Comparison Of The Effect Of Low Discrepancy Sequences Using Monte Carlo Method To Price American Options

The pricing problem of American options has puzzled people for a long time until the Least Squares Monte Carlo(LSM)simulation algorithm proposed by Longstaff and Schwartz in 2001 that the pricing problem of American options was solved,this method has become the standard method for pricing American options by Monte Carlo simulation.We need to generate a large number of random numbers that obey the standard normal distribution when we use the LSM algorithm.However,the sequences brought by programs always are predictable,periodic and high-discrepancy.Such sequences are not ideal for MC method.In this paper,we use Halton,Sobol and Faure 3 low-discrepancy sequences to generate random numbers which obey uniform distribution,then use Box ―― Muller algorithm to transform them into standard normal distribution.Halton,Sobol are often used in various fields,however,there is no detailed demonstration about advantages and disadvantages of the 3 sequences,their respective applications and the improvement effects on efficiency of LSM.Therefore,this paper not only compares the 3 sequences to illustrate the advantages,disadvantages and applicability of each sequences,but also compares the improvement effects of Quasi Monte Carlo simulation method of the 3 sequences on the stability and variance reduction of pricing results.Based on the example of option pricing,this paper shows how different sequences can improve the algorithm.The conclusions can also be applied to graphics rendering,computer simulation and other fields.