Volatility Parametrizations with Random Coefficients: Analytic Flexibility for Implied Volatility Surfaces

Paper: arXiv 2411.04041

Authors: Nicola F. Zaugg, Leonardo Perotti, Lech A. Grzelak

Abstract

It is a market practice to express market-implied volatilities in some parametric form. The most popular parametrizations are based on or inspired by an underlying stochastic model, like the Heston model (SVI method) or the SABR model (SABR parametrization). Their popularity is often driven by a closed-form representation enabling efficient calibration. However, these representations indirectly impose a model-specific volatility structure on observable market quotes. When the market’s volatility does not follow the parametric model regime, the calibration procedure will fail or lead to extreme parameters, indicating inconsistency. This article addresses this critical limitation - we propose an arbitrage-free framework for letting the parameters from the parametric implied volatility formula be random. The method enhances the existing parametrizations and enables a significant widening of the spectrum of permissible shapes of implied volatilities while preserving analyticity and, therefore, computation efficiency. We demonstrate the effectiveness of the novel method on real data from short-term index and equity options, where the standard parametrizations fail to capture market dynamics. Our results show that the proposed method is particularly powerful in modeling the implied volatility curves of short expiry options preceding an earnings announcement, when the risk-neutral probability density function exhibits a bimodal form.

Complexity vs Empirical Score

• Math Complexity: 7.0/10
• Empirical Rigor: 7.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces an innovative parameter randomization framework to overcome structural limitations in classical implied volatility parametrizations like SVI and SABR. It provides strong analytical foundations while validating performance empirically on challenging market regimes like short-term earnings announcements, supported by open-source code.

Research Flowchart

  flowchart TD
    A[Research Goal: Address limitations of standard IV parametrizations via random coefficients] --> B[Data Inputs: Short-term equity/index options & Pre-earnings announcement data]
    B --> C[Methodology: Introduce Random Coefficient Framework to SVI/SABR models]
    C --> D[Computation: Derive closed-form IV formulas & enforce arbitrage-free conditions]
    D --> E[Model Calibration & Testing against standard parametrizations]
    E --> F[Key Outcomes: Captures complex/bimodal PDFs, eliminates extreme parameters, maintains efficiency]

Cited References (31)

1. Arbitrage-free SVI volatility surfaces (2014)
2. Managing smile risk (2002)
3. An arbitrage-free interpolation of class C2 for option prices (2021)
4. No arbitrage global parametrization for the eSSVI volatility surface (2022)
5. Robust calibration and arbitrage-free interpolation of SSVI slices (2019)
6. Implied volatility surface: Construction methodologies and characteristics (2011) — link
7. The stochastic collocation Monte Carlo sampler: highly efficient sampling from expensive distributions (2019)
8. Lognormal-mixture dynamics and calibration to market volatility smiles (2002)

1 of 31 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.