Poisson and Exponential Distributions
The Poisson distribution models how many rare events occur in a period, like large moves or trade arrivals. Learn the formula, examples and its limits.
The Poisson distribution describes how many times an event happens in a fixed period when events occur independently at a constant average rate. It is used to model the number of trades arriving in a second, the number of large price jumps in a year, the number of defaults in a portfolio or the number of times a stop is hit. It is simple, with a single parameter, and it is the starting point for many models of event arrivals in markets.
The formula#
P(k events) = (λ^k × e^(-λ)) / k!- λ (lambda): the average number of events in the period.
- k: the number of events (0, 1, 2, ...).
The mean and variance of a Poisson distribution are both λ.
Worked examples#
Relationship to other distributions#
| Distribution | Relationship |
|---|---|
| Binomial | Poisson approximates the binomial when the number of trials is large and the probability small (λ = n × p). See Binomial and Bernoulli Distributions |
| Exponential | The time between Poisson events follows an exponential distribution with mean 1 / λ |
| Normal | For large λ, the Poisson looks approximately normal |
Uses in finance#
| Use | Example |
|---|---|
| Jump models | Merton's jump diffusion model (1976) adds Poisson distributed price jumps to option pricing |
| Credit risk | Counting defaults in a portfolio, with intensity models |
| Market microstructure | Arrival of orders and trades. See The Order Book and Market Depth |
| Operational risk | Frequency of losses from errors or system failures. See Operational and Model Risk |
| Insurance and catastrophe bonds | Frequency of disasters |
Overdispersion and clustering#
Real market events usually show overdispersion: the variance of counts is larger than the mean, because events cluster. Large moves come in bursts during crises; orders arrive in waves after news. Models that handle this include:
- Negative binomial distribution: allows variance larger than the mean.
- Hawkes processes: self exciting processes where each event raises the chance of more events soon after, widely used for order flow and jump clustering.
Limits#
- Constant rate assumption: event rates change with market conditions.
- Independence assumption: events often trigger more events.
- Rare event estimates: with few historical events, λ itself is uncertain.
Estimating the rate#
The simplest estimate of λ is the number of events divided by the length of the observation period. With few events, that estimate is uncertain: if you saw 3 large moves in 2 years, the true yearly rate could plausibly be anywhere from under 1 to about 4. Using longer histories, or pooling similar assets, gives more reliable rates.
Frequently asked questions#
What is the Poisson distribution?#
A distribution describing the number of times an event occurs in a fixed period, assuming events happen independently at a constant average rate.
How is the Poisson distribution used in trading?#
To model counts of rare events such as large price jumps, defaults and order arrivals, and as a building block in jump diffusion and microstructure models.
What are the limits of the Poisson model in markets?#
Market events cluster and their rates change over time, so counts are often more variable than a Poisson model predicts.
Next, learn why extreme moves happen so often in Fat Tails.
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