American Option Value as the Maximum of Exercise and Continuation Value
Summary
The note asks why an American derivative's value one period before maturity is the maximum of immediate exercise value and discounted expected maturity payoff under the risk-neutral measure. It assumes an arbitrage-free, complete market with a deterministic bond as numeraire and a unique equivalent martingale measure.
The answer explains the holder's choice: exercise now to receive the current payoff, or keep the option and receive its maturity payoff later. The continuation value is the risk-neutral conditional expectation discounted by the period's interest factor. Taking the larger value reflects the right to choose between those alternatives. This explanation illustrates the one-step backward induction principle, but it does not derive the full multi-period exercise problem or address practical model estimation and market frictions.
Key ideas
- An American option can be exercised before maturity, so its holder compares immediate exercise with waiting.
- The continuation value is the discounted risk-neutral conditional expectation of the later payoff.
- The one-period value is the larger of exercise value and continuation value.
- The explanation assumes an arbitrage-free complete market and deterministic financing as stated in the question.
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# Pricing of American Deriviatives
# Pricing of American Deriviatives
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 by Gordon (score 2, accepted)
https://quant.stackexchange.com/a/22221
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*}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.