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A Black--Scholes inequality: applications and generalisation

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  • Michael R. Tehranchi

Abstract

The space of call price functions 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.

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  • Michael R. Tehranchi, 2017. "A Black--Scholes inequality: applications and generalisation," Papers 1701.03897, arXiv.org, revised Aug 2019.
    • Handle: RePEc:arx:papers:1701.03897
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    References listed on IDEAS

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    3. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
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