Pricing European Option Under Stochastic Volatility Model With Dividend

This paper concerns about pricing European option under stochastic volatility mod-el when stock pays dividend.We deduct the option price in both the PDE and probabilityrepresentation. Stochastic volatility model considers the volatility of return of risky assetas a random variable,which in this paper is considered as the solution of one stochasticdiferential equation. The random factor efects the volatility is diference to the randomfactor efects the return of risky asset. Under Black-Scholes Model,the volatility of returnof risky asset is constant,therefore,the market is complete, in which the value of deriva-tive can be complicated by the riskless asset and the underlying.However,according tothe data,volatility is not constant.Under stochastic volatility model,because we introducethe new random factor,it is more similar to the real world.But stochastic volatility mod-el has not complexity,that is,the value of derivative can not be only complicated by theriskless asset and the underlying.The pricing problem is more complicated under the s-tochastic volatility model.Unlike in Black-Scholes Model,We can not use the method ofcomplicating to obtain the option price in probability representation.In this paper,to solvethis problem,we first deduct the option price in PDE representation using the no-risk arbi-trage,luckily we can use the method of no-risk arbitrage in stochastic volatility model,thenobtain the probability representation by Feynman-Kac formula.we also consider risky as-set as stock and it pays dividend,the result is similar to the no-dividend situation,which issame in Black-Scholes Model.At the end of the paper,we briefly discuss how to use theresults to price European option in practice,we mainly consider the problem of parameterestimation.