)

Consequently the likelihood function is given by

119871 (120579 | 119884119879) = (

1

21205871205902Δ)1198792

times exp[ minus1

21205902Δ

119879

sum119894=1

(119883119894minus119883119894minus1minus120572 (120583minus119883

119894minus1) Δ

119883120574

119894minus1

)

2

]

(28)

Taking the natural logarithm on both sides it follows that

log 119871 = minus119879

2log (21205871205902Δ)

minus1

21205902Δ[

119879

sum119894=1

(119883119894minus 119883119894minus1

minus 120572 (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

]

(29)

Now consider the problem of maximizing the likelihoodfunction given by

120597 log (119871)120597120579

= 0 ie 120579 = Arg120579

max (log (119871)) (30)

The estimation process leads to the estimated parameters120572 and 120583 are the solutions of which the next equation systemwhere it is assumed that the value of 120574 is known Consider

119860 minus 119864 (1 minus 120572Δ) minus 120572120583Δ119861 = 0

119862 minus 119861 (1 minus 120572Δ) minus 120572120583Δ119863 = 0(31)

where

119860 =

119879

sum119894=1

119883119894119883119894minus1

1198832120574

119894minus1

119861 =

119879

sum119894=1

119883119894minus1

1198832120574

119894minus1

119862 =

119879

sum119894=1

119883119894

1198832120574

119894minus1

119863 =

119879

sum119894=1

1

1198832120574

119894minus1

119864 =

119879

sum119894=1

(119883119894minus1

119883120574

119894minus1

)

2

(32)

Thus the estimated parameters are given by

=119864119863 minus 119861

2 minus 119860119863 + 119861119862

(119864119863 minus 1198612) Δ

120583 =119860 minus 119864 (1 minus Δ)

119861Δ

(33)

Additionally the parameter 120590 is determined by an equa-tion independent of the system of (31) and is completelydetermined by the estimated parameters

= radic1

119879Δ

119879

sum119894=1

(119883119894minus 119883119894minus1

minus (120583 minus 119883119894minus1) Δ

119883120574

119894minus1

)

2

(34)

This estimation scheme has several important qualitiesFirst you come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simple and does not requiresophisticated routines to perform calculations very extensive

5 Simulations

This sectionwill present comparisons of the results with thosepublished by other authors using the techniques describedabove Comparisons are made with the estimate based onsimulated data sets to analyze statistical properties of theproposed estimators

Yu andPhillips [1] present the results of a simulation studywith the Monte Carlo technique in comparing their methodwith Nowmanrsquos [2] To make this comparison are suggestedthree scenarios to resolve data monthly weekly and daily Inall cases the basic pattern of square root is defined as

119889119883119905= (120582 + 120573119883

119905) 119889119905 + 120590119883

120574

119905119889119861119905

with 120582 gt 0 120573 lt 0 120590 gt 0 120574 = 05(35)

Note that with appropriate substitution of this modelparameters are equivalent to those described in (1) (Theequivalence of models (1) and (35) is satisfied by having120582 = 120572120583 and 120573 = minus120572 Originally in Yu and Phillips [1] theparameter 120582 in (35) is defined as 120572 but was changed to avoidconfusion with the model (1))

The specific parameters used in each stage are shown inTable 2

To perform the first comparison parameters are set asspecified in Table 2 and obtained a series simulatedThen youset the value to 120574 = 05 and proceed to the estimation by themethod of Section 4 to obtain 120573 and This procedure wasrepeated 1000 times (as inYu andPhillips [1]) Some statisticalestimators are presented in Tables 3 4 and 5

Tables 3 4 and 5 show that the method proposed byNowman [2] gives good estimates for the parameters 120590 and120574 as manifested by Yu and Phillips [1] it becomes clearthat a starting point for the method is defined by the latterFurthermore through the resulting estimate of the parameter120590 our method becomes much more accurate than Nowman[2] introducing a bias less than 02 for the monthly dataseries and a small value for cases of weekly and daily series

Journal of Probability and Statistics 7

Table 2 Monte Carlo study parameters

Monthly Weekly DailySize 500 1000 2000120582 072 3 6120573 minus012 minus05 minus1120590 06 035 025120574 05 05 05

In the case of 120582 and 120573 are parameters observed inefficientperformance (as discussed Yu and Phillips [1 page 220]) inthe method of Nowman 120582 yielding bias of 87 47 and64 for data monthly weekly and daily respectively Alsofor 120573 the bias present in the estimate is 94 46 and54 for the respective monthly weekly and daily series Themethod proposed byYu andPhillips has a better performancethan Nowman measured from the bias in the parametersreducing it by 15 8 and 16 for the case of 120582 and 5 6and 6 in the case of 120573 for the respective monthly weeklyand daily series Our method has a performance similar tothat of Yu and Phillips for daily data series reducing the biasregarding Nowman method in 15 for 120582 and 4 for 120573 Forweekly data series for reducing 120582 bias regarding the methodof Nowman is 7 whereas for 120573 is 6 as reduction Yu andPhillips method

An important point to note in the results obtained byour method is the condition that the parameter 120574 is knowna priori This is necessary in order to find closed formulas forthe estimators

In the approach to the model proposed by Nowman andYu-Phillips 119889119883

119905= (120582 + 120573119883

119905)119889119905 + 120590119883

120574

119905119889119861119905is implied to have

particular difficulty in estimating the parameter of speed ofreversion To make this condition clear Table 6 shows theresults obtained by the method of Durham and Gallant [4]and our method to estimate the parameters of the model119889119883119905= 120572(120583 + 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905with 120572 gt 0 120583 gt 0 120590 gt 0

120574 = 05 The results are the average of 512 estimations ofmonthly series with 1000 observations

The results shown in Table 6 show that the errors inestimation of 120583 and 120590 are small (less than 1) for bothmethods On the other hand the estimated speed of reversionhas an error of about 10 resulting in Durham and Gallantwhile ourmethod has an error close to 7 From this analysisit is clear that the parameter 120572 has a higher error in estimatingthan the other parameter estimates

In this regard the estimation error of 120572 in Yu [3] providesan analysis of the bias in the parameter estimation inherentreversion total observation time which is sampled in thecontinuous process for the special case where the long-termaverage 120583 is zero Reference is made to the fact that the bias inparameter depends asymptotically on total observation timewhich of the process and not the number of observationsmade in this period

Figures 1 2 and 3 show the results of the estimationof the parameters for different observation horizons withdifferent sampling frequencies (The results are obtained asthe average of the individual estimates on 1000 simulated

