Transcript 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.

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 MeanReverting 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 MeanReverting 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!