Monte Carlo Valuation of American Options

Abstract

An American option is a contract giving its holder the right to buy (call option) or sell (put option) one unit of an underlying security of value S for a prearranged amount. This right can be exercised at any time prior to the expiration date T. In contrast, a European option can be exercised only at the expiry. Define the amount paid to the holder of an American option at the moment of exercise, the payoff, as Ψ (S, t) ≥ 0; a standard contract is a put option where Ψ = max(K − S, 0) and K is the strike price. The discounted exercise value of the option is Z(t) = Ψ (t) / B(t), where B(t) is the value at time t of $1 invested in a riskless money market account at t = 0. American option valuation can be characterised as an optimal stopping problem. The time 0 value of an American option is given by $$V(0) = \mathop {\sup }\limits_{0\tau T} E\left[ {Z\left( \tau \right)} \right]$$ (1) where the supremum is taken over all the possible stopping times τ less than the expiration date T, and the expectation is taken over the risk-neutral probability density. This is the primal problem.