Computational Finance – Pricing The American Option

Recall that the American option has strike K and maturity T and gives the holder the right to exercise at any time in [0, T ]. The American option is not straightforward to price in the Monte Carlo framework that we have discussed. The reason is that the derivative cash flow function f(S, t) is not well defined. The problem is that we cannot compute the derivative cash flow until we know how the American option is going to be exercised. If, on the other hand, we knew the optimal exercise strategy, then it would be a straightforward task, using Monte Carlo, to obtain the expected discounted cashflows, and hence the price. Lets first define what an exercise strategy is. Denote an exercise strategy by π(S, t), which is a binary valued function of two variables, the price and the time. The exercise strategy π(S, t) specifies whether to exercise or not at the state (S, t),