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Path dependent volatility

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  • Paolo Foschi
  • Andrea Pascucci

Abstract

We propose a general class of non-constant volatility models with dependence on the past. The framework includes path-dependent volatility models such as that by Hobson&Rogers and also path dependent contracts such as options of Asian style. A key feature of the model is that market completeness is preserved. Some empirical analysis, based on the comparison with the performance of standard local volatility and Heston models, shows the effectiveness of the path dependent volatility.
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Suggested Citation

  • Paolo Foschi & Andrea Pascucci, 2008. "Path dependent volatility," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 31(1), pages 13-32, May.
    • Handle: RePEc:spr:decfin:v:31:y:2008:i:1:p:13-32
    DOI: 10.1007/s10203-007-0076-6
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    References listed on IDEAS

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    1. Robert C. Merton, 2005. "Theory of rational option pricing," World Scientific Book Chapters, in: Sudipto Bhattacharya & George M Constantinides (ed.), Theory Of Valuation, chapter 8, pages 229-288, World Scientific Publishing Co. Pte. Ltd..
    2. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    3. Andrea Pascucci, 2008. "Free boundary and optimal stopping problems for American Asian options," Finance and Stochastics, Springer, vol. 12(1), pages 21-41, January.
    4. Andrea Pascucci & Marco Di Francesco, 2005. "On the complete model with stochastic volatility by Hobson and Rogers," Finance 0503013, University Library of Munich, Germany.
    5. Andrea Pascucci & Paolo Foschi, 2005. "Calibration of the Hobson&Rogers model: empirical tests," Finance 0509020, University Library of Munich, Germany.
    6. Carl Chiarella & Oh-Kang Kwon, 2000. "A Complete Stochastic Volatility Model in the HJM Framework," Research Paper Series 43, Quantitative Finance Research Centre, University of Technology, Sydney.
    7. Rama Cont, 2006. "Model Uncertainty And Its Impact On The Pricing Of Derivative Instruments," Mathematical Finance, Wiley Blackwell, vol. 16(3), pages 519-547, July.
    8. 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 637-654, May-June.
    9. David G. Hobson & L. C. G. Rogers, 1998. "Complete Models with Stochastic Volatility," Mathematical Finance, Wiley Blackwell, vol. 8(1), pages 27-48, January.
    10. Carol Alexander & Leonardo M. Nogueira, 2006. "Hedging Options with Scale-Invariant Models," ICMA Centre Discussion Papers in Finance icma-dp2006-03, Henley Business School, University of Reading.
    11. Rama Cont, 2006. "Model uncertainty and its impact on the pricing of derivative instruments," Post-Print halshs-00002695, HAL.
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    Citations

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    Cited by:

    1. Marcel Nutz & Andr'es Riveros Valdevenito, 2023. "On the Guyon-Lekeufack Volatility Model," Papers 2307.01319, arXiv.org, revised Jul 2024.
    2. Jazaerli, Samy & F. Saporito, Yuri, 2017. "Functional Itô calculus, path-dependence and the computation of Greeks," Stochastic Processes and their Applications, Elsevier, vol. 127(12), pages 3997-4028.
    3. Sekine, Jun, 2008. "Marginal distribution of some path-dependent stochastic volatility model," Statistics & Probability Letters, Elsevier, vol. 78(13), pages 1846-1850, September.
    4. Andrea Pascucci, 2008. "Free boundary and optimal stopping problems for American Asian options," Finance and Stochastics, Springer, vol. 12(1), pages 21-41, January.
    5. Ofelia Bonesini & Giorgia Callegaro & Martino Grasselli & Gilles Pag`es, 2023. "From elephant to goldfish (and back): memory in stochastic Volterra processes," Papers 2306.02708, arXiv.org, revised Jan 2025.
    6. Marcel Nutz & Andrés Riveros Valdevenito, 2024. "On the Guyon–Lekeufack volatility model," Finance and Stochastics, Springer, vol. 28(4), pages 1203-1223, October.
    7. Mauro Rosestolato & Tiziano Vargiolu & Giovanna Villani, 2013. "Robustness for path-dependent volatility models," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 36(2), pages 137-167, November.
    8. Carey, Alexander, 2008. "Natural volatility and option pricing," MPRA Paper 6709, University Library of Munich, Germany.
    9. Foschi, Paolo & Pascucci, Andrea, 2009. "Calibration of a path-dependent volatility model: Empirical tests," Computational Statistics & Data Analysis, Elsevier, vol. 53(6), pages 2219-2235, April.
    10. Jun Sekine, 2008. "A Note On The Risk-Premium Process In An Equilibrium," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 11(07), pages 705-716.
    11. Jim Gatheral & Paul Jusselin & Mathieu Rosenbaum, 2020. "The quadratic rough Heston model and the joint S&P 500/VIX smile calibration problem," Papers 2001.01789, arXiv.org.

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    More about this item

    Keywords

    Option pricing; Kolmogorov equations; Volatility modeling; CO 2 ; 35K65; 91B28;
    All these keywords.

    JEL classification:

    • G1 - Financial Economics - - General Financial Markets
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics

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