Explicit Option Pricing Formula for Mean-Reverting Asset

Anatoliy Swishchuk

Math & Comp Finance Lab

Dept of Math & Stat, U of C

MITACS Project Meeting

McMaster University, Hamilton

November 12, 2005

Outline• Mean-Reverting Models (MRM): Deterministic

vs. Stochastic• MRM in Finance Markets: Variances or

Volatilities (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

Motivations for the Work

• Paper: Javaheri, Wilmott and Haug (2002) ”GARCH and Volatility Swaps”, Wilmott Magazine, Jan Issue (they applied PDE approach 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 to calculate numerically the solution)

Mean-Reversion Effect• Guitar String Analogy: if we pluck the guitar

string, the string will revert to its place of equilibrium

• To measure how quickly this reversion back to the equilibrium location would happen we had to pluck the string

• Similarly, the only way to measure mean reversion is when the variances of asset prices in financial markets and asset prices in energy markets get plucked away from their non-event levels and we observe them go back to more or less the levels they started from

The Mean-Reverting Deterministic Process

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

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 so quickly as to bring the variable back to its equilibrium within a single day

• The value of 365/a gives us an idea of how quickly the variable takes to get back to the equilibrium-in days

Mean-Reverting Stochastic Process

Mean-Reverting Models in Financial Markets

• Stock (asset) Prices follow geometric Brownian motion

• The Variance of Stock Price follows Mean-Reverting Models

• Example: Heston Model

Mean-Reverting Models in Energy Markets

• Asset Prices follow Mean-Reverting Stochastic Processes

• Example: Pilipovic Model

Mean-Reverting Models in Energy Markets

CTM for Martingales

CTM for SDEs. I.

CTM for SDEs. II.

Connection between phi_t and phi_t^(-1)

Idea of Proof. I.

Idea of Proof. II.

Mean-Reverting Model

Solution of MRM by CTM

Solution of GBM Model (to compare)

Properties of

Explicit Expression for

Explicit Expression for

Explicit Expression for S(t)

Properties of

Properties of

Properties of Eta(t). II.

Properties of MRM S(t). I.

Dependence of ES(t) on T

Dependence of ES(t) on S_0 and T

Properties of MRM S(t). II.

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

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

Drawback of One-Factor Mean-Reverting Models

• The long-term mean L remains fixed over time: needs to be recalibrated on a continuous basis in order to ensure that the resulting curves are marked to market

• The biggest drawback is in option pricing: results in a model-implied volatility term structure that has the volatilities going to zero as expiration time increases (spot volatilities have to be increased to non-intuitive levels so that the long term options do not lose all the volatility value-as in the marketplace they certainly do not)

European Call Option for MRM.I.

European Call Option. II.

Expression for y_0 for MRM

Expression for C_T in the case of MRM

C_T=BS(T)+A(T)

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

Expression for BS(T)

Expression for A(T).I.

Expression for A(T).II.

Characteristic (moment generating) function of Eta(T):

Expression for A(T). II.

European Call Option for MRM(Explicit Formula)

Boundaries for C_T

European Call Option for MRM in Risk-Neutral World

Boundaries for MRM in Risk-Neutral World

Dependence of C_T on T

Paper may be found on the following web page(E-Yellow Series Listing,

Dept of Math & Stat, U of C, Calgary, AB):

http://www.math.ucalgary.ca/research/preprint.php

Future work . I: Analytical Approach (Integro – Partial DE)

(Joint Working Paper with T. Ware)

Future Work .II: Probabilistic Approach (Change of Time Method).

The End

Thank You for Your Attention and Time!