Mathematical Principles and Quantitative Finance · 01. November 2023

The Crank-Nicolson Method Simply Explained

The Crank-Nicolson Method is a powerful numerical technique frequently employed in quantitative finance to solve partial differential equations (PDEs). It plays a crucial role in areas such as option pricing and interest rate modeling.

In financial mathematics, continuous time and space are divided into discrete intervals or grid points. This discretization allows us to approximate complex differential equations with finite differences.

One of the distinctive features of the Crank-Nicolson method is its combination of implicit and explicit time-stepping. It calculates the option or derivative price at the midpoint between two time steps. Essentially, it considers both the current time step (explicit) and the next time step (implicit) in its calculations.

The method sets up a tridiagonal system of linear equations at each time step. This system is formed by relating the option values at different grid points. Solving these equations efficiently can be accomplished using methods like the Thomas algorithm.

To obtain accurate results, the Crank-Nicolson method incorporates boundary and initial conditions. These conditions are essential for calculating the option price at each time step, ensuring the model aligns with real-world scenarios.One of the notable advantages of the Crank-Nicolson method is its stability and convergence properties. It's less susceptible to numerical instability compared to purely explicit methods, making it a reliable choice for various financial modeling tasks.

Consider a European call option with the following parameters:

Current stock price ($ S_0 $): $100
Strike price ($ K $): $105
Time to expiration ($ T $): 1 year
Volatility ($ \sigma $): 20%
Risk-free interest rate ($ r $): 5%

1. Discretization

Divide time into discrete intervals (e.g., monthly time steps) and create a grid of stock prices (e.g., from $80 to $120) with discrete intervals.

2. Initialize Boundary Conditions

Set the option value at maturity ($ T $) based on the option's payoff:

$ \text{Payoff} = \max(S - K, 0) $

3. Iterative Process

Calculate option values at each time step, moving backward in time. This involves setting up and solving tridiagonal systems of equations that consider both explicit (the current time step) and implicit (the next time step) components.

4. Continuation Value

At each time step, calculate the continuation value by considering potential future option values, averaging values from the previous and current time steps. The general Crank-Nicolson formula for finite differences is:

$ \frac{V_{i}^{n+1} - V_{i}^{n}}{\Delta t} = \frac{1}{2} \left[ \mathcal{L} V^{n+1} + \mathcal{L} V^{n} \right] $

Where:

$ V_{i}^{n} $: Option value at grid point $ i $ and time step $ n $.
$ \Delta t $: Time step size.
$ \mathcal{L} $: Differential operator representing the PDE (e.g., the Black-Scholes equation).

5. Convergence

Repeat the process until convergence is achieved, i.e., the option values no longer change significantly between iterations.

6. Final Option Price

The final computed option price at the initial time step represents the fair market value of the European call option.

For example, solving this system for the given parameters may yield a fair price of $5.85 for the European call option.

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