explicit option pricing formula for mean-reverting asset anatoliy swishchuk math & comp finance...
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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
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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
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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)
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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
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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 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
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Mean-Reverting Stochastic Process
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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
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Mean-Reverting Models in Energy Markets
• Asset Prices follow Mean-Reverting Stochastic Processes
• Example: Pilipovic Model
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Mean-Reverting Models in Energy Markets
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CTM for Martingales
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CTM for SDEs. I.
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CTM for SDEs. II.
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Connection between phi_t and phi_t^(-1)
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Idea of Proof. I.
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Idea of Proof. II.
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Mean-Reverting Model
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Solution of MRM by CTM
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Solution of GBM Model (to compare)
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Properties of
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Explicit Expression for
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Explicit Expression for
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Explicit Expression for S(t)
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Properties of
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Properties of
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Properties of Eta(t). II.
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Properties of MRM S(t). I.
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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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Properties of MRM S(t). II.
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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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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)
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European Call Option for MRM.I.
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European Call Option. II.
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Expression for y_0 for MRM
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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 A(T).I.
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Expression for A(T).II.
Characteristic (moment generating) function of Eta(T):
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Expression for A(T). II.
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European Call Option for MRM(Explicit Formula)
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Boundaries for C_T
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European Call Option for MRM in Risk-Neutral World
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Boundaries for MRM in Risk-Neutral World
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Dependence of C_T on T
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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
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Future work . I: Analytical Approach (Integro – Partial DE)
(Joint Working Paper with T. Ware)
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Future Work .II: Probabilistic Approach (Change of Time Method).
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The End
Thank You for Your Attention and Time!