Utility Maximization When Shorting American Options

Abstract

An investor initially shorts a divisible American option $f$ and dynamically trades stock $S$ to maximize her expected utility. The investor faces the uncertainty of the exercise time of $f$, yet by observing the exercise time she would adjust her dynamic trading strategy accordingly. We thus investigate the robust utility maximization problem $V(x)=\sup_{(H,c)}\inf_{\eta}\mathbb{E}[U(x+H\cdot S-c(\eta(f)-p))]$, where $H$ is the dynamic trading strategy for $S$, $c$ represents the amount of $f$ the investor initially shorts, $\eta$ is the liquidation strategy for $f$, and $p$ is the initial price of $f$. We mainly consider two cases: In the first case, the investor shorts a fixed amount of $f$; i.e., without loss of generality, $c=1$ and $p=0$; in the second case, she statically trades $f$; i.e., $c$ can be any nonnegative number. We first show that in both cases $V(x)=\sup_{(H,c)}\inf_{\tau}\mathbb{E}[U(x+H\cdot S-c(f_\tau-p))]=\inf_{\rho}\sup_{(H,c)}$ $\mathbb{E}[U(x+H\cdot S-c(f_\rho-p))]$, where $\tau$ is a pure stopping time, $\rho$ is a randomized stopping time, and $H$ satisfies a certain nonanticipation condition. Then in the first case (i.e., $c=1$), we show that when $U$ is exponential, $V(x)=\inf_{\tau}\sup_{H}\mathbb{E}[U(x+H\cdot S-f_\tau)]$; for general utility, this equality may fail, yet it can be recovered if we in addition let $\tau$ be adapted to $H$ in a certain sense. Finally, in the second case ($c\in[0,\infty)$) we obtain a duality result for the robust utility maximization on an enlarged space.