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Option Pricing Under Ornstein-Uhlenbeck Stochastic Volatility: A Linear Model

Author

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  • GIACOMO BORMETTI

    (Centro Studi Rischio e Sicurezza, Istituto Universitario di Studi Superiori, V.le Lungo Ticino Sforza 56, Pavia, 27100, Italy;
    Istituto Nazionale di Fisica Nucleare – Sezione di Pavia, via Bassi 6, Pavia, 27100, Italy)

  • VALENTINA CAZZOLA

    (Centro Studi Rischio e Sicurezza, Istituto Universitario di Studi Superiori, V.le Lungo Ticino Sforza 56, Pavia, 27100, Italy;
    Dipartimento di Fisica Nucleare e Teorica, Università degli Studi di Pavia, via Bassi 6, Pavia, 27100, Italy;
    Istituto Nazionale di Fisica Nucleare – Sezione di Pavia, via Bassi 6, Pavia, 27100, Italy)

  • DANILO DELPINI

    (Dipartimento di Fisica Nucleare e Teorica, Università degli Studi di Pavia, via Bassi 6, Pavia, 27100, Italy;
    Istituto Nazionale di Fisica Nucleare – Sezione di Pavia, via Bassi 6, Pavia, 27100, Italy)

Abstract

We consider the problem of option pricing under stochastic volatility models, focusing on the linear approximation of the two processes known as exponential Ornstein-Uhlenbeck and Stein-Stein. Indeed, we show they admit the same limit dynamics in the regime of low fluctuations of the volatility process, under which we derive the exact expression of the characteristic function associated to the risk neutral probability density. This expression allows us to compute option prices exploiting a formula derived by Lewis and Lipton. We analyze in detail the case of Plain Vanilla calls, being liquid instruments for which reliable implied volatility surfaces are available. We also compute the analytical expressions of the first four cumulants, that are crucial to implement a simple two steps calibration procedure. It has been tested against a data set of options traded on the Milan Stock Exchange. The data analysis that we present reveals a good fit with the market implied surfaces and corroborates the accuracy of the linear approximation.

Suggested Citation

  • Giacomo Bormetti & Valentina Cazzola & Danilo Delpini, 2010. "Option Pricing Under Ornstein-Uhlenbeck Stochastic Volatility: A Linear Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 13(07), pages 1047-1063.
    • Handle: RePEc:wsi:ijtafx:v:13:y:2010:i:07:n:s0219024910006108
    DOI: 10.1142/S0219024910006108
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    References listed on IDEAS

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    1. Alexander Lipton, 2001. "Mathematical Methods for Foreign Exchange:A Financial Engineer's Approach," World Scientific Books, World Scientific Publishing Co. Pte. Ltd., number 4694, February.
    2. Stefano Galluccio & Yann Le Cam, 2005. "Implied Calibration of Stochastic Volatility Jump Diffusion Models," Finance 0510028, University Library of Munich, Germany.
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    Cited by:

    1. Kallinterakis, Vasileios & Liu, Fei & Pantelous, Athanasios A. & Shao, Jia, 2020. "Pricing inefficiencies and feedback trading: Evidence from country ETFs," International Review of Financial Analysis, Elsevier, vol. 70(C).

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