Itô's Lemma and Stochastic Calculus › Worked Examples

Let's apply Itô's Lemma to several important examples that arise frequently in financial modeling. These examples illustrate both the mechanics of applying the formula and the surprising results that emerge from stochastic calculus.

Consider $f(W_t) = W_t^2$ where $W_t$ is standard Brownian motion. Here $dW_t = dW_t$ (the process is Brownian motion itself with $\mu = 0$ and $\sigma = 1$). This simple example clearly demonstrates how Itô's Lemma differs from ordinary calculus.

Computing the partial derivatives:

$$\frac{\partial f}{\partial x} = 2W_t, \quad \frac{\partial^2 f}{\partial x^2} = 2, \quad \frac{\partial f}{\partial t} = 0$$

where:

• $f(x, t) = x^2$: the function being differentiated

• $W_t$: the stochastic process (Brownian motion)

Applying Itô's Lemma with $\mu = 0$ and $\sigma = 1$:

$$\begin{aligned} d(W_t^2) &= \left( 0 + 0 + \frac{1}{2} \cdot 1^2 \cdot 2 \right) dt + 1 \cdot 2W_t \, dW_t && \text{(substitute partials into Itô formula)} \\ &= dt + 2W_t \, dW_t && \text{(simplify)} \end{aligned}$$

where:

• $d(W_t^2)$: differential of the squared Brownian motion

• $dt$: contribution from the Itô correction term

• $2W_t \, dW_t$: contribution from the standard chain rule

Notice that the ordinary chain rule would give $d(W_t^2) = 2W_t \, dW_t$, missing the $dt$ term entirely. The extra $dt$ term comes from the Itô correction and reflects the fact that Brownian motion is constantly fluctuating. Even when $W_t = 0$, the squared process is increasing on average. This may seem counterintuitive at first, but it makes sense: when the Brownian motion passes through zero, it doesn't stay there. The constant oscillations around zero still contribute positive values to the squared process.

We can verify this result and gain further insight by integrating both sides:

$$\begin{aligned} W_T^2 &= \int_0^T dt + 2\int_0^T W_t \, dW_t \\ &= T + 2\int_0^T W_t \, dW_t \end{aligned}$$

where:

• $W_T^2$: terminal squared value of Brownian motion

• $T$: total time horizon (integral of the deterministic drift $dt$)

• $\int_0^T W_t \, dW_t$: stochastic integral term

• $W_T$: value of Brownian motion at time $T$

Rearranging gives us an explicit formula for the stochastic integral:

$$\int_0^T W_t \, dW_t = \frac{1}{2}(W_T^2 - T)$$

where:

• $\int_0^T W_t \, dW_t$: stochastic integral of Brownian motion with respect to itself

• $W_T^2$: squared terminal value of Brownian motion

Compare this to ordinary calculus, where $\int_0^T x \, dx = \frac{1}{2}T^2$. The $-T/2$ correction term appears because stochastic integrals behave differently from ordinary integrals. This example showcases how the Itô correction manifests in integral form.

This is the most important example for finance because it connects the geometric Brownian motion model of asset prices to the lognormal distribution. Suppose $S_t$ follows geometric Brownian motion:

$$dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$$

where:

• $S_t$: asset price

• $\mu$: drift parameter

• $\sigma$: volatility parameter

• $dW_t$: Brownian increment

Let $f(S_t) = \ln(S_t)$. We want to find the dynamics of the log-price, which will reveal important properties about the distribution of future prices. Computing derivatives with respect to $S$:

$$\frac{\partial f}{\partial S} = \frac{1}{S_t}, \quad \frac{\partial^2 f}{\partial S^2} = -\frac{1}{S_t^2}, \quad \frac{\partial f}{\partial t} = 0$$

where:

• $f(S)$: natural logarithm function

• $S_t$: geometric Brownian motion process

Notice that the second derivative is negative, indicating that the logarithm function is concave. This concavity will cause the Itô correction to subtract from the drift rather than add to it.

