On the implied volatility of European and Asian call options under the stochastic volatility Bachelier model

Paper: arXiv 2308.15341

Authors: Elisa Alòs, Eulalia Nualart, Makar Pravosud

Abstract

In this paper we study the short-time behavior of the at-the-money implied volatility for European and arithmetic Asian call options with fixed strike price. The asset price is assumed to follow the Bachelier model with a general stochastic volatility process. Using techniques of the Malliavin calculus such as the anticipating Ito’s formula we first compute the level of the implied volatility when the maturity converges to zero. Then, we find a short maturity asymptotic formula for the skew of the implied volatility that depends on the roughness of the volatility model. We apply our general results to the SABR and fractional Bergomi models, and provide some numerical simulations that confirm the accurateness of the asymptotic formula for the skew.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper relies heavily on advanced mathematical techniques like Malliavin calculus and anticipating Itô formulas to establish short-time implied volatility asymptotics under the Bachelier model. While mathematically strong and novel in extending rough volatility concepts to arithmetic Asian options, it lacks extensive empirical backtesting, relying primarily on basic numerical simulations for validation.

Research Flowchart

  flowchart TD
    A[Goal: Analyze Short-Time ATM Implied Volatility<br>for European & Asian Options in Bachelier Model] --> B[Inputs: General Stochastic Volatility Models<br>SABR & Fractional Bergomi]
    B --> C[Methodology: Apply Malliavin Calculus<br>& Anticipating Itô's Formula]
    C --> D[Compute Short-Maturity Asymptotics:<br>Volatility Level & Skew]
    D --> E[Run Numerical Simulations]
    E --> F[Outcomes: Formula Accounts for Volatility Roughness<br>& Validated via Simulations]