| title |
author |
institution |
email |
Unwind Value |
Keith A. Lewis |
KALX, LLC |
|
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 $f = 1%$.