Option pricing under the normal SABR model with Gaussian quadratures

TL;DR

This paper introduces a Gaussian quadrature method for accurately pricing options under the normal SABR model, improving precision and avoiding arbitrage issues compared to traditional asymptotic approximations.

Contribution

The paper develops a Gaussian quadrature integration scheme for the normal SABR model, providing highly accurate, arbitrage-free option prices with only 49 evaluation points.

Findings

Achieves highly accurate option prices and deltas.

Uses only 49 quadrature points for computation.

Ensures arbitrage-free pricing.

Abstract

The stochastic-alpha-beta-rho (SABR) model has been widely adopted in options trading. In particular, the normal ($\beta=0$) 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.