LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
The Bachelier model writes a forward price as a starting value plus volatility times a Brownian motion, with no drift. That forward is a martingale: the best guess of the next value, given what is known now, is the current value. A call on it pays the positive part of the forward minus the strike at expiration. The lesson prices the call as the expected value of that payoff and reduces the expectation to the normal distribution function and the normal density, evaluated at one threshold built from the forward, the strike, volatility, and time.
Why it matters
The algebra is short next to the geometric-Brownian-motion solution, and the lesson presents that shortness as the attraction. The forward is arithmetic Brownian motion without a drift. Once the expectation is in terms of a standard normal variable, ordinary normal probabilities finish the price. A put is left as an exercise with the same moves. A following session codes the paths. This one stays on the expectation.
The idea in plain words
Time runs from zero to a horizon T. For a concrete reading, the lesson treats the whole life as one step, so the time increment equals T. Integrate the stochastic equation. The Brownian motion at zero is zero, and the forward at T equals the forward at zero plus volatility times the Brownian value at T. Written one step at a time, the next forward is the current forward plus volatility times the next Brownian increment. That increment is normal with variance equal to the step.
To see the martingale property, take the conditional expectation of the next forward given the information at the current time. Substitute the solution. The Brownian increment has mean zero, so it drops out, and the expectation equals the forward you already have. The current forward is the lesson's best prediction of the next one.
At expiration the call payoff is the positive part of the forward minus the strike, the hockey-stick shape. That expression is deterministic once T is known, and T is not known in advance. The price today is the expectation of the hockey stick. Substitute the solution for the forward. Write the Brownian value at T as the square root of T times a standard normal variable Z, scaled by volatility. Group the terms as the starting forward minus the strike, minus the scaled Z, all inside the positive part.
An indicator replaces the positive part so the expectation can be split. The indicator equals one when Z is at most the threshold Y, and equals zero otherwise. The lesson's threshold is the starting forward minus the strike, divided by volatility times the square root of T. The piece that does not involve a random variable comes out as that grouped forward gap times the probability that Z lands at or below Y, which is the normal distribution function at Y. What remains is an expectation of Z times the indicator.
A normal-density integral finishes the second piece. The lesson says that integral equals minus the normal density at Y, and it leaves the substitution for the reader. The resulting price uses the starting forward, the strike, the normal distribution function, and the normal density. An Overleaf note is mentioned as a longer reference, without a URL read aloud.
How to use it
Set Y to the forward gap divided by volatility times the square root of time. Weight the forward gap by the normal distribution function at Y, and carry the volatility-times-square-root-of-T scale through the density piece the lesson identifies as minus the density at Y. A put repeats the argument. The lesson does not write that formula.
What usually goes wrong
Adding a drift to the forward breaks the martingale step the lesson uses. The conditional expectation would then shift by the drift, and the prediction would no longer be the current forward. The model under discussion is driftless.
Using variance where the formula needs a standard deviation mis-scales Y. The Brownian variance over the life is T. The standard deviation is the square root of T. Y divides by volatility times that standard deviation.
The positive part and the indicator are the same gate. Dropping the indicator and taking a raw expectation of the forward minus the strike prices an instrument that can pay a negative amount. The call does not.
The density step is left as a check. A lognormal call formula answers a different model.
Questions people ask
What is the forward at expiration?
The starting forward plus volatility times the Brownian motion at expiration. The value at time zero of that Brownian motion is zero.
Why is the forward a martingale?
The next increment has mean zero, so the conditional expectation of the next forward is the forward you have now.
What ingredients does the call price use?
The starting forward, the strike, the normal distribution function, and the normal density, meeting at the threshold Y built from the forward gap, volatility, and the square root of time.