Curtailing the Range for Lattice and Grid Methods

Abstract

Numerical option pricing methods discretize a continuous-time continuous-state state space and approximate the solution to the continuous problem with the results of local computations on all of the nodes of the discrete lattice. This procedure converges to the correct solution as the time and price intervals go to zero, but at the cost of larger and larger numbers of calculations. Many of the calculations on a fine lattice are irrelevant to valuing the derivative. They may occur at stock prices that have virtually no probability of being reached. Or the stock price may be reachable from the initial price, but be so far away from the strike price that the option is either (almost) certain to be out of the money at maturity, meaning it is worthless now, or (almost) certain to be exercised at maturity, meaning it can be priced as a forward now, with no need for any further calculations. Eliminating the superfluous calculations at these nodes can greatly improve the performance of a lattice valuation technique, and the possible increase in computation speed becomes much greater as the number of stochastic variables in the lattice increases. Here, Andricopoulos et al., show how to curtail the price range for a numerical valuation technique and demonstrate the substantial improvement in speed that it can produce.