Volatility Forecasts for Option Valuations

Transcription

1 Volatility Forecasts for Option Valuations Louis H. Ederington University of Oklahoma Wei Guan University of South Florida St. Petersburg July 2005 Contact Info: Louis Ederington: Finance Division, Michael F. Price College of Business, University of Oklahoma, 205A Adams Hall, Norman, OK 73019, USA. Phone: (405) Wei Guan: College of Business, University of South Florida St. Petersburg, 140 Seventh Avenue, St. Petersburg, FL 33701, USA. Phone: (727) Comments may be sent to the authors at or

2 Abstract Volatility Forecasts for Option Valuations We document several problems with GARCH type model predictions over the multi-day horizons common to option valuations and value-at-risk models. One, GARCH model forecasts of the return standard deviation - the most common volatility measure and the most appropriate for option valuation and value-at-risk models - are positively biased. Two, the bias is especially severe following high volatility days. Three, in forecasting volatility over longer horizons, the GARCH model puts too little weight on older observations relative to the more recent observations. That is older observations are more important in forecasting volatility next month than in forecasting volatility tomorrow while the GARCH procedure treats them equally at both horizons. We present a simple unbiased regression estimator of the standard deviation of returns which avoids these problems. We find it forecasts better out-of-sample than GARCH, EGARCH, and historical volatility across a wide variety of markets and forecast horizons.

3 Volatility Forecasts for Option Valuations I. Introduction This paper explores the problems that arise when time-series models estimated from daily (or higher frequency) data, the GARCH(1,1) model in particular, are used to forecast volatility over the longer horizons common to option valuations and longer term value-at-risk measures and shows that a simple regression based alternative forecasts better than the main extant forecasting models over these longer horizons. In addition to time-series model forecasts, implied volatilities calculated from option prices provide volatility forecasts over multi-day horizons and theoretically should reflect all available information, including time-series information. However much evidence indicates that this is not 1 the case. Moreover implied volatilities cannot simultaneously be used to price the derivatives from whose prices they are calculated and are only available for specific time horizons. Consequently, time-series models remain the major source of volatility forecasts. While a large number of time-series volatility forecasting models have been proposed in the financial econometrics literature, in practice three dominate: (1) the GARCH(1,1) model, (2) the exponentially weighted moving average model employed by Riskmetrics, and (3) the historical variance or standard deviation. Of these, we focus particular attention on the properties of long-run GARCH forecasts. Since the Riskmetrics model may be viewed as a special case of the GARCH(1,1) model without mean reversion and with fixed coefficients, most of our findings for GARCH apply to this model also. Compared with the GARCH model, the historical standard deviation would appear decidedly inferior since, despite the overwhelming evidence of volatility persistence, it (1) gives equal weight to old and recent observations in forecasting future volatility tomorrow, (2) imposes an arbitrary cutoff on the past data considered, and (3) ignores mean reversion. However, studies comparing the forecasting ability of the two are roughly split on 2 whether GARCH or the historical standard deviation predicts better. Our findings help explain why this is the case.

4 While GARCH models generate volatility forecasts for the very next period or observation (normally the next day), uses in option pricing (and longer horizon value-at-risk measures) commonly require volatility forecasts for much longer periods of weeks, months, or even years. These are generally obtained by successive forward substitution. That is the volatility forecast for day t+1 is used together with the GARCH parameters to forecast volatility on day t+2, the forecast for day t+2 is used to forecast volatility for day t+3, etc.. These are then summed to obtain forecast volatility for the period from t+1 through t+n, sometimes referred to as integrated volatility 3 Andersen et al (2005). Most evaluations of GARCH models in the econometrics literature have focused on their ability to forecast volatility at short horizons (generally the next period). As Christoffersen and Diebold (2000) have observed, much less is known about volatility forecastability at longer horizons. An exception to this statement are explorations of the long memory property of many financial series, e.g., Ding, Granger, and Engle (1993), Ding and Granger (1996) and Andersen and Bollerslev (1997) among others. These studies find that absolute and squared returns are more correlated at long lags than exponential models, such as GARCH(1,1), predict they should be - leading to the development of alternatives, such as FIGARCH. Exploring the properties of multi-day volatility forecasts generated from daily data for twelve different markets, we document two other - and we would argue more serious - problems with GARCH(1,1) forecasts over longer horizons. One, GARCH forecasts of volatility over longer periods are biased upward. The bias is more serious when volatility is measured in terms of the standard deviation of returns - the most common measure and the most appropriate for option valuation and value-at-risk models. Whether volatility is measured in terms of the standard deviation or variance, the bias is most severe when GARCH predicts considerably higher than normal volatility. We find that this bias arises because, since GARCH forecasts are a function of past squared return observations, a single extremely volatile day leads to a big jump in the GARCH forecast of future volatility which is rarely fully realized. 2

