 The pricing of derivatives on assets with quadratic volatilityChristian Zuhlsdorff Applied Mathematical Finance, 2001, vol. 8, issue 4, 235-262 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 guarantees 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. Keywords: Strong Solutions; Stochastic Differential Equation; Option Pricing; Quadratic Volatility; Implied Volatility; Smiles; Frowns (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:taf:apmtfi:v:8:y:2001:i:4:p:235-262 Ordering information: This journal article can be ordered from DOI: 10.1080/13504860210127271 Access Statistics for this article Applied Mathematical Finance is currently edited by Professor Ben Hambly and Christoph Reisinger More articles in Applied Mathematical Finance from Taylor & Francis Journals |
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