Markov Chain Monte Carlo Analysis of

Option Pricing Models

Matthew C. Pollard∗

Tom Smith (supervisor)

Department of Finance and Applied Statistics,

Australian National University

6th August 2007

Abstract

We use the Markov Chain Monte Carlo (MCMC) method to investigate a largeclass of continuous-time option pricing models. These include: constant-volatility,stochastic volatility, price jump-diffusions and volatility jump-diffusions. Esti-mation of these models is difficult or impossible by usual statistical methods.Consequently, there is little consensus which model is best. We propose a newBayesian method for estimation and model selection, the “Mixture Model MCMC”algorithm. Formal model selection is performed by posterior model probabilities.Probabilities for each model are calculated using historic returns and option pricesover the S&P 500 index. The method allow for joint inference of the physical andrisk-neutral distribution of parameters.

∗Email: matthewcpollard@berkeley.edu,I would like to thank Tom Smith, Doug Foster and Michael Martin for their invaluable advice, criticismand cuban cigars.

1

1 Introduction

Continuous-time models are the centerpiece of modern financial theories. Such modelsare specified as solutions to stochastic differential equations that model the randomevolution of variables in time. Consider, for example, the following model for equityprices (Merton, 1971):

M1 :dSt

St= µdt + σdWt + d

Nt∑j=1

ξj

.

Here, St is the equity price, µ and σ are drift and volatility constants, dWt is a Brownianmotion increment, Nt is a Poisson jump process with intensity λ, and ξ are jumps drawnfrom a normal distribution with parameters µJ and σ2

J .

Empirical investigation of continuous-time models involves answering two funda-mental questions:

1. What are the values of unknown parameters and state variables?

2. Is the model a good description of reality?

The first question concerns estimation. Given a time series of data Y, we wish toextract as much information about the parameters Θ = {µ, σ, λ, µJ , σJ} and hiddenstate variables X = {N(t), ξ} as the sample contains. The conditional distributionp(Θ, X|Y ) is the sufficient statistic: any parameter estimate may be obtained by in-tegrating this distribution. However, estimation of p(Θ, X|Y ) is generally impossibleusing classical methods in all but the simplest models.

The second question concerns model adequacy. Given the choice between severalalternative models, we wish to pick the model that best describes reality. The modelM1 is one of many popular equity price models. Another is the Black-Scholes (1973)model:

M2 :dSt

St= µdt + σdWt.

Formal tests comparing M1 to M2 is not simple in continuous-time. The modelsspecify price distributions not actual prices, and observed prices are only a sample pathalong the true distribution. Classical model selection methods are either inapplicableor have low power. A natural and attractive alternative is the probability the modelis true given the data. Formally, this is p(Mk|Y ) and is called the posterior modelprobability. However, computing posterior model probabilities for models such as M1

and M2 is hard.This thesis proposes a Bayesian solution to (1) and (2). It presents a new Markov

Chain Monte Carlo algorithm that estimates both p(Θ, X|Y ) and p(Mk|Y ). Thealgorithm, the “Mixture Model MCMC”, is a Bayesian method designed for model se-lection. The algorithm simultaneously estimates the parameter space Θ = (Θ1, ...,Θk),the state space X = (X1, ..., Xk) and the model space M = (M1, ...,Mk) by sampling

2

Figure 1: Left: S&P 500, log-prices and returns. Middle: Merton model simulation,log-prices and returns. Right: Black-Scholes model simulation, log-prices and returns.

from the mixture model

M∗ = p(M1|Y )M1 + p(M2|Y )M2 + ... + p(Mk|Y )Mk. (1)

The method builds on the work of Jones (2000) and Eraker, Johannes and Polson(2001), by directly incorporating Bayesian model selection into the MCMC simulation.

The major contribution of the thesis is in performing in option pricing model selec-tion. A large class of popular models shall be estimated and selected. These include:constant-volatility, stochastic volatility, jump-diffusions, jump-diffusions with stochas-tic volatility, and stochastic volatility with volatility jumps. Posterior probabilities foreach model are calculated using S&P 500 options prices and index returns, and weselect the most probable option pricing model. We shall also test whether mixturemodels specified by (1) outperform the best stand alone model. The stability of pos-terior model probabilities shall also be investigated with particular focus on periods ofmarket distress, such as the 1987 crash and the 1997 Russian Ruble devaluation.