5 Two, GARCH(1,1) (like many other volatility forecasting models) places the same relative weights on returns on days t and t-1 in forecasting volatility on day t+s whether s is 1 (tomorrow) or 20 (next month). In other words, in the GARCH procedure, today s volatility receives the same weighting relative to volatility yesterday or last week in forecasting volatility a month from now as 4 it does in forecasting volatility tomorrow. We show that the parameters which best forecast volatility over longer periods vary with the forecast horizon. Specifically, while today s return may be considerably more important than yesterday s or last week s in forecasting volatility tomorrow, the difference in relative importance is much less in forecasting volatility over the next month. While evidence of bias and inappropriate weightings of older versus recent observations is presented for GARCH(1,1), we expect the same criticisms to apply to most GARCH class models and to Riskmetric s exponentially weighted moving average model since most share the 5 characteristics which lead to bias and inaccurate forecasts for GARCH(1,1). We find that the simple, non-linear regression model suggested in Ederington and Guan (2005a) alleviates these problems and forecasts volatility better than the GARCH(1,1) model, EGARCH and historical volatility over a large variety of markets and horizons on an out-of-sample basis. This model is structurally similar to the GARCH(1,1) model in that the weights attached to past returns decline exponentially but is based on absolute rather than squared return observations and allows parameter values to vary with the forecast horizon. Our data set consists of daily data on one equity index market (the S&P 500), two interest rates (3 month T-Bills and 1 year T-Notes), two exchange rates (Yen/Dollar and Dollar/Pound), two commodities (crude oil and gold), and five individual equities from the Dow-Jones index (Caterpillar, Disney, Dupont, GE, and Walmart). The paper is organized as follows. The next section briefly reviews the volatility forecasting literature with emphasis on long horizon forecasts and the GARCH(1,1) model. The issue of how the relative weights attached to recent and older observations vary or should vary with the forecast horizon is explored in Section III with results presented in Section IV. The bias in GARCH 3

6 forecasts of volatility over multi-day horizons is documented in V. In section VI, we show that this bias is due primarily to the substantial impact of extremely volatile days on the GARCH forecast. In VII, we present a simple unbiased alternative which allows relative weights on past observations to vary with the forecast horizon. In section VIII, we demonstrate that our model forecasts multi-day volatility better than GARCH(1,1), EGARCH, and historical volatility. Section IX concludes the paper. II. GARCH Volatility Forecasts Over Multi-period Horizons Despite criticism and a surfeit of suggested alternatives, the GARCH(1,1) model remains the workhorse of volatility forecasting - along with the historical standard deviation. It is the benchmark against which each new volatility forecasting model is measured and the only model aside from the historical standard deviation to make its way into derivatives textbooks. The model is: (1) where r t is the surprise log return (i.e., r t=r-e t-1(r t), where R t=ln(p t/p t-1) and P t is the asset price at time t) and v is the variance of r. t 6 t Many uses of volatility forecasts, such as option valuation, require volatility estimates over a much longer horizon than that from t to t+1. Typically the GARCH model is estimated using daily data, so t+1 represents the next day, but for option valuation purposes, what is required is a volatility estimate over the life of the option which may expire months in the future. To generate the required long-horizon volatility forecast, estimated forecast volatilities in subsequent periods (normally days so we will use this term henceforth) out to the expiration date are generated by leading equation 1 2 and substituting forecast variances on the right hand side. Since E t(r t+1 ) = v t+1, successive forward substitution yields the expression for the expected volatility at time t+k based on the forecast for t+1: 4

7 (2) While v t+1 and v t+k are point volatility estimates, option valuation requires a volatility forecast for the 7 entire period until the option expires, not a single day. To generate such forecasts, it is normally assumed that returns are independent so the forecast variance for the entire period (sometimes referred to as integrated volatility ) is obtained by summing the forecast variances for each day. 8 Let V t+s represent average volatility over the period from t to t+s, i.e.,. Summing equation 2 from k=1 to s and dividing by s yields the forecast average volatility over the future period from t+1 through t+s, V t+s: (3) where. Several criticisms have been levied against the GARCH(1,1) model. One is that unlike alternatives, such as EGARCH, it treats positive and negative shocks equally while there is evidence 9 (particularly in stock markets) that negative shocks are more persistent. A second is the usual assumption for estimation that log-returns are normally distributed while fat tails are the norm. Most relevant to our paper is the criticism that the GARCH(1,1) model s memory is too short, e.g., Engle and Bollerslev (1986), Ding and Granger (1996), Baille, Bollerslev, and Mikkelsen (1996), and Bollerslev and Mikkelsen (1996). From equation 2 it follows that. If 1 + is considerably less than 1.0, this impact rapidly approaches zero while there is evidence that its impact lasts longer. III. Forecast Horizon and the Relative Importance of Past Observations The first issue we consider is how the relative importance of recent versus older observations in predicting future volatility depends on the forecast horizon. Suppose at the end of trading on a 5

