# 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
SFB 303 Homepage

11.03.1999, Webmaster