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Multivariate Option Pricing with Copulas

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  • Umberto Cherubini
  • Elisa Luciano

Abstract

In this paper we suggest the adoption of copula functions in order to price multivariate contingent claims. Copulas enable us to imbed the marginal distributions extracted from vertical spreads in the options markets in a multivariate pricing kernel. We prove that such kernel is a copula function, and that its super -replication strategy is represented by the Fréchet bounds. As applications, we provide prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, we provide no-arbitrage pricing bounds, as well as the values consistent with independence of the underlying assets. As a final reference value, we use a copula function calibrated on historical data.

Suggested Citation

  • Umberto Cherubini & Elisa Luciano, 2002. "Multivariate Option Pricing with Copulas," ICER Working Papers - Applied Mathematics Series 05-2002, ICER - International Centre for Economic Research.
    • Handle: RePEc:icr:wpmath:05-2002
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    References listed on IDEAS

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    1. Joshua Rosenberg, 1999. "Semiparametric Pricing of Multivariate Contingent Claims," New York University, Leonard N. Stern School Finance Department Working Paper Seires 99-028, New York University, Leonard N. Stern School of Business-.
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    8. Margrabe, William, 1978. "The Value of an Option to Exchange One Asset for Another," Journal of Finance, American Finance Association, vol. 33(1), pages 177-186, March.
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    Cited by:

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    3. Taizhong Hu & Alfred Müller & Marco Scarsini, 2002. "Some Counterexamples in Positive Dependence," ICER Working Papers - Applied Mathematics Series 28-2003, ICER - International Centre for Economic Research, revised Jul 2003.
    4. Y. Malevergne & D. Sornette, 2003. "Testing the Gaussian copula hypothesis for financial assets dependences," Quantitative Finance, Taylor & Francis Journals, vol. 3(4), pages 231-250.
    5. Müller, Alfred & Scarsini, Marco, 2005. "Archimedean copulæ and positive dependence," Journal of Multivariate Analysis, Elsevier, vol. 93(2), pages 434-445, April.
    6. Yanqin Fan & Xiaohong Chen & Andrew Patton, 2004. "(IAM Series No 003) Simple Tests for Models of Dependence Between Multiple Financial Time Series, with Applications to U.S. Equity Returns and Exchange Rates," FMG Discussion Papers dp483, Financial Markets Group.
    7. Antonio Lijoi & Igor Prünster & Stephen G. Walker, 2004. "On rates of convergence for posterior distributions in infinite–dimensional models," ICER Working Papers - Applied Mathematics Series 24-2004, ICER - International Centre for Economic Research.
    8. van den Goorbergh, R.W.J., 2004. "Essays on optimal hedging and investment strategies and on derivative pricing," Other publications TiSEM 4b4b16af-8621-463f-bbfa-0, Tilburg University, School of Economics and Management.
    9. Enrico Diecidue & Fabio Maccheroni, 2002. "Coherence without Additivity," ICER Working Papers - Applied Mathematics Series 10-2002, ICER - International Centre for Economic Research.
    10. Antonio Lijoi & Igor Prünster & Stephen G. Walker, 2004. "Contributions to the understanding of Bayesian consistency," ICER Working Papers - Applied Mathematics Series 13-2004, ICER - International Centre for Economic Research.
    11. Chen, Xiaohong & Fan, Yanqin, 2006. "Estimation of copula-based semiparametric time series models," Journal of Econometrics, Elsevier, vol. 130(2), pages 307-335, February.
    12. Castagnoli, Erio & Maccheroni, Fabio & Marinacci, Massimo, 2002. "Insurance premia consistent with the market," Insurance: Mathematics and Economics, Elsevier, vol. 31(2), pages 267-284, October.
    13. Renault, Jerome & Scarlatti, Sergio & Scarsini, Marco, 2005. "A folk theorem for minority games," Games and Economic Behavior, Elsevier, vol. 53(2), pages 208-230, November.
    14. Umberto Cherubini & Elisa Luciano, 2003. "Pricing Vulnerable Options With Copulas," Journal of Risk Finance, Emerald Group Publishing Limited, vol. 5(1), pages 27-39, April.
    15. van den Goorbergh, R.W.J. & Genest, C. & Werker, B.J.M., 2003. "Multivariate Option Pricing Using Dynamic Copula Models," Discussion Paper 2003-122, Tilburg University, Center for Economic Research.
    16. Salvatore Modica & Marco Scarsini, 2003. "The convexity-cone approach to comparative risk and downside risk," ICER Working Papers - Applied Mathematics Series 01-2003, ICER - International Centre for Economic Research.
    17. Chen, Xiaohong & Fan, Yanqin & Patton, Andrew J., 2004. "Simple tests for models of dependence between multiple financial time series, with applications to U.S. equity returns and exchange rates," LSE Research Online Documents on Economics 24681, London School of Economics and Political Science, LSE Library.
    18. van den Goorbergh, R.W.J. & Genest, C. & Werker, B.J.M., 2003. "Multivariate Option Pricing Using Dynamic Copula Models," Other publications TiSEM 86ec50af-0fb6-4782-b2dd-d, Tilburg University, School of Economics and Management.
    19. Giannopoulos, Kostas, 2008. "Nonparametric, conditional pricing of higher order multivariate contingent claims," Journal of Banking & Finance, Elsevier, vol. 32(9), pages 1907-1915, September.
    20. Antonio Lijoi & Igor Prünster & Stephen G. Walker, 2004. "On consistency of nonparametric normal mixtures for Bayesian density estimation," ICER Working Papers - Applied Mathematics Series 23-2004, ICER - International Centre for Economic Research.
    21. Yukihiro Tsuzuki, 2012. "On the Optimal Super- and Sub-Hedging Strategies," CARF F-Series CARF-F-300, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Aug 2013.

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    Keywords

    option pricing; basket options; copula functions; non-normal returns;
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