IDEAS home Printed from https://ideas.repec.org/a/wsi/ijtafx/v22y2019i01ns0219024919500080.html
   My bibliography  Save this article

Determination Of The Lévy Exponent In Asset Pricing Models

Author

Listed:
  • GEORGE BOUZIANIS

    (Department of Computing, Goldsmiths College, University of London, New Cross, London SE14 6NW, UK)

  • LANE P. HUGHSTON

    (Department of Computing, Goldsmiths College, University of London, New Cross, London SE14 6NW, UK)

Abstract

We consider the problem of determining the Lévy exponent in a Lévy model for asset prices given the price data of derivatives. The model, formulated under the real-world measure ℙ, consists of a pricing kernel {πt}t≥0 together with one or more non-dividend-paying risky assets driven by the same Lévy process. If {St}t≥0 denotes the price process of such an asset, then {πtSt}t≥0 is a ℙ-martingale. The Lévy process {ξt}t≥0 is assumed to have exponential moments, implying the existence of a Lévy exponent ψ(α) = t−1log𝔼(eαξt) for α in an interval A ⊂ ℝ containing the origin as a proper subset. We show that if the prices of power-payoff derivatives, for which the payoff is HT = (ζT)q for some time T > 0, are given at time 0 for a range of values of q, where {ζt}t≥0 is the so-called benchmark portfolio defined by ζt = 1/πt, then the Lévy exponent is determined up to an irrelevant linear term. In such a setting, derivative prices embody complete information about price jumps: in particular, the spectrum of the price jumps can be worked out from current market prices of derivatives. More generally, if HT = (ST)q for a general non-dividend-paying risky asset driven by a Lévy process, and if we know that the pricing kernel is driven by the same Lévy process, up to a factor of proportionality, then from the current prices of power-payoff derivatives we can infer the structure of the Lévy exponent up to a transformation ψ(α) → ψ(α + μ) − ψ(μ) + cα, where c and μ are constants.

Suggested Citation

  • George Bouzianis & Lane P. Hughston, 2019. "Determination Of The Lévy Exponent In Asset Pricing Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(01), pages 1-18, February.
    • Handle: RePEc:wsi:ijtafx:v:22:y:2019:i:01:n:s0219024919500080
    DOI: 10.1142/S0219024919500080
    as

