Analytic solutions for pricing American style options

While many studies focus on determining an optimal exercise boundary or exploring the early exercise premium between American and European options, this work instead considers a scenario in which the option writer consumes a portion of the wealth process when the holder fails to exercise optimally. Under this framework, we propose a modified Black–Scholes partial differential equation for pricing American options, enabling their values to be expressed through European-like closed-form solutions. Numerical comparisons between our method and several established approaches indicate that the resulting option values are closely aligned. This is particularly evident when the option initially enters the exercise region and subsequently returns to the continuation region. In contrast to existing approaches, our method not only yields a closed-form solution but also offers a more straightforward and transparent way to price American options.