Published online by Cambridge University Press: 07 October 2011
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).
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.