Pricing Exotic Option With Credit Risk Of Jump Diffusion Under Stochastic Interest Rates

The classical Black-Scholes model plays an important role in pricing financederivatives, but the assumption that market interest rates are fixed and the market isno friction does not match the actual situation. Exotic options pricing with credit riskunder stochastic interest rates was studied, and the corresponding numerical analysiswas given in this paper. The main results were as follows:(1) Three exotic options pricing with credit risk under stochastic interest rates werestudied. Firstly, The expression of exotic options was obtained by Risk-Neutral valu-ation when stochastic differential equations of the underlying assets, interest rates anddefault intensity are given. Then, based on Girsanov theorem, an equivalent probabilitymeasure is introduced, while under the new probability measure, there is only a ran-dom variable of the underlying asset. Finally, based on the nature of Brownian motionand Bayesian law, the analytical solutions of exotic options are given, and numericalanalysis is presented.(2) Binary options pricing with credit risk and stochastic interest rates under Jumpdiffusion process was studied, and we assumed w was not0. Firstly, The expressionof binary options was obtained by Risk-Neutral valuation when stochastic differentialequations of the underlying assets, interest rates and default intensity are given. Then,based on Girsanov theorem, an equivalent probability measure is introduced, whileunder the new probability measure, there is only a random variable of the underlyingasset. Finally, based on the nature of Brownian motion and Bayesian law, the analyticalsolution of chooser option is given, and numerical analysis is presented.