Barrier style contracts under Lévy processes once again


In this paper we present new pricing formulas for some Barrier style contracts of European type when the underlying process is driven by an important class of Lévy processes, which includes CGMY model, generalized hyperbolic Model and Meixner Model, when no symmetry properties are assumed, complementing in this way previous findings in Fajardo (J Bank Financ 53:179–187, 2015). Also, we show how to implement our new formulas.

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Fig. 1


  1. 1.

    \(\Pi (\{0\})\) could be defined as 0. Here I follow Cont and Tankov (2004).

  2. 2.

    Equivalently for all t, see Theorem 25.17 in Sato (1999).

  3. 3.

    More precisely, \(\frac{d\tilde{\mathbb Q}_t}{d{\mathbb Q}_t}=e^{X_t-(r-\delta )t},\;\;t\ge 0\).

  4. 4.

    The Lévy models of interest satisfy it.

  5. 5.

    A payoff function is a nonnegative Borel function on \(\mathbb R\).

  6. 6.

    Here we use the same denomination used in example 5.15 by Carr and Lee (2009).

  7. 7.

    This method is widely used to estimate Lévy processes, see for example Ramezani and Zeng (2007).


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I would like to thank the comments of an anonymous referee who helps to improve the present version of the paper. Also, I would like to thank seminar participants at EMAp/FGV, SBFin 2015, SAET 2016 and Barcelona Workshop on Mathematical Finance. Financial support from CNPq of Brazil is also acknowledge.

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Correspondence to José Fajardo.

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Fajardo, J. Barrier style contracts under Lévy processes once again. Ann Finance 14, 93–103 (2018).

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  • Skewness
  • Lévy processes
  • Absence of symmetry
  • Barrier contracts

JEL Classification

  • C52
  • G12