Details

In this detailed description, the symbols have the following meanings:

is the current time

is the maturity date

is the stock price at time

is the strike price

is the stock dividend yield

is the risk-free interest rate

is the stock volatility

is the cumulative distribution function of the standard normal distribution

is the moving free boundary

is the critical boundary

Let

,

,

and be a non-negative continuous function of time. Consider a contract whose value at time is given by:

,

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

, for for all .

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

Subject to the value matching condition, the method in [1] proposes to numerically approximate the critical asset price at time by a recursive procedure. This method requires solving integral equations, where is the number of time steps. The method in [2] proposes to evaluate analytically the integrals, assuming that the remains constant within each time subinterval, instead of employing a numerical technique (e.g. the composite Simpson's rule) to approximate the integrals. The method in [3] further expands this idea by assuming that is an exponential function within each time subinterval. In this Demonstration, the method in [2] derives as a special case of the method in [3], where the exponential function's exponent is set to zero.

Following the regularized incomplete beta function, the temporal point of the nonuniform time mesh is obtained by , where , and .

References

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

[2] J. Huang, M. G. Subrahmanyam and G. G. Yu, "Pricing and Hedging American Options: A Recursive Integration Method," The Review of Financial Studies, 9(1), 1996 pp. 277–300. www.jstor.org/stable/2962372.

[3] N. Ju, "Pricing an American Option by Approximating Its Early Exercise Boundary as a Multipiece Exponential Function," The Review of Financial Studies, 11(3), 1998 pp. 627–646. www.jstor.org/stable/2646012.