Some Exotic Options Pricing In Jump-Diffusion Models

With the fast development of modern information technology and the integration of the whole world,many new financial derivate securities and new markets emerge in endlessly. As an important derivate security,option is the "super weapon" of asset hedging for investors with widely research.Nowadays,pricing options under jump-diffusion models is a very hot topic in option pricing research. By using stochastic calculus theories and martingale method, we establish several option pricing formulas under jump-diffusion models and obtain some pratical results in this paper.In detail, we have main contribution in this paper as follows:1.In need of finance engineering,a transform of usual European option-exponential option is introduced. Firstly we establish the economical market under jump-diffusion model. Using martingale method and Girsanov theorem,we obtain the pricing formula of exponential options when the interest rate and volatility are determin-istic. Secondly the relationship between exponential call and put options are also obtained.2.Another new kind of option-reload option is introduced. The valuation of this option is divided into the sum of two European contingent claim with different expire time. we also use the martingale method and conditional expectation , the explicit pricing formula of European reload options with dividend under jump-diffusion models is gained.3.By using the hedging strategy,the stochastic differential equations with terminal condition about expomential European option and exchange option is derived in jump-diffusion models . Then we transfer them into simple heat-conducting equations in physics .Finally we get the explicit pricing formula of exchange option with martingale method.4.Pricing options with stochastic interest rate under jump-diffusion models is also a difficult work in option pricing research. Introducing the stochastic life into the model on the basis of stochastic interest rate,we derive the pricing formula of European call option in jump-diffusion models with stochastic life when the interest rate obeys the Hull-White model.