0 10 20 30 40 50 60 70 80 9014

145

15

155

16

165

17

175

18

185

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120572 = 15

Figure 1 120572 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

0 10 20 30 40 50 60 70 80 900497

04975

0498

04985

0499

04995

05

05005

0501

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120583 = 05

Figure 2 120583 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

series in 4 sampling scenarios (a) bimonthly series with 500observations (b) monthly series with 1000 observations (c)weekly series with 4333 observations and (d) daily series with21000 observations The reference model for the simulationswas 119889119883

119905= 120572(120583 minus 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905 with 120572 = 15 120583 = 05

120574 = 05 and 120590 = 04)Figure 1 shows the evolution of the estimate of 120572 for

observations from 10 to 83 years Behavior of the estimatesin the four scenarios can be established empirically by thedependence of the bias on the estimate and the horizonof observation as they have more years of observation

8 Journal of Probability and Statistics

Table 3 Comparison of monthly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 13440 minus02332 12330 minus02275 06173 04919 12797 minus02200 05989Bias 06240 minus01132 05130 minus01075 00173 minus00081 05597 minus01000 minus00011Var 05897 01713 05732 01589 00099 00071 05406 00149 00004MSE 09791 01841 08363 01705 00102 00071 08533 00249 00004The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 500observations each The real values of the parameters are 120582 = 072 120573 = minus012 120590 = 06 and 120574 = 05

Table 4 Comparison of weekly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 44090 minus07320 41650 minus07011 03762 04925 42038 minus07022 03499Bias 14090 minus02320 11650 minus02011 00262 minus00075 12038 minus02022 minus00001Var 40663 01109 38608 01030 00172 00357 35427 00992 00001MSE 60855 01647 52180 01435 00179 00358 49883 01400 00001The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 3 120573 = minus05 120590 = 035 and 120574 = 05

Table 5 Comparison of daily data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 98250 minus15360 88440 minus14750 02521 04970 89542 minus14933 02500Bias 38250 minus05360 28440 minus04750 00021 minus00030 29542 minus04933 00000Var 191959 05219 178220 04798 00244 00746 155818 04334 00000MSE 338266 08092 259100 07054 00244 00746 242938 06763 00000The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 60 120573 = minus10 120590 = 025 and 120574 = 05

Table 6 Comparison of Durham and Gallant and proposed methods

Parameter(lowast) Real Bias(a) Bias(b) Real Bias(a) Bias(b)

120572 05 0048900 0037700 05 0059730 0021400120583 006 0000610 0000127 006 minus0000310 minus0000021120590 015 0000200 0000032 022 0000120 minus0000046120572 05 0043780 0032400 04 0045890 0032200120583 006 0000050 0000016 006 0000770 0000046120590 003 0000010 minus0000003 015 0000080 minus0000025The data are obtained with 512 simulations monthly series (119889119905 = 112) with 1000 observations each and the parameters are listed as real(a)Results of the method proposed by Durham and Gallant(b)Results of the method proposed in this paper(lowast)In Durham and Gallant [4] 120583 = 120579

1 120572 = 120579

2 and 120590 = 120579

3

bias decreases with no evidence of a significant differencebetween the estimates different frequencies

From Figure 2 it is observed that after few observationsvirtually independent of the year of observation estimationsfor 120583 are obtained with a bias of less than 1 showingthe efficiency of the estimation of the long-term averagereversion processes average In the case of estimation of 120590Figure 3 shows that the bias in the parameter estimates is

more dependent on the number of observations which is thetotal time of observation of the processThis is reflected in thefact that the estimates are in series with a higher sampling ratereaches a lower bias for the same observation horizon thanlower sample rate and therefore have fewer observations Onthe other hand as for the case of the estimate of 120583 the biaspresent in the estimates of 120590 is practically negligible (less than1) in most scenarios

Journal of Probability and Statistics 9

0 10 20 30 40 50 60 70 80 900386

0388

039

0392

0394

0396

0398

04

0402

0404

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120590 = 04

Figure 3 120590 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

6 Conclusions and Comments

This paper presents a new alternative for Gaussian estimationmodels of mean reversion of one factor under the Eulernumerical approximation that includes among others somemodels of interest rates in the short term

Monte Carlo simulations show a good approximation ofthe diffusion term parameter but less approximation in atrend term parameterThe estimation results have almost thesame order as that in Nowmanrsquos [2] and in some cases betterthan that in Durham and Gallantrsquos [4] It should be notedthat the proposed methodology outperforms the estimate of120590 compared to the other alternatives presented These resultsbecome more relevant since in practice the estimation ofvolatility is extremely useful in the evaluation of financialderivatives in which the trend term loses significance whenit is absorbed by the processes of change measurement andvaluation risk neutral

The estimate based on this new method though doesnot eliminate the bias in estimation of the reversion rateparameter for a small sample is in some cases comparablewith the results in Yu and Phillips [1] if samples have longertime horizons

This estimation scheme has several important qualitiesFirst we come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simpler and does not requiresophisticated routines to perform extensive calculations

References

[1] J Yu and P C B Phillips ldquoA Gaussian approach for continuoustime models of the short-term interest raterdquo The EconometricsJournal vol 4 no 2 pp 210ndash224 2001

[2] K Nowman ldquoGaussian estimation of single-factor continuoustime models of the term structure of interest ratesrdquoThe Journalof Finance vol 52 pp 1695ndash1703 1997

[3] J Yu ldquoBias in the estimation of themean reversion parameter incontinuous time modelsrdquo Journal of Econometrics vol 169 no1 pp 114ndash122 2012

[4] G B Durham and A R Gallant ldquoNumerical techniques formaximum likelihood estimation of continuous-time diffusionprocessesrdquo Journal of Business amp Economic Statistics vol 20 no3 pp 297ndash338 2002

[5] L ValdiviesoW Schoutens and F Tuerlinckx ldquoMaximum like-lihood estimation in processes of Ornstein-Uhlenbeck typerdquoStatistical Inference for Stochastic Processes vol 12 no 1 pp 1ndash192009

[6] O Elerian S Chib and N Shephard ldquoLikelihood inference fordiscretely observed nonlinear diffusionsrdquo Econometrica vol 69no 4 pp 959ndash993 2001

[7] A R Pedersen ldquoConsistency and asymptotic normality ofan approximate maximum likelihood estimator for discretelyobserved diffusion processesrdquo Bernoulli vol 1 no 3 pp 257ndash279 1995

[8] T Koulis and A Thavaneswaran ldquoInference for interest ratemodels using Milsteinrsquos approximationrdquo The Journal of Math-ematical Finance vol 3 pp 110ndash118 2013

[9] G J Jiang and J L Knight ldquoA nonparametric approach to theestimation of diffusion processes with an application to a short-term interest rate modelrdquo Econometric Theory vol 13 no 5 pp615ndash645 1997

[10] D Florens-Zmirou ldquoOn estimating the diffusion coefficientfrom discrete observationsrdquo Journal of Applied Probability vol30 no 4 pp 790ndash804 1993

