The Pricing Of Options In A Stochastic Volatility Model With Poisson Jump

As a financial derivative, option is trading based on the right to buy or sell anunderlying asset. The buyer pays a premium to have the right to buy or sell anunderlying asset at a specified price in the future. If the price of the underlying assetmoves against the expected direction, the buyer could forfeit exercising the option andlose the premium. If the price moves as expected, the buyer exercises the option andearns unlimited yield minus limited premium. Options with leverage effectivelyreduce cost and increase earnings, which attracts investors in financial markets.However, the premium (i.e. the option price) is very important. Overvalued premiumreduces liquidity while undervalued premium has negative effects on options riskmanagement.Option pricing has already been discussed frequently in history. The most classicone is Black-Scholes option pricing model. However, a lot of data suggests thatassuming the volatility as constant in BS model is not accurate. The price volatility ofthe underlying stock normally shows high peak and fat tail phenomenon. In addition,BS model cannot explain volatility smile. While stochastic volatility models of theOU type possess authentic capability of capturing some stylized features of financialtime series[12]. In another aspect, the stock price may jump at some time. Empiricalanalysis shows that the stock price process driven by Levy process can be suitable todiscribe the high kurtosis and negative skewness[2]. So we construct continuous timestochastic volatility models with poisson jump for financial assets where the volatilityprocesses are OU process, and the range of jump is a CIR process.This article introduces the common stochastic processes, such as Levy processes,Poisson process, OU process, CIR process, etc. It also describes their characteristicsbriefly. Then it also introduces the change of measure of the Poisson process and the Brownian motion to achieve option pricing under the risk-neutral measure. Next, this essay presents several classic option pricing models, including BS model, a model with stock price volatility as one OU process, and a model with both stock price and volatility involving related stochastic Poisson jump process. It gives research results of these models.Based on the three models as mentioned, the paper discusses the stock price of the underlying asset with Poisson jump whose coefficient is a CIR process, the stochastic volatility is an OU process. Then discuss in the incomplete market,how to make change of measure, and under the risk neutral measure, we give the PDE that the option price satisfies.Consider a stochastic volatility model: dSt=μStdt+√vtStdwt1+utSt-dNt, dvt=αvtdt+σdwt2,(4.0.1) dut=β(η-ut)dt+δ√utdwt3, where S(t) is the security price, Nt is a poisson process defined on the probability space (Ω,F,P), the intensity of Nt is λ,M(t)=N(t)-λt is the compensated poisson process,wt1,wt,wt,Nt are mutual independence, ut is an CIR process which is used to describe the range of the jump, the CIR process can guarantee that ut stay positive.To guarantee that there is no arbitrage opportunity, we should assume λ is a positive value, then exists the risk neutral measure: P(A)=∫AZ(T)dP,A∈F,and This risk neutral measure is unique,under this measure,and on the probabilityspace(Ω,F,P), the compensated poisson process M(t)=N(t)-λt is amartingale.Then(4.0.1)can be written as dS(t)=rS(t)dt+s(t-)u(t)dM(t)+√vtS(t)dwt1, dvt=αvtdt+σdwt2,(4.2.6) dut=β(η-ut)dt+δ√utdwt3.The processes wt1,wt2,wt3,Mt are independence,and under the measure P,N(t)is a poisson process whose intensity is λ.With regard to the new model under the new probability measure,we apply the Ito formula with jump,we can get the SDE that f=f(t,St,vt,ut)satisfied.Then we use the martingale property of e-rt f(t,St,vt,ut),and different the discount of f and let the coefficient function of dt be zero,so the main result of this paper is as followingTheorem.1.Suppose V=f(t,St,vt,ut) is the option price at time t,then as the certain function of t,V=f(t,St,vt,ut)satisfies the following equation:The boundary conditions are: f(t,S,∞,u)=S,...