Getting Started with Itô’s Lemma for Stochastic Finance

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Stochastic calculus gives us the tools to work with randomness in motion. Classic calculus fails when the underlying variable follows a Brownian path. With Itô’s Lemma and its extensions, we analyze asset prices, derive dynamics, and simulate outcomes under uncertainty. Let’s look at Itô’s Lemma and examine how we apply them to equities and interest rates.

Let X_t follow an Itô process:

This handles a single process. Most practical models involve multiple variables. For two stochastic processes Xt and Yt, each with its own drift, volatility, and a possible correlation ρ. Brownian motion has infinite variation. That makes Itô calculus a second-order theory, unlike classical first-order differentials.

Applications in Equities and Interest Rates

Let S_t be an equity price following geometric Brownian motion:

Apply Itô’s Lemma to ln⁡(S_t):

This result shows that log returns are normally distributed. It also allows us to simulate S_t directly:

This forms the basis for derivative pricing and portfolio modeling.

Interest rate models often use mean-reverting processes. A common form is the Ornstein-Uhlenbeck process:

Apply Itô’s Lemma to f(r_t) = r_t², and you get a dynamic for the variance. Apply it to bond prices P(t, T), and you get the PDE governing fixed-income instruments. These dynamics shape yield curves and forward rates.

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Generating Standard Normal Random Variables

Simulating stochastic processes begins with normal random variables. To simulate Brownian motion, we draw:

Then construct an increment ΔW over a small time step Δt as:

This allows numerical schemes such as Euler-Maruyama:

By simulating thousands of paths, we approximate expectations under stochastic models. This is the basic concept behind Monte Carlo simulation.

The Steady State Distribution

Some stochastic processes settle into a stable distribution. These processes are called ergodic. Over time, they forget their initial state.

The Ornstein-Uhlenbeck process gives us:

As t→∞, the distribution of r_t approaches:

This is the steady state. It gives insight into the long-run behavior of interest rates or volatility.

If a process lacks a steady state (like geometric Brownian motion), the variance grows with time. That reflects unbounded uncertainty. If a process has a steady state, its risk is contained.

Steady states guide portfolio construction and scenario analysis. They help quantify long-run expectations and define convergence targets in simulation.

Stochastic calculus extends classical tools into a world ruled by uncertainty. Itô’s Lemma and its extensions allow precise modeling of returns, rates, and risk. We produce standard normal draws to simulate paths. We study long-run behavior through steady state distributions.

The formulas matter, but the insight matters more. Every derivative model, every Monte Carlo path, every hedging ratio starts with this calculus. To work with financial motion, you must learn to work with noise. And noise, it turns out, has structure.

import numpy as np
import matplotlib.pyplot as plt

# Set up consistent matplotlib style
plt.rcParams.update({
    "font.family": "serif",
    "axes.spines.top": False,
    "axes.spines.right": False
})

# Ito's Lemma for log(S)
def ito_log_gbm(S0, mu, sigma, T, steps):
    dt = T / steps
    t = np.linspace(0, T, steps + 1)
    W = np.cumsum(np.random.normal(0, np.sqrt(dt), size=steps))
    W = np.insert(W, 0, 0)
    ln_S = np.log(S0) + (mu - 0.5 * sigma**2) * t + sigma * W
    S = np.exp(ln_S)
    return t, S, ln_S

# Simulate Ornstein-Uhlenbeck process
def simulate_ou(r0, mu, theta, sigma, T, steps):
    dt = T / steps
    r = np.zeros(steps + 1)
    r[0] = r0
    for i in range(steps):
        dr = theta * (mu - r[i]) * dt + sigma * np.sqrt(dt) * np.random.normal()
        r[i+1] = r[i] + dr
    return np.linspace(0, T, steps + 1), r

# Generate standard normal from Brownian increment
def simulate_standard_normal_from_bm(T, steps):
    dt = T / steps
    dW = np.random.normal(0, np.sqrt(dt), size=steps)
    Z = dW / np.sqrt(dt)  # Standardized to N(0,1)
    return Z

# Steady-state distribution for OU
def steady_state_ou_pdf(x, mu, theta, sigma):
    var = sigma**2 / (2 * theta)
    return (1 / np.sqrt(2 * np.pi * var)) * np.exp(-(x - mu)**2 / (2 * var))

# Simulate GBM and log-returns using Ito
t, S, ln_S = ito_log_gbm(S0=100, mu=0.05, sigma=0.2, T=1, steps=1000)

plt.figure(figsize=(10, 4))
plt.plot(t, S, label="GBM Price")
plt.title("GBM Simulation")
plt.xlabel("Time")
plt.ylabel("Value")
plt.savefig("ito_gbm_logprice.png")
plt.show()

# Simulate Ornstein-Uhlenbeck Process
t_ou, r = simulate_ou(r0=0.03, mu=0.05, theta=0.7, sigma=0.02, T=5, steps=1000)

plt.figure(figsize=(10, 4))
plt.plot(t_ou, r, label="OU Process (Interest Rate)")
plt.title("Ornstein-Uhlenbeck Simulation")
plt.xlabel("Time")
plt.ylabel("Rate")
plt.savefig("ou_process.png")
plt.show()

# Generate and plot standard normal
Z = simulate_standard_normal_from_bm(T=1, steps=10000)

plt.figure(figsize=(10, 4))
plt.hist(Z, bins=50, density=True, alpha=0.6, label="Simulated Standard Normals")
x = np.linspace(-4, 4, 200)
plt.plot(x, (1/np.sqrt(2*np.pi)) * np.exp(-x**2 / 2), label="PDF N(0,1)", linestyle='--')
plt.title("Standard Normal from Brownian Increments")
plt.xlabel("Z")
plt.ylabel("Density")
plt.savefig("standard_normal.png")
plt.show()

# Plot steady-state distribution
x_vals = np.linspace(-0.02, 0.10, 300)
pdf_vals = steady_state_ou_pdf(x_vals, mu=0.05, theta=0.7, sigma=0.02)

plt.figure(figsize=(10, 4))
plt.plot(x_vals, pdf_vals, label="Steady-State PDF")
plt.title("Steady-State Distribution of OU Process")
plt.xlabel("Rate")
plt.ylabel("Density")
plt.savefig("ou_steady_state.png")
plt.show()