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Exact Superreplication Strategies for a Class of Derivative Assets

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  • Joel Vanden
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    Abstract

    A superreplicating hedging strategy is commonly used when delta hedging is infeasible or is too expensive. This article provides an exact analytical solution to the superreplication problem for a class of derivative asset payoffs. The class contains common payoffs that are neither uniformly convex nor concave. A digital option, a bull spread, a bear spread, and some portfolios of bull spreads or bear spreads, are all included as special cases. The problem is approached by first solving for the transition density of a process that has a two-valued volatility. Using this process to model the underlying asset and identifying the two volatility values as σmin and σmax, the value function for any derivative asset in the class is shown to solve the Black-Scholes-Barenblatt equation. The subreplication problem and several related extensions, such as option pricing with transaction costs, calculating superreplicating bounds, and superreplication with multiple risky assets, are also addressed.

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    File URL: http://www.tandfonline.com/doi/abs/10.1080/13504860500117560
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    Bibliographic Info

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

    Volume (Year): 13 (2006)
    Issue (Month): 1 ()
    Pages: 61-87

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    Handle: RePEc:taf:apmtfi:v:13:y:2006:i:1:p:61-87

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

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    Related research

    Keywords: Superreplication; subreplication; uncertain volatility; Black-Scholes-Barenblatt equation; transaction costs;

    References

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    1. M. Avellaneda & A. Levy & A. ParAS, 1995. "Pricing and hedging derivative securities in markets with uncertain volatilities," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 73-88.
    2. 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.
    3. Marco Avellaneda & Antonio ParAS, 1996. "Managing the volatility risk of portfolios of derivative securities: the Lagrangian uncertain volatility model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 3(1), pages 21-52.
    4. Edirisinghe, Chanaka & Naik, Vasanttilak & Uppal, Raman, 1993. "Optimal Replication of Options with Transactions Costs and Trading Restrictions," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 28(01), pages 117-138, March.
    5. Y. M. Kabanov & M. Safarian, 1995. "On Leland's Strategy of Option Pricing with Transaction Costs," SFB 373 Discussion Papers 1995,65, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    6. Boyle, Phelim P & Vorst, Ton, 1992. " Option Replication in Discrete Time with Transaction Costs," Journal of Finance, American Finance Association, vol. 47(1), pages 271-93, March.
    7. Tiziano Vargiolu & Silvia Romagnoli, 2000. "Robustness of the Black-Scholes approach in the case of options on several assets," Finance and Stochastics, Springer, vol. 4(3), pages 325-341.
    8. Avellaneda Marco & ParaS Antonio, 1994. "Dynamic hedging portfolios for derivative securities in the presence of large transaction costs," Applied Mathematical Finance, Taylor & Francis Journals, vol. 1(2), pages 165-194.
    9. 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.
    10. Leland, Hayne E, 1985. " Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
    11. T. J. Lyons, 1995. "Uncertain volatility and the risk-free synthesis of derivatives," Applied Mathematical Finance, Taylor & Francis Journals, vol. 2(2), pages 117-133.
    12. Bergman, Yaacov Z & Grundy, Bruce D & Wiener, Zvi, 1996. " General Properties of Option Prices," Journal of Finance, American Finance Association, vol. 51(5), pages 1573-1610, December.
    13. Jouini Elyes & Kallal Hedi, 1995. "Martingales and Arbitrage in Securities Markets with Transaction Costs," Journal of Economic Theory, Elsevier, vol. 66(1), pages 178-197, June.
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    Cited by:
    1. Nicole Branger & Antje Mahayni, 2011. "Tractable hedging with additional hedge instruments," Review of Derivatives Research, Springer, vol. 14(1), pages 85-114, April.

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