Brownian Motion and Itô Calculus
Brownian motion and stochastic calculus provide the mathematical framework for modeling random continuous processes. Itô calculus is essential for derivative pricing and quantitative finance.
Key Points
- Standard Brownian motion has independent increments and continuous paths.
- Itô's lemma extends the chain rule to stochastic processes.
- Stochastic integrals are defined as limits of simple processes.
Formulas
Itô's lemma
$$df(t, X_t) = \frac{\partial f}{\partial t} dt + \frac{\partial f}{\partial x} dX_t + \frac{1}{2}\frac{\partial^2 f}{\partial x^2} (dX_t)^2$$
Geometric Brownian motion
$$dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$$
Itô isometry
$$\mathbb{E}\left[\left(\int_0^T X_t \, dW_t\right)^2\right] = \mathbb{E}\left[\int_0^T X_t^2 \, dt\right]$$
Code Example
import numpy as np
T = 1.0
N = 1000
dt = T / N
dW = np.random.randn(N) * np.sqrt(dt)
W = np.cumsum(dW)
# Simulate geometric Brownian motion
S = 100 * np.exp(np.cumsum((0.05 - 0.5*0.2**2)*dt + 0.2*dW))
print(S[-1])