Stochastic differential equations and GBM

Stochastic differential equations describe how processes evolve under combined deterministic drift and Brownian noise. They are the continuous-time analogue of difference equations — and the standard modelling language for nearly every continuous-time finance object: stock prices, interest rates, volatilities, currencies.

General Itô SDE

dX(t) = μ(t, X) dt + σ(t, X) dW(t)

μ is the drift coefficient (may depend on time and state); σ is the diffusion coefficient (likewise). Standard solvability conditions (Lipschitz continuity, linear growth) guarantee existence and uniqueness of a strong solution given an initial condition.

Geometric Brownian motion

dS = μ S dt + σ S dW
S(t) = S(0) exp((μ - σ²/2) t + σ W(t))

The canonical model for stock prices. Constant percentage drift μ; constant percentage volatility σ; positivity preserved (S > 0 always). Black-Scholes' workhorse. Empirical caveats: real volatility is stochastic and clustered (motivating Heston and SABR); real drift is not constant (motivating regime models).

Ornstein-Uhlenbeck process

dX = θ(μ - X) dt + σ dW

Mean-reverting Gaussian process. X drifts toward μ at speed θ; the noise dW perturbs it. Used for interest rates (Vasicek), spreads, mean-reverting strategy P&L, FX deviations from PPP. Closed-form solution:

X(t) = μ + (X(0) - μ) e^{-θt} + σ ∫_0^t e^{-θ(t-s)} dW(s)

E[X(t)] = μ + (X(0) - μ) e^{-θt}
Var[X(t)] = (σ²/(2θ)) (1 - e^{-2θt})
Var[X(∞)] = σ²/(2θ)   (stationary)

CIR / Square-root process

dX = θ(μ - X) dt + σ √X dW

Cox-Ingersoll-Ross (1985). Mean-reverting but with state-dependent volatility — diffusion grows as √X, so X never goes negative (provided Feller condition 2θμ ≥ σ² holds). Used for short-rate models, variance processes (Heston), and any positive mean-reverting state variable.

Heston stochastic volatility

dS = μ S dt + √v · S dW_1
dv = κ(θ - v) dt + σ_v √v dW_2
dW_1 · dW_2 = ρ dt

Two-factor model: price and variance. Variance is CIR; price follows lognormal with stochastic vol. Semi-closed-form option pricing via the characteristic function (Heston 1993). The 'Heston smile' captures volatility skew empirically observed in equity markets.

Jump-diffusion (Merton 1976)

dS / S = μ dt + σ dW + (J - 1) dN
N: Poisson process with intensity λ
J: jump size, typically lognormal

Adds discrete jumps to GBM. Better captures heavy-tailed daily returns and the volatility smile at short maturities. Closed-form European option price (sum of Black-Scholes prices weighted by Poisson probabilities).

Numerical SDE solution

• Euler-Maruyama: X(t + Δt) ≈ X(t) + μ(t, X) Δt + σ(t, X) √Δt · Z. Order-1/2 strong convergence; easy to implement.
• Milstein: adds (1/2)σ ∂σ/∂x ((ΔW)² - Δt). Order-1 strong convergence; better for state-dependent vol.
• Predictor-corrector and Runge-Kutta variants: useful for stiff or near-singular SDEs.

Choosing a model

GBM is the baseline; everyone starts there. Heston / SABR for vol-surface fitting. CIR or Vasicek for short rates. Jump-diffusion for short-dated options on liquid stocks. Models with multiple regimes or rough paths for advanced research. The choice depends on what you're trying to price and the data you have.