QMF07TalkSwish

8/7/2019 QMF07TalkSwish

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Explicit Option Pricing Formulafor a Mean-Reverting Asset in

Energy MarketsAnatoliy Swishchuk

Mathematical & ComputationalFinance Lab

Dept of Math & Stat, Universityof Calgary, Calgary, AB, Canada

QMF 2007 ConferenceSydney, Australia

December 12-15, 2007

This research is supported by MITACS and NSERC


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Outline

Mean-Reverting Models (MRM): Deterministicvs. Stochastic

MRM in Finance Markets: Variances orVolatilities (Not Asset Prices)

MRM in Energy Markets: Asset Prices

Change of Time Method (CTM) Mean-Reverting Model (MRM)

Option Pricing Formula

Drawback of One-Factor Models Future Work


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Motivations for the Work Paper: Javaheri, Wilmott and Haug (2002)

GARCH and Volatility Swaps, WilmottMagazine, Jan Issue(they applied PDEapproach to find a volatility swap for MRM and

asked about the possible option pricing formula Paper: Bos, Ware and Pavlov (2002) On a

Semi-Spectral Method for Pricing an Option on a

Mean-Reverting Asset, Quantit. Finance J.(PDE approach, semi-spectral method tocalculate numerically the solution)


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Wilmott, Javaheri & Haug (2002)Model

Wilmott, Javaheri & Haug (GARCH and

Volatility Swaps, Wilmott Magazine, 2002)-volatility swap for

-continuous-time GARCH(1,1) model


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M. Yors Results M. Yor On some exponential functions of

Brownian motion, Adv. In Applied Probab., v. 24,No. 3, (1992), 509-531-started the research forthe integral of an exponential Brownian motion

H. Matsumoto, M. Yor Exponential Functionalsof Brownian motion, I: Probability laws at fixedtime, Probability Surveys, v. 2 (2005), 312-347-

there is still no closed form probability densityfunction, while the best result is a function with adouble integral.


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Mean-Reversion Effect

Guitar String Analogy: if we pluck the guitarstring, the string will revert to its place of

equilibrium To measure how quickly this reversion back tothe equilibrium location would happen we had topluck the string

Similarly, the only way to measure meanreversion is when the variances of asset pricesin financial markets and asset prices in energy

markets get plucked away from their non-eventlevels and we observe them go back to more orless the levels they started from


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The Mean-Reverting Deterministic

Process


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Mean-Reverting Plot (a=4.6,L=2.5)


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Meaning of Mean-Reverting Parameter

The greater the mean-reverting parameter

value, a, the greater is the pull back to the

equilibrium level For a daily variable change, the change in time,

dt, in annualized terms is given by 1/365

If a=365, the mean reversion would act soquickly as to bring the variable back to itsequilibrium within a single day

The value of 365/a gives us an idea of howquickly the variable takes to get back to theequilibrium-in days


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Mean-Reverting Stochastic Process


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Mean-Reverting Models in Financial

Markets

Stock (asset) Prices followgeometric Brownian motion

The Variance of Stock Pricefollows Mean-Reverting Models

Example: Heston Model


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Mean-Reverting Models inEnergy Markets

Asset Prices follow Mean-Reverting

Stochastic Processes Example: Continuous-Time GARCH

Model (or Pilipovic One-Factor Model)


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Mean-Reverting Models in Energy

Markets


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Change of Time: Definition and Examples

Change of Time-change time from t to a non-negative process T(t) with non-decreasing samplepaths

Example1 (Subordinator): X(t) and T(t)>0 aresome processes, then X(T(t)) is subordinated toX(t); T(t) is change of time

Example 2(Time-Changed Brownian Motion):M(t)=B(T(t)), B(t)-Brownian motion

Example 3(Product Process):


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Time-Changed Brownian Motion by

Bochner

Bochner (1949) (Diffusion Equation and

Stochastic Process, Proc. N.A.S. USA, v.35)-introduced the notion of change oftime (CT) (time-changed Brownian motion)

Bochner (1955) (Harmonic Analysis andthe Theory of Probability, UCLA Press,

176)-further development of CT


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Change of Time: First Intro into Financial

Economics

Clark (1973) (A

Subordinated StochasticProcess Model with FixedVariance for SpeculativePrices, Econometrica,

41, 135-156)-introducedBochners (1949) time-changed Brownianmotion into financialeconomics:

He wrote down a model

for the log-price M as

M(t)=B(T(t)),

where B(t) is Brownianmotion, T(t) is time-change (B and T are

independent)


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Change of Time: Short History. I.

