### References

[1] For example: Gavin Jackson, Why market volatility is growing more intense, Financial Times, 14 September 2015; D. Sornette and S. von der Becke, Crashes and high-frequency trading, UK government Foresight report August 2011.

[2] We might have expected a event to occur once in a thousand years for normal data, since allowing for 262 business days in a year, we find , where is the normal cumulative distribution function. But if we estimate the volatility using a 33-day moving average the frequency of recorded shocks is a little higher than we expect because of fluctuations in the estimated volatility. For a five-year volatility the frequency is closer to the expected one.

[3] For financial assets such as stock indices, the futures returns are very close to the excess total returns from holding the asset (“excess” meaning over the risk-free rate, and “total” meaning including dividends for stocks and coupon payments for bonds). We therefore use futures returns back to the start of futures trading, and the excess returns from holding the index assets before this date.

[4] In fact, a t-distribution with 5 to 10 degrees of freedom is a better model of market returns, as shown in Figure 4.