Probability Distribution

A probability distribution is a mathematical description of the possible values of a random variable and how likely each is. It may be given as a table, a formula or a curve, and it accounts for all outcomes: the probabilities sum, or integrate, to one. The function that assigns probabilities to outcomes is called the probability distribution function, a general term that covers the probability mass function of a discrete variable and the probability density function of a continuous one.

Distributions are the basic language of uncertainty in finance. A statement about the risk of a portfolio, the value of an option or the chance that a strategy loses money is a statement about a distribution of future outcomes. The choice of which distribution to assume, and how well it matches reality, lies behind most quantitative models and many of their failures.

A discrete distribution applies when the variable can take only separate values, such as the number of winning trades in a week or the number of defaults in a loan book. Its probability mass function gives the probability of each value directly.

A continuous distribution applies when the variable can take any value in a range, as with a return or a price. The probability of any single exact value is zero, so the distribution is described by a probability density function. The density at a point is not itself a probability. Probabilities are areas under the curve between two points. In both cases the cumulative distribution function gives the probability that the variable is less than or equal to a stated value. Reading it in reverse gives a quantile: the value below which a stated share of outcomes falls.

A distribution is commonly summarised by a few numbers. The expected value, or mean, is its centre of gravity. For a discrete variable it is:

The variance measures spread around the mean:

• $x_i$ is the $i$-th possible value of the variable.
• $p_i$ is the probability of that value.

Suppose a trade gains 200 with probability 0.4 and loses 100 with probability 0.6. The expected value is $200 \times 0.4 - 100 \times 0.6 = 20$. The variance is $(200 - 20)^{2} \times 0.4 + (-100 - 20)^{2} \times 0.6 = 12{,}960 + 8{,}640 = 21{,}600$, and its square root, the standard deviation, is about 147. The trade has a positive expected value but loses more often than it wins, and the spread of outcomes is large compared with the average.

Two further measures describe shape. Skewness shows whether one tail is longer than the other, and kurtosis shows how heavy the tails are. The median and mode are alternative measures of the centre.

• Normal. The bell-shaped normal distribution is fully described by its mean and standard deviation. It is the default model for returns over short periods, and its standardised form is the standard normal distribution.
• Lognormal. The lognormal distribution is used for prices, which cannot be negative. It underlies the standard option pricing formula.
• Student's t. Student's t-distribution resembles the normal but has heavier tails. It is used in tests on small samples and as a more realistic model of returns.
• Binomial. This discrete distribution counts successes in a fixed number of independent trials and is the basis of tree models for options.
• Poisson. This discrete distribution counts events occurring at random over an interval, such as order arrivals, defaults or price jumps.
• Uniform. Every value in a range is equally likely. It is the starting point for generating random numbers in simulation.

Risk measurement is a question about the lower tail of a distribution. Value at risk is a quantile of the distribution of portfolio losses: the loss that will be exceeded only with a stated small probability. Option pricing integrates the option's payoff over the distribution of the underlying price at expiry. Statistical tests compare an observed statistic with the distribution it would follow if a hypothesis were true. Monte Carlo simulation draws repeatedly from assumed distributions to build up the distribution of a complex outcome that cannot be calculated directly.

Empirical distributions, built directly from observed data and usually viewed as a histogram, are used alongside theoretical ones. Comparing the two shows where a model fits and where it does not.

Models are tractable because they assume a particular distribution, and they fail where that assumption is wrong. Returns on financial assets have fat tails: extreme moves occur far more often than a normal distribution predicts. A risk model built on normality will report that a severe loss is a once-in-many-centuries event when losses of that size in fact occur every few years. Distributions of returns are also not fixed. Volatility rises and falls, and correlations between assets shift, so a distribution estimated from one period may describe the next poorly.

A further limit is that a distribution describes risk that can be quantified. It presumes that the set of possible outcomes and their probabilities are known or can be estimated. Events with no precedent in the data are absent from any fitted distribution, which is why stress tests based on hypothetical scenarios are used to supplement statistical measures.