American Option Prices, Exercise, and Martingale Behavior

# Hedging and (non-) martingality of American option


# Hedging and (non-) martingality of American option












The American option remains a tricky thing to understand (at least for me), for I was reading this excellent question and corresponding answer by @Kevin. I'd like to understand the issue from a more practitioner oriented point of view, with as little maths as possible.

Let's suppose then we know that the stock price, which is a tradable, evolves according to a simple Black-Scholes model under the pricing measure $\mathbb Q$: $$ dS = r S dt + \sigma S dW $$ where $r,\sigma$ are all constants. Just to be clear, let's assume there is no model misspecification.

Suppose I sold $U(S_t,K,t,T)$ which denotes the correct price of an American put option. According to the QFSE question and answer I linked to above, $U$ is not a martingale.

However, since the only market risk factor is $S$ which is tradable, I can delta hedge this option perfectly (assuming no frictions etc), and if the buyer behaves rationally/optimally then even though $U$ is not a martingale there is no arbitrage.

Is my understanding correct?

The second question I have, and where I now have the most doubt is: as long as $S$ is in the continuation region (eg not deep in the money) then $U$ satisfies the BS PDE and so in this case, even though in general $U$ is not a martingale, we can write $$ E^\mathbb Q [ dU ] = r U dt $$ where $dU$ is the infinitesimal change in the option price.

Is this right?

## Answer by Andrea (score 2, accepted)

https://quant.stackexchange.com/a/81673

From the linked question

$$U_t=\max\left\{\frac{\xi_t}{B_t},\mathbb{E}^\mathbb{Q}[U_{t+1}|\mathcal{F}_t]\right\}\geq \mathbb{E}^\mathbb{Q}[U_{t+1}|\mathcal{F}_t]$$

And so $U_t$ (the price) is a supermartingale.

But the value of your portfolio is something else: as soon as you exercise, you jump on a completely different portfolio which will evolve as a martingale (it depends on settlement, cash or physical).

Price and Value of American Options are not the same thing because tomorrow option gives you less rights than today's, that is where the "super" comes from.

The price of an American option is a supermartingale, the value (including rational exercise) is a martingale like everything else.

For European Options, they are the same thing, but here, the value depends on the start date, since optimal exercise could have already happened in the past.