Choosing an Exit Threshold with Brownian Motion and Bankruptcy Risk

Summary

The document frames an exit decision for a share whose price follows a driftless Brownian motion and which may become worthless at an independent exponential bankruptcy time. A trader chooses a price threshold and sells at the first time the share exceeds it, receiving the discounted threshold price only if the crossing occurs before bankruptcy. The proposed analysis relates this event to the running maximum of Brownian motion and invokes the reflection principle to express a crossing probability in terms of the normal distribution.

The answer then sets up expected discounted proceeds by integrating over the bankruptcy time and the threshold hitting time, suggesting that the threshold should maximize this expected value. It does not carry out that maximization or give an optimal threshold. The displayed probability uses a fixed time horizon, whereas the bankruptcy horizon is random; the subsequent integral is the relevant direction for incorporating that uncertainty. The setup also assumes the stated price process and does not address dividends, drift, transaction costs, or other market features.

Key ideas

• The proposed strategy sells when a driftless Brownian price first crosses a chosen threshold.
• The share pays the threshold price only if the crossing precedes an exponentially distributed bankruptcy time.
• The reflection principle connects a fixed-horizon crossing probability to the distribution of Brownian motion's running maximum.
• Expected discounted proceeds can be expressed by integrating over bankruptcy and threshold-crossing times.
• The document sets up an optimization problem but does not derive the maximizing threshold.

Tags

# Optimal Choice of exceeding time


# Optimal Choice of exceeding time












Suppose you hold a share from company $Z$ whose vaue at time $t$ is $S_0+\sigma B_t$ where $B_t$ is Brownian Motion and $\sigma$ denotes some volatility. Now lets assume that company $Z$ may go bankrupt at some expoentially-distributed random variabl $T$ with mean $1/\lambda$. Now you plan to sell your share at the first time $H$ that the price exceeds $a$, i.e $H=\inf\{t: S_0+\sigma B_t>a\}$. If $H<T$ the vaue to you is $a\exp(-rH)$, otherwise it is worth nothing.

Do you have any idea how I can derive the optimal choice of $a$ ?

My intuitive way to solve this exercise is to first check what is the probability that $H<T$, i.e $P(H<T)$ Now I think I can use the Reflection Principle, so I define $S_t=\sup (S_0+\sigma B_t)$, then the Principle states that $P(S_T>a)=P(H<T)$. I think the solution to the problem is to find the maximal $a$ such that $P(S_T>a)$, but I do not know how to compute $P(S_T>a)$.

## Answer by jensa (score 2, accepted)

https://quant.stackexchange.com/a/9545

It's been quite a while since I did this stuff, but I'll add my input. Please correct me if appropriate.

$\{H < T\} = \{ \sup_{0\leq s \leq T} (S_{0} + \sigma B_{s}) > a \} = \{\sup_{0 \leq s \leq T} B_s > \frac{a-S_0}{\sigma}\}$.

Set $\mu := \frac{a-S_0}{\sigma}$ and $M_{T} := \sup_{0 \leq s \leq T} B_{s}$.

Then, $P(\{H < T\} = P(\{M_T > \mu \}) = 2\left(1 - \Phi\left(\frac{\mu}{\sqrt{T}}\right)\right)$.

Seek to maximize $V(a) := E\{ae^{-rH}1_{\{H < T\}}\} = a \int_{0}^{\infty}\lambda e^{-\lambda x} \int_{0}^{x}e^{-ry} \frac{d}{d\xi}\left(2 \Phi\left(\frac{\mu}{\sqrt{\xi}}\right)-1\right)(y) \, dy \, dx$.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.