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The Pricing of Derivatives on Assets with Quadratic Volatility

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Author Info
Christian Zühlsdorff
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 garantees 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.

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Publisher Info
Paper provided by University of Bonn, Germany in its series Bonn Econ Discussion Papers with number bgse5_2002.

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Length: 29
Date of creation: Jan 2002
Date of revision:
Handle: RePEc:bon:bonedp:bgse5_2002

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Postal: Bonn Graduate School of Economics, University of Bonn, Adenauerallee 24 - 26, 53113 Bonn, Germany
Fax: +49 228 73 9221
Web page: http://www.bgse.uni-bonn.de/index.php?id=494

For technical questions regarding this item, or to correct its listing, contact: (Daniel Park).

Related research
Keywords: strong solutions; stochastic differential equation; option pricing; quadratic volatility; implied volatility; smiles; frowns;

Find related papers by JEL classification:
G12 - Financial Economics - - General Financial Markets - - - Asset Pricing
G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

This paper has been announced in the following NEP Reports:

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  1. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166. [Downloadable!] (restricted)
  2. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-54, May-June. [Downloadable!] (restricted)
  3. Leif Andersen, Jesper Andreasen, 2000. "Volatility skews and extensions of the Libor market model," Applied Mathematical Finance, Taylor and Francis Journals, vol. 7(1), pages 1-32, March. [Downloadable!] (restricted)
  4. Miltersen, Kristian R & Sandmann, Klaus & Sondermann, Dieter, 1997. " Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Journal of Finance, American Finance Association, vol. 52(1), pages 409-30, March. [Downloadable!] (restricted)
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  5. Sven Rady, 1997. "Option pricing in the presence of natural boundaries and a quadratic diffusion term (*)," Finance and Stochastics, Springer, vol. 1(4), pages 331-344. [Downloadable!] (restricted)
  6. K. Sandmann & Sandmann, K., 1995. "The Direct Approach to Debt Option Pricing," Discussion Paper Serie B 212, University of Bonn, Germany. [Downloadable!]
  7. Beniamin Goldys, 1997. "A note on pricing interest rate derivatives when forward LIBOR rates are lognormal," Finance and Stochastics, Springer, vol. 1(4), pages 345-352. [Downloadable!] (restricted)
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