8 Tuesday, you are forecasting volatility for (1) tomorrow (Wednesday), and (2) Wednesday a week or month forward. Given what we know about volatility persistence, today s volatility should be more important than Monday s or last Friday s in predicting tomorrow s volatility. But is it much more important than Monday s or Friday s volatility in predicting volatility a week or month forward? In the GARCH model, the relative importance of past observations is the same whether forecasting volatility for tomorrow, a week, month, or year hence while we hypothesize that differences in relative importance between past observations should decline as the forecast horizon lengthens. 2 Consider how GARCH forecasts depend on past realizations of r. Since,, and successive substitution back to time t-j yields the alternative expression of the GARCH(1,1) model in equation 1: (4) where. As equation 4 makes clear, in the GARCH(1,1) forecast of v t+1, j the squared return deviation at time t-j receives a weight of 1 and (assuming <1) the weights decline exponentially. Substituting equation 4 into equation 2, yields: (5) As equation 5 makes clear, while the absolute weights differ with the horizon k, in the GARCH(1,1) model, the relative weights on past squared surprise returns decline exponentially whether 2 k-1 j forecasting volatility for tomorrow or for the distant future. Since v t+k / r t-j = 1( 1+ ), the relative partials for successive past observations are: 6

9 (6) So the relative weights assigned to successive past observations are in the ratio regardless of the forecast horizon, k, and how far in the past, j. It is instructive to note that the weights decline at quite different exponential weights going backward and forward. Considering the impact of a single past squared return shock, r forward volatility forecasts m days apart: 2 t-j, on (7) Considering the impact of observations m days apart on the volatility forecast for the same future day: (8) While the expression in equation 7 is the well-known short memory property, our concern is with the expression in equation 8. Note that the weights going backward (equation 8) decline much faster than those going forward (equation 7). For instance, for daily observations on the S&P 500 index from 1/3/1968 to 12/31/2002, GARCH estimates of 1 and are.0690 and.9228 respectively 10 so ( 1+ ) =.9210 and the reversion of forward forecasts to the mean is fairly slow. At the same 10 2 time, =.4478 so r t-10 (that is an observation 10 days or two weeks ago) receives a weight only % of that attached to r t in forecasting any future volatility. The same results are obtained for integrated volatility. Summing equation 5 from k=1 to s and dividing by s yields the forecast average volatility from t+1 through t+s, V t+s: (9) 7

10 where (10) It is easily seen from equation 9, that the relative impact of r and r on the integrated m volatility from t to t+s is in the same ratio. That is: 2 2 t-j t-j-m (11) If the weight attached to r is X, the weight attached to r is X irrespective of the horizon s. 2 2 m 10 t t-m We hypothesize that the GARCH parameter estimates which maximize the likelihood of generating observed returns for the very next period, e.g., the next day if the model is estimated from daily data, do not forecast volatility very well over longer horizons. Specifically, we hypothesize that better long horizon forecasts are obtained if equation 9 is altered to allow to vary with the forecast horizon s: (12) We further hypothesize that the s which yields the best volatility forecast will increase with the horizon s so that: (13) The basic idea is simple and intuitive. Suppose at the end of trading on Tuesday we want to forecast volatility for both tomorrow (Wednesday) and a week hence. Among other information, we know volatility today (Tuesday) and Tuesday a week ago. For forecasting tomorrow s volatility, today s volatility is likely much more important than volatility a week ago. For forecasting volatility next Wednesday (a week from tomorrow), we expect the difference in the importance of these two past volatilities to be smaller. We hypothesize that as the forecast horizon lengthens, better volatility 8

11 forecasts are obtained by decreasing the weights on the most recent observations while increasing the weights on older observations relative to the parameter estimates generated by the GARCH procedure. If correct, our argument could explain a surprising finding in past studies. The most common volatility alternative to GARCH for option valuation purposes (and the model presented most often in derivatives texts) is the historical variance or standard deviation. Conceptually, the historical variance would appear decidedly inferior to the GARCH(1,1) model in that it weights all included past squared surprise returns, r 2 t-j, equally (i.e., =1) instead of attaching more weight to more recent observations ( <1). Also it imposes an arbitrary cutoff, J, on how many past returns are included; that is all observations after t-j receive the same weight and all before t-j receive a zero weight. It is hard to believe that the informativeness of the return on day t-j is equal to that of day t while there is no information in knowing the return on day t-j-1. Yet many studies find that historical volatility forecasts future volatility better than GARCH(1,1). In a survey of 39 such studies, Poon and Granger (2003) report that 22 find that historical volatility (including however weighted measures) forecasts actual volatility better while 17 find that GARCH forecasts better. Our hypothesis provides a possible explanation. If the GARCH(1,1) estimate of is too low when forecasting far beyond t+1, then the historical variance imposition of =1 may be better at long horizons. To explore whether volatility forecasts can be improved by varying with the forecast horizon, we use OLS regressions to obtain estimates of s (also s and s) in equation 12. Letting AV(s) t represent the actual (or ex post) realized variance over the period from t+1 through t+s. i.e.,, equation 12 is estimated by applying OLS to the equation pair: (14) 9

