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+---
+title: Unwind Value
+author: Keith A. Lewis
+institution: KALX, LLC
+email: [email protected]
+---
+
+Let $S_t = S_0\exp(\mu t + \sigma B_t - \sigma^2t/2)$ be geometric Brownian motion
+and $\overline{S}_t = \max_{0\le u\le t} S_u$ be its running maximum.
+
+Let $t_0,\ldots,t_n$ be permitted _unwind_ times, $T$ a _horizon_ time, and $L$ a draw-down limit.
+Define the unwind (stopping) time $\tau = \min\{t_j\mid \overline{S}_{t_j} - S_{t_j} > L\}$
+and $\tau\wedge T = \min\{\tau, T\}$.
+
+Calculate $P(\tau < T)$ (unwind before horizon), $fE[e^{-r(\tau\wedge T)}(\tau\wedge T)]$
+(expected discounted time to the earlier of unwind or horizon) where $f$ is a fee and $r$ is the risk-free rate.
+
+Input variables: $S_0$, $r$, $\sigma$, $(t_j)$, $T$, $L$, $IR$ (information ratio)
+
+Dependent variables: $\mu = IR \sigma$.
+
+Typically the $t_j$ are end of month dates and $T$ is 1 or 2 years.
+
+Create a table for $IR\in\{0.5, 1, 1.5\}$ and $L = \{5, 7.5, 10\}$ with $S_0 = 100$,
+$\sigma = 5\%$, $r = 4.5\%$, and $f = 1\%$.