- hist30

hist60 •

hist120-

hist240

Figure 3.1 Historic volatilities of the FTSE 100 index, 1984-1995.

Of course, if one really believes in the assumption of constant volatility that underlies this estimation method one should always use as long a history as possible, so that sampling errors are reduced. Long-term predictions should be unaffected by short-term phenomena such as volatility clustering (§4.1), so it is appropriate to take the average over a long historic period. Short-term predictions should reflect current market conditions, which means that only the immediate past returns should be used.

Extreme events are just as important to current estimates, whether they occurred yesterday or at any other time in the

This apparently sensible approach hides a major problem: extreme events are just as important to current estimates, whether they occurred yesterday or at any other time in the averaging period. Even just one unusual return will affect the -day historic volatility or correlation to the same extent for exactly n days following that day, although a conditional volatility or correlation may have

averaging period long ago returned to normal levels.

3.1.2 Historic Volatility in Financial Markets

Figure 3.1 illustrates equally weighted volatility estimates of different periods for the FTSE 100 index. Daily squared returns are averaged over the last n observations for n = 30, 60, 120 and 240 days, and this variance is transformed to an annualized volatility using (1.1). The 240-day volatility of the FTSE jumped up to 26% the day after Black Monday and it stayed at that level for almost a whole year because that one, huge squared return had exactly the same weight in the average for 240 days. Exactly 240 days after the event the large return fell out of the moving average, and so the volatility forecast returned to its normal level of around 13%. But nothing of note happened in

Jun-96

Dec-96

Jun-97

Dec-97

Jun-98

Dec-98

- 5day-yr1 - 25day-yr1 --50day-yr1

Figure 3.2 Historic volatilities of South African rand 1-year swap rates.

the markets on the 240th day after 17 October 1987, or 30 or 60 or 120 days after Black Monday. The drastic fall in volatility on those dates is just a ghost of Black Monday, an artefact of the method. Certainly it is no reflection of the real market conditions.

Short term equally weighted averages have more pronounced ghost features because an extreme event is being averaged over just a few observations, but at least these ghost features last for a relatively short period of time: they last for as long as the averaging period.

Following an extreme event there will be very noticeable differences between volatility estimates obtained from equally weighted averages of different lengths. On the other hand, if the market has been quite stable for some time, there will be little difference between historic volatility estimates of different lengths.

Figure 3.2 illustrates this. Historic 5-day, 25-day and 50-day volatilities of the South African rand 1-year swap rate show a number of ghost effects of the swap rate jumps in October 1997 and January 1998. Following both of these there are enormous differences between the 5-day, 25-day and 50-day estimates. For example, during the month of December 1997, the 25-day volatility had returned to more normal levels around 6%, but the 50-day volatility recorded 23% because it was still affected by the one event in October

From June 1998 to the end of the data period the South African fixed income market could be characterized by high volatility without any really extreme market moves. The marked discrepancies between the 25-day and the 50-day estimates were not so noticeable at this time. Of course, the 5-day historic volatility is still jumping around all over the place at the end of the data period, because there is such a large sampling error when the sample size is only 5.

If the market has been quite stable for some time, there will be little difference between historic volatility estimates of different lengths

1997.

3.1.3 Historic Correlation in Energy Markets

In the earlier days of the KCBOT contract, the use of 30-day historic correlations would have substantially overestimated, and then underestimated, the correlation that should be used to calculate the proxy hedge ratio

Following just one event that affects both markets in a similar way, the -day historic correlation will remain high for exactly n days. This makes -day historic correlation estimates appear more stable as n increases. Figures 3.3 and 3.4 illustrate this point with historic correlation estimates in energy markets.3

In the natural gas market some futures contracts have a relatively low volume of trade and some traders may be inclined to use a more liquid proxy, if it is highly correlated. Consider the Kansas City Western KCBOT, and the NYMEX natural gas prompt future contracts. The NYMEX has always traded at a higher price. When the KCBOT was first introduced both volume and volatility were very low relative to the NYMEX, but as the KCBOT contract trading volume increased the spread decreased substantially. In the last few years of the sample shown in Figure 3.3, their correlation was very high and stable: although the KCBOT closes later, trading on this contract after the NYMEX has closed is still very thin. Spread volatility is now relatively low and consequently correlations are much larger.

However close in price two futures contracts appear to be, caution should always be exercised when historic correlation estimates are used for the proxy hedge. During the second half of the data period the 30-day correlations shown in Figure 3.3 were indeed very similar to the exponentially weighted average correlations, with X = 0.94 (§3.2.1). However, in the earlier days of the KCBOT contract, the use of 30-day historic correlations would have substantially overestimated, and then underestimated, the correlation that should be used to calculate the proxy hedge ratio.

On 26 March 1996 the price of the future increased quite sharply and then remained more or less around this new level. The spread between the two contracts narrowed significantly; the 30-day historic correlation between the two futures prices rose from about 0.85 to about 0.93 and stayed around this level for exactly 30 days afterwards. Then 30 days later, on 9 May 1996, the 30-day correlation fell from 0.94 to 0.74, although nothing of note happened in the market on that day. The apparent drop in correlation was just an artefact of the equal weighting of historical data.

Figure 3.3 compares the 30-day historic correlation, which remains a standard measure of correlation in energy markets, with an exponentially weighted moving average correlation measure. The exponentially weighted moving average more accurately reflects what is happening between the two markets. After the spread decreased sharply on 26 March 1996 it gradually rose again, peaking in the summer of 1996 before falling back to very low levels. While the correlation rose sharply on 26 March 1996, it should have declined gradually

3A fuller discussion of correlation in energy markets is given in Alexander (1999a). Many thanks to Enron for providing the data used in these examples.