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The distribution of the maximum of a variance gamma process and path-dependent option pricing

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  • Roman Ivanov

Abstract

Although numerical procedures often supply a required accuracy, closed-form expressions allow one to escape any accumulation of errors. In this paper, we discuss the possibility of obtaining explicit results for a variance gamma process. We derive the exact distribution of the maximum of the variance gamma process over a finite interval of time and establish the prices of path-dependent options including digital barrier, fixed-strike lookback, and lookback options. The obtained formulas are based on values of hypergeometric functions. Copyright Springer-Verlag Berlin Heidelberg 2015

Suggested Citation

  • Roman Ivanov, 2015. "The distribution of the maximum of a variance gamma process and path-dependent option pricing," Finance and Stochastics, Springer, vol. 19(4), pages 979-993, October.
    • Handle: RePEc:spr:finsto:v:19:y:2015:i:4:p:979-993
    DOI: 10.1007/s00780-015-0277-8
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    References listed on IDEAS

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    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
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    3. Dilip B. Madan & Frank Milne, 1991. "Option Pricing With V. G. Martingale Components1," Mathematical Finance, Wiley Blackwell, vol. 1(4), pages 39-55, October.
    4. Peter Carr & Roger Lee & Liuren Wu, 2012. "Variance swaps on time-changed Lévy processes," Finance and Stochastics, Springer, vol. 16(2), pages 335-355, April.
    5. Dilip B. Madan & Frank Milne, 1991. "Option Pricing With V. G. Martingale Components," Working Paper 1159, Economics Department, Queen's University.
    6. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    7. Madan, Dilip B & Seneta, Eugene, 1990. "The Variance Gamma (V.G.) Model for Share Market Returns," The Journal of Business, University of Chicago Press, vol. 63(4), pages 511-524, October.
    8. Patrizia Semeraro, 2006. "A Multivariate Time-Changed Lévy Model for Financial Applications," ICER Working Papers - Applied Mathematics Series 10-2006, ICER - International Centre for Economic Research.
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    Full references (including those not matched with items on IDEAS)

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    More about this item

    Keywords

    Variance gamma process; Distribution of maximum; Path-dependent options; Exact formula; Hypergeometric function; 60G51; 60G70; 60J75; 33C20; C02; D46; D53; G12;
    All these keywords.

    JEL classification:

    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • D46 - Microeconomics - - Market Structure, Pricing, and Design - - - Value Theory
    • D53 - Microeconomics - - General Equilibrium and Disequilibrium - - - Financial Markets
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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