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A Simple Model for Option Pricing with Jumping Stochastic Volatility

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  • Stefano Herzel

    (Ist. Mat. Gen. Fin., Università di Ferugia, Facoltà di Economia, Via A. Pascoli 1, 06100 Perugia, Italy)

Abstract

This paper proposes a simple modification of the Black–Scholes model by assuming that the volatility of the stock may jump at a random time τ from a valueσato a valueσb. It shows that, if the market price of volatility risk is unknown, but constant, all contingent claims can be valued from the actual priceC0, of some arbitrarily chosen "basis" option. Closed form solutions for the prices of European options as well as explicit formulas forvegaanddeltahedging are given. All such solutions only depend onσa,σbandC0. The prices generated by the model produce a "smile"-shaped curve of the implied volatility.

Suggested Citation

  • Stefano Herzel, 1998. "A Simple Model for Option Pricing with Jumping Stochastic Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 1(04), pages 487-505.
    • Handle: RePEc:wsi:ijtafx:v:01:y:1998:i:04:n:s0219024998000266
    DOI: 10.1142/S0219024998000266
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    Cited by:

    1. M. Montero, 2004. "Partial derivative approach for option pricing in a simple stochastic volatility model," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 42(1), pages 141-153, November.
    2. Xin‐Jiang He & Song‐Ping Zhu, 2018. "On full calibration of hybrid local volatility and regime‐switching models," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 38(5), pages 586-606, May.
    3. Chilarescu, Constantin & Viasu, Iana Luciana, 2011. "Phénomènes financiers et mélange de lois : Une nouvelle méthode d’estimation des paramètres," MPRA Paper 33909, University Library of Munich, Germany.

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