# Kim's Method with Nonuniform Time Grid for Pricing American Options

Kim's Method with Nonuniform Time Grid for Pricing American Options

This Demonstration shows Kim's method [1] 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.

*

B

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 [3]. 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.

*

B

h

0<h≤1

h=d/d

t

i

t

i-1

h=1

h

h

h