Details

M. Broadie and J. Detemple [3] developed an analytical formula to estimate American capped call options. In order to approximate an American call payoff , they use the value of a capped call written on the same asset [1]. If the price of the underlying asset is , when the payoff of a capped call option is , where is the strike price and is the cap. The payoff is the same as a standard American option, except that the cap limits the maximum possible payoff. Since the policy of exercising when the asset price reaches the constant cap is an admissible policy for the American option, for any .

Hence a lower bound is still obtained after optimizing over :

.

The determination of is a simple univariate differentiable optimization problem for any given . In this Demonstration, Mathematica's built-in function NMaximize is applied.

The evaluation of the capped call payoff function derivative with respect to the cap, while the underlying asset price approaches the cap from below, leads to an exercise boundary , where is the American call optimal early exercise boundary at time . The boundary is the solution to the equation , where .

In this Demonstration, the Newton–Raphson technique is applied to solve . Moreover, M. Broadie and J. Detemple [1] prove that an upper bound for the American call price is obtained, after replacing the optimal early exercise boundary function with in Kim's integral equation [2]. The integral in Kim's equation represents the early exercise premium and is approximated by Simpson's rule.

References

[1] 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.

[2] I. J. Kim, “The Analytic Valuation of American Options,” The Review of Financial Studies, 3(4), 1990 pp. 547–572.

[3] M. Broadie and J. Detemple, "American Capped Call Options on Dividend Paying Assets," The Review of Financial Studies, 8(1), 1995 pp. 161–191.