  We demonstrate the effectiveness of an adaptive explicit Euler method for the
approximate solution of the Cox-Ingersoll-Ross model. This relies on a class of
path-bounded timestepping strategies which work by reducing the stepsize as
solutions approach a neighbourhood of zero. The method is hybrid in the sense
that a convergent backstop method is invoked if the timestep becomes too small,
or to prevent solutions from overshooting zero and becoming negative. Under
parameter constraints that imply Feller's condition, we prove that such a
scheme is strongly convergent, of order at least 1/2. Control of the strong
error is important for multi-level Monte Carlo techniques. Under Feller's
condition we also prove that the probability of ever needing the backstop
method to prevent a negative value can be made arbitrarily small. Numerically,
we compare this adaptive method to fixed step implicit and explicit schemes,
and a novel semi-implicit adaptive variant. We observe that the adaptive
approach leads to methods that are competitive in a domain that extends beyond
Feller's condition, indicating suitability for the modelling of stochastic
volatility in Heston-type asset models.