[11] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 pp 177ndash186 1977

[12] M Brennan and E Schwartz ldquoAnalyzing convertible bondsrdquoJournal of Financial and Quantitative Analysis vol 15 pp 907ndash929 1980

[13] J Cox J Ingersoll and S Ross ldquoAn analysis of variable rate loancontractsrdquoThe Journal of Finance vol 35 pp 389ndash403 1980

[14] K Chan F Karolyi F Longstaff and A Sanders ldquoAn empiricalcomparison of alternative models of short term interest ratesrdquoThe Journal of Finance vol 47 pp 1209ndash1227 1992

[15] A Lari-Lavassani A Sadeghi and A Ware ldquoMean revertingmodels for energy option pricingrdquo Electronic Publications of theInternational Energy Credit Association 2001 httpwwwiecanet

[16] D Pilipovic Energy Risk Valuing andManaging Energy Deriva-tives McGraw-Hill New York NY USA 2007

[17] G Courtadon ldquoThe Pricing of options on default-free bondsrdquoJournal of Financial and Quantitative Analysis vol 17 pp 75ndash100 1982

[18] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[19] K Dunn and J McConnell ldquoValuation of GNMA Mortgage-backed securitiesrdquoThe Journal of Finance vol 36 no 3 pp 599ndash616 1981

[20] K Ramaswamy and S Sundaresan ldquoThe valuation of floating-rate instrumentsrdquo Journal of Financial Economics vol 17 pp251ndash272 1986

[21] S Sundaresan Valuation of Swaps Working Paper ColumbiaUniversity 1989

10 Journal of Probability and Statistics

[22] F Longstaff ldquoThe valuation of options on yieldsrdquo Journal ofFinancial Economics vol 26 no 1 pp 97ndash122 1990

[23] G Constantinides and J Ingersoll ldquoOptimal bond trading withpersonal taxesrdquo Journal of Financial Economics vol 13 pp 299ndash335 1984

[24] G Jiang and J Knight ldquoFinite sample comparison of alternativeestimators of Ito diffusion processes a Monte Carlo studyrdquoTheJournal of Computational Finance vol 2 no 3 1999

[25] L Giet andM LubranoBayesian Inference in Reducible Stochas-tic Differential Equations Document de Travail No 2004ndash57GREQAM University drsquoAix-Marseille 2004

[26] A R Bergstrom ldquoGaussian estimation of structural parametersin higher order continuous time dynamic modelsrdquo Economet-rica vol 51 no 1 pp 117ndash152 1983

[27] A Bergstrom ldquoContinuous time stochastic models and issuesof aggregation over timerdquo in Handbook of Econometrics ZGriliches and M D Intriligator Eds vol 2 Elsevier Amster-dam The Netherlands 1984

[28] A Bergstrom ldquoThe estimation of parameters in nonstationaryhigher-order continuous-time dynamic modelsrdquo EconometricTheory vol 1 pp 369ndash385 1985

[29] A Bergstrom ldquoThe estimation of open higher-order continuoustime dynamic models with mixed stock and flow datardquo Econo-metric Theory vol 2 pp 350ndash373 1986

[30] A Bergstrom Continuous Time Econometric Modelling OxfordUniversity Press Oxford UK 1990

[31] P C B Phillips and J Yu ldquoCorrigendum to lsquoA Gaussianapproach for continuous time models of short-term interestratesrsquordquoThe Econometrics Journal vol 14 no 1 pp 126ndash129 2011

[32] M J D Powell ldquoAn efficient method for finding the minimumof a function of several variables without calculating deriva-tivesrdquoThe Computer Journal vol 7 pp 155ndash162 1964

[33] V S Huzurbazar ldquoThe likelihood equation consistency and themaxima of the likelihood functionrdquo Annals of Eugenics vol 14pp 185ndash200 1948

[34] A Kohatsu-Higa and S Ogawa ldquoWeak rate of convergence foran Euler scheme of nonlinear SDErsquosrdquoMonte Carlo Methods andApplications vol 3 no 4 pp 327ndash345 1997

[35] V Bally andD Talay ldquoThe law of the Euler scheme for stochasticdifferential equations error analysis with Malliavin calculusrdquoMathematics and Computers in Simulation vol 38 no 1ndash3 pp35ndash41 1995

[36] A Zellner ldquoBayesian analysis of regression error termsrdquo Journalof the American Statistical Association vol 70 no 349 pp 138ndash144 1975

[37] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing England UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 7

Table 2 Monte Carlo study parameters

Monthly Weekly DailySize 500 1000 2000120582 072 3 6120573 minus012 minus05 minus1120590 06 035 025120574 05 05 05

In the case of 120582 and 120573 are parameters observed inefficientperformance (as discussed Yu and Phillips [1 page 220]) inthe method of Nowman 120582 yielding bias of 87 47 and64 for data monthly weekly and daily respectively Alsofor 120573 the bias present in the estimate is 94 46 and54 for the respective monthly weekly and daily series Themethod proposed byYu andPhillips has a better performancethan Nowman measured from the bias in the parametersreducing it by 15 8 and 16 for the case of 120582 and 5 6and 6 in the case of 120573 for the respective monthly weeklyand daily series Our method has a performance similar tothat of Yu and Phillips for daily data series reducing the biasregarding Nowman method in 15 for 120582 and 4 for 120573 Forweekly data series for reducing 120582 bias regarding the methodof Nowman is 7 whereas for 120573 is 6 as reduction Yu andPhillips method

An important point to note in the results obtained byour method is the condition that the parameter 120574 is knowna priori This is necessary in order to find closed formulas forthe estimators

In the approach to the model proposed by Nowman andYu-Phillips 119889119883

119905= (120582 + 120573119883

119905)119889119905 + 120590119883

120574

119905119889119861119905is implied to have

particular difficulty in estimating the parameter of speed ofreversion To make this condition clear Table 6 shows theresults obtained by the method of Durham and Gallant [4]and our method to estimate the parameters of the model119889119883119905= 120572(120583 + 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905with 120572 gt 0 120583 gt 0 120590 gt 0

120574 = 05 The results are the average of 512 estimations ofmonthly series with 1000 observations

The results shown in Table 6 show that the errors inestimation of 120583 and 120590 are small (less than 1) for bothmethods On the other hand the estimated speed of reversionhas an error of about 10 resulting in Durham and Gallantwhile ourmethod has an error close to 7 From this analysisit is clear that the parameter 120572 has a higher error in estimatingthan the other parameter estimates

In this regard the estimation error of 120572 in Yu [3] providesan analysis of the bias in the parameter estimation inherentreversion total observation time which is sampled in thecontinuous process for the special case where the long-termaverage 120583 is zero Reference is made to the fact that the bias inparameter depends asymptotically on total observation timewhich of the process and not the number of observationsmade in this period

