1. LEARNING OBJECTIVES

By the end of this lesson, you will be able to:

  • Define stochastic processes and classify them by state space and time space.

  • Construct Brownian Motion (Wiener process) from a simple random walk and prove its key properties.

  • Understand the properties of Brownian Motion (continuity, independent increments, normal increments, quadratic variation).

  • Define and apply Ito’s Lemma to transform stochastic differential equations.

  • Derive the solution to Geometric Brownian Motion (GBM) using Ito’s Lemma.

  • Apply Ito’s Lemma to derive the Black-Scholes partial differential equation.

  • Understand the relationship between Brownian Motion and the heat equation.

  • Simulate Brownian Motion and Geometric Brownian Motion paths.

2. STOCHASTIC PROCESSES – DEFINITION AND CLASSIFICATION

2.1 Definition

A stochastic process is a collection of random variables {X_t, t ∈ T} indexed by time t in some set T. For each fixed ω ∈ Ω, the function t → X_t(ω) is called a sample path or trajectory.

2.2 Classification

By State Space:

  • Discrete State: The process takes on a countable number of values (e.g., number of defaults, stock price movements in ticks).

  • Continuous State: The process takes on values in a continuous interval (e.g., stock price, interest rate).

By Time Space:

  • Discrete Time: T = {0, 1, 2, …} (e.g., daily returns, weekly volatility).

  • Continuous Time: T = [0, ∞) (e.g., Brownian Motion, diffusion processes).

2.3 Key Properties of Stochastic Processes

  1. Adaptedness: X_t is F_t-measurable (the value at time t is known given the information up to time t).

  2. Markov Property: The future depends only on the present, not on the past:

    P(X_{t+s} ∈ A | F_t) = P(X_{t+s} ∈ A | X_t)

  3. Martingale Property: E[X_t | F_s] = X_s for s ≤ t.

  4. Stationarity: The distribution of X_t is invariant under time shifts.

  5. Independent Increments: X_{t+s} – X_t is independent of F_t.

2.4 Financial Application – Asset Price Processes

Asset prices are modelled as continuous-time stochastic processes with:

  • Drift: The expected return (μ).

  • Volatility: The standard deviation of returns (σ).

  • Random Shocks: Brownian Motion (W_t).

3. BROWNIAN MOTION (WIENER PROCESS) – CONSTRUCTION AND PROPERTIES

3.1 Definition

A stochastic process {W_t, t ≥ 0} is a Brownian Motion (Wiener process) if:

  1. W_0 = 0 (starts at zero).

  2. Independent Increments: For any 0 ≤ t_1 < t_2 < … < t_n, the increments:

    W_{t_2} – W_{t_1}, W_{t_3} – W_{t_2}, …, W_{t_n} – W_{t_{n-1}}
    are independent.

  3. Normally Distributed Increments: For any 0 ≤ s < t:

    W_t – W_s ~ N(0, t – s)

  4. Continuity: The sample paths t → W_t are continuous almost surely.

3.2 Construction from a Random Walk

Let S_n = Σ_{i=1}^n ξ_i, where ξ_i = ±1 with equal probability (simple symmetric random walk).

Define the scaled process:

W_t^{(n)} = (1/√n) * S_{⌊nt⌋}

As n → ∞, W_t^{(n)} converges in distribution to Brownian Motion W_t.

3.3 Key Properties of Brownian Motion

  1. Mean: E[W_t] = 0.

  2. Variance: Var(W_t) = t.

  3. Covariance: For s ≤ t:

    Cov(W_s, W_t) = s

  4. Self-Similarity: For any scaling factor c > 0:

    {W_{ct}, t ≥ 0} = √c * {W_t, t ≥ 0} (in distribution).

  5. Time Reversal: {W_t – W_T, 0 ≤ t ≤ T} = {W_t, 0 ≤ t ≤ T} (in distribution).

  6. Fractal Dimension: Brownian Motion paths have fractal dimension 1.5 (they are nowhere differentiable).

3.4 Quadratic Variation of Brownian Motion

For a partition 0 = t_0 < t_1 < … < t_n = T, the quadratic variation is:

[W]T = lim{n→∞} Σ_{i=1}^n (W_{t_i} – W_{t_{i-1}})² = T

Implications:

  • The quadratic variation of Brownian Motion is deterministic (equal to T).

