The Hartman–Watson distribution is an absolutely continuous probability distribution which arises in the study of Brownian functionals. It is named after Philip Hartman and Geoffrey S. Watson, who encountered the distribution while studying the relationship between Brownian motion on the n-sphere and the von Mises distribution. Important contributions to the distribution, such as an explicit form of the density in integral representation and a connection to Brownian exponential functionals, came from Marc Yor.

The Hartman-Watson distribution determines the joint distribution of the time integral of a geometric Brownian motion and its terminal value. This relation underlies its applications in financial mathematics. Notable applications are pricing Asian options in the Black-Scholes model and European options in stochastic volatility models with volatility following a geometric Brownian motion, such as the SABR model.

The Hartman–Watson distributions are the probability distributions ${\displaystyle (\mu _{r})_{r>0}}$, which satisfy the following relationship between the Laplace transform and the modified Bessel function of first kind:

where ${\displaystyle I_{\nu }(r)}$ denoted the modified Bessel function defined as

The unnormalized density of the Hartman-Watson distribution is

for ${\displaystyle r>0,\;t>0}$.

It satisfies the equation

The density of the Hartman-Watson distribution is defined on ${\displaystyle \mathbb {R} _{+}}$ and given by