Quasi-monte Carlo Methods In Pricing And Sensitivity Analysis Of American Options

In mathematical finance,the pricing and sensitivity analysis of derivatives have always been the core issues of concern,and such problems often involve high-dimensional integrals.For high-dimensional integration,the Monte Carlo(MC)method avoids the curse of dimensionality,but its shortcoming is that the convergence rate is slow.Many variance reduction techniques have been developed to improve the computational efficiency of the MC method.The quasi-Monte Carlo(QMC)method is a deterministic simulation method.It has a higher order of convergence rate than the MC method.In this thesis,by suitably using path generation methods and dimension reduction techniques,the QMC methods are successfully applied to the pricing and sensitivity analysis of American options,and better estimators compared to the MC methods are obtained.In addition,a new approach to construct control variates which further improves the computational efficiency of the QMC method is also proposed.The pricing and sensitivity analysis of American options are challenging problems in financial engineering due to the involved optimal stopping time problem,which can be solved by using dynamic programming.We can get lower and upper bounds by MC simulation to ensure that the true price falls into a valid confidence interval.Progress has been made in using MC simulation to obtain both the lower and upper bounds of American option prices as well as the estimates of American option Greeks.However,there are few works on pricing American options using QMC methods,especially to compute the upper bound.And the applicability of QMC methods on estimating American option Greeks is open.In this thesis,we propose using QMC simulation to replace MC simulation for computing both the lower bound by the least-squares Monte Carlo method and the upper bound by the duality approach.We also propose efficient QMC methods combined with the generalized infinitesimal perturbation analysis approach for estimating American option Greeks.Moreover,we propose to suitably use path generation contructions and dimension reduction techniques for further efficiency improvements.We perform numerical experiments on American options under the Black-Scholes model and the variance gamma model,in which the options have the path-dependent feature or are written on multiple underlying assets.We find that QMC simulation in combination with appropriate path generations and dimension reduction techniques can significantly increase the efficiency in computing both the lower and upper bounds,resulting in better estimates and tighter confidence interval of the true prices,and significantly reduce the variance of the estimators of American option Greeks.As a variance reduction technique,control variates can also be used in American option pricing problems.However,the main idea of commonly used control variates uses the information at the exercise time only.In this thesis,we present a new approach to construct control variates for American option pricing.The unbiasedness and consistency of new control variate estimators in MC simulation are proven.We also combine QMC methods with control variates and path generations for further efficiency improvements.We perform numerical experiments on American put options and American-Asian call options under the Black-Scholes model.We find that new control variates in combination with QMC simulation and appropriate path generations can significantly increase the efficiency in American option pricing.