The paper is structured as follows. Section two provides a short review of MCMCand option pricing literature. Section three presents an overview of MCMC theory andoption pricing models. It provides an example of using the method to estimate theBlack-Scholes model using both equity and option prices. Section four discusses themethods used to estimate continuous time models and the priors used for sampling,and section five describes the option and index return data. Section seven presents theresults of fitting option pricing models using returns data exclusively. Section eightoutlies future work and section nine concludes.

3

2 Literature Review

MCMC model selection of option pricing models lies at a confluence of three disjointacademic literatures: option pricing models, MCMC methods and Bayesian modelselection.

The first equity price model was proposed by Bachlier in his thesis Theorie de laSpeculation (1900) in which prices followed a Brownian motion. Osborne (1959) firstproposed the geometric Brownian motion (M2). Following the work of Merton (1969,1971) and Black and Scholes (1973), geometric Brownian motion became the de riguermodel in mathematical finance. However, discovery of fat-tailed (leptokurtotic) returndistributions and the “volatility smile” in Black-Scholes implied volatilities (Bakashiet al., 1997) suggests the geometric Brownian motion is inadequate. Formal empiricaltests typically reject the model (Sundaresan, 2000).

Many alternative models have been proposed with the general feature of generatingreturn distributions with excess-kurtosis. These include: (1) a chaos theory, fractalBrownian motion model, see Mandelbrot (1963) and Rogers (1997); (2) time-changedBrownian motions, see Clarke (1973) and Heyde (2000), (3) Levy processes (Germanet al., 2001), (4) numerical “implied binomial trees”, see Derman and Kani (1994) andDupire (1994) and (5) ARCH models, see Engle (1995). Each approach significantlydeparts from the geometric Brownian motion model specification.

Other alternative models build directly upon GBM by adding additional terms thestochastic differential equation. These include: (1) jump-diffusions, where prices locallyfollow GBM but experience occasional jumps, see Merton (1976) and Kou (2000); (2)stochastic volatility, where prices follow GBM with a stochastic instantaneous volatilityterm, see Hull and White (1987); (3) the constant elasticity of variance (CEV) model,see Cox and Ross (1976); (4) stochastic-volatility jump-diffusions. where (1) and (2)are combined, see Heston (1993) and Duffie et al. (2000) and (5) stochastic-volatilitywith jumps and jump-diffusions, where volatility undergoes jumps between diffusionand prices experience jumps, see Eraker (2004).

Bayesian statistics is routinely hindered by hard or intractable integration prob-lems. The development of Markov Chain Monte Carlo methods provides an attractive,numerical way around integration. The two popular algorithms, the Gibbs samplerand Metropolis-Hastings (Gelfand and Smith, 1990; Besag and Green, 1993; Smithand Roberts, 1993), are extensively used by Bayesian statisticians (Clyde and George,2004). Following MCMC, research in Bayesian model selection quickly took off; see, forexample, George and McCulloch, 1993; Smith and Kohn, 1996; Geweke, 1996; Clyde,DeSimone and Parmigiani, 1996.

Markov Chain Monte Carlo methods have grown popularity in finance followingthe work of Carlin, Polson and Stoffer (1992), who demonstrate the MCMC approachfor estimation of state-space models. MCMC as an estimation tool for asset pricingmodel was popularised by Johannes, Kumar and Polson (1998) and Eraker, Johannesand Polson (2000), who perform estimation of stochastic jump diffusion models.

4

3 Theory

The thesis uses results and tools from a number of different academic fields. These are:MCMC methods, Bayesian model selection, options theory and stochastic differentialcalculus. Given of the large scope, this section attempts to tie these ideas together andgives theory for how we intend to solve problem (1) and (2).

3.1 Overview of MCMC

Markov Chain Monte Carlo is a conditional simulation methodology that generatesrandom samples from a given target distribution. In our case, this target is p(Θ, X|Y )for estimation and p(Mk|Y ) for model selection. The fundamental idea behind MCMC,the Hammersley-Clifford theorem, is that any joint distribution can be characterized byits complete set of conditional distributions. Knowledge of p(Θ|X, Y ) and p(X|Θ, Y )completely characterizes the joint distribution p(Θ, X|Y ).

