Without the discoveries of the Japanese mathematician Kiyoshi Itô (伊藤 清, 1915–2008), the Black–Scholes–Merton model — and therefore the modern derivatives market — could not exist.

Before moving on to Itô’s discovery, it’s helpful to understand the generalized Wiener process, and even before that, to get clear on the Markov and Wiener processes (you can review these using the provided links).

Generalized Wiener Process

A generalized Wiener process is essentially a Wiener process with an added trend.
In an ordinary Wiener process, the variable’s expected change over time is zero, and its variance increases linearly with time (equal to 1 per unit of time).
Also, the changes are not autocorrelated — they don’t depend on past values.

In the generalized Wiener process, however, a trend (drift) appears.
In a stochastic process, the expected rate of change of the mean is called the drift rate, and the rate of change of the variance is called the variance rate.

Graphically, it looks like this:

The generalized Wiener process is described by the formula:
dx = a*dt + b*dz

This expression describes the infinitesimal change of a variable over infinitely small time intervals. (Recall that d is the same as Delta, only representing an infinitesimally small amount.)

In discrete form, for finite time intervals, the same process can be written as: Δx=a*Δt+b*ε* √Δt
where ε is a random variable drawn from a standard normal distribution.

Let’s return to the continuous version:

dx = a*dt + b*dz

The right-hand side has two parts — the trend and the Wiener process.
If there were no random component (no Wiener process), the movement of x would be deterministic, meaning the expected rate of change of x would simply be a:

dx = a*dt, – Drift Rate, – a = dx/dt.

Integrating over a small time interval gives: x =x0​+at

The second component, b*dz, adds noise/variability to the deterministic trend.
Its magnitude is determined by b times the Wiener process.

Since the variance rate of the Wiener process per unit time is 1, the variance rate of b*dz is b2.

On the graph, you can see that as the curve rises, the magnitude of fluctuations increases as well.

Itô Process

An Itô process is a type of generalized Wiener process in which ( a ) and ( b ) are functions of both time and the underlying variable:

dx = a (x, t) dt + b (x, t) dz


Itô’s Lemma

Itô proved that if x (e.g., a stock price) follows an Itô process, and G is a function of ( x ) and ( t ) (e.g., an option), then G also follows an Itôs process — with its drift rate and variance rate determined by a specific formula:


and where the ( dz ) term is the same Wiener process as in the original equation.

In short, this not-so-intuitive and mathematically challenging formula gave finance professionals the ability to formally link the value of a stock and the value of its derivative (option).

To make Itô’s Lemma clearer, one needs to first understand how a stock price evolves over time — a topic that will be discussed in the next section.

Source: Options, Futures & Other Derivatives, John C. Hull