Probability Theory Fundamentals › Key Probability Distributions in Finance

Certain probability distributions appear repeatedly in quantitative finance, and each encodes specific assumptions about how uncertainty manifests. Understanding their properties helps you recognize when to apply each one and interpret model outputs correctly. Choosing the right distribution is a fundamental modeling decision.

The normal (Gaussian) distribution is common in financial modeling because it is tractable and because the Central Limit Theorem gives normal approximations to properly centered and normalized sums. In the classical case of iid observations with finite positive variance, these sums converge in distribution to a standard normal. Extensions to non-identical or dependent observations require additional conditions; independence and finite variance alone are not sufficient for arbitrary sequences. That theorem does not say that returns must be normal merely because trading aggregates many decisions. Normal models can approximate the center of some return distributions, while their tails and dependence often require separate treatment.

A normal random variable $X \sim \mathcal{N}(\mu, \sigma^2)$ with $\sigma>0$ has density:

$$f_X(x) = \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-\frac{(x - \mu)^2}{2\sigma^2}\right)$$

where:

• $f_X(x)$: probability density at value $x$, representing the relative likelihood of observing a value near $x$
• $\mu$: mean (center) of the distribution, where the peak occurs
• $\sigma$: standard deviation (spread), controlling the width of the bell curve
• $\sigma^2$: variance, the square of the standard deviation
• $\exp(\cdot)$: exponential function ($e$ raised to the given power)
• $\pi$: mathematical constant (≈ 3.14159)
• The term $(x - \mu)^2 / (2\sigma^2)=\tfrac{1}{2}z^2$ is one-half the squared standardized distance $z=(x-\mu)/\sigma$
• The normalization factor $1/(\sigma\sqrt{2\pi})$ ensures the total area under the curve equals 1

The formula's structure reveals important intuition: the exponential of a negative quadratic creates the bell shape. Values near the mean make the exponent close to zero, while the exponent's magnitude grows quadratically with standardized distance and drives the density toward zero symmetrically. The density's slope magnitude does not increase forever; it peaks one standard deviation from the mean and then tends back toward zero in the tails.

The parameters are:

• $\mu$: Mean, the center of the distribution. In finance, this represents expected return.
• $\sigma^2$: Variance, measuring spread. Its square root $\sigma$ is the standard deviation, often called volatility.

The standard normal distribution ($\mu = 0$, $\sigma = 1$) is especially important. Any normal variable with $\sigma>0$ can be standardized:

$$Z = \frac{X - \mu}{\sigma} \sim \mathcal{N}(0, 1)$$

where:

• $Z$: standardized random variable (also called z-score)
• $X$: original normal random variable
• $\mu$: mean of $X$
• $\sigma$: standard deviation of $X$
• $\mathcal{N}(0, 1)$: standard normal distribution with mean 0 and variance 1

The standardization formula shifts the distribution so the mean becomes zero (by subtracting $\mu$) and scales it so the standard deviation becomes one (by dividing by $\sigma$). This transformation preserves shape while relocating the distribution. The standardized value $Z$ tells us how many standard deviations $X$ lies from its mean; a $Z$ of 2 means two standard deviations above average.

This transformation allows us to use standard normal tables or functions to compute probabilities for any normal distribution. We only need to tabulate one distribution rather than infinitely many. When a financial analyst computes that a return is "two sigma below average," they're implicitly using this standardization: they've converted the return to a z-score to assess how unusual it is.

While returns may be approximated as normal in some settings, a normal price model assigns positive probability to negative prices. A log-normal model avoids that result by restricting the modeled price to strictly positive values. This is a property of the model, not an exceptionless market fact: common stock can become worthless or be cancelled. Modeling the logarithm of a positive price as normal produces the log-normal distribution.

If $\ln(X) \sim \mathcal{N}(\mu, \sigma^2)$ with $\sigma>0$, then $X$ follows a nondegenerate log-normal distribution. Under the common model in which period log returns $\ln(1+r_t)$ are jointly normal (including iid normal log returns), their sum is normal; with positive sum variance, the compounded price ratio is nondegenerate log-normal. If the summed log-return variance is zero, the ratio is deterministic: the zero-variance boundary of this model. Marginal normality of each period alone is not sufficient.

Under the usual nondegenerate model with nonzero volatility, the Black-Scholes stock price at a fixed positive future horizon is log-normal, which keeps the modeled price positive while preserving normal log-return calculations. For derivative valuation, the relevant distribution is the model's risk-neutral distribution; it should not be confused with an estimate of the stock's real-world return distribution. Within this nondegenerate model, percentage gains are unbounded above while simple-return losses are bounded at 100%.

Several other distributions appear frequently in quantitative finance, each suited to particular modeling contexts:

• Uniform distribution: Has constant density on a bounded interval, so subintervals of equal length have equal probability. Each individual point still has probability zero. It is used for random number generation and some Monte Carlo methods.
• Exponential distribution: Models waiting times under a constant-hazard assumption, which can be used as a simplified model for trades or defaults. Its "memoryless" property means that, conditional on the event not yet occurring, the distribution of the remaining waiting time does not depend on how long we have already waited.
• Poisson distribution: Models counts over fixed intervals in a homogeneous Poisson process, with constant arrival intensity, independent increments, and simple arrivals (no simultaneous batches). If the intensity is $\lambda$ events per unit time, a window of length $\tau$ has Poisson mean $\lambda\tau$. This can be a simplified model for trades or credit events, not a general law for actual market arrivals.
• Student's t-distribution: Similar to normal but with heavier tails. A fitted t model can represent heavier marginal tails than a fitted normal model; whether it fits extreme returns better depends on the asset, horizon, parameters, dependence structure, and validation data. The t-distribution has a parameter (degrees of freedom) that controls tail heaviness, allowing it to interpolate between normal-like behavior and fat-tailed behavior.
• Chi-squared distribution: For an iid normal sample with $n\geq2$ and $\sigma^2>0$, the usual sample variance $S^2=\sum_{i=1}^{n}(X_i-\bar X)^2/(n-1)$ satisfies $(n-1)S^2/\sigma^2\sim\chi^2_{n-1}$. Here $X_i$ is observation $i$, $\bar X$ is the sample mean, and $\sigma^2$ is the common population variance. Other chi-squared tests rely on their own finite-sample or asymptotic assumptions, so this is not a generic law for volatility uncertainty.