Finite Difference Methods for Option Pricing › The Crank-Nicolson Scheme

The Crank-Nicolson scheme represents a middle ground between explicit and implicit methods. It averages the spatial derivatives at times $t_n$ and $t_{n+1}$, achieving second-order accuracy in time while maintaining unconditional stability. This combination of accuracy and stability makes Crank-Nicolson the method of choice for most practical applications.

Instead of evaluating spatial derivatives entirely at $t_n$ (implicit) or $t_{n+1}$ (explicit), Crank-Nicolson uses the average of both evaluations. This averaging can be motivated by the trapezoidal rule for numerical integration: when integrating the time derivative, we approximate the integral of the spatial operator by averaging its values at the beginning and end of the time interval. The scheme uses:

$$\frac{\partial^2 V}{\partial S^2} \approx \frac{1}{2}\left[\frac{V_{j+1}^n - 2V_j^n + V_{j-1}^n}{(\Delta S)^2} + \frac{V_{j+1}^{n+1} - 2V_j^{n+1} + V_{j-1}^{n+1}}{(\Delta S)^2}\right]$$

where:

• $V_{\cdot}^n$: values at current time step
• $V_{\cdot}^{n+1}$: values at next time step
• $\Delta S$: price step size

and similarly for the first derivative. This averaging can be viewed as taking a half-step with the explicit scheme and a half-step with the implicit scheme. The explicit part uses known information from the future time step, while the implicit part ensures stability by coupling values at the current time step.

Substituting the approximations into the Black-Scholes PDE:

$$\frac{V_j^{n+1} - V_j^n}{\Delta t} + \frac{1}{2}\left(\mathcal{L}_j V^n + \mathcal{L}_j V^{n+1}\right) = 0$$

where:

• $\mathcal{L}_j V$: spatial operator at node $j$, equal to $\frac{1}{2}\sigma^2 S_j^2 \frac{\partial^2 V}{\partial S^2} + rS_j\frac{\partial V}{\partial S} - rV$
• $V^n, V^{n+1}$: option values at current and next time steps

The spatial operator $\mathcal{L}_j$ encapsulates all the terms involving spatial derivatives and the discounting term. Applying this operator at both time levels and averaging creates the characteristic Crank-Nicolson structure. Rearranging terms by time step:

$$-\alpha_j V_{j-1}^n + (1 + \beta_j) V_j^n - \gamma_j V_{j+1}^n = \alpha_j V_{j-1}^{n+1} + (1 - \beta_j) V_j^{n+1} + \gamma_j V_{j+1}^{n+1}$$

where:

• LHS: implicit terms at current time $t_n$
• RHS: explicit terms at next time $t_{n+1}$
• $\alpha_j, \beta_j, \gamma_j$: Crank-Nicolson coefficients

The left side involves unknown values at $t_n$, forming a tridiagonal system just as in the implicit scheme. The right side uses known values at $t_{n+1}$, which we can compute explicitly before solving the system. The coefficients are:

$$\begin{aligned} \alpha_j &= \frac{\Delta t}{4}\left(\sigma^2 j^2 - rj\right) \\ \beta_j &= \frac{\Delta t}{2}\left(\sigma^2 j^2 + r\right) \\ \gamma_j &= \frac{\Delta t}{4}\left(\sigma^2 j^2 + rj\right) \end{aligned}$$

where:

• $\Delta t$: time step
• $\sigma$: volatility
• $r$: risk-free rate
• $j$: spatial index

Notice that these coefficients are exactly half of the corresponding implicit scheme coefficients, which reflects the averaging of the spatial operator between two time levels. The left side involves unknown values at $t_n$, while the right side uses known values at $t_{n+1}$. This again leads to a tridiagonal system at each time step.

The Crank-Nicolson price is consistent with the other methods. By averaging the explicit and implicit approaches, it maintains stability while improving accuracy, effectively reducing the bias introduced by time discretization.

The Crank-Nicolson scheme achieves $O((\Delta t)^2)$ accuracy in time, compared to $O(\Delta t)$ for both the explicit and implicit schemes. This means that halving the time step reduces the time discretization error by a factor of four rather than two. The improved accuracy comes from the averaging, which cancels the leading-order error terms in the Taylor expansion. For most applications, Crank-Nicolson provides the best balance of accuracy, stability, and computational efficiency.