Paper: arXiv 2405.11329

Authors: Qiang Liu, Shuxin Guo

Abstract

On April 22, 2020, the CME Group switched to Bachelier pricing for a group of oil futures options. The Bachelier model, or more generally the arithmetic Brownian motion (ABM), is not so widely used in finance, though. This paper provides the first comprehensive survey of options pricing under ABM. Using the risk-neutral valuation, we derive formulas for European options for three underlying types, namely an underlying that does not pay dividends, an underlying that pays a continuous dividend yield, and futures. Further, we derive Black-Scholes-Merton-like partial differential equations, which can in principle be utilized to price American options numerically via finite difference.

Complexity vs Empirical Score

  • Math Complexity: 6.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents rigorous mathematical derivations for closed-form option pricing under Arithmetic Brownian Motion (ABM) and highlights common pitfalls in risk-neutral valuation. However, it lacks empirical backtesting, real-world data validation, or calibration against the 2020 negative oil futures events that inspired the study.

Research Flowchart

  flowchart TD
    A[Goal: Comprehensive Survey & Valuation of Options under Arithmetic Brownian Motion] --> B[Core Methodology: Risk-Neutral Valuation Framework]
    
    B --> C[Inputs & Underlying Assets]
    C --> C1[1. Non-dividend Paying]
    C --> C2[2. Continuous Dividend Yield]
    C --> C3[3. Commodity Futures]

    B --> D[Derivation & Analytical Modeling]
    
    D --> E[Outcome 1: Analytical European Option Pricing Formulas]
    D --> F[Outcome 2: BSM-like PDEs for Numerical American Option Valuation]