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Hedging with Stochastic and Local Volatility

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Author Info

  • Carol Alexander

    ()
    (ICMA Centre, University of Reading)

  • Leonardo M. Nogueira

    ()
    (ICMA Centre, University of Reading)

Abstract

We derive the local volatility hedge ratios that are consistent with a stochastic instantaneous volatility and show that this ‘stochastic local volatility’ model is equivalent to the market model for implied volatilities. We also show that a common feature of all Markovian single factor stochastic volatility models, (log)normal mixture option pricing models and ‘sticky delta’ models is that they predict incorrect dynamics for implied volatility. As a result they over-hedge the Black-Scholes model in the presence of a market skew and this explains the poor delta hedging performance of these models reported in the literature. Whilst the traditional ‘sticky tree’ local volatility models do not possess this unfortunate property, they cannot be used for pricing without exogenous and ad hoc smoothing of results. However the stochastic local volatility framework allows one to extend a good pricing model into a good hedging model. The theoretical results are supported by an empirical analysis of the hedging performance of seven models, each with different volatility characteristics, on the SP500 index skew.

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Bibliographic Info

Paper provided by Henley Business School, Reading University in its series ICMA Centre Discussion Papers in Finance with number icma-dp2004-10.

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Length: 47 pages
Date of creation: Jul 2004
Date of revision: Dec 2004
Handle: RePEc:rdg:icmadp:icma-dp2004-10

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Related research

Keywords: Local volatility; stochastic volatility; implied volatility; hedging; dynamic delta hedging; volatility dymamics;

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References

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  1. Schroder, Mark Douglas, 1989. " Computing the Constant Elasticity of Variance Option Pricing Formula," Journal of Finance, American Finance Association, vol. 44(1), pages 211-19, March.
  2. H. Berestycki & J. Busca & I. Florent, 2002. "Asymptotics and calibration of local volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 61-69.
  3. Marco Avellaneda & Craig Friedman & Richard Holmes & Dominick Samperi, 1997. "Calibrating volatility surfaces via relative-entropy minimization," Applied Mathematical Finance, Taylor & Francis Journals, vol. 4(1), pages 37-64.
  4. Bates, David S., 2000. "Post-'87 crash fears in the S&P 500 futures option market," Journal of Econometrics, Elsevier, vol. 94(1-2), pages 181-238.
  5. Matthias Fengler & Wolfgang Härdle & Christophe Villa, 2003. "The Dynamics of Implied Volatilities: A Common Principal Components Approach," Review of Derivatives Research, Springer, vol. 6(3), pages 179-202, October.
  6. 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.
  7. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-43.
  8. Bernard Dumas & Jeff Fleming & Robert E. Whaley, 1998. "Implied Volatility Functions: Empirical Tests," Journal of Finance, American Finance Association, vol. 53(6), pages 2059-2106, December.
  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. Heath, David & Jarrow, Robert & Morton, Andrew, 1992. "Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation," Econometrica, Econometric Society, vol. 60(1), pages 77-105, January.
  11. Alexander, Carol, 2004. "Normal mixture diffusion with uncertain volatility: Modelling short- and long-term smile effects," Journal of Banking & Finance, Elsevier, vol. 28(12), pages 2957-2980, December.
  12. Peter Carr & Katrina Ellis & Vishal Gupta, 1998. "Static Hedging of Exotic Options," Journal of Finance, American Finance Association, vol. 53(3), pages 1165-1190, 06.
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Cited by:
  1. Abdelkoddousse Ahdida & Aurélien Alfonsi, 2013. "A Mean-Reverting SDE on Correlation matrices," Post-Print hal-00617111, HAL.

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