American Options (Early Exercise)

American Options and Early Exercise

American options can be exercised at any time up to expiry, unlike European options which can only be exercised at expiry.

Why Early Exercise Matters

For American puts, early exercise can be optimal when the option is deep in-the-money. Exercising early allows you to:

• Receive the intrinsic value immediately
• Invest the proceeds at the risk-free rate

For American calls on non-dividend-paying stocks, early exercise is never optimal (you'd throw away the time value).

Pricing with the Binomial Tree

The binomial tree handles American options naturally. At each node, compare:

1. Continuation value: the discounted expected value of holding
2. Intrinsic value: the payoff from exercising now

$V_j = \max(\text{intrinsic}, e^{-r\Delta t}(p \cdot V_j^{\text{up}} + (1-p) \cdot V_j^{\text{down}}))$

For a put, intrinsic value = max(K - S, 0).

The option holder will exercise early if the intrinsic value exceeds the continuation value.

Early Exercise Premium

The early exercise premium is the difference between the American and European prices:

$\text{Early Exercise Premium} = \text{American Price} - \text{European Price}$

This is always ≥ 0 (American options are worth at least as much as European).