Applying Itô's Lemma with $\mu_S = \mu S_t$ and $\sigma_S = \sigma S_t$:

$$\begin{aligned} d(\ln S_t) &= \left( 0 + \frac{\mu S_t}{S_t} + \frac{1}{2} \sigma^2 S_t^2 \cdot \left( -\frac{1}{S_t^2} \right) \right) dt + \frac{\sigma S_t}{S_t} \, dW_t && \text{(apply Itô's Lemma)} \\ &= \left( \mu - \frac{1}{2}\sigma^2 \right) dt + \sigma \, dW_t && \text{(simplify terms)} \end{aligned}$$

where:

• $d(\ln S_t)$: differential of the log-price process

• $\mu - \frac{1}{2}\sigma^2$: drift of the log-price

• $\sigma$: volatility of the log-price

This is an main result. Even though $S_t$ follows geometric Brownian motion (a complicated multiplicative process with state-dependent coefficients), its logarithm follows simple arithmetic Brownian motion with constant drift $\mu - \frac{1}{2}\sigma^2$ and constant volatility $\sigma$. This step to additive dynamics makes GBM tractable.

Integrating from $0$ to $T$:

$$\begin{aligned} \ln S_T - \ln S_0 &= \left( \mu - \frac{1}{2}\sigma^2 \right) T + \sigma W_T \\ \ln \frac{S_T}{S_0} &= \left( \mu - \frac{1}{2}\sigma^2 \right) T + \sigma W_T \end{aligned}$$

where:

• $S_T, S_0$: asset prices at times $T$ and $0$

• $\mu, \sigma$: drift and volatility parameters

• $W_T$: Brownian motion value at $T$

Since $W_T \sim \mathcal{N}(0, T)$, we can immediately determine the distribution of log-returns:

$$\ln \frac{S_T}{S_0} \sim \mathcal{N}\left( \left( \mu - \frac{1}{2}\sigma^2 \right) T, \sigma^2 T \right)$$

where:

• $\mathcal{N}(\cdot)$: normal distribution

• $\ln(S_T/S_0)$: log-return over the period $[0, T]$

This proves that log-returns are normally distributed under the GBM model. Taking the exponential of both sides yields the explicit solution to the GBM SDE:

$$S_T = S_0 \exp\left( \left( \mu - \frac{1}{2}\sigma^2 \right) T + \sigma W_T \right)$$

where:

• $S_T$: asset price at time $T$

• $S_0$: initial asset price

• $\mu$: expected return parameter

• $\sigma$: volatility parameter

• $W_T$: value of Brownian motion at time $T$

The term $-\frac{1}{2}\sigma^2$ is called the Itô drift correction. It explains why the expected log-return $(\mu - \frac{1}{2}\sigma^2)$ is less than the expected percentage return $\mu$. This correction arises from the concavity of the logarithm function combined with the volatility of the underlying process. It carries into portfolio theory and long-term wealth accumulation, as we shall see in later chapters.

Consider $Y_t = e^{\sigma W_t - \frac{1}{2}\sigma^2 t}$. This exponential form appears frequently in risk-neutral pricing and change of measure arguments. The specific form of the exponent, with its $-\frac{1}{2}\sigma^2 t$ term, is carefully chosen to produce a special property.

Let $g(W_t, t) = e^{\sigma W_t - \frac{1}{2}\sigma^2 t}$, or equivalently, $g(w, t) = e^{\sigma w - \frac{1}{2}\sigma^2 t}$.

