Prices and Sensitivities of Barrier and First-Touch Digital Options in Lévy-Driven Models

Abstract

We present a fast and accurate FFT-based method of computing the prices and sensitivities of barrier options and first-touch digital options on stocks whose log-price follows a Lévy process. The numerical results obtained via our approach are demonstrated to be in good agreement with the results obtained using other (sometimes fundamentally different) approaches that exist in the literature. However, our method is computationally much faster (often, dozens of times faster). Moreover, our technique has the advantage that its application does not entail a detailed analysis of the underlying Lévy process: one only needs an explicit analytic formula for the characteristic exponent of the process. Thus our algorithm is very easy to implement in practice. Finally, our method yields accurate results for a wide range of values of the spot price, including those that are very close to the barrier, regardless of whether the maturity period of the option is long or short.

Efficient Option Pricing Under Levy Processes, with CVA and FVA

“…For all s and x ≤ h, V s + 1 (x) = 1, and V 0 (x) = 0, x > h. By Lemma 2.3 in Boyarchenko and Levendorskiȋ [48], we have, for…”

“…These problems can be solved in closed form that is amenable to very fast numerical calculations using the operator form of the WienerHopf factorization method developed in a series of works by Boyarchenko and Levendorskiȋ [30,48,50,51]. The maturity period of the claim is divided into N subintervals, using points 0 = t 0 < t 1 < · · · < t N = T, and each sub-period [t s , t s + 1 ] is replaced with an exponentially distributed random maturity period with mean s = t s + 1 − t s .…”

“…Section 2.2) and φ ± q (ξ ) are the Wiener-Hopf factors of q(q + ψ(ξ )) −1 , defined by E ± q (e iξ x ) = φ ± q (ξ )e iξ x . In this subsection and the next three subsections and Appendix A, we reproduce the detailed prescription from Boyarchenko and Levendorskiȋ [48,51] for calculation of φ ± q , and refer the reader to Boyarchenko and Levendorskiȋ [48,51], Sato [52] for background on the WienerHopf factorization. Note that if X is a process of finite variation with positive (resp., negative) drift, then the measure g + (y)dy has an atom at 0 (resp., the measure g − (y)dy has an atom at 0).…”

“…The obvious analogue of Remark 4.5 is valid here as well, the only difference being that in order to calculate the convolution coefficients c + ℓ and c 0 +,ℓ −c 1 +,ℓ , we must first calculate the array of values of φ + q (ξ ) on a suitable grid in the ξ -space using the algorithm of Section A.1.1 (where "suitable" means "suitable for the application of the refined iFFT technique of Boyarchenko and Levendorskiȋ [48,51]). …”

“…The algorithms are similar to the ones in [48] with natural modifications to account for the nonlinearity of the stream g(V) and kinks. The underlying is modeled as S t = e X t , where X t is a Lévy process of exponential type (λ − , λ + ), with λ − < −1 < 0 < λ + .…”