Mean Reversion & Jumps

Geometric Brownian Motion is a beautiful wanderer — but it wanders away forever. Its log price has no anchor, no home, no reason to come back. That’s fine for a stock index that genuinely trends. But an enormous slice of finance does the opposite: interest rates, credit spreads, implied volatility, commodity prices, the spread in a pairs trade — these things get yanked back toward some level. Push them away and a restoring force reels them in. And there’s a second thing GBM can’t do: its paths are perfectly continuous, so they glide. Real prices don’t always glide — sometimes they gap, lurching 20% overnight on a crash, an earnings bomb, or a currency de-peg. This lesson installs the two missing pieces: mean reversion (a price with a home) and jumps (a price that can teleport).

Mean reversion: the Ornstein–Uhlenbeck process

Analogy. Picture a spring with a weight on the end, or a dog on a leash. Pull the weight far from rest and the spring yanks back hard; nudge it slightly and the pull is gentle. The further you stray, the stronger the tug home. That restoring force, plus a bit of random jostling, is mean reversion.

The mathematical version is the Ornstein–Uhlenbeck (OU) process, written as a stochastic differential equation:

$dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t$

Read it piece by piece — every symbol is intuitive:

• $\theta$ (theta) is the long-run mean: the level $X$ is pulled toward — the spring’s resting position.
• $X_t$ is where the process is right now.
• $(\theta - X_t)$ is the gap between home and now. If $X_t$ is above $\theta$, this is negative, so the drift pushes down; if below, it pushes up. The force always points home.
• $\kappa$ (kappa) is the reversion speed — the spring’s stiffness. Big $\kappa$ means a fierce, fast pull back; small $\kappa$ means a lazy, slow drift home.
• $\sigma\,dW_t$ is the same Brownian noise from GBM — the random jostle that keeps knocking $X$ off its perch.

The whole thing is a tug-of-war between a deterministic pull toward $\theta$ and random shocks that scatter $X$ away. When the noise wins for a moment, $X$ strays; the pull then reels it back. The result is a path that bounces around $\theta$ in a stable band instead of marching off to infinity.

Crank the κ slider and watch the path hug the mean line; drop it toward zero and the path loosens into the faint random walk that drifts away and never comes home.

Half-life: how fast it snaps back

Knowing that a process reverts isn’t enough — a trader needs to know how fast. The clean summary number is the half-life: the time it takes for a deviation from $\theta$ to decay halfway back, on average (ignoring new shocks). Because the pull decays the gap exponentially at rate $\kappa$, the half-life is:

$t_{1/2} = \frac{\ln 2}{\kappa}$

Worked example. Suppose a credit spread reverts with $\kappa = 0.5$ per year. Then

$t_{1/2} = \frac{\ln 2}{0.5} = \frac{0.693}{0.5} \approx 1.39 \text{ years.}$

A shock takes about 1.4 years to half-decay — sluggish. Now take a fast-reverting quantity with $\kappa = 5$ per year:

$t_{1/2} = \frac{0.693}{5} \approx 0.14 \text{ years} \approx 51 \text{ days.}$

Same formula, a much tighter leash. Bigger $\kappa$ ⇒ shorter half-life ⇒ faster snap-back.

The stationary band

Here is the structural difference from a random walk. A pure random walk’s variance grows without bound — wait long enough and it can be anywhere. The OU process instead settles into a finite long-run variance:

$\text{Var}_\infty = \frac{\sigma^2}{2\kappa}.$

The pull caps how far the noise can drag $X$. With $\sigma = 0.4$ and $\kappa = 0.5$, the stationary variance is $0.16 / 1.0 = 0.16$, a standard deviation of $0.4$ around $\theta$. Double $\kappa$ to $1.0$ and the variance halves to $0.08$ — a tighter band. Stronger reversion ⇒ smaller stationary spread. That bounded band, not the drift, is what makes mean-reversion tradeable: you know roughly how wide the rubber band stretches before it snaps back.

Where OU fits — and where it doesn’t

The OU process is the right tool wherever a quantity has a genuine anchor:

• Short-term interest rates. The classic Vasicek model is OU applied to the short rate: a policy-anchored level $\theta$, a reversion speed $\kappa$, Gaussian shocks. (Its cousin the CIR model adds a tweak so the rate can’t go negative.)
• Credit spreads. Risk premia widen in panics and compress in calm — they breathe around a level rather than trending forever.
• Implied volatility. Vol spikes in a crash, then decays back toward a long-run average. Classic reversion.
• Commodity prices. A marginal-cost-of-production floor and demand ceiling tether prices like oil or natural gas to an anchor.
• The spread in a pairs / stat-arb trade. This is the big one. Take two related stocks; their individual prices trend (GBM-like), but a well-chosen spread between them is stationary and mean-reverting. Trade the spread: short it when stretched high, long it when stretched low, and let the pull do the work.

The pitfall — and it’s a costly one. Do not slap OU onto a single stock’s raw price. An individual equity has no fixed price anchor: a company that compounds value will see its price trend upward indefinitely, and there’s no law dragging it back to last year’s level. Modeling a stock price as mean-reverting will have you “buying the dip” forever as a falling stock keeps falling — you mistook a trend for a deviation. Mean reversion lives in spreads, ratios, and rates — not in raw equity levels.

This is exactly why cointegration matters: two individually trending (non-stationary) series can have a linear combination — their spread — that is stationary and mean-reverting. Find that cointegrating spread and you’ve found something OU genuinely fits, even though neither leg on its own does.

