Article2000

MEAN-REVERTING STOCHASTIC VOLATILITY

Jean‐Pierre Fouque · George Papanicolaou

176 citations0/0 files storedInternational Journal of Theoretical and Applied Finance

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Stochastic processes and financial applications Financial Markets and Investment Strategies Financial Risk and Volatility Modeling

Abstract

We present derivative pricing and estimation tools for a class of stochastic volatility models that exploit the observed "bursty" or persistent nature of stock price volatility. An empirical analysis of high-frequency S&P 500 index data confirms that volatility reverts slowly to its mean in comparison to the tick-by-tick fluctuations of the index value, but it is fast mean-reverting when looked at over the time scale of a derivative contract (many months). This motivates an asymptotic analysis of the partial differential equation satisfied by derivative prices, utilizing the distinction between these time scales. The analysis yields pricing and implied volatility formulas, and the latter is used to "fit the smile" from European index option prices. The theory identifies the important group parameters that are needed for the derivative pricing and hedging problem for European-style securities, namely the average volatility and the slope and intercept of the implied volatility line, plotted as a function of the log-moneyness-to-maturity-ratio. The results considerably simplify the estimation procedure, and the data produces estimates of the three important parameters which are found to be stable within periods where the underlying volatility is close to being stationary. These segments of stationarity are identified using a wavelet-based tool. The remaining parameters, including the growth rate of the underlying, the correlation between asset price and volatility shocks, the rate of mean-reversion of the volatility and the market price of volatility risk can be roughly estimated, but are not needed for the asymptotic pricing formulas for European derivatives. The extension to American and path-dependent contingent claims is the subject of future work.

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References

1991Conditional Heteroskedasticity in Asset Returns: A New Approach 1993A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options 1987The Pricing of Options on Assets with Stochastic Volatilities 1997Empirical Performance of Alternative Option Pricing Models 1994Implied Binomial Trees 1991Stock Price Distributions with Stochastic Volatility: An Analytic Approach 2002Bayesian Analysis of Stochastic Volatility Models 1998Implied Volatility Functions: Empirical Tests

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