Abstract: The Heston model is one of the most well-known stochastic volatility models. It models jointly the evolution of the price process of an investment asset and the stochastic variance of the asset's log-returns. The variance process is given as a Cox-Ingersoll-Ross process, also known as mean-reverting square root process, a process used widely in financial and other applications. While the Heston model provides tractability to a certain extent, its numerical treatment is well-known to be very challenging. Key components in the model are the variance process and its time integral conditioned on the variance values at the end points of the integral. We derive a new series representation for the latter quantity. For this we explore the connection of the CIR process to squared Bessel processes and bridges. The new representation has the advantage that truncation errors decay exponentially, and that building blocks of this series are random variables that are largely independent of the model parameters. Based on this new representation, we derive high-accuracy direct inversion methods that enable the efficient sampling of the Heston model. This talk is based on joint work with SJA Malham and J Shen.