Optimal Stopping Rules and Their Applications to Pricing American Options with Heteroskedastic Returns

Abstract

In financial engineering, Black-Scholes formula is a commonly used method of pricing derivatives. However, in Black-Scholes model, the parameter representing volatility was originally assumed to be constant, which is, now believed incorrect. Conditional heteroskedastic models have been largely investigated since Engle (1982) introduced his renowned ARCH model. In this talk, instead of MLE, we adopt the empirical likelihood estimation to estimate the parameters emerging in the GARCH, IGARCH, EGARCH and TGARCH models. Finally, we adopt Snell's envelope to obtain a reasonable price, by Monte-Carlo Simulations, for American options with the underlying assets driven by a stochastic di_erence equation disturbed by a heterogenous noise as below Sn = Sn-1 + μSn-1h + σn-1Sn-1Wn;h; for 1≦n≦N;where the volatility σn satisfies a GARCH model, Nh = 1, and Wn;h is a sequence of i.i.d. random variable with zero mean and variance of h.