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# Kim's Method for Pricing American Options

This Demonstration shows Kim's method [1] for pricing American options. A European financial option is an instrument that allows its holder the right to buy or sell an equity at a future maturity date for a fixed price called the "strike price." An American option allows its holder to exercise the contract at any time up to the maturity date, and because of this it is worth more than the European option, by an amount called the "early exercise premium." For the American call's holder, the early exercise becomes optimal when the underlying asset price exceeds a critical boundary , above which the intrinsic value of the option becomes greater than its holding value.
According to Kim's method, the valuation of the American option derives from an integral expression of the early exercise premium as a function of the optimal exercise boundary plus the value of the European option. The plot shows the optimal boundary approach, using either the trapezoidal rule (blue dashed line) or Simpson's rule (red line) to approximate the early exercise premium integral. Both approximation techniques use the same time discretization (from 4 to 50 time steps). The table shows the American call price depending on the integral approximation technique and time discretization.

### 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 or Simpson's rule 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.

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