# American Capped Call Options with Exponential Cap

American Capped Call Options with Exponential Cap

This Demonstration shows the maximization process of an American capped call option with an exponential cap (or barrier) [4]. 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 >max{X,r·X/δ} and , its value never exceeds the value of the standard American call. Thus, identifying the parameters and that maximize the capped call payoff function (,,a), 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 (t) for the American call optimal exercise boundary . Finally, after replacing with (t) in Kim's integral equation [1], 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.

L(t)=·

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0

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a≥0

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C

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L(t)

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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.

n

The upper graph shows the exponential function (blue line) that maximizes the capped call price, the function (t) (red line with dots) that approaches the optimal exercise boundary , and the optimal exercise boundary approach (0) 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.

L(t)

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t=0

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.

t=0

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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.