The prediction of periods of high volatility in exchange markets

Eur. Phys. J. B 20, 581–584 (2001) THE EUROPEAN PHYSICAL JOURNAL B EDP Sciences c Società Italiana di Fisica Springer-Verlag 2001 The prediction of periods of high volatility in exchange markets C. Windsora and A. Thyagaraja UKAEA Fusion, D3, Culham Science Centre, OX14 3DB, UK Received 11 October 2000 Abstract. A statistical connection is identified between the current spread in a market over a given time period and the drift of the market during previous time periods. It is shown that periods of high spread are likely to be preceded by periods with relatively large market drifts. Several markets, including the UK pound per US Dollar, US Dollar per Yen, UK pound per Euro, and the UK FT100 index have been analysed from 1991 to 2000 over variable periods of weeks, months and quarters. Within each period, i the natural logarithm of the daily end-of-trade market value has been least squares fitted to a linear regression line, and evaluations made of the regression line slope µi , the direct spread si with respect to the mean value, and the regression spread ri of the deviations from the regression line. Significant correlations have been observed between the current monthly direct spread si for each period i and the absolute value of the drifts |µi−j | evaluated j periods earlier. This correlation coefficient is as high as 0.746 for a period of one quarter (j = 1) and appears to die away after around 9 months for quarterly averages, after around 4 months for monthly averages and after around 2 months for weekly averages. PACS. 89.65.Gh Economics, business, and financial markets 1 Introduction In many dealing situations it is desirable to perform trades at a time when the current market volatility is relatively low. The volatility of an index may be defined as the standard deviation, σ, of the natural logarithm of the index changes over a given time period. This definition is linked to the random walk model of index movements, with its quasi-Gaussian distribution of the daily market changes. However volatility is usually quoted on an annual basis, so that the standard deviation of thep market changes p over any period of Ndays trading days is σ Ndays / Nyear where Nyear = 261 is the mean number of trading days in a year. Our paper deals with periods of weeks, months and quarters. Within each period i, the daily end-of-trade market values were least-squares fitted to a linear regression line as illustrated in Figure 1, and estimates made of the root-mean-square deviation from the mean value which we shall call the direct spread si , the deviation from the regression line which we shall call the regression spread ri , and the slope of the regression line µi . Within the random walk model, the mean values of both spreads are related to the volatility. This relationship with volatility was investigated by simulating many long random walks with a given value of σ and zero overall drift, evaluating the spreads and slope within each period, and averaging them to fine the mean. For exama e-mail: [email protected] ple the mean direct spread appears close to the value σ(Ndays /2πNyear )1/2 . The regression spread appears to be about 2/3 of this value. The magnitude of the slope is correlated with the direct spread. However with simulated random-walks there is no significant correlation either between the magnitude of the slope and the regression spread in any one period, or between either spread and values of the slope in previous periods. Such correlations are shown in real data, and can be used to help estimate current volatility. Figure 1 illustrates the monthly averages for the UK Pounds per Euro rate. 2 Market drift, direct spread and regression spread Detailed investigations have been made for several exchange markets, including the GBP per USD, GBP per EUR and USD per JPY. Other markets such as the UK, US, EU, Swiss and Japanese interest rates and the Dow Jones, FTSE 100, Nikkei and DAX indices have also been analysed. Indices from 1991 to 2000 have been converted into natural logarithms and normalised on 1/1/00. The process of conversion to natural logarithms means that spreads and drifts from any market index are equal in percentage terms and so are directly comparable. They have then been averaged over variable periods of weeks, months, quarters and years. Figure 2 shows the UK Pounds per US Dollar exchange rate as a function of time over monthly 582 The European Physical Journal B 0.06 loge(UK Pounds per Euro) 0.05 0.04 0.03 0.02 0.01 0.00 98.0 98.2 98.4 98.6 98.8 99.0 99.2 99.4 99.6 99.8 100.0 Decimal Year Fig. 1. The natural logarithm of the exchange index (feint) giving the UK pounds per Euro, normalised on 1/1/00, and converted from the German Mark at 1 EUR = 1.95583 DEM before 1/1/99. The daily values within each monthly period have been fitted to a linear regression line (full) together with a vertical line denoting the spread with respect to the regression line. 0.008 Spread Drift/3 Average Spread Average Drift 0.007 Spread or drift/3 0.006 0.005 0.004 0.003 0.002 0.001 0 95 95.5 96 96.5 97 97.5 98 Decimal Year 98.5 99 99.5 100 Fig. 2. The monthly regression spreads ri (dark circles) and drifts |µi | (feint crosses) of the exchange index between the UK pound and US dollar. Average values defined later in the text are shown by continuous and dashed lines respectively. Around 1997 a period of generally high spread is preceded by a period of relatively high drift. periods from 1995 to 2000. There is much scatter in the individual monthly values, but the weighted average values, detailed later and shown by the lines, show significant correlations. Most usefully for prediction purposes, there is some indication that periods of high spread are preceded by periods of high drift. Thus around 1997 in the centre of the figure, the dashed grey line showing the average drift rises rapidly around half a year previous to the rise in the average volatility. The same effect is present in the monthly data points. 