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General Black-Scholes models accounting for increased market volatility from hedging strategies

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  • K. Ronnie Sircar
  • George Papanicolaou
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    Abstract

    Increases in market volatility of asset prices have been observed and analysed in recent years and their cause has generally been attributed to the popularity of portfolio insurance strategies for derivative securities. The basis of derivative pricing is the Black-Scholes model and its use is so extensive that it is likely to influence the market itself. In particular it has been suggested that this is a factor in the rise in volatilities. A class of pricing models is presented that accounts for the feedback effect from the Black-Scholes dynamic hedging strategies on the price of the asset, and from there back onto the price of the derivative. These models do predict increased implied volatilities with minimal assumptions beyond those of the Black-Scholes theory. They are characterized by a nonlinear partial differential equation that reduces to the Black-Scholes equation when the feedback is removed. We begin with a model economy consisting of two distinct groups of traders: reference traders who are the majority investing in the asset expecting gain, and programme traders who trade the asset following a Black-Scholes type dynamic hedging strategy, which is not known a priori, in order to insure against the risk of a derivative security. The interaction of these groups leads to a stochastic process for the price of the asset which depends on the hedging strategy of the programme traders. Then following a Black-Scholes argument, we derive nonlinear partial differential equations for the derivative price and the hedging strategy. Consistency with the traditional Black-Scholes model characterizes the class of feedback models that we analyse in detail. We study the nonlinear partial differential equation for the price of the derivative by perturbation methods when the programme traders are a small fraction of the economy, by numerical methods, which are easy to use and can be implemented efficiently, and by analytical methods. The results clearly support the observed increasing volatility phenomenon and provide a quantitative explanation for it.

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    File URL: http://www.tandfonline.com/doi/abs/10.1080/135048698334727
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    Bibliographic Info

    Article provided by Taylor & Francis Journals in its journal Applied Mathematical Finance.

    Volume (Year): 5 (1998)
    Issue (Month): 1 ()
    Pages: 45-82

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    Handle: RePEc:taf:apmtfi:v:5:y:1998:i:1:p:45-82

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    Web page: http://www.tandfonline.com/RAMF20

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    Related research

    Keywords: Black-scholes Model; Dynamic Hedging; Feedback Effects; Option Pricing; Volatility;

    References

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    1. Gerard Gennotte and Hayne Leland., 1989. "Market Liquidity, Hedging and Crashes," Research Program in Finance Working Papers RPF-184, University of California at Berkeley.
    2. Eric Renault & Nizar Touzi, 1996. "Option Hedging And Implied Volatilities In A Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 6(3), pages 279-302.
    3. Brennan, Michael J & Schwartz, Eduardo S, 1989. "Portfolio Insurance and Financial Market Equilibrium," The Journal of Business, University of Chicago Press, vol. 62(4), pages 455-72, October.
    4. 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.
    5. Hans Föllmer & Martin Schweizer, 1993. "A Microeconomic Approach to Diffusion Models For Stock Prices," Mathematical Finance, Wiley Blackwell, vol. 3(1), pages 1-23.
    6. Bick, Avi, 1987. "On the Consistency of the Black-Scholes Model with a General Equilibrium Framework," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(03), pages 259-275, September.
    7. 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.
    8. Frey, Rüdiger & Alexander Stremme, 1995. "Market Volatility and Feedback Effects from Dynamic Hedging," Discussion Paper Serie B 310, University of Bonn, Germany.
    9. Frey, Rüdiger, 1996. "The Pricing and Hedging of Options in Finitely Elastic Markets," Discussion Paper Serie B 372, University of Bonn, Germany.
    10. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
    11. Gregory Duffee & Paul Kupiec & Patricia White, 1990. "A primer on program trading and stock price volatility: a survey of the issues and the evidence," Finance and Economics Discussion Series 109, Board of Governors of the Federal Reserve System (U.S.).
    12. Hull, John C & White, Alan D, 1987. " The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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    Cited by:
    1. Celso Brunetti & Alessio Caldarera, 2006. "Asset Prices and asset Correlations in Illiquid Markets," Computing in Economics and Finance 2006 331, Society for Computational Economics.
    2. K. Ronnie Sircar & George Papanicolaou, 1999. "Stochastic volatility, smile & asymptotics," Applied Mathematical Finance, Taylor & Francis Journals, vol. 6(2), pages 107-145.
    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.

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