American Capped Call Options with Exponential Cap
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This Demonstration shows the maximization process of an American capped call option with an exponential cap (or barrier) . Because the capped call must be instantly exercised if the underlying asset price rises above a price, determined by an exponential function over time, , where and , its value never exceeds the value of the standard American call. Thus, identifying the parameters and that maximize the capped call payoff function , we obtain a lower bound for the American call price . Moreover, the evaluation of the capped call payoff function partial derivatives with respect to the cap parameters and , while the underlying asset price approaches from below, provides a lower approach for the American call optimal exercise boundary . Finally, after replacing with in Kim's integral equation , an upper bound for the American call price is obtained. Thus the capped call option is really a tool used to bracket the pricing of the commonly traded American option.[more]
The table shows the upper and lower bounds for the theoretical American call price. Simpson's rule is applied to approximate Kim's integral, so if you increase the time discretization you get a more accurate (tighter) upper bound.
The upper graph shows the exponential function (blue line) that maximizes the capped call price, the function (red line with dots) that approaches the optimal exercise boundary , and the optimal exercise boundary approach at (black dashed line). For the capped call's holder, the early exercise is not optimal while the asset price moves in the white area.
The lower graph monitors the maximization process of the capped call payoff function at , depending on the cap parameters and . The orange line shows the consecutive steps (red dots) of the maximization process, using Mathematica's built-in function FindMaximum.
Lastly, this Demonstration does not use Mathematica's built-in function FinancialDerivative, which may be applied on a variety of financial instruments, including several types of barrier and capped power options.[less]
M. Broadie and J. Detemple  specified a three-step procedure for valuing American capped call options when the cap is described by the function . Based on this procedure, S.-L. Chung, M.-W. Hung, and J.-Y. Wang  approximated an American call payoff by maximizing the value of a call with exponential cap written on the same asset. If the price of the underlying asset is when , the payoff of the capped call option is , where is the strike price and is the cap's constant growth rate. The payoff is the same as a standard American option, except that the cap function limits the maximum possible payoff. Since the policy of exercising when the asset price reaches the exponential cap is an admissible policy for the American option, for any .
Hence a lower bound is still obtained after optimizing over . Denote the optimal solution as
The determination of is a bivariate differentiable optimization problem for any given . With the formulas of derivatives and , the optimal solution should satisfy . In this Demonstration, Mathematica's built-in function FindMaximum is applied to maximize the payoff function and determine the optimal solution .
Following the same idea of M. Broadie and J. Detemple , the evaluation of the capped call payoff function derivatives with respect to and , while the underlying asset price approaches 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 and .
In this Demonstration, Mathematica's built-in function FindMinimum is applied to minimize . Following the method of M. Broadie and J. Detemple , an upper bound for the American call price is obtained after replacing the optimal early exercise boundary function with in Kim's integral equation . The integral in Kim's equation represents the early exercise premium and is approximated by Simpson's rule.
Compared to the method that uses a constant cap over time , the method with the exponential cap function  leads to tighter bounds for the theoretical American call price.
 I. J. Kim, "The Analytic Valuation of American Options," The Review of Financial Studies, 3(4), 1990 pp. 547–572.
 M. Broadie and J. Detemple, "American Capped Call Options on Dividend-Paying Assets," The Review of Financial Studies, 8(1), 1995 pp. 161–191.
 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.
 S.-L. Chung, M.-W. Hung, and J.-Y. Wang, "Tight Bounds on American Option Prices," Journal of Banking and Finance, 34(1), 2010 pp. 77–89.