May 19, 2026
While a simple Wiener process relies on constant drift and constant volatility coefficients, real-world asset prices operate in changing economic environments. To capture this complexity, we expand our framework to an Itô process.
An Itô process is a generalized Wiener process(1.1) where the drift parameter $a(S, t)$ and the volatility parameter $b(S, t)$ are no longer fixed constants, but dynamic functions of the current stock price $S$ and time $t$:
$$dS = a(S, t) \, dt + b(S, t) \, dz$$
To approximate this continuous-time process over a very small discrete time interval $\Delta t$, we assume that the drift and variance rates remain constant at their initial values calculated at time $t$. This allows us to write the change in the stock price as:
$$\Delta S = a(S, t) \, \Delta t + b(S, t) \epsilon \sqrt{\Delta t}$$
where $\epsilon \sim N(0, 1)$.
In standard calculus, if we want to find the total change $df$ in a continuous function $f(x)$ caused by a small change $dx$, we rely on a standard Taylor series expansion. For deterministic variables, we safely ignore second-order terms like $(dx)^2$ because they become infinitesimally small much faster than $dx$ as the time interval shrinks toward zero.
However, in stochastic calculus—where variables are driven by continuous random shocks—this assumption breaks down. Because the random component of a process scales with $\sqrt{dt}$, squaring that random change yields a term directly proportional to $dt$ since $(\sqrt{dt})^2 = dt$. This means that the second-order volatility effect does not vanish; instead, it leaves a permanent mark on the expected path of the function.
While an Itô process allows parameters to vary dynamically, we must choose functional forms for drift and variance that accurately mirror how real financial markets operate.
It is tempting to model stock prices using a simple generalized Wiener process with a constant absolute drift rate ($a$) and a constant variance rate ($b$). However, this approach fails to capture a core economic reality: the expected percentage return required by investors is independent of the stock's absolute price level.
If an investor requires a 10% expected return per year when a stock is trading at \$100, they will still require a 10% expected return if that same stock splits, grows, or drops to \$10 (ceteris paribus).
Therefore, modeling stock price movements with a constant absolute dollar increase is inappropriate. Instead, we assume that the expected percentage return (the absolute expected drift divided by the current stock price) remains constant. For simplicity throughout this framework, the asset is assumed to be a non-dividend-paying stock.
If $S$ is the stock price at time $t$, and $\mu$ is the constant expected rate of return, the absolute expected drift rate must scale proportionally with the stock price:
$$\text{Expected Drift Rate} = \mu S$$
Thus, in a very short time interval $\Delta t$, the expected absolute increase in the stock price is $\mu S \, \Delta t$.
A similar logic applies to market uncertainty. An investor is generally just as uncertain about the percentage return when a stock is at \$100 as they are when it is at \$10.
This means that the standard deviation of the absolute price change over a short period $\Delta t$ should also be directly proportional to the stock price itself. We express this absolute variability as:
$$\text{Absolute Volatility Component} = \sigma S \, dz$$
where $\sigma$ represents the volatility of the stock price, and $\sigma^2$ represents its variance rate.
Combining our proportional drift and proportional volatility components yields the standard Itô process for stock price behavior, widely known as Geometric Brownian Motion (GBM):
$$dS = \mu S \, dt + \sigma S \, dz\tag{2.1}$$
By dividing both sides by the current stock price $S$, we can view this process purely in terms of the instantaneous percentage return:
$$\frac{dS}{S} = \mu \, dt + \sigma \, dz $$
Where:
To simulate asset price paths or estimate parameters using historical daily stock data, we must translate the continuous-time equations into a discrete-time framework. Over a small discrete time interval $\Delta t$, the change in the stock price from $S$ to $S + \Delta S$ is expressed by approximating equation (3.5):
$$\frac{\Delta S}{S} = \mu \Delta t + \sigma \epsilon \sqrt{\Delta t} $$
Alternatively, we can express this as the absolute dollar change ($\Delta S$) by multiplying through by the current stock price $S$:
$$\Delta S = \mu S \Delta t + \sigma S \epsilon \sqrt{\Delta t} $$
where $\epsilon$ represents a random variable drawn from a standard normal distribution, $\epsilon \sim N(0, 1)$.
