Option Pricing Based On Brownian And Poisson Motion

Black-Scholes option pricing formula is built on lots of the complete financialmarket assumptions, but the real world economy is not complete. How to expandresearch for options pricing is one of the important work according to the practice. Inthe dissertation, using the principle of no-arbitrage hedge, stochastic process, as well asthe option pricing formula, we research the multi-asset option pricing with transactioncosts, and the European option pricing with time delay on the jump-difusion market.In the dissertation, our works are following:1!In this dissertation, We use the CEV (Constant Elasticity of Variance) pro-cess to improve the constant volatility in the stock price model, which is driven bythe Brownian motion and Poisson process at the same time. Then we construct themathematical model of stock prices under the CEV process. In this model, we assumethat the stock pay continuous dividend, derivative the partial diferential equations forthe multi-assets of option pricing, while the option pay transaction costs in proportion.2!Arriojas, Hu and Mohammed et al derived a Black-Scholes formula for a calloption value in the market with stock price having time-delay. Based on their researchframework, we introduce the jump-difusion process, constructed the mathematicalmodel of stock prices having time-delay under the jump-difusion process. By adjustingthe drift we get the assets discount process is a martingale. Then, we obtain a closedform solution and the partial diferential equations of the European option price by useof the option pricing theory over a measurable interval of the payof function, while thestock pay continuous dividend. And based on the conclusion, we expand to study theoption pricing problem with stochastic interest rate.Through these studies, we can fond that:1!The power index of stock prices isrelevant to the selection of volatility elasticity α in the pricing model. The Poissonintensity parameter λ afects the transaction costs, which decrease with a larger λparameter.2!The closed form solution and the partial diferential equations of theEuropean option price can only get over the measurable interval of the payof function,while the stock pay continuous dividend. And the option price not only relevant to thecurrent stock price, but also influenced by the stock prices of the past time. The efectsof the jump process is a complex problem. But from the partial diferential equations,when there is a jump process, the option price should consider the risk premium, itssize depends on the density and strength of the jump process.