Lévy Processes

Real prices don't just wander smoothly — they also gap. An earnings surprise, a flash crash, a central bank announcement: the price is at one level, and then, almost instantly, it's somewhere else, with no continuous path connecting the two. Brownian motion — the standard building block for smooth, continuous randomness — has no way to represent a jump; by construction its paths never break. Lévy processes are the broader family that includes Brownian motion as one special case but also allows sudden discontinuous jumps, giving quants a way to model markets that mostly drift quietly and occasionally lurch.

The shared property: independent, stationary increments

What ties the whole family together is one structural rule: the change in the process over any time interval doesn't depend on where the process has been before that interval (independent increments), and the statistical character of a change over an interval of a given length is the same no matter when that interval starts (stationary increments) — a jump-or-drift over any given hour looks the same, statistically, whether it's the first hour of trading or the last. Brownian motion satisfies this with purely continuous, Gaussian increments. A Poisson process — counting the arrival of discrete events like large trades or news hits — also satisfies it, but with increments that are jumps, not smooth drift. A Lévy process is anything built from this same independent-and-stationary-increments rule, and the Lévy-Khintchine decomposition is the formal result saying every such process is some mix of three ingredients: a steady drift, a continuous Brownian component, and a jump component describing how often jumps of various sizes occur.

ingredient	what it captures	pure example
drift	steady, predictable trend	a straight line
Brownian part	continuous, unpredictable wobble	ordinary Brownian motion
jump part	sudden discontinuous moves	a Poisson jump process

Two named examples matter most for finance. Merton's jump-diffusion model bolts occasional, normally-sized jumps (arriving at a Poisson rate) onto ordinary Brownian motion — a direct, intuitive combination of "smooth drift most of the time, discrete shocks sometimes." Variance Gamma and other "pure jump" processes go further, dropping the continuous Brownian piece entirely and building the whole path out of infinitely many tiny jumps plus occasional larger ones — a model built from the observation that even a normal trading day, zoomed in far enough, looks more like a flurry of small discrete trades than a smooth continuous line.

The explorer above draws continuous Brownian-style paths — the smooth end of the Lévy family. Imagine occasionally letting one path jump instantly to a new level mid-simulation, at random moments, and you have the intuitive picture of a jump-diffusion path: smooth Brownian wandering, interrupted now and then by a sudden, unconnected step.

A rough sense of the difference

Take a stock at $100. Under pure Brownian motion, simulating one trading day produces a path that ends up somewhere nearby, having wobbled continuously the whole way — an ending price of $70 by the close of a single ordinary day is, under the model's own assumptions, astronomically unlikely, because a continuous path has to physically travel through every intermediate price on the way down, and normal-sized daily wobbles don't cover that much ground. Add a modest jump component — say, a small probability each day of an independent, separately-distributed jump on top of the usual continuous wobble — and a $70 close becomes a rare but explicitly modeled possibility rather than something the model has essentially ruled out by construction. That's the entire practical difference: not a better fit to the "normal" 95% of days, which Brownian motion already handles fine, but a way of not pretending the other 5% away.

Where this matters in practice

• Fatter tails, more honestly. Pure Brownian-motion models like Black-Scholes systematically underprice the risk of large, sudden moves because continuous paths simply cannot produce them; adding a jump component is the direct fix, and it's part of why jump-diffusion and other Lévy-based models tend to fit observed option-implied volatility smiles better than pure Brownian models.
• Crash and gap risk. Overnight gaps, earnings jumps, and flash crashes are exactly the kind of event a pure-diffusion model treats as near-impossible but a jump-augmented Lévy model can assign a realistic, calibratable probability to.
• Credit and default modeling. A firm's value jumping to zero at an unpredictable default time is naturally a jump event, not a continuous drift to zero — Lévy-process machinery underlies several structural credit models for exactly this reason.
• The cost is complexity. Pricing and hedging under a jump-diffusion or pure-jump model is materially harder than under Black-Scholes — the market isn't "complete" the same way (jumps can't be perfectly hedged with just the underlying and cash the way continuous diffusion can), and closed-form pricing formulas are rarer, pushing more of the work onto numerical methods.