Numerical Methods For American Option Pricing

In recent decades, establish the option pricing theory is one of the most importantdevelopments in the financial field. Compared with European option, Americanoptions can be exercised at any time up to the maturity dates, then they have morechance to gain profits in practice. They are widely used in the financial field.Therefore American option pricing is the core question of the option pricing theory. Itis significance to study the numerical methods for American option pricing.Modern option pricing theory is based on Black-Scholes differential equationmodel. Under the assumption of efficient market hypothesis, the asset price followsthe geometric brown movement, the Black-Scholes differential equation of Europeanoption is obtained by using a series of technique to eliminate the randomness. Thispaper considers that market risk is neutral. The model of American Option Pricingwith dividends is obtained by modifying the Black-Scholes analysis.In general, there is no analytical formula for the valuation of American options.Several numerical methods for Black-Scholes differential equation of Americanoption pricing model with dividends is proposed in this paper.The paper is organized as follows. The part of this introduction briefly introducessome basic concepts of financial derivatives and option knowledge. In addition, thispart also introduces some numerical methods for American option pricing. Thesecond part educes the derivation process of the differential equation of Black-Scholesand optimal exercise boundary in detail. In the third part of this paper, By usingFront-fixing transformation and property of free boundary, a free boundary problemof American call option on a continuous dividend paying can be transformed into intoa fixed domain problem with nonlinear parameters. A three-level the finite differencescheme is proposed for solving this nonlinear problem. We employ the two-steppredictor-corrector technique to obtain first level. Numerical experiments show thatthis method is a more efficient algorithm compared with the binary tree method. Thefourth part a high order compact difference scheme is proposed for solving this nonlinear problem. The predictor-corrector technique is used to compute the firstlevel. Numerical experiments show that the high order compact difference schemegive better accuracy than the finite difference scheme. The relevant conclusions aremade in final part.