Details

The boundary derives from the maximized payoff function [2] for an American capped call.

The table shows, at :

• The American call according to the "lower-upper bound approximation (LUBA)" [3], an empirical approximation method deriving from a weighted average of the option's upper (UB) and lower (LB) bounds,

• The American call according to the "lower bound approximation (LBA)" [3], an empirical approximation method deriving from the option's lower bound (LB),

• The American call upper bound (UB) [3], deriving from Kim's integral equation [1] where is replaced by ,

• The American call lower bound (LB) [3], deriving from the maximized payoff function of an American capped call.

M. Broadie and J. Detemple [2] developed an analytical formula to estimate American capped call options and they proved that their value could never exceed the value of American call options [3]. Conclusively, the maximized value of an American capped call option is a lower bound (LB) approach for an American call option. The LB approach leads to a boundary , where is the American call optimal early exercise boundary function over time. Moreover, M. Broadie and J. Detemple use Kim's integral equation [1] for the early exercise premium; after replacing the optimal early exercise boundary function with in the integral equation, the result is an upper bound (UB) for the American call. Finally, after examining a large number of options and using regression analysis, they developed two empirical methods (LBA and LUBA) to get more accurate approximations for the American call option.

References

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

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

[3] M. Broadie and J. Detemple, "American Option Valuation: New Bounds, Approximations, and a Comparison of Existing Methods," Review of Financial Studies, 9(4), 1996 pp. 1211–1250.