Numerical inversion of Laplace transforms: a survey of techniques with applications to derivative pricing

On an Integral Arising in Mathematical Finance

“…Solving the PDE must be done numerically since no analytical solution satisfying the necessary boundary and terminal conditions is known. Secondly, one may attempt to numerically invert the Laplace transform of the price derived by Geman and Yor, [8], [2], [7], [6]. A fast and reasonably efficient third approach is to use the well known approximation for the Asian price derived by moment matching.…”

Laplace transform identities for diffusions, with applications to rebates and barrier options

“…Probably the first and most widely cited of these is Geman and Yor [12], where an expression was obtained for the Laplace transform of the price of an Asian option on a security following a geometric Brownian motion. The numerical aspects of inverting this particular expression have been the focus of a number of studies, including Craddock et al [6], Fu et al [10], Geman and Eydeland [11] and Shaw [26,Chap. 10].…”

“…Nevertheless, one cannot be sure, since there are no guaranteed error bounds-a drawback of all inversion algorithms. Furthermore, Craddock et al [6] have reported that inversion schemes, in general, appear quite sensitive to model parameters. This apparent lack of robustness, together with the computational effort required and the absence of error bounds, make us hesitant to endorse numerical Laplace transform inversion unreservedly as a practical technique for valuing exotic options.…”

“…In order to interpret the above expressions, note that the Laplace transforms on the left-hand sides of (5), (6) and (8) are of the form L α {f (t)} =f (α), while the Laplace transforms on the right-hand sides of (7) and (8) are of the form L γ {f (s)} =f (γ).…”

“…by Davidov and Linetsky [7,8] and Fu et al [10], to invert Laplace transforms associated with derivative valuation problems. We also draw the reader's attention to the comparative study of numerical schemes for Laplace transform inversion in Craddock et al [6], which focused specifically on applications to derivative pricing. Figure 2 presents surface and contour plots of the pricing function R 10,50 (· , ·) for a rebate with maturity T = 10 years and reference level z = 50, using the same parameter values as before.…”