Levy Processes In Option Pricing

Black-Scholes models a stock price as a path that wiggles continuously, never teleporting from one level to another. Real stock prices don't always cooperate — an earnings surprise, a takeover announcement, or a flash crash can move a price several percent in seconds, a gap the continuous model has no way to represent except as an enormous, improbable string of small moves happening all at once. Levy processes are the mathematical family built to allow for exactly this: random paths that combine ordinary continuous wiggling with the possibility of sudden, discrete jumps.

Named after the mathematician Paul Levy, they're not one specific model but a broad category — Brownian motion (the smooth process behind Black-Scholes) is technically the simplest member of the family, with the jump part switched off. Add jumps back in, of varying sizes and frequencies, and you get richer models: the Merton jump-diffusion model bolts occasional normally-sized jumps onto ordinary Brownian motion; the variance gamma and CGMY models go further, replacing the continuous part entirely with pure jump activity, so the whole path is built from an infinite number of tiny jumps rather than a mix of smooth drift and occasional shocks.

Why option pricers reach for them

The practical draw is fatter, more realistic tails. Black-Scholes' lognormal distribution systematically underprices the chance of a large one-day move, which is exactly why deep out-of-the-money options trade at implied volatilities higher than at-the-money ones in real markets — the volatility skew. A model that allows genuine jumps produces fat tails naturally, without needing to fudge the volatility input strike by strike to compensate, which is what a pure Black-Scholes desk effectively does when it quotes a different implied volatility for every strike on the same underlying.

What this means in practice

Levy-based pricers are common on desks trading short-dated options, where the gap-risk from an overnight jump is a large share of total risk, and on desks pricing exotic payoffs sensitive to the exact path of jumps, such as barrier options that can be knocked in or out by a sudden move. The tradeoff is complexity: Levy models generally don't have simple closed-form prices the way Black-Scholes does, and are usually priced through characteristic-function or Fourier-transform methods instead of a single clean formula.

Practice

1. Why does a model that allows for jumps naturally produce a volatility skew, without adjusting the volatility input by strike the way a Black-Scholes desk has to?
2. Ordinary Brownian motion is a Levy process with its jump component switched off — what does that suggest about how Black-Scholes relates to jump-diffusion models like Merton's?