Quasi-Monte Carlo methods in finance.
详细信息
文摘
The classic error bounds for quasi-Monte Carlo approximation follow the Koksma-Hlawka inequality based on the assumption that the integrand has finite variation. Unfortunately, not all functions have this property. In particular, integrands for common applications in finance, such as option pricing, do not typically have bounded variation. In contrast to this lack of theoretical precision, quasi-Monte Carlo methods perform quite well empirically. This research provides some theoretical justification for these observations. We present new error bounds for a broad class of option pricing problems using quasi-Monte Carlo approximation in one and multiple dimensions. The method for proving these error bounds uses a recent result of Niederreiter (2003) and does not require bounded variation or other smoothness properties.;This research also provides numerical experiments on application of quasi-Monte Carlo methods. We apply quasi-Monte Carlo sequences in a duality approach to value American options. We compare the results and computational effort using different low discrepancy sequences. The results demonstrate the value of sequences using expansions of irrationals.;We then apply quasi-Monte Carlo simulation to LIBOR market models based on the Brace-Gatarek-Musiela/Jamshidian framework. We price an exotic interest-rate derivative, in particular, a European payer swaption and compare the results using pseudo random sequences and different low discrepancy sequences. We also tested randomized quasi-Monte Carlo methods and apply importance sampling to quasi-Monte Carlo simulation.