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?


1 Answer 1


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 $S_c(t)$).

  • $\begingroup$ it is the $f$ that is problematic , where did that come from? I am familiar with the rest of the formula. Anywhere to derive it from or should I just take it as given? $\endgroup$
    – k b
    Commented Apr 1, 2017 at 9:43
  • $\begingroup$ What do you mean where did it come from? First passage time problems have been studied for various processes, including BM, GBM, etc. A general approach to solving them is through the use of the Kolmogorov Backward Equation. $\endgroup$
    – Alex C
    Commented Apr 1, 2017 at 15:13
  • $\begingroup$ would like to know how it is derived , but i guess i should just take it as given, ? are there any alternative representation of this formula? $\endgroup$
    – k b
    Commented Apr 1, 2017 at 16:35
  • $\begingroup$ In a few simple cases there may be a formula for first passage times, but the usual case will give a PDE which will not have an analytical solution. So the formula above looks nice, but the "big mess" is hidden inside the computation of f, as you can imagine. To say "it is easy to solve once we have the 1st passage time f" would be a bit misleading in my opinion. $\endgroup$
    – Alex C
    Commented Apr 1, 2017 at 16:54
  • $\begingroup$ How would one go about to derive this formula? $\endgroup$
    – k b
    Commented Apr 1, 2017 at 22:19

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