Tighter 'uniform bounds for Black-Scholes implied volatility' and the applications to root-finding

Paper: arXiv 2302.08758

Authors: Jaehyuk Choi, Jeonggyu Huh, Nan Su

Abstract

Using the option delta systematically, we derive tighter lower and upper bounds of the Black-Scholes implied volatility than those in Tehranchi [SIAM J. Financ. Math. 7 (2016), 893-916]. As an application, we propose a Newton-Raphson algorithm on the log price that converges rapidly for all price ranges when using a new lower bound as an initial guess. Our new algorithm is a better alternative to the widely used naive Newton-Raphson algorithm, whose convergence is slow for extreme option prices.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper presents analytical mathematical proofs to tighten analytical bounds on Black-Scholes implied volatility using option delta, demonstrating a clever improvement over Tehranchi (2016). While mathematically sound and clear in its algorithmic formulation for root-finding, it lacks extensive empirical numerical testing against large real-world option datasets.

Research Flowchart

  flowchart TD
    A[Research Goal: Derive tighter implied volatility bounds & improve convergence] --> B[Inputs: Option prices, strikes, risk-free rate, & option delta]
    B --> C[Methodology: Apply option delta systematically]
    C --> D[Outcome 1: Tighter lower & upper bounds for Black-Scholes IV]
    D --> E[Computational Step: Newton-Raphson on log price using new lower bound initial guess]
    E --> F[Outcome 2: Rapidly converging algorithm for all price ranges, outperforming naive NR]