The Price Of The Derivatives Under Stochastic And Jump-diffusion Model

With the fast development of information technology and the integration of the whole world,many new financial derivate securities and new markets emerge. As an important derivate security,option is the "super weapon" of asset hedging for investors with widely research. But it often assumes the no-risk interest rate is no-stochastic, volatility of stocks is certain and the price process of stocks is continuous,also it neglects the default risk and stochastic life;A lot of financial data and statistics show that on-risk interest rate often have uncertainted features, adding jump-diffusion or stochastic volatility may improve the B-S model getting better results .Because there are no third party-currency options for privately written options, it will exist default risk Some options may come to an end and be canceled prematurely .That is, the contract may have a stochastic lifeIn detail, we are going to model some examples of our lives on the basis of B-S model,using stochastic calculus theories and martingale method. By choosing different numeraire and corresponding probability measure,we have main contributions in this paper as follows:1. we price the mortgage insurance under Vasicek interest rate and jump-diffusion model first. In another way ,we consider establishing stochastic volatility and Vasicekinterest rate. Both we get the pricing formula2. In order to generalize the Black-Scholes model,there are to ways. The first one assumes the volatility follows a stochastic process. The other one introduces jumps into the return process. So, we consider both elements to model exchange option,using different numeraire and corresponding probability ,we obtain its pricing easily.3. Another new kind of option-reload options is introduced. The explicit pricing formula of it under jump-diffusion models and stochastic life is gained.4. A transform of usual European option-exponential option is introduced. We assume jump-diffusion models,stochastic interest rate(maybe) and default risk for pricing exponential by multiplying an ingredient. Bying the changs of numeraire and corresponding probability,we get the explicit pricing formula.