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The Pricing of Derivatives on Assets with Quadratic Volatility

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  • Zühlsdorff, Christian

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.

Suggested Citation

  • Zühlsdorff, Christian, 2002. "The Pricing of Derivatives on Assets with Quadratic Volatility," Bonn Econ Discussion Papers 5/2002, University of Bonn, Bonn Graduate School of Economics (BGSE).
    • Handle: RePEc:zbw:bonedp:52002
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    References listed on IDEAS

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    More about this item

    Keywords

    strong solutions; stochastic differential equation; option pricing; quadratic volatility; implied volatility; smiles; frowns;
    All these keywords.

    JEL classification:

    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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