The Pricing of Derivatives on Assets with Quadratic Volatility
University of Bonn - Institute of Statistics
SFB 303 Working Paper No. B - 451
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 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.
Number of Pages in PDF File: 10
JEL Classification: G12, G13
Date posted: November 17, 1999