What is the Crank-Nicolson scheme and why is it preferred for option pricing?

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AcadiFi · Accepted Answer

Crank-Nicolson averages the explicit and implicit schemes, evaluating spatial derivatives at both the current and next time step with equal weight. If explicit uses time j values and implicit uses time j+1 values, CN uses 0.5*(j) + 0.5*(j+1). This gives second-order accuracy in both space and time, O(dS^2) + O(dt^2), versus first-order in time for explicit and implicit. It's unconditionally stable like implicit, but also much more accurate. For pricing a 9-month American put on Oakhaven Utilities strike $52, CN with 200 space nodes and 400 time steps typically achieves 4-decimal accuracy where explicit would need 5000+ time steps for stability and similar precision. The drawback: CN can produce oscillations when payoffs have kinks (at-the-money regions). Rannacher smoothing fixes this by taking the first 2-4 steps with fully implicit before switching to CN, damping the oscillations without losing overall accuracy. ```mermaid graph LR A[Explicit
fast but unstable] --> C[Crank-Nicolson] B[Implicit
stable but O dt] --> C C --> D[Second-order
accurate] ```

What is the Crank-Nicolson scheme and why is it preferred for option pricing?

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