  We propose an extension of the Cox-Ross-Rubinstein (CRR) model based on
$q$-binomial (or Kemp) random walks, with application to default with logistic
failure rates. This model allows us to consider time-dependent switching
probabilities varying according to a trend parameter on a non-self-similar
binomial tree. In particular, it includes tilt and stretch parameters that
control increment sizes. Option pricing formulas are written using $q$-binomial
coefficients, and we study the convergence of this model to a Black-Scholes
type formula in continuous time. A convergence rate of order $O(N^{-1/2})$ is
obtained.
