An efficient convergent lattice algorithm for European Asian options

Abstract

Financial options whose payoff depends critically on historical prices are called path-dependent options. Their prices are usually harder to calculate than options whose prices do not depend on past histories. Asian options are popular path-dependent derivatives, and it has been a long-standing problem to price them efficiently and accurately. No known exact pricing formulas are available to price them under the continuous-time Black-Scholes model. Although approximate pricing formulas exist, they lack accuracy guarantees. Asian options can be priced numerically on the lattice. A lattice divides the time to maturity into n equal-length time steps. The option price computed by the lattice converges to the option value under the Black-Scholes model as n->~. Unfortunately, only subexponential-time algorithms are available if Asian options are to be priced on the lattice without approximations. Efficient approximation algorithms are available for the lattice. The fastest lattice algorithm published in the literature runs in O(n^3^.^5)-time, whereas for the related PDE method, the fastest one runs in O(n^3) time. This paper presents a new lattice algorithm that runs in O(n^2^.^5) time, the best in the literature for such methods. Our algorithm exploits the method of Lagrange multipliers to minimize the approximation error. Numerical results verify its accuracy and the excellent performance.