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The pricing of derivatives on assets with quadratic volatility

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  • Christian Zuhlsdorff

Abstract

The basic model of financial economics is the Samuelson model of geometric Brownian motion because of the celebrated Black-Scholes formula for pricing the call option. The asset's volatility is a linear function of the asset value and the model guarantees positive asset prices. In this paper, it is shown that the pricing partial differential equation can be solved for level-dependent volatility which is a quadratic polynomial. If zero is attainable, both absorption and negative asset values are possible. Explicit formulae are derived for the call option: a generalization of the Black-Scholes formula for an asset whose volatiliy is affine, the formula for the Bachelier model with constant volatility, and new formulae in the case of quadratic volatility. The implied Black-Scholes volatilities of the Bachelier and the affine model are frowns, the quadratic specifications imply smiles.

Suggested Citation

  • Christian Zuhlsdorff, 2001. "The pricing of derivatives on assets with quadratic volatility," Applied Mathematical Finance, Taylor & Francis Journals, vol. 8(4), pages 235-262.
    • Handle: RePEc:taf:apmtfi:v:8:y:2001:i:4:p:235-262
    DOI: 10.1080/13504860210127271
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    References listed on IDEAS

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    Cited by:

    1. Antoine Jacquier & Martin Keller-Ressel, 2015. "Implied volatility in strict local martingale models," Papers 1508.04351, arXiv.org.
    2. Yishen Li & Jin Zhang, 2004. "Option pricing with Weyl-Titchmarsh theory," Quantitative Finance, Taylor & Francis Journals, vol. 4(4), pages 457-464.
    3. Leif Andersen, 2011. "Option pricing with quadratic volatility: a revisit," Finance and Stochastics, Springer, vol. 15(2), pages 191-219, June.

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