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

,

,

,

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":

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 or

Simpson's rule is employed to approximate the integral.

[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.