Crank-Nicolson scheme
Quantitative Finance glossary · Numerical Methods & Optimization
Definition
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).
Crank-Nicolson scheme in plain English
Some financial prices are governed by an equation that describes how value spreads and changes over time (a partial differential equation), and to solve it on a computer you march forward in tiny time steps on a grid. You can estimate each step using where things are now (an "explicit" guess, simple but can blow up) or where they'll be next (an "implicit" guess, stable but heavier). Crank-Nicolson just averages the two, which turns out to be both accurate and rock-solid stable no matter how big your steps. It's the go-to method for standard option pricing — though for payoffs with a sudden jump (like an all-or-nothing bet) it can wobble, which practitioners fix by taking a couple of the sturdier implicit steps first before switching over.
Related Quantitative Finance terms
Quantitative Finance practice questions · Quantitative Finance flashcards · Quantitative Finance study guide