The Pricing of Derivatives on Assets with Quadratic Volatility
AbstractThe 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.
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Bibliographic InfoPaper provided by University of Bonn, Germany in its series Discussion Paper Serie B with number 451.
Date of creation: Mar 1999
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option pricing; quadratic volatility; volatility smiles;
Find related papers by JEL classification:
- G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
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