Pricing Of American Call Option Under Lévy Model With Stochastic Volatility

We focus on the pricing of American call option problem under Levy model with stochastic volatility.Option pricing is one of the important contents in the Modern Theory of Finance.Option price include several parameters: price,strike price,interest rate,time to expiration,volatility and so on. Volatility is the critical one,it is used to describe the fluctuation of option price in a short while.In the B-S model.volatility is assumed as a constant[1],but in reality,it is often seemed as a random progress.There are several ways to describe volatility,one of the ways is to assume the volatility satisfy another stochastic differential equation,it is called stochastic volatility.Hull and White extend the constant volatility to stochastic volatility[11] ,reference [24] give the American call option pricing with stochastic volatility.In my thesis,I extend the Brow-nian process in reference [24] to Levy process,give the partial differential equation the European option satisfied under Levy process with stochastic volatility by equivalent martingale measure,discuss the problem of the partial differential equation American call option satisfied on dividend-paying and placing stocks under Levy process with stochastic volatility,and also conclude that the optimal exercising time of American call options can only be at the time immediately before payment of the dividend or expiration time.I extend the conclusion of reference[ 11 ] and reference[24] on the price of underlying assets.