Consider the following algorithm: given two initial draws, X(0) and Θ(0), drawX(1) ∼ p(X|Θ(0), Y ) and then Θ(1) ∼ p(Θ|X(1), Y ); iteratively continuing this step gen-erates a sequence of random variables {X(j),Θ(j)}J

j=1. This sequence is a Markov Chainand has an attractive property: the equilibrium distribution of the chain converges top(Θ, X|Y ) under a number of metrics. If the conditional distributions p(Θ|X, Y ) andp(X|Θ, Y ) are known and can be directly drawn from, the algorithm described is calleda Gibbs sampler. If one or both of the conditional distributions cannot be drawn from,the Metropolis-Hastings algorithm applies. This is a two step procedure using rejec-tion sampling. First a candidate value for Θ(j+1)

∗ is drawn from some known and easilysampled distribution F . Second, Θ(j+1)

∗ is accepted with a probability given by theratio of likelihood functions. If accepted, Θ(j) = Θ(j)

∗ , otherwise Θ(j) = Θ(j−1). Thesame process proceeds for X before reiterating. Formally,

Step 1 : Draw Θ(j+1)∗ from F

Step 2 : Accept Θ(j+1)∗ with probability p(Θ(j+1)

∗ ,Θ(j)).

Step 3 : Set Θ(j+1) = Θ(j+1)∗ if accept, Θ(j+1) = Θ(j) if reject,

where

p(Θ(j+1)∗ ,Θ(j)) = min

(1,

P (Y |Θ(j+1)∗ , X(j))

P (Y |Θ(j), X(j)

).

Updating X proceeds identically:

Step 5 : Draw X(j+1)∗ from G

Step 6 : Accept X(j+1)∗ with probability p(X(j+1)

∗ , X(j))

where

p(X(j+1)∗ , X(j)) = min

(1,

P (Y |Θ(j+1), X(j+1)∗ )

P (Y |Θ(j+1), X(j)

).

Gibbs and Metropolis-Hastings algorithms provides random samples from the tar-

5

Figure 2: Example of MCMC simulation. Top: Markov chain for {a(j), b(j)}Jj=1. Bot-

tom: estimated distributions p(a|Y ), p(b|Y ).

get distribution. These random samples can be used for parameter and state variableestimation using the Monte Carlo method. For example, a point estimate of Θi maybe obtained by the marginal mean,

E(Θi|Y ) =∫

p(Θi|Y )dΘi ≈1J

J∑j=1

Θ(j)i .

The Monte Carlo method is essentially a method for approximating integrals. It worksby roughly mapping out a probability space by discrete random blocks of equal mea-sure, the random sample, and suitably summing over these points.

3.2 Overview of Price Time-Series Models

All the continuous-time models that shall be investigated are subsets of the followinggeneral model. Let St denote an equity price and Xt denote a vector of state variables.St solves the SDE,

St+1 = St +∫ t+1

tµS(Ss, Xs,Θ)ds +

∫ t+1

tσS(Ss, Xs,Θ)dWs +

Nt+1∑j=Nt

ξj , (2)

where dWt is a standard Brownian increment, Nt is a point process for jump times, ξ arejumps and µS(.) and σS(.) are suitable functions (bounded and Lipschitz-continuous).

6

The state variable Xt solves the SDE

Xt+1 = Xt +∫ t+1

tµX(Xs,Θ)ds +

∫ t+1

tσX(Xs,Θ)dW ∗

s +Mt+1∑k=Mt

ζk, (3)

where µX and σX are (possibly) different functions to µS and σS and dW ∗t is another

Brownian increment that may be correlated to dWt.

All continuous-time asset price models that have been proposed may be specifiedby (2) and (3) by specific choices for µS , µX , σS , σY . The simplest possible example is

St+1 = St +∫ t+1

tµSsdt +

∫ t+1

tσSsdWs.

There are no jumps, no state variables and constant instantaneous trend µ and volatil-ity σ. This is the Black-Scholes model.

A more complex model with volatility Vt as a state variable is

St+1 = St +∫ t+1

tµSsdt +

∫ t+1

t

√VtSsdWs,

Vt+1 = Vt +∫ t+1

tκ(µv − Vs)ds +

∫ t+1

tσv

√VsdW ∗

s ,

and Wt, W ∗t have correlation ρ. This is the square-root stochastic volatility model first

proposed by Heston (1993). A third example the Merton model M1 considered in theintroduction.

