Comparison Study Of Numerical Methods For American Options Pricing

Option contracts are widely used by almost all the major financial institutions andinvestors, so as to speculate the stock market trends or to control the level of risk.American option is the most widely traded option nowadays. However, pricingAmerican options，even in the standard case of a lognormal process for the underlyingasset, is still an active research area. Pricing Americm options is much more complexthan pricing European options. There is generally no pricing method like theBlack-Scholes formula, and thus numerical computing methods must be used. Americanoptions can be exercised at any time up to the maturity date, which refers to the problemof finding the optimal execution time.This paper attempts to price American options by seeking the optimal stopping rule.Necessary theoretical preparations of American options pricing are elaborated fromChapter1to Chapter4. The paper discusses how to simulate a stock price path, theapplication of Monte Carlo Simulation, and the theory of finding the optimal stoppingrule. Three pricing methods are discussed and compared respectively in Chapter5,including using the binomial tree model, making use of the Black-Scholes formula andusing Monte Carlo simulation to obtain an exercise boundary. And the pricingprinciples and numerical examples are given for each method.Each of the three methods discussed in this paper has its own disadvantages andshortcomings. The binomial tree model is easy to understand, and applies to bothAmerican options pricing and European options pricing. We can obtain accuratetheoretical price by using the binomial tree model even when there is a great number oftime intervals. But the binomial tree model only considers a discrete set of stock pricevalues at each time period. On one hand, we can only obtain approximation solutions tothe option price when the number of time intervals is small, which sacrifices theprecision. On the other hmd，the large number of time intervals leads to the pressure ofcalculation and high computational complexity. Therefore, the binomial tree method issuitable for the case when the number of exercise opportunity is limited, and doesn’tapply to other types of options. The Black-Scholes formula can be used to get a price,which cm be further used as the exercise boundary to price the American option. Thismethod is simple to understand and easy to implement, but lacks a solid theoretical foundation. The pricing principle of this method determines that the option price canusually be underestimated. The accuracy of this method should be further explored andexamined in the future. Monte Carlo simulation is used to price American options bydealing with a large amount of stock price paths. The optimal execution boundary isfirst estimated, which can then be used to estimate the option price. Using Monte Carlosimulation in American options pricing has a solid theoretical foundation and theaccuracy of the results can be guaranteed. Yet a large amount of simulated paths need tobe stored at one time, which m^es the method computationally memory-intensive if wewant to further improve the accuracy. Overall, the Monte Carlo simulation method hasobtained considerable development and progress in recent years, and is a quite effectivenumerical method for pricing financial derivative securities. More developments andapplications will be further improved in the future.