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Dynamic Hedging in Incomplete Markets: A Simple Solution

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  • Suleyman Basak

    ()

  • Georgy Chabakauri

    ()

Abstract

Despite much work on hedging in incomplete markets, the literature still lacks tractable dynamic hedges in plausible environments. In this article, we provide a simple solution to this problem in a general incomplete-market economy in which a hedger, guided by the traditional minimum-variance criterion, aims at reducing the risk of a non-tradable asset or a contingent claim. We derive fully analytical optimal hedges and demonstrate that they can easily be computed in various stochastic environments. Our dynamic hedges preserve the simple structure of complete-market perfect hedges and are in terms of generalized \Greeks," familiar in risk management applications, as well as retaining the intuitive features of their static counterparts. We obtain our time-consistent hedges by dynamic programming, while the extant literature characterizes either static or myopic hedges, or dynamic ones that minimize the variance criterion at an initial date and from which the hedger may deviate unless she can pre-commit to follow them. We apply our results to the discrete hedging problem of derivatives when trading occurs infrequently. We determine the corresponding optimal hedge and replicating portfolio value, and show that they have structure similar to their complete market counterparts and reduce to generalized Black-Scholes expressions when specialized to the Black-Scholes setting. We also generalize our results to richer settings to study dynamic hedging with Poisson jumps, stochastic correlation and portfolio management with benchmarking.

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Bibliographic Info

Paper provided by Financial Markets Group in its series FMG Discussion Papers with number dp680.

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Date of creation: May 2011
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Handle: RePEc:fmg:fmgdps:dp680

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Web page: http://www.lse.ac.uk/fmg/

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  1. Anderson, Ronald W & Danthine, Jean-Pierre, 1980. " Hedging and Joint Production: Theory and Illustrations," Journal of Finance, American Finance Association, vol. 35(2), pages 487-98, May.
  2. Jér�me B. Detemple & René Garcia & Marcel Rindisbacher, 2003. "A Monte Carlo Method for Optimal Portfolios," Journal of Finance, American Finance Association, vol. 58(1), pages 401-446, 02.
  3. Joost Driessen & Pascal J. Maenhout & Grigory Vilkov, 2009. "The Price of Correlation Risk: Evidence from Equity Options," Journal of Finance, American Finance Association, vol. 64(3), pages 1377-1406, 06.
  4. Bossaerts, P. & Hillion, P., 1995. "Local Parametric Analysis of Hedging in Discrete Time," Discussion Paper 1995-23, Tilburg University, Center for Economic Research.
  5. GEORGE W. EVANS & BRUCE McGOUGH, 2007. "Optimal Constrained Interest-Rate Rules," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 39(6), pages 1335-1356, 09.
  6. Andrea Buraschi & Paolo Porchia & Fabio Trojani, 2010. "Correlation Risk and Optimal Portfolio Choice," Journal of Finance, American Finance Association, vol. 65(1), pages 393-420, 02.
  7. Costa, O. L. V. & Paiva, A. C., 2002. "Robust portfolio selection using linear-matrix inequalities," Journal of Economic Dynamics and Control, Elsevier, vol. 26(6), pages 889-909, June.
  8. Bekaert, Geert & Harvey, Campbell R, 1995. " Time-Varying World Market Integration," Journal of Finance, American Finance Association, vol. 50(2), pages 403-44, June.
  9. Jaksa Cvitanic & Fernando Zapatero, 2004. "Introduction to the Economics and Mathematics of Financial Markets," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262532654, December.
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Cited by:
  1. Ankirchner, Stefan & Schneider, Judith C. & Schweizer, Nikolaus, 2014. "Cross-hedging minimum return guarantees: Basis and liquidity risks," Journal of Economic Dynamics and Control, Elsevier, vol. 41(C), pages 93-109.
  2. Suleyman Basak & Georgy Chabakauri, 2010. "Dynamic Mean-Variance Asset Allocation," Review of Financial Studies, Society for Financial Studies, vol. 23(8), pages 2970-3016, August.
  3. Alexandru Badescu & Robert J. Elliott & Juan-Pablo Ortega, 2012. "Quadratic hedging schemes for non-Gaussian GARCH models," Papers 1209.5976, arXiv.org, revised Dec 2013.
  4. Badescu, Alexandru & Elliott, Robert J. & Ortega, Juan-Pablo, 2014. "Quadratic hedging schemes for non-Gaussian GARCH models," Journal of Economic Dynamics and Control, Elsevier, vol. 42(C), pages 13-32.
  5. Lioui, Abraham, 2013. "Time consistent vs. time inconsistent dynamic asset allocation: Some utility cost calculations for mean variance preferences," Journal of Economic Dynamics and Control, Elsevier, vol. 37(5), pages 1066-1096.

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