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Perfect option hedging for a large trader

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  • RØdiger Frey

    ()
    (Department of Mathematics, ETH ZØrich, ETH-Zentrum, CH-8092 ZØrich, Switzerland)

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    Abstract

    Standard derivative pricing theory is based on the assumption of agents acting as price takers on the market for the underlying asset. We relax this hypothesis and study if and how a large agent whose trades move prices can replicate the payoff of a derivative security. Our analysis extends prior work of Jarrow to economies with continuous security trading. We characterize the solution to the hedge problem in terms of a nonlinear partial differential equation and provide results on existence and uniqueness of this equation. Simulations are used to compare the hedging strategies in our model to standard Black-Scholes strategies.

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    Bibliographic Info

    Article provided by Springer in its journal Finance and Stochastics.

    Volume (Year): 2 (1998)
    Issue (Month): 2 ()
    Pages: 115-141

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    Handle: RePEc:spr:finsto:v:2:y:1998:i:2:p:115-141

    Note: received: April 1996; final version received: April 1997
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    Web page: http://www.springerlink.com/content/101164/

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    Related research

    Keywords: Option pricing; Black-Scholes model; hedging; large trader; feedback effects;

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    References

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    Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
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    1. Rüdiger Frey & Alexander Stremme, 1997. "Market Volatility and Feedback Effects from Dynamic Hedging," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 351-374.
    2. Jarrow, Robert A., 1992. "Market Manipulation, Bubbles, Corners, and Short Squeezes," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 27(03), pages 311-336, September.
    3. Jarrow, Robert A., 1994. "Derivative Security Markets, Market Manipulation, and Option Pricing Theory," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 29(02), pages 241-261, June.
    4. Gerard Gennotte and Hayne Leland., 1989. "Market Liquidity, Hedging and Crashes," Research Program in Finance Working Papers RPF-192, University of California at Berkeley.
    5. Grossman, Sanford J, 1988. "An Analysis of the Implications for Stock and Futures Price Volatility of Program Trading and Dynamic Hedging Strategies," The Journal of Business, University of Chicago Press, vol. 61(3), pages 275-98, July.
    6. Eckhard Platen & Martin Schweizer, 1998. "On Feedback Effects from Hedging Derivatives," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 67-84.
    7. Hans Föllmer & Martin Schweizer, 1993. "A Microeconomic Approach to Diffusion Models For Stock Prices," Mathematical Finance, Wiley Blackwell, vol. 3(1), pages 1-23.
    8. Holthausen, Robert W. & Leftwich, Richard W. & Mayers, David, 1987. "The effect of large block transactions on security prices: A cross-sectional analysis," Journal of Financial Economics, Elsevier, vol. 19(2), pages 237-267, December.
    9. Yaacov Z. Bergman & Bruce D. Grundy & Zvi Wiener, . "Theory of Rational Option Pricing: II (Revised: 1-96)," Rodney L. White Center for Financial Research Working Papers 11-95, Wharton School Rodney L. White Center for Financial Research.
    10. Grossman, Sanford J & Zhou, Zhongquan, 1996. " Equilibrium Analysis of Portfolio Insurance," Journal of Finance, American Finance Association, vol. 51(4), pages 1379-1403, September.
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    Cited by:
    1. François-Heude, Alain & Yousfi, Ouidad, 2013. "On the liquidity of CAC 40 index options Market," MPRA Paper 47921, University Library of Munich, Germany, revised 01 Jul 2013.
    2. Vathana Ly Vath & Mohamed Mnif & Huyên Pham, 2007. "A model of optimal portfolio selection under liquidity risk and price impact," Finance and Stochastics, Springer, vol. 11(1), pages 51-90, January.
    3. Bertram Düring & Michel Fournié & Ansgar Jüngel, 2001. "High order compact finite difference schemes for a nonlinear Black-Scholes equation," CoFE Discussion Paper 01-07, Center of Finance and Econometrics, University of Konstanz.
    4. Jarrow, Robert & Protter, Philip, 2005. "Large traders, hidden arbitrage, and complete markets," Journal of Banking & Finance, Elsevier, vol. 29(11), pages 2803-2820, November.
    5. Peter Bank & Dietmar Baum, 2004. "Hedging and Portfolio Optimization in Financial Markets with a Large Trader," Mathematical Finance, Wiley Blackwell, vol. 14(1), pages 1-18.
    6. Umut Çetin & H. Soner & Nizar Touzi, 2010. "Option hedging for small investors under liquidity costs," Finance and Stochastics, Springer, vol. 14(3), pages 317-341, September.
    7. Rossella Agliardi & Ramazan Gençay, 2012. "Hedging through a Limit Order Book with Varying Liquidity," Working Paper Series 12_12, The Rimini Centre for Economic Analysis.
    8. Simona Sanfelici, 2007. "Calibration of a nonlinear feedback option pricing model," Quantitative Finance, Taylor & Francis Journals, vol. 7(1), pages 95-110.
    9. Gill, Ryan & Lee, Kiseop & Song, Seongjoo, 2007. "Computation of estimates in segmented regression and a liquidity effect model," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 6459-6475, August.
    10. U. �etin & R. Jarrow & P. Protter & M. Warachka, 2006. "Pricing Options in an Extended Black Scholes Economy with Illiquidity: Theory and Empirical Evidence," Review of Financial Studies, Society for Financial Studies, vol. 19(2), pages 493-529.
    11. Bertram Düring & Michel Fournié & Ansgar Jüngel, 2004. "Convergence of a high-order compact finite difference scheme for a nonlinear Black-Scholes equation," CoFE Discussion Paper 04-02, Center of Finance and Econometrics, University of Konstanz.
    12. Umut Cetin & L.C.G. Rogers, 2007. "Modeling liquidity effects in discrete time," LSE Research Online Documents on Economics 2844, London School of Economics and Political Science, LSE Library.
    13. Ku, Hyejin & Lee, Kiseop & Zhu, Huaiping, 2012. "Discrete time hedging with liquidity risk," Finance Research Letters, Elsevier, vol. 9(3), pages 135-143.
    14. Alexandre Roch, 2011. "Liquidity risk, price impacts and the replication problem," Finance and Stochastics, Springer, vol. 15(3), pages 399-419, September.
    15. Bank, Peter & Baum, Dietmar, 2002. "Hedging and portfolio optimization in illiquid financial markets," SFB 373 Discussion Papers 2002,53, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    16. Mattias Jonsson & Jussi Keppo, 2002. "Option pricing for large agents," Applied Mathematical Finance, Taylor & Francis Journals, vol. 9(4), pages 261-272.

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