Subleading correction to the Asian options volatility in the Black-Scholes model

Paper: arXiv 2407.05142

Authors: Dan Pirjol

Abstract

The short maturity limit $T\to 0$ for the implied volatility of an Asian option in the Black-Scholes model is determined by the large deviations property for the time-average of the geometric Brownian motion. In this note we derive the subleading $O(T)$ correction to this implied volatility, using an asymptotic expansion for the Hartman-Watson distribution. The result is used to compute subleading corrections to Asian options prices in a small maturity expansion, sharpening the leading order result obtained using large deviations theory. We demonstrate good numerical agreement with precise benchmarks for Asian options pricing in the Black-Scholes model.

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 provides a rigorous mathematical derivation of the O(T) subleading correction for Asian option implied volatility in the Black-Scholes model using the Hartman-Watson distribution. While mathematically impressive and clear, it relies heavily on analytical expansions with empirical validation limited to numerical benchmark checks rather than real-world market data backtesting.

Research Flowchart

  flowchart TD
    A["Research Goal: Derive $O(T)$ subleading correction<br>to Asian options implied volatility ($T\to 0$)"] --> B["Inputs: Black-Scholes model &<br>Geometric Brownian Motion"]
    B --> C["Methodology: Apply Asymptotic Expansion<br>to Hartman-Watson Distribution"]
    C --> D["Compute small-maturity expansion<br>beyond Large Deviations Theory"]
    D --> E["Validate via numerical benchmarks<br>for Asian options pricing"]
    E --> F["Outcomes: Sharpened Asian option prices<br>& high-precision implied volatility formula"]