1. Analysis of Financial Data Spring 2012 Lecture 9: Volatility Priyantha Wijayatunga Department of Statistics, Umeå University Priyantha.wijayatunga@stat.umu.se Course homepage: http://www8.stat.umu.se/kursweb/vt012/staa2st017mom2/ http://www8.stat.umu.se/kursweb/vt012/staa2st017mom2/

2. Volatility Volatility is about how much the time series changes. Measures of volatility are measures of dispersion. A slightly more statistical definition of volatility: A measure of the tendency of a market or an asset to vary within a time period. The most frequently used measure of volatility is the standard deviation of an asset’s relative price changes Sequence plot indicates: –Average returns show fairly small changes over time (at least after day 100). –Variances over time more unstable. Accordingly, constant of variance is questionable.

3. Volatility of logreturns Kurtosis is less than 3 and skewness is close to zero Statistics DIFF(logPt,1) NValid253 Missing1 Mean-,0012 Std. Deviation,02528 Skewness,191 Std. Error of Skewness,153 Kurtosis2,227 Std. Error of Kurtosis,305 Percentiles25-,0135 50,0000 75,0122

4. Normal Distribution for Volatility Normal distribution is not completely satisfactory as a model for log-returns (in many cases): Distribution of asset returns commonly have “long”/“fat”/”heavy” tails – i.e. positive excess kurtosis. We need distributions that put more probability mass in the tails than that the normal distribution does. Practical consequence: random samples from such a distribution probably contain more “extreme” values (interesting from an investors’ point of view – since log-returns are considered as a random sample from the model distribution) More extreme values – can make (or loose…) more money

5. Models and Methods for Volatility Direct modeling of volatility: treat only those methods that are based on historical (log)returns. In particular, the following methods and models are used: –Moving Averages method –ARCH models

6. Moving Average Method Used for short-term prediction The basic idea behind moving averages: Any large irregular component at any point in time will exert a smaller effect if the observation at that point is averaged by its immediate neighbors. In classical time series analysis, moving averages are used for extraction of the seasonal components. Note: The moving averages are not models! A moving average for some time period is the arithmetic mean of the values in that time period and those close to it. Moving averages have many applications in finance. For instance, technical analysis make use of moving averages (calls them stochastics ) of returns to build signals to trade. see e.g. http://en.wikipedia.org/wiki/Technical_analysishttp://en.wikipedia.org/wiki/Technical_analysis

7. 3-day MA and General MA As an example, we define a 3-day moving average, MA(3) A general m-day Moving average, MA(m)

8. 3-day MA and 7-day MA

9. MA(m) The smaller the m is, the more variability of the MA is observed. Less smoothing in the time series The larger the time span, the smoother the plot. In other words, the variability of the MA may be adjusted by choosing m. (often the choice is somewhat arbitrary) What can the best moving average? We would like to estimate the future value of returns. We are also interested in estimation of volatility (variance or standard deviation). When is it appropriate to forecast the value of Y t + 1 by using a moving average ?

10. Linear Predictors The best predictor of Y t+1 given Y t =y t, Y t-1 =y t−1,..., is = E(Y t+1 |Y t,Y t−1,...). With a linear model, the best linear predictor = b 0 + b 1 Y t + b 2 Y t−1 + · · · + b p Y t−p. When b 1 = · · · = b p = 0, we have = E(Y t+1 ) = μ. This means that in this case we should use as predictor for Y t+1. In this case, it may not be a good idea to use the moving averages for prediction

11. However, the best predictor above is valid only under the assumption of stationarity When stationarity does not hold it may be preferable to use the moving average method as a simple predictor for Y t+1, e.g. because the moving average is not a model, and thus does not make any stationarity assumption MA:s are very simple but in a way also quite crude, and you need to choose m, which in principle is arbitrary… According to the various empirical studies, the constant the variance of asset log returns is questionable!! The moving averages are therefore used in this context. One objective is to “smooth” out the irregular components of asset log returns in order to allow us to see a clearer picture of the underlying regularities in a time series (if there are any…). Linear Predictors

12. MA of Squared Logreturns Recall that we have defined the sample variance of the returns as We have seen that the average log returns are often very close to zero. So if we assume that the log returns have zero (sample) mean we obtain a simpler sample variance Note that s 2 in the second expression is just an average: it is the average of the squared log returns, t = 1, 2, 3,.... Instead of taking the average over all the observed log returns we can compute a moving average of squared log returns:

13. Then in the same way as before, we can make plots of the moving averages of squared returns. Squared log-returns Their moving averages MA of Squared Logreturns

14. Predicting Time Series We have seen that: The moving average of the log returns can be useful to predict next log return Y t+1 (or equivalently estimating E(Y t+1 |y t, y t-1,…) when the conditional mean of the log returns is a function of time (non-stationarity in the mean). Similarly, the moving average of the squared log returns can be useful to predict the conditional variance of Y t+1 given y t, y t-1,… (an estimate of Var(Y t+1 |y t,y t-1,…)) if it is a function of the time t. Changing variance is often observed in the data as groups (clusters) of different variability/volatility.

