Topic 15 – PDE / Finite Differences

1. Why PDEs appear in option pricing

Under diffusion models (e.g. Black–Scholes),

• The option price $V(t, S)$ solves a backward PDE: $$\frac{\partial V}{\partial t} - \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} - r S \frac{\partial V}{\partial S} * r V = 0$$

• With:

  • Terminal condition at $T$: payoff (e.g. $V(T,S)=\max(S-K,0)$),
  • Boundary conditions for small/large $S$.

Idea:

“Instead of simulating paths forward, we solve a PDE backward in time on a grid in $(t,S)$-space.”

When this is useful:

• Low dimensional (1–2 underlying factors),
• Need early exercise (American options),
• Want smooth prices and Greeks,
• Sometimes as a benchmark for other methods.

2. Finite differences – basic idea

Finite differences = approximate derivatives with local differences on a grid.

• Build a grid:
  • Time: $ t_0 < t_1 < < t_N = T $,
  • Space (e.g. price): $ S_0, S_1, , S_M $.
• Replace derivatives with discrete approximations:
  • $\partial V/\partial t \approx (V^{n+1}_i - V^n_i)/\Delta t$,
  • $\partial^2 V/\partial S^2 \approx (V^n_{i+1} - 2V^n_i + V^n_{i-1}) / \Delta S^2$,
  • etc.

Then:

• PDE becomes a linear system linking values at different grid points.
• You step backward in time from maturity to today.

Schemes:

• Explicit:
  • New value expressed directly from old values.
  • Simple but can be numerically unstable unless time step is very small.
• Implicit / Crank–Nicolson:
  • Need to solve a linear system each time step.
  • More costly per step but unconditionally stable (or much more stable).

3. Stability and the CFL intuition

For explicit schemes you have a stability condition (CFL-type):

• Roughly: $$\Delta t \le C \cdot \frac{\Delta S^2}{\sigma^2 S^2}$$ (detail depends on discretisation and region).

Intuition:

“If you walk too far in time relative to your spatial resolution, the numerical solution blows up or oscillates.”

Practical consequences:

• Explicit FD is fine for didactic or very coarse problems.
• For real desks:
  • Use implicit or Crank–Nicolson schemes,
  • Or more advanced solvers (ADI, operator splitting) in 2D+.

4. Where PDE / FD shines vs Monte Carlo

Good for:

• 1D or 2D early-exercise products:
  • American options,
  • Callable/puttable bonds,
  • Some convertible structures.
• Getting smooth Greeks:
  • Differentiating a PDE solution is often cleaner than MC Greeks.

Less good for:

• High dimension (>2–3 underlyings),
• Very complex path-dependence (storage, swing with state constraints),
• Where MC or tree/LSMC methods are more natural.

Typical pattern:

“Use PDE/FD as the workhorse for 1D early-exercise and as a benchmark for other methods; use MC when dimension or path-dependence explodes.”

5. Interview soundbite

“Finite differences solve the pricing PDE backward on a grid in time and underlying space. They’re ideal for low-dimensional problems with early exercise, like American options, because you can enforce exercise conditions at each time step. The trade-off is numerical complexity: you must choose grids and schemes carefully for stability and accuracy. In practice we use implicit or Crank–Nicolson schemes and treat PDE prices as benchmarks and calibration anchors for more approximate or higher-dimensional methods.”