Please use this identifier to cite or link to this item: http://hdl.handle.net/10419/78398
Authors: 
Zühlsdorff, Christian
Year of Publication: 
2002
Series/Report no.: 
Bonn Econ Discussion Papers 5/2002
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 garantees 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.
Subjects: 
strong solutions
stochastic differential equation
option pricing
quadratic volatility
implied volatility
smiles
frowns
JEL: 
G12
G13
Document Type: 
Working Paper

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