Applying Itô's Lemma to Stock Price Processes

Why Ordinary Calculus Fails: The Role of Quadratic Variation

To understand why we need a new rule, let's consider the Taylor series expansion for a function $f(X_t)$, where $X_t$ is some process:

In ordinary calculus, the $(d{X}_t{)}^2$ term is of a higher order than $d{X}_t$ and vanishes as the time step goes to zero. However, when $X_t$ is a stochastic process driven by Brownian motion, this is no longer true.

The key insight, which you may recall from your probability studies, is that a Wiener process has a non-zero quadratic variation. Informally, over a small time interval $dt$, the squared increment of a Wiener process, $(d{W}_t{)}^2$, is not random but deterministic and equal to $dt$. All higher-order terms, like $(d{W}_t{)}^3$, do vanish.

This means that when we expand a function of an Itô process, the second-order term $(d{X}_t{)}^2$ does not disappear and must be retained. The following video provides an excellent explanation of why this is the case.

Itô's Lemma: The Chain Rule for Stochastic Processes

With the insight about quadratic variation, we can now state the lemma. Let $X_t$ be an Itô process following the SDE:

And let $f(x,t)$ be a twice-differentiable function. Then the process $Y_t=f(X_t,t)$ also follows an SDE, given by Itô's Lemma:

Let's dissect this formula:

• $\frac{\partial f}{\partial t}dt+a(X_t,t)\frac{\partial f}{\partial x}dt$: This is the new drift term. It includes the change due to time passing ($\partial f/\partial t$) and the drift contributed by the underlying process $X_t$.
• $b(X_t,t)\frac{\partial f}{\partial x}d{W}_t$: This is the new diffusion term. The randomness is inherited from $X_t$.
• $\frac{1}{2}b(X_t,t{)}^2\frac{{\partial }^{2}f}{\partial x^2}dt$: This is the Itô correction term. It is a non-random drift component that arises purely from the volatility of the underlying process. This term has no equivalent in ordinary calculus and is a direct consequence of the $(d{W}_t{)}^2=dt$ rule.

Applying Itô's Lemma to Geometric Brownian Motion

Now we'll use this powerful tool to achieve our main objective: finding the SDE for the natural logarithm of a stock price, $\ln (S_t)$, where $S_t$ follows GBM. This is the classic application of Itô's Lemma in finance.

Recall the SDE for $S_t$:

Here, our function is $f(S_t)=\ln (S_t)$. This function does not depend explicitly on time $t$, so $\partial f/\partial t=0$. The process is $S_t$, with drift coefficient $a(S_t,t)=\mu S_t$ and diffusion coefficient $b(S_t,t)=\sigma S_t$.

The following document provides a concise and clear walkthrough of the derivation.

As the derivation shows, we first find the partial derivatives of our function $f(s)=\ln (s)$:

• $\frac{\partial f}{\partial s}=\frac{1}{s}$
• $\frac{{\partial }^{2}f}{\partial s^2}=-\frac{1}{s^2}$

Plugging these, along with $a=\mu S_t$ and $b=\sigma S_t$, into the Itô's Lemma formula gives:

Simplifying this expression yields a remarkably clean result:

This is a pivotal result. The SDE for $\ln (S_t)$ describes a process with a constant drift $\mu -\frac{1}{2}{\sigma }^2$ and a constant volatility $\sigma$. This is a much simpler process—a generalized Wiener process—than the original GBM for $S_t$, where the drift and diffusion coefficients depended on $S_t$ itself. This simplification is precisely why we perform this transformation.

The image below provides a great visual summary of what we just accomplished. It shows that while other transformations of the GBM process $S$ still result in complex processes, the log-transformation (labeled #3) simplifies it to a process with constant coefficients, which in turn implies a normal distribution for its values.

A Practical Example: The Volatility Drag

Let's solidify this with a numerical example. Consider a stock whose price follows GBM with an expected return $\mu =10\%$ and volatility $\sigma =20\%$. The SDE for the price $S_t$ is:

What is the SDE for its logarithm, $X_t=\ln (S_t)$? Using our derived formula, the new drift is:

The volatility remains $\sigma =0.20$. Therefore, the SDE for the log-price is:

Notice that the expected return of the stock is 10%, but the drift of its logarithm is only 8%. This reduction of $\frac{1}{2}{\sigma }^2$ is often called "volatility drag." It's a direct consequence of the Itô correction term and is a crucial, non-intuitive feature of stochastic models. It implies that volatility doesn't just create uncertainty around the drift; it actively reduces the compound growth rate of the asset's logarithm.

Conclusion

In this lesson, we've unlocked the primary computational tool of continuous-time finance, Itô's Lemma. We saw how it extends the familiar chain rule to the stochastic world by incorporating a correction term for quadratic variation.

Key Takeaways:

• Itô's Lemma is the "chain rule" for functions of stochastic processes. It is necessary because the squared increment of a Wiener process, $(d{W}_t{)}^2$, is on the order of $dt$ and cannot be ignored.
• The lemma adds a second-derivative term to the standard chain rule: $\frac{1}{2}{b}^2\frac{{\partial }^{2}f}{\partial x^2}dt$.
• Applying Itô's Lemma to the function $f(S_t)=\ln (S_t)$ for a stock following GBM transforms the complex SDE for $S_t$ into a simple SDE for $\ln (S_t)$ with constant coefficients.
• The resulting SDE for the log-price is: $d(\ln S_t)=(\mu -\frac{1}{2}{\sigma }^2)dt+\sigma d{W}_t$. The term $\frac{1}{2}{\sigma }^2$ represents a "volatility drag" on the drift of the log-price.

We are now perfectly positioned for our next lesson. Having derived this simple SDE for the log-price, we can easily integrate it to find an explicit formula for $\ln (S_T)$ and, by extension, for $S_T$. This will allow us to determine the probability distribution of future stock prices—the famous lognormal distribution—and begin making quantitative predictions.