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The price of an American put option can be written as the following optimal stopping problem: $V(0) = \mathop {\sup }\limits_{\tau \in \mathcal{T}} {\mathbb{E}^\mathbb{Q}}\left[ {{e^{ - r\tau }}\max [K - S(\tau ),0]} \right]$, where $\mathcal{T}$ is the set of all stopping (exercise) times. Assume no-dividend case.

If I look at ${\mathbb{E}^\mathbb{Q}}\left[ {{e^{ - r\tau }}\max [K - S(\tau ),0]} \right]$, then this is a price of a Black-Scholes European option maturing at $\tau$.

To solve the American option pricing problem using a common sense logic - why can't I just compute Black-Scholes price for each $\tau$-maturity option and then take a maximum price?

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  • $\begingroup$ The various options maturing at different $\tau$ are not independent of each other: if $S(\tau_1)$ is large then $S(\tau_2)$ for $\tau_2\approx \tau_1$ is probably also large and vice versa. You cannot treat the problem as choosing the best among independent assets (options). Instead the $S(\tau)$ belong to a single random trajectory of stock price. $\endgroup$ – noob2 Mar 28 at 18:33
  • $\begingroup$ Thanks @noob2, that I perfectly understand. The true solution is the "recursive" dynamic programming. But I don't see this in the general formula. Which element in the general formula have I overlooked that would bring in the dependency? $\endgroup$ – user2743931 Mar 28 at 18:46
  • $\begingroup$ All I am saying is you left out the SDE $\frac{dS}{S}=r dt + \sigma dW$ that ties the $S(\cdot)$ values togther. Which is obvious, I know. $\endgroup$ – noob2 Mar 28 at 23:08
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You could but there are difficulties associated with this approach. The main one is that $\tau$ is stochastic, ie it is different for different paths of $S$, so the standard Black-Scholes formula does not apply. For example some $\tau$s you need to check are of the form $\tau =\inf\{t : S(t) <B\}$ in which case you need to value a barrier option with barrier $B$, and other choices for $\tau$ that you need to check that are even more complicated

Having said that, there are methods for valuing American options that are in spirit along the lines to what you are saying, where one looks for the maximum value over all stopping times, suitably parameterized. The calculation for each particular stopping time is more involved that the Black-Scholes formula, of course, as I explained. Recent work along these lines is, for example, this

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