Optimal Stopping in the Stock Market When the Future is Discounted

Abstract

In the random walk stock market model, a stock is purchased at price $x$ and is sold at time $t$ for the price $x + S_t$ where $S_t = \sum^t_0 X_i, X_i$ is the price change during the $i$th epoch, and $X_1, X_2, \cdots$ are i.i.d. random variables with $\mu = E(X_1) > 0$ and finite $\sigma^2 = E(X^2_1) - \mu^2 > 0$. Discounting the future by a factor of $\gamma$ per epoch, $0 < \gamma < 1$, a selling or stopping policy $t$ has expected payoff or utility $u(t) = E\{\gamma^t(x + S_t)\}$. This article determines second order asymptotic properties of the optimal selling policy $s$, the first passage time of $S_n$ across a straight line boundary $c$, whose utility is equal to the value $V(x) = \sup_tu(t)$ of the stock purchased at price $x$. Specifically, as $\gamma \rightarrow 1$, renewal theory is utilized to evaluate the limiting distribution of $s, E(s), V(x)$, and the first passage boundary $c$ up to second order terms.