Efficient evaluation of double-barrier options and joint cpdf of a L\'evy process and its two extrema

TL;DR

This paper introduces a rapid and precise method for pricing double barrier options in Le9vy models using Wiener-Hopf factorization, achieving high accuracy with minimal computation time.

Contribution

The paper develops a novel, fast numerical approach for evaluating double barrier options and joint distributions in Le9vy models, utilizing sinh-deformations and the Wiener-Hopf technique.

Findings

Achieves precision of 10^{-15} in seconds

Provides explicit algorithms for no-touch options and digitals

Demonstrates fundamental challenges in barrier option pricing methods

Abstract

In the paper, we develop a very fast and accurate method for pricing double barrier options with continuous monitoring in wide classes of L\'evy models; the calculations are in the dual space, and the Wiener-Hopf factorization is used. For wide regions in the parameter space, the precision of the order of $10^{-15}$ is achievable in seconds, and of the order of $10^{-9}-10^{-8}$ - in fractions of a second. The Wiener-Hopf factors and repeated integrals in the pricing formulas are calculated using sinh-deformations of the lines of integration, the corresponding changes of variables and the simplified trapezoid rule. If the Bromwich integral is calculated using the Gaver-Wynn Rho acceleration instead of the sinh-acceleration, the CPU time is typically smaller but the precision is of the order of $10^{-9}-10^{-6}$, at best. Explicit pricing algorithms and numerical examples are for no-touch options, digitals (equivalently, for the joint distribution function of a L\'evy process and its supremum and infimum processes), and call options. Several graphs are produced to explain fundamental difficulties for accurate pricing of barrier options using time discretization and interpolation-based calculations in the state space.