Chapter 2: FAQs 201
How Good is the Assumption of Normal
Distributions for Financial Returns?
Short Answer
The answer has to be ‘it depends.’ It depends on the
timescale over which returns are measured. For stocks
over very short timescales, intraday to several days,
the distributions are not normal, they have fatter tails
and higher peaks than normal. Over longer periods
they start to look more normal, but then over years or
decades they look lognormal.
It also depends on what you mean by ‘good.’ They are
very good in the sense that they are simple distributions
to work with, and also, thanks to theCentral Limit
Theorem, sensible distributions to work with since there
are sound reasons why they might appear. They are also
good in that basic stochastic calculus and Ito’s lemmaˆ
assume normal distributions and those concepts are
bricks and mortar to the quant.
Example
In Figure 2-11 is the probability density function for
the daily returns on the S&P index since 1980, scaled
to have zero mean and standard deviation of one, and
also the standardized normal distribution. The empirical
peak is higher than the normal distribution and the tails
are both fatter.
On 19th October 1987 the SP500 fell 20.5%. What is the
probability of a 20% one-day fall in the SP500? Since we
are working with over 20 years of daily data, we could
argue that empirically there will be a 20% fall in the
SPX index every 20 years or so. To get a theoretical
estimate, based on normal distributions, we must first
estimate the daily standard deviation for SPX returns.