15. Moving Averages If the volatility is stable through time, then the moving average estimates obtained with different values of m will be similar (at least for large enough m). A large log return has a long effect on the volatility estimate if m is large, and a short effect if m is small. After m days the effect of a large return disappears. Moving averages are quite naive estimators of the volatility and are probably best used for descriptive purposes. For instance, to illustrate that how the volatility varies over time.

16. However, moving average estimates are sometimes used as predictions of future volatility. Then, a rule of thumb is: long term prediction of volatility should be based on large window moving average estimate ( large m) and short term prediction of volatility should be based on narrow windows ( small m). How small is a good question and a critical issue! It will depend on how often one believes that the variance is changing through time. Moving Averages

17. An improvement of the naive moving average estimates of the volatility are obtained by weighting the squared log returns differently: recent squared log returns should be weighted stronger than squared log returns observed earlier to them. Exponentially weighted moving average estimates of the volatility are constructed as: Improved Moving Averages

18. For large t, the sum of the weights can be shown to be close to one. We have then a weighted average of the squared returns. EWMA estimates are based on a weighted average of the whole history of squared returns in contrast with MA which are based on a “window” of chosen size. Exponentially Weighted Moving Averages

19. EWMA: squared log-returns

20. EWMA We may rewrite a formula for the EWMA estimate as This formula is useful for recursive computation and to understand to what extent the current observation is influencing the EWMA compared to the past. determines the reaction of the volatility estimate to the last observed market event (log return). The smaller λ is, the stronger the influence of the latest observation on the volatility estimate. determines the persistence of historical volatility: when λ is large, a large volatility yesterday implies a large volatility today independently of what does happen today.

21. EWMA Estimates The EWMA estimates of the volatility are commonly used as predictors of tomorrow’s volatility, or for longer horizons. Use the following rule of thumb: larger λ for long range predictions and smaller λ for short range predictions. Again: choice of λ somewhat arbitrary…

22. RiskMetrics RiskMetrics™ data are produced by J.P. Morgan (www.riskmetrics.com). The data consist of volatility estimates of returns of many financial assets, such as exchange rates, interest rates, equity indices and some key commodities. RiskMetrics data are widely used to evaluate risks in the financial markets. The volatilities are estimated with EWMA method using = 0.94 for all assets.

23. h–day Return Often an investor is interested in the h−day log return (and the variance of that), that is a prediction of Var(Y t+h |Y t,Y t−1,...) where Y t+h = logX t+h − logX t. Notice that Y t+h = Y t+1 + · · · + Y t+h−1. Assuming that the log returns are independent (market efficiency) we can write Var (Y t+h |Y t, Y t-1,...) = Var (Y t+1 |Y t, Y t-1,...) + · · · + Var (Y t+h−1 |Y t, Y t-1,...). Hence, a prediction of Var (Y t+h |Y t, Y t-1,...) can be obtained by predicting the variances of the one-day log returns on the right hand side. It is common practice to use EWMA to predict all these one- day returns, using the same (although there are probably better ways…).

24. Then the same estimator is used as prediction for all the one-day log return variances giving Var (Y t+h |Y t, Y t-1,...) = Thus, the corresponding standard deviation is StDev(Y t+h |Y t, Y t-1,...) = This is called the ”square root of time rule”. Suggested improvement: if enough data is available - work directly with h−day log returns, to which the EWMA estimator is applied. h–day Return

25. MA Methods Moving average and EWMA estimates are not model based. They are methods which try to track the changes in volatility observed in the returns. Another way to tackle the problem of estimating and forecasting volatilities is to build models for the conditional variance, Var (Y t |Y t-1,Y t-2,...), in the same way that we did for the conditional expectation, E(Y t |Y t-1,Y t-2,...) (linear autoregressions). Such models for the Conditional Variance