Proof of non-convergence of the short-maturity expansion for the SABR model

TL;DR

This paper proves that the short-maturity expansion of option prices in the uncorrelated SABR model is asymptotic and does not converge for any positive maturity, with specific convergence limits in certain parameter regimes.

Contribution

It demonstrates the non-convergence of the short-maturity expansion in the SABR model by analyzing the analyticity of a key integral function and identifying the asymptotic nature of the series.

Findings

The $T$-series expansion is asymptotic and non-convergent for any $T>0$.

In certain limits, the implied volatility expansion has a finite convergence radius.

The analyticity of the payoff function $g(u)$ is established in a specific complex strip.

Abstract

We study the convergence properties of the short maturity expansion of option prices in the uncorrelated log-normal ($\beta=1$) SABR model. In this model the option time-value can be represented as an integral of the form $V(T) = \int_{0}^\infty e^{-\frac{u^2}{2T}} g(u) du$ with $g(u)$ a "payoff function" which is given by an integral over the McKean kernel $G(s,t)$. We study the analyticity properties of the function $g(u)$ in the complex $u$-plane and show that it is holomorphic in the strip $|\Im(u) |< \pi$. Using this result we show that the $T$-series expansion of $V(T)$ and implied volatility are asymptotic (non-convergent for any $T>0$). In a certain limit which can be defined either as the large volatility limit $\sigma_0\to \infty$ at fixed $\omega=1$, or the small vol-of-vol limit $\omega\to 0$ limit at fixed $\omega\sigma_0$, the short maturity $T$-expansion for the implied volatility has a finite convergence radius $T_c = \frac{1.32}{\omega\sigma_0}$.