We consider the approximation of one-dimensional stochastic differential equations
(SDEs) with non-Lipschitz drift or diffusion coefficients. We present a modi-
fied explicit Euler-Maruyama discretisation scheme that allows us to prove strong
convergence, with a rate. Under some regularity and integrability conditions, we
obtain the optimal strong error rate. We apply this scheme to SDEs widely used
in the mathematical finance literature, including the Cox-Ingersoll-Ross (CIR), the
3/2 and the Ait-Sahalia models, as well as a family of mean-reverting processes
with locally smooth coefficients. We numerically illustrate the strong convergence
of the scheme and demonstrate its efficiency in a multilevel Monte Carlo setting.