day between 0 and 50 cents up or down. Twenty-seven per cent (95 % minus 68 %) of the days (69 days) it will shift between 50 cents and a dollar up or down. There will be around 13 days where the Future will move more than 1 dollar during the day.
In the financial markets where a Future trades at 50 (hence a mean of 50), a standard deviation of
THE IMPACT OF VOLATILITY AND TIME ON THE STANDARD DEVIATION
Volatility is the measure of the variation of a financial asset over a certain time period. An asset with high volatility displays sharp directional moves and large intraday moves; one can think of times when exchanges experience turbulent moments, when for instance geopolitical issues arise and investors seem to be panicking a bit. With low volatility one could think of the infamous summer lull; at some stage markets hardly move for days, volumes are very low and people are not investing during their summer holidays.
For T, the square root of time to maturity is represented in an annualised form, meaning that when maturity is in 3 months time, T will be ¼ (year), where its square root is ½.
So with a Future (F) trading at 50.00, volatility (
Obviously, when volatility and/or time to maturity increase the range gets larger (in essence an increase in the standard deviation); a decrease in volatility and/or time to maturity will result in a smaller range (a decrease in the standard deviation).
The dashed line in Chart 2.2 (10 % volatility and 1 year to maturity) displays the distribution with a standard deviation of:
Chart 2.2 shows the probability distributions for the Future, currently trading at 50, at two different volatility levels and three different times to maturity.
This is exactly the same as the distribution for the Future with a volatility of 20 % and a maturity of 3 months (the earlier example). In the standard deviation formula the volatility has doubled from 10 % to 20 %. With regards to T, 3 months' maturity is one quarter of a year. By taking the square root of a quarter, it means that the T component has actually been halved. So by halving the standard deviation for the T component and multiplying it by 2 for the
It is important to realise that the surface of the different distributions has the same size every time. The total chance has to be kept at 100 % all the time. Maturity can be shorter, though the height of the chart will be higher then, in order to keep the surface at the same size. A higher volatility (with unchanged time to maturity) will result in a much broader/wider area in the Future; to compensate for that (keeping the total size of the charts at the same level) the height of the chart/distribution will be lower.
In conclusion: the effect on the standard deviations is linear with regards to volatility moves and has a square root function with regards to time changes. This feature will come back several times when discussing the Greeks. One just needs to recall how the charts will change (keeping size/surface constant) with changes in volatility and changes in time; it will help to explain other features in option theory as well.
Chapter 3
Volatility
Volatility is the measure of the variation (or the dispersion) of the returns (profits/losses) of a Future over a certain period of time.
One could say: the riskier the asset, the higher the volatility (think of market crashes). The lower an asset's risk, the lower its volatility (think of the summer lull). So, in highly volatile markets one could expect large moves of the Future where at low volatile markets there might be days where the Future hardly moves.
THE PROBABILITY DISTRIBUTION OF THE VALUE OF A FUTURE AFTER ONE YEAR OF TRADING
In option trading volatility is expressed on an annualised basis. It is a calculation of the daily returns based on a full year's expectation of the combined returns. The annualised volatility predicts the probability of the outcome of the value of a Future after one year of trading (usually 256 trading days).
The probability is based on the Gaussian distribution. With low volatility (for example, 10 % as depicted in Chart 3.1), one could expect the Future, which initially started at 50, to settle somewhere between 40 and 60 after one year of trading. (Here a 95 % confidence level has been applied, being 95 % of all probable occurrences, and hence 2 standard deviations of 10 %.) If the volatility is twice as high (20 %), the range for the Future to settle after one year of trading would (almost) double as well, now between 30 and 70.
Chart 3.1 Probability distribution at 10 % volatility
When volatility is at 40 %, the range (almost) doubles again.
NORMAL DISTRIBUTION VERSUS LOG-NORMAL DISTRIBUTION
Charts 3.1, 3.2 and 3.3 show that the distribution range for the Future to settle after one year of trading would double with double volatility, however the word “almost” has been added between brackets. This is the result of the convention in the financial markets to apply a log-normal distribution rather than a normal distribution.
Chart 3.2 Probability distribution at 20 % volatility
Chart 3.2 Probability distribution at 40 % volatility
The application of a log-normal scale serves two purposes:
a. the Future cannot become negative;
b. the Future, not being able to go below zero, could in fact increase towards many times the initial value.
A Future, now trading at 50, can lose a maximum of $50 but could easily gain several hundreds of dollars. On a logarithmic scale the impact of an asset going up from 50 to 100 is equivalent to an asset going up from 100 to 200 – or, mathematically expressed in logarithmic returns:
