On Option Pricing By Using Importance Sampling Monte Carlo Simulation

In this paper, we investigate the Importance Sampling Monte Carlo method thatis used to reduce the variance for pricing financial derivatives driven by high-dimensionalGaussian vectors. Suggested by the important results in the paper of P. Glasserman, ameasure transformation is adopted, then the equations determining the optimal drift coef-ficients in the measure transformation are obtained, which make the variance in the sim-ulation corresponding to the new measure reach minimal. The existence and uniquenessof the optimal drift coefcients are proved, and an efcient Newton Raphson algorithm isused to find the optimal drift coefcients. We illustrate the efectiveness of our methodwith several options, such as Asian options under geometry Brownian motion and theHeston model respectively, Straddle option and Basket option. Additionally, a suitablestratified sampling technique is also used to achieve significant variance reduction. Theone merits of the method in this paper is that we do not need any smoothness require-ment for payof function, even if we do not have the explicit expression of the payof,the another is that the optimal drift coefcients are unique, which make the variance insimulations reaching true minimal.One thing worth mentioning is the efect of reducing variance by using ImportanceSampling with drift, in the multi-modal problems, is not as expected. We need to developnew densities of Importance Sampling, such as mixed normal density, t-density and scaledstandard normal density(introducing a second parameterσ, that is standard deviation),these are all what we want to research. Of course, if the underlying asset is a Le′vy process,we will also need to consider Importance Sampling with drift under Le′vy process so as toreduce the variance of Monte Carlo simulation.This paper is arranged as followings. Section1introduces the fundamental knowl-edge of Monte Carlo method. Section2cites how to generate sample paths and simulatethe value and the variance of options. In section3, we construct the model and discussits feasibility.Section4illustrates the efectiveness of our method via numerical examplesfor several cases of option pricing.In the last section, we summarize this paper and givefollow-up study suggestions.