Front-tracking Finite Difference Methods For The Valuation Of Options Under Stochastic Interest Rate

An option is a contract which gives the buyer(the owner or holder of the option) the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on or before a specified date. Pricing the value of options is an important part in options trading.The long history of the theory of option pricing began in 1900 when the French mathematician Louis Bachelier deduced an option pricing formula based on the assumption that stock prices follow a Brownian motion with zero drift. Black and Scholes deduced B-S model base on no-arbitrage theory, which assumes constant rate and the stock price follows geometric Brownian motion. Then Merton adjusted it to an option valuation model with dividend warrants, which calls Black-Scholes-Merton model.Interest rate is one of the most basic factors which have a great influence on the option pricing. The article will introduce several types of the one-factor short-rate models, such as Merton model、Vasicek model and CIR model.This article will use the finite difference algorithm to solve the problems of fixing solution, and set up an for the stochastic interest rate model Vasicek and CIR.We make a series of conversions for No-arbitrage American option pricing model2 21() 02 t ss sV(10)sS V(10)r-D SV-r V(28)then set up an explicit difference scheme to estimate the option price and find the optimal exercise boundary of American option.We use MATLAB to set up a numerical simulation for American put option pricing model, for example, giving different values to rate r. From the picture, we get that the finite difference algorithm is feasible and interest rate exactly has a great influence on the option pricing.