Figures 1 2 and 3 show the results of the estimationof the parameters for different observation horizons withdifferent sampling frequencies (The results are obtained asthe average of the individual estimates on 1000 simulated

0 10 20 30 40 50 60 70 80 9014

145

15

155

16

165

17

175

18

185

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120572 = 15

Figure 1 120572 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

0 10 20 30 40 50 60 70 80 900497

04975

0498

04985

0499

04995

05

05005

0501

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120583 = 05

Figure 2 120583 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

series in 4 sampling scenarios (a) bimonthly series with 500observations (b) monthly series with 1000 observations (c)weekly series with 4333 observations and (d) daily series with21000 observations The reference model for the simulationswas 119889119883

119905= 120572(120583 minus 119883

119905)119889119905 + 120590119883

120574

119905119889119861119905 with 120572 = 15 120583 = 05

120574 = 05 and 120590 = 04)Figure 1 shows the evolution of the estimate of 120572 for

observations from 10 to 83 years Behavior of the estimatesin the four scenarios can be established empirically by thedependence of the bias on the estimate and the horizonof observation as they have more years of observation

8 Journal of Probability and Statistics

Table 3 Comparison of monthly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 13440 minus02332 12330 minus02275 06173 04919 12797 minus02200 05989Bias 06240 minus01132 05130 minus01075 00173 minus00081 05597 minus01000 minus00011Var 05897 01713 05732 01589 00099 00071 05406 00149 00004MSE 09791 01841 08363 01705 00102 00071 08533 00249 00004The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 500observations each The real values of the parameters are 120582 = 072 120573 = minus012 120590 = 06 and 120574 = 05

Table 4 Comparison of weekly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 44090 minus07320 41650 minus07011 03762 04925 42038 minus07022 03499Bias 14090 minus02320 11650 minus02011 00262 minus00075 12038 minus02022 minus00001Var 40663 01109 38608 01030 00172 00357 35427 00992 00001MSE 60855 01647 52180 01435 00179 00358 49883 01400 00001The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 3 120573 = minus05 120590 = 035 and 120574 = 05

Table 5 Comparison of daily data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 98250 minus15360 88440 minus14750 02521 04970 89542 minus14933 02500Bias 38250 minus05360 28440 minus04750 00021 minus00030 29542 minus04933 00000Var 191959 05219 178220 04798 00244 00746 155818 04334 00000MSE 338266 08092 259100 07054 00244 00746 242938 06763 00000The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 60 120573 = minus10 120590 = 025 and 120574 = 05

Table 6 Comparison of Durham and Gallant and proposed methods

Parameter(lowast) Real Bias(a) Bias(b) Real Bias(a) Bias(b)

120572 05 0048900 0037700 05 0059730 0021400120583 006 0000610 0000127 006 minus0000310 minus0000021120590 015 0000200 0000032 022 0000120 minus0000046120572 05 0043780 0032400 04 0045890 0032200120583 006 0000050 0000016 006 0000770 0000046120590 003 0000010 minus0000003 015 0000080 minus0000025The data are obtained with 512 simulations monthly series (119889119905 = 112) with 1000 observations each and the parameters are listed as real(a)Results of the method proposed by Durham and Gallant(b)Results of the method proposed in this paper(lowast)In Durham and Gallant [4] 120583 = 120579

1 120572 = 120579

2 and 120590 = 120579

3

bias decreases with no evidence of a significant differencebetween the estimates different frequencies

From Figure 2 it is observed that after few observationsvirtually independent of the year of observation estimationsfor 120583 are obtained with a bias of less than 1 showingthe efficiency of the estimation of the long-term averagereversion processes average In the case of estimation of 120590Figure 3 shows that the bias in the parameter estimates is

more dependent on the number of observations which is thetotal time of observation of the processThis is reflected in thefact that the estimates are in series with a higher sampling ratereaches a lower bias for the same observation horizon thanlower sample rate and therefore have fewer observations Onthe other hand as for the case of the estimate of 120583 the biaspresent in the estimates of 120590 is practically negligible (less than1) in most scenarios

Journal of Probability and Statistics 9

0 10 20 30 40 50 60 70 80 900386

0388

039

0392

0394

0396

0398

04

0402

0404

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120590 = 04

Figure 3 120590 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

6 Conclusions and Comments

This paper presents a new alternative for Gaussian estimationmodels of mean reversion of one factor under the Eulernumerical approximation that includes among others somemodels of interest rates in the short term

Monte Carlo simulations show a good approximation ofthe diffusion term parameter but less approximation in atrend term parameterThe estimation results have almost thesame order as that in Nowmanrsquos [2] and in some cases betterthan that in Durham and Gallantrsquos [4] It should be notedthat the proposed methodology outperforms the estimate of120590 compared to the other alternatives presented These resultsbecome more relevant since in practice the estimation ofvolatility is extremely useful in the evaluation of financialderivatives in which the trend term loses significance whenit is absorbed by the processes of change measurement andvaluation risk neutral

The estimate based on this new method though doesnot eliminate the bias in estimation of the reversion rateparameter for a small sample is in some cases comparablewith the results in Yu and Phillips [1] if samples have longertime horizons

This estimation scheme has several important qualitiesFirst we come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simpler and does not requiresophisticated routines to perform extensive calculations

References

[1] J Yu and P C B Phillips ldquoA Gaussian approach for continuoustime models of the short-term interest raterdquo The EconometricsJournal vol 4 no 2 pp 210ndash224 2001

[2] K Nowman ldquoGaussian estimation of single-factor continuoustime models of the term structure of interest ratesrdquoThe Journalof Finance vol 52 pp 1695ndash1703 1997

[3] J Yu ldquoBias in the estimation of themean reversion parameter incontinuous time modelsrdquo Journal of Econometrics vol 169 no1 pp 114ndash122 2012

[4] G B Durham and A R Gallant ldquoNumerical techniques formaximum likelihood estimation of continuous-time diffusionprocessesrdquo Journal of Business amp Economic Statistics vol 20 no3 pp 297ndash338 2002

[5] L ValdiviesoW Schoutens and F Tuerlinckx ldquoMaximum like-lihood estimation in processes of Ornstein-Uhlenbeck typerdquoStatistical Inference for Stochastic Processes vol 12 no 1 pp 1ndash192009

[6] O Elerian S Chib and N Shephard ldquoLikelihood inference fordiscretely observed nonlinear diffusionsrdquo Econometrica vol 69no 4 pp 959ndash993 2001

[7] A R Pedersen ldquoConsistency and asymptotic normality ofan approximate maximum likelihood estimator for discretelyobserved diffusion processesrdquo Bernoulli vol 1 no 3 pp 257ndash279 1995

