This paper illustrates how to construct an unconditionally stable finite-difference lattice consistent with the equity option volatility smile. In particular, the paper shows how to extend the method of forward induction on Arrow-Debreu securities to generate local instantaneous volatilities in implicit and semi-implicit (Crank-Nicholson) lattices. The technique developed in the paper provides a highly accurate fit to the entire volatility smile and offers excellent convergence properties and high flexibility of asset- and time-space partitioning. Contrary to standard algorithms based on binomial trees, our approach is well suited to price options with discontinuous payouts (e.g. knock-out and barrier options) and does not suffer from problems arising from negative branching probabilities.