It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

The coefficients in the stochastic differential equation that the short interest rate follows are of vital importance in the subsequent modelling of bond prices and other interest rate products. Empirical tests have previously been performed by various authors who compare a variety of popular short‐rate models. Most recently, Ahn and Gao compared their model with affine‐drift models and showed that their model with a non‐linear drift function outperforms the others. This paper compares the model developed by Goard, which is a time‐dependent generalization of the Ahn–Gao model, with the Ahn–Gao model itself. It is found that the time‐dependent model using a second‐order Fourier series in time, outperforms the Ahn–Gao model for all data sets considered.

In this article, we investigate and compare the performance of various one-factor diffusion models in their ability to capture the behaviour of Brent crude oil prices. New proposed models, which have a three-quarters power in the diffusion term, are found to outperform all other popular models tested. Analytic solutions for futures prices under the new models are found and used to calibrate market prices. Results from the calibration show that one of the new three-quarters models with a mean-reverting property outperforms other popular models in fitting and forecasting futures prices.

This paper demonstrates how the infinite‐dimensional part of the classical symmetry algebra of linear partial differential equations can be used to find solutions to initial value problems. We can formulate a secondary initial value problem whose solution, when augmented to a pre‐specified infinitesimal Lie point symmetry, as a one‐parameter superposition, produces a symmetry that is compatible with the original initial condition. This secondary initial value problem may be easier to solve than the original. If not, we can iterate the process to generate a third or subsequent initial value problem.