[8] T Koulis and A Thavaneswaran ldquoInference for interest ratemodels using Milsteinrsquos approximationrdquo The Journal of Math-ematical Finance vol 3 pp 110ndash118 2013

[9] G J Jiang and J L Knight ldquoA nonparametric approach to theestimation of diffusion processes with an application to a short-term interest rate modelrdquo Econometric Theory vol 13 no 5 pp615ndash645 1997

[10] D Florens-Zmirou ldquoOn estimating the diffusion coefficientfrom discrete observationsrdquo Journal of Applied Probability vol30 no 4 pp 790ndash804 1993

[11] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 pp 177ndash186 1977

[12] M Brennan and E Schwartz ldquoAnalyzing convertible bondsrdquoJournal of Financial and Quantitative Analysis vol 15 pp 907ndash929 1980

[13] J Cox J Ingersoll and S Ross ldquoAn analysis of variable rate loancontractsrdquoThe Journal of Finance vol 35 pp 389ndash403 1980

[14] K Chan F Karolyi F Longstaff and A Sanders ldquoAn empiricalcomparison of alternative models of short term interest ratesrdquoThe Journal of Finance vol 47 pp 1209ndash1227 1992

[15] A Lari-Lavassani A Sadeghi and A Ware ldquoMean revertingmodels for energy option pricingrdquo Electronic Publications of theInternational Energy Credit Association 2001 httpwwwiecanet

[16] D Pilipovic Energy Risk Valuing andManaging Energy Deriva-tives McGraw-Hill New York NY USA 2007

[17] G Courtadon ldquoThe Pricing of options on default-free bondsrdquoJournal of Financial and Quantitative Analysis vol 17 pp 75ndash100 1982

[18] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[19] K Dunn and J McConnell ldquoValuation of GNMA Mortgage-backed securitiesrdquoThe Journal of Finance vol 36 no 3 pp 599ndash616 1981

[20] K Ramaswamy and S Sundaresan ldquoThe valuation of floating-rate instrumentsrdquo Journal of Financial Economics vol 17 pp251ndash272 1986

[21] S Sundaresan Valuation of Swaps Working Paper ColumbiaUniversity 1989

10 Journal of Probability and Statistics

[22] F Longstaff ldquoThe valuation of options on yieldsrdquo Journal ofFinancial Economics vol 26 no 1 pp 97ndash122 1990

[23] G Constantinides and J Ingersoll ldquoOptimal bond trading withpersonal taxesrdquo Journal of Financial Economics vol 13 pp 299ndash335 1984

[24] G Jiang and J Knight ldquoFinite sample comparison of alternativeestimators of Ito diffusion processes a Monte Carlo studyrdquoTheJournal of Computational Finance vol 2 no 3 1999

[25] L Giet andM LubranoBayesian Inference in Reducible Stochas-tic Differential Equations Document de Travail No 2004ndash57GREQAM University drsquoAix-Marseille 2004

[26] A R Bergstrom ldquoGaussian estimation of structural parametersin higher order continuous time dynamic modelsrdquo Economet-rica vol 51 no 1 pp 117ndash152 1983

[27] A Bergstrom ldquoContinuous time stochastic models and issuesof aggregation over timerdquo in Handbook of Econometrics ZGriliches and M D Intriligator Eds vol 2 Elsevier Amster-dam The Netherlands 1984

[28] A Bergstrom ldquoThe estimation of parameters in nonstationaryhigher-order continuous-time dynamic modelsrdquo EconometricTheory vol 1 pp 369ndash385 1985

[29] A Bergstrom ldquoThe estimation of open higher-order continuoustime dynamic models with mixed stock and flow datardquo Econo-metric Theory vol 2 pp 350ndash373 1986

[30] A Bergstrom Continuous Time Econometric Modelling OxfordUniversity Press Oxford UK 1990

[31] P C B Phillips and J Yu ldquoCorrigendum to lsquoA Gaussianapproach for continuous time models of short-term interestratesrsquordquoThe Econometrics Journal vol 14 no 1 pp 126ndash129 2011

[32] M J D Powell ldquoAn efficient method for finding the minimumof a function of several variables without calculating deriva-tivesrdquoThe Computer Journal vol 7 pp 155ndash162 1964

[33] V S Huzurbazar ldquoThe likelihood equation consistency and themaxima of the likelihood functionrdquo Annals of Eugenics vol 14pp 185ndash200 1948

[34] A Kohatsu-Higa and S Ogawa ldquoWeak rate of convergence foran Euler scheme of nonlinear SDErsquosrdquoMonte Carlo Methods andApplications vol 3 no 4 pp 327ndash345 1997

[35] V Bally andD Talay ldquoThe law of the Euler scheme for stochasticdifferential equations error analysis with Malliavin calculusrdquoMathematics and Computers in Simulation vol 38 no 1ndash3 pp35ndash41 1995

[36] A Zellner ldquoBayesian analysis of regression error termsrdquo Journalof the American Statistical Association vol 70 no 349 pp 138ndash144 1975

[37] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing England UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Journal of Probability and Statistics

Table 3 Comparison of monthly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 13440 minus02332 12330 minus02275 06173 04919 12797 minus02200 05989Bias 06240 minus01132 05130 minus01075 00173 minus00081 05597 minus01000 minus00011Var 05897 01713 05732 01589 00099 00071 05406 00149 00004MSE 09791 01841 08363 01705 00102 00071 08533 00249 00004The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 500observations each The real values of the parameters are 120582 = 072 120573 = minus012 120590 = 06 and 120574 = 05

Table 4 Comparison of weekly data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 44090 minus07320 41650 minus07011 03762 04925 42038 minus07022 03499Bias 14090 minus02320 11650 minus02011 00262 minus00075 12038 minus02022 minus00001Var 40663 01109 38608 01030 00172 00357 35427 00992 00001MSE 60855 01647 52180 01435 00179 00358 49883 01400 00001The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 3 120573 = minus05 120590 = 035 and 120574 = 05

Table 5 Comparison of daily data series with Nowman Yu-Phillips and proposed methods

Nowman Yu Phillips Nowman-Yu Phillips Proposed120582 120573 120582 120573 120590 120574 120582 120573 120590

Mean 98250 minus15360 88440 minus14750 02521 04970 89542 minus14933 02500Bias 38250 minus05360 28440 minus04750 00021 minus00030 29542 minus04933 00000Var 191959 05219 178220 04798 00244 00746 155818 04334 00000MSE 338266 08092 259100 07054 00244 00746 242938 06763 00000The results of the methods of Nowman and Yu Phillips are taken into account Yu and Phillips [1] Our results were obtained with 1000 simulations with 1000observations each The real values of the parameters are 120582 = 60 120573 = minus10 120590 = 025 and 120574 = 05

Table 6 Comparison of Durham and Gallant and proposed methods

Parameter(lowast) Real Bias(a) Bias(b) Real Bias(a) Bias(b)

