We propose a novel Monte Carlo-based estimator for digital options with assets modelled by a stochastic differential equation (SDE) that are solved approximately using a time-stepping scheme like Euler-Maruyama or Milstein. The new estimator is based on repeated, hierarchical path splitting as shown in (Asmussen and Glynn, 2007). Using the fact that paths of an SDE solution sharing parts of a Brownian path are conditionally independent, we show that the new estimator has an improved strong convergence rate as the size of the time-step decreases. We also show that the computational complexity of Multilevel Monte Carlo (MLMC) using the new estimator is similar to the complexity of a classical MLMC estimator when applied to smoother options. In particular, when using the Euler-Maruyama scheme, we show that the computational complexity to approximate the value of a digital option with root-mean-square error, e, is O(e^{-2-n}) for any positive n. This is an improvement over the computational complexity of a classical Monte Carlo estimator, O(e^{-3}), and a classical MLMC estimator, O(e^{-5/2}), when approximating the value of a digital option. Using the Milstein scheme or combining with an antithetic estimator (Giles and Szpruch, 2014) reduces the computational complexity further to the canonical complexity O(e^{-2}).