Pricing American Options with the Lower-Upper Bound Approximation (LUBA) Method
This Demonstration shows the lower and upper bound approximation methods  for an American call option.[more]
The upper graph shows the lower approach (red line) for the early exercise boundary , and its approximation using Kim's method (black dashed line). For the American call's holder, the early exercise becomes optimal when the asset price exceeds , where the intrinsic value of the option becomes greater than its holding value. An American capped call option is automatically exercised if the underlying asset rises above a predetermined price, which is called the "cap price".
The lower graph locates the cap price that maximizes the payoff function of an American capped call and shows the approximations that correspond to the top table. The blue curve represents the payoff function of the American capped call depending on the cap price . The horizontal black line is the reference for the American call, according to Mathematica's built-in function FinancialDerivative.
Adjust the "zoom out" level accordingly, in order to achieve the best visualization.[less]
The boundary derives from the maximized payoff function  for an American capped call.
The table shows, at :
• The American call according to the "lower-upper bound approximation (LUBA)" , 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)" , an empirical approximation method deriving from the option's lower bound (LB),
• The American call upper bound (UB) , deriving from Kim's integral equation  where is replaced by ,
• The American call lower bound (LB) , deriving from the maximized payoff function of an American capped call.
M. Broadie and J. Detemple  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 . 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  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.
 I. Kim, “The Analytic Valuation of American Options,” Review of Financial Studies, 3(3), 1990 pp. 547–572.
 M. Broadie and J. Detemple, "American Capped Call Options on Dividend Paying Assets," 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," Review of Financial Studies, 9(4), 1996 pp. 1211–1250.