The Pricing of Derivatives on Assets with Quadratic Volatility
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 volatility is a linear function of the asset value and the model guarantees positive asset prices. We show that the the pricing PDE can be solved if the volatility function is a quadratic polynomial and give explicit formulas for the call option: a generalization of the Black-Scholes formula for an asset whose volatility is affine, a formula for the Bachelier model with constant volatility and a new formula in the case of quadratic volatility. The implied Black-Scholes volatilities of the Bachelier and the affine model are frowns, the quadratic specifications also imply smiles.
|Date of creation:||Mar 1999|
|Contact details of provider:|| Postal: Bonn Graduate School of Economics, University of Bonn, Adenauerallee 24 - 26, 53113 Bonn, Germany|
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References listed on IDEAS
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- Rady, Sven, 1994. "The Direct Approach to Debt Option Pricing," Munich Reprints in Economics 3404, University of Munich, Department of Economics.
- 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 409-430, March.
- 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.
- K. Sandmann & Sandmann, K., 1995. "The Direct Approach to Debt Option Pricing," Discussion Paper Serie B 212, University of Bonn, Germany.
- 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.
- 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 637-654, May-June. Full references (including those not matched with items on IDEAS)
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