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Why American Option Exercise Cannot Be Priced by Maximizing Black-Scholes Values

Article Quant Q&A · Author: user2743931
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Summary

The note asks whether an American put can be valued by calculating Black-Scholes prices for options with different maturities and selecting the largest. The response explains the key obstacle: an exercise time in the optimal stopping formulation is generally stochastic and depends on the path followed by the underlying price. A standard Black-Scholes European formula does not directly price that path-dependent exercise rule.

For example, exercising when the asset first crosses a barrier creates a stopping time tied to a barrier event, requiring valuation methods that account for that behavior. American option methods can still search for the best exercise policy over a parameterized set of stopping rules, but each candidate policy requires more involved valuation than a plain European formula. The discussion assumes a non-dividend-paying underlying and offers no specific numerical method or comparison of implementation approaches.

Key ideas

  • American option value is formulated as the maximum expected discounted payoff over admissible exercise times.
  • Exercise times can depend on the realized path of the underlying price, making them stochastic.
  • A European Black-Scholes price for a fixed maturity does not capture a path-dependent exercise policy.
  • American option valuation can optimize over parameterized stopping rules, but evaluating each rule may require more complex methods.

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Full text
# American options and stopping times


# American options and stopping times












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?

## Answer by piterbarg (score 8, accepted)

https://quant.stackexchange.com/a/63055

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

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.