Ito's Lemma and Stochastic Differential Equations

Ito's lemma is the stochastic analogue of the chain rule for differentiation. Due to the nonzero quadratic variation of Brownian motion, there appears an additional "correction" second-order term.

Ito Integral

The integral ∫₀^T f_t dW_t for adapted processes fₜ. It cannot be defined pointwise (W_t is nowhere differentiable). It is defined as the L²-limit of step processes.

Properties: Martingale: E[∫₀^T f dW] = 0. Ito isometry: E[(∫₀^T f dW)²] = E[∫₀^T f² dt].

Ito's Lemma

Let dX = b dt + σ dW, f ∈ C²(ℝ). Then: df(X_t) = f'(X_t) dX_t + (1/2)f''(X_t) d[X,X]_t = [f'(X_t)b + (1/2)f''(X_t)σ²] dt + f'(X_t)σ dW_t.

Key point: The additional term (1/2)f''σ² dt is the "quadratic correction." This distinguishes stochastic calculus from deterministic calculus.

Multidimensional Ito's lemma: For dXᵢ = bᵢ dt + Σⱼ σᵢⱼ dWⱼ: df = Σᵢ ∂f/∂xᵢ dXᵢ + (1/2)Σᵢ,ⱼ ∂²f/∂xᵢ∂xⱼ dXᵢ dXⱼ. Rule: dt·dt = 0, dW·dt = 0, dWᵢ dWⱼ = ρᵢⱼ dt.

SDEs and Geometric Brownian Motion

SDE: dX_t = b(X_t, t)dt + σ(X_t, t)dW_t. The solution is an adapted continuous process.

Geometric Brownian Motion (GBM): dS = μS dt + σS dW. Solution: S_t = S₀ exp((μ-σ²/2)t + σW_t). Black-Scholes model for stock prices. ln(S_t/S₀) ~ N((μ-σ²/2)t, σ²t).

Task: (a) Apply Ito's lemma to f(W_t) = W_t². Obtain: d(W_t²) = 2W_t dW_t + dt. Hence E[W_t²] = t. (b) For GBM: find E[S_t] and Var[S_t]. Why is ln S normal, but S is lognormal? (c) Ornstein-Uhlenbeck process: dX = -θX dt + σ dW. Find the solution via Ito's lemma for f(X_t, t) = e^{θt}X_t.

Stochastic Differential Equations

SDE: dX_t = b(X_t, t)dt + σ(X_t, t)dW_t. Conditions for existence of a unique solution (Lipschitz in X, linear growth): |b(x,t)| + |σ(x,t)| ≤ C(1+|x|). Weak vs. strong solution: strong exists under the Lipschitz condition; weak is more general.

Examples of SDEs in finance: CIR model for interest rates: dr = κ(θ−r)dt + σ√r dW. Heston: dV = κ(θ−V)dt + σ√V dW_V. Complexity: nonlinearity, requirement for positivity of the process. Euler-Maruyama discretization: X_{n+1} = X_n + b(X_n)Δt + σ(X_n)ΔW_n, strong convergence order 1/2.

Girsanov's Theorem and the Risk-Neutral Measure

Novikov's Lemma: If E[exp(1/2 ∫₀ᵀ θ_s²ds)] < ∞, then the process dQ/dP = exp(-∫θdW − 1/2∫θ²dt) is a martingale and defines a new measure Q. Under Q: Ŵ_t = W_t + ∫₀ᵗ θ_s ds is a Brownian motion.

Option pricing: Change the measure so that e^{-rt}S_t is a Q-martingale (risk-neutral measure). Then the price of the option = e^{-rT}·E^Q[payoff]. The Black-Scholes formula follows from Girsanov's theorem for GBM.

Black-Scholes Partial Differential Equation

From the replicating portfolio: ∂V/∂t + 1/2σ²S²∂²V/∂S² + rS∂V/∂S − rV = 0. Boundary conditions: V(S,T) = (S−K)⁺ for a call. Connection to the heat equation: variable change S = Ke^x, t = T−τ/r, v = Ve^{αx+βτ} → standard heat equation.

Greeks: Sensitivity of Options

Delta (Δ): ∂V/∂S = N(d₁) for a call. The share of the asset in the replicating portfolio. Gamma (Γ): ∂²V/∂S² = N'(d₁)/(Sσ√T). Convexity of price—profit from volatility. Theta (Θ): ∂V/∂t—time decay. Vega (ν): ∂V/∂σ = S·N'(d₁)·√T. Sensitivity to volatility. For a delta-neutral position: Γ·σ²S²/2 + Θ = 0 (P&L neutrality).

Implied Volatility and the Volatility Smile

Implied volatility (IV): σ_impl = BS⁻¹(C_market)—the volatility at which the BS price equals the market price. Volatility smile: IV depends on K and T—a violation of the constant σ assumption. Reasons: heavy tails in actual returns, investor risk aversion. Local volatility models (Dupire): σ = σ(S,t) replicates the smile. Stochastic volatility (Heston): dV = κ(θ−V)dt + ξ√V dW_V, corr(W_S, W_V) = ρ. Semi-analytical formula via characteristic functions.

Interest Rates and Yield Curve Models

Vasicek model: dr = κ(θ−r)dt + σdW. Explicit solution: r_t = θ + (r₀−θ)e^{−κt} + σ∫e^{−κ(t−s)}dW_s. Normal distribution → allows for negative rates. CIR model: dr = κ(θ−r)dt + σ√r dW. Square root → rate ≥ 0 when 2κθ > σ² (Feller condition). Zero-coupon bonds: P(r,T) = A(T)e^{−B(T)r}—affine structure.

Risk-Neutral Measure and Fundamental Theorems of Asset Pricing

FTAP-1 (First Fundamental Theorem): No-arbitrage market ↔ equivalent martingale measure Q exists. FTAP-2: Complete market ↔ Q is unique. Incomplete market: range of admissible prices (superhedging). Karatzas-Shreve theorem: in a complete market any admissible contingent claim can be replicated. Practice: CDS (Credit Default Swap), structured products—are priced via the risk-neutral measure.

Numerical Example: Ito's Lemma for the Log Price

Problem: S_t satisfies dS=μS dt+σS dW. Apply Ito's lemma to f(S)=ln(S). Numbers: μ=0.12, σ=0.25, dt=1/252.

Step 1: Ito's lemma: df=(∂f/∂t+μS·∂f/∂S+σ²S²/2·∂²f/∂S²)dt+σS·∂f/∂S·dW. For f=ln(S): ∂f/∂S=1/S, ∂²f/∂S²=−1/S², ∂f/∂t=0.

Step 2: d(ln S)=(μS·(1/S)+σ²S²/2·(−1/S²))dt+σS·(1/S)dW=(μ−σ²/2)dt+σdW.

Step 3: Numbers: drift=(0.12−0.03125)/252=0.08875/252≈0.000352 per day. Volatility=0.25/√252≈0.01575 per day. Daily log-return: X~N(0.000352, 0.01575²).

Step 4: Without Ito correction (naively): expected ln-return=μ·dt=0.000476. Correct: (μ−σ²/2)·dt=0.000352. Difference σ²/2=0.03125 per year—critically important: without the correction, the model would overstate log-returns. This is the basis for deriving the Black-Scholes formula.