Analysis of financial data Anders Lundquist Spring 2010

Outline, week 3 Volatility Method of moving averages ARCH models Value-at-risk

Volatility Volatility is what we have called dispersion in the beginning of the course. 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. Probably the most frequently used measure of volatility is the standard deviation of an asset’s relative price changes

Time plot – Nordea log-returns

Nordea log-returns

Normality

Numerical summaries Mean Minimum Q1 Median Q3 Maximum -0, , , , , ,14910 StDev Range IQR 0, , ,03732 Skewness Kurtosis 0,55 2,09

Stationarity of mean

Stationarity of variance

Normal distr. as a model? Normal distribution 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. The distribution puts more probability mass in the tails than 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

Volatility Sequence plot indicates some : – Average returns show fairly small changes over time (at least after day 100). – Variances over time more unstable. Accordingly, the stationarity of variance is questionable.

Findings on asset (log-)returns Positive excess kurtosis of daily returns of the market indexes and stocks. The mean of daily returns close to zero. Variance seems to be varying - high for some periods and low for other periods. Variance (standard deviation) at least does not diverge to infinity – in other words, volatility varies within some boundaries. Conclusion: We need the alternative models for asset returns in order to take the varying volatility into account.

There is no common definition of volatility. Depends on the situation at hand – Model (log-returns, Black-Scholes formula for option pricing) – Data (Options, stock prices, interest rates, currency exchange rates etc.) – Sampling strategy (e.g. volatility of returns within a day can not be estimated from daily data; we need within-day data to estimate daily volatility. Furthermore, the volatility of stock returns consists of within-day variation and variation between trading days ). – We will casually ignore the within-day variations…

Volatility is important because (for instance): – It is often used as a risk measure. – When considering a fund’s volatility, an investor may find it difficult to decide which fund will provide the optimal risk-reward combination. Usually, the standard deviation is used as proxy for volatility, which in turn serves as a measure of risk. – Option traders would like to predict the volatility of the price process until the execution, in order to asses their profit opportunity.

Implied volatility – belongs to option pricing theory and is a forward-looking estimate based on a market consensus. – Based on continuous time models for option pricing (e.g. B-S). Historic volatility (Statistical volatility)– is a regular volatility that looks backward. It is based on discrete time models and (or) historical data.

The implied volatility is the estimated volatility of an assets’s price. This is a variable in option-pricing formulas showing the extent to which the return of the underlying asset will fluctuate between now and the option’s expiration. Investors like to focus on the promise of high returns, but they should also ask how much risk they must assume in exchange for these returns. In addition to known factors such as market price, interest rate, expiration date, and strike price, implied volatility is used in calculating an option’s premium. Implied volatility can be derived from a model such as the Black-Scholes Model.

If it is assumed that the option prices are described by a model, such as Black-Scholes, then the volatility (standard deviation) may be estimated from the historically observed prices. Then, the volatility is implied by the model in question. Log-normal assumption for the asset returns used for derivation of implied volatility. Empirical studies show that the implied volatility of an asset’s return may be different from the actual volatility. Based on a model, implied volatility brings only information that is already contained in the option prices.

Often, volatility is calculated as the annualized standard deviation of historical returns. Annual Volatility = (100A)% A is annualizing factor calculated as A =, if there are 252 trading days during the year.

From now on, we will reduce our study to direct modeling of volatility. In other words, we will treat only those methods that are based on historical returns. In particular, the following methods (models) will be described: – Moving Averages method – ARCH models

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!

MA 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.

MA, 3-day As an example, we define a 3-day moving average, MA(3)

General MA A m-day Moving average, MA(m)

Nordea returns, and a 10-day MA

MA(10) vs. MA(30)

The smaller m is, the more variability of the MA is observed. Correspondingly, 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) Natural question: What can the moving averages be used for?

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 ?

We have earlier seen that the best linear 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, this means that = +  1 Y t +  2 Y t−1 + · · · +  p Y t−p. When  1 = · · · =  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 is not a good idea to use the moving averages for prediction

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 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 stationarity of the variance of asset returns is questionable. The moving averages are therefore used in this context. One objective is to “smooth” out the irregular components of asset returns in order to allow us to see a clearer picture of the underlying regularities in a time series (if there are any…).

Recall that we have defined the sample variance of the returns as We have seen that the average returns y are often very close to zero. So if we assume that the 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 square returns, t = 1, 2, 3,.... Instead of taking the average over all the observed returns we can compute a moving average of square returns:

Then in the same way as before, we can make plots of the moving averages of squared returns. First the squared log-returns only

MA(10) and MA(60) of squared log- returns

We have seen that: The moving average of the returns can be useful to predict Y t+1 (or equivalently estimating E(Y t+1 )) when the mean of the returns μ is a function of time (non-stationarity in the mean). Similarly, the moving average of the squared returns can be useful to predict the variance of Y t+1 (estimate Var (Y t+1 )) if the variance  2 is a function of the time t. Changing variance is often observed in the data as groups (clusters) of different variability/volatility.

If the volatility is stable/stationary through time, then the moving average estimates obtained with different values of m will be similar (at least for large enough m). A large 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 the volatility does vary over time.

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.

An improvement of the naive moving average estimates of the volatility are obtained by weighting the squared returns differently: current squared returns should be weighted stronger than squared returns observed long ago. Exponentially weighted moving average estimates of the volatility are constructed as:

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.

EWMA, Nordea squared log-returns  = 0.8 (red), 0.94 (black)

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.

There are two terms: determines the reaction of the volatility estimate to the last observed market event (return). The smaller is, the stronger the influence of the latest observation on the volatility estimate. determines the persistence of historical volatility: when  is large, a large volatility yesterday implies a large volatility today independently of what does happen today.

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 for long range predictions and smaller for short range predictions. Again: choice of somewhat arbitrary…

RiskMetrics RiskMetrics™ data are produced by J.P. Morgan ( The data consist of volatility estimates of returns of many financial assets, such as exchange rates, interest reates, 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.

Often an investor is interested in the h−day 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 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 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…).

Then the same estimator is used as prediction for all the one-day return variances giving Var (Y t+h |Y t, Y t-1,...) = I 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 returns, to which the EWMA estimator is applied.

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 will be discussed next …