Itô's Lemma and Stochastic Calculus › Stochastic Differential Equations

Building on our understanding of Brownian motion from the previous chapter, we now formalize how to describe the continuous-time evolution of random processes. We need to express how a quantity changes when it involves both predictable trends and random fluctuations. A stochastic differential equation (SDE) provides exactly this capability by expressing the instantaneous change in a process as the sum of a deterministic component and a random component. This dual structure captures the essential nature of financial markets, where prices exhibit both systematic tendencies and random fluctuations.

The differential notation $dX_t$ represents the infinitesimal change in $X_t$ over an infinitesimally small time interval $dt$. This notation provides intuition and leads to correct results when following the rules of stochastic calculus, despite lacking traditional rigor. The notation can be made rigorous through the theory of stochastic integration, but for practical purposes, working with differentials and following consistent rules yields reliable answers.

The two terms in an SDE have distinct interpretations:

• Drift term ($\mu \, dt$): The expected change in the process over time. This is the deterministic trend component that would remain if we removed all randomness. You can think of it as the underlying direction in which the process is being pushed. If we observed many sample paths and averaged them, the drift would be what remains.

• Diffusion term ($\sigma \, dW_t$): The random fluctuation around the drift. Since $dW_t$ has variance $dt$, the diffusion coefficient $\sigma$ controls the magnitude of random shocks. Larger values of $\sigma$ mean the process experiences more substantial random perturbations at each instant.

The simplest SDE has constant coefficients:

where:

• $X_t$: value of the process at time $t$

• $\mu$: constant drift rate

• $\sigma$: constant volatility

• $dt$: infinitesimal time step

• $dW_t$: infinitesimal Brownian increment

This is arithmetic Brownian motion, also called Brownian motion with drift. The process drifts upward at rate $\mu$ while fluctuating randomly with volatility $\sigma$. Because both the drift and diffusion coefficients are constants that do not depend on the current level of $X_t$, the process can wander anywhere on the real line. As we discussed in the previous chapter, this process can take negative values, making it unsuitable for modeling asset prices directly. After all, stock prices cannot become negative, yet arithmetic Brownian motion places no lower bound on the values it can reach.

A more realistic model for asset prices is geometric Brownian motion (GBM), which we introduced conceptually in the previous chapter. The key innovation in GBM is that both the drift and diffusion scale with the current price level:

where:

• $S_t$: asset price at time $t$

• $\mu$: constant percentage drift

• $\sigma$: constant percentage volatility

• $dt$: infinitesimal time increment

• $dW_t$: Brownian motion increment

Here, both the drift and diffusion are proportional to the current price level. This proportionality fundamentally changes the behavior of the process and ensures several desirable properties:

• The process remains strictly positive (prices cannot go negative) because the multiplicative structure prevents the process from crossing zero

• Percentage returns, not absolute returns, have constant volatility, which matches empirical observations about financial markets

• The model exhibits the multiplicative dynamics observed in real markets, where a 10% gain followed by a 10% loss does not return you to your starting point

We can factor out $S_t$ to see that the percentage change follows a simple form:

where:

• $dS_t/S_t$: instantaneous percentage return

• $\mu$: drift rate

• $dt$: infinitesimal time increment

• $\sigma$: volatility parameter

• $dW_t$: Brownian motion increment

This equation says the instantaneous percentage return has drift $\mu$ and volatility $\sigma$. This means volatility is usually a percentage of price, not an absolute dollar amount. A 20% volatility means that daily percentage moves have a standard deviation related to this 20% annual figure, regardless of whether the stock trades at $10 or$1,000 This matches the stylized facts of financial returns we covered in Chapter 1 of this part.