Mean-Reverting Stochastic Volatility

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.