This is where we relate everything we’ve just said to finance. That should intuitively make sense as over time, the change of the stock price is based on some overall trend (the Constant A part) and an element of randomness (the Constant B part and randomness part).Ĭonstant A and Constant B are usually derived by analyzing historical market data. Which means the change in the stock price = current stock price multiplied by some constant value over time +Ĭurrent stock price + change due to randomness multiplied by another constant. Let S be stock price.Įxplanation: Change in S = Constant A * Current S * change in time + Constant B * Current S * change due to randomness as modeled by GBM In this case, we try to link the equation to finance. Let’s replace X (a regular variable) with S (stock price) so that you can visualize this better. More info on the derivation of Itô’s lemma: Derivation of Itô’s lemma by Math PartnerĪ variation of Itô’s lemma that uses GBM is:īefore we explain it. Which means the change in the value of a variable = some constant value over time + change due to randomness multiplied by another constant. This equation takes into account Brownian motion.Įxplanation: Change in X = Constant A * change in time + Constant B * change due to randomness as modeled by Brownian motion. The main equation in Itô calculus is Itô’s lemma. The main aspects of stochastic calculus revolve around Itô calculus, named after Kiyoshi Itô. Without a smooth curve, we can’t draw those slope lines productively.

We can keep zooming in but we will not be able to find a smooth curve.

If we zoom in, we see that it looks… somewhat the same.

Stochastic calculus with infinitesimals