    Download full text from publisher

    File URL: http://www.worldscientific.com/doi/abs/10.1142/S0219024919500080
    Download Restriction: Access to full text is restricted to subscribers
    File URL: https://libkey.io/10.1142/S0219024919500080?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    2. Dilip B. Madan & Frank Milne, 1991. "Option Pricing With V. G. Martingale Components1," Mathematical Finance, Wiley Blackwell, vol. 1(4), pages 39-55, October.
    3. Eberlein, Ernst & Keller, Ulrich & Prause, Karsten, 1998. "New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model," The Journal of Business, University of Chicago Press, vol. 71(3), pages 371-405, July.
    4. Dilip B. Madan & Frank Milne, 1991. "Option Pricing With V. G. Martingale Components," Working Paper 1159, Economics Department, Queen's University.
    5. Dickson, David C. M. & Waters, Howard R., 1993. "Gamma Processes and Finite Time Survival Probabilities," ASTIN Bulletin, Cambridge University Press, vol. 23(2), pages 259-272, November.
    6. 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.
    7. José Fajardo, 2018. "Barrier style contracts under Lévy processes once again," Annals of Finance, Springer, vol. 14(1), pages 93-103, February.
    8. Breeden, Douglas T & Litzenberger, Robert H, 1978. "Prices of State-contingent Claims Implicit in Option Prices," The Journal of Business, University of Chicago Press, vol. 51(4), pages 621-651, October.
    9. Heston, Steven L, 1993. "Invisible Parameters in Option Prices," Journal of Finance, American Finance Association, vol. 48(3), pages 933-947, July.
    10. Dorje C. Brody & Lane P. Hughston & Ewan Mackie, 2011. "General Theory of Geometric L\'evy Models for Dynamic Asset Pricing," Papers 1111.2169, arXiv.org, revised Jan 2012.
    11. Ragnar Norberg, 2004. "Vasiček Beyond The Normal," Mathematical Finance, Wiley Blackwell, vol. 14(4), pages 585-604, October.
    12. Küchler, Uwe & Tappe, Stefan, 2014. "Exponential stock models driven by tempered stable processes," Journal of Econometrics, Elsevier, vol. 181(1), pages 53-63.
    13. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters, in: Anastasios G Malliaris & William T Ziemba (ed.), THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78, World Scientific Publishing Co. Pte. Ltd..
    14. Friedrich Hubalek & Carlo Sgarra, 2006. "Esscher transforms and the minimal entropy martingale measure for exponential Levy models," Quantitative Finance, Taylor & Francis Journals, vol. 6(2), pages 125-145.
    15. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    16. Leif Andersen & Alexander Lipton, 2013. "Asymptotics For Exponential Lévy Processes And Their Volatility Smile: Survey And New Results," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(01), pages 1-98.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Jean-Philippe Aguilar, 2021. "The value of power-related options under spectrally negative Lévy processes," Review of Derivatives Research, Springer, vol. 24(2), pages 173-196, July.
    2. Lane P. Hughston & Leandro Sánchez-Betancourt, 2020. "Pricing with Variance Gamma Information," Risks, MDPI, vol. 8(4), pages 1-22, October.
    3. Lane P. Hughston & Leandro S'anchez-Betancourt, 2020. "Pricing with Variance Gamma Information," Papers 2003.07967, arXiv.org, revised Sep 2020.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. 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.
    2. Li, Minqiang, 2008. "Price Deviations of S&P 500 Index Options from the Black-Scholes Formula Follow a Simple Pattern," MPRA Paper 11530, University Library of Munich, Germany.
    3. Laura Ballotta, 2009. "Pricing and capital requirements for with profit contracts: modelling considerations," Quantitative Finance, Taylor & Francis Journals, vol. 9(7), pages 803-817.
    4. Ballotta, Laura, 2005. "A Lévy process-based framework for the fair valuation of participating life insurance contracts," Insurance: Mathematics and Economics, Elsevier, vol. 37(2), pages 173-196, October.
    5. Liuren Wu, 2006. "Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns," The Journal of Business, University of Chicago Press, vol. 79(3), pages 1445-1474, May.
    6. Winston Buckley & Sandun Perera, 2019. "Optimal demand in a mispriced asymmetric Carr–Geman–Madan–Yor (CGMY) economy," Annals of Finance, Springer, vol. 15(3), pages 337-368, September.
    7. Yacine Aït-Sahalia & Jean Jacod, 2012. "Analyzing the Spectrum of Asset Returns: Jump and Volatility Components in High Frequency Data," Journal of Economic Literature, American Economic Association, vol. 50(4), pages 1007-1050, December.
    8. Don M. Chance & Eric Hillebrand & Jimmy E. Hilliard, 2008. "Pricing an Option on Revenue from an Innovation: An Application to Movie Box Office Revenue," Management Science, INFORMS, vol. 54(5), pages 1015-1028, May.
    9. repec:dau:papers:123456789/1392 is not listed on IDEAS
    10. Kao, Lie-Jane & Wu, Po-Cheng & Lee, Cheng-Few, 2012. "Time-changed GARCH versus the GARJI model for prediction of extreme news events: An empirical study," International Review of Economics & Finance, Elsevier, vol. 21(1), pages 115-129.
    11. Saralees Nadarajah & Bo Zhang & Stephen Chan, 2014. "Estimation methods for expected shortfall," Quantitative Finance, Taylor & Francis Journals, vol. 14(2), pages 271-291, February.
    12. Carr, Peter & Wu, Liuren, 2004. "Time-changed Levy processes and option pricing," Journal of Financial Economics, Elsevier, vol. 71(1), pages 113-141, January.
    13. Vladimir K. Kaishev & Dimitrina S. Dimitrova, 2009. "Dirichlet Bridge Sampling for the Variance Gamma Process: Pricing Path-Dependent Options," Management Science, INFORMS, vol. 55(3), pages 483-496, March.
    14. Fiorani, Filo, 2004. "Option Pricing Under the Variance Gamma Process," MPRA Paper 15395, University Library of Munich, Germany.
    15. Svetlana Boyarchenko & Sergei Levendorskiĭ, 2019. "Sinh-Acceleration: Efficient Evaluation Of Probability Distributions, Option Pricing, And Monte Carlo Simulations," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(03), pages 1-49, May.
    16. Colino, Jesús P., 2008. "New stochastic processes to model interest rates : LIBOR additive processes," DES - Working Papers. Statistics and Econometrics. WS ws085316, Universidad Carlos III de Madrid. Departamento de Estadística.
    17. Jingzhi Huang & Liuren Wu, 2004. "Specification Analysis of Option Pricing Models Based on Time- Changed Levy Processes," Finance 0401002, University Library of Munich, Germany.
    18. Chan, Tat Lung (Ron), 2019. "Efficient computation of european option prices and their sensitivities with the complex fourier series method," The North American Journal of Economics and Finance, Elsevier, vol. 50(C).
    19. Khaled Salhi, 2017. "Pricing European options and risk measurement under exponential Lévy models — a practical guide," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(02n03), pages 1-36, June.
    20. Sun, Qi & Xu, Weidong, 2015. "Pricing foreign equity option with stochastic volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 437(C), pages 89-100.
    21. Olivier Le Courtois, 2018. "Some Further Results on the Tempered Multistable Approach," Post-Print hal-02312142, HAL.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:wsi:ijtafx:v:22:y:2019:i:01:n:s0219024919500080. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Tai Tone Lim (email available below). General contact details of provider: http://www.worldscinet.com/ijtaf/ijtaf.shtml .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.