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 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.

For the specific pattern of nonuniform time discretization where , we can calculate the length of the first time step , according to the formula . Then, all lengths derive as: , .

● Derivation of trapezoidal approximation, using nonuniform time grid:

Given that , we get:

.

● Derivation of Simpson's approximation, using nonuniform time grid:

Using theLagrange polynomial interpolation (Wolfram MathWorld) [4] for in and integrating with substitution , we get:

.

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] 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.

[3] J. D. Evans, R. Kuske, and J. B. Keller, "American Options on Assets with Dividends Near Expiry," Mathematical Finance, 12(3), 2002 pp. 219–237. doi:10.1111/1467-9965.02008.

[4] J. Rokne, "Explicit Calculation of the Lagrangian Interval Interpolating Polynomial," Computing, 9(2), 1972 pp. 149–157. doi:10.1007/BF02236964.