In Section 11.1, we present two variance-reduction techniques in the context of multiscale stochastic volatility models. In the same context, we apply in Section 11.2 the perturbation method to Merton's problem of finding an optimal portfolio allocation using power utility functions. We present in Section 11.3 an application to forward-looking estimation of stock betas using skews of implied volatilities.

Application to Variance Reduction in Monte Carlo Computations

Monte Carlo methods are natural and essential tools in computational finance. Examples include pricing and hedging financial instruments with complex structure or high dimensionality (Glasserman, 2003). Variancereduction techniques play a crucial role in making Monte Carlo simulations practical in terms of computational time. There are many different such techniques, and here we concentrate on two of them: importance sampling and control variate, applied to simulations of multiscale stochastic volatility dynamics presented in this book. The objective is to sample from the full stochastic volatility model using the approximations derived in the previous chapters as tools to speed up the convergence of the Monte Carlo estimates. We give a brief description of the methods, and refer to the two papers of Fouque and Han (2004b, 2007) for more details and numerical illustrations.

Importance Sampling

Our starting point is the stochastic volatility dynamics given under a riskneutral pricing measure ℙ* by (4.1), and the quantity of interest is the price of a European option given by the expectation (4.2).