Finite-difference time-stepping scheme that averages explicit (forward) and implicit (backward) Euler steps. Second-order accurate in both Δ t and Δ x (O(Δ t^2 + Δ x^2)) and unconditionally stable. The workhorse for solving the Black-Scholes and Heston PDEs in the absence of barriers or discontinuous payoffs; non-smooth payoffs (digitals, barriers) induce oscillations, mitigated by Rannacher start-up (two implicit-Euler steps before switching to CN).