Feb-96 Aug-96 Feb-97 Aug-97 Feb-98 Aug-98 Feb-99

-EWMA Corr --30dayCorr

Figure 3.3 EWMA and 30-day correlation of NYMEX and KCBOT prompt futures.

again until the summer of 1996 before rising again to more comfortable levels.

Spot-future correlations are meaningful only for assets with uncertain carry costs or assets that cannot be stored or shorted. Otherwise the spot-future relationship is deterministic: Future Price - Spot Price + Carry Cost. Spot-future correlation can be far lower in commodity markets than in financial markets, because of transportation costs, storage constraints, and other logistic problems. Figure 3.4 shows correlations between West Texas Intermediate (WTI) spot and futures crude oil prices calculated using equal weighting over 3 months, 6 months, 1 year and 2 years. These correlations are generally quite stable over the latter part of the period and substantially less than 1. However, increased volatility during the Gulf war induced a marked change in spot-future correlations.

Spot-future correlation can be far lower in commodity markets than in financial markets, because of transportation costs, storage constraints, and other logistic problems

The ghost effects of the Gulf war on correlation measures is less intense, but longer-lasting, as the averaging period increases. On 17 January 1991 when spot and future prices dropped from about $32 to about $22 overnight with the outbreak of war, equally weighted correlations increased substantially by an amount in inverse proportion to the length of average. On 18 January 1991 the 3-month correlation rose from 0.8 to 0.91, staying above 0.9 until 17 April 1991 when it jumped down from 0.94 to 0.83. The 2-year correlation jumped from 0.81 to 0.86, staying at around this level for exactly 2 years, long after the other averages had returned to more realistic levels. But nothing special happened on 17 April 1991, or on 18 July 1991, long after the outbreak of the Gulf war. The sharp declines in correlation measures on these dates are just an artefact of the estimation method.

The sharp declines in correlation measures on these dates are just an artefact of the estimation method

Energy producers that are exposed to many commodities may wish to hedge revenues with basket options. There is no need to pay for greater protection than the risk one is exposed to. If the exposure is to the sum of several commodities.

. - -,-,-,-,-,-1-,-1-

Jun-90 Jun-91 Jun-92 Jun-93 Jun-94 Jun-95 Jun-96 Jun-97 Jun-98

-3 months 6 months --1 year -2 years Figure 3.4 Equally weighted correlations of WTI crude oil spot and futures.

The lower the buying individual put options can be more of a gamble than a hedge. Not only correlation, the smaller this, basket options are generally far cheaper than buying options on individual the basket volatility and markets. This is because basket volatility is related to the volatility of the cheaper the option individual options as 2+>, = ( + cyy)2 - 2(1 - p)axay, so the basket volatility is less than the sum of individual volatilities unless p = 1. The lower the correlation, the smaller the basket volatility and the cheaper the option.

Figure 3.5 shows some historic correlation measures between NYMEX prompt futures on crude oil and natural gas. The long-term correlations are in the region of 0.1-0.2, so long-term basket options on natural gas and crude oil should be relatively cheap. However, among other factors, differences in settlement dates and procedures across different markets produce highly unstable short-term correlations. For example, the 30-day correlation in Figure 3.5 can fluctuate between 0.3 and -0.3 in the course of a few days. Even though they may be cheaper, there will be far more uncertainty when hedging with short-term basket options because their prices will be much more variable.

3.1.4 When and How Should Historic Estimates Be Used?

One should ask whether it is appropriate to measure long-term volatility and correlation with extreme events in the data set

In what circumstances, if at all, should one consider applying equally weighted volatility or correlation measures? Long-term volatility could be forecast with this method, but only when one assumes that the past is an accurate reflection of the future. How long a look-back period should be used? A long-term volatility based on all the FTSE 100 index data in Figure 3.1 is around 15%, but if Black Monday were excluded and the data period started in 1988 then long-term volatility forecasts would be lower, at about 13%>.

Since market events on a single day have such a prolonged effect on long-term volatility and correlations that are calculated using equally weighted moving averages, one should ask whether it is appropriate to measure long-term

volatility and correlation with extreme events in the data set. Should these be filtered out first?

One cannot give an objective answer to this question. Equally weighted averages can give some idea of the possible range for long-term volatility or correlation, with and without extreme market moves. The decision about which of these forecasts to use depends on ones subjective beliefs about the possible occurrence of extreme events during the risk horizon. Users may define their beliefs about long-term volatility by a probability distribution over the range that seems plausible from historic calculations, and this distribution should really be carried through the rest of the analysis. For example, instead of a single mark-to-market (MtM) value of a long-term option, use the mean MtM, with an MtM standard error based on the distribution of long-term volatility (§5.3.1).

Are equally weighted averages at all successful for short-term volatility forecasting? Most of the empirical evidence indicates that the historical method is not very effective for short-term horizons (§5.1). Although shorter averages are supposed to capture more of the clustering in volatility, the equal weighting does not properly account for the dynamic properties of forcef int° a returns. The historic model is essentially a static model which should not be forced into a time-varying framework with little regard for the consequences.

The historic model is essentially a static model which should not be time-

varying framework with little regard for the consequences

3.2 Exponentially Weighted Moving Averages

An exponentially weighted moving average (EWMA) puts more weight on the more recent observations, and thus takes some account of the dynamic ordering in returns. When an EWMA is applied to squared returns the