Feller (1966) (An Introduction to ProbabilityTheory, vol. II, NY: Wiley)-introduced subordinatedprocesses X(T(t)) with Markov process X(t) and T(t) as aprocess with independent increments (i.e., Poissonprocess); T(t) was called randomized operational time

Johnson (1979) (Option Pricing When theVariance Rate is Changing, working paper, UCLA)-introduced time-changed SVM in continuous time

Johnson & Shanno (1987) (Option PricingWhen the Variance is Changing, J. of Finan. & Quantit.Analysis, 22, 143-151)-studied the pricing of optionsusing time-changing SVM


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Change of Time: Short History. II.

Ikeda & Watanabe (1981) (SDEs and DiffusionProcesses, North-Holland Publ. Co)-introduced andstudied CTM for the solution of SDEs

Barndorff-Nielsen, Nicolato & Shephard(2003)(Some recent development in stochastic volatility

modelling)-review and put in context some of theirrecent work on stochastic volatility (SV) modelling,including the relationship between subordination and SV(random time-chronometer)

Carr, Geman, Madan & Yor (2003) (SV for LevyProcesses, mathematical Finance, vol.13)-usedsubordinated processes to construct SV for LevyProcesses (T(t)-business time)


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CT and Embedding Problem Embedding Problem was first terated by Skorokhod

(1965)-sum of any sequence of i.r.v. with mean zero andfinite variation could be embedded in Brownian motion(BM) using stopping time

Dambis (1965) and Dubis and Schwartz (1965)-every

continuous martingale could be time-changed BM Huff (1969)-every processes of pathwise bounded

variation could be embedded in BM

Monroe (1972)-every right continuous martingale couldbe embedded in a BM

Monroe (1978)-local martingale can be embedded in BM


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Change of Time:

Simplest (Martingale) Case


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Change of Time:

Ito Integrals Case


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Change of Time: SDEs Case


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Geometric Brownian Motion SVM


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Change of Time Method


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Connection between phi_t and phi_t^(-1)


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Solution for GBM Equation

Using Change of Time


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Explicit Expression for


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Mean-Reverting SV Model


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Solution of MRM by CTM


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Comparison: Solution of GBM & MRM

-GBM

-MRM


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Explicit Expression for S(t)

where


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Properties of


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Properties of


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Properties of eta(t)


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Properties of Eta(t). II.


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Mean Value of MRM S(t)


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Dependence of ES(t) on T


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Dependence of ES(t) on S_0 and T


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Variance for S(t)


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Dependence of Variance of S(t) on S_0 and T


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Dependence of Volatility of S(t) on S_0 and T


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European Call Option for MRM.I.


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European Call Option. II.

E i f C T i th f


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Expression for C_T in the case of

MRM

C_T=BS(T)+A(T)


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Expression for C_T=BS(T)+A(T).II.


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Expression for BS(T)


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Expression for y_0 for MRM


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Expression for A(T).I.

Moment generating) function of


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Moment generating) function of

Eta(T)


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Expression for A(T)

European Call Option for MRM


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European Call Option for MRM

(Explicit Formula)

European Call Option for MRM in Risk-Neutral


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European Call Option for MRM in Risk-Neutral

World


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D d f C T T


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Dependence of C_T on T

Comparison of Three


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Comparison of Three

Solutions

Heston Model Mean-Reverting Model

Black-Scholes Model

Comparison: Heston Model


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(1993)


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Explicit Solution for CIR Process: CTM

Comparison: Solutions to the Three


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Comparison: Solutions to the Three

Models

-GBM

-MRM

-Heston model

Summary


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Summary

-martingale

-martingale

-sum of two martingales

1.

2.

3.

GBM Model

Mean-Reverting Model

Heston Model


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Problem

-explicit expression ?

We know all the moments at this moment,though

To calculate an option price for Heston model, for example

Drawback of One-Factor Mean-


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Reverting Models The long-term mean L remains fixed over time:

needs to be recalibrated on a continuous basisin order to ensure that the resulting curves aremarked to market

The biggest drawback is in option pricing: results

in a model-implied volatility term structure thathas the volatilities going to zero as expirationtime increases(spot volatilities have to beincreased to non-intuitive levels so that the longterm options do not lose all the volatility value-asin the marketplace they certainly do not)


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Future Work Change of Time

Method for Two-FactorContinuous-TimeGARCH Model


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The End

Thank You for YourAttention and Time!

[email protected]

http://wwww.math.ucalgary.ca/~aswish/

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