Mean-Reverting Stochastic Volatility

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 con rms that volatility reverts slowly to its mean in comparison to the tick-by-tick uctuations 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 di erential equation satis ed by derivative prices, utilizing the distinction between these time scales.The analysis yields pricing and implied volatility formulas, and the latter is used to " t the smile" from European index option prices. The theory identi es 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 identi ed 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.

Path-dependent option pricing: the path integral partial averaging method

“…Recent work has shown that a path-independent option price, in a stochastic volatility environment, can be approximated by pricing a more complex path dependent option in the usual Black-Scholes framework. The volatility risk is included by an extra continuous path-dependent payout function which has the same form as (2.5) [45,46]. This work suggests that the computational framework presented here might also be used to price path dependent options consistent with the market implied volatility.…”

Microeconomic co-evolution model for financial technical analysis signals

“…maxfd; d 0 g. The relationship H ¼ d þ 1 2 [63] allows to deal with a fBm z H ðtÞ, H ¼ d þ 1 2 in order to fix y, g, z, H such that y þ gf ðtÞ þ zz H ðtÞ (13) has the same self-similarity exponent and the same range of RðtÞ. Whilst the contribution off ðtÞ is due to the local agent interactions, the contribution of z H ðtÞ can be interpreted either like the contribution of noise traders, in the case of uncorrelated signal, or like the presence of fundamentalist traders in the market whether a mean reverting process occurs [22]. This gives an interesting perspective about the level of SOC that can be masked by either noise or fundamentalists traders.…”

“…Microeconomic models of financial markets rank in complexity from the simplest models, typically considering the interaction of two main types of agents-the fundamentalists and the chartists [18][19][20][21] either without market information or not caring about the fundamentals, thus creating white noise, while mean reversion effects [22] can be accounted due to the activity of fundamentalists. The first question to be raised is whether a microeconomic approach can be found based on insight about the mechanism of the formation of financial quantities.…”

Stochastic Volatility: Option Pricing using a Multinomial Recombining Tree

“…We plan to address this problem using a systematic statistical analysis in a subsequent article. For now, the reader is directed to current work in (Fouque et al, 2000b), (Nielsen and Vestergaard, 2000) or (Bollerslev and Zhou, 2002).…”