3 The correlations between historic regression drift and spread For the purpose of predicting spread it is appropriate to make scatter plots of the direct spreads si or regression spreads ri during a given period i against the absolute value of the drift |µi−j |, as it was j periods earlier. Figure 3 shows the direct spread si against |µi−j |, for the GBP per USD index, but this time for quarterly periods, with a prior period of j = 1 quarter. In fact for any C. Windsor and A. Thyagaraja: The prediction of periods of high volatility in exchange markets 583 Table 1. The parameters of the fitted linear relationship between the current regression period’s direct spread si or its regression spread ri and the drift µi−j over the prior monthly or quarterly period j periods before. The correlation coefficient Cs or Cr would be 1 for a perfect positive correlation and 0 for a random distribution. The predictive ratios (Rs or Rr ) is 1 minus the spread measured from the regression line ri divided by the overall direct spread soo or regression spread roo . Index Prior Direct Period Slope Ratio Corr Slope Ratio Corr (Ms ) (Rs ) (Cs ) (Mr ) (Rr ) (Cr ) 1 quarter 4.915 0.170 0.747 3.310 0.130 0.702 2 quarters 4.549 0.147 0.723 2.668 0.090 0.643 3 quarters 5.248 0.201 0.776 3.661 0.177 0.557 4 quarters 3.115 0.067 0.600 1.938 0.047 0.551 5 quarters 2.232 0.033 0.505 1.788 0.039 0.526 6 quarters 1.665 0.023 0.463 1.908 0.056 0.576 1 month 0.972 0.087 0.639 0.390 0.068 0.601 2 months 0.186 0.004 0.279 0.146 0.009 0.368 3 months 0.445 0.018 0.433 0.318 0.045 0.544 4 months 0.331 0.010 0.377 0.345 0.053 0.566 5 months 0.132 0.002 0.236 0.215 0.020 0.447 6 months 0.297 0.008 0.354 0.148 0.009 0.372 USD perYEN 1 quarter −1.168 0.017 −0.142 0.174 0.002 0.248 GBP per USD Regression GBP per EUR 1 quarter −1.385 0.020 −0.095 0.241 0.002 0.246 USD per YEN 1 month 0.463 0.019 0.439 0.674 0.079 0.625 GBP per EUR 1 month 0.405 0.061 0.585 0.284 0.007 0.347 0.020 0.018 Current quarterly spread 0.016 0.014 0.012 0.010 0.008 0.006 0.004 USD/GBP 0.002 0.000 0.00 0.05 0.10 0.15 0.20 0.25 Previous quarterly drift Fig. 3. The correlation between quarterly direct spreads si for a particular quarter and the drifts |µi−j | for the GBP/USD exchange index measured one quarter earlier j = 1. The line shows the least squares fit. The parameters of the least squares fitted lines and the correlation coefficients for several times j, periods and indices are given in Table 1. period j > 0 the degree of correlation is similar no matter whether the direct spread si or the regression spread ri is used. This is useful in that the direct spread is most directly related to the volatility. The exact results are sensitive to the starting date within the correlation period, but appear to be significant for all starting dates. In most cases there appears to be a significant linear relation- ship between spread and prior drift so that a regression fit may again be made as shown, and estimates made of the direct slope Ms , regression slope Mr and the correlation coefficients Cs or Cr . The results for several indices and prior periods are shown in Table 1. It is seen that as the prior period increases, the slope tends to drop, along with the correlation coefficient. The predictive power 584 The European Physical Journal B 0.25 0.20 Predictive ratio (R) Quarters 0.15 0.10 Months 0.05 Weeks 0.00 1 2 3 4 5 6 Number of prior periods Fig. 4. The predictive ratio (Rs ) given by the spread r from the regression line between the overall spread si for a particular period and drifts |µi−j | for j periods previous, expressed as a fraction of the overall spread soo , and plotted against the number of prior periods j. of the correlations depends on the difference between the spread from the regression line rs or rr and the overall spread soo , or roo . We define predictive ratios Rs = 1 − sr /soo and Rr = 1 − rr /roo . Although there is considerable scatter, the regression spread rr generally rises with the number of prior periods j towards the overall spread roo , so that the predictive ratios Rs and Rr move towards zero. 4 The influence of the prior time period in predicting future spread Figure 4 shows the predictive ratio (Rs ) plotted against the number of prior periods for periods of quarters, months and weeks. The predictive ratio appears to drop off significantly as the averaging period decreases. However some of the statistical information can be recovered by combining the drift data from many previous periods. A simple method of allowing for this extended correlation is to define an exponential average drift over a defined number of periods. It suggests that for quarterly periods, a simple average over 3 quarters is appropriate giving a predictive ratio, enhanced over that seen in any one quarter, Rs = 0.28. For monthly periods an average over 8 months with a weighting factor of each month over the previous one of 0.7 gives the much lower but still significant predictive ratio Rs = 0.073. The statistics for weekly periods are very poor, with all periods having a correlation coefficient below 0.5 and a predictive ratio below 0.03. However an average over 52 weeks with a weighting factor of 0.89 gives the slightly lower predictive ratio Rs = 0.06. More advanced statistical approaches, such as principal component analysis could improve these results. 5 Conclusions There is a significant correlation between both the direct spread si of the market values measured with respect to the mean measured over a period i, and the regression spread ri with respect to a regression line, to the absolute value of the drifts |µi−j | measured j prior periods previously. In general periods of high volatility are preceded by periods of high market drift, either upward or downward. After any appreciable market drift, the future market will be characterised by increased uncertainty while the price settles leading to an enhanced volatility. The correlations are more marked in non-trending markets, such as the GBP per USD exchange rate, than in say the USD per JPY rate. The correlations are strongest when longer periods, such as quarters, are used when the volatility estimate may be improved by as much as 20%. However the predictive power from shorter periods may be enhanced by suitable averaging. The authors are most grateful to Jessica James from Bank 1 for supplying the data used in the analysis and for invaluable discussions. The UKAEA authors acknowledge the support from the UK Department of Trade and Industry. References 1. M.T. Leung, H. Daouk, A. Chen, Int. J. Forcasting, 16, 173 (2000).