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Hedging with Stochastic and Local Volatility

Author

Listed:
  • 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.

Suggested Citation

  • Carol Alexander & Leonardo M. Nogueira, 2004. "Hedging with Stochastic and Local Volatility," ICMA Centre Discussion Papers in Finance icma-dp2004-10, Henley Business School, Reading University, revised Dec 2004.
  • Handle: RePEc:rdg:icmadp:icma-dp2004-10
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    File URL: http://www.icmacentre.ac.uk/pdf/discussion/DP2004-11.pdf
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    References listed on IDEAS

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    4. 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 327-343.
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    Cited by:

    1. Abdelkoddousse Ahdida & Aurélien Alfonsi, 2013. "A Mean-Reverting SDE on Correlation matrices," Post-Print hal-00617111, HAL.

    More about this item

    Keywords

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

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • C16 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Econometric and Statistical Methods; Specific Distributions

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