12 Since OLS minimizes the sum of the squared residuals, this procedure finds the equation parameter estimates which minimize the in-sample root mean squared volatility forecast errors. Thus we can test whether the parameters which yield the best linear unbiased estimates of the variance (assuming the classical least squares assumptions are met) differ from the GARCH parameter estimates and vary with the forecast horizon. One advantage of multi-period volatility forecasts, is that the ex-post variance of returns, AV(s) t, for a multi-period horizon is much less noisy than that for a single period allowing a regression estimation. IV. Results: The Forecast Horizon and Parameter Estimates. To examine how the parameters in equation 12 which yield the best volatility forecasts estimates (in terms of root mean squared errors of the in-sample variance forecasts) vary with the forecast horizon s and how these compare with the GARCH parameter estimates of the same equation, equations 1 and 14 are estimated using daily data for our twelve financial series. Data sources, time periods, and descriptive statistics are reported in Table 1. Daily log returns are defined as R t = ln(p t/p t-1) and daily return deviations or surprises are defined as r t = Rt- where is measured as the mean of R t over the entire data period. GARCH(1,1) and regression estimates of the equation 12 parameters are reported in Table 2 for horizons s of 10, 20, 40, 80, and 160 days. We are particularly interested in Panel A where estimates of s are reported. As shown there, the results confirm our argument that better volatility forecasts are obtained by increasing the relative weight on older versus more recent observations as the forecast horizon lengthens. With the single exception of the S&P500 index at the 10 day horizon, the values for beta which minimize the sum of squared differences between actual and forecast variances are all greater than the GARCH(1,1) estimates. Moreover, in 40 of the 48 pairs in Table 2, the regression estimates of s increase as the horizon s lengthens. The implied differences in the weights attached to older observations are substantial. 2 Consider, for instance, the weight attached to r t-20 (or approximately one month ago) relative to that 10

13 2 attached to today s volatility, r t,in generating forecasts for volatility over the period from t+1 to t+s. For the GARCH model, the average estimated over our twelve markets is.9085 which translates 2 2 into a relative weight on r t-20 of.147 or less than one sixth the weight given r t. This is the same regardless of the forecast horizon. By comparison, at the ten day horizon, the average OLS estimate 2 of s over our twelve markets is.9472 which translates into a relative weight on r t-20 of.338 or over double the GARCH weight. The average s estimate of.968 at the 20 day horizon implies a relative 2 2 weight for r t-20 of.524 or a little over half the weight on r t. The average s estimate of.981 at the day horizon implies a relative weight for r t-20 of.679 or a little over two-thirds the weight on r t 2 and over four times the GARCH relative weight on r t-20. For older observations, the weighting differences are even more stark. Consider, the weight 2 2 attached to r t-41 (or approximately two months earlier) relative to that attached to r t in generating the volatility forecasts. Again using the average estimates across the twelve markets, the implied relative weights are.0196 for GARCH,.108 for the least squares model at a 10 day horizon,.265 at a 20 days horizon, and.452 at a 160 day horizon. So while volatility two months or more ago generally has virtually no impact on the GARCH forecasts, there is evidence that it contains considerable information about likely volatility at the longer horizons. GARCH and regression estimates of the parameters and in equation 12 are reported in Panels B and C of Table 2 respectively for various time horizons. represents the percentage weight 2 attached to the most recent observation of r while indicates the degree of mean reversion. Unlike, s GARCH estimates of the parameters s and s vary with the time horizon s as expressed in equation 10. Specifically for given GARCH parameters, 0, 1,and, s increases and s decreases as the horizon s over which volatility is being forecast rises. Given the finding that the regression estimates of the decay parameter s tend to exceed the GARCH estimates, we expect the regression estimates of s to be lower. In other words, since the regression estimates of the parameters in equation 12 put relatively more weight on older 12 observations than GARCH, the absolute weight on the most recent observations should be less. As 11 s s

14 shown in Panel B of Table 2, that is the case. For the same horizon, the regression estimates of s are consistently and substantially lower than the GARCH estimates. As the forecast horizon lengthens, both estimates of s decline (except for the regression estimates in three markets when moving from 80 to 160 day horizons) but (except for T-Notes) the regression estimates tend to decline more rapidly. On average, the GARCH estimates of s at a 160 day horizon are 60.8% of those at a 10 day horizon. The regression estimates at the 160 day horizon average 29.8% of the 10 day estimates. The implication is that as the volatility forecast horizon lengthens, the regression model puts progressively less weight on recent observations and more on older observations. Estimates of the intercept s (the mean regression parameter) at different horizons are reported in Panel C. While there is no strong, consistent pattern, in general, regression estimates of s tend to exceed the GARCH estimates at short horizons and to be lower at long horizons. V. The Bias In GARCH Multi-period Forecasts V.1. Variance or Standard Deviation We next examine whether GARCH forecasts of volatility over multi-day horizons are biased. For this analysis, it is potentially important whether volatility is measured in terms of the standard deviation or variance of returns since, by Jensen s inequality, an unbiased predictor of the variance will tend to over-estimate the standard deviation and an unbiased predictor of the standard deviation will tend to under-estimate the variance. GARCH(1,1) models the variance and in the previous section we examined which parameters best forecast the variance of returns. However, the standard deviation is the much more common measure in the volatility forecasting literature. Also, Poon and Granger (2003) argue that econometrically the standard deviation is the better measure since the variance is more susceptible to outliers and deviations from normality. Since we focus on the use of volatility forecasts for option pricing and value-at-risk models, there are additional reasons to prefer the standard deviation. For near-the-money options, the Black- 12

