The Adaptive Mesh Model: A New Approach to Efficient Option Pricing

Exact closed-form valuation equations for traded derivative securities are rare. Numerical approximation, most commonly with Binomial and Trinomial lattice models, gives exact valuation in the limit, but convergence is non-monotonic and often slow, due to 'distribution error' and 'truncation error.' This paper explains how truncation error arises and describes the Adaptive Mesh Model (AMM), a new approach that sharply reduces it by grafting one or more small sections of fine high-resolution lattice onto a tree with coarser time and price steps. Three different AMM structures are presented, one for pricing ordinary options, one for barrier options, and one for computing delta and gamma efficiently. The AMM approach can be adapted to a wide variety of contingent claims, yielding significant improvement in efficiency with very little increase in computational effort. For some common problems, including calculating delta, accuracy increases by several orders of magnitude relative to the standard models with no measurable increase in execution time at all.