Building arbitrage-free implied volatility surface by entropy minimization

17 Pages Posted: 17 May 2024

See all articles by Claude Muller

Claude Muller

Natixis

Aymane Fardadi

University of Paris-Saclay - CentraleSupélec

Date Written: May 16, 2024

Abstract

In this paper, we are interested in the building of arbitrage-free implied volatility surfaces based on relative entropy minimization, with a special focus on the numerical issues and their resolution. In the exotic pricing world, working with arbitrage-free implied volatility surfaces (in short AFIVS) is a prerequisite for the use of arbitrage-free models calibrated on them, like LV or LSV models. Moreover, as a requirement of the regulator, each implied volatility surface must reprice its quoted market vanilla prices. AFIVS are also needed for market-makers on vanillas.
We show that for any maturity of a surface, an arbitrage-free implied volatility slice can be deduced from a distribution, solving a strictly convex unconstrained minimization. The function to minimize depends on a set of european call price bounds, and an unspecified prior measure. With a particular well-chosen prior, we provide all the numerical tricks and formulas leading to an implementation of the minimization problem which has proved its robustness and efficiency within a high-frequency use.

Keywords: entropy minimization, Kullback-Leibler divergence, optimization, BFGS algorithm, smile, skew

Suggested Citation

Muller, Claude and Fardadi, Aymane, Building arbitrage-free implied volatility surface by entropy minimization (May 16, 2024). Available at SSRN: https://ssrn.com/abstract=4830934 or http://dx.doi.org/10.2139/ssrn.4830934

Aymane Fardadi

University of Paris-Saclay - CentraleSupélec ( email )

Moulon Plateau 3 Joliot Curie Street
Gif-sur-Yvette, 91190
France

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