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

International Journal of Theoretical and Applied Finance (IJTAF), 2010

Posted: 8 Jun 2010

See all articles by Mitya Boyarchenko

Mitya Boyarchenko

University of Michigan - Department of Mathematics

Sergei Levendorskii

Calico Science Consulting

Date Written: December 1, 2009

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.

Keywords: Option pricing, greeks, barrier options, first-touch digitals, Lévy processes, Fast Fourier transform; Carr's randomization; KoBoL processes; CGMY model; Normal Inverse Gaussian processes, Variance Gamma processes, Wiener–Hopf factorization

Suggested Citation

Boyarchenko, Mitya and Levendorskii, Sergei Z., Prices and Sensitivities of Barrier and First-Touch Digital Options in Lévy-Driven Models (December 1, 2009). International Journal of Theoretical and Applied Finance (IJTAF), 2010 . Available at SSRN: https://ssrn.com/abstract=1620241

Mitya Boyarchenko (Contact Author)

University of Michigan - Department of Mathematics ( email )

530 Church Street
2074 East Hall
Ann Arbor, MI 48109
United States

Sergei Z. Levendorskii

Calico Science Consulting ( email )

Austin, TX
United States

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