  • This is a fundamental result for stochastic calculus. It implies that (dW_t)² = dt.

3.5 The Covariation of Two Brownian Motions

If W_t^{(1)} and W_t^{(2)} are two Brownian motions with correlation ρ, then:

dW_t^{(1)} dW_t^{(2)} = ρ dt

3.6 Brownian Motion as a Martingale

Brownian Motion W_t is a martingale:

E[W_t | F_s] = W_s for s ≤ t

Proof:

E[W_t | F_s] = E[W_s + (W_t – W_s) | F_s] = W_s + E[W_t – W_s] = W_s

4. ITO’S LEMMA – THE CHAIN RULE FOR STOCHASTIC CALCULUS

4.1 The Ito Integral

Before Ito’s Lemma, we need the Ito integral. For an adapted process θ_t satisfying E[∫_0^T θ_t² dt] < ∞, the Ito integral is:

∫_0^T θ_t dW_t

This is defined as the limit of Riemann sums:

∫0^T θ_t dW_t = lim{n→∞} Σ_{i=1}^n θ_{t_{i-1}} (W_{t_i} – W_{t_{i-1}})

Key Properties of the Ito Integral:

  1. Martingale: ∫_0^T θ_t dW_t is a martingale.

  2. Ito Isometry: E[(∫_0^T θ_t dW_t)²] = E[∫_0^T θ_t² dt].

  3. Zero Mean: E[∫_0^T θ_t dW_t] = 0.

4.2 The Ito Lemma (Univariate Version)

Let X_t be an Ito process:

dX_t = μ_t dt + σ_t dW_t

Let f(t, x) be a twice continuously differentiable function. Then:

df(t, X_t) = (∂f/∂t + μ_t ∂f/∂x + (1/2) σ_t² ∂²f/∂x²) dt + σ_t ∂f/∂x dW_t

Derivation (Taylor Expansion):

Expand f(t + dt, X_t + dX_t) using a Taylor series up to second order:

df = (∂f/∂t) dt + (∂f/∂x) dX_t + (1/2) (∂²f/∂x²) (dX_t)² + (∂²f/∂x∂t) dX_t dt + …

Substitute dX_t = μ_t dt + σ_t dW_t:

(dX_t)² = μ_t² (dt)² + 2 μ_t σ_t dt dW_t + σ_t² (dW_t)²

Using the rules:

  • (dt)² = 0

  • dt dW_t = 0

  • (dW_t)² = dt

We get:

(dX_t)² = σ_t² dt

Substituting back into the Taylor expansion:

df = (∂f/∂t) dt + (∂f/∂x)(μ_t dt + σ_t dW_t) + (1/2)(∂²f/∂x²) σ_t² dt

Collecting terms:

df = (∂f/∂t + μ_t ∂f/∂x + (1/2) σ_t² ∂²f/∂x²) dt + σ_t ∂f/∂x dW_t

4.3 The Ito Lemma (Multivariate Version)

Let X_t = (X_t^{(1)}, X_t^{(2)}, …, X_t^{(n)}) be an n-dimensional Ito process:

dX_t^{(i)} = μ_i(t, X_t) dt + Σ_{j=1}^m σ_{ij}(t, X_t) dW_t^{(j)}

For a function f(t, X_t):

df = (∂f/∂t + Σ_i μ_i ∂f/∂x_i + (1/2) Σ_{i,j} (σ σ^T){ij} ∂²f/∂x_i∂x_j) dt + Σ{i,j} σ_{ij} ∂f/∂x_i dW_t^{(j)}

5. GEOMETRIC BROWNIAN MOTION (GBM)

5.1 Definition

Geometric Brownian Motion is defined by the SDE:

dS_t = μ S_t dt + σ S_t dW_t

Where:

  • μ is the drift (expected return).

  • σ is the volatility (standard deviation of returns).

