Black-Scholes Is Solved Two Ways: By Formula and by Finite Differences

When a financial derivative needs a price, the Black-Scholes model is one of the classic tools people turn to. This paper walks through that model from the ground up, starting with the market context and the calculus ideas needed to follow the math. It then derives the Black-Scholes-Merton equation and solves it in two different ways: an analytical solution using variable separation and a numerical solution using finite differences. That side-by-side comparison is the paper’s main point, showing how the same pricing problem can be attacked with exact algebra or with a computational grid. The author closes by explaining how Black-Scholes is used today, and adds appendices with economics concepts and code scripts so readers can practice the ideas themselves.

A stock price can feel like a story with no script. One day it rises, then it slips, then it jumps again. The Black-Scholes model tries to write that story in math. This paper takes that classic idea and makes it readable. It starts with the market setting, then walks through the calculus needed to follow the model. The surprise is not that Black-Scholes exists. It is that the same pricing problem can be handled in two very different ways. One path uses a clean analytical solution. The other uses a numerical grid, called finite differences, to approximate the answer step by step.

Why the same option can be priced two ways

Black-Scholes is the first widely used model for option pricing. An option is a contract that gives the right, but not the duty, to buy or sell later at a set price. The model turns that problem into a parabolic partial differential equation, which is a math law that links change over time with change across prices. The paper does not stop at the formula’s fame. It shows how the equation can be solved in two forms. One solution uses variable separation, a way to split one hard equation into simpler pieces. The other uses finite differences, a numerical method that replaces smooth change with many small steps on a grid. The paper’s goal is to help readers see both routes clearly.

From random motion to a pricing law

The model starts with random motion. The paper reviews the Wiener process, Brownian motion, geometric Brownian motion, and random walk. Brownian motion is the drifting, erratic movement seen in tiny particles. In finance, it becomes a way to describe price noise. Geometric Brownian motion keeps prices positive while letting them wander. The paper also introduces Itô’s lemma, a calculus rule for functions that depend on a random path. That rule helps turn the price path into the Black-Scholes-Merton equation. In plain terms, the model asks how today’s price, risk, time, and uncertainty combine into a fair value for the option.

Why this matters beyond the classroom

That two-way split is the paper’s real value. An exact formula is elegant, but a numerical grid is often easier to adapt. The finite-difference path shows how a computer can work through the same pricing problem when the math gets messy. That makes the model useful not just as a historic result, but as a living tool. The appendix with code scripts points in the same direction. It lets readers try the ideas rather than only read about them. For a general reader, that matters because finance often looks sealed behind symbols. This paper opens the door and shows the moving parts.

Back to the same surprise

The surprising part is still the same one from the start. A pricing problem that sounds fixed and formal can be approached in two very different ways. That is useful because not every option problem stays neat enough for a closed-form answer, a formula you can write down in one line. The next test for this teaching approach is simple: can a reader move from the appendix notes and code scripts to a working Black-Scholes calculation on their own? The paper gives the pieces. It then hands over the puzzle, one grid point at a time.