Error bound for the asymptotic expansion of the Hartman-Watson integral

TL;DR

The paper addresses the problem of obtaining a provable, uniform bound on the error of the leading term in the short-time asymptotics of the Hartman-Watson distribution in the regime $rt=\rho$ fixed as $t\to 0$. It develops a saddle point expansion for the transformed integral, identifies the leading coefficient $G(\rho)$, and uses Watson's lemma to derive the leading term $\theta(\rho/t,t)$. The main contribution is a uniform bound $|\vartheta(t,\rho)| \le \frac{1}{70} t$ for all $\rho>0$, with a stronger bound $\vartheta_{\max}(t)$ involving exponential-integral and erfc terms; the paper also provides exact results for $\rho=1$ and demonstrates the behavior of the error term both for small and large arguments. These results improve the reliability of analytic approximations for numerical evaluation and have practical impact on short-maturity option pricing models, including Asian options and related stochastic-volatility settings.

Abstract

This note gives a bound on the error of the leading term of the $t\to 0$ asymptotic expansion of the Hartman-Watson distribution $θ(r,t)$ in the regime $rt=ρ$ constant. The leading order term has the form $θ(ρ/t,t)=\frac{1}{2πt}e^{-\frac{1}{t} (F(ρ)-π^2/2)} G(ρ) (1 + \vartheta(t,ρ))$, where the error term is bounded uniformly over $ρ$ as $|\vartheta(t,ρ)|\leq \frac{1}{70}t$.

Figures (4)

• Figure 1: Integration contours for $I(\rho,t)$ in the $\xi$ complex plane for the application of the asymptotic expansion. The red dots show the saddle points. Left: contour for $0<\rho<1$. The contour passes through the saddle points $B(\xi=-x_1+i\pi)$ and $A(\xi=x_1+i\pi)$. Middle: contour for $\rho>1$. The contour passes through the saddle point $S(\xi=i y_1)$. Right: the contour for $\rho=1$, passing through the saddle point $S(\xi=i\pi)$.
• Figure 2: Plot of $\delta(\tau,1)$ for $\rho=1$. The dashed line is $-\frac{1}{35} \tau$.
• Figure 3: Plot of $\delta(\tau)$ vs $\tau$ for several values of $\rho$. Left: two extreme values of $\rho$ in the $[0,1]$ interval. Right: several values of $\rho$ larger than 1. The dashed line shows the bound $-\frac{1}{35}\tau$.
• Figure 4: Plot of $\delta'(0, \rho)$ vs $\rho$. The dashed line is at $-\frac{1}{35}$ and corresponds to the minimum value of $\delta'(0,\rho)$ which is reached at $\rho=1$.

Theorems & Definitions (7)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof
• Remark 1