Paper: arXiv 2506.09760

Authors: Roberto Baviera, Michele Domenico Massaria

Abstract

In April 2020, the Chicago Mercantile Exchange temporarily switched the pricing formula for West Texas Intermediate oil market options from the Black model to the Bachelier model. In this context, we introduce an additive Bachelier model that provides a simple closed-form solution and a good description of the implied volatility surface. This new additive model exhibits several notable mathematical and financial properties. It ensures the no-arbitrage condition, a critical requirement in highly volatile markets, while also enabling a parsimonious synthesis of the volatility surface. The model features only three parameters, each with a clear financial interpretation: the volatility term structure, the vol-of-vol, and a parameter for modelling skew. Model calibration can follow a cascade procedure: first, it accurately replicates the term structures of forwards and At-The-Money volatilities observed in the market; second, it fits the smile of the volatility surface. The proposed model also supports efficient pricing of path-dependent exotic options via Monte Carlo simulation, using a straightforward and computationally efficient approach. Overall, this model provides a robust and parsimonious description of the oil option market during the exceptionally volatile first period of the Covid-19 pandemic.

Complexity vs Empirical Score

  • Math Complexity: 7.0/10
  • Empirical Rigor: 6.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper introduces an innovative additive Bachelier model specifically designed to address negative oil prices during the COVID-19 period. It combines strong mathematical rigor—such as analytic Fourier transform option formulas—with an empirical application to WTI option data and a practical cascade calibration scheme.

Research Flowchart

  flowchart TD
    A[Research Goal: Model WTI Oil Options with Negative Prices during COVID-19] --> B[Input Data: Market Forwards & ATM Volatility Surface]
    B --> C[Additive Bachelier Model Framework]
    C --> D[Two-Step Cascade Calibration Procedure]
    D --> E[1. Match Forward & ATM Volatility Term Structures]
    D --> F[2. Fit Volatility Smile & Price Path-Dependent Exotics via Monte Carlo]
    E & F --> G[Outcomes: Parsimonious 3-Parameter Model with No-Arbitrage & Closed-Form Solutions]

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1 of 109 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.