First-Passage Probabilities and the American Put Exercise Boundary
Summary
The document interprets an American put valuation expression that integrates discounted exercise payoffs against the probability density of first reaching an exercise boundary. For a proposed exercise curve, the discounted payoff at each time is weighted by the probability that the stock first hits that curve then; integrating across times gives the value of that stopping strategy.
The critical exercise boundary is described as the curve that maximizes this value across candidate strategies, with the resulting maximum identified as the put price. This explains the formula as an optimization characterization of early exercise. The account does not provide a procedure for finding the optimal boundary, so the expression is more definitional than computational. It also gives no numerical example or evidence beyond the conceptual interpretation.
Key ideas
- A candidate exercise curve defines a first-hit stopping strategy.
- The discounted payoff is weighted by the first-passage probability at each time.
- Integrating those contributions gives the value of the candidate strategy.
- The critical boundary maximizes value over candidate exercise curves.
- The characterization does not itself explain how to compute the optimal boundary.
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# First passage probability in american option pricing
# First passage probability in american option pricing
In an article i recently read (The American Put Option and Its Critical Stock Price by David S. Bunch and Herb Johnson link) the authors presented this formula as something very general and as common knowledge
> $$P = \mathop {\max }\limits_{{S_c}} \int\limits_0^T {{e^{ - rt}}(X - S_c)} fdt,\quad (S > {S_C})$$ where $P, r, T, X,$ and $Sc$ are the American put price, risk-free rate, time to maturity, exercise price, and critical stock price, respectively.Let S be the current stock price (at time $ t= 0).$ $f$, is the first-passage probability,
However i cant recall that i have seen this formula AND $f$ in the same formula, what am I missing? Where did this formula come from?
## Answer by Alex C (score 2)
https://quant.stackexchange.com/a/33397
This is more or less the definition of the Critical Exercise Boundary in its relation to the put price.
Assume $S_0(t)$ is an arbitrary exercise curve from 0 to T. Then $e^{-rt}(X-S_0)$ is the discounted payoff of exercising at time t. This is then multiplied by the probability of reaching $S_0$ for the first time at time t, which is f, to get an expected value. We integrate over all possible times from 0 to T to get the overall value of this exercise strategy.
We can try this again for another hypothesized exercise strategy $S_1(t)$, then $S_2(t)$ and so on, each time getting a different value of the integral. The best curve we call $S_c(t)$, i.e the one which produces the largest value of this integral. This value is also the put price P.
In other words this equation describes the optimization process through which the critical exercise boundary is obtained: you have to select a curve, such that if you exercise the first time the curve is hit, you get the best possible value. (But of course it is more definitional than computational: it doesn't tell you how to find such an $S_c(t)$).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.