The Regularization Method for a Degenerate Parabolic Variational Inequality Arising from American Option Valuation

Option trading forms part of our financial markets. A traded option gives to its owner the right to buy (call option) or to sell (put option) a fixed quantity of assets of a specified stock at a fixed price (exercise or strike price). There are two major types of traded options. One is the American option that can be exercised at any time prior to its expiry date, and the other option, which can only be exercised on the expiry date, is called the European option. It was shown by Black and Scholes (cf. [3]) that the value of an European option is governed by a second order parabolic differential equation with respect to time and the underlying stock price. This is now referred to as the Black-Scholes equation. The value of an American option is governed by a more complex mathematical model due to the flexibility on exercise date. It can be shown that American option pricing is determined by a linear complementarity problem involving the BlackScholes differential operator and a constraint on the value of the option (cf., for example, [20, 19]). This complementarity problem can also be formulated as a variational inequality (cf. [19]). The Black-Scholes equation is a degenerate partial differential as its coefficients of the first and second order spatial derivatives vanish as the underlying stock price approaches zero. A popular method of removing this difficulty is to introduce a new variable and transform the Black-Scholes equation into a heat equation defined on the whole real number set. This technique is used in many existing papers such as [1, 10, 20]. In this case, the degeneracy point is transformed to −∞. However, when solve the resulting heat equation numerically,