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Valuing a Perpetual American Put with a Brownian Hitting-Time Model

Article Quant Q&A · Author: user40884
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Summary

The document asks how to value a perpetual American put in the Black–Scholes–Merton framework when exercise occurs the first time the underlying reaches a chosen threshold below the strike. It sets up the discounted payoff and attempts to transform its expectation using an exponential Brownian likelihood ratio, then asks how to interpret the resulting hitting probability and determine the optimal threshold.

The discussion highlights that evaluating a perpetual option requires the probability of ever reaching the exercise boundary under the appropriate measure, with discounting handled consistently. The proposed change-of-measure argument and probability expression are not resolved in the document, so it does not establish a valid valuation formula or optimal boundary. Its main value is as a setup for studying first-passage times, measure changes, and optimal exercise; the assumptions are limited to a classical perpetual put in the stated BSM setting.

Key ideas

  • The payoff is discounted until the underlying first reaches an exercise threshold.
  • The threshold must be chosen to maximize the expected payoff.
  • A Brownian measure change is proposed to relate the expectation to a hitting probability.
  • The document leaves the hitting-probability calculation and optimal threshold unresolved.

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# American Perpetual Put Option


# American Perpetual Put Option












I want to compute the expected payoff of a (classical) perpetual American put option in the Black-Scholes-Merton (BSM) framework with an optimal strategy of exercising the option at time $\tau=\inf\{t:S_{t}=S^{*}\}$. In here $K,r>0,\sigma>0$ and $W_{t}$ are the strike price, risk-free rate, implied volatility and a Brownian motion according to the BSM framework. $S^{*}<K$ has to be determined by maximizing the expected payoff using this optimal strategy. The expected payoff is as following $$\mathbb{E}[e^{-r\tau}(K-S^{*})^{+}1_{\tau<\infty}]$$ My attempt was as following:

$e^{-r\tau}=\frac{S_{0}}{S^{*}}e^{\sigma W_{t}-\frac{\sigma^{2}t}{2}}$ and $(K-S^{*})^{+}=(K-S^{*})$, thus $\mathbb{E}[e^{-r\tau}(K-S^{*})^{+}1_{\tau<\infty}]=(K-S^{*})\frac{S_{0}}{S^{*}}\mathbb{E}[e^{\sigma W_{t}-\frac{\sigma^{2}t}{2}}1_{\tau<\infty}]$. Now, we have a Radon-Nikodym derivative changing or tilting the measure with $\sigma$ according to the Cameron-Martin theorem $$\frac{dP_{\sigma}}{dP}=Z_{\sigma}(t)=e^{\sigma W_{t}-\frac{\sigma^{2}t}{2}}$$ If we interpret the function $1_{\tau<\infty}$ as $t$ will ever hit $\tau$ we can write the expected payoff as following $$P_{\sigma}(\sigma W_{t}-\frac{\sigma^{2}t}{2}=\log(\frac{S^{*}}{S_{0}}))$$ wherein standard Brownian motion has the same law as Brownian motion with a drift $\sigma$ under the risk-neutral measure. The following equation is given $$P(B_{t}+\gamma t=\eta)=e^{2\eta\gamma}$$ and given the last equation the expected pay off should be set equal to $\frac{S^{*}}{S_{0}}^{2r/\sigma^{2}}$.

However, I fail to interpret the last step correctly and cannot find the right solution. Can you help me with the interpretation and equation?

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