This allows us to express the distribution of the discrete return compactly as:
$$\frac{\Delta S}{S} \sim \phi(\mu \, \Delta t, \sigma^2 \Delta t) \tag{2.2}$$
The mathematical tool used to calculate the differential of a function of a stochastic process is known as Itô’s Lemma. It can be viewed as the stochastic equivalent of the chain rule in standard calculus and is foundational for pricing derivatives on financial statements.
If we have a continuous, differentiable function $G(S, t)$ that depends on the asset price $S$ and time $t$ (such as a stock option), Itô’s Lemma states that $G$ also follows a process driven by the same underlying random shocks:
$$dG = \left( a \frac{\partial G}{\partial S} + \frac{\partial G}{\partial t} + \frac{1}{2} b^2 \frac{\partial^2 G}{\partial S^2} \right) dt + b \frac{\partial G}{\partial S} \, dz\tag{2.3}$$
Note: See the Appendix for the complete mathematical derivation of Itô’s Lemma. This formal relationship shows that the change in the derivative value $G$ over an infinitesimal time step $dt$ is normally distributed:
$$dG \sim N\left( \left( a \frac{\partial G}{\partial S} + \frac{\partial G}{\partial t} + \frac{1}{2} b^2 \frac{\partial^2 G}{\partial S^2} \right) dt, \,\, b^2 \left(\frac{\partial G}{\partial S}\right)^2 dt \right)$$
For students evaluating financial instruments, this proves a vital economic reality: a derivative's expected change depends not only on the drift of the underlying stock ($a$) and time decay ($\frac{\partial G}{\partial t}$), but also explicitly on the asset's variance ($b^2$) via the second derivative ($\frac{\partial^2 G}{\partial S^2}$). This extra variance term is known as the Itô correction factor.
While the absolute change $dS$ is normally distributed over an instantaneous step, we need to find how the stock price behaves over a long time horizon $T$. Because $S$ changes continuously, we cannot simply add up $dS$ steps linearly over a long period. Instead, we use Itô’s Lemma to analyze the process followed by the natural logarithm of the stock price.
$$G = \ln S$$
Let our function be defined as $G(S, t) = \ln S$. We begin by calculating its partial derivatives with respect to $S$ and $t$:
First derivative with respect to $S$: $$\frac{\partial G}{\partial S} = \frac{1}{S}$$
Second derivative with respect to $S$: $$\frac{\partial^2 G}{\partial S^2} = -\frac{1}{S^2}$$
First derivative with respect to $t$: $$\frac{\partial G}{\partial t} = 0$$
From our stock price model $dS = \mu S \, dt + \sigma S \, dz$, we identify our drift and volatility coefficients to substitute into Itô's Lemma:
Recall the general formula for Itô’s Lemma established in the previous section (2.1):
$$dG = \left( a \frac{\partial G}{\partial S} + \frac{\partial G}{\partial t} + \frac{1}{2} b^2 \frac{\partial^2 G}{\partial S^2} \right) dt + b \frac{\partial G}{\partial S} \, dz$$
Substituting our partial derivatives and process coefficients into this formula gives:
$$dG = \left( (\mu S) \frac{1}{S} + 0 + \frac{1}{2} (\sigma S)^2 \left( -\frac{1}{S^2} \right) \right) dt + (\sigma S) \frac{1}{S} \, dz$$
We simplify the terms inside the brackets by canceling out the stock price variable $S$:
$$dG = \left( \mu - \frac{1}{2} \sigma^2 \right) dt + \sigma \, dz$$
Since $G = \ln S$, we can write the continuous-time process for the log stock price as:
$$d(\ln S) = \left( \mu - \frac{1}{2} \sigma^2 \right) dt + \sigma \, dz$$
Because the drift coefficient $\left( \mu - \frac{1}{2} \sigma^2 \right)$ and the volatility coefficient $\sigma$ are entirely constant, $G =\ln S$ follows the generalized Wiener process.
