The Fourier Approach Of The American Option Pricing Under Jump-diffusion Processes

American option is the most frequently traded options in the financial mar-ket, while the stock option is well studied in the literature. Empirical research hasfound that the stock price movement is not simply follow the Geometry BrowianMotion, but takes on a characteristic of high peak and fat tail. So the traditionalB-S-M model is no long give a good answer, this can be done by introducing theJump-Di?usion model. The American option pricing under the Jump-Di?usionmodel is a tough problem in the option pricing theorem. This is duo to thecharacter of early exercise in American option and nonlocal problem caused byjump. Now the mostly used method is the finite element method based on thevariational inequalities. Since the finite element method is sophisticated in itselfand is hard to program on the computer, this thesis is trying to use the extendedfourier transform method to provide another perspective and ideas to overcomethe di?culties met in pricing American options under Jump-Di?usion processes.The fist chapter of this thesis simply refer the situation of study in the valua-tion of American options under the Jump-Di?usion processes. And we comparedthe main methods in recently usage in this filed. Then we provide the methodadopted in this paper, and present the main work in our thesis.The second chapter is the basis of the following content. In the beginning,we introduced the important concepts and formulations in financial mathematicssuch as arbitrage free pricing theorem, martingale, Ito? lemma etc. Then wecarefully introduced B-S-M model and Jump-Di?usion model. Based on these,we get the main point this thesis is going to solve—the free boundary problem ofAmerican options. After introduced all the mathematical models above, we thenexplained the most important method used in this paper—the extended fouriertransform method , and present the main ideas of Cauchy residual theorem andcontour integration in complex variable functions.The third chapter is the primary chapter of this thesis. At first, we comparedanother fourier approach in pricing American options, and then put forward thekey assumption used in this paper. Under the condition of the key assumption, weuse the extended fourier transform method to price the American put options inthe Jump-Di?usion model, and take the contour integration to obtain the Cauchyprincipal value integral representation of option's valuation. In the end of this chapter we identically changed the representation to another form, which is easierto explain and numerically implemented.The last chapter of this thesis is to test and verify the key assumption andresults proposed and obtained in chapter three. We first use iterative methodto obtain the free boundary, and then make use of this boundary to get theprice of American options. Finally, we compared some results available with ournumerical results, and concludes the chapter.