Stochastic Calculus of Ito and the Black-Scholes Formula

Financial mathematics in continuous time is based on the stochastic calculus of Ito. The Black-Scholes-Merton formula (1973)—the first analytical result for derivative pricing—earned the 1997 Nobel Prize and transformed the financial industry.

Brownian Motion and Ito Processes

Standard Brownian motion (Wiener process) W_t: Four properties: W₀ = 0, trajectories are continuous, increments are independent (W_t − W_s ⊥ W_s − W₀ for 0 ≤ s < t), increments are normally distributed: W_t − W_s ~ N(0, t−s).

Key relation: (dW_t)² = dt (in the sense of convergence in probability). This distinguishes stochastic calculus from deterministic: the "square of an infinitesimal" does not vanish.

Ito process: dX_t = μ(X_t, t)dt + σ(X_t, t)dW_t. μ — drift (deterministic component), σ — diffusion (random component). Examples: Geometric Brownian Motion (GBM): dS = μS dt + σS dW — stock price model. CIR process: dr = κ(θ−r)dt + σ√r dW — short rate model (always positive).

Ito's lemma: For a twice differentiable function f(X_t, t):

df = (∂f/∂t + μ·∂f/∂X + σ²/2·∂²f/∂X²)dt + σ·∂f/∂X dW_t

Key point: the second order term σ²/2·∂²f/∂X² enters the drift via (dW)² = dt. Deterministic calculus: df = (∂f/∂t + μ·∂f/∂X)dt — no second order. Stochastic: Ito's correction compensates for nonlinearity.

Example: f(S) = ln S. df = (−σ²/2)dt + σ dW → ln S_T = ln S₀ + (μ−σ²/2)T + σW_T → S_T = S₀ exp((μ−σ²/2)T + σW_T) — formula for GBM.

Derivation of the Black-Scholes Equation

Model: dS = μS dt + σS dW. V(S, t) — price of a European option.

Delta hedging strategy: Portfolio Π = V − Δ·S (long option, short Δ shares). By Ito's lemma: dΠ = dV − Δ dS = (∂V/∂t + μS·∂V/∂S + σ²S²/2·∂²V/∂S²)dt + σS·∂V/∂S dW − Δ(μS dt + σS dW). For Δ = ∂V/∂S (delta): random terms (dW) vanish → dΠ is deterministic. No-arbitrage: dΠ = r·Π·dt.

Black-Scholes equation:

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

Analytical solution (European call):

C₀ = S₀N(d₁) − Ke^{−rT}N(d₂) d₁ = [ln(S₀/K) + (r + σ²/2)T]/(σ√T), d₂ = d₁ − σ√T

Clarification: N(d₂) — (risk-neutral) probability of option exercise. N(d₁) — delta hedge. Ke^{−rT}N(d₂) — present value of the strike to be paid. S₀N(d₁) — present value of the share received (upon exercise).

Option Greeks

Δ (delta) = ∂C/∂S = N(d₁): sensitivity to the underlying asset price. When S → K (at-the-money): Δ ≈ 0.5. Interpretation: for every +1 ruble increase in S, the option rises by Δ rubles. Delta-neutral portfolio: removes dependence on market direction.

Γ (gamma) = ∂²C/∂S² = N'(d₁)/(Sσ√T): “acceleration”—how quickly Δ changes as S changes. High Γ → unstable delta → frequent rebalancing. Maximum at-the-money.

Θ (theta) = ∂C/∂t < 0: “time decay”—the option loses value over time, all else equal. Always negative for a long position (time works against the option buyer).

Vega = ∂C/∂σ = S₀N'(d₁)√T: sensitivity to volatility. Increase in σ → increase in option price (greater chance of being in-the-money). “Volatility trading” — a separate strategy.

Numerical Example

S₀ = 100, K = 100, T = 0.5, r = 5%, σ = 20%.

d₁ = [ln(1) + (0.05 + 0.02)·0.5]/(0.2·√0.5) = [0 + 0.035]/0.1414 = 0.2475. d₂ = 0.2475 − 0.1414 = 0.1061.

N(0.2475) ≈ 0.5977, N(0.1061) ≈ 0.5423. C₀ = 100·0.5977 − 100·e^{−0.025}·0.5423 = 59.77 − 52.99 = 6.78.

Δ = 0.5977. Γ = N'(0.2475)/(100·0.2·0.7071) = 0.388/14.14 ≈ 0.0274. Vega = 100·0.388·0.7071 ≈ 27.4 (increase in σ by 1% → price increase by 0.274).

Assignment: S₀=100, K=100, T=0.5, r=5%, σ=20%. (1) Calculate C₀ via BS. (2) Find Δ, Γ, Vega. (3) If σ_impl=25% — how does the price change? (4) Construct a delta hedging strategy: buy 1 option, sell Δ shares. After 1 day S=102: compute new Δ, rebalance. What is the P&L of the position? (5) Implement BS calculation in Python for a grid S ∈ [80,120], plot the price profile and all Greeks.