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Linear stochastic volatility models

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  • Jacek Jakubowski
  • Maciej Wisniewolski

Abstract

In this paper we investigate general linear stochastic volatility models with correlated Brownian noises. In such models the asset price satisfies a linear SDE with coefficient of linearity being the volatility process. This class contains among others Black-Scholes model, a log-normal stochastic volatility model and Heston stochastic volatility model. For a linear stochastic volatility model we derive representations for the probability density function of the arbitrage price of a financial asset and the prices of European call and put options. A closed-form formulae for the density function and the prices of European call and put options are given for log-normal stochastic volatility model. We also obtain present some new results for Heston and extended Heston stochastic volatility models.

Suggested Citation

  • Jacek Jakubowski & Maciej Wisniewolski, 2009. "Linear stochastic volatility models," Papers 0909.4765, arXiv.org, revised May 2013.
    • Handle: RePEc:arx:papers:0909.4765
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    References listed on IDEAS

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    1. Y. Maghsoodi, 2007. "Exact Solution Of A Martingale Stochastic Volatility Option Problem And Its Empirical Evaluation," Mathematical Finance, Wiley Blackwell, vol. 17(2), pages 249-265, April.
    2. Marc Romano & Nizar Touzi, 1997. "Contingent Claims and Market Completeness in a Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 7(4), pages 399-412, October.
    3. Hull, John C & White, Alan D, 1987. "The Pricing of Options on Assets with Stochastic Volatilities," Journal of Finance, American Finance Association, vol. 42(2), pages 281-300, June.
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