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Explicit implied volatilities for multifactor local-stochastic volatility models


  • Matthew Lorig
  • Stefano Pagliarani
  • Andrea Pascucci


We consider an asset whose risk-neutral dynamics are described by a general class of local-stochastic volatility models and derive a family of asymptotic expansions for European-style option prices and implied volatilities. Our implied volatility expansions are explicit; they do not require any special functions nor do they require numerical integration. To illustrate the accuracy and versatility of our method, we implement it under five different model dynamics: CEV local volatility, quadratic local volatility, Heston stochastic volatility, $3/2$ stochastic volatility, and SABR local-stochastic volatility.

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  • Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Explicit implied volatilities for multifactor local-stochastic volatility models," Papers 1306.5447,, revised Nov 2014.
  • Handle: RePEc:arx:papers:1306.5447

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    References listed on IDEAS

    1. Jan Baldeaux & Alexander Badran, 2014. "Consistent Modelling of VIX and Equity Derivatives Using a 3/2 plus Jumps Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(4), pages 299-312, September.
    2. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Pricing approximations and error estimates for local L\'evy-type models with default," Papers 1304.1849,, revised Nov 2014.
    3. Leif Andersen, 2011. "Option pricing with quadratic volatility: a revisit," Finance and Stochastics, Springer, vol. 15(2), pages 191-219, June.
    4. Pagliarani, Stefano & Pascucci, Andrea, 2011. "Analytical approximation of the transition density in a local volatility model," MPRA Paper 31107, University Library of Munich, Germany.
    5. Martin Forde & Antoine Jacquier, 2011. "Small-Time Asymptotics for an Uncorrelated Local-Stochastic Volatility Model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 18(6), pages 517-535, April.
    6. 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. Matthew Lorig, 2014. "Indifference prices and implied volatilities," Papers 1412.5520,, revised Sep 2015.
    2. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "A Taylor series approach to pricing and implied vol for LSV models," Papers 1308.5019,
    3. Matthew Lorig & Ronnie Sircar, 2015. "Portfolio Optimization under Local-Stochastic Volatility: Coefficient Taylor Series Approximations & Implied Sharpe Ratio," Papers 1506.06180,
    4. Weston Barger & Matthew Lorig, 2016. "Approximate pricing of European and Barrier claims in a local-stochastic volatility setting," Papers 1610.05728,, revised Apr 2017.
    5. Ankush Agarwal & Ronnie Sircar, 2017. "Portfolio Benchmarking under Drawdown Constraint and Stochastic Sharpe Ratio," Working Papers hal-01388399, HAL.
    6. Tim Leung & Matthew Lorig, 2015. "Optimal Static Quadratic Hedging," Papers 1506.02074,, revised Nov 2015.
    7. Ankush Agarwal & Ronnie Sircar, 2016. "Portfolio Benchmarking under Drawdown Constraint and Stochastic Sharpe Ratio," Papers 1610.08558,
    8. Akihiko Takahashi & Toshihiro Yamada, 2014. "This paper proposes a unified method for precise estimates of the error bounds in asymptotic expansions of an option price and its Greeks (sensitivities) under a stochastic volatility model. More gene," CARF F-Series CARF-F-347, Center for Advanced Research in Finance, Faculty of Economics, The University of Tokyo, revised Sep 2014.

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