以最高价卖出股票
Selling a stock at the ultimate maximum
摘要：
假设股票价格$Z=(Z_t)_{0\leq t\leq T}$服从具有漂移$μ\in\mathbb{R}$和波动率$σ>0$的几何布朗运动，并使$M_t=\max_{0\leq s\leq t}Z_s$为$t\in[0,T]$，我们考虑最优预测问题\[V_1=\inf_{0\leqτ\leq T}\mathsf{E}\biggl(\frac{M_T}{Z_τ}\biggr)\quadand\quad V_2=\sup_{0\leqτ\leq T}\mathsf{E}\biggl(\frac{Z_τ}{M_T}\biggr),\]，其中下确界和上确界取$Z$的所有停时$τ$。我们表明，在第一个问题中，以下策略是最优的：如果$μ\leq0$立即停止;如果$μ\in (0,σ^2)$一旦$M_t/Z_t$达到指定的时间函数就停止;如果$μ\geqσ^2$等到最后一次$T$。通过对比，我们证明了在第二个问题中以下策略是最优的：如果$μ\leqσ^2/2$立即停止，如果$μ>σ^2/2$等待，直到最后一次$T$。这两种解决方案都支持并强化了人们普遍持有的金融观点，即"一个人应该卖掉坏股票，保留好股票。证明的方法利用抛物自由边界问题和当地的时间-空间微积分技术。由此产生的不平等现象本身就很不寻常和有趣，因为它们涉及未来，因此具有预测性。
英文摘要：
Assuming that the stock price $Z=(Z_t)_{0\leq t\leq T}$ follows a geometric Brownian motion with drift $μ\in\mathbb{R}$ and volatility $σ>0$, and letting $M_t=\max_{0\leq s\leq t}Z_s$ for $t\in[0,T]$, we consider the optimal prediction problems \[V_1=\inf_{0\leqτ\leq T}\mathsf{E}\biggl(\frac{M_T}{Z_τ}\biggr)\quadand\quad V_2=\sup_{0\leqτ\leq T}\mathsf{E}\biggl(\frac{Z_τ}{M_T}\biggr),\] where the infimum and supremum are taken over all stopping times $τ$ of $Z$. We show that the following strategy is optimal in the first problem: if $μ\leq0$ stop immediately; if $μ\in (0,σ^2)$ stop as soon as $M_t/Z_t$ hits a specified function of time; and if $μ\geqσ^2$ wait until the final time $T$. By contrast we show that the following strategy is optimal in the second problem: if $μ\leqσ^2/2$ stop immediately, and if $μ>σ^2/2$ wait until the final time $T$. Both solutions support and reinforce the widely held financial view that ``one should sell bad stocks and keep good ones.'' The method of proof makes use of parabolic free-boundary problems and local time--space calculus techniques. The resulting inequalities are unusual and interesting in their own right as they involve the future and as such have a predictive element.
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