The Homotopy Analysis Method On Options Pricing

The explosive development of derivative financial instruments continues to provide new possibilities and increasing flexibility to manage finance and risk in a way specifically tailored to the needs of individual investors or firms. However, no matter how advanced financial instruments are, they are still exposed to high risks. As a result, we must understand and master these financial derivative products more vividly and concretely. This paper takes American options which is the most common derivative on the financial markets for an example, using the homotopy analysis method to solve the partial differential equation—in order to solve approximate solution of American option, this paper will deal with the free boundary problem which is the most important one in American options pricing being transferred into a fixed boundary problem.The dissertation consists of three chapters. The main content of this paper is arranged as follows:Chapter One introduces the bases of the paper, including the background, significance, and recent research of the problem.In chapter2, it makes American call option model dimensionless, transforms it into a linear parabolic quadratic partial differential equations, and uses the homotopy analysis method to be combined with the financial meaning and the partial differential equation condition, and then constructs the homotopy equation and gives the initial guess of the solution at the same time—the free boundary problem is transformed into an number of fixed-boundary problem.In the third chapter, before solving the problem, we discuss the convergence of the method, and obtain the reasonable conditions of the problem. According to the conditions, we solve the problem through numerical experiments. Finally, we analyze the relationship among different parameters (including exercise price, volatility, risk-free interest rate, dividend.etc.) and the premium of American options.