Pricing of American Deriviatives

Question

Reading the book by Andrea Pascucci "PDE and Martingale Method in Option Pricing" I am struggling with a very simple issue. Suppose we want to find the price of an American derivative $X$ in an arbitrage-free and complete market. Let $\mathbb{Q}$ be then the (unique) equivalent martingale measure with numeraire $B$ (the deterministic bond) and let $X_t$ be the value of the American derivative a time $t$ ($X_t$ is thus, in the most general formulation, a $\mathcal{F}_t$-adapted stochastic process). Let $H$ be the no-arbitrage price of $X$. Clearly it must be $H_T=X_T$. At time $t=T-1$ the price is determined as

$$ H_{T-1} = \max\left(X_{T-1},\frac{1}{1+r}\,\mathbb{E}^{\mathbb{Q}}\left[ X_T\mid\mathcal{F}_{T-1}\right]\right). \quad(1) $$

I clearly understand that $\frac{1}{1+r}\,\mathbb{E}^{\mathbb{Q}}\left[ X_T\mid\mathcal{F}_{T-1}\right]$ is the no-arbitrage price at time $T-1$ of an European derivative with maturity $T$ and payoff $X_T$, but which is the no-arbitrage argument behind equation (1) ?

Answer 1

For an American option, you have the right to exercise at any intermediate time. Then, at time $T-1$, if you exercise your option, you obtain the payoff $X_{T-1}$. However, if you wait to exercise at the maturity $T$, your value is $\frac{1}{1+r}\mathbb{E}^Q\left(X_T \mid \mathscr{F}_{T-1} \right)$. Your option value at time $T-1$ is the maximum of these two values, that is, \begin{align*} \max\left(X_{T-1}, \, \frac{1}{1+r}\mathbb{E}^Q\left(X_T \mid \mathscr{F}_{T-1} \right) \right). \end{align*}