How to Calculate Options Prices and Their Greeks. Ursone Pierino

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.

      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:

. This is why the downside is somewhat limited

(showing a 25 to 50 move being equivalent to a 50 to 100 move). In this way a $25 range on the downside is equivalent to a $50 range on the upside. Also a 10 to 50 range will be equivalent to a move from 50 to 250, a scenario where there is $40 on the downside versus $200 on the upside. When not applying the ln sign, the relationship will be clear as well:

or

.