QM 6 Statistical Distributions for Financial Asset Prices and Returns

A random variable is a rule that attaches a number to every outcome of an experiment. In finance these are often called stochastic variables, a reminder that their values cannot be known in advance. They come in two kinds. A discrete random variable has a countable set of separate outcomes, such as heads or tails on a fair coin, each with probability 0.50, or the faces of a fair six-sided die, each with probability one-sixth. A continuous random variable can take any value inside a range, such as a real number drawn between 0 and 1. When a data series has very many observations, its distribution tends to look continuous, which is why daily stock returns are modelled as continuous even though any single reading is a discrete number.

Whatever its type, a random variable X is fully described by two non-negative functions. The cumulative distribution function, or CDF, gives the probability that X is at or below a chosen level x.

The CDF never decreases as x rises, it approaches 0 far to the left, and it approaches 1 far to the right, and it applies to both discrete and continuous variables. For a continuous variable the probability density function, or PDF, is the slope of the CDF: it is proportional to the chance of landing in a narrow band around a value, and the probability of falling inside an interval is the area under the PDF across that interval. The whole area beneath a PDF equals 1.

For a discrete variable the counterpart of the density is the probability mass function, or PMF, which gives the probability that X equals a specific value exactly. Summing the PMF over an interval gives the interval probability, and summing it over every outcome gives 1. Because a discrete CDF jumps at each outcome and is flat between them, it is a step function rather than a smooth curve and is not differentiable in the usual sense.

Two dice: a worked distribution

Rolling a pair of fair dice gives six times six, or 36, equally likely combinations. Each possible sum from 2 to 12 arises with its own frequency: a sum of 12 happens in only one way, while a sum of 10 happens in three ways, namely (4, 6), (5, 5), and (6, 4), so its probability is three divided by 36, about 8.33 percent. Dividing every frequency by 36 produces the PMF, and running totals produce the CDF. The chance of a sum of 10 or less is 91.67 percent.

PMF: probability mass function. CDF: cumulative distribution function.

Assessing an asset means measuring both its expected reward and its risk. Unconditional measures do this without assuming any particular market state. The unconditional mean, also called the expected value, weights each possible outcome by its probability and adds them up, the average that would emerge over endlessly many trials.

The unconditional variance measures how far outcomes spread around the mean, and the standard deviation is its square root. The covariance measures the direction and strength of the linear link between two variables, though a large covariance figure alone does not prove a strong relationship, since its size depends on the units of the two variables.

The two-dice case

Using the PMF above, the expected sum of two dice is 7. Squaring each sum, weighting by its probability and adding gives an expected square of 54.84, so the variance is 54.84 minus 7 squared, which is 5.84, and the standard deviation is the square root of 5.84, about 2.42. That standard deviation puts every outcome on a common scale: a roll of 12 sits (12 minus 7) divided by 2.42, or 2.07, standard deviations above the mean. For a single die the expected value is (1 plus 2 plus 3 plus 4 plus 5 plus 6) divided by 6, which is 3.5. Because two separate dice are independent, the expected value of their product equals the product of their expected values, 3.5 times 3.5, or 12.25, so their covariance is 12.25 minus 12.25, which is 0. Independence forces the covariance to zero.

The law of large numbers

As a random and representative sample grows, its sample mean settles toward the unconditional mean. This is the law of large numbers, and it is the foundation of statistical inference, the practice of drawing conclusions about a population from a sample. Later readings build on it for hypothesis testing, confidence intervals, regression, and Bayesian analysis. A simulation of 10,000 dice throws makes the point: the running sample mean hovers near the theoretical 7 and the running sample variance near 35 divided by 6, about 5.83.

Modelling prices and returns means choosing a distribution whose shape matches the process at work. This section covers the three most useful discrete families, and the next covers the continuous ones. Getting the choice right matters, since the assumed distribution drives every forecast built on top of it.

Discrete uniform distribution

A discrete uniform distribution has a finite set of equally spaced outcomes, each equally likely. The faces of a fair die are the classic case. With outcomes running from a minimum a to a maximum b, the mean and variance are as follows.

For a single die, a is 1 and b is 6, so the mean is (1 plus 6) divided by 2, which is 3.5, and the variance is (6 squared minus 1) divided by 12, which is 35 divided by 12, about 2.92.

Binomial distribution

The binomial distribution counts how many successes occur across a set count of independent trials, each one succeeding with probability p and failing with probability 1 minus p. It extends the Bernoulli distribution, which describes one such trial on its own. It fits any binary process, for example whether an investment pays off or whether a stock ticks up or down over a set number of periods, and it underlies the binomial option pricing model, which breaks a continuous price path into small up-or-down steps. In that setting a stock reaching two up moves and one down move over three steps, from a start of 100 with an up factor of 1.1 and a down factor of 0.9, is worth 100 times 1.1 squared times 0.9, which is 108.90.

When n is large and both n times p and n times (1 minus p) exceed 5, the binomial is well approximated by a normal distribution with the same mean and variance. The shape is symmetric when p equals 0.5, right skewed when p is above 0.5, and left skewed when p is below 0.5, and larger n damps the skew.

