Topic:  Monte Carlo and Quasi-Monte Carlo Methods and Applications

Speaker: Yongzeng Lai(Associate Professor, Department of Mathematics, Wilfrid Laurier University, Canada)

Time: 16:00-17:30, Dec. 13 (Fri.), 2013

Venue: C.S Lam Conference Room, Lingnan Hall

Language: English + Chinese

 

Abstract

The option sensitivities or Greeks are important in financial hedging and risk management. However, values of option Greeks are even harder to obtain than values of options themselves. It is difficult to deal with options with discontinuous payoff functions by the conventional finite central difference method. While the likelihood ratio method is helpless when the density functions do not exist or it is hard to find. A third approach-the Malliavin calculus method, was proposed about a decade ago and showed its advantages over the other two methods for exotic options. The main idea of this method is to express the Greeks in terms of option payoff functions multiplied by weight functions depending on Malliavin derivatives. 

 

This paper discusses simulation of option Greeks for multi-asset options under the subordinated Brownian motion model by Malliavin calculus combined with Monte Carlo and quasi-Monte Carlo methods. By using the chain rule, integration by parts, the reflection principle, etc. from Malliavin calculus, as well as the tower property of conditional expectation, we are able to express Greeks in terms of the expectations of the option payoff functions multiplied by the weights involving Malliavin derivatives for multi-asset options in both path independent and path dependent cases. In the one asset case, the formulas recover those found in the literature. Numerical results show that the Malliavin calculus method is usually more efficient than the finite difference method for options with non-smooth payoffs. The superiority of the former over the latter is much more significant when combined with quasi-Monte Carlo methods. For example, when simulating \Gamma_{11} or \Gamma_{12} of a basket type down-and-out option or a corridor option, the efficiencies reach up to more than ten thousand times in a two-asset case and more than one thousand times in a six-asset case.

 

Prof. Lai's CV/UploadFiles/xsbg/2013/12/201312101150359810.pdf