15 Scholes option price is an approximately linear function of the standard deviation and a non-linear function of the variance for most pricing models and strikes. The derivative of the Black-Scholes call or put price with respect to the standard deviation (vega) is where S is the underlying asset price, is the dividend or interest rate on the underlying asset, t is the time to expiration, n(.) is the normal density function,, K is the strike and r is the risk-free rate. If, which for normal times to maturity and the volatilities in our data set occurs when S is slightly below the strike K, d=0 and and the option price is a linear function of. For S close to (but not equal to) it is approximately linear. Hence, if the pricing model is correct, an unbiased estimator of the standard deviation leads to unbiased estimates of near-the-money option values (and approximately unbiased for other strikes and models) while an unbiased estimator of the variance does not. Consequently, the standard deviation is the standard volatility measure in options markets. Likewise value-at-risk measures are a linear function of the standard deviation so an unbiased estimator of the standard deviation yields unbiased VaR values while an unbiased measure of the variance does not. For these reasons, we focus most of our attention on the forecast standard deviation, though many of our results are stronger for the variance. V.2. The Bias in GARCH Forecasts Evidence on whether GARCH(1,1) forecasts of the realized standard deviation and variance of returns are biased is presented in Table 3 for forecast horizons s of 10 and 40 trading days. Letting AVol(s) t represent the actual realized volatility of returns over the period from t+1 through t+s. i.e., and FVol(s) t represent the GARCH forecast of volatility for that period, we report the mean of FVol(s) - AVol(s) for volatility measured in terms of both the standard deviation and variance of t t 13 returns of returns. FVol(s) t is calculated using GARCH model parameters estimated using the entire data set as reported in Table 2. Purely out-of-sample results will be reported in the next section. For ease of interpretation and comparison across markets, the mean prediction error is presented as a 13

16 percentage of AVol(s) t. The mean over or under prediction is reported in columns 2 (10 day horizon) and 3 (40 day horizon) for the standard deviation of returns and in columns 6 and 7 for the variance. We also present tests of the null hypothesis that the mean GARCH forecasts are unbiased. Since our data overlap, the prediction errors are highly correlated. In other words, if our first 10 day period is from Monday through Friday a week later, the second is from Tuesday through Monday and the two observations share 9 of 10 days in common. Failure to correct for the serial correlation created by this overlap results in z-values which are severely biased upward. Consequently in testing whether the mean prediction error is zero, we calculate the standard deviation of the mean forecast difference incorporating all (s-1)! covariances. As reported in Table 3, there is weak evidence that GARCH tends to over-estimate the variance of returns and much stronger evidence that it tends to over-estimate the standard deviation. In 20 of the 24 market/horizons reported in Table 3, the mean difference between the variance predicted by GARCH and the actual ex post variance is positive. However, due to the overlapping data and the high variation in the realized variance, this mean difference is usually insignificant. Results also vary widely across markets. In the T-Bill market GARCH overestimates the variance over the next 40 days by 38.6% on average, while for crude oil and GE stock it actually tends to slightly underestimate the realized variance. GARCH s tendency to over-estimate the variance appears somewhat stronger at the longer horizon but these differences are generally insignificant. There is much stronger evidence that GARCH tends to over estimate the standard deviation of returns and this is true at both the 10 and 40 day horizons. In all 12 markets at the 10-day horizon and 11 of the 12 at the 40-day horizons, the mean prediction error is positive and in 18 of the 24 market/horizons it is significantly different from zero at the 1% level. It seems clear that GARCH is an upwardly biased estimator of the standard deviation. Is this bias consistent across the board or does it follow a particular pattern? A standard test for bias and one heavily used in testing whether implied volatilities are unbiased is to estimate the regression: AVol(s) t = 0 + 1FVol(s) t + e t. The null that the forecasting model is unbiased implies 14

17 0=0 and 1=1. It is surprising that this test which has been applied repeatedly to implied volatility forecasts, e.g., Canina and Figlewski (1993), Christensen and Prabhala (1998), Jorion (1995), Szakmary et al (2003), and Ederington and Guan (2005b), has not to our knowledge been applied to time series models. Results for estimations of this equation are presented in Table 4 for both the standard deviation and variance measures of volatility. Since due to the serial correlation induced by the overlapping observations OLS standard errors are severely biased downward, we apply the Hansen- Hodrick (1980) correction. The results are striking. Regardless of whether volatility is measured in terms of the variance or standard deviation and whether measured over 10 or 40 days, in every market and. Generally is significantly greater than zero at the.01 level and is significantly less than 1 at the.01 level. The null that GARCH yields unbiased forecasts of either the standard deviation of returns or the variance of returns is decidedly rejected. Our finding that and matches typical findings for implied volatility. Indeed the estimates of 1 for GARCH in Table 4 are somewhat less than those usually estimated for implied volatility. While results of such estimations have been widely used to demonstrate bias in implied volatility forecasts, we find it highly surprising that no one has asked whether time series forecasts fare any better. It is worth noting that whereas in general (Table 3) we find much stronger evidence for bias in GARCH estimates of the standard deviation than of the variance, if bias is measured in terms of the deviation of 1 from 1.0, the GARCH forecast of the variance is more severely biased than its forecast of the standard deviation. Nonetheless, it is also clear that (at least in-sample) the GARCH forecast does have predictive power. In most cases 2 is significantly greater than zero and the adjusted R s are reasonably high. When volatility is measured in terms of the standard deviation, the average 2 adjusted R is.310 for the 10-day horizon and.334 for the 40-day horizon. This is in contrast to 15

