This paper studies the pricing and hedging of continuously sampled arithmetic average rate options. We derive a new analytical approximate formula for pricing and hedging the arithmetic average rate options. The correction to the analytical approximate formula is governed by a partial differential equation (PDE) with smooth coefficients and zero initial condition, enabling it to be evaluated accurately by a numerical method. Numerical experiments show that the error of our semi-analytical method (ie, analytical approximation with the correction) is of the order of 10–7 for the grid size used in this paper, and the CPU time required for the numerical computation is only one second for a short-tenor option and 22 seconds for a long-tenor option. The accuracy can be improved further by reducing the grid size in a trade-off with CPU time. Our method is more accurate than any other method reported in the literature and it is faster than other PDE methods. With the error well controlled, our results can be used as a benchmark to justify the error computed by other approximation methods, including Monte Carlo simulation.