3.3 From Equity Price to Option Price

The models so-far considered specify the dynamics of prices St and state variablesXt. Given knowledge of Θ and (St, Xt), they completely describe the predictive jointdistribution p(ST , XT ). Now consider an option over the equity St with price Ct =f(St, Xt,Θ). The Fundamental Theorem of Arbitrage Pricing asserts the existence ofsome probability measure Q such that

f(St, Xt,Θ) = EQ

(e−∫ T

tr(Xs)ds(ST −K)+|St

)

in the case of a call option where (x)+ = max(0, x). This holds regardless of thedynamics of St and Xt, and under market completeness, Q is unique.

Now we are ready to specify a complete option pricing model. This consists ofthree equations: (1) the equity price model, (2) the state variable models, and (3) thediscounted expectation over p(St, Xt). Specifically,

St+1 = St +∫ t+1

tµS(Ss, Xs,Θ)ds +

∫ t+1

tσS(Ss, Xs,Θ)dWs +

Nt+1∑j=Nt

ξj ,

Xt+1 = Xt +∫ t+1

tµX(Xs,Θ)ds +

∫ t+1

tσX(Xs,Θ)dW ∗

s +Mt+1∑k=Mt

ζk,

7

Ct = f(St, Xt,Θ) + εt.

An error is included in the final equation since option prices are measured with noise.An important example of this is the bid-ask spread.

3.4 Example: Estimating the Black-Scholes model with MCMC

We consider the simplest application of MCMC for option price models: estimatingBlack-Scholes model for a call option. The full model is

St+1 = St +∫ t+1

tµSsdt +

∫ t+1

tσSsdWs

Ct = BS(σ, St) + εt

where BS(.) = StΦ(d1)−Ke−r(T−t)Φ(d2), the Black-Scholes formula.The target of estimation is p(Θ|Y ). The parameters are Θ = (µ, σ). Since there

are no state variables, X is dropped from this distribution. The observed data isY = (S, C, R), where S, C and R are time series of equity prices, option prices andlog-returns up to time t.

The Hammersley-Clifford theorem states that knowledge of p(µ|Y ) and p(σ|Y )fully characterize P (µ, σ|S, C). MCMC iteratively draws from these two conditionaldistributions, that is, given µ(j) and (σ2)(j), the algorithm updates by drawing from

µ(j+1) ∼ p(µ|(σ2)(j), Y ) and (σ2)(j+1) ∼ p(σ2|µ(j+1), Y )

The above densities must be known either explicitly or up to a ratio of likelihoodfunctions before we can use MCMC. Bayes rule decomposes the densities to:

p(µ|σ2, S, C,R) ∝ p(S, C, R|µ, σ2)p(µ)

p(σ2|µ, S, C, R) ∝ p(S, C, R|µ, σ2)p(σ2)

where p(µ) and p(σ2) are prior distribution. We assume independence of µ and σ2.

µ does not appear in the Black-Scholes formula and is µ is independent of C andS: p(µ|σ2, S, C,R) = p(µ|σ2, R). Thus,

p(µ|σ2, S, C,R) ∝ p(Y |µ, σ2)p(µ)

The log-returns Y are normally distributed under the model. If µ has a prior normaldistribution as well, p(µ|σ2, S, C,R) is normal and proposals for µ(j) can be sampleddirectly using the Gibbs sampler.

For volatility, invoking Bayes rule a second time yields

p(σ2|µ, S, C, R) ∝ p(C|µ, σ2)p(Y |µ, σ2)p(σ2)

This distribution cannot be directly sampled from as the posterior does not belong to

8

some recognizable distribution. The likelihood p(C|σ2, S) is extremely complicated:

p(C|σ2, S) ∝T∏

t−1

p(Ct|σ2, St) ∝T∏

t−1

exp

(−1

2

(Ct −BS(σ, St)

σc

)2)

where BS(σ, St) is non-linear in σ and St.

