Properties Of Option Prices In Jump-diffusion Model

In last century 70's, some scholars started their investigation about properties of some option prices in certain conditions, and got corresponding success. However most of these papers are studied in diffusion model, and actural prices of underlying assets around us are always having the character of jumps because of uncertainty and frequency of accidents. So we study the properties of European option prices and American option prices in the present paper in the condition of jump-diffusion model, and generalize properties of option prices in [1] and [2] from diffusion model to jump-diffusion model. We introduce the concept that is called "volatility time" in the article, and in the case of a convex contract function, we show monotonicity and convexity in underlying assets for European option prices. And the properties for European option prices are time decay and both monotonicity and continuity in the volatility. Similarly, the same properties can be proved for American option prices on the base of European option prices by Bermudan option prices.In section 1, we simply introduce option and mathematical models. In section 2, we show convexity and monotonicity in underlying assets for both European option prices and American option prices. In section 3, we show monotonicity in volatility for both European option prices and American option prices. In section 4,we show continuity in the volatility and time decay for both option prices.