Summary:
Descriptive statistics are tools that allow traders to better understand and comprehend
data in an easy and effective manner.
Instead of presenting the height measurements of 100 men, I could instead offer that
the average height of these 100 men is 5 feet 8 inches.
By providing one descriptive statistic, I have characterized a quality of the entire
sample of height measurements.
I can take this process further by revealing that the standard deviation of heights is 3 inches.
By calculating descriptive statistics on market prices, market returns, and market
volume, we can learn much about the nature of recent price movement.
These descriptive statistics will become the building blocks for our quantitative
trading systems.
The mean of a series, more commonly referred to as the average, is a measure of
central location. The mean is the sum of the values in a distribution divided by the
number of data points in the distribution.
We may wish to measure how widely values spreadacross a distribution. Do the values clump closely around a central point or are they distributed widely? The most popular methods used to measure the dispersion
of values are variance and standard deviation.
Variance, often represented by 2, measures how wide the spread of values
span from the mean.
if a company has a larger variance than another company it means the returns of that
company vary more widely and are more volatile.
Correlation is another important descriptive statistic. It measures the strength of a
relationship between two series.
The correlation statistic is calculated by multiplying the difference
of one series from its mean by the corresponding difference of another series
from its mean, taking the average product, and then dividing by the product of the
standard deviation of both series.
In Figure 2.5, y increases as x does. This indicates a positive correlation
between x and y. In Figure 2.6, a different relationship exists. As x increases, y
decreases, indicating negative correlation.
it is important to perform your analysis using returns rather than prices.
Standard deviation is a popular method of measuring dispersion, primarily due
to its properties under certain circumstances, specifically those associated with
a normal distribution.
normal distribution, For example, we know that roughly 68.26 percent of
the values in a normal distribution fall between ±1 standard deviation of the mean,
95.44 percent fall between ±2 standard deviations, and 99.74 percent between ±3
standard deviations
explain how volatility varies over time. These models are called GARCH, for Generalized Auto
Regressive Conditional Heteroskedasticity. A GARCH process exists when
volatility itself changes over time, wandering back and forth around a long-term
average.
We start by calculating the standard deviation of the first 20 returns. On the
next day, we drop the first return from our calculation, add the 21st return, and
recalculate the standard deviation of returns. The following day we drop the second
return and add the 22d return in our calculation of standard deviation. And so
on. Each value of the 20-day standard deviation will have 19 common return
points as the value before and value after. In this sense, the calculation “rolls” with
each day, hence the expression “rolling volatility.”
volatility scales with the square root of time, we multiply our daily standard deviation by the square root of trading days in a year (typically 252 for equity markets). The result of this adjustment is an annualized standard deviation, or volatility.
This property of volatility is common to almost all markets. Periods of
high volatility are often followed by further periods of high volatility, slowly
decreasing to more normal levels over time. Similarly, periods of low volatility are
often followed by further periods of low volatility, eventually returning to normal
levels over time.GARCH cannot or does not predict
market returns or prices, volatility has important implications
for generating trading signals as well as managing the risk of portfolios
If we know a market’s annualized volatility, we can calculate the daily risk of
being long or short.
The daily standard deviation of being long or short the S&P
500 is equal to the value of the portfolio multiplied by the annualized standard
deviation divided by the square root of 252 (trading days in year)
lognormal distribution. This is somewhat similar to the normal
distribution in its shape, except the left- and right-hand side of the distribution
is not symmetrical.The lognormal distribution
of prices is used commonly and is the basis for the Black-Scholes option pricing
formula.
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