Instead, the Metropolis-Hastings can be used. The driver distribution F is arbitrarybut must be positive. The steps are

Step 1 : Draw (σ2)(j+1)∗ from F

Step 2 : Accept (σ2)(j+1)∗ with probability p((σ2)(j+1)

∗ , (σ2)(j)∗ )

where

p(Θ(j+1)∗ ,Θ(j)) = min

(1,

P (C|(σ2)(j+1)∗ , S)

P (C|(σ2)(j), S

).

3.5 Bayesian Model Selection

Suppose we have K competing models M1, ...,MK and wish to the select likeliest one.The model selection criterion to be used is the posterior model probability, p(Mk|Y ).Using Bayes theorem, this equals

p(Mk|Y ) =p(Y |Mk)p(Mk)∑k p(Y |Mk)p(Mk)

(4)

where p(Mk) is the prior belief of likelihood and

p(Y |Mk) =∫

p(Y |Θk,Mk)p(Θk|Mk)dΘk.

This integral is hard to compute in all but the simplest of cases. As before, MCMCwill be used to avoid it. There are two approaches that we will consider:

Separate Model Method For each model k = 1, 2, ...,K, we run separate MCMCalgorithms to generate a samples N realizations of parameters and state variables:(Θ(1)

k X(1)k ), ..., (Θ(N)

k X(N)k ). For each pair, the likelihood function for data Y is calcu-

lated:L(Y ; Θ(i)

k , X(i)k ) = p(Y |Mk,Θ

(i)k , X

(i)k )

and we estimate p(Y |Mk) by

p(Y |Mk) ≈1N

N∑i=1

L(Y ; Θ(i)k , X

(i)k ).

Plugging this estimate into (4) yields p(Mk|Y ).

Mixture Model Method We add another Markov variable, M, to the MCMC algo-rithm, and proceed to directly estimate the mixture model M∗ =

∑Kk=1 p(Mk|Y )Mk.

9

For each loop, a new model M(j+1) is proposed by drawing independently from theprior distribution p(M = Mk) :

Step 1 : Draw M(j+1) from p(M)

Step 2 : Accept M(j+1) with probability p(M(j+1),M(j))

where

p(M(j+1),M(j)) = min

(1,

p(Y |M(j+1),Θ(j+1), X(j+1)

p(Y |M(j),Θ(j), X(j)

).

The remaining steps for updating Θ and X proceed similarly, with acceptance prob-abilities now conditioned on M = M(j). The output of the algorithm is a sequence(M(1),M(2), ...,M(J)) and we can estimate p(Mk|Y ) by

p(Mk|Y ) ≈ 1J

J∑j=1

I(Mj = k).

We are interested to see the mixture model M∗ performs better than any stand-alone Mk. From a Bayesian perspective, this is tautological: the mixture model has ahigher posterior probability than any Mk by construction. Other measures, such asRMSE, will be used in investigating this.

10

4 Methodology

4.1 Candidate Models

The following option pricing models are considered. These are:

MBS : ”Black-Scholes Model”,dSt

St= µdt + σdWt

MM : ”Merton Model”,dSt

St= µdt + σdWt + d

Nt∑j=1

ξj

.

Nt ∼ Poisson(λJ), ξj ∼ N (µJ , σ2J)

MK : ”Kou Model”,dSt

St= µdt + σdWt + d

Nt∑j=1

ξj

.

Nt ∼ Poisson(λJ), ξj ∼ DE(µJ , σ2J)

MESV : ”Exponential SV”,dSt

St= µdt + σeVtdWt,

dVt = κ(µV − Vt) + σV dW ∗t ,

MSV : ”Heston Square Root”,dSt

St= µdt + σ

√VtdWt,

dVt = κ(µV − Vt) + σV

√VtdW ∗

t ,

Corr(Wt,W∗t ) = ρ

MSV J : ”Heston with Price Jumps”,dSt

St= µdt + σ

√VtdWt + d

Nt∑j=1

ξj

,

dVt = κ(µV − Vt) + σV dW ∗t ,

Corr(Wt,W∗t ) = ρ, Nt ∼ Pois(λ), ξj ∼ N (µS , σ2

S)

MSV DJ : ”Heston, Price and Stoc. Jumps”,dSt

St= µdt + σ

√VtdWt + d

Nt∑j=1

ξj

,

dVt = κ(µV − Vt) + σV dW ∗t + d

Mt∑j=1

ζj

,

Corr(Wt,W∗t ) = ρS , Nt ∼ Pois(λS), ξj ∼ N (µS , σ2

S)