Since the total change in the Wiener process over a horizon $T$ behaves as $z_T - z_0 = \epsilon \sqrt{T}$ where $\epsilon \sim N(0,1)$, we find that the total long-term change in the log stock price is:
$$\ln S_T - \ln S_0 \sim N\left( \left( \mu - \frac{1}{2} \sigma^2 \right) T, \,\, \sigma^2 T \right) \tag{2.4}$$
Alternatively, adding $\ln S_0$ to both sides defines the final distribution of the log stock price at time $T$:
$$\ln S_T \sim N\left( \ln S_0 + \left( \mu - \frac{1}{2} \sigma^2 \right) T, \,\, \sigma^2 T \right)$$
The formula for Itô’s Lemma is highly structured. While it resembles a standard Taylor expansion from regular calculus, the core breakthrough lies in how random volatility creates a unique non-vanishing term: $\frac{1}{2} b^2 \frac{\partial^2 G}{\partial S^2}$. Here is the logical step-by-step derivation for students.
First, we assume our asset price $S$ follows an Itô stochastic process driven by random volatility over time:
$$dS = a \, dt + b \, dz$$
Where:
Our goal is to track $dG$, which represents the infinitesimal change in a separate derivative function $G(S, t)$ whose valuation depends completely on this underlying stock price $S$ and time $t$.
In mathematics, when looking for the small total change $dG$ of a multivariable function $G(S, t)$, we use a multivariate Taylor expansion. Writing this out up to the second-order derivatives yields:
$$dG = \frac{\partial G}{\partial S}dS + \frac{\partial G}{\partial t}dt + \frac{1}{2}\frac{\partial^2 G}{\partial S^2}(dS)^2 + \frac{1}{2}\frac{\partial^2 G}{\partial t^2}(dt)^2 + \frac{\partial^2 G}{\partial S \partial t}dS\,dt + \dots$$
In standard calculus, as the time increment approaches zero ($dt \to 0$), all squared or cross-product terms like $(dt)^2$, $dS\,dt$, and $(dS)^2$ shrink so fast that they are treated as zero and ignored. However, in stochastic calculus, the random noise alters this behavior completely.
Let us square our asset process equation from Step 1 to see how $(dS)^2$ behaves:
$$(dS)^2 = (a\,dt + b\,dz)^2 = a^2(dt)^2 + 2ab\,dt\,dz + b^2(dz)^2$$
As the time step becomes infinitesimally small, we must perform an analysis on the order of magnitude of each component. Recall that a Wiener process shock $dz$ scales proportionally to the square root of time ($\sqrt{dt}$). This gives us the following properties:
This leads us to the formal Itô Multiplication Table rules:
Now, we take these surviving rules and substitute them back into our Taylor expansion from Step 2. Eliminating the terms that vanished to zero leaves us with a streamlined expression:
$$dG = \frac{\partial G}{\partial S}dS + \frac{\partial G}{\partial t}dt + \frac{1}{2}\frac{\partial^2 G}{\partial S^2}(dS)^2$$
Next, we expand this by inserting the full definition of our stock price process ($dS = a\,dt + b\,dz$) and our proven identity for the squared term ($(dS)^2 = b^2 dt$):
$$dG = \frac{\partial G}{\partial S}(a\,dt + b\,dz) + \frac{\partial G}{\partial t}dt + \frac{1}{2}\frac{\partial^2 G}{\partial S^2}(b^2 dt)$$
Distributing the terms yields:
$$dG = \frac{\partial G}{\partial S}a\,dt + \frac{\partial G}{\partial S}b\,dz + \frac{\partial G}{\partial t}dt + \frac{1}{2}\frac{\partial^2 G}{\partial S^2}b^2 dt$$
As a final step, we gather all the deterministic parts containing the common factor $dt$ together into a single bracket, and place the stochastic random shock ($dz$) neatly at the end:
$$dG = \left( a \frac{\partial G}{\partial S} + \frac{\partial G}{\partial t} + \frac{1}{2}b^2 \frac{\partial^2 G}{\partial S^2} \right)dt + b \frac{\partial G}{\partial S}\,dz$$