A Black–Scholes inequality: applications and generalisations

Michael R. Tehranchi ()
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Michael R. Tehranchi: University of Cambridge

Finance and Stochastics, 2020, vol. 24, issue 1, No 1, 38 pages

Abstract: Abstract The space of call price curves has a natural noncommutative semigroup structure with an involution. A basic example is the Black–Scholes call price surface, from which an interesting inequality for Black–Scholes implied volatility is derived. The binary operation is compatible with the convex order, and therefore a one-parameter sub-semigroup gives rise to an arbitrage-free market model. It is shown that each such one-parameter semigroup corresponds to a unique log-concave probability density, providing a family of tractable call price surface parametrisations in the spirit of the Gatheral–Jacquier SVI surface. An explicit example is given to illustrate the idea. The key observation is an isomorphism linking an initial call price curve to the lift zonoid of the terminal price of the underlying asset.

Keywords: Semigroup with involution; Implied volatility; Peacock; Lift zonoid; Log-concavity; 60G44; 91G20; 60E15; 26A51; 20M20 (search for similar items in EconPapers)
JEL-codes: D52 D53 G12 (search for similar items in EconPapers)
Date: 2020
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Citations: View citations in EconPapers (4)

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DOI: 10.1007/s00780-019-00410-6

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