Corr(Nt,Mt) = ρV , Mt ∼ Pois(λV ), ζ ∼ G(α, β)

11

All models are nested within MSV DJ ; for example, set κ = 0, σV = 0 and λV = λS = 0to recover MBS . Our benchmark model is the Black-Scholes. Two Bayesian modelselection statistics are calculated for each model: p(Mk|Y ) and mean Deviance =−2log p(Y |Mk). Lower deviance indicates better fit. Diagnostic checks on residualsare also carried out. Let Yt = log(St); the residuals are defined by

Yt − Yt−1 − µ− Jtξt√σVt

= εt ≈ N (0, 1).

Under a properly specified model, the residuals should be normal and homoscedasticwith variance of 1.

4.2 Time-Discretization

The basis for our MCMC estimation is a Euler time-discretization of the continuous-time models. Suppose we have the process

dXt = µ(Xt, t)dt + σ(Xt, t)dWt;

the Euler discretization is given by

Xt∆ −X(t−1)∆ = µ(X(t−1)∆)∆ + σ(X(t−1)∆)∆ε

where ∆ is the discretization resolution and ε ∼ N (0, 1). The SVDJ model is discretizedas

Yt∆ − Y(t−1)∆ = µ∆ +√

σVt∆∆εy + ξt∆Jyt∆

Vt∆ − V(t−1)∆ = κ(µV − V(t−1)∆)∆ + σV

√V(t−1)∆∆εV + ζt∆JV

t∆.

The magnitude of error introduced by Euler discretization is Op(∆). Higher orderapproximations using the kth partial derivative ∂kσ(.)

∂Xkt

correction terms reduce the error

order to Op(∆(k+1)/2).

4.3 Priors

The estimation framework is Bayesian and prior beliefs must be specified for Θ, X andM. The research intent is to see which model(s) perform best given the same informa-tion set for estimation. Therefore, we approach with apriori ignorance and the priorsare uninformative. Where ever possible, we use conjugate prior distributions whichprovide closed-form and easy to simulate posteriors. For example, in estimating theBlack-Scholes model, the conjugate priors for µ and σ2 are normal and inverse-gamma.This makes the likelihood function p(Y |µ, σ2) have normal form. A comprehensive listof the priors used can be found in the appendix.

For model selection, we also approach with apriori ignorance. Each model is equally

12

likely:

p(Mk) =1K

∀k.

The posterior model probability under this prior reduces to

p(Mk|Y ) =p(Y |Mk)∑n

j=1 p(Y |Mj).

5 Data

The dataset consists of daily returns over the S&P500 index and traded S&P500 optionprices. The options are European in exercise style, include both calls and puts, andhave various maturities and strike prices. Option prices at each observation are set asthe mid-point between the bid and ask spread.

The period considered is 1996 to 2006. This period includes a number of interestingmarket events. These include the 1997 Russian bond default and collapse of LTCM, aswell as the rise and fall of the the dot.com bubble. From a statistical perspective, thisperiod features several large intra-day price movements and volatility clusters. Thesample contains n = 2550 trading days and N = 4580 option prices. Option prices areprovided by OptionMetrics.

6 Results

We first consider and estimate the candidate models using returns observations only.At present, model estimation using both option prices and returns is not yet completed.The stochastic volatility with jumps are currently experimental and are not includedin this paper.

6.1 Unconditional Distribution of Returns

Three models for the unconditional distribution of returns, rt, are considered,

MN : rt ∼ Normal(µ, σ2)

MT : rt ∼ Students T (µ, σ2, k)

MDE : rt ∼ DoubleExponetial(µ, σ2)

The priors are µ ∼ N (0, 10), σ2 ∼ IG(2.5, 0.025) and k ∼ χ210. Table (1) lists the means

and 95% credibility sets of posterior distributions of µ, σ2 and k, and the posteriormodel probabilities. The fitted distributions are overlaid with a kernel-density estimateof rt in figure

We see overwhelming support for the double exponential model, which has a poste-rior probability of (almost) 1. The Student’s-t model is preferable over normal, beingmore probable by several orders of magnitude.

