  The present work is devoted to strong approximations of a generalized
A\"{i}t-Sahalia model arising from mathematical finance. The numerical study of
the considered model faces essential difficulties caused by a drift that blows
up at the origin, highly nonlinear drift and diffusion coefficients and
positivity-preserving requirement. In this paper, a novel explicit Euler-type
scheme is proposed, which is easily implementable and able to preserve
positivity of the original model unconditionally, i.e., for any time step-size
$h >0$. A mean-square convergence rate of order $0.5$ is also obtained for the
proposed scheme in both non-critical and general critical cases. Our work is
motivated by the need to justify the multi-level Monte Carlo (MLMC) simulations
for the underlying model, where the rate of mean-square convergence is required
and the preservation of positivity is desirable particularly for large
discretization time steps. Numerical experiments are finally provided to
confirm the theoretical findings.
