A Simple Model for Option Pricing with Jumping Stochastic Volatility

Stefano Herzel

International Journal of Theoretical and Applied Finance (IJTAF), 1998, vol. 01, issue 04, 487-505

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.

Date: 1998
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DOI: 10.1142/S0219024998000266

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