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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:

  1. 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.

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

Thanks.

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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).

  1. 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.

  2. 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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