Quantitative model · Stochastic processes
Stochastic-process simulation across diffusion, jumps, volatility, and regime switching
The model runs five process classes under a common one-year setup. The comparison shows what each specification can and cannot represent.
Decision summary
The process determines which risks a scenario set can represent
Interpretation
A simulation is a conditional experiment, not a forecast. GBM, jumps, stochastic volatility, mean reversion, and regimes encode different hypotheses. They therefore produce different tails and dependencies.
Decision context
Select the process according to the risk being evaluated. Budget sampling and discretization error before using simulated outcomes for pricing, stress, or allocation.
Intended analytical use
Quantitative developers, risk and derivatives analysts, model validators, and researchers use it to study stochastic dynamics.
Principal limitation
The browser explorer uses 50 one-year paths and Euler-type discretization with illustrative parameters. A fixed seed makes a run repeatable. It does not make the scenario representative.
Process
Simulation
50 paths · 252 steps · T = 1y
Euler-Maruyama discretization
Box-Muller normals, mulberry32 seed
Simulated process paths
View representative mean-path data
Terminal-value distribution
View terminal distribution summary
| Minimum | 25th percentile | Median | Mean | 75th percentile | Maximum |
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Process notes
Geometric Brownian motion
Applications. The default model for equity prices. It is the substrate of Black-Scholes and of most textbook portfolio mathematics.
Captured dynamics. Proportional randomness (returns, not price changes, are the natural unit), positivity of prices, and compounding. That includes the subtle \(-\tfrac{1}{2}\sigma^2\) drag separating median from mean growth.
Omitted dynamics. 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
Applications. Short-rate models (Vasicek), commodity convenience yields, volatility itself, and the spread dynamics behind statistical arbitrage.
Captured dynamics. 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.
Omitted dynamics. Gaussian increments allow negative values, fatal for nominal rates near zero and tolerable for spreads. The reversion speed is assumed constant. 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
Applications. Pricing short-dated options where crash risk dominates; credit and event-risk modeling; any setting where the smile is steep at short maturities.
Captured dynamics. Discontinuity. A Poisson process \(N_t\) with intensity \(\lambda\) fires jumps of lognormal size \(J\). That produces 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.
Omitted dynamics. 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 and leaves clustering unexplained.
Process notes
Heston stochastic volatility
Applications. The workhorse stochastic-volatility model for equity index options. Its characteristic function is known in closed form, so whole smiles calibrate quickly.
Captured dynamics. Variance as a mean-reverting random process (CIR dynamics), volatility clustering, and the leverage effect through \(\rho < 0\). Prices fall as volatility rises, which generates the implied-vol skew. Simulation uses a full-truncation Euler scheme. That scheme floors \(v_t\) at zero inside the square roots, so the discretized variance cannot go negative.
Omitted dynamics. No jumps, so very short-dated smiles are too flat. One volatility factor, so the term structure of skew is too rigid. Calibrated parameters also drift over time, which indicates residual misspecification. The case study treats this in depth.
Process notes
Regime-switching GBM
Applications. Tactical allocation, risk-regime dashboards, and econometric business-cycle work in the Hamilton (1989) tradition.
Captured dynamics. Long low-volatility intervals with positive drift, interrupted by high-volatility intervals with weaker returns. The two-state model reproduces fat tails and clustering as an emergent mixture, with no exotic distributions required. Each path carries its own hidden regime sequence.
Omitted dynamics. 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 represents regime-dependent return behavior without identifying a causal mechanism.