18 Christoffersen and Diebold (2000) who find GARCH has little ability to forecast volatility more than 10 or 20 days ahead. Measured in terms of the variance, predictive ability is considerably lower. This is partially expected since the ex-post variance will have more noise but even at the 40 day 2 horizon the adjusted R is.206 for the variance and.334 for the standard deviation. The results in Table 4 imply that the GARCH model tends to underestimate volatility when predicted volatility is low and/or over-estimate when predicted volatility high. In fact the bias is almost entirely the latter - that GARCH over-estimates when predicted volatility is high. To examine more closely how GARCH s forecast bias varies with forecast volatility, in each market we rank the observations on the basis of the GARCH forecast of the standard deviation of returns and assign the observations to twenty ranked bi-deciles. For each bi-decile, we calculate the mean forecast error FE=FSD-ASD, where FSD is the GARCH forecast of the standard deviation of returns and ASD is the actual standard deviation and then express this as a percentage of the mean forecast volatility:. Means of averaged across our twelve markets for a forty-day horizon are graphed in Figure 1 for the twenty FSD bi-deciles. As shown in Figure 1, despite the positive intercepts in Table 4, there is no evidence that GARCH tends to underestimate volatility in any bidecile. Indeed as shown by results for the bottom bi-decile, when predicted volatility is quite low, the over-prediction tends to be slightly more than when only moderately low volatility is predicted - although not significantly so in most markets. As shown in Figure 1, GARCH s over-prediction tendency is particularly strong in the top bidecile. When GARCH forecasts very high volatility, it tends to substantially over-predict. Across our twelve markets, the over-prediction of the standard deviation in the top bi-decile averages 21.2%. For the other nineteen bi-deciles, it averages 5.3% and in no other bi-decile does it exceed 10.5%. For the variance, the mean bias in the top bi-decile is even higher. In summary, we find a substantial bias in GARCH(1,1) forecasts of volatility over multi-day horizons. GARCH tends to consistently over-predict the standard deviation of returns - the most appropriate volatility measure for option valuation and value-at-risk measures and the most common 16

19 volatility measure in the literature. While there is evidence of a positive bias at all levels of predicted volatility, this bias is much stronger when very high volatility is predicted. When volatility is measured in terms of the variance, there is weak evidence that GARCH tends to over-predict in general and stronger evidence that it tends to over-predict when the predicted variance is high. VI. The Squared Return Issue What causes GARCH s to substantially over-predict volatility when predicted volatility is high? We hypothesize it is because the GARCH forecast of volatility over a future period is a linear function of past squared return innovations so that a single highly volatile day leads to an unreasonably high forecast. Earlier, we saw that in most of our markets, the GARCH estimations place most of their weight on squared returns in the last few days. This leads to the question whether a single highly volatile day leads to an inordinate jump in the GARCH forecast leading to overestimation. To examine whether GARCH tends to over-forecast volatility following highly volatile days, we first identify the 1% and 5% of days with the largest absolute return deviations. For instance in the S&P 500 market, the average r is but on 82 of the 8239 days r exceeds.0299 and on t 412 days it exceeds Consistent with volatility persistence, volatility is indeed much higher than normal following these high volatility days. For instance, on average across our twelve markets, the standard deviation of log returns over the 10 days following one of the 1% most volatile days is about 93% higher than the average 10 day standard deviation. Nonetheless, the GARCH model generally predicts still higher volatility. We examine how these high volatility days impact the GARCH forecasts and whether GARCH tends to over-predict volatility for the succeeding period. Results are shown in Table 5 for volatility forecasts over 10 day and 40 day horizons following the 1% most volatile days and in Table 6 following the 5% most volatile days. In columns 2-4, we report how these high volatility days impact the forecast. We report the mean percentage increase in the volatility forecast measured 17 t

