Stochastic modeling of financial derivatives.

摘要

The Coupled Additive Multiplicative Noises (CAM) model is introduced as a stochastic volatility process to extend the classical Black-Scholes model. The fast Fourier transform (FFT) method is used to compute the values of the probability density function of the underlying assets under the CAM model, as well as the price of European call options. We discuss four dierent discretization schemes for the CAM model: the Euler scheme, the simplied weak Euler scheme, the order 2 weak Taylor scheme and the stochastic Adams-Bashforth scheme. A martingale control variate method for pricing European call options is developed, and its advantages in terms of variance reduction are investigated numerically. We also develop Monte Carlo methods for estimating the sensitivities of the European call options under the CAM model.