The Optimal Stopping Problem Based On Lévy Process And The Research Of Bermudan Option

Actually, Bermuda option is a special case of American option, it is not yet common in our country, such that there is still not so many research about it. However, because of the gradual development of the Bennuda option in our country, it is necessary to complete its research. Due to the positive effect of the application of optimal stopping theory on option pricing problem, our article is also with a view to optimal stopping theory to study the problem, and we will divide the study of Bermuda options into two different parts:the option pricing problem and the perform strategy of option. It is studied that the Levy process describes better to the characteristics of financial data, and it prompted our article to describe the logarithmic return of the underlying asset prices with Levy process.First, our article solved the Bermuda option pricing problem by changing it to optimal stopping problem based on no arbitrage and the risk neutral theory. Second, take consideration about the holders, we research the rules of the options to perform or not by using the backward recursive system, and also prove it. Third, by the builded optimal stopping problem of the Bermuda options, we obtain that the Bermuda option price is equal to the maximum expect of the discounted revenue on the all feasible points under the martingale probability measure. Thus, by using Radon-Nikodym derivative, we realized the transform of probability measure, find the equivalent martingale probability measure. And given the form of discounted prices of the underlying asset under the equivalent martingale probability measure; Finally, we take a numerical calculation of Bermudan option with specific Levy process -- NIG-Levy process, for the convenience of calculation and the unity of principle, by choosing the Esscher-transform, we find the equivalent martingale measure and give the Bermudan option price.