Details

In this detailed description, the symbols have the following meanings: : current time; : maturity date; : stock price at time ; : strike price; : stock dividend yield; : risk-free interest rate; : stock volatility; : the cumulative distribution function of the standard normal distribution; : the moving free boundary; : the optimal boundary.

Consider the class of contracts consisting of a European call option and a sure flow of payments that are paid at the rate

for ,

where

,

,

and is a non-negative continuous function of time. Each member of the class of contracts is parametrized by . The value of the contract at time is

,

where denotes the value at time of a European call option on with strike price and maturity . The optimal exercise boundary for the American call option is obtained by solving the "value matching condition":

, for for all .

The value of the American call option is then given by .

Subject to the "value matching condition," the critical asset price at time can be numerically approximated by a computationally intensive recursive procedure. This method requires solving integral equations, where is the number of time steps. Each time the integral equation is solved, either the trapezoidal rule (Wolfram MathWorld) or Simpson's rule (Wolfram MathWorld) is employed to approximate the integral.

References

[1] I. J. Kim, “The Analytic Valuation of American Options,” Review of Financial Studies, 3(4), 1990 pp. 547–572. www.jstor.org/stable/2962115.

[2] M. Broadie and J. Detemple, "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods," The Review of Financial Studies, 9(4), 1996 pp. 1211–1250. doi:10.1093/rfs/9.4.1211.