Finite Difference Schemes and Convergence Analysis

Setup

The Black-Scholes PDE as a Parabolic Problem

The Black-Scholes PDE for a European option price $C(t, S)$ is:

$\frac{\partial C}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 C}{\partial S^2} + rS \frac{\partial C}{\partial S} - rC = 0, \qquad (t, S) \in [0, T) \times (0, \infty).$

This is a parabolic PDE (heat equation type) in the variables $t$ and $S$. It is solved backwards in time from the terminal condition $C(T, S) = g(S)$ (the payoff function).

Log-Spot Transformation

The variable coefficient $\sigma^2 S^2$ in the second-order term introduces numerical difficulty. Setting $x = \ln S$ and reversing time $\tau = T - t$ (time to expiry), the PDE becomes:

$\frac{\partial C}{\partial \tau} = \frac{1}{2}\sigma^2 \frac{\partial^2 C}{\partial x^2} + \left(r - \frac{1}{2}\sigma^2\right)\frac{\partial C}{\partial x} - rC,$

with constant coefficients. This is the canonical form used in finite difference implementations. It reduces to a heat equation modulo first-order and zero-order terms.

Grid Setup

Discretise on a uniform grid in $x$ and $\tau$:

• $x_i = x_{\min} + i \Delta x$, for $i = 0, 1, \ldots, M$.
• $\tau_j = j \Delta\tau$, for $j = 0, 1, \ldots, N$; so $\tau_N = T$.

Let $C_i^j \approx C(x_i, \tau_j)$. Boundary conditions:

• At $x_{\min}$ (far-left, $S \approx 0$): $C_i^j = 0$ for a call.
• At $x_{\max}$ (far-right, $S \to \infty$): $C_i^j = e^{x_i} - K e^{-r\tau_j}$ for a call (forward price minus discounted strike).

Initial condition ($\tau = 0$): $C_i^0 = g(e^{x_i})$ (the payoff at expiry).

Explicit Scheme (FTCS)

Discretisation

Forward in time, central in space (FTCS — Forward Time, Centred Space):

$\frac{C_i^{j+1} - C_i^j}{\Delta\tau} = \frac{\sigma^2}{2} \frac{C_{i+1}^j - 2C_i^j + C_{i-1}^j}{\Delta x^2} + \left(r - \frac{\sigma^2}{2}\right)\frac{C_{i+1}^j - C_{i-1}^j}{2\Delta x} - rC_i^j.$

Define $\lambda = \frac{\sigma^2 \Delta\tau}{2\Delta x^2}$, $\nu = \frac{(r - \sigma^2/2)\Delta\tau}{2\Delta x}$. Rearranging:

$C_i^{j+1} = (\lambda - \nu) C_{i-1}^j + (1 - 2\lambda - r\Delta\tau) C_i^j + (\lambda + \nu) C_{i+1}^j.$

This is explicit: each new value $C_i^{j+1}$ is computed directly from the previous time level. No linear system to solve. Cost per time step: $O(M)$.

Stability: Von Neumann Analysis

Assume $C_i^j = \xi^j e^{i\alpha x_i}$ for some complex amplitude $\xi$. Substituting into the FTCS stencil (considering only the diffusion term for clarity):

$\xi = 1 + 2\lambda(\cos(\alpha\Delta x) - 1).$

Stability requires $|\xi| \leq 1$ for all frequencies $\alpha$. Since $\cos(\alpha\Delta x) - 1 \in [-2, 0]$:

$|\xi|_{\max} = |1 - 4\lambda| \leq 1 \implies \lambda \leq \frac{1}{2} \implies \frac{\sigma^2 \Delta\tau}{\Delta x^2} \leq 1.$

This is the CFL (Courant-Friedrichs-Lewy) condition for the parabolic problem. In physical variables: $\Delta t \leq \Delta x^2 / \sigma^2$. The time step is quadratically constrained by the spatial step — a serious limitation for fine grids.

Accuracy

Truncation error: Taylor-expanding each term shows the FTCS scheme is $O(\Delta\tau) + O(\Delta x^2)$: first-order in time, second-order in space. The overall convergence rate is $O(\Delta\tau) + O(\Delta x^2)$.

Implicit Scheme (BTCS)

Discretisation

Backward in time, central in space (BTCS):

$\frac{C_i^{j+1} - C_i^j}{\Delta\tau} = \frac{\sigma^2}{2} \frac{C_{i+1}^{j+1} - 2C_i^{j+1} + C_{i-1}^{j+1}}{\Delta x^2} + \left(r - \frac{\sigma^2}{2}\right)\frac{C_{i+1}^{j+1} - C_{i-1}^{j+1}}{2\Delta x} - rC_i^{j+1}.$

Rearranging into standard tridiagonal form:

$-(\lambda - \nu) C_{i-1}^{j+1} + (1 + 2\lambda + r\Delta\tau) C_i^{j+1} - (\lambda + \nu) C_{i+1}^{j+1} = C_i^j.$

This is a tridiagonal linear system of size $(M-1)\times(M-1)$. Solved in $O(M)$ operations by the Thomas algorithm (forward elimination + back substitution).

Stability

Von Neumann analysis on the BTCS scheme gives:

$\xi = \frac{1}{1 + 2\lambda(1 - \cos(\alpha\Delta x))}.$

Since the denominator is always $\geq 1$, we have $|\xi| \leq 1$ for all $\alpha$ and all $\lambda > 0$. The BTCS scheme is unconditionally stable: no constraint between $\Delta\tau$ and $\Delta x$. Large time steps are permitted. The accuracy remains $O(\Delta\tau) + O(\Delta x^2)$.

Crank-Nicolson Scheme

Discretisation

The $\theta$-method averages the explicit and implicit discretisations with weights $\theta$ and $1-\theta$:

$\frac{C_i^{j+1} - C_i^j}{\Delta\tau} = \theta \cdot \mathcal{L}^{j+1} C_i + (1-\theta) \cdot \mathcal{L}^j C_i,$

where $\mathcal{L}$ is the finite difference operator for the spatial terms. Crank-Nicolson sets $\theta = 1/2$: