Pricing Bond And Options Under Jump-Diffusion Combined Model Within Two-factor Market Structure

In recent years, Financial markets have developed rapidly. The researches of the theory and application of the Financial mathematics have also been developed rapidly. Many scholars have made significant contributions on the pricing derivatives. However, many studies are based on the traditional single-factor interest rates model, and many empirical study shows that the traditional interest rate models (such as CIR, Vasicek) does not accurately describe the real market structure. B.H.Lin and S.K.Yeh[25] developed jump-diffusion Vasicek model. J.C.Hull and A.White[23] employed two-factor term structure model. For the option pricing, The classical Black-Scholes model has been unable to fully capture the observational data of asset return. Many scholars made improvements. It focused on in two cases : First, the Jump-Diffusion model was first introduced by R.Merton[13] who assumes that the jump risks is non-systematic and the height of jump is still subject to normal distribution and gives European call option prices formula similar to the Black-Scholes model. Second, the introduction of stochastic volatility model, assuming that the instantaneous volatility of the stock is related to another stochastic process. Such as E.M.Stein and Stein[28],Hull and White[18] and R.Schobel and J.W.Zhu[29]. Later, combining the advantages of Jump-diffusion model and stochastic volatility model,Chen and Scott[26], Duffie and Kan[27] and Deng[34] researched European option pricing in two-factor CIR model.However, many facts have demonstrated that the jump risks also exist in the market structure, especially ,the effects of the jumps of the interest rate variable on pricing derivatives(for example [25,30,31]).As discussed in T.G.Andersenet al.[38],G.Bates[39] and D. Bakshi et al.[40],the most reasonable model of stock prices would include stochastic volatility,stochastic interest rates and Jump-diffusion.In this paper, we develop a combined model where includes stochastic volatility,stochastic interest rate with jump risks and jump-diffusion model within two-factor framework. Thus in the risk-neutral market, the risk-free rate r follows two-factor CIR model with jump risk: α_j,θ_j,σ_j are positive constants. {(W₁ (t),W₂(t))':t∈[0,T]}is 2-dimensional Brownian Motion. {( N₁ (t ), N₂(t ))': t∈[0, T]}is 2-dimensional Poisson process whose intensity parameters areλ₁ > 0,λ₂> 0,respectively. Assume that W₁ , W₂ independent with N₁ , N₂, m₁ , m₂ and N₁ , N₂ independent with m1 , m2 . m1 , m2 are random variables sequence. Y_1j , Y_2j stand for their height of jump, and are defined by:In the model we discuss the bond option and stock option pricing. The main results are:As an introduction, Chapter 1 introduces the history and current situation of financial mathematics, option pricing research,and the basis of the topics.The final presentation is the main contents of the thesis.Chapter 2 leads in the two-factor market structure model. In this model, we employs the fourier inversion transform, partial differential equations and Feynman-Kac equation to study zero-coupon bond and bond with coupons pricing, as well as European zero-coupon bond option and bond with coupons options pricing,and attains their solutions.And gives some numerical examples.Chapter 3 introduces two-factor market structure combined Jump-diffusion model ,and employs the fourier inversion transform, partial differential equations and Feynman-Kac equation to price European stock option and attains the solutions.It extends the results of L.O.Scott[22] and made the data analysis.Chapter 4 is summary and outlook.