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A Black–Scholes inequality: applications and generalisations

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  • Michael R. Tehranchi

    (University of Cambridge)

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.

Suggested Citation

  • Michael R. Tehranchi, 2020. "A Black–Scholes inequality: applications and generalisations," Finance and Stochastics, Springer, vol. 24(1), pages 1-38, January.
    • Handle: RePEc:spr:finsto:v:24:y:2020:i:1:d:10.1007_s00780-019-00410-6
    DOI: 10.1007/s00780-019-00410-6
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    References listed on IDEAS

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    Cited by:

    1. Claude Martini & Arianna Mingone, 2021. "Explicit no arbitrage domain for sub-SVIs via reparametrization," Papers 2106.02418, arXiv.org.
    2. Arianna Mingone, 2022. "No arbitrage global parametrization for the eSSVI volatility surface," Papers 2204.00312, arXiv.org.
    3. Arianna Mingone, 2022. "Smiles in delta," Papers 2209.00406, arXiv.org.
    4. Claude Martini & Arianna Mingone, 2020. "No arbitrage SVI," Papers 2005.03340, arXiv.org, revised May 2021.

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    More about this item

    Keywords

    Semigroup with involution; Implied volatility; Peacock; Lift zonoid; Log-concavity;
    All these keywords.

    JEL classification:

    • D52 - Microeconomics - - General Equilibrium and Disequilibrium - - - Incomplete Markets
    • D53 - Microeconomics - - General Equilibrium and Disequilibrium - - - Financial Markets
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates

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