Sample statistics in Binary options trading

Sample statistics try to capture the general properties of the shape of a frequency distribution of data. Generally, these summary statistics are referred to as ‘moments’. The mean or expected value is the 1st moment and measures the central tendency of a data set.

The variance (or standard deviation) is the 2nd moment and measures the dispersion of the data points around the mean. A measure of the symmetry (or non-symmetry) of the distribution is known as skewness, which is the 3rd moment. Finally, the degree of ‘peakness’ of the distribution and the presence of ‘fat tails’ is measured by the kurtosis (i.e. 4th moment). In addition to these summary statistics for individual random variables (e.g. the return on AT&T stocks), we are also interested in co-movements between two (or more) variables. For example, do the monthly returns of AT&T stocks tend to rise or fall with the monthly returns of Microsoft’s stock? You can of course trade that stock via binary options platforms which are quite interesting and innovative.

The strength of this co-movement can be measured by the covariance or correlation coefficient between the two stock returns. As a first approximation, random variables such as stock returns are usually assumed to have population means, variances, skewness and so on that are constant (over time) and their correlation is also often assumed to be constant over time. Sample statistics using real data (e.g. arithmetic mean) are then used to provide esti¬mates of these ‘population parameters’.

The arithmetic mean is the most widely used statistical measure of central tendency of a binary option data series. (Other measures of central tendency include the median - the middle value - and the mode - the value that occurs the most times.) Suppose we have a data series consisting of n stock returns R,(t = 1,2,, n), then the sample (arithmetic) mean is:

The geometric mean return is:

S = V(1 + D,)( 1 +R2)---( 1 + R„) - 1            

Some recent study reports the FTSE100 stock price index at the end of June for the last six years, togeth¬er with the annual (holding  period) returns (e.g. return for 2007 is 13.28% = (6607.9/5833.42 - 1) X100%). Using equations [18] and [19] the mean returns are:

 Year (June)        Binary FTSE100  Returns

2002       4656.36

2003       4031.17 -13.43%

2004       4464.07 10.74%

2005       5113.16 14.54%

2006       5833.42 14.86%

2007       6607.90 13.28%

Arithmetic mean: R = (l/5)(-13.43% + 10.74% + 14.54% + 14.86% + 13.28%)= 39.99/5 = 7.998%

Geometric mean: g = ^(1 — 0.1343)(1 + 0.1074)- --(1 + 0.1328) - 1 = 1,4287(a2) - 1 = 7.396%

The median is that return of binary options that lies in the middle position when the returns are ordered by size from lowest to highest. In our example, given the returns {-13.43%, 10.74%, 13.28%, 14.54%, 14.86%} the median is 13.28%. The mode is the observation that occurs most often- with our limitation of only six data points this would be 15% (to the nearest per cent).