Module 68 — Stochastic Calculus & Financial Mathematics

Brownian motion and its properties, geometric Brownian motion as the stock price model, Itô's lemma (the chain rule of stochastic calculus), Black-Scholes PDE derivation, and risk-neutral pricing — the mathematical backbone of derivatives pricing.

Learning Objectives

Understand Brownian motion and why it models randomness in financial markets
Apply geometric Brownian motion to model stock price dynamics
Use Itô's lemma to find the dynamics of functions of stochastic processes
Follow the derivation of the Black-Scholes PDE
Understand risk-neutral pricing and the change of measure

1. Brownian Motion (Wiener Process)

What Brownian Motion Is

Brownian motion (W_t) is a continuous-time stochastic process named after botanist Robert Brown, who observed pollen grains moving randomly in water (1827). Mathematically formalized by Norbert Wiener.

Properties:

W₀ = 0 (starts at zero)
W_t has independent increments: (W_t − W_s) is independent of W_s for t > s
W_t − W_s ~ N(0, t − s): increments are normally distributed with variance proportional to time
Paths are continuous but nowhere differentiable (can't take the ordinary derivative)

The key intuition: Over a small time interval dt, the change in W is:

dW = ε × √dt     where ε ~ N(0,1)

Volatility scales with the square root of time — this is why a 1-day VaR multiplied by √252 gives an annual volatility estimate.

Quadratic Variation

Unlike ordinary functions where (dt)² is negligible, for Brownian motion:

(dW)² = dt   (exactly, not approximately)

This is the crucial difference from ordinary calculus and is the source of Itô's correction term.

2. Geometric Brownian Motion (GBM)

The Stock Price Model

Black and Scholes (1973) and Merton (1973) assumed stock prices follow Geometric Brownian Motion:

dS = μ × S × dt + σ × S × dW

Where:
S = stock price
μ = drift (expected return)
σ = volatility
dW = Brownian motion increment

Why "geometric"? The drift and diffusion terms are proportional to S (not absolute). This ensures:

S can never become negative (prices can't go below zero)
Percentage returns are normally distributed (log-returns are normal)

From GBM to the Log-Normal Price Distribution

Applying Itô's lemma (next section) to f(S) = ln(S):

d(ln S) = (μ − σ²/2) dt + σ dW

Integrating from 0 to T:
ln(S_T) − ln(S₀) = (μ − σ²/2)T + σ W_T

S_T = S₀ × exp[(μ − σ²/2)T + σ√T × ε]    where ε ~ N(0,1)

This shows: S_T is log-normally distributed — the log of the terminal price is normally distributed.

3. Itô's Lemma

The Problem: Chain Rule Doesn't Work

In ordinary calculus, if f(x) is a function of x and x changes by dx, then:

df = f'(x) × dx

For stochastic processes, this fails because (dW)² = dt is non-negligible. The correction is Itô's lemma.

Itô's Lemma (Single Variable)

For a smooth function f(S, t) where S follows a general Itô process:

df = (∂f/∂t + μS × ∂f/∂S + ½σ²S² × ∂²f/∂S²) dt + σS × ∂f/∂S × dW

The key extra term: ½σ²S² × ∂²f/∂S² This is the "Itô correction" — it arises from the (dW)² = dt property and has no analogue in ordinary calculus.

Applying Itô's Lemma: Deriving Log Returns

Let f(S) = ln(S):

∂f/∂t = 0
∂f/∂S = 1/S
∂²f/∂S² = −1/S²

Applying Itô's lemma:
d(ln S) = [0 + μS × (1/S) + ½σ²S² × (−1/S²)] dt + σS × (1/S) dW
         = [μ − σ²/2] dt + σ dW

The −σ²/2 term (the Itô correction) is why the expected value of ln(S_T) is (μ − σ²/2)T, not μT.

4. The Black-Scholes PDE

The Replication Argument

Black, Scholes, and Merton's key insight: an option can be replicated by a continuously rebalanced portfolio of stock and cash (the "delta-hedged portfolio"). If replication is perfect, the replicating portfolio and the option must have the same value to avoid arbitrage.

The hedged portfolio:

Π = V(S, t) − Δ × S

Where:
V = option value (unknown)
Δ = hedge ratio (units of stock sold short)

Change in portfolio:

dΠ = dV − Δ × dS

Apply Itô's lemma to V(S, t), substitute GBM for dS, and choose Δ = ∂V/∂S (the "delta") to eliminate the random dW term.

