A Fast and Accurate Numerical Approach for Pricing American-Style Power Options

In this paper, we present a fast and accurate numerical approach applied to specific American-style derivatives, namely American power call and put options, whose main feature is that the underlying asset is raised to a power. The study is set in the Black–Scholes framework, and we consider continuously paying dividends assets and arbitrary positive values for the power. It is important to note that although a log-normal process raised to a power is again log-normal, the resulting change in variables may lead to a negative dividend rate, and this case remains largely understudied in the literature. We derive closed-form formulas for the perpetual options’ optimal boundaries and for the fair prices. For finite maturities, we approximate the optimal boundary using some first-hitting properties of the Brownian motion. As a consequence, we obtain the option price quickly and with relatively high accuracy—the error is at the third decimal position. We further provide a comprehensive analysis of the impact of the parameters on the options’ value, and discuss ordinary European and American capped options. Various numerical examples are provided.