LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A European call’s strike and expiry are known the day you buy it. The share price at expiry is not. The payoff is a function of that unknown price, so a price today has to come from a model of how the share can move between now and expiry. Arithmetic Brownian motion supplies one such model and can print negative prices. Geometric Brownian motion stays positive and is the process behind the Black-Scholes European formulas.
Why it matters
The call gives the holder a choice, not a duty, to buy the share at strike K at time T. Both K and T are fixed when the contract is opened. The payoff at T is the maximum of the terminal share price minus K, and zero. On a chart with the terminal price on the horizontal axis, the payoff is zero to the left of K and rises one-for-one to the right. That picture is fully known as a function. It is useless as a number until you say what the terminal price might be.
The idea in plain words
The lecture’s first process is arithmetic Brownian motion, the model Louis Bachelier used in 1900. In the form they write, the change in the modeled price is a drift times a small time step, plus a volatility times the change in a standard Brownian motion. Brownian motion here is the random piece, with independent increments centered at zero. Drop the drift and the change is only the volatility times that random piece. Integrating from 0 to T gives a terminal value equal to the start value plus volatility times the Brownian increment over the whole horizon. That formula is a distribution of terminal prices, and the lecture says a European call or put can be priced in closed form from it.
The defect is sign. An arithmetic path can cross below zero. Shares generally cannot. The lecture notes that some traded assets have in fact printed negative prices, and that a Bachelier model or a displaced model can be the better description there. For ordinary shares, a negative print is a modeling problem, not a market fact you want to allow by accident.
Geometric Brownian motion changes the increment so it scales with the current price. The lecture writes the percentage change as a drift times the time step plus a volatility times the Brownian increment, which is the same statement as a price change equal to drift times price times time, plus volatility times price times the Brownian increment. Because the shock multiplies the current price, the path the lecture describes does not go negative. That process is the one used to reach the Black-Scholes partial differential equation and then the pricing formulas for European calls and puts. Black-Scholes, in this article, means those constant-volatility European prices and the equation they come from, not a claim that markets follow the equation.
How to use it
Start every pricing sketch by writing the payoff at T and marking that you do not know the input. Then choose a process and say what it implies for the sign of the price. If the underlier can be negative, do not reject arithmetic motion just because textbooks prefer the geometric one. The lecture leaves that door open. If the underlier is a share, the geometric non-negativity is the reason they give for switching.
What usually goes wrong
Pricing from a single drawn path treats one finish above K as the forecast. The lecture’s whole point is a cloud of finishes. One path cannot be the distribution.
Questions people ask
What is known on the day the call is bought?
The strike and the expiry. The share price at expiry is not known, which is why a single payoff number cannot be written down yet.
Why can arithmetic Brownian motion be a problem for shares?
Its solution can be negative. The lecture treats negative share prices as the wrong model for stocks, while noting that some other assets have traded below zero, where this model or a displaced one may fit.
What does geometric Brownian motion change?
The random shock scales with the current price, so the lecture’s version does not produce negative prices. That process is the one they connect to the Black-Scholes equation and the European call and put formulas.