When should I use the Poisson distribution in finance?

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AcadiFi · Accepted Answer

The Poisson distribution models the count of independent events occurring at a constant average rate lambda over a fixed interval. Its PMF is P(X=k) = e^(-lambda) * lambda^k / k!, with mean and variance both equal to lambda. In finance it fits rare-event counts: defaults in a pool, operational loss incidents, trading halts, extreme price jumps.

Consider Parabolic Credit Bank tracking defaults in a 500-name high-yield portfolio with average annual default rate of 3.2%. Expected defaults lambda = 500 * 0.032 = 16. Probability of exactly 20 defaults is e^(-16) * 16^20 / 20! = 5.6%. For jump-diffusion stock models (Merton 1976), the number of jumps in [0,T] follows Poisson(lambda*T). Operational risk capital under Basel II uses compound Poisson: total loss = sum over Poisson-distributed count of events, each with a severity distribution.

Key assumption: events are independent. When defaults cluster (contagion), Poisson underestimates tail risk; analysts then shift to negative binomial or Cox processes with stochastic lambda. Always check variance-to-mean ratio; if it exceeds 1.3, Poisson is misspecified.

When should I use the Poisson distribution in finance?

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