20 as where FSD t-1, t+s-1 is the GARCH standard deviation forecast (the square root of the variance forecasts from equations 9 and 10) on day t-1 for the period t to t+(s- 1) where s=1, 10, or 40. Day t is the high volatility day. We include the increase in the forecast standard deviation for the next day (i.e., s=1) for comparison. As reported in Table 5, the mean percentage increase in the forecast standard deviation for the 10-day periods following one of the 1% highest volatility days ranges from 18% for GE to 74% for T-Bill rates and averages 36.6% for our twelve markets. For the 40 day forecast, the mean percentage increase in the forecast standard deviation ranges from 16.7% for GE to 61.4% for T- Bills and averages 32.1% for our twelve markets. Obviously, in terms of the forecast variance, the mean percentage increases are much larger. Note that in most markets, the mean percentage increase in the forecast standard deviation for the 10-day horizon is not much less than that for the next day and the mean percentage increase in the 40-day forecast is not much below that for 10-days. In general following a high volatility day, the GARCH model predicts a substantial increase in volatility and projects this increase far into the future. In other words, the mean reversion is fairly slow. On average the increase in the 10-day volatility forecast is 95.7% of the increase in the 1-day forecast and the increase in the 40-day forecast averages 85.0% of the increase in the 1-day forecast. Although it is often argued that GARCH s memory is too short, over these horizons, it is fairly long. In columns 5-6, we report the mean forecast error (again expressed as a percentage of realized volatility) for the 10 and 40 day horizons respectively. T-values for tests of the null that the mean prediction error is zero are reported in columns 8 and 9. The degree of over-prediction varies widely across our twelve markets but is always positive and in most markets is substantial and 14 significant. On average across our twelve markets, the GARCH model overestimates the 10-day standard deviation of returns by 23.5% and the 40-day standard deviation by 31.4%. In general, the over-prediction percentage is even worse if measured in terms of the variance. For instance, on 18

21 average following one of the 1% most volatile days, the predicted variance over the 10 day horizon is more than double the actual realized variance of returns. Consistent with our argument that the GARCH model puts too much emphasis on the most recent volatilities in forecasting volatility over longer periods, the over-prediction percentage tends to be worse at the longer horizon. For every series except Walmart, the mean over-prediction percentage is greater for the 40-day horizon than for the 10-day following one of the 1% most volatile days - averaging 31.4% at the 40 day horizon versus 23.5% at the ten day horizon. 15 Results following the 5% most volatile days are reported in Table 6. As one would expect, the degree of over-prediction is less. At the 10-day horizon, the average percentage over-prediction of actual volatility across our twelve markets is 15.0% versus 23.5% following the 1% most volatile days. At the 40 day horizon the mean over-prediction is 16.7% following the 5% most volatile and 31.4% following the 1% most volatile. Nonetheless, in every market except GE at the 10-day horizon, the null that GARCH does not over-predict is rejected at the.01 level. Since the figures in Table 5 apply to only 1% of our observations, one might question how important this tendency to over-predict. Consequently in Table 7 we examine whether this tendency to over-predict persists long after one of the 1% most volatility days. Designating the highly volatile day as t, in Table 5 we examine predicted and actual volatilities for the period t+1 through t+10 and t+40. In Table 7, we present statistics on forecasts for the period t+5 through t+14 and t+44. For these, day t is the high volatility day but data through day t+4 is used to form the GARCH forecast. Consequently, the high volatility day, t, receives a lower weight in formulating the GARCH forecast from t+5 to t+14 (or t+44) than it did in calculating volatility from t+1 to t+10 (t+40). For instance, the average estimated beta across our twelve markets is For this value, the weight on the day t volatility for forecasts from t+5 is about 68% of the weight given to day t in forecasts for volatility from day t+1 in Table 5. Consequently, the GARCH forecasts for the period from t+5 to t+14 tend to be lower than those for the period from t+1 to t+10. On the other hand, actual ex-post volatility also tends to be lower from t+5 through t+14 than for t through t+10. Ditto for the 40 day horizon. 19

22 As reported in Table 7, the tendency to over-predict actual volatility following a high volatility day persists fairly long after the high volatility day itself. In all but one of our market/horizons (Walmart at the 40 day horizon), the null that the GARCH does not over-predict volatility from t+5 to t+14 is rejected at the 1% as is the null that it does not over-predict from t+5 to t+44. In many of our markets the percentage over-prediction from t+5 to t+14 (or t+44) is actually greater than that from t+1 to t+10 (t+40). Indeed while the average percentage over-prediction from t+1 to t+10 was 23.5% across our twelve markets, it is 30.3% from t+5 to t+14. The average overprediction is 32.3% from t+5 to t+44 versus 30.4% from t+1 to t+40. Next we examine whether the impact of extreme volatility days on the GARCH forecast accounts for the bias documented in Table 4 above by estimating the regression: where ASD(s) is the actual standard deviation of returns from t+1 through t+s, FSD(s) is the t GARCH forecast of the standard deviation for the same period and D t is a measure of the impact of extreme events on the GARCH forecast defined as follows. Letting D(j) t=1 if the absolute surprise return at t-j is in the top 1%, and 0 otherwise we define (15) t where is the GARCH estimate of beta. Since in the GARCH(1,1) model, day t-j s impact on the j GARCH forecast is proportional to we weight the lagged dummies for extremely volatile days by. For example, if today s volatility (and only today s) is in the top 1%, D t =1; if volatility five 5 days ago (and only then) was in the top1%, D t= <1; and if both days were among the top 1%, 5 D t=1+. To the extent these extremely volatility days are responsible for the bias in GARCH forecasts, we expect to observe 2<0. If these 1% days are responsible for much of the bias documented in Table 4, then compared to the Table 4 regressions without D t, estimates of 1 should be higher and those of 0 lower. 20

