Barrier Option Pricing Using Adjusted Transition Probabilities

Abstract

While closed-form valuation formulas like Black–Scholes are the Holy Grail of options modeling, numerical approximation in a lattice model is the workhorse of real world pricing for exotics, such barrier options. But the workhorse has to work harder for some options than for others, particularly when the price of the underlying starts off near the barrier. Pricing errors are relatively large when the barrier falls in the middle between two price nodes. A variety of techniques has been advanced in the literature to mitigate the problem for special cases, but they are easily foiled by multiple, nonlinear, or discontinuous barriers. This article deals with the problem in a more systematic fashion and offers a general solution. The trouble with standard lattice models is that the total probability of passing from one node near the barrier, but not beyond it, to another such node in the next time step includes some paths that would breach the barrier along the way. Barone-Adesi, Fusari, and Theal show how performance can be greatly improved by adjusting for those paths.