Study On Option Pricing With Stochastic Volatility

As a financial derivatives, Options can be used to hedge market risk. The pricing problem of options has always been the core issue of options trading, as well as hot and difficult problem in financial field. Traditional Black-Scholes option pricing model is not only a great achievement in the field of finance, but also has profound impact on the entire field of economics. Classic Black-Scholes model is as follows:The stochastic differential method is applied to the pricing problem for underling assets without derived paying, and one can obtain the price function of any derivative securities satisfies the following of partial differential equations with certain boundary value condition:Solving partial differential equations one can obtain Black-Scholes pricing formula.The pricing formula for European call option:The pricing formula for European put option is as following:After that, many scholars investigate Black-Scholes model based on the stochastic volatility model, such as Heston model, Hull-White model, Stein-Stein model. Notice that Logistic model can characterize the phenomenon of S-shaped growth bounded growth curve. We investigate the option under a class of stochastic volatility with Logistic equation.We consider the following stochastic volatility model in this paper: where k,γ,σ,α are constants, and one has:We obtain the parabolic partial differential equation that price function of European call option satisfies as follows:The boundary value condition is:We also use Cosmol software to simulate the numerical solution of the type of this qualitative analysis for numerical solution is also give in this paper.