A note on the option price and 'Mass at zero in the uncorrelated SABR model and implied volatility asymptotics'

TL;DR

This paper enhances the accuracy of option pricing and mass at zero estimation in the uncorrelated SABR model by employing Gauss-Hermite quadrature and integrating CEV prices, avoiding small-strike asymptotics.

Contribution

It introduces a more precise numerical method for option pricing in the uncorrelated SABR model using advanced quadrature and direct integration techniques.

Findings

Improved numerical accuracy with Gauss-Hermite quadrature.

Accurate, arbitrage-free option prices across all strikes.

Avoids reliance on small-strike asymptotic approximations.

Abstract

Gulisashvili et al. [Quant. Finance, 2018, 18(10), 1753-1765] provide a small-time asymptotics for the mass at zero under the uncorrelated stochastic-alpha-beta-rho (SABR) model by approximating the integrated variance with a moment-matched lognormal distribution. We improve the accuracy of the numerical integration by using the Gauss--Hermite quadrature. We further obtain the option price by integrating the constant elasticity of variance (CEV) option prices in the same manner without resorting to the small-strike volatility smile asymptotics of De Marco et al. [SIAM J. Financ. Math., 2017, 8(1), 709-737]. For the uncorrelated SABR model, the new option pricing method is accurate and arbitrage-free across all strike prices.