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Esscher transform and the duality principle for multidimensional semimartingales

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  • Ernst Eberlein
  • Antonis Papapantoleon
  • Albert N. Shiryaev

Abstract

The duality principle in option pricing aims at simplifying valuation problems that depend on several variables by associating them to the corresponding dual option pricing problem. Here, we analyze the duality principle for options that depend on several assets. The asset price processes are driven by general semimartingales, and the dual measures are constructed via an Esscher transformation. As an application, we can relate swap and quanto options to standard call and put options. Explicit calculations for jump models are also provided.

Suggested Citation

  • Ernst Eberlein & Antonis Papapantoleon & Albert N. Shiryaev, 2008. "Esscher transform and the duality principle for multidimensional semimartingales," Papers 0809.0301, arXiv.org, revised Nov 2009.
    • Handle: RePEc:arx:papers:0809.0301
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    1. Hans U. Gerber & Hlias S. W. Shiu, 1996. "Martingale Approach To Pricing Perpetual American Options On Two Stocks," Mathematical Finance, Wiley Blackwell, vol. 6(3), pages 303-322, July.
    2. Ernst Eberlein & Antonis Papapantoleon & Albert Shiryaev, 2008. "On the duality principle in option pricing: semimartingale setting," Finance and Stochastics, Springer, vol. 12(2), pages 265-292, April.
    3. José Fajardo & Ernesto Mordecki, 2006. "Pricing Derivatives On Two-Dimensional Lévy Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 9(02), pages 185-197.
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    Cited by:

    1. Michail Anthropelos & Michael Kupper & Antonis Papapantoleon, 2015. "An equilibrium model for spot and forward prices of commodities," Papers 1502.00674, arXiv.org, revised Jan 2017.
    2. Fred Espen Benth & Giulia Di Nunno & Asma Khedher & Maren Diane Schmeck, 2015. "Pricing of Spread Options on a Bivariate Jump Market and Stability to Model Risk," Applied Mathematical Finance, Taylor & Francis Journals, vol. 22(1), pages 28-62, March.
    3. Alessandro Gnoatto, 2017. "Coherent Foreign Exchange Market Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(01), pages 1-29, February.
    4. Ballotta, Laura & Deelstra, Griselda & Rayée, Grégory, 2017. "Multivariate FX models with jumps: Triangles, Quantos and implied correlation," European Journal of Operational Research, Elsevier, vol. 260(3), pages 1181-1199.
    5. Alev{s} v{C}ern'y & Johannes Ruf, 2020. "Simplified stochastic calculus via semimartingale representations," Papers 2006.11914, arXiv.org, revised Jan 2022.
    6. Holger Fink & Stefan Mittnik, 2021. "Quanto Pricing beyond Black–Scholes," JRFM, MDPI, vol. 14(3), pages 1-27, March.
    7. Ilya Molchanov & Michael Schmutz, 2009. "Exchangeability type properties of asset prices," Papers 0901.4914, arXiv.org, revised Apr 2011.
    8. Molchanov, Ilga & Schmutz, Michael & Stucki, Kaspar, 2012. "Invariance properties of random vectors and stochastic processes based on the zonoid concept," DES - Working Papers. Statistics and Econometrics. WS ws122014, Universidad Carlos III de Madrid. Departamento de Estadística.
    9. Ledenyov, Dimitri O. & Ledenyov, Viktor O., 2015. "Wave function method to forecast foreign currencies exchange rates at ultra high frequency electronic trading in foreign currencies exchange markets," MPRA Paper 67470, University Library of Munich, Germany.
    10. Rheinländer, Thorsten & Schmutz, Michael, 2013. "Self-dual continuous processes," Stochastic Processes and their Applications, Elsevier, vol. 123(5), pages 1765-1779.
    11. Michail Anthropelos & Michael Kupper & Antonis Papapantoleon, 2018. "An Equilibrium Model for Spot and Forward Prices of Commodities," Mathematics of Operations Research, INFORMS, vol. 43(1), pages 152-180, February.
    12. Laura Ballota & Griselda Deelstra & Grégory Rayée, 2015. "Quanto Implied Correlation in a Multi-Lévy Framework," Working Papers ECARES ECARES 2015-36, ULB -- Universite Libre de Bruxelles.
    13. Černý, Aleš & Ruf, Johannes, 2023. "Simplified calculus for semimartingales: Multiplicative compensators and changes of measure," Stochastic Processes and their Applications, Elsevier, vol. 161(C), pages 572-602.
    14. Roman V. Ivanov, 2023. "On the Stochastic Volatility in the Generalized Black-Scholes-Merton Model," Risks, MDPI, vol. 11(6), pages 1-23, June.
    15. Thorsten Rheinlander & Michael Schmutz, 2012. "Self-dual continuous processes," Papers 1201.6516, arXiv.org.
    16. Fred Espen Benth & Hanna Zdanowicz, 2016. "Pricing And Hedging Of Energy Spread Options And Volatility Modulated Volterra Processes," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 19(01), pages 1-22, February.
    17. Svetlozar Rachev & Frank J. Fabozzi & Boryana Racheva-Iotova & Abootaleb Shirvani, 2017. "Option Pricing with Greed and Fear Factor: The Rational Finance Approach," Papers 1709.08134, arXiv.org, revised Mar 2020.
    18. Rüschendorf Ludger & Wolf Viktor, 2015. "Cost-efficiency in multivariate Lévy models," Dependence Modeling, De Gruyter, vol. 3(1), pages 1-16, April.
    19. Ernst Eberlein & Zorana Grbac & Thorsten Schmidt, 2010. "Discrete tenor models for credit risky portfolios driven by time-inhomogeneous L\'evy processes," Papers 1006.2012, arXiv.org, revised Apr 2013.
    20. Lorenzo Torricelli, 2016. "Valuation of asset and volatility derivatives using decoupled time-changed Lévy processes," Review of Derivatives Research, Springer, vol. 19(1), pages 1-39, April.
    21. Roman V. Ivanov & Katsunori Ano, 2016. "On exact pricing of FX options in multivariate time-changed Lévy models," Review of Derivatives Research, Springer, vol. 19(3), pages 201-216, October.
    22. Ernst Eberlein & Kathrin Glau & Antonis Papapantoleon, 2008. "Analysis of Fourier transform valuation formulas and applications," Papers 0809.3405, arXiv.org, revised Sep 2009.
    23. Fred Espen Benth & Hanna Zdanowicz, 2014. "Pricing and hedging of energy spread options and volatility modulated Volterra processes," Papers 1409.5801, arXiv.org.
    24. Thorsten Rheinlander & Michael Schmutz, 2012. "Quasi self-dual exponential L\'evy processes," Papers 1201.5132, arXiv.org.

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