Finance & Options

Lesson 11 of 12

American Options and Early Exercise

Exercising early is an optimal stopping problem: a deep in-the-money put is sometimes worth exercising before expiry, and a call on a non-dividend stock never is.

The right to stop early

An American option can be exercised at any time up to expiry, and a European one only at expiry. The extra right can't hurt, so an American option is worth at least the European one and at least its intrinsic value g(S)g(S), the payoff from exercising now, such as (K−S)+(K - S)^+ for a put.

On a binomial tree the rule from risk-neutral pricing (7.0.1) changes in one place. At each node the holder compares cashing in with holding on, and the option is worth the larger:

Vn(s)=max⁡{g(s), q Vn+1(us)+(1−q) Vn+1(ds)1+r},q=1+r−du−d.V_n(s) = \max\Big\{ g(s),\ \frac{q\,V_{n+1}(us) + (1-q)\,V_{n+1}(ds)}{1+r} \Big\}, \qquad q = \frac{1+r-d}{u-d}.

Here rr is the interest rate per step, so 1+r1+r plays the role of erΔte^{r\Delta t}. You fill the tree backwards from expiry, where VN(s)=g(s)V_N(s) = g(s).

The holder's choice is a stopping time: a rule that decides whether to exercise using only the prices seen so far. The American price is the value of the best such rule,

V0=max⁡τ≤NEQ[(1+r)−τg(Sτ)],V_0 = \max_{\tau \le N} E^Q\big[(1+r)^{-\tau} g(S_\tau)\big],

and the backward max above finds it without listing every rule. The seller charges this amount and hedges as if the holder will play perfectly. If the holder exercises at a worse time, the hedge is left with money to spare.

The secretary problem and deciding when to quit a game have the same shape: at each step, is what I get now worth more than what I expect from waiting?