120572 05 0048900 0037700 05 0059730 0021400120583 006 0000610 0000127 006 minus0000310 minus0000021120590 015 0000200 0000032 022 0000120 minus0000046120572 05 0043780 0032400 04 0045890 0032200120583 006 0000050 0000016 006 0000770 0000046120590 003 0000010 minus0000003 015 0000080 minus0000025The data are obtained with 512 simulations monthly series (119889119905 = 112) with 1000 observations each and the parameters are listed as real(a)Results of the method proposed by Durham and Gallant(b)Results of the method proposed in this paper(lowast)In Durham and Gallant [4] 120583 = 120579

1 120572 = 120579

2 and 120590 = 120579

3

bias decreases with no evidence of a significant differencebetween the estimates different frequencies

From Figure 2 it is observed that after few observationsvirtually independent of the year of observation estimationsfor 120583 are obtained with a bias of less than 1 showingthe efficiency of the estimation of the long-term averagereversion processes average In the case of estimation of 120590Figure 3 shows that the bias in the parameter estimates is

more dependent on the number of observations which is thetotal time of observation of the processThis is reflected in thefact that the estimates are in series with a higher sampling ratereaches a lower bias for the same observation horizon thanlower sample rate and therefore have fewer observations Onthe other hand as for the case of the estimate of 120583 the biaspresent in the estimates of 120590 is practically negligible (less than1) in most scenarios

Journal of Probability and Statistics 9

0 10 20 30 40 50 60 70 80 900386

0388

039

0392

0394

0396

0398

04

0402

0404

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120590 = 04

Figure 3 120590 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

6 Conclusions and Comments

This paper presents a new alternative for Gaussian estimationmodels of mean reversion of one factor under the Eulernumerical approximation that includes among others somemodels of interest rates in the short term

Monte Carlo simulations show a good approximation ofthe diffusion term parameter but less approximation in atrend term parameterThe estimation results have almost thesame order as that in Nowmanrsquos [2] and in some cases betterthan that in Durham and Gallantrsquos [4] It should be notedthat the proposed methodology outperforms the estimate of120590 compared to the other alternatives presented These resultsbecome more relevant since in practice the estimation ofvolatility is extremely useful in the evaluation of financialderivatives in which the trend term loses significance whenit is absorbed by the processes of change measurement andvaluation risk neutral

The estimate based on this new method though doesnot eliminate the bias in estimation of the reversion rateparameter for a small sample is in some cases comparablewith the results in Yu and Phillips [1] if samples have longertime horizons

This estimation scheme has several important qualitiesFirst we come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simpler and does not requiresophisticated routines to perform extensive calculations

References

[1] J Yu and P C B Phillips ldquoA Gaussian approach for continuoustime models of the short-term interest raterdquo The EconometricsJournal vol 4 no 2 pp 210ndash224 2001

[2] K Nowman ldquoGaussian estimation of single-factor continuoustime models of the term structure of interest ratesrdquoThe Journalof Finance vol 52 pp 1695ndash1703 1997

[3] J Yu ldquoBias in the estimation of themean reversion parameter incontinuous time modelsrdquo Journal of Econometrics vol 169 no1 pp 114ndash122 2012

[4] G B Durham and A R Gallant ldquoNumerical techniques formaximum likelihood estimation of continuous-time diffusionprocessesrdquo Journal of Business amp Economic Statistics vol 20 no3 pp 297ndash338 2002

[5] L ValdiviesoW Schoutens and F Tuerlinckx ldquoMaximum like-lihood estimation in processes of Ornstein-Uhlenbeck typerdquoStatistical Inference for Stochastic Processes vol 12 no 1 pp 1ndash192009

[6] O Elerian S Chib and N Shephard ldquoLikelihood inference fordiscretely observed nonlinear diffusionsrdquo Econometrica vol 69no 4 pp 959ndash993 2001

[7] A R Pedersen ldquoConsistency and asymptotic normality ofan approximate maximum likelihood estimator for discretelyobserved diffusion processesrdquo Bernoulli vol 1 no 3 pp 257ndash279 1995

[8] T Koulis and A Thavaneswaran ldquoInference for interest ratemodels using Milsteinrsquos approximationrdquo The Journal of Math-ematical Finance vol 3 pp 110ndash118 2013

[9] G J Jiang and J L Knight ldquoA nonparametric approach to theestimation of diffusion processes with an application to a short-term interest rate modelrdquo Econometric Theory vol 13 no 5 pp615ndash645 1997

[10] D Florens-Zmirou ldquoOn estimating the diffusion coefficientfrom discrete observationsrdquo Journal of Applied Probability vol30 no 4 pp 790ndash804 1993

[11] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 pp 177ndash186 1977

[12] M Brennan and E Schwartz ldquoAnalyzing convertible bondsrdquoJournal of Financial and Quantitative Analysis vol 15 pp 907ndash929 1980

[13] J Cox J Ingersoll and S Ross ldquoAn analysis of variable rate loancontractsrdquoThe Journal of Finance vol 35 pp 389ndash403 1980

[14] K Chan F Karolyi F Longstaff and A Sanders ldquoAn empiricalcomparison of alternative models of short term interest ratesrdquoThe Journal of Finance vol 47 pp 1209ndash1227 1992

[15] A Lari-Lavassani A Sadeghi and A Ware ldquoMean revertingmodels for energy option pricingrdquo Electronic Publications of theInternational Energy Credit Association 2001 httpwwwiecanet

[16] D Pilipovic Energy Risk Valuing andManaging Energy Deriva-tives McGraw-Hill New York NY USA 2007

[17] G Courtadon ldquoThe Pricing of options on default-free bondsrdquoJournal of Financial and Quantitative Analysis vol 17 pp 75ndash100 1982

[18] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[19] K Dunn and J McConnell ldquoValuation of GNMA Mortgage-backed securitiesrdquoThe Journal of Finance vol 36 no 3 pp 599ndash616 1981

[20] K Ramaswamy and S Sundaresan ldquoThe valuation of floating-rate instrumentsrdquo Journal of Financial Economics vol 17 pp251ndash272 1986

[21] S Sundaresan Valuation of Swaps Working Paper ColumbiaUniversity 1989

10 Journal of Probability and Statistics

[22] F Longstaff ldquoThe valuation of options on yieldsrdquo Journal ofFinancial Economics vol 26 no 1 pp 97ndash122 1990

[23] G Constantinides and J Ingersoll ldquoOptimal bond trading withpersonal taxesrdquo Journal of Financial Economics vol 13 pp 299ndash335 1984

[24] G Jiang and J Knight ldquoFinite sample comparison of alternativeestimators of Ito diffusion processes a Monte Carlo studyrdquoTheJournal of Computational Finance vol 2 no 3 1999