The remaining terms must equal the risk-free rate (to avoid arbitrage):

dΠ = r × Π × dt

This gives the Black-Scholes PDE:

∂V/∂t + ½σ²S² × ∂²V/∂S² + rS × ∂V/∂S − rV = 0

Solving the Black-Scholes PDE

With boundary condition: V(S, T) = max(S − K, 0) for a call option.

The solution is the famous Black-Scholes formula:

C = S₀ × N(d₁) − K × e^(-rT) × N(d₂)

Where:
d₁ = [ln(S₀/K) + (r + σ²/2)T] / (σ√T)
d₂ = d₁ − σ√T

N(·) = cumulative standard normal distribution function

Interpreting the Formula

N(d₂): The risk-neutral probability that the option expires in the money (S_T > K). N(d₁): The delta of the option — the hedge ratio. K × e^(-rT) × N(d₂): Present value of the expected exercise price payment. S₀ × N(d₁): Present value of expected stock receipt on exercise.

5. Risk-Neutral Pricing

The Change of Measure

The most powerful idea in mathematical finance is the change of measure (Girsanov's theorem). It says: we can change the probability measure so that all assets grow at the risk-free rate. Under this "risk-neutral measure" (Q), all assets earn the risk-free rate r, and option pricing simplifies to:

V₀ = e^(-rT) × E^Q[payoff at T]

Why this works: If we can perfectly hedge an option, the expected return on the hedge portfolio equals r under no arbitrage. The "real world" drift μ disappears — it doesn't matter for pricing, only for hedging.

Risk-Neutral Pricing for the European Call

Under Q measure, the stock price process becomes:

dS = r × S × dt + σ × S × dW^Q

S_T = S₀ × exp[(r − σ²/2)T + σ√T × ε^Q]    where ε^Q ~ N(0,1) under Q

Call price = e^(-rT) × E^Q[max(S_T − K, 0)]
           = e^(-rT) × [S₀ × e^(rT) × N(d₁) − K × N(d₂)]
           = S₀ × N(d₁) − K × e^(-rT) × N(d₂)

This reproduces the Black-Scholes formula — confirming that risk-neutral pricing and the PDE approach are equivalent.

6. Extensions and Limitations

Beyond Black-Scholes

The Black-Scholes model assumes:

Constant volatility (σ is fixed)
Log-normal returns (no fat tails)
Continuous trading (no jumps)
No transaction costs
Constant interest rates

Real markets violate all these assumptions. Extensions:

Stochastic volatility models (Heston model): σ itself follows a mean-reverting stochastic process. Produces the "volatility smile" (implied volatility varying with strike) observed in real options markets.

Jump-diffusion models (Merton, Kou): Add a Poisson jump process to GBM to capture sudden large moves. Better for equity indices and currencies.

Local volatility models (Dupire): Volatility is a deterministic function of (S, t): σ(S, t). Exactly calibrates to all observed option prices.

Rough volatility models (Bergomi, Gatheral): Recent research shows volatility is rougher (more fractal) than standard stochastic volatility models capture. Active research area.

Self-Assessment

FERROQUANT Capital's options desk is pricing a 3-month European call on BIQAI Group equity. Current price: PKR 150, strike: PKR 160, 3-month risk-free rate: 12% annualized, implied volatility: 30% annualized.

(a) Calculate d₁ and d₂ using the Black-Scholes formula. (b) Find N(d₁) and N(d₂) from a standard normal table (approximate to 4 decimal places). (c) Calculate the call price. (d) What is the delta of this call? What does it mean for hedging? (e) If BIQAI Group announces a profit warning and implied volatility rises to 45%, qualitatively describe what happens to the call price and why.
Explain in plain language why the Itô correction term (−σ²/2) appears in the log-return process. What would go wrong if you tried to use ordinary calculus instead? Specifically: what would the expected log-return be under ordinary calculus vs Itô's lemma, and why does the difference matter for option pricing?
A junior quant at FERROQUANT argues: "We should use the real-world drift μ = 20% (our equity return expectation) instead of the risk-free rate r = 12% when pricing options — we're giving away money by using the lower discount rate." Explain clearly why this argument is wrong, using the concept of risk-neutral pricing and the delta-hedging argument.