23 Results are shown in Table 8 where we repeat the results from the regression without D t for easier comparison. In every market, and in most it is significantly less than 0 at the.01 level implying that part of the bias in GARCH volatility forecasts is due to over-prediction following these extreme days. As expected, accounting for the impact of these extreme days on the GARCH forecast eliminates much of the bias documented in Table 4. In every market, adding D t to the regression raises the estimate of 1 and lowers that for 0. Indeed, only in the two foreign exchange markets (where D t is insignificant) is still significantly less than 1.0 and significantly greater than 0 at the 5% level. Across our twelve markets averages.892 when D is included in the regression t versus.737 without D. Considering the relative crudeness of the D variable (in that it considers t only observations in the top 1% and treats all observations within the top 1% the same regardless of size), it appears possible that virtually all the bias in the regression test is attributable to GARCH s tendency to overpredict following extremely volatile days. We also defined a variable structurally like D t but for observations when one or more of the absolute return on days t-19 through t was in the top 5% but not the top 1%. In most markets, this variable was insignificant and had little impact on the other coefficients indicating that most of the bias is indeed due to the impact of the extremely volatile (top 1%) days on the GARCH forecast. In summary, we find that GARCH tends to over-predict the ex-post standard deviation of returns and that this bias is much greater following highly volatile days. t VII. An Alternative Volatility Forecasting Model We have shown that GARCH(1,1) tends to over-predict the ex-post standard deviation of returns and that this bias is especially high following highly volatile days. We have also shown that GARCH tends to put too little weight on older observations when predicting volatility over long horizons. While we present evidence only for GARCH(1,1) we expect these traits to be shared by most GARCH type models such as EGARCH and the Riskmetrics model. Virtually all are 21

24 formulated in terms of the variance so are likely to be biased predictors of the standard deviation. Like GARCH(1,1) they put the same relative weights on old and recent observations whether forecasting volatility for tomorrow, next week or next month. Like GARCH(1,1), the Riskmetrics model is based on squared return observations so extreme days are likely to have an inordinate impact on the forecasts. This leads us to search for a forecasting model which is unbiased and puts less weight on extremely volatile days. Our analysis of the problems with the GARCH(1,1) model suggests several attributes such a model should have. One, it should be unbiased. More specifically, since the most appropriate measure of volatility over multi-day horizons for most uses is the standard deviation, it should be an unbiased predictor of the standard deviation of returns. Two, the model should be constructed so that highly volatile single days do not lead to substantial over-prediction of volatility. Three, the relative importance of successive observations should be allowed to vary with the forecast horizon since the evidence indicates that, vis-a-vis older observations, very recent observations are more important in predicting volatility in the near future than in the more distant future. One of the models suggested in Ederington and Guan (2005a) which we now develop more thoroughly meets these requirements. This model is structurally similar to the GARCH(1,1) model in that the weights on past volatilities decline exponentially and it incorporates mean reversion but (1) models the standard deviation directly, (2) is based on absolute, not squared, surprise returns, and (3) is estimated using linear regression and the ex-post standard deviation of returns. Specifically we model the standard deviation of returns from t+1 to t+s, SD(s) t as: (16) This is structurally identical to the GARCH expression in equation 9 except for the following: (1) the standard deviation replaces the variance on the left-hand side of the equation, (2) the absolute return, 2 rt-j, replaces r t-j on the right-hand side, (3) the coefficient s is allowed to change with the horizon s and no form is imposed on and, and (4) the addition of the term. s s 22

25 By formulating the model in terms of absolute, rather than squared returns, the impact of extreme days on the volatility estimate is reduced but this introduces a complication. While 2 2 E(r )=, under quite general conditions, there is no general expression for E( r ). However, if log returns r t are normally distributed with mean, then where r t = Rt-, and where W j =1. Based on this, we include the term in equation 16. Obviously, this means that the forecasting model is based on the presumption that log returns are approximately normally distributed. Of course, normality is generally assumed in obtaining maximum likelihood estimations of the GARCH model. Hence our assumption of normality is not imposing an additional restriction relative to the usual GARCH estimations. Moreover, if the volatility estimates are to be used for option valuation purposes, normality is assumed by most option pricing models (e.g., Black-Scholes and Barone-Adesi-Whaley). An advantage of forecasting volatility for multi-day horizons is that we can measure the standard deviation and variance for these periods. Hence we estimate the parameters in equation 16 by regressing the ex post standard deviation from t+1 to t+s on the rt-j from j=0 to J (setting 16 J=250). An advantage is that (assuming the classical least squares assumptions are met) the resulting predictions should be unbiased - both in general (as in Table 3) and in terms of the regression test (as in Table 4). We term this the ARLS model where LS refers to the fact that it is estimated using ordinary least squares, R refers to the fact that the coefficients on older observations are restricted to an exponential decay, and A to the fact that it is based on past absolute returns. Estimates of s, s, and s for s=40 days are reported in Table 9. As shown there, as compared with the GARCH estimates in Table 2, the least squares estimates of s for the ARLS model tend to be considerably higher and those of s considerably lower. This means that (as compared with GARCH(1,1)) older observations receive considerably more weight relative to the most recent observation in forecasting future volatility. For instance, the average of across our twelve markets is.963 for the model in equation 16 versus.908 for the GARCH model. Thus in 23