Convex Volatility Interpolation

30 Pages Posted: 23 May 2024 Last revised: 25 Jun 2024

Date Written: May 26, 2024


This article introduces a new method for calibrating implied volatility surfaces, termed Convex Volatility Interpolation (CVI). CVI uses a parameterization of the volatility surface calibrated using quadratic programming (QP) with linear constraints. Its dual parameterization in cubic spline and B-spline spaces maps a set of intuitive parameters to the weights of basis functions. 

As CVI has no restrictions on the number of parameters, it can fit any volatility surface. The method works consistently across all underlyings without the need for hyperparameter tuning, relying on dimensionless numbers for the parameterization and for the relative weights of the different terms in the objective function. The objective function includes various terms that penalize deviations from the mid-price or those beyond the bid-ask, as well as a regularization term to smooth the smile. 

Since CVI is model-free, it relies entirely on the optimization process to enforce the absence of static arbitrage. CVI eliminates butterfly and calendar spread arbitrage across all strikes and expiries, including in the tails. While no-calendar-spread-arbitrage constraints are linear in variance space, no-butterflyarbitrage constraints are only linear in the tails in that space. Notably, this paper details the derivation and linearization of no-butterfly-arbitrage constraints within the CVI cubic spline parameter space. 

This study also highlights the effectiveness of Clarabel, a state-of-the-art, open-source convex optimization solver, in handling the CVI optimization problem. For most underlyings, the calibration time is measured in hundredths of a second, and in tenths of a second for the most liquid ones. As an example, Clarabel can fit the S&P 500 volatility surface, calibrated from 14,500 volatility bids and asks across 46 expiries, involving two iterations of the CVI's QP problem with 20 parameters per expiry, in just 0.2 seconds.

Keywords: volatility surface, volatility interpolation, option pricing, convex optimization, quadratic programming

JEL Classification: G13, C61, C63, G17

Suggested Citation

Deschâtres, Fabrice, Convex Volatility Interpolation (May 26, 2024). Available at SSRN: or

Fabrice Deschâtres (Contact Author)

affiliation not provided to SSRN

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