Jumps: when prices gap

Mean reversion fixes the anchor problem. Jumps fix a different one. Pure diffusion — GBM and OU alike — has continuous paths: between any two instants the price passes through every value in between, never skipping. That smoothness is mathematically convenient and physically false. A pure-diffusion model cannot produce an overnight −20% gap, because to get from 100 to 80 it would have to trade through 99, 98, … and there simply is no trading overnight. Yet crashes, earnings surprises, takeover announcements, and currency de-pegs gap prices all the time.

The jump-diffusion (Merton) model patches this by adding a second engine on top of the diffusion:

$dS_t = \mu S_t\,dt + \sigma S_t\,dW_t + S_{t^-}\,(J - 1)\,dN_t.$

Don’t sweat the notation — the structure is simple. The first two terms are ordinary GBM: smooth drift and diffusion. The new third term is the jump engine:

• $N_t$ is a Poisson process with rate $\lambda$ (lambda): a clock that fires at random times, $\lambda$ times per year on average. Most of the time nothing happens and the price is just smooth GBM.
• When the clock fires, the price is instantly multiplied by a random jump factor $J$ (say 0.85 for a −15% gap, or 1.10 for a +10% pop).

So jump-diffusion is “GBM, but every so often the price teleports.” Calm continuous wandering, punctuated by rare, sudden gaps.

Toggle the chart between diffusion only and with jumps to see exactly what the jump component bolts on: the smooth path is what GBM gives you; the cliffs are what only jumps can produce.

Worked example. With jump rate $\lambda = 2$ per year, you expect about 2 jumps in a year (the Poisson mean is just $\lambda$). The probability of no jump at all in a year is

$P(0 \text{ jumps}) = e^{-\lambda} = e^{-2} \approx 0.135,$

about 13.5%. So even at two-jumps-a-year, roughly one year in seven passes with no gap at all — and the chance of two or more jumps is $1 - e^{-2}(1 + 2) \approx 1 - 0.406 = 0.594$, almost 60%. Jumps are rare per-event but, stacked over a year, very much in play.

Why jumps matter: fat tails & gap risk

Adding jumps isn’t cosmetic — it repairs the single biggest lie a Gaussian/GBM model tells about risk.

Fat tails and skew. Stacking rare large jumps on top of smooth diffusion makes extreme moves far more likely than a normal distribution allows, and (if down-jumps are bigger or more frequent than up-jumps) makes the left tail heavier — negative skew. That’s precisely the shape of real returns. It’s also why the volatility smile exists: option markets price in jump/crash risk that the smooth Black–Scholes (pure-GBM) model ignores, so implied vol bends upward at the wings. Jump-diffusion was one of the first models to explain the smile rather than paper over it.

Gap risk breaks hedging. Delta-hedging assumes you can continuously rebalance as the price moves smoothly. But you can’t trade inside a gap — when the price teleports from 100 to 80, there is no 90 at which to adjust. A delta-hedged book that’s perfectly neutral against small moves can take a brutal loss across a jump, because the hedge was calibrated for the smooth world that just stopped existing for a second.

Risk implication. Value-at-Risk, Greeks, and stress numbers computed under pure GBM systematically understate tail and gap risk. The model literally cannot imagine the crash, so it reports a portfolio as safer than it is. Many blow-ups trace back to a thin-tailed model meeting a fat-tailed world.

The pitfall. Jumps fix the tails but cost you estimation pain: a jump model adds parameters — the rate $\lambda$, the jump-size mean, the jump-size spread — and jumps are rare by definition, so a short data sample contains only a handful of them (or none). Calibrating $\lambda$ from five years of data when the true rate is one big jump per decade is statistical guesswork. Richer model, hungrier for data, and the data is exactly what rare events refuse to provide.

Choosing a model

You now have three building blocks: GBM (trending, continuous), OU (anchored, continuous), and jump-diffusion (gaps on top of either). Picking the right one is mostly about answering two questions: does this quantity have an anchor? and can it gap?

Two judgment calls worth internalizing. First, anchor beats everything: if the quantity has a level it returns to, start from OU, not GBM — modeling a spread as a trendless wanderer throws away your entire edge. Second, gaps demand jumps: anything with crash, de-peg, or binary-event risk needs the jump term, or your risk numbers are fiction. And when no clean closed-form process fits — a basket of jumpy, reverting, correlated, path-dependent things — you stop hunting for a formula and simulate, which is what the Monte Carlo toolkit is for.

Putting it together

GBM walks away forever, but huge swaths of finance — rates, spreads, vol, commodities, pairs spreads — get pulled home. The Ornstein–Uhlenbeck process, $dX_t = \kappa(\theta - X_t)\,dt + \sigma\,dW_t$, is that spring: a restoring drift toward the long-run mean $\theta$ with stiffness $\kappa$, plus noise. Its half-life $\ln 2 / \kappa$ says how fast shocks decay, and its finite stationary variance $\sigma^2/(2\kappa)$ is the bounded band that makes reversion tradeable — but only on spreads, rates, and ratios, never a raw stock price. Pure diffusion can’t gap, so the jump-diffusion (Merton) model bolts a Poisson stream of sudden jumps onto GBM, with $P(\text{no jump in a year}) = e^{-\lambda}$. Jumps create the fat tails, skew, and gap risk that smooth models miss — which is why pure-GBM VaR and delta-hedges fail across a crash. Choose by asking anchor? (→ OU) and can it gap? (→ jumps); when nothing fits, simulate.

Next up — we put these processes to work, calibrating them to real data and simulating whole portfolios of anchored, jumpy, correlated assets. The building blocks are in place; now we make them earn their keep.