1 Volatility Modeling Using Intraday Data Elements of Financial Risk Management Chapter Peter Christoffersen Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen Overview Our goal is to harness the information in intraday prices for computing daily volatility • Let us estimate the mean of returns using a long sample of daily observations: • When estimating the mean of returns only the first and the last observations matter as all the intermediate terms cancel out • When estimating the mean, we need a long time span of data Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Overview • The start and end points S0 and ST will be the same irrespective of the sampling frequency of returns • Consider now instead estimating variance on a sample of daily returns: • In the variance estimator, the intermediate prices not cancel out • All return observations now matter because they are squared before they are summed in the average Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Overview • Now we have price observations at the end of every hour instead of every day and the market for the asset at hand is open 24 hours a day • Now we have 24.T observations to estimate 2 and we can get a much more precise estimate than when using just the T daily returns • Implication of this high-frequency sampling is that just as we can use 21 daily prices to estimate a monthly volatility we can also use 24 hourly observations to estimate a daily volatility Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Realized Variance: Four Stylized Facts • Assume that we are monitoring an asset that trades 24 hours per day and that it is extremely liquid so that bid-ask spreads are virtually zero and new information is reflected in the price immediately • Let m be the number of observations per day on an asset If we have 24 hour trading and 1-minute observations, then m = 24*60 = 1,440 • Let the jth observation on day t+1 be denoted St+j/m Then the closing price on day t+1 is St+m/m = St+1, and the jth 1minute return is Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Realized Variance: Four Stylized Facts • With m observations daily, we can calculate an estimate of the daily variance from the intraday squared returns simply as • Here we not divide the sum of squared returns by m If we did we would get a 1-minute variance • This is the total variance for a 24-hour period • Here we not subtract the mean of the 1-minute returns as it is so small that it will not impact the variance estimate Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Realized Variance: Four Stylized Facts • The top panel of Figure 5.1 shows the time series of daily realized S&P 500 variance computed from intraday squared returns • The bottom panel shows the daily close-to-close squared returns of S&P 500 • The squared returns in the bottom panel are much more jagged and noisy when compared with the realized variances in the top panel • Figure 5.1 illustrates the first stylized fact of RV: RVs are much more precise indicators of daily variance than are daily squared returