Monte Carlo Methods For Option Pricing

Option is the basic financial derivatives, option pricing is one of the most important research fields in financial engineering, partial differential equation method, martingale method and numerical methods are mainly used methods in option pricing. The B-S option pricing model is the successful example of partial differential equation pricing method, the B-S equation and B-S formula have become necessary contents for every financial engineering textbook now. The theory basis for Monte Carlo method is Probability and Statistics. Monte Carlo method, binary tree and finite difference method are all numerical methods. The essence of Monte Carlo pricing method is predicting the average payoff of option through simulating sample path of underlying asset, and then getting the estimate of option price. The biggest comparable advantage in Monte Carlo method is its error converge rate is O(n^-1/2) and this rate will not be effected by the dimension of problem, so Monte Carlo method is efficient for pricing of high dimension options.This thesis is to give a research of the theory of Monte Carlo method and its application in option pricing; some results file a gap in the literature. The main contents are:1. Give a brief review of background knowledge about option, abstract the common methods in option pricing, introduce the research history of Monte Carlo pricing method, note the valuable reference paper in this field.2. Introduce the generalization of Monte Carlo method, make emphasis in error estimate and work efficiency evaluation. Methods for generating normal variates were summarized as well.3. Give the theory basis for application of Monte Carlo method in option pricing: risk-neutral pricing principle, briefly discuss the key technology and implement process in different situations. Give the derivation of integral representations of option price, sequentially provide a basis for discussion of quasi-Monte Carlo method. Finally conclude the advantage of Monte Carlo pricing method and its applicable option variety. 4. Describe some variance reduction methods in Monte Carlo method with detailed examples in option pricing field, and deduce the optimal coefficient b^*.5. Make clear efficiency of Quasi-Monte Carlo method and make comparison between Quasi-Monte Carlo and Monte Carlo method in their difference of error estimation, summarize issues being worth attention in the usage of Quasi-Monte Carlo pricing method. Evaluation the uniformity of low discrepancy sequences with knowledge of (t,m, d)-net and (t, d)-sequence and point out start points selection and high dimension clustering effect. Note the new research aspects: randomized Quasi-Monte Carlo and effective dimension reduction method, the later consist Brown bridge and principle components.