Option pricing under the normal SABR model with Gaussian quadratures

Paper: arXiv 2301.02797

Authors: Jaehyuk Choi, Byoung Ki Seo

Abstract

The stochastic-alpha-beta-rho (SABR) model has been widely adopted in options trading. In particular, the normal ($β=0$) SABR model is a popular model choice for interest rates because it allows negative asset values. The option price and delta under the SABR model are typically obtained via asymptotic implied volatility approximation, but these are often inaccurate and arbitrageable. Using a recently discovered price transition law, we propose a Gaussian quadrature integration scheme for price options under the normal SABR model. The compound Gaussian quadrature sum over only 49 points can calculate a very accurate price and delta that are arbitrage-free.

Complexity vs Empirical Score

• Math Complexity: 7.5/10
• Empirical Rigor: 4.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper introduces a mathematically elegant Gaussian quadrature integration scheme for the normal SABR model to resolve asymptotic approximation inaccuracies. While the numerical and theoretical development is strong, empirical validation on real market option chains and extensive backtesting are lacking.

Research Flowchart

  flowchart TD
    A["Research Goal: Arbitrage-Free Option Pricing<br>& Delta in Normal SABR Model"] --> B["Inputs: Normal SABR Parameters<br>(β=0, Forward, Strike, Vol, Nu, Rho, Expiry)"]
    B --> C["Apply Price Transition Law<br>(Exact Integral Formulation)"]
    C --> D["Numerical Integration:<br>Compound Gaussian Quadrature (49 Points)"]
    D --> E["Calculate Option Price & Delta"]
    E --> F["Outcomes: High Accuracy, Arbitrage-Free<br>& Superior to Asymptotic Implied Volatility"]