13

6.2 Continuous-Time Models for Returns

6.2.1 Jump Diffusion Models

We consider Merton and Kou jump-diffusion models against the Black-Scholes alterna-tive. Jumps have normal distribution under the Merton model and double-exponentialunder the Kou model; and in both jumps arrive with constant intensity λ.

Table (2) shows the posterior means and 95% credibility sets for µJ , σ2J and λ and

p(Mk|Y ) in Black-Scholes (MBS), Merton (MM ) and Kou (MK) models. Figure (5)displays the estimated probability of daily jump for the two models and the S&P500returns. Both models exhibit clustering in the jump times, particularly during the1999-2004 period of high volatility. However, the models specify that jump times followa Poisson process, therefore durations between subsequent jumps is independent arejointly independent. This suggests misspecification of both models.

There is little difference between Merton and Kou jump time estimates. The modelspick the same set of days for jumps with p(Jt > 0.5). However, the Kou modeldistinguishes jumps and non-jumps more sharply than the Merton model. This isevident from the jump rate parameter: Kou λ = 1.38% versus Merton λ = 7.80%, andgraphically: there are far fewer jumps with low probability (p(Jt < 0.1)) in the Kouplot.

The posterior probabilities for the models are shown in table (2). Given thetwo jump-diffusion alternatives, the the Black-Scholes model is highly improbable,p(M1|Y ) ≈ 0. The Kou model is more probable than the Merton Model by factor 1.5.

6.2.2 Stochastic Volatility Models

We consider the exponential stochastic volatility model MESV and the Heston square-root model MSV against the Black-Scholes alternative. Table (3) shows the posteriormeans for the parameters and the posterior model probabilities . We see strong ev-idence for SV against BS and ESV, who both have posterior probabilities near zero.Both models have approximately residuals, as seen in figure (6). . The two volatilityfits are surprisingly different in fit: the exponential stochastic volatility fit is noticeablysmoother.

7 Work Remaining

Model inference using both returns and option prices is currently under progress. Oneproblem we currently face is the computational burden. Each of the models consideredhas a closed form expression for option prices up to a complex-valued integration:

C(t, τ) = StΠ1(t, τ, S, V ) + e−rT KΠ2(t, τ, S, V ),

where

Πj(t, τ, St, Vt) =12

+1π

∫ ∞0

Re

[e−iξln(K)fj(t, τ, St, Vt; ξ)

iξ

]dξ.

14

Model µ σ2 k Deviance p(Mk|Y )MN 4.26e-4 1.12e-4 1.49e-267

MT 4.96e-4 1.46e-4 5.59 1.38e-07

MDE 5.49e-4 1.43e-4 0.999

Table 1: Posterior means and 95% credibility sets of the unconditional distributionmodel parameters.

Model µ σ2 µJ σ2J λ Deviance p(Mk|Y )

MBS 4.29e-4 1.33e-4 -18612 2.04e-5

MM 3.95e-4 1.07e-4 0.0183 9.65e-03 0.0780 -19146 0.389

MK 3.94e-4 9.97e-5 -0.0335 0.929 0.0138 -19344 0.611

Table 2: Posterior means and 95% credibility sets of the Black-Scholes, Merton andKou jump diffusion model.

Model µ σ2 µV σ2V κ Deviance p(Mk|Y )

MBS 8.51e-4 1.71e-4 -6067 1.268e-82

MESV 8.52e-4 4.92e-5 -1.01 1.31 0.00906 -6367 2.29e-4

MH 8.54e-4 2.50e-4 0.488 1.24 0.241 -6520 1.00

Table 3: Posterior means of parameters in the Black-Scholes (BS), exponential sto-chastic volatility (ESV ) and Heston stochastic volatility (H) models.

−0.06 −0.04 −0.02 0.00 0.02 0.04 0.06

010

2030

4050

Den

sity

Figure 3: Unconditional distribution of daily returns (black) and fitted models, normal(red), student’s-t (blue) and double exponential (indigo).

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Figure 4: Top: S&P 500 daily returns, Middle: probability of jump, p(Jt|M2), Mertonmodel, Bottom: probability of jump, p(Jt|M3), Kou model.

Figure 5: Estimated volatility Vt from the SV Heston model (red) and the ESV Expo-nential model (black).

