SFB 303 Discussion Paper No. B - 451

Author: Zühlsdorff, Christian
Title: The Pricing of Derivatives on Assets with Quadratic Volatility
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 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.
Keywords: option pricing, quadratic volatility, volatility smiles
JEL-Classification-Number: G13
Creation-Date: March 1999
URL: ../1999/b/bonnsfb451.pdf

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