Asian Option Pricing Based On Stochastic Volatility And Stochastic Interest Rates

Option is the core of risk management in modern financial markets. Since1973,after Black and Scholes set up the famous Black-Scholes option pricing model, thetheory of option pricing has been developed rapidly. In recent years, in theinternational financial derivative markets, trading of derivative products besides thewell-known European options, and Asian options is a representative of them. Over theyears, numerous economic scholars and researchers make much further research inAsian option pricing, but these studies mostly regard the volatility of asset returns andrisk-free rate as the constants.This paper studies Asian option pricing with volatility and interest rates arestochastic, as follows:(1) The underlying asset price follows a geometric Brownian motion, firstcalculate the characteristic function of the logarithm of the path variables, with theapplication of risk-neutral pricing principle, get the geometric average Asian calloption pricing formula, and then using Fast Fourier Transform (FFT) to calculate anapproximation of the value of this option, and finally compared to the option value ofBlack-Scholes equation that was solved directly, the fast Fourier transform methodproved to be effective.(2) In the CIR stochastic volatility model, using Feynman-Kac formula, thismodel will convert seeking logarithmic characteristic function of the path variable ofAsian option to solving the backward parabolic equations Cauchy problem, derive acall option pricing formula of geometric Average Asian under risk neutral conditions.(3) On the basis of CIR stochastic volatility model, stochastic interest rateprocess instead of a constant risk-free interest rate, first find the correspondingcharacteristic function, and then use the principle of risk-neutral pricing to calculatepricing formula of Asian call option.