Take the 2-minute tour ×
Quantitative Finance Stack Exchange is a question and answer site for finance professionals and academics. It's 100% free, no registration required.

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

share|improve this question

1 Answer 1

The model here is the binomial option pricing model, so the second term in the brackets is 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 option 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 $E(V_{n+1})\leq G_n$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.