Finite Difference Method For Pricing Multi-asset American Put Option

Option is a kind of financial derivatives, it is always one of the focus of the financial sector. Its huge "leverage effect" helps people to own a right to protect their assets from depreciation by paying a bit "premium". The "subprime mortgage crisis", swept across the United States from 2007, makes people realize the importance of wealth management, at the same time, the option has already been popular in Europe and the United States.2015 is a significant year for our national financial markets, the first officially stock options listed on February 9th, that means the application range of options spread all around the world, so the pricing problem should be solved.In this paper, we concern on the valuation of two-asset American put option. At first, using the hedge theory, we obtain the Black-Scholes equation (B-S equation). Furthermore, we transform this problem to a non-linear parabolic problem on a bounded domain by using a penalty function and perfectly match layer technique (PML technique). At last, the finite difference method is used to discrete it. The remainder of this paper is organized as follows:In the first part, this paper mainly introduces the definition and classification of options. Furthermore, starting from the historical background of option, this paper provides the phas-es of option pricing problem, one-asset European option is given by B-S equation, but for American style, we simply introduce several numerical methods.The second part mainly obtains the multi-dimensional put option pricing B-S equation by using no risk hedge principle and high-dimensional Ito formula: where P is the put option price, r is the risk-free interest rate, qi and σi are the dividend rate, volatility for i-th underlying asset, respectively, ρij is the correlation coefficient between Si and Sj.In the third part, we mainly transform the two-dimension B-S equation into a problem defined on a bounded domain, which can be used numerical method to solve. This paper is realized by using following four steps:(1)Analyzing the original equation, we obtain the equivalent linear complementary problem: The termination condition is: where α1 and α2 are positive constants with α1+α2= 1.(2) Coefficient transformation to simplify coefficient is just like So we get The initial condition is: where Payoff function is:(3)Penalty method:a penalty term will be added to obtain a 2-dimension parabolic problem:(4)PML technique:the unbounded domain should be truncated into a bounded one, Ω={(x,y);-L1-δ1<x<L1+δ1,-L2-δ2<y<L2+δ2],we get whereIn the fourth part, we apply the finite element method to solve the resulting problem, the numerical simulations are provided to test the performance of the proposed method.