Computing derivatives:

$$\begin{aligned} \frac{\partial g}{\partial t} &= -\frac{1}{2}\sigma^2 e^{\sigma w - \frac{1}{2}\sigma^2 t} = -\frac{1}{2}\sigma^2 Y_t \\ \frac{\partial g}{\partial w} &= \sigma e^{\sigma w - \frac{1}{2}\sigma^2 t} = \sigma Y_t \\ \frac{\partial^2 g}{\partial w^2} &= \sigma^2 e^{\sigma w - \frac{1}{2}\sigma^2 t} = \sigma^2 Y_t \end{aligned}$$

where:

• $g(w, t)$: the function defining the martingale

• $Y_t$: value of the exponential martingale

• $\sigma$: volatility parameter

Applying Itô's Lemma (with $dW_t = dW_t$, so $\mu = 0$, $\sigma_W = 1$):

$$\begin{aligned} dY_t &= \left( -\frac{1}{2}\sigma^2 Y_t + 0 + \frac{1}{2} \cdot 1^2 \cdot \sigma^2 Y_t \right) dt + 1 \cdot \sigma Y_t \, dW_t && \text{(apply Itô's Lemma)} \\ &= \sigma Y_t \, dW_t && \text{(drift terms cancel)} \end{aligned}$$

where:

• $dY_t$: change in the martingale process

• $\sigma Y_t$: volatility term proportional to the process level

The process $Y_t$ has zero drift! This means $Y_t$ is a martingale, which is a process whose expected future value equals its current value. The careful choice of $-\frac{1}{2}\sigma^2 t$ in the exponent exactly cancels the Itô correction that would otherwise appear from the convexity of the exponential function. This cancellation is not coincidental; it is precisely engineered to eliminate the drift.

In the next chapter, this exponential martingale provides the mathematical mechanism for changing from the real-world probability measure to the risk-neutral measure, which is essential for derivative pricing.

Sometimes we need the dynamics of a product $X_t Y_t$, for example when computing the dynamics of wealth or hedged portfolio value. The product rule for Itô processes differs from the ordinary product rule by an additional term:

$$d(X_t Y_t) = X_t dY_t + Y_t dX_t + dX_t dY_t$$

where:

• $d(X_t Y_t)$: differential of the product

• $X_t, Y_t$: two Itô processes

• $dX_t \, dY_t$: quadratic covariation term

The extra term $dX_t \, dY_t$ has no analog in ordinary calculus. It arises from the cross-term in the Taylor expansion and survives because of the non-zero quadratic variation of Brownian motion.

Using the multiplication rules that encode the behavior of infinitesimal products:

• $dt \cdot dt = 0$ (second-order in time, negligible)

• $dt \cdot dW_t = 0$ (mixed term, order $(\Delta t)^{3/2}$, negligible)

• $dW_t \cdot dW_t = dt$ (quadratic variation of Brownian motion)

If $dX_t = \mu_X dt + \sigma_X dW_t$ and $dY_t = \mu_Y dt + \sigma_Y dW_t$ (driven by the same Brownian motion), then:

$$dX_t \cdot dY_t = \sigma_X \sigma_Y dt$$

where:

• $\sigma_X, \sigma_Y$: diffusion coefficients of $X$ and $Y$

The full product rule becomes:

$$\begin{aligned} d(X_t Y_t) &= X_t (\mu_Y dt + \sigma_Y dW_t) + Y_t (\mu_X dt + \sigma_X dW_t) + \sigma_X \sigma_Y dt && \text{(substitute differentials)} \\ &= (\mu_X Y_t + \mu_Y X_t + \sigma_X \sigma_Y) dt + (\sigma_X Y_t + \sigma_Y X_t) dW_t && \text{(group drift and diffusion terms)} \end{aligned}$$

where:

• $d(X_t Y_t)$: differential of the product process

• $(\sigma_X Y_t + \sigma_Y X_t) dW_t$: interaction term from covariance term $\sigma_X \sigma_Y$ in the drift represents the correlation between the two processes. When both processes are driven by the same Brownian motion and both have positive diffusion coefficients, this term adds to the drift of the product. This effect captures the idea that correlated random movements reinforce each other when considering the product.