Limit proxy methods for fast Monte-Carlo Greeks

© 2012 Dr. Jiun Hong ChanThis thesis presents new Monte-Carlo techniques for the pricing and Greeks computations of exotic derivatives in the LIBOR market model and the Heston stochastic volatility model. These new techniques allow the rapid computation of Greeks even when pay-offs are discontinuous and underlying densities post-discretization are singular. Chapter 3 introduces a new class of numerical schemes known as quasi mean-shifted proxy simulation schemes for discretizing processes driven by Brownian motions. This is a generalization of the partial proxy simulation scheme developed by Fries and Joshi. Under this class of numerical schemes, Greeks for financial products with discontinuous pay-offs can be evaluated efficiently via finite difference approximations with the main constraint being that the discontinuities at each step must be determined by a one-dimensional function: the proxy constraint function. A specific quasi mean-shifted proxy simulation scheme known as the minimal partial proxy scheme is constructed in this chapter. In Chapters 4 - 5, we show that, for any numerical scheme that belongs to the class of quasi mean-shifted proxy simulation schemes, the pathwise adjoint method can be used to compute price sensitivities even when pay-offs are discontinuous and underlying densities post-discretization are singular. Using this result, new Monte-Carlo techniques known as the pathwise partial proxy method and the pathwise minimal partial proxy method are developed. We also consider linearizing the non-linear proxy constraint functions in order to reduce the computational complexity. In Chapters 6 - 7, we shift our focus to the Heston stochastic volatility model. Specifically, we present three new discretization schemes for the Heston stochastic volatility model - two schemes for simulating the variance process and one scheme for simulating the integrated variance process. These new schemes evolve the Heston process accurately over long steps without the need to sample the intervening values. Hence, prices of financial derivatives can be evaluated rapidly using our new schemes. An efficient approach to computing the first and second order price sensitivities in the Heston model is also presented