Finite Difference Option Pricing

There's a beautiful fact hiding under the Black-Scholes formula: every option's value obeys a single equation, a partial differential equation (PDE) relating how the price changes with time to how it changes with the stock price. For a plain European option you can solve that equation with pen and paper. But for most real options, American, barrier, dividend-heavy, no clean solution exists. Finite difference methods solve the equation the way engineers solve heat flow or fluid flow: cover the problem with a grid and turn the smooth equation into arithmetic on the grid points.

The equation to solve

The Black-Scholes PDE says that at every point, the option's value $V$ satisfies

$$\frac{\partial V}{\partial t} + \tfrac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0.$$

Don't be scared by the symbols, each term is an option Greek you already know. $\partial V/\partial t$ is theta, how the value bleeds with time. $\partial^2 V/\partial S^2$ is gamma, the curvature of the payoff, scaled by the variance $\sigma^2 S^2$. $\partial V/\partial S$ is delta, scaled by the risk-free drift $rS$. And $-rV$ is the discounting. The equation just says these forces balance out to zero at every price and time. (The Feynman-Kac theorem is what connects this PDE to the discounted-expectation picture of pricing.)

Turning the equation into a grid

We build a rectangular grid. One axis is the stock price $S$, chopped into levels from 0 up to some large ceiling. The other axis is time, chopped from today out to expiry. At each grid point we want the option's value.

Two edges give us free answers. At expiry (the right edge) the value is simply the payoff, which we know exactly, a call at each price level is $\max(S - K, 0)$. At the price boundaries (top and bottom), the value is also known: a call is worthless when $S = 0$ and behaves like the stock minus the discounted strike when $S$ is huge. Starting from those known edges, we sweep backward in time, filling each earlier column from the column just after it. Every derivative in the PDE becomes a difference between neighbouring grid points, hence "finite difference."

Three schemes, one trade-off

How you connect a node to its neighbours defines the scheme, and there are three classic choices:

Scheme	How it computes each node	Trade-off
Explicit	Directly from three known nodes in the next column	Simple, but blows up unless the time step is tiny
Implicit	By solving a small system of equations per column	Always stable, a bit more work
Crank-Nicolson	Averages explicit and implicit	Stable and more accurate; the practitioner default

The explicit scheme is essentially a trinomial tree: each earlier node is a probability-weighted blend of three later nodes. That makes it wonderfully intuitive, but it's only stable if the time step is small enough relative to the price step, otherwise tiny numerical errors snowball into garbage.

Worked example: setting up the grid

Price a call with strike $K = 100$. Pick price levels $S = 0, 20, 40, 60, 80, 100, 120, 140$ and a few time steps out to expiry. Fill in the expiry column first from the payoff $\max(S - K, 0)$:

$S$	0	20	40	60	80	100	120	140
value at expiry	0	0	0	0	0	0	20	40

Those are exact, no model needed. Now take one backward step with the explicit scheme. The value at an interior node one step earlier is a weighted blend of the node above, level, and below it in the expiry column:

$$V_{\text{earlier}} \approx \frac{1}{1 + r\,\Delta t}\big[\,p_u V_{\text{up}} + p_m V_{\text{mid}} + p_d V_{\text{down}}\,\big],$$

where the weights $p_u, p_m, p_d$ come from $\sigma$, $r$, and the grid spacing and behave just like the up/middle/down probabilities of a trinomial tree. Apply this at every interior node to fill the previous column, then repeat column by column until you reach today's price. The value sitting at $S_0$ in the first column is your option price. For an American option you add one line at each node, take the larger of this computed value and the immediate exercise payoff, which is exactly what makes finite difference (and trees) the go-to for early-exercise problems.

Common pitfalls

• Instability from the explicit scheme. The most common beginner error: a big time step makes the answer oscillate wildly. Shrink $\Delta t$ or switch to implicit/Crank-Nicolson.
• A grid ceiling that's too low. If the top price level isn't far enough above the strike, the boundary condition contaminates the answer. Rule of thumb: extend the price grid to three or four standard deviations past the strike.
• Getting boundary conditions wrong. The whole sweep is anchored on the payoff and the two edges. A wrong edge condition, common with barrier options, quietly corrupts every node.
• Over-refining. Doubling both the price and time grids roughly quadruples the work. Crank-Nicolson gets you accuracy with a coarser grid, use it before you throw more nodes at the problem.