[25] L Giet andM LubranoBayesian Inference in Reducible Stochas-tic Differential Equations Document de Travail No 2004ndash57GREQAM University drsquoAix-Marseille 2004

[26] A R Bergstrom ldquoGaussian estimation of structural parametersin higher order continuous time dynamic modelsrdquo Economet-rica vol 51 no 1 pp 117ndash152 1983

[27] A Bergstrom ldquoContinuous time stochastic models and issuesof aggregation over timerdquo in Handbook of Econometrics ZGriliches and M D Intriligator Eds vol 2 Elsevier Amster-dam The Netherlands 1984

[28] A Bergstrom ldquoThe estimation of parameters in nonstationaryhigher-order continuous-time dynamic modelsrdquo EconometricTheory vol 1 pp 369ndash385 1985

[29] A Bergstrom ldquoThe estimation of open higher-order continuoustime dynamic models with mixed stock and flow datardquo Econo-metric Theory vol 2 pp 350ndash373 1986

[30] A Bergstrom Continuous Time Econometric Modelling OxfordUniversity Press Oxford UK 1990

[31] P C B Phillips and J Yu ldquoCorrigendum to lsquoA Gaussianapproach for continuous time models of short-term interestratesrsquordquoThe Econometrics Journal vol 14 no 1 pp 126ndash129 2011

[32] M J D Powell ldquoAn efficient method for finding the minimumof a function of several variables without calculating deriva-tivesrdquoThe Computer Journal vol 7 pp 155ndash162 1964

[33] V S Huzurbazar ldquoThe likelihood equation consistency and themaxima of the likelihood functionrdquo Annals of Eugenics vol 14pp 185ndash200 1948

[34] A Kohatsu-Higa and S Ogawa ldquoWeak rate of convergence foran Euler scheme of nonlinear SDErsquosrdquoMonte Carlo Methods andApplications vol 3 no 4 pp 327ndash345 1997

[35] V Bally andD Talay ldquoThe law of the Euler scheme for stochasticdifferential equations error analysis with Malliavin calculusrdquoMathematics and Computers in Simulation vol 38 no 1ndash3 pp35ndash41 1995

[36] A Zellner ldquoBayesian analysis of regression error termsrdquo Journalof the American Statistical Association vol 70 no 349 pp 138ndash144 1975

[37] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing England UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Probability and Statistics 9

0 10 20 30 40 50 60 70 80 900386

0388

039

0392

0394

0396

0398

04

0402

0404

dt = 16 and n = 500

dt = 112 and n = 1000dt = 152 and n = 4333

dt = 1252 and n = 21000

120590 = 04

Figure 3 120590 estimation results 119889119905 = 16 112 152 and 1252 119899 = 5001000 4333 and 21000 respectively

6 Conclusions and Comments

This paper presents a new alternative for Gaussian estimationmodels of mean reversion of one factor under the Eulernumerical approximation that includes among others somemodels of interest rates in the short term

Monte Carlo simulations show a good approximation ofthe diffusion term parameter but less approximation in atrend term parameterThe estimation results have almost thesame order as that in Nowmanrsquos [2] and in some cases betterthan that in Durham and Gallantrsquos [4] It should be notedthat the proposed methodology outperforms the estimate of120590 compared to the other alternatives presented These resultsbecome more relevant since in practice the estimation ofvolatility is extremely useful in the evaluation of financialderivatives in which the trend term loses significance whenit is absorbed by the processes of change measurement andvaluation risk neutral

The estimate based on this new method though doesnot eliminate the bias in estimation of the reversion rateparameter for a small sample is in some cases comparablewith the results in Yu and Phillips [1] if samples have longertime horizons

This estimation scheme has several important qualitiesFirst we come to a closed formula for Gaussian estimationwithout numerical optimization methods Second the com-putational implementation is simpler and does not requiresophisticated routines to perform extensive calculations

References

[1] J Yu and P C B Phillips ldquoA Gaussian approach for continuoustime models of the short-term interest raterdquo The EconometricsJournal vol 4 no 2 pp 210ndash224 2001

[2] K Nowman ldquoGaussian estimation of single-factor continuoustime models of the term structure of interest ratesrdquoThe Journalof Finance vol 52 pp 1695ndash1703 1997

[3] J Yu ldquoBias in the estimation of themean reversion parameter incontinuous time modelsrdquo Journal of Econometrics vol 169 no1 pp 114ndash122 2012

[4] G B Durham and A R Gallant ldquoNumerical techniques formaximum likelihood estimation of continuous-time diffusionprocessesrdquo Journal of Business amp Economic Statistics vol 20 no3 pp 297ndash338 2002

[5] L ValdiviesoW Schoutens and F Tuerlinckx ldquoMaximum like-lihood estimation in processes of Ornstein-Uhlenbeck typerdquoStatistical Inference for Stochastic Processes vol 12 no 1 pp 1ndash192009

[6] O Elerian S Chib and N Shephard ldquoLikelihood inference fordiscretely observed nonlinear diffusionsrdquo Econometrica vol 69no 4 pp 959ndash993 2001

[7] A R Pedersen ldquoConsistency and asymptotic normality ofan approximate maximum likelihood estimator for discretelyobserved diffusion processesrdquo Bernoulli vol 1 no 3 pp 257ndash279 1995

[8] T Koulis and A Thavaneswaran ldquoInference for interest ratemodels using Milsteinrsquos approximationrdquo The Journal of Math-ematical Finance vol 3 pp 110ndash118 2013

[9] G J Jiang and J L Knight ldquoA nonparametric approach to theestimation of diffusion processes with an application to a short-term interest rate modelrdquo Econometric Theory vol 13 no 5 pp615ndash645 1997

[10] D Florens-Zmirou ldquoOn estimating the diffusion coefficientfrom discrete observationsrdquo Journal of Applied Probability vol30 no 4 pp 790ndash804 1993

[11] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 pp 177ndash186 1977

[12] M Brennan and E Schwartz ldquoAnalyzing convertible bondsrdquoJournal of Financial and Quantitative Analysis vol 15 pp 907ndash929 1980

[13] J Cox J Ingersoll and S Ross ldquoAn analysis of variable rate loancontractsrdquoThe Journal of Finance vol 35 pp 389ndash403 1980

[14] K Chan F Karolyi F Longstaff and A Sanders ldquoAn empiricalcomparison of alternative models of short term interest ratesrdquoThe Journal of Finance vol 47 pp 1209ndash1227 1992

[15] A Lari-Lavassani A Sadeghi and A Ware ldquoMean revertingmodels for energy option pricingrdquo Electronic Publications of theInternational Energy Credit Association 2001 httpwwwiecanet

[16] D Pilipovic Energy Risk Valuing andManaging Energy Deriva-tives McGraw-Hill New York NY USA 2007

