American Option Value and the Early Exercise Rule

Summary

The note explains the value of an American option in a binomial model as the larger of its immediate exercise payoff and its discounted expected value from holding the option. The continuation value uses risk-neutral probabilities for the next-period option values. Comparing the two alternatives captures the holder’s ability to exercise at each step.

The accompanying explanation says that choosing less than this maximum would leave value unused and give the option writer an advantage relative to optimal exercise. It also describes exercising at the first time the immediate payoff is at least as large as the value of waiting. This is a compact intuition rather than a worked arbitrage construction: it does not give the requested numerical example, and its claim about the writer’s gain is not developed formally. The rule is presented within the binomial pricing framework, so applying it requires a correctly specified model and risk-neutral continuation values.

Key ideas

• An American option’s value is the greater of its exercise payoff and its continuation value.
• The continuation value is the discounted risk-neutral expectation of next-period option values.
• Early exercise is optimal when the immediate payoff is at least as high as the value of waiting.
• The explanation gives intuition but does not provide the requested explicit arbitrage example.

Tags

# Intuition behind American Option pricing


# Intuition behind American Option pricing












The price of an American option is given by $$V_n = \max\left(G_n,\frac{pV_{n +1}H^d + qV_{n + 1}H^u}{1 + r}\right)$$ where p, q are the risk neutral probabilities.

I have two questions:

- How can one intuitively see that this must be the formula to avoid arbitrage? If possible cite a trivial example showing arbitrage if one does not take the maximum of these two values.

- How to intuitively see that the ideal time to exercise the option is $\min\{n: V_n = G_n\}$

Thanks.

## Answer by emcor (score 5)

https://quant.stackexchange.com/a/14551

The model here is the binomial option pricing model, so the second term in the brackets represents the expected future value of the option (under riskneutral probabilities).

- The aim of the option holder is always to maximize the value of his option. He can at any point sell the option at the fair market price $E(V_{n+1})$ or exercise it to get $G_n$. So if he would not choose the maximum of the two, the option seller would have an implicit gain by not having the American option optimally exercised and hence arbitrage.

- The optimal time to exercise the option is when the future value is not higher than the current payoff (so there is no value in waiting further), so you exercise soon as this is the case. Note that $V_n=G_n$ is when $E(V_{n+1})\leq G_n$.

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