5.2 Solution Using Ito’s Lemma

We want to solve for S_t. Take f(S) = ln(S):

∂f/∂t = 0
∂f/∂S = 1/S
∂²f/∂S² = -1/S²

Applying Ito’s Lemma:

d(ln S_t) = (0 + μ S_t * (1/S_t) + (1/2) σ² S_t² * (-1/S_t²)) dt + σ S_t * (1/S_t) dW_t
d(ln S_t) = (μ – (1/2) σ²) dt + σ dW_t

Integrating from 0 to t:

ln(S_t) – ln(S_0) = (μ – (1/2) σ²) t + σ W_t

Exponentiating:

S_t = S_0 * exp( (μ – (1/2) σ²) t + σ W_t )

5.3 Properties of GBM

  1. Log-Normal Distribution: S_t is log-normally distributed.

  2. Mean: E[S_t] = S_0 e^{μt}.

  3. Variance: Var(S_t) = S_0² e^{2μt} (e^{σ²t} – 1).

  4. No Negative Prices: S_t > 0 for all t.

5.4 Financial Application – Stock Price Modelling

GBM is the standard model for stock prices in the Black-Scholes framework. The drift under the risk-neutral measure Q becomes r (the risk-free rate):

dS_t = r S_t dt + σ S_t dW_t^Q

The solution under Q is:

S_t = S_0 * exp( (r – (1/2) σ²) t + σ W_t^Q )

6. ITO’S LEMMA AND THE BLACK-SCHOLES PDE

6.1 Derivation of the PDE

Let V(t, S_t) be the price of a European option. Applying Ito’s Lemma:

dV = (∂V/∂t + μ S ∂V/∂S + (1/2) σ² S² ∂²V/∂S²) dt + σ S ∂V/∂S dW_t

6.2 Constructing a Risk-Free Portfolio

We construct a portfolio Π that contains one option and a short position of Δ shares:

Π = V – Δ S

We choose Δ = ∂V/∂S (delta hedging) to eliminate the dW_t term.

The portfolio differential is:

dΠ = dV – Δ dS

Substitute dV and dS:

dΠ = (∂V/∂t + μ S ∂V/∂S + (1/2) σ² S² ∂²V/∂S² – Δ μ S) dt + (σ S ∂V/∂S – Δ σ S) dW_t

With Δ = ∂V/∂S, the dW_t term cancels:

dΠ = (∂V/∂t + (1/2) σ² S² ∂²V/∂S²) dt

6.3 The Risk-Free Rate

Since Π is risk-free, it must earn the risk-free rate r:

dΠ = r Π dt = r (V – Δ S) dt

Equating the two expressions:

∂V/∂t + (1/2) σ² S² ∂²V/∂S² = r (V – S ∂V/∂S)

Rearranging gives the Black-Scholes PDE:

∂V/∂t + (1/2) σ² S² ∂²V/∂S² + r S ∂V/∂S – r V = 0

7. BROWNIAN MOTION AND THE HEAT EQUATION

7.1 The Heat Equation

The heat equation (diffusion equation) is:

∂u/∂t = (1/2) ∂²u/∂x²

7.2 Fundamental Solution

The fundamental solution of the heat equation is the Gaussian kernel:

u(t, x) = [1 / √(2πt)] * exp( -x² / (2t) )

7.3 Relationship with Brownian Motion

The transition density of Brownian Motion is the fundamental solution of the heat equation:

P(W_t ∈ dx) = [1 / √(2πt)] * exp( -x² / (2t) ) dx

This is why Brownian Motion is also called the “heat equation process.”

7.4 The Black-Scholes PDE as a Heat Equation

By a change of variables, the Black-Scholes PDE can be transformed into the heat equation. This is how the Black-Scholes formula is derived.

8. PRACTICAL IMPLEMENTATION

A. Simulating Brownian Motion Paths:

python
import numpy as np
import matplotlib.pyplot as plt

def simulate_brownian(T, n_steps, n_paths):
    dt = T / n_steps
    dW = np.random.normal(0, np.sqrt(dt), (n_paths, n_steps))
    W = np.cumsum(dW, axis=1)
    return W

# Parameters
T = 1.0
n_steps = 252
n_paths = 100

# Simulate
W = simulate_brownian(T, n_steps, n_paths)

# Plot
t = np.linspace(0, T, n_steps)
plt.figure(figsize=(10, 6))
for i in range(min(10, n_paths)):
    plt.plot(t, W[i, :], alpha=0.6)
plt.xlabel('Time')
plt.ylabel('W_t')
plt.title('Simulated Brownian Motion Paths')
plt.show()

