A Fully Coupled Solution Algorithm for Pricing Options with Complex Barrier Structures

Abstract

The Black–Scholes option pricing equation was a major intellectual achievement, both for academics and for practitioners, but it rests upon a variety of restrictive conditions on the option’s payoff. More complicated structures are priced with numerical methods based on discretizing the state space in either a lattice framework or a discrete approximation to the fundamental partial differential equation (PDE). Both approaches use a fixed step size in the time dimension and converge asymptotically to the continuous-time solution as the step size goes to zero. But for many payoff structures, “asymptotically” entails an enormous number of calculations to achieve acceptable accuracy. Many techniques have been advanced over the years to speed up convergence for particular cases. The main problems with the standard numerical methods arise because of discontinuities in the option’s payoff structure, such as the abrupt change in value when a knock-out option hits its barrier, that interact with the discreteness of the state space approximation.

This article presents a powerful new approach that allows much greater flexibility in constructing the approximating lattice in time and price space, and therefore a potentially huge improvement in computational efficiency. The trick is to abandon the fixed time step and to treat the PDE as a fully coupled system that is solved simultaneously in both the price and time dimensions. As Zhu and de Hoog show, even with fixed time and price steps, this allows the approximation to retain much greater accuracy with larger step sizes than current methods. But the new approach allows the lattice to be structured much more flexibly so that more points can be placed in the regions that are critical for valuation, while leaving less dense coverage where it doesn’t matter. In illustrative examples, the same degree of accuracy is achieved more than 10 times faster with the fully coupled system than with a normal Crank–Nicolson approach.