Common distributions

A small zoo of distributions

The normal distribution $N(\mu,\sigma^2)$ is the bell curve described by its mean and variance. About 68% of the probability lies within one $\sigma$ of the mean, 95% within 1.96 $\sigma$ and 99.7% within three. Standardising, $Z=(X-\mu)/\sigma$ , turns any normal into $N(0,1)$ , whose cumulative distribution $\Phi$ is tabulated. Daily returns are often modelled as normal, which is a useful but imperfect first approximation.

\ln S\sim N(m,s^{2})\ \Rightarrow\ E[S]=e^{m+s^{2}/2},\qquad P(N=k)=\frac{\lambda^{k}e^{-\lambda}}{k!}

(1)

A price cannot go below zero, so its log is modelled as normal and the price itself is lognormal: skewed to the right, with mean $e^{m+s^2/2}$ larger than the median $e^{m}$ . A Poisson variable counts events in a fixed interval when they arrive independently at a constant average rate $\lambda$ : trades in a minute, defaults in a year. The Student- $t$ distribution looks like a normal with heavier tails; its degrees of freedom $\nu$ control the heaviness, with small $\nu$ giving far more extreme values and $\nu\to\infty$ recovering the normal.

Tail numbers and a lognormal meanA daily return is

N(0.05\%,\ 1.2\%^2)

. Its 95% one-day value at risk is the loss exceeded only 5% of the time:

1.645\cdot1.2\%-0.05\%=1.924\%

. For a log price

N(0.05,\ 0.2^2)

starting from 1, the expected price is

e^{0.05+0.02}\approx1.0725

, above the median

e^{0.05}\approx1.0513

. With

\lambda=3

trades per minute, the chance of a quiet minute with none is

e^{-3}\approx0.0498

Normal: symmetric; use $z$ -scores. Good for sums, rough for single returns.
Lognormal: positive and right-skewed, for prices. Poisson: counts of independent events.
Student- $t$ : heavy tails; fits returns far better than the normal in the extremes.

Why 'once in 10,000 years' keeps happeningA daily move of five standard deviations has probability about

3\times10^{-7}

under a normal model, once in roughly 13,000 years of trading days. Real equity markets have produced several such days in a few decades. The normal curve is a poor guide to extreme days, which is why risk models use heavier-tailed distributions such as the Student-

t

Lab · what's graded

Pick the distribution20 pts
A normal probability10 pts
A value at risk15 pts
A Poisson probability15 pts
A lognormal mean15 pts
Why the Student-t15 pts
Normal models and crashes10 pts

A small zoo of distributions

\ln S\sim N(m,s^{2})\ \Rightarrow\ E[S]=e^{m+s^{2}/2},\qquad P(N=k)=\frac{\lambda^{k}e^{-\lambda}}{k!}

(1)

Tail numbers and a lognormal meanA daily return is

N(0.05\%,\ 1.2\%^2)

. Its 95% one-day value at risk is the loss exceeded only 5% of the time:

1.645\cdot1.2\%-0.05\%=1.924\%

. For a log price

N(0.05,\ 0.2^2)

starting from 1, the expected price is

e^{0.05+0.02}\approx1.0725

, above the median

e^{0.05}\approx1.0513

. With

\lambda=3

trades per minute, the chance of a quiet minute with none is

e^{-3}\approx0.0498

Normal: symmetric; use $z$ -scores. Good for sums, rough for single returns.
Lognormal: positive and right-skewed, for prices. Poisson: counts of independent events.
Student- $t$ : heavy tails; fits returns far better than the normal in the extremes.

Why 'once in 10,000 years' keeps happeningA daily move of five standard deviations has probability about

3\times10^{-7}

t

Lab · what's graded

Pick the distribution20 pts
A normal probability10 pts
A value at risk15 pts
A Poisson probability15 pts
A lognormal mean15 pts
Why the Student-t15 pts
Normal models and crashes10 pts

A small zoo of distributions

Lab · what's graded

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Common distributions

A small zoo of distributions

Lab · what's graded

Log in to run your code.