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Hedging for the Long Run



In the years following the publication of Black and Scholes [7], numerous alternative models have been proposed for pricing and hedging equity derivatives. Prominent examples include stochastic volatility models, jump diffusion models, and models based on Levy processes. These all have their own shortcomings, and evidence suggests that none is up to the task of satisfactorily pricing and hedging extremely long-dated claims. Since they all fall within the ambit of risk-neutral pricing, it is thus natural to speculate that their deficiencies are (at least in part) attributable to the modelling constraints imposed by the risk-neutral approach itself. To investigate this idea, we present a simple two-parameter model for a diversifed equity accumulation index. Although our model does not admit an equivalent risk-neutral probability measure, it nevertheless fulfils a minimal no-arbitrage condition for an economically viable financial market. Furthermore, we demonstrate that contingent claims can be priced and hedged, without the need for an equivalent change of probability measure. Convenient formulae for the prices and hedge ratios of a number of standard European claims are derived, and a series of hedge experiments for extremely long-dated claims on the S&P 500 total return index are conducted. Our model serves also as a convenient medium for illustrating and clarifying several points on asset price bubbles and the economics of arbitrage.

Suggested Citation

  • Eckhard Platen & Hardy Hulley, 2008. "Hedging for the Long Run," Research Paper Series 214, Quantitative Finance Research Centre, University of Technology, Sydney.
  • Handle: RePEc:uts:rpaper:214

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    References listed on IDEAS

    1. Stein, Elias M & Stein, Jeremy C, 1991. "Stock Price Distributions with Stochastic Volatility: An Analytic Approach," Review of Financial Studies, Society for Financial Studies, vol. 4(4), pages 727-752.
    2. Jun Liu, 2004. "Losing Money on Arbitrage: Optimal Dynamic Portfolio Choice in Markets with Arbitrage Opportunities," Review of Financial Studies, Society for Financial Studies, vol. 17(3), pages 611-641.
    3. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    4. Bates, David S, 1996. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options," Review of Financial Studies, Society for Financial Studies, vol. 9(1), pages 69-107.
    5. Eckhard Platen, 2001. "Arbitrage in Continuous Complete Markets," Research Paper Series 72, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. Shleifer, Andrei & Vishny, Robert W, 1997. " The Limits of Arbitrage," Journal of Finance, American Finance Association, vol. 52(1), pages 35-55, March.
    7. Schroder, Mark Douglas, 1989. " Computing the Constant Elasticity of Variance Option Pricing Formula," Journal of Finance, American Finance Association, vol. 44(1), pages 211-219, March.
    8. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    9. 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.
    10. 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.
    11. repec:dau:papers:123456789/1392 is not listed on IDEAS
    12. Peter Carr & Helyette Geman, 2002. "The Fine Structure of Asset Returns: An Empirical Investigation," The Journal of Business, University of Chicago Press, vol. 75(2), pages 305-332, April.
    13. Helyette Geman & P. Carr & D. Madan & M. Yor, 2003. "Stochastic Volatility for Levy Processes," Post-Print halshs-00144385, HAL.
    14. Yacine Ait-Sahalia, 2002. "Maximum Likelihood Estimation of Discretely Sampled Diffusions: A Closed-form Approximation Approach," Econometrica, Econometric Society, vol. 70(1), pages 223-262, January.
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    Cited by:

    1. Zhi Jun Guo & Eckhard Platen, 2012. "The Small And Large Time Implied Volatilities In The Minimal Market Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(08), pages 1-23.
    2. Jan Baldeaux & Eckhard Platen, 2013. "Liability Driven Investments under a Benchmark Based Approach," Research Paper Series 325, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Travis Fisher & Sergio Pulido & Johannes Ruf, 2015. "Financial Models with Defaultable Num\'eraires," Papers 1511.04314,, revised Oct 2017.
    4. Hardy Hulley, 2009. "Strict Local Martingales in Continuous Financial Market Models," PhD Thesis, Finance Discipline Group, UTS Business School, University of Technology, Sydney, number 19, january-d.
    5. Kevin Fergusson & Eckhard Platen, 2015. "Less Expensive Pricing and Hedging of Long-Dated Equity Index Options When Interest Rates are Stochastic," Research Paper Series 357, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. Baldeaux Jan & Ignatieva Katja & Platen Eckhard, 2014. "A tractable model for indices approximating the growth optimal portfolio," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 18(1), pages 1-21, February.
    7. Travis Fisher & Sergio Pulido & Johannes Ruf, 2017. "Financial Models with Defaultable Numéraires," Working Papers hal-01240736, HAL.
    8. Johannes Ruf, 2010. "Hedging under arbitrage," Papers 1003.4797,, revised May 2011.
    9. Eckhard Platen, 2008. "The Law of Minimal Price," Research Paper Series 215, Quantitative Finance Research Centre, University of Technology, Sydney.
    10. Xavier Warin, 2016. "The Asset Liability Management problem of a nuclear operator : a numerical stochastic optimization approach," Papers 1611.04877,
    11. Eckhard Platen, 2009. "Real World Pricing of Long Term Contracts," Research Paper Series 262, Quantitative Finance Research Centre, University of Technology, Sydney.
    12. Ralph Rudd & Thomas A. McWalter & Joerg Kienitz & Eckhard Platen, 2018. "Quantization Under the Real-world Measure: Fast and Accurate Valuation of Long-dated Contracts," Papers 1801.07044,, revised Jan 2018.
    13. Daniel Fernholz & Ioannis Karatzas, 2010. "On optimal arbitrage," Papers 1010.4987,
    14. repec:wsi:afexxx:v:12:y:2017:i:02:n:s2010495217500105 is not listed on IDEAS

    More about this item


    long-dated claims; risk-neutral pricing; real-world pricing; arbitrage; minimal market model; squared Bessel processes; hedge simulations; asset price bubbles;

    JEL classification:

    • G10 - Financial Economics - - General Financial Markets - - - General (includes Measurement and Data)
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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