Primer on Measuring Spread Volatility Lets begin by asking the question: Does statistical arbitrage generate higher returns when volatility is high or when it is low? Absent any stock-specic events, higher interstock (spread) volatility should generate greater returns from a well calibrated model. Figure 6.6 shows the average local volatility (20-day moving window) for pairwise spreads for stocks in the S&P 500 index from 1995 through 2003. Two years of outstanding returns for statistical arbitrage were 2000 and 2001. Both were years of record high spread volatility; 2000 higher in spread volatility and statistical arbitrage return than 2001nicely supporting the ceteris paribus answer. But 1999 was the worst year for statistical arbitrage return in a decade while spread volatility was equally high. There were many 0.15 0.10 0.05 199503 199703 199903 200103 200303 FIGURE 6.6 Average local standard deviation of spreads

InterstockVolatility stock-specic events, principally earnings related, with uniformly negative impact on return in 1999. So noticeable, widespread, and troubling were these events that the SEC eventually passed Regulation Fair Disclosure (Reg. FD) to outlaw the activities. Using a local estimate of volatility, what picture is obtained from representative spread series? What can we infer from the spread volatility chart in Figure 6.6 using the sample local volatility reference patterns? Figure 6.7 illustrates local volatility (using an equally weighted, 20-point window) for two sample spread series. The top panel, (a), shows the spread series, the center panel, (b), the local volatility estimates. There is nothing surprising here, the calculation being a measure of variation about a constant line segment of the curves in the top frame. Noteworthy is the observation that the average level of local volatility is similar for the two series.

What happens when a different measure of local is used? The bottom panel, Figure 6.7(c), illustrates the situation for a 60-point window: The striking feature now is the higher level of volatility indicated for the greater amplitude spread. (While we continue to couch the presentation in terms of a spread, the discussion applies equally to any time series.) Once again, there is no surprise here. The 60-point window captures almost a complete cycle of the greater amplitude seriesthe estimated volatility would be constant if precisely a full cycle was capturedand, hence, the local volatility estimate reects the amplitude of the series. In the previous case, the shorter window was reecting only part of the slower moving series variation. Which estimate of volatility reects reversion return opportunity? Here the answer is easy.

Now consider what picture would emerge if an average over a set of such series were examined, each such series mixed with its own noise on both amplitude and frequency. Properly cautioned, what can be inferred from Figure 6.6? Before attempting an answer, the archetypal example analyses clearly advise looking at local volatility estimates from a range of windows (or local weighting schemes)it does seem advisable to concentrate on evidence from shorter intervals and focus on average levels of local volatility; mundane variation in the estimate may be little more than artifact. May be. 109 110