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Black Scholes for Portfolios of Options in Discrete Time: the Price is Right, the Hedge is wrong

Author

Listed:
  • Bas Peeters

    () (Faculty of Economics and Business Administration, Vrije Universiteit Amsterdam)

  • Cees L. Dert

    () (Faculty of Economics and Business Administration, Vrije Universiteit Amsterdam)

  • André Lucas

    () (Faculty of Economics and Business Administration, Vrije Universiteit Amsterdam)

Abstract

Taking a portfolio perspective on option pricing and hedging, we show that within the standard Black-Scholes-Merton framework large portfolios of options can be hedged without risk in discrete time. The nature of the hedge portfolio in the limit of large portfolio size is substantially different from the standard continuous time delta-hedge. The underlying values of the options in our framework are driven by systematic and idiosyncratic risk factors. Instead of linearly (delta) hedging the total risk of each option separately, the correct hedge portfolio in discrete time eliminates linear (delta) as well as second (gamma) and higher order exposures to the systematic risk factor only. The idiosyncratic risk is not hedged, but diversified. Our result shows that preference free valuation of option portfolios using linear assets only is applicable in discrete time as well. The price paid for this result is that the number of securities in the portfolio has to grow indefinitely. This ties the literature on option pricing and hedging closer together with the APT literature in its focus on systematic risk factors. For portfolios of finite size, the optimal hedge strategy makes a trade-off between hedging linear idiosyncratic and higher order systematic risk.

Suggested Citation

  • Bas Peeters & Cees L. Dert & André Lucas, 2003. "Black Scholes for Portfolios of Options in Discrete Time: the Price is Right, the Hedge is wrong," Tinbergen Institute Discussion Papers 03-090/2, Tinbergen Institute.
    • Handle: RePEc:tin:wpaper:20030090
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    References listed on IDEAS

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    1. Fama, Eugene F & French, Kenneth R, 1992. "The Cross-Section of Expected Stock Returns," Journal of Finance, American Finance Association, vol. 47(2), pages 427-465, June.
    2. Ross, Stephen A., 1976. "The arbitrage theory of capital asset pricing," Journal of Economic Theory, Elsevier, vol. 13(3), pages 341-360, December.
    3. Leland, Hayne E, 1985. "Option Pricing and Replication with Transactions Costs," Journal of Finance, American Finance Association, vol. 40(5), pages 1283-1301, December.
    4. Rubinstein, Mark, 1984. "A Simple Formula for the Expected Rate of Return of an Option over a Finite Holding Period," Journal of Finance, American Finance Association, vol. 39(5), pages 1503-1509, December.
    5. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    6. Gilster, John E., 1990. "The Systematic Risk of Discretely Rebalanced Option Hedges," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 25(4), pages 507-516, December.
    7. Antonio S. Mello & Henrik J. Neuhaus, 1998. "A Portfolio Approach to Risk Reduction in Discretely Rebalanced Option Hedges," Management Science, INFORMS, vol. 44(7), pages 921-934, July.
    8. Stephen A. Ross, 2013. "The Arbitrage Theory of Capital Asset Pricing," World Scientific Book Chapters, in: Leonard C MacLean & William T Ziemba (ed.), HANDBOOK OF THE FUNDAMENTALS OF FINANCIAL DECISION MAKING Part I, chapter 1, pages 11-30, World Scientific Publishing Co. Pte. Ltd..
    9. Boyle, Phelim P & Vorst, Ton, 1992. "Option Replication in Discrete Time with Transaction Costs," Journal of Finance, American Finance Association, vol. 47(1), pages 271-293, March.
    10. Peter Carr & Katrina Ellis & Vishal Gupta, 1998. "Static Hedging of Exotic Options," Journal of Finance, American Finance Association, vol. 53(3), pages 1165-1190, June.
    11. Harry Markowitz, 1952. "Portfolio Selection," Journal of Finance, American Finance Association, vol. 7(1), pages 77-91, March.
    12. John Lintner, 1965. "Security Prices, Risk, And Maximal Gains From Diversification," Journal of Finance, American Finance Association, vol. 20(4), pages 587-615, December.
    13. Y.M. Kabanov & D.O. Kramkov, 1998. "Asymptotic arbitrage in large financial markets," Finance and Stochastics, Springer, vol. 2(2), pages 143-172.
    14. Fama, Eugene F. & French, Kenneth R., 1993. "Common risk factors in the returns on stocks and bonds," Journal of Financial Economics, Elsevier, vol. 33(1), pages 3-56, February.
    15. William F. Sharpe, 1964. "Capital Asset Prices: A Theory Of Market Equilibrium Under Conditions Of Risk," Journal of Finance, American Finance Association, vol. 19(3), pages 425-442, September.
    16. Boyle, Phelim P. & Emanuel, David, 1980. "Discretely adjusted option hedges," Journal of Financial Economics, Elsevier, vol. 8(3), pages 259-282, September.
    17. 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 637-654, May-June.
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    More about this item

    Keywords

    Option Hedging; Discrete Time; Portfolio Approach; Preference Free Valuation; Hedging Errors; Arbitrage Pricing Theory;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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