LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
Brownian motion starts at zero. Its increments over disjoint intervals are independent, the increment over a window is normal with variance equal to the window, and almost every path is continuous and nowhere differentiable. Because the increments are independent, the position at a horizon T is just the increment from zero to T. A European call price is the expected discounted payoff of a stock model built on that law.
Why it matters
A normal density is not a probability at a point. Five hundred draws start to trace the bell. Landing below zero has probability 50 percent. Landing below -2 is about 2.4 percent, the area. The mean minimizes average error. Variance is the expected squared distance from it. A standard normal has mean zero and variance one. Keep drawing and the sample moments approach those values. Code an event as 0 or 1 and the average approaches its probability, so the histogram fills in the curve.
In a market you do not see that curve. The lecture's animation moves the center and the spread while a single histogram piles up. A volatile stretch and a bull stretch are not the same unobserved law. A stochastic process is a random variable at each index. Connect normal draws and you have one path. Look across many paths at one date and the cross-section returns to the normal that produced that date. Change the seed and the path changes. The dots were distributions.
The idea in plain words
Brownian motion is not a normal draw with a frozen variance. Replay it and the bell flattens. The lecture's picture starts with tails near -20 and 20 and ends near -60 and 60. Uncertainty grows with the clock, which is what a longer step is for. You can still price off the terminal law without storing every wiggle, because independent increments make the net change from zero to T a normal with variance T. A fractional version does not offer that shortcut. It needs the path's memory, so the draw at T is not a function of T alone. Either way, enough simulated paths make the empirical terminal histogram approach the theoretical one when the theoretical one exists.
Louis Bachelier's 1900 dissertation is the lecture's first financial use: arithmetic Brownian motion for an option. The stock is a drift plus a scaled Brownian shock, so each date is normal, with the mean moved and the variance resized. Tell the model the terminal stock and it tells you the call payoff. The price today is the expectation of that payoff, discounted. Black–Scholes has a closed form for a different stock equation. Heston has a fast transform. The arithmetic model can be averaged by simulation. It can also be averaged by sampling the terminal normal directly, which is faster and, in the lecture, converges to the same theoretical call price. The payoff diagram is that terminal sample. The running average of discounted payoffs settles on the model price.
How to use it
Use the terminal law when the contract only cares about the stock at expiry and the increments really are independent. Simulate the path when the contract cares about the route, or when the process, like a fractional one, has no terminal shortcut. Do not confuse the model price with a claim that stocks follow arithmetic Brownian motion. The lecture's desk hypothetical is the opposite: you still quote, and you would rather quote with an explicit, wrong-in-a-known-way model plus adjustments for credit and liquidity than with no model. Arithmetic Brownian motion will not carry a volatility skew or a term structure. Using it against a listed surface can be a trade against yourself. The engine underneath the later models is still this process.
What usually goes wrong
Reading a density height as a probability. Treating a piled-up histogram as the law, after the unobserved center has moved. Simulating an entire arithmetic path when only the terminal normal was required. And, for a process with memory, sampling a terminal normal that the process does not have. The lecture's last warning is aimed at the remark that models are false so they should not be used. The job is a decision under uncertainty. The model is how the decision is scored.
Questions people ask
Why does the bell flatten?
Variance equals elapsed time. A short horizon is a tight normal. The lecture's long horizon reaches about -60 to 60, against about -20 to 20 at the start.
When can you skip the path?
When increments are independent, the value at T is the single increment from zero to T. Fractional Brownian motion does not have that property.
What does the call average converge to?
The theoretical price under arithmetic Brownian motion, whether you walk the paths or draw the terminal normal and discount the payoff. That is a model price, not a market price.