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The skewed multifractal random walk with applications to option smiles

Author

Listed:
  • Benoit Pochard

    (Centre de mathematiques appliquees, Ecole Polytechnique, Palaiseau, FRANCE)

  • Jean-Philippe Bouchaud

    (Science & Finance, Capital Fund Management
    CEA Saclay;)

Abstract

We generalize the construction of the multifractal random walk (MRW) due to Bacry, Delour and Muzy to take into account the asymmetric character of the financial returns. We show how one can include in this class of models the observed correlation between past returns and future volatilities, in such a way that the scale invariance properties of the MRW are preserved. We compute the leading behaviour of q-moments of the process, that behave as power-laws of the time lag with an exponent zeta_q=p-2p(p-1) lambda^2 for even q=2p, as in the symmetric MRW, and as zeta_q=p(1-2p lambda^2)+1-alpha (q=2p+1), where lambda and alpha are parameters. We show that this extended model reproduces the `HARCH' effect or `causal cascade' reported by some authors. We illustrate the usefulness of this skewed MRW by computing the resulting shape of the volatility smiles generated by such a process, that we compare to approximate cumulant expansions formulas for the implied volatility. A large variety of smile surfaces can be reproduced.

Suggested Citation

  • Benoit Pochard & Jean-Philippe Bouchaud, 2002. "The skewed multifractal random walk with applications to option smiles," Science & Finance (CFM) working paper archive 0204047, Science & Finance, Capital Fund Management.
  • Handle: RePEc:sfi:sfiwpa:0204047
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    Citations

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    Cited by:

    1. Christian Walter, 2020. "Sustainable Financial Risk Modelling Fitting the SDGs: Some Reflections," Sustainability, MDPI, vol. 12(18), pages 1-28, September.
    2. Jaume Masoliver & Josep Perello, 2006. "Multiple time scales and the exponential Ornstein-Uhlenbeck stochastic volatility model," Quantitative Finance, Taylor & Francis Journals, vol. 6(5), pages 423-433.
    3. Segnon, Mawuli & Lux, Thomas, 2013. "Multifractal models in finance: Their origin, properties, and applications," Kiel Working Papers 1860, Kiel Institute for the World Economy (IfW Kiel).
    4. Subbotin, Alexandre, 2009. "Volatility Models: from Conditional Heteroscedasticity to Cascades at Multiple Horizons," Applied Econometrics, Russian Presidential Academy of National Economy and Public Administration (RANEPA), vol. 15(3), pages 94-138.
    5. Masaaki Fukasawa & Tetsuya Takabatake & Rebecca Westphal, 2022. "Consistent estimation for fractional stochastic volatility model under high‐frequency asymptotics," Mathematical Finance, Wiley Blackwell, vol. 32(4), pages 1086-1132, October.
    6. Saâdaoui, Foued, 2023. "Skewed multifractal scaling of stock markets during the COVID-19 pandemic," Chaos, Solitons & Fractals, Elsevier, vol. 170(C).
    7. Rudy Morel & Gaspar Rochette & Roberto Leonarduzzi & Jean-Philippe Bouchaud & St'ephane Mallat, 2022. "Scale Dependencies and Self-Similar Models with Wavelet Scattering Spectra," Papers 2204.10177, arXiv.org, revised Jun 2023.
    8. Lisa Borland & Jean-Philippe Bouchaud & Jean-Francois Muzy & Gilles Zumbach, 2005. "The Dynamics of Financial Markets -- Mandelbrot's multifractal cascades, and beyond," Science & Finance (CFM) working paper archive 500061, Science & Finance, Capital Fund Management.
    9. Alexander Subbotin & Thierry Chauveau & Kateryna Shapovalova, 2009. "Volatility Models: from GARCH to Multi-Horizon Cascades," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00390636, HAL.
    10. L. Borland & J. -Ph. Bouchaud, 2005. "On a multi-timescale statistical feedback model for volatility fluctuations," Papers physics/0507073, arXiv.org.
    11. Benoit Pochard & Jean-Philippe Bouchaud, 2003. "Option pricing and hedging with minimum expected shortfall," Science & Finance (CFM) working paper archive 500029, Science & Finance, Capital Fund Management.
    12. Lisa Borland & Jean-Philippe Bouchaud, 2005. "On a multi-timescale statistical feedback model for volatility fluctuations," Science & Finance (CFM) working paper archive 500059, Science & Finance, Capital Fund Management.
    13. Zoltan Eisler & Janos Kertesz, 2004. "Multifractal model of asset returns with leverage effect," Papers cond-mat/0403767, arXiv.org, revised May 2004.

    More about this item

    JEL classification:

    • G1 - Financial Economics - - General Financial Markets
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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