Abstract
We consider the problem of pricing American options using the generalized Black–Scholes model. The generalized Black–Scholes model is a modified form of the standard Black–Scholes model with the effect of interest and consumption rates. In general, because the American option problem does not have an exact closed-form solution, some type of approximation is required. A simple numerical method for pricing American put options under the generalized Black–Scholes model is presented. The proposed method corresponds to a free boundary (also called an optimal exercise boundary) problem for a partial differential equation. We use a transformed function that has Lipschitz character near the optimal exercise boundary to determine the optimal exercise boundary. Numerical results indicating the performance of the proposed method are examined. Several numerical results are also presented that illustrate a comparison between our proposed method and others.
Highlights
Hedging and pricing are important issues in derivative securities
McKean [3] and Van Moerbeke [4] proved that the valuation of American options constitutes a free boundary problem and they studied the properties of the free boundary
The main contribution of this paper is the development of a numerical method for finding the optimal exercise boundary of American put options under the generalized Black–Scholes model
Summary
Hedging and pricing are important issues in derivative securities. European and American options can be exercised only on the expiration date and at any time until the expiration date, respectively. The assumptions of the Black–Scholes [1] and Alghalith [10] models are adopted to develop a numerical method that uses the transformed function to value American put options. The main contribution of this paper is the development of a numerical method for finding the optimal exercise boundary of American put options under the generalized Black–Scholes model. After determining the optimal exercise boundary, we calculate the value of the American put options by applying finite difference method (FDM) and use the Crank–Nicolson method in time discretization.
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