Kim's Method with Nonuniform Time Grid for Pricing American Options
This Demonstration shows Kim's method  for pricing American options using a nonuniform time grid. 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.[more]
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, using an iterative algorithm via backward induction. Both approximation techniques use the same time discretization (from 4 to 50 time steps). The coefficient () determines the rate of length change between two successive time steps: . For the time grid becomes uniform. As decreases, the time grid becomes denser close to expiry, where the optimal exercise boundary is singular . Thus, for a relative small number of time steps, the coefficient may help an analyst to look for a better approximation of the optimal exercise boundary. The table shows the American call price depending on the integral approximation technique, time steps, and the coefficient. The grid lines show the time discretization pattern.[less]
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
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:
 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.
 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.
 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.
 J. Rokne, "Explicit Calculation of the Lagrangian Interval Interpolating Polynomial," Computing, 9(2), 1972 pp. 149–157. doi:10.1007/BF02236964.