Читать книгу How to Calculate Options Prices and Their Greeks - Ursone Pierino - Страница 5

The “Bell” curve or Gaussian distribution, called the normal standard distribution, displays how data/observations will be distributed in a specific range with a certain probability. Think of the height of a population; let's assume a group of people where 95 % of all the persons are between 1.10 m and 1.90 m, implying a mean of 1.50 m . Looking at Chart 2.1, one can see that 95 % of the observations are within 2 standard deviations on either side of the mean (on the chart at 0.00), totalling 4 standard deviations. So 0.80 m (the difference between 1.90 and 1.10) represents 4 standard deviations, resulting in a standard deviation of 0.20 m.

Chart 2.1 Normal probability distribution

With a mean of 1.50 m and a standard deviation of 0.20 m one could say that there is a likelihood of 68 % for the people to have a height between 1.30 and 1.70 m, a high likelihood of 95 % for people to have a height between 1.10 and 1.90 m and almost certainty, around 99.7 %, for people to have a height between 0.90 and 2.10 m. Or to say it differently; hardly any person is taller than 2.10 m or smaller than 0.90 m.

STANDARD DEVIATION IN A FINANCIAL MARKET

The same could be applied to the daily returns of a Future in a financial market. According to its volatility it will have a certain standard deviation. When, for instance, a Future which is trading at 50 with a daily standard deviation of 1 %, one could say that with 256 trading days in a year (365 days minus the weekends and some holidays), in 68 % of these days, being 174 days, the Future will move during each trading 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 (or simply ) will be applied, where σ stands for volatility and stands for the square root of time to maturity (expressed in years).

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 () at 20 % and maturity (T) 3 months (¼ year), the standard deviation will be: . This implies that when 2 standard deviations are applied, the Future at maturity will be expected to be somewhere between 40.00 and 60.00 with 95 % certainty (sometimes called the confidence level).

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: , being 10 % times the square root of 1 times 50, which makes $5. When applying 3 standard deviations, the 35/65 range can be calculated (being 3 × 5 = 15 points lower and higher compared to the current Future level) where the 99.7 % probability applies.

Chart 2.2

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 component, the outcome will be exactly the same probability distribution.

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.