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Coupling index and stocks

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  • Benjamin Jourdain
  • Mohamed Sbai

Abstract

In this paper, we are interested in continuous-time models in which the index level induces feedback on the dynamics of its composing stocks. More precisely, we propose a model in which the log-returns of each stock may be decomposed into a systemic part proportional to the log-returns of the index plus an idiosyncratic part. We show that, when the number of stocks in the index is large, this model may be approximated by a local volatility model for the index and a stochastic volatility model for each stock with volatility driven by the index. This result is useful from a calibration perspective: it suggests that one should first calibrate the local volatility of the index and then calibrate the dynamics of each stock. We explain how to do so in the limiting simplified model and in the original model.

Suggested Citation

  • Benjamin Jourdain & Mohamed Sbai, 2012. "Coupling index and stocks," Quantitative Finance, Taylor & Francis Journals, vol. 12(5), pages 805-818, March.
    • Handle: RePEc:taf:quantf:v:12:y:2012:i:5:p:805-818
    DOI: 10.1080/14697681003785959
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    References listed on IDEAS

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    1. Nicole Branger & Christian Schlag, 2004. "Why is the Index Smile So Steep?," Review of Finance, European Finance Association, vol. 8(1), pages 109-127.
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    6. Longstaff, Francis A & Schwartz, Eduardo S, 2001. "Valuing American Options by Simulation: A Simple Least-Squares Approach," The Review of Financial Studies, Society for Financial Studies, vol. 14(1), pages 113-147.
    7. Pierre Cizeau & Marc Potters & Jean-Philippe Bouchaud, 2000. "Correlation structure of extreme stock returns," Papers cond-mat/0006034, arXiv.org, revised Jan 2001.
    8. 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.
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    Cited by:

    1. Carole Bernard & Oleg Bondarenko & Steven Vanduffel, 2021. "A model-free approach to multivariate option pricing," Review of Derivatives Research, Springer, vol. 24(2), pages 135-155, July.
    2. Frédéric Abergel & Rémy Tachet Des Combes & Riadh Zaatour, 2017. "Nonparametric model calibration for derivatives," Post-Print hal-01686114, HAL.
    3. Frédéric Abergel & Rémy Tachet Des Combes & Riadh Zaatour, 2017. "Nonparametric model calibration for derivatives," Post-Print hal-01399542, HAL.
    4. Huy N. Chau & Duy Nguyen & Thai Nguyen, 2024. "On short-time behavior of implied volatility in a market model with indexes," Papers 2402.16509, arXiv.org, revised Mar 2025.

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