[17] G Courtadon ldquoThe Pricing of options on default-free bondsrdquoJournal of Financial and Quantitative Analysis vol 17 pp 75ndash100 1982

[18] J C Cox J E Ingersoll Jr and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[19] K Dunn and J McConnell ldquoValuation of GNMA Mortgage-backed securitiesrdquoThe Journal of Finance vol 36 no 3 pp 599ndash616 1981

[20] K Ramaswamy and S Sundaresan ldquoThe valuation of floating-rate instrumentsrdquo Journal of Financial Economics vol 17 pp251ndash272 1986

[21] S Sundaresan Valuation of Swaps Working Paper ColumbiaUniversity 1989

10 Journal of Probability and Statistics

[22] F Longstaff ldquoThe valuation of options on yieldsrdquo Journal ofFinancial Economics vol 26 no 1 pp 97ndash122 1990

[23] G Constantinides and J Ingersoll ldquoOptimal bond trading withpersonal taxesrdquo Journal of Financial Economics vol 13 pp 299ndash335 1984

[24] G Jiang and J Knight ldquoFinite sample comparison of alternativeestimators of Ito diffusion processes a Monte Carlo studyrdquoTheJournal of Computational Finance vol 2 no 3 1999

[25] L Giet andM LubranoBayesian Inference in Reducible Stochas-tic Differential Equations Document de Travail No 2004ndash57GREQAM University drsquoAix-Marseille 2004

[26] A R Bergstrom ldquoGaussian estimation of structural parametersin higher order continuous time dynamic modelsrdquo Economet-rica vol 51 no 1 pp 117ndash152 1983

[27] A Bergstrom ldquoContinuous time stochastic models and issuesof aggregation over timerdquo in Handbook of Econometrics ZGriliches and M D Intriligator Eds vol 2 Elsevier Amster-dam The Netherlands 1984

[28] A Bergstrom ldquoThe estimation of parameters in nonstationaryhigher-order continuous-time dynamic modelsrdquo EconometricTheory vol 1 pp 369ndash385 1985

[29] A Bergstrom ldquoThe estimation of open higher-order continuoustime dynamic models with mixed stock and flow datardquo Econo-metric Theory vol 2 pp 350ndash373 1986

[30] A Bergstrom Continuous Time Econometric Modelling OxfordUniversity Press Oxford UK 1990

[31] P C B Phillips and J Yu ldquoCorrigendum to lsquoA Gaussianapproach for continuous time models of short-term interestratesrsquordquoThe Econometrics Journal vol 14 no 1 pp 126ndash129 2011

[32] M J D Powell ldquoAn efficient method for finding the minimumof a function of several variables without calculating deriva-tivesrdquoThe Computer Journal vol 7 pp 155ndash162 1964

[33] V S Huzurbazar ldquoThe likelihood equation consistency and themaxima of the likelihood functionrdquo Annals of Eugenics vol 14pp 185ndash200 1948

[34] A Kohatsu-Higa and S Ogawa ldquoWeak rate of convergence foran Euler scheme of nonlinear SDErsquosrdquoMonte Carlo Methods andApplications vol 3 no 4 pp 327ndash345 1997

[35] V Bally andD Talay ldquoThe law of the Euler scheme for stochasticdifferential equations error analysis with Malliavin calculusrdquoMathematics and Computers in Simulation vol 38 no 1ndash3 pp35ndash41 1995

[36] A Zellner ldquoBayesian analysis of regression error termsrdquo Journalof the American Statistical Association vol 70 no 349 pp 138ndash144 1975

[37] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing England UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Journal of Probability and Statistics

[22] F Longstaff ldquoThe valuation of options on yieldsrdquo Journal ofFinancial Economics vol 26 no 1 pp 97ndash122 1990

[23] G Constantinides and J Ingersoll ldquoOptimal bond trading withpersonal taxesrdquo Journal of Financial Economics vol 13 pp 299ndash335 1984

[24] G Jiang and J Knight ldquoFinite sample comparison of alternativeestimators of Ito diffusion processes a Monte Carlo studyrdquoTheJournal of Computational Finance vol 2 no 3 1999

[25] L Giet andM LubranoBayesian Inference in Reducible Stochas-tic Differential Equations Document de Travail No 2004ndash57GREQAM University drsquoAix-Marseille 2004

[26] A R Bergstrom ldquoGaussian estimation of structural parametersin higher order continuous time dynamic modelsrdquo Economet-rica vol 51 no 1 pp 117ndash152 1983

[27] A Bergstrom ldquoContinuous time stochastic models and issuesof aggregation over timerdquo in Handbook of Econometrics ZGriliches and M D Intriligator Eds vol 2 Elsevier Amster-dam The Netherlands 1984

[28] A Bergstrom ldquoThe estimation of parameters in nonstationaryhigher-order continuous-time dynamic modelsrdquo EconometricTheory vol 1 pp 369ndash385 1985

[29] A Bergstrom ldquoThe estimation of open higher-order continuoustime dynamic models with mixed stock and flow datardquo Econo-metric Theory vol 2 pp 350ndash373 1986

[30] A Bergstrom Continuous Time Econometric Modelling OxfordUniversity Press Oxford UK 1990

[31] P C B Phillips and J Yu ldquoCorrigendum to lsquoA Gaussianapproach for continuous time models of short-term interestratesrsquordquoThe Econometrics Journal vol 14 no 1 pp 126ndash129 2011

[32] M J D Powell ldquoAn efficient method for finding the minimumof a function of several variables without calculating deriva-tivesrdquoThe Computer Journal vol 7 pp 155ndash162 1964

[33] V S Huzurbazar ldquoThe likelihood equation consistency and themaxima of the likelihood functionrdquo Annals of Eugenics vol 14pp 185ndash200 1948

[34] A Kohatsu-Higa and S Ogawa ldquoWeak rate of convergence foran Euler scheme of nonlinear SDErsquosrdquoMonte Carlo Methods andApplications vol 3 no 4 pp 327ndash345 1997

[35] V Bally andD Talay ldquoThe law of the Euler scheme for stochasticdifferential equations error analysis with Malliavin calculusrdquoMathematics and Computers in Simulation vol 38 no 1ndash3 pp35ndash41 1995

[36] A Zellner ldquoBayesian analysis of regression error termsrdquo Journalof the American Statistical Association vol 70 no 349 pp 138ndash144 1975

[37] X Mao Stochastic Differential Equations and ApplicationsHorwood Publishing England UK 2nd edition 2008

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Journal ofApplied Mathematics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ProbabilityandStatistics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Advances in

Mathematical Physics

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

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Volume 2014

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Stochastic AnalysisInternational Journal of

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OptimizationJournal of

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Volume 2014

International Journal of

Combinatorics

OperationsResearch

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in

DecisionSciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of