Stochastic processes

Random Walk, Three Ways

One stream of noise feeds three models. GBM drifts forever, Ornstein-Uhlenbeck mean-reverts, Merton jump diffusion adds the fat tails the others miss. Toggle which models you watch; the paths draw live.

Geometric Brownian Motion end —

Ornstein-Uhlenbeck end —

Merton Jump Diffusion end —

Shared parameters

μ — drift0.08

σ — volatility0.20

T — horizon (years)1.00

N — paths60

Ornstein-Uhlenbeck

θ — reversion speed2.0

μ_OU — long-run mean100

Merton Jump Diffusion

λ — jumps / year2.0

m — mean log-jump-0.05

s — jump sd0.10

How to use

The panels share the same random noise — only the SDE structure differs. Toggle models with the tabs up top; the grid reflows to show just what you select. Drag the shared μ, σ, T, then tune θ, μ_OU for OU and λ, m, s for jumps. Each change replays the animated paths.

The three SDEs

Geometric Brownian Motion (drift + diffusion, no memory):

dS = μ·S·dt + σ·S·dW

Ornstein-Uhlenbeck (mean-reverting to μ_OU at rate θ):

dX = θ·(μ_OU − X)·dt + σ·dW

Merton Jump Diffusion (GBM plus Poisson jumps with intensity λ and lognormal size):

dS/S = (μ − λκ)·dt + σ·dW + (J − 1)·dN

where N is a Poisson process with rate λ, J has ln(J) ~ N(m, s²), and κ = exp(m + s²/2) − 1 is the mean jump size. The −λκ term is the compensator: it keeps the expected return equal to μ after adding the jumps.

What the same seed reveals

The three panels share the same Brownian increments and the same Poisson draws. Differences are purely structural:

GBM drifts and never returns. Long-run mean is undefined — the process wanders.
OU is the value-investor's universe. Prices wander but a restoring force pulls them back to μ_OU. Used for interest rates (Vasicek), commodity spreads, statistical arbitrage.
Jump diffusion matches GBM most of the time but adds occasional discontinuities. Captures the fat tails GBM misses — at the cost of two more parameters that are notoriously hard to estimate from data.

When each is the right model

GBM: equity index over long horizons, foreign exchange in calm regimes, the no-arbitrage default when nothing better is known.

OU: short-term interest rates (Vasicek 1977), pair-trading spreads, volatility itself, commodity prices around marginal-cost equilibrium.

Jump diffusion: single-name equities (earnings, M&A), credit spreads, anything where information arrives in discrete packets.

None of the three captures volatility clustering — that needs stochastic volatility (Heston) or GARCH.

Citations

Merton, R.C. (1976). "Option pricing when underlying stock returns are discontinuous." Journal of Financial Economics, 3, 125–144.
Uhlenbeck, G.E. & Ornstein, L.S. (1930). "On the theory of the Brownian motion." Physical Review, 36, 823.
Vasicek, O. (1977). "An equilibrium characterization of the term structure." Journal of Financial Economics, 5, 177–188.
Hull, J.C. Options, Futures, and Other Derivatives, 11th ed., Ch. 14 (Wiener processes), Ch. 27 (Jumps in asset prices).

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