A simplified Wiener–Hopf factorization method for pricing double barrier options under Lévy processes

We developed a new method to price double barrier options under pure non-Gaussian Lévy processes admitting jumps of unbounded variation. In our approach, we represent the underlying process as a sequence of spectrally positive, negative, and again positive jumps, with spectrally positive jumps corresponding to half of the time moment. This rule is applied to the increments of the Lévy process at exponentially distributed time points. The justification for the convergence of the method in time is carried out using the interpretation of time randomization as the numerical Laplace transform inversion with the Post-Widder formula. The main algorithm consists of the recurrent calculation of the sufficient simple expectations of the intermediate price function depending on the position of the extremum of spectrally positive or negative part of the underying Lévy process at a randomized time moment. It corresponds to a sequence of problems for integro-differential equations on an interval, each of which is solved semi-explicitly by the Wiener–Hopf method. We decompose the characteristic function of a randomized spectrally positive (negative) process as the product of two Wiener–Hopf factors, the first of which corresponds to the exponential distribution and the second is calculated as a ratio. The parameter of the exponential distribution is numerically found as the only root of the factorizing operator symbol using Newton’s method. Thus, by combining the sequential application of the Wiener–Hopf operators in explicit form with the characteristic function of our interval, we obtain the solution to each auxiliary problem. The main advantage of the suggested methods is that being very simple for programming it makes it possible to avoid dealing with Wiener–Hopf matrix factorization or solving systems of coupled nontrivial Wiener–Hopf equations that require application of tricky approximate factorization formulas.