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Prices Of Barrier And First-Touch Digital Options In Lévy-Driven Models, Near Barrier

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  • MITYA BOYARCHENKO

    ()
    (Department of Mathematics, University of Michigan, 530 Church Street, 2074 East Hall, Ann Arbor, MI 48109-1043, USA)

  • MARCO DE INNOCENTIS

    ()
    (Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, UK; RiskCare Ltd, 22 Cousin Lane, London, EC4R 3TE, UK)

  • SERGEI LEVENDORSKIĬ

    ()
    (Department of Mathematics, University of Leicester, University Road, Leicester, LE1 7RH, UK)

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    Abstract

    We calculate the leading term of asymptotics of the prices of barrier options and first-touch digitals near the barrier for wide classes of Lévy processes with exponential jump densities, including the Variance Gamma model, the KoBoL (a.k.a. CGMY) model and Normal Inverse Gaussian processes. In the case of processes of infinite activity and finite variation, with the drift pointing from the barrier, we prove that the price is discontinuous at the boundary. This observation can serve as the basis for a simple robust test of the type of processes observed in real financial markets. In many cases, we calculate the second term of asymptotics as well. By comparing the exact asymptotic results for prices with those of Carr's randomization approximation, we conclude that the latter is very accurate near the barrier. We illustrate this by including numerical results for several types of Lévy processes commonly used in option pricing.

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    Bibliographic Info

    Article provided by World Scientific Publishing Co. Pte. Ltd. in its journal International Journal of Theoretical and Applied Finance.

    Volume (Year): 14 (2011)
    Issue (Month): 07 ()
    Pages: 1045-1090

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    Handle: RePEc:wsi:ijtafx:v:14:y:2011:i:07:p:1045-1090

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    Related research

    Keywords: Option pricing; barrier options; first-touch digitals; one-touch options; Lévy processes; Carr's randomization; KoBoL processes; CGMY model; Normal Inverse Gaussian processes; Variance Gamma processes; Wiener-Hopf factorization; asymptotics;

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