Overview.

Jumps in daily return occur significantly more often and of a greater magnitude than a normal distribution would suggest. How can this be modelled?

Realised confidence level distribution vs normal distribution.

This distribution was taken from up to seven years data from FSTE 100 components. You can see that normal distribution has a close approximation for up to 90% confidence level however above this, high factors of the standard deviation occurred.

Note A +/- 5x standard deviation return or higher should occur in less once per 100 years across all components under a normal distribution however this has occurred just less than once per year per component. More dramatically, a return of +/- 7.5 standard deviations should occur less than 1 per billion years across all components, however this has actually occurred 100 times in the last seven years. (It could be more as the data is only 92% complete)

Trading strategies.

Clearly out of the money put/calls need to take this risk into account. High returns/jumps which should occur one per year under a normal distribution, occur on average nine times per year, and when they do they have a dramatic effect on return. However some components have had a low or below expected number of jumps over the last 7 years, which may reduce the risk of in-the-money options.

Graphs comparing historical and normal distribution.

The historical and normal distributions are similar up to a confidence level of about 0.75. Note: both graphs are normalised as a ration of the standard deviation so it is not surprising that the graphs meet around 0.68. However you can see the already the tail diverges.

Note A confidence level of 0.75 suggests that one day a week might experience a return which larger than expected under a normal distribution.

In the follow chart, you could characterise this divergence as a fat tail or a kurtosis.

Note A confidence level of 0.953 suggests that one day a month might experience a return about 1.4x expected under a normal distribution. While statically significant, the real impact of this ratio would need to quantified.

However, in this chart you can see the historical tail is rapidly diverging with the normal distribution.

Note At a confidence level of 0.996, there is likely to be a jump of about 2x expected once per year.

This chart shows some extreme jumps. (It also shows artefacts a smaller number of samples for this range.)
It would be worth further investigation to determine the true natural of these to determine if they were a) real, b) only occurring in one instrument, c) if these were associated with news events.

Note At a confidence level of 0.9996, a portfolio of 10 instruments will see one instrument get a 3.1x return once per year. At a confidence level of 0.99996, a portfolio of 100 instruments will see one instrument get a 10.7x return once per year.

Symmetry of low and high return.