The Valuation Of Options For Jump Processes And The Choice Of Numeraire

The pricing of option is the core of mathematical finance. On the one hand, the valuation formula to the option is sometimes ideal finance service to investor. On the other hand, the stockjobbers who sell varieties of stocks, can use the formula not only to save time, specialists, and employees, but also to creat excessive profits. So, to make certain the price of the option is of much importance. Jump processes are able to describe rare incidents which have an influence on the prices of assets in market, which accord with financial statistical rules more conformably. So, this paper will study the pricing of options when stock processes are drived by jump processes.To begin with, we point changes of martingale measures can be replaced by the choice of numeriare, and improve the price process of the replicatable contingent claims is independent of the changes of numeriare, for a market under semimartingale model. And, we give the valuation of the options in a stochastic interest rate model, by choosing a special numeriare. Secondly, we introduce the Levy processes and the geometric Levy market model in which the stock price process is governed by a Levy process. Then, we obtain decomposed forms of a Levy process and expression of the market model. Thirdly, we discuss the pricing method of Asian option in the model, removing the path dependency by a portfolio and using the changes of numeraire techniques. Finally, we prove the existence of the equivalent probability martingale measure in the model mentioned above, and consequently, the market is a fair one.