Exponential Distribution

Motivation: why this matters in quant finance

The exponential distribution models waiting time under a constant event rate: time to default, time to the next trade, time to the next jump, or time to an operational loss. It is the waiting-time partner of the Poisson distribution, which counts how many events occur in a window.

Its signature property is memorylessness. Survival up to today does not change the distribution of the remaining wait. That is mathematically clean and financially strong; real hazard rates often move with time, credit quality, and market state.

Definition

A nonnegative random variable

T

follows

\operatorname{Exp}(\lambda)

f(t)=\lambda e^{-\lambda t}, \qquad t\geq0,

where $\lambda>0$ is the rate. The CDF and survival function are

F(t)=1-e^{-\lambda t}, \qquad \overline{F}(t)=e^{-\lambda t}.

Key Properties

The moments are

\mathbb{E}[T]=\frac{1}{\lambda}, \qquad \operatorname{Var}(T)=\frac{1}{\lambda^2}.

Memorylessness is the identity

\mathbb{P}(T>s+t\mid T>s)=\mathbb{P}(T>t).

The hazard rate is constant:

h(t)=\frac{f(t)}{\overline{F}(t)}=\lambda.

The MGF is

M_T(s)=\frac{\lambda}{\lambda-s}, \qquad s<\lambda.

In Quant Finance

In reduced-form credit modelling, a constant-intensity default time has

\mathbb{P}(\tau>T)=e^{-\lambda T}.

With recovery $R$ , a rough spread approximation is

s\approx\lambda(1-R).

If events arrive as a Poisson process with rate $\lambda$ , then inter-arrival times are independent exponentials. Order flow, jump times, and some operational-risk models start from this count/waiting-time link.

For independent credits with $\tau_i\sim\operatorname{Exp}(\lambda_i)$ ,

\min_i\tau_i\sim\operatorname{Exp}\left(\sum_i\lambda_i\right).

The first-to-default rate is the sum of the independent intensities.

Worked Example: First Default

For three independent credits with hazards $1\%$ , $2\%$ , and $5\%$ , the first default rate is $0.08$ . The probability of at least one default within five years is

1-e^{-0.08\cdot5}\approx0.330.

The riskiest name matters, but the basket rate is not the maximum rate; it is the sum.

Common Confusions and Pitfalls

$\lambda$ is a probability. No. It is a rate and can exceed 1.

Memorylessness is realistic by default. It is a modelling baseline, not an empirical guarantee.

The exponential models returns. It models nonnegative waiting times, not signed returns.

Where This Goes Next

Poisson Distribution: counts events whose waiting times are exponential.
Uniform Distribution: inverse transform gives $T=-\ln U/\lambda$ .
Chi-Squared and Related Distributions: $\chi^2_2$ is exponential with rate $1/2$ .

References

Bertsekas and Tsitsiklis, Introduction to Probability, 2nd ed., Ch. 3 Sec. 3.1-3.2 and Ch. 4 Sec. 4.4.
Mosteller, Fifty Challenging Problems in Probability, Problems 27-30, for rare-event and Poisson-limit reasoning.
James, Witten, Hastie, and Tibshirani, An Introduction to Statistical Learning, 2nd ed., Ch. 4, for event probability modelling and generalized linear model context.