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Figure 6: Quantile-Quantile plots comparing the SV and ESV residual quantiles to thenormal distribution.

At each update of the MCMC chain, the price of an option given the current parameterstate must be calculated. This has a tremendous multiplying effect on the time requiredfor each update. Presently, it takes 1-3 minutes to fit a model using returns alone.Including option prices, the time skyrockets to around 10 to 30 hours depending onthe model.

Another difficulty is that option prices contain information on the risk-neutraldistribution of parameters, not the actual distribution implied by returns. Our currentsolution involves introducing a pair of parameters, one risk-neutral and one physical,into each model and using option prices to update the risk-neutral parameter alone.This unfortunately increases the computational burden somewhat.

8 Conclusion

We consider a large set of popular option pricing models that specify stochastic dif-ferential equations for returns. The models include: Black-Scholes, Merton and Koujump-diffusion, exponential and square-root stochastic volatility, square-root stochasticvolatility with price jumps and square-root stochastic volatility with price and volatil-ity jumps. These more complex models are difficult to estimate without resorting tocomplicated statistical methods. We show how to use Markov Chain Monte Carlomethods to estimate each model and use Bayesian posterior model probabilities forformal model selection. We find that the square-root stochastic volatility with pricejumps model performs best against the alternatives in modelling daily returns fromthe S&P 500 index.

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References

[1] Merton, R., “Option Pricing When Underlying Stock Returns are Discontinuous”,Journal of Financial Economics, 3, 125-14, 1976.

[2] Black, F., & M. Scholes. “The Pricing of Options and Corporate Liabilities.”Jour-nal of Political Economy, 81, 637-654 , 1973.

[3] Osborne, M., “Brownian motion in the stock market,” Operations Research 7,145-173, 1959.

[4] Sundaresan, S., “Continuous-time methods in finance: A review and an assess-ment,” Journal of Finance, 55, 1569-1622, 2000.

[5] Bakshi, G., Cao, C. & Chen, Z., “Empirical performance of alternative optionpricing models,” Journal of Finance, 52, 2003-2049, 1997.

[6] Duffie, D., “Dynamic Asset Pricing Theory,” 2nd edn. Princeton University Press,Princeton, 1995.

[7] Bollerslev, T., “Generalized Autoregressive Conditional Heteroskedasticity,” Jour-nal of Econometrics, 31, 307-327, 1986.

[8] Mandelbrot, B., “The variation of certain speculative prices”, Journal of Business,36, 394-419, 1963.

[9] Merton, R, “Option pricing when underlying stock returns are discontinuous,”,Journal of Financial Economics, 3, 224-244, 1976a.

[10] Merton, R, “The impact on option pricing of specification error in the underlyingstock price returns,”, Journal of Finance, 31, 333-350, 1976b.

[11] Kou, S., “A jump diffusion model for option pricing,” Management Science 48,1086-1101, 2002.

[12] Press, J., “A Compound Events Model of Security Prices,” Journal of Business,40, 317-335, 1967.

[13] Ramezani, C. & Zeng, Y., “Maximum likelihood estimation of asymmetric jumpdiffusion processes: application to security prices.”, working paper, University ofWisconsin, Madison, 1998.

[14] Andersen, L. & Andreasen, J., “Volatility Skews and Extensions of the LiborMarket Model”, Applied Mathematical Finance, Vol 7, 1-32, 2000.

[15] Naik, V., “Option valuation and hedging strategies with jumps in the volatility ofasset returns,” Journal of Finance, 48, 1969-1984, 1993.

[16] Bakshi, G & Cao, C. “Disentangling the contribution of return-jumps andvolatility-jumps on individual equity option prices,” working paper, Smith Busi-ness School, University of Maryland, 2004.

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A Priors

Where ever possible we choose standard conjugate priors that allow for direct samplingfrom the conditional posteriors.

µ ∼ N (0, 1) σ2 ∼ IG(2.5, 0.025)µV ∼ N (0, 1) σ2

V ∼ IG(2.5, 0.025)κ ∼ B(1.5, 20) µJ ∼ N (0, 1)σ2

J ∼ IG(5.0, 20) µJV ∼ G(20, 10)λJ = λV ∼ B(1, 5) % ∼ U(−1, 1)

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