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Dynamic Hedging in Incomplete Markets: A Simple Solution

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  • Basak, Suleyman
  • Chabakauri, Georgy

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.

Suggested Citation

  • Basak, Suleyman & Chabakauri, Georgy, 2011. "Dynamic Hedging in Incomplete Markets: A Simple Solution," CEPR Discussion Papers 8402, C.E.P.R. Discussion Papers.
  • Handle: RePEc:cpr:ceprdp:8402
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    Cited by:

    1. Dilip B. Madan & Yazid M. Sharaiha, 2015. "Option overlay strategies," Quantitative Finance, Taylor & Francis Journals, vol. 15(7), pages 1175-1190, July.
    2. Tak Kuen Siu & Robert J. Elliott, 2019. "Hedging Options In A Doubly Markov-Modulated Financial Market Via Stochastic Flows," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 22(08), pages 1-41, December.
    3. 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.
    4. Atmaz, Adem & Basak, Suleyman, 2019. "Option prices and costly short-selling," Journal of Financial Economics, Elsevier, vol. 134(1), pages 1-28.
    5. PeiLin Hsieh & Robert Jarrow, 2019. "Volatility Uncertainty, Time Decay, and Option Bid-Ask Spreads in an Incomplete Market," Management Science, INFORMS, vol. 65(4), pages 1833-1854, April.
    6. Paskalis Glabadanidis & Leon Zolotoy, 2013. "Benchmark replication portfolio strategies," Journal of Asset Management, Palgrave Macmillan, vol. 14(2), pages 95-110, April.
    7. 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.
    8. 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.
    9. Kamma, Thijs & Pelsser, Antoon, 2022. "Near-optimal asset allocation in financial markets with trading constraints," European Journal of Operational Research, Elsevier, vol. 297(2), pages 766-781.
    10. Alexandre Carbonneau & Fr'ed'eric Godin, 2021. "Deep Equal Risk Pricing of Financial Derivatives with Multiple Hedging Instruments," Papers 2102.12694, arXiv.org.
    11. Min Dai & Hanqing Jin & Steven Kou & Yuhong Xu, 2021. "A Dynamic Mean-Variance Analysis for Log Returns," Management Science, INFORMS, vol. 67(2), pages 1093-1108, February.
    12. 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.
    13. Niu, Baozhuang & Chu, Lap-Keung & Ni, Jian & Wang, Junwei, 2018. "Buy now and price later: Supply contracts with time-consistent mean–variance financial hedgingAuthor-Name: Li, Qiang," European Journal of Operational Research, Elsevier, vol. 268(2), pages 582-595.
    14. 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.
    15. Alavi Fard, Farzad & He, Jian & Ivanov, Dmitry & Jie, Ferry, 2019. "A utility adjusted newsvendor model with stochastic demand," International Journal of Production Economics, Elsevier, vol. 211(C), pages 154-165.

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    More about this item

    Keywords

    benchmarking; correlation risk; derivatives; discrete hedging; hedging; incomplete markets; minimum-variance criterion; Poisson jumps; risk management; time-consistency;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D81 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Criteria for Decision-Making under Risk and Uncertainty
    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions

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