Poisson distribution

The Poisson distribution models the number of independent events that occur in a fixed span of time or space, arriving at a constant average rate lambda per unit. It suits rare, countable events such as loan defaults in a portfolio or order arrivals in a trading system. The chance of observing k events across an interval of length t appears below, and both the mean and the variance equal lambda over one unit of time.

The Bernoulli, binomial, and Poisson families are linked: the binomial is the sum of n Bernoulli trials, and the Poisson is the limit of the binomial as n grows without bound and p shrinks toward zero with n times p held at lambda.

Continuous uniform distribution

The continuous uniform distribution treats every value in an interval from a to b as equally likely, often over the range 0 to 1. Its mean and variance are shown below, its skewness is zero, and its kurtosis is 1.8.

Its usefulness is practical. Because its CDF is a straight line, a draw from a uniform distribution can be fed through the inverse CDF of any target distribution to generate a draw from that target, which is the engine of Monte Carlo simulation. For instance, a uniform draw of 0.333 maps to a two-dice sum of 6, since 0.333 lies between the CDF values 0.2778 and 0.4167 for that outcome. Repeating this thousands of times builds a full range of possible outcomes, each with its probability, and is a standard way to simulate asset price paths.

Normal distribution

The normal, or Gaussian, distribution is the symmetric bell-shaped curve fixed entirely by its mean and its variance, written N(mu, sigma squared). Values cluster near the mean, and sums of many independent variables tend toward it even when the parts are not normal, which is why it anchors so much of financial modelling and risk management. It is symmetric, so its mean, median, and mode coincide, its skewness is 0, and its kurtosis is 3. The standard normal is the special case with mean 0 and variance 1. Two spans are worth memorising: about 68 percent of the data lies within one standard deviation of the mean, and about 95 percent lies within two.

Log-normal distribution

A variable is log-normal when its natural logarithm is normal, which makes it positively skewed and strictly positive. That fits asset prices well, since a price cannot fall below zero, and it is the assumption behind the Black-Scholes option pricing model, in which log returns, the natural log of the price ratio, are taken to be normal. Log returns have practical advantages over simple returns: they cannot imply a loss beyond 100 percent, they add across periods, and the product of independent log-normal variables is itself log-normal, matching how returns compound. If X is normal with mean mu and variance sigma squared, then Y equal to e raised to X is log-normal with the mean and variance below.

Logistic distribution

The logistic distribution is symmetric and bell-shaped like the normal but with heavier tails, so it assigns more weight to extreme outcomes. It is defined by a location parameter mu, which is also its mean, and a scale parameter s tied to its spread. Its skewness is 0, its kurtosis is 4.2, and its excess kurtosis is 1.2. It is central to logistic regression, a workhorse for binary classification problems such as predicting success or failure, and it appears throughout machine learning.

Skewness and excess kurtosis for the discrete uniform and the binomial depend on their parameters.

A joint distribution describes two random variables at once, which is how an investor gauges the dependence between asset returns. The joint bivariate CDF gives the probability that the first variable is at or below one level while the second is at or below another. When the two variables are independent, the joint distribution factors into the product of the individual distributions.

The marginal distribution isolates one variable on its own, recovered by integrating the joint density over the other variable, or by reading the row and column totals of a discrete table. It is what matters when an investor wants to study a single asset within a portfolio. For discrete variables the joint and marginal picture is usually laid out in a contingency table.

The last row and column are the marginal distributions of firm type and of return.

From the totals, the marginal probability that a randomly chosen firm is non-technology is 400 divided by 500, or 80 percent. The joint probability that a firm is both a technology firm and posts a return above 10 percent is 60 divided by 500, which is 12 percent.

Conditional expectations

A conditional expectation, written E[X given Y equals y], is the expected value of X once we know the value of Y. It is how new information is folded into a forecast: for a discrete variable it sums the outcomes of X weighted by their probabilities conditional on Y.

The total probability rule for expectations rebuilds the unconditional mean by weighting each conditional expectation by the probability of its condition, across a set of mutually exclusive and exhaustive scenarios.

The aim of modelling markets is to hold a set of possible future outcomes with attached probabilities, then revise them coherently as information arrives. Bayes theorem is the formal tool for that revision. A prior, or a priori, probability is the belief held before the new data, an ex-ante figure. A posterior, or a posteriori, probability is the belief after the data has been taken in, an ex-post figure derived from the prior.

The theorem follows from a simple fact: the probability that events A and B both occur can be written two ways, as the chance of A given B times the chance of B, or as the chance of B given A times the chance of A. Setting the two equal and rearranging gives the rule.

Return to the index of 500 firms. The empirical probabilities read off the table include the chance a firm is a technology firm, 0.20, the chance a technology firm returns above 10 percent, 0.60, and the same for a non-technology firm, 0.25. The overall chance of a return above 10 percent is 0.60 times 0.20 plus 0.25 times 0.80, which is 0.32. Applying the theorem, the chance that a firm is a technology firm given a return above 10 percent is 0.60 times 0.20 divided by 0.32, which is 0.375. Learning that the return beat 10 percent lifts the probability from the prior 0.20 to 0.375.