# American Capped Call Options with Constant Cap

American Capped Call Options with Constant Cap

This Demonstration shows the maximization process of an American capped call option with a constant cap (or barrier). Because the capped call must be instantly exercised if the underlying asset price rises above a predetermined price , which is called the "cap" or "cap price", its value never exceeds the value of the standard American call. Thus, identifying the cap that maximizes the capped call payoff function (,L), we obtain a lower bound for the American call price. Moreover, the evaluation of the capped call payoff function derivative with respect to the cap, while the underlying asset price approaches the cap from below, provides a lower approach for the American call optimal exercise boundary [1]. Finally, after replacing with (t) in Kim's integral equation [2], 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

L

C

0

S

0

LB

*

L

t

B

t

B(t)

*

L

UB

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 increasing the time discretization gives a more accurate (tighter) upper bound.

n

The upper graph shows the cap (blue line) that maximizes the capped call price, the function (t) (red dots) that approaches the optimal exercise boundary , and the optimal exercise boundary approach at (orange dashed line). For the capped call's holder, the early exercise is not optimal while the asset price moves in the blue area.

L

*

L

B(t)

*

L

t=0

The lower graph shows the cap price (black dot) that maximizes the payoff function of the capped call at . The black curve represents the payoff function of the American capped call, depending on the cap price at .

t=0

t=0

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.