Jump Processes and Poisson Arrivals in Option Pricing

Jump-diffusion models add Poisson-driven discontinuities to a price process because continuous models understate large single-day moves.

Jump processes add discontinuous moves to a price model by combining continuous Brownian diffusion with a Poisson-driven jump component, and they exist because real asset prices sometimes move in a single instant by an amount that a purely continuous process assigns almost no probability to. The Merton jump-diffusion model, the standard starting point, adds a compound Poisson process to geometric Brownian motion, where jumps arrive at a constant rate and each jump size is drawn from its own distribution.

A Poisson process counts the number of events that have occurred by time t, where events arrive independently of each other at a constant average rate, usually written as lambda, meaning lambda events are expected per unit of time. The waiting time between consecutive events follows an exponential distribution, and the probability of exactly k events occurring in an interval of length t follows the Poisson distribution with parameter lambda times t. The defining feature that makes it useful for modeling rare jumps is that the probability of more than one event in a very short interval is negligible relative to the probability of exactly one, so jumps can be treated as isolated…

In the jump-diffusion setting, each arrival of the Poisson process triggers a jump in the log price by a random amount, typically drawn from a normal distribution with its own mean and variance, distinct from the continuous diffusion's drift and volatility. The combined process is: the differential of the logarithm of S equals a drift term times dt, plus sigma times dW, plus the jump size times the differential of the Poisson counting process.

Geometric Brownian motion, discussed at length in Geometric Brownian Motion and Its Limitations, assigns vanishingly small probability to a large move over a short interval, because a normal distribution's tails decay very quickly. Real markets show large single-day moves far more often than a purely continuous, constant-volatility model would predict, particularly around earnings announcements and macroeconomic surprises. Adding a jump component directly raises the probability of a large short-horizon move without requiring an unrealistically high level of continuous volatility to match observed option prices.

This is one of the most direct explanations for the volatility smile. Since a jump-diffusion model naturally produces more extreme outcomes than a pure diffusion, options struck away from the current price come out more expensive relative to at-the-money options once jumps are included, matching the pattern observed in real markets and discussed fully in Volatility Smile, Skew, and Implied Volatility.