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Discontinuous payoff option pricing by Mellin transform: A probabilistic approach

Author

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  • Gzyl, H.
  • Milev, M.
  • Tagliani, A.

Abstract

The Mellin transform technique is applied for solving the Black-Scholes equation with time-dependent parameters and discontinuous payoff. We show that the option pricing is equivalent to recovering a probability density function on the positive real axis based on its moments, which are integer or fractional Mellin transform values. Then the Mellin transform can be effectively inverted from a collection of appropriately chosen fractional (i.e. non-integer) moments by means of the Maximum Entropy (MaxEnt) method. An accurate option pricing is guaranteed by previous theoretical results about MaxEnt distributions constrained by fractional moments. We prove that typical drawbacks of other numerical techniques, such as Finite Difference schemes, are bypassed exploiting the Mellin transform properties. An example involving discretely monitored barrier options is illustrated and the accuracy, efficiency and time consuming are discussed.

Suggested Citation

  • Gzyl, H. & Milev, M. & Tagliani, A., 2017. "Discontinuous payoff option pricing by Mellin transform: A probabilistic approach," Finance Research Letters, Elsevier, vol. 20(C), pages 281-288.
    • Handle: RePEc:eee:finlet:v:20:y:2017:i:c:p:281-288
    DOI: 10.1016/j.frl.2016.10.011
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    References listed on IDEAS

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    1. Liming Feng & Vadim Linetsky, 2008. "Pricing Discretely Monitored Barrier Options And Defaultable Bonds In Lévy Process Models: A Fast Hilbert Transform Approach," Mathematical Finance, Wiley Blackwell, vol. 18(3), pages 337-384, July.
    2. Kim, Jerim & Kim, Jeongsim & Joo Yoo, Hyun & Kim, Bara, 2015. "Pricing external barrier options in a regime-switching model," Journal of Economic Dynamics and Control, Elsevier, vol. 53(C), pages 123-143.
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    Cited by:

    1. Amirhossein Sobhani & Mariyan Milev, 2017. "A Numerical Method for Pricing Discrete Double Barrier Option by Lagrange Interpolation on Jacobi Node," Papers 1712.01060, arXiv.org, revised Feb 2018.
    2. Ahmadian, D. & Farkhondeh Rouz, O. & Ivaz, K. & Safdari-Vaighani, A., 2020. "Robust numerical algorithm to the European option with illiquid markets," Applied Mathematics and Computation, Elsevier, vol. 366(C).

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    More about this item

    Keywords

    Barrier options; Black-Scholes equation; Discontinuous payoff; Fractional moments; Maximum entropy; Mellin transform;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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