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Deterministic implied volatility models

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  • P. Balland

Abstract

In this paper, we characterize two deterministic implied volatility models, defined by assuming that either the per-delta or the per-strike implied volatility surface has a deterministic evolution. Practitioners have recently proposed these two models to describe two regimes of implied volatility (see Derman (1999 Risk 4 55-9)). In an arbitrage-free sticky-delta model, we show that the underlying asset price is the exponential of a process with independent increments under the unique risk neutral measure and that any square-integrable claim can be replicated up to a vanishing risk by trading portfolios of vanilla options. This latter result is similar in nature to the quasi-completeness result obtained by Bjork et al (1997 Finance Stochastics 1 141-74) for interest rate models driven by Levy processes. Finally, we show that the only arbitrage-free sticky-strike model is the standard Black-Scholes model.

Suggested Citation

  • P. Balland, 2002. "Deterministic implied volatility models," Quantitative Finance, Taylor & Francis Journals, vol. 2(1), pages 31-44.
    • Handle: RePEc:taf:quantf:v:2:y:2002:i:1:p:31-44
    DOI: 10.1088/1469-7688/2/1/303
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    Cited by:

    1. Jose Corcuera & Joao Guerra, 2010. "Dynamic complex hedging in additive markets," Quantitative Finance, Taylor & Francis Journals, vol. 10(9), pages 1023-1037.
    2. Juho Kanniainen & Martin Magris, 2018. "Option market (in)efficiency and implied volatility dynamics after return jumps," Papers 1810.12200, arXiv.org.
    3. repec:hum:wpaper:sfb649dp2005-020 is not listed on IDEAS
    4. Fengler, Matthias R. & Härdle, Wolfgang Karl & Mammen, Enno, 2005. "A dynamic semiparametric factor model for implied volatility string dynamics," SFB 649 Discussion Papers 2005-020, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    5. Martin Magris & Perttu Barholm & Juho Kanniainen, 2017. "Implied volatility smile dynamics in the presence of jumps," Papers 1711.02925, arXiv.org, revised May 2020.
    6. Toby Daglish & John Hull & Wulin Suo, 2007. "Volatility surfaces: theory, rules of thumb, and empirical evidence," Quantitative Finance, Taylor & Francis Journals, vol. 7(5), pages 507-524.
    7. Seungho Yang & Jaewook Lee, 2014. "Do affine jump-diffusion models require global calibration? Empirical studies from option markets," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 111-123, January.
    8. L. Rogers & M. Tehranchi, 2010. "Can the implied volatility surface move by parallel shifts?," Finance and Stochastics, Springer, vol. 14(2), pages 235-248, April.
    9. Fengler, Matthias R. & Härdle, Wolfgang & Mammen, Enno, 2003. "Implied volatility string dynamics," SFB 373 Discussion Papers 2003,54, Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes.
    10. Quaye, Enoch & Tunaru, Radu, 2022. "The stock implied volatility and the implied dividend volatility," Journal of Economic Dynamics and Control, Elsevier, vol. 134(C).

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