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The pricing of derivatives on assets with quadratic volatility

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  • Christian Zuhlsdorff
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    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.

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    File URL: http://www.tandfonline.com/doi/abs/10.1080/13504860210127271
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    Bibliographic Info

    Article provided by Taylor & Francis Journals in its journal Applied Mathematical Finance.

    Volume (Year): 8 (2001)
    Issue (Month): 4 ()
    Pages: 235-262

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    Handle: RePEc:taf:apmtfi:v:8:y:2001:i:4:p:235-262

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    Web page: http://www.tandfonline.com/RAMF20

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    Related research

    Keywords: Strong Solutions; Stochastic Differential Equation; Option Pricing; Quadratic Volatility; Implied Volatility; Smiles; Frowns;

    References

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    1. Miltersen, K. & K. Sandmann & D. Sondermann, 1994. "Closed Form Solutions for Term Structure Derivatives with Log-Normal Interest Rates," Discussion Paper Serie B 308, University of Bonn, Germany.
    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.
    3. K. Sandmann & Sandmann, K., 1995. "The Direct Approach to Debt Option Pricing," Discussion Paper Serie B 212, University of Bonn, Germany.
    4. Rady, Sven, 1994. "The Direct Approach to Debt Option Pricing," Munich Reprints in Economics 3404, University of Munich, Department of Economics.
    5. 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.
    6. Leif Andersen & Jesper Andreasen, 2000. "Volatility skews and extensions of the Libor market model," Applied Mathematical Finance, Taylor & Francis Journals, vol. 7(1), pages 1-32.
    7. 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.
    8. Freddy Delbaen & Walter Schachermayer, 1994. "Arbitrage And Free Lunch With Bounded Risk For Unbounded Continuous Processes," Mathematical Finance, Wiley Blackwell, vol. 4(4), pages 343-348.
    9. 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.
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    Cited by:
    1. 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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