# Verify properties
print(f"E[W_T] = {np.mean(W[:, -1]):.4f} (expected 0)")
print(f"Var[W_T] = {np.var(W[:, -1]):.4f} (expected {T:.4f})")

B. Simulating Geometric Brownian Motion:

python
def simulate_gbm(S0, mu, sigma, T, n_steps, n_paths):
    dt = T / n_steps
    dW = np.random.normal(0, np.sqrt(dt), (n_paths, n_steps))
    W = np.cumsum(dW, axis=1)
    t = np.linspace(0, T, n_steps)
    S = S0 * np.exp((mu - 0.5 * sigma**2) * t + sigma * W)
    return S, t

# Parameters
S0 = 100
mu = 0.08
sigma = 0.20
T = 1.0
n_steps = 252
n_paths = 100

# Simulate
S, t = simulate_gbm(S0, mu, sigma, T, n_steps, n_paths)

# Plot
plt.figure(figsize=(10, 6))
for i in range(min(10, n_paths)):
    plt.plot(t, S[i, :], alpha=0.6)
plt.xlabel('Time')
plt.ylabel('S_t')
plt.title('Simulated GBM Paths')
plt.show()

# Verify properties
print(f"E[S_T] = {np.mean(S[:, -1]):.4f} (expected {S0 * np.exp(mu * T):.4f})")
print(f"Var[S_T] = {np.var(S[:, -1]):.4f}")

C. Verifying Ito’s Lemma:

python
def black_scholes_call(S, K, T, r, sigma):
    d1 = (np.log(S / K) + (r + 0.5 * sigma**2) * T) / (sigma * np.sqrt(T))
    d2 = d1 - sigma * np.sqrt(T)
    from scipy.stats import norm
    return S * norm.cdf(d1) - K * np.exp(-r * T) * norm.cdf(d2)

# Compute Delta and Gamma (derivatives of the option price)
S0 = 100
K = 100
T = 1.0
r = 0.03
sigma = 0.20
eps = 0.01

V = black_scholes_call(S0, K, T, r, sigma)
V_up = black_scholes_call(S0 + eps, K, T, r, sigma)
V_down = black_scholes_call(S0 - eps, K, T, r, sigma)

# Delta = dV/dS
delta = (V_up - V_down) / (2 * eps)

# Gamma = d²V/dS²
gamma = (V_up - 2 * V + V_down) / (eps**2)

print(f"Option Price: {V:.4f}")
print(f"Delta: {delta:.4f}")
print(f"Gamma: {gamma:.4f}")

# Verify Ito's Lemma: dV = Delta * dS + 0.5 * Gamma * (dS)^2
dS = 0.10
dV_approx = delta * dS + 0.5 * gamma * dS**2
dV_actual = black_scholes_call(S0 + dS, K, T, r, sigma) - V
print(f"dS = {dS:.2f}")
print(f"Actual dV: {dV_actual:.4f}")
print(f"Approx dV: {dV_approx:.4f}")
print(f"Error: {dV_actual - dV_approx:.6f}")

D. The Quadratic Variation of Brownian Motion:

python
# Compute quadratic variation empirically
W = simulate_brownian(T=1.0, n_steps=252, n_paths=1)[0]
quadratic_var = np.sum(np.diff(W)**2)
print(f"Quadratic Variation: {quadratic_var:.4f} (expected {T:.4f})")

9. SUMMARY FOR THE FINANCE PRACTITIONER

  • Stochastic Processes model the evolution of random variables over time. Asset prices are stochastic processes.

  • Brownian Motion is the building block of continuous-time finance. It has independent, normally distributed increments and continuous paths.

  • Quadratic Variation of Brownian Motion is deterministic (equal to t). This is captured by the rule (dW_t)² = dt.

  • Ito’s Lemma is the chain rule for stochastic calculus. It is used to derive the dynamics of functions of Ito processes.

  • Geometric Brownian Motion is the standard model for stock prices. The solution is log-normal: S_t = S_0 * exp((μ – σ²/2)t + σW_t).

  • The Black-Scholes PDE is derived using Ito’s Lemma and delta hedging. It is the foundation of option pricing.

 
 
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