Unlike European options, American options can be exercised at any time up to and including the expiration date. This early exercise feature makes American options more difficult to price than European ones. In this chapter, we present risk-neutral pricing of American options.

**Definition 44.1** (Reward Function). A reward function is any function *g* : (0, ∞) × [0, *T*] → **R** that is continuous and satisfies the linear growth condition

for some constants *K*_{1} and *K*_{2}.

**Definition 44.2** (American Option). In a Black-Scholes market {(*X*^{(0)}_{t}, *X*_{t}) : *t* ≥ 0}, an American option with reward function *g* and maturity *T* is a derivative that pays the amount *g*(*X*_{t}, *t*) when exercised at any time *t* [0, *T*].

The reward function for an American call option with strike *K* is *g*(*X*_{t}, *t*) = (*X*_{t} − *K*)^{+}. The reward function for an American put option with strike *K* is *g*(*X*_{t}, *t*) = (*K* − *X*_{t})^{+}.

**Definition 44.3** (Consumption Strategy). Let {(*X*^{(0)}_{t}, *X*_{t}) : *t* ≥ 0} be the Black-Scholes market given in Definition 42.2. A consumption strategy is a stochastic process {*A*_{t} : 0 ≤ *t* ≤ *T*} satisfying the following conditions:

(a) It is adapted to the underlying filtration {_{t} : 0 ≤ *t* ≤

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