Module 04 · Stochastic Processes
Stochastic Process Explorer
Five canonical models of randomness in markets, simulated live with a seeded generator. Each process is a hypothesis about how uncertainty moves — pick one, perturb its parameters, and watch the hypothesis draw itself.
Process
Simulation
50 paths · 252 steps · T = 1y
Euler-Maruyama discretization
Box-Muller normals, mulberry32 seed
Terminal-value distribution across paths, with the mean marked.
Process notes
Geometric Brownian motion
Where it is used. The default model for equity prices; the substrate of Black-Scholes and of most textbook portfolio mathematics.
What it captures. Proportional randomness (returns, not price changes, are the natural unit), positivity of prices, and compounding — including the subtle \(-\tfrac{1}{2}\sigma^2\) drag that separates median from mean growth.
What it misses. Everything the rest of this page exists for: volatility is constant, returns are Gaussian and independent, and nothing ever jumps. Real markets violate all three, persistently.
Process notes
Ornstein-Uhlenbeck
Where it is used. Short-rate models (Vasicek), commodity convenience yields, volatility itself, and the spread dynamics behind statistical arbitrage.
What it captures. Mean reversion — the pull toward \(\theta\) at speed \(\kappa\), with a stationary distribution \(\mathcal{N}(\theta, \sigma^2/2\kappa)\). The half-life of a shock is \(\ln 2 / \kappa\), a genuinely useful number in trading.
What it misses. Gaussian increments allow negative values (fatal for nominal rates near zero, tolerable for spreads), the reversion speed is assumed constant, and the level it reverts to must actually exist — many "mean-reverting" spreads are regimes in disguise, and OU is silent about the regime change that ends the trade.
Process notes
Merton jump-diffusion
Where it is used. Pricing short-dated options where crash risk dominates; credit and event-risk modeling; any setting where the smile is steep at short maturities.
What it captures. Discontinuity. A Poisson process \(N_t\) with intensity \(\lambda\) fires jumps of lognormal size \(J\), producing genuine fat tails and — with \(\mu_J < 0\) — the negative skew that equity markets exhibit. It explains why deep out-of-the-money puts cost "too much" under Black-Scholes.
What it misses. Jump risk here is diversifiable and independent of the diffusion; in reality crashes arrive precisely when volatility is already elevated, and jump intensity is anything but constant. Volatility between jumps is still flat — Merton fixes the tails while leaving clustering unexplained.
Process notes
Heston stochastic volatility
Where it is used. The workhorse stochastic-volatility model for equity index options, chosen largely because its characteristic function is known in closed form and whole smiles can be calibrated quickly.
What it captures. Variance as a mean-reverting random process (CIR dynamics), volatility clustering, and — through \(\rho < 0\) — the leverage effect: prices fall as volatility rises, generating the implied-vol skew. Simulated here with a full-truncation Euler scheme, which floors \(v_t\) at zero inside the square roots so the discretized variance cannot go negative.
What it misses. No jumps, so very short-dated smiles are too flat; a single volatility factor, so the term structure of skew is too rigid; and calibrated parameters drift over time, quietly confessing misspecification. The case study treats this in depth.
Process notes
Regime-switching GBM
Where it is used. Tactical allocation, risk-regime dashboards, and econometric business-cycle work in the Hamilton (1989) tradition.
What it captures. The blunt empirical fact that markets have moods: long quiet stretches with drift, punctuated by high-volatility episodes with poor returns. Even this two-state caricature reproduces fat tails and clustering as an emergent mixture — no exotic distributions required; each path carries its own hidden regime sequence.
What it misses. The chain's transition probabilities are fixed and the regimes are labeled in hindsight; real regime identification is only obvious after the fact. The model describes the moods without explaining the psychology — which is, candidly, most of financial modeling in one sentence.