Pricing And Properties Of Options Under Jump-Diffusion Market With Default Risk

We first summarize the pricing and property of options under the simpler jump-diffusion market with default risk,i.e.,with fixed jump intensity.We use the facts that the price follow the certain parabolic integral-differential equation to study the convex-ity and monotonicity of option price in the model with jump.And we give the convex conditions.And then use the preservation of convexity to derive the option value' s monotonicity on the different parameters of the,such as volatility,jump size and jump intensity.Then in a more general case,we study the "jump to the default model" in the pricing formula and the conditions under which the option price is the unique classical solution.In particular,finding an exact condition to match the option price at the default boundary with the payment in the recycling rules.We finally summarize the spatial convexity of the option price in this model and the relationship between the convexity and the monotonicity of the parameters.