The Pricing of Derivatives on Assets with Quadratic Volatility

Arbeitspapier

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.

Angaben zum Objekt

• Sprache: Englisch

Verweise und Beziehungen

• Erschienen in: Series: Bonn Econ Discussion Papers ; No. 5/2002

Klassifikation und Themen

• Klassifikation: Wirtschaft ; Asset Pricing; Trading Volume; Bond Interest Rates ; Contingent Pricing; Futures Pricing; option pricing
• Thema: strong solutions ; stochastic differential equation ; option pricing ; quadratic volatility ; implied volatility ; smiles ; frowns

Beteiligte, Orts- und Zeitangaben

• Ereignis: Geistige Schöpfung
• (wer): Zühlsdorff, Christian
• Ereignis: Veröffentlichung
• (wer): University of Bonn, Bonn Graduate School of Economics (BGSE)
• (wo): Bonn
• (wann): 2002

Weitere Informationen

• Handle: http://hdl.handle.net/10419/78398
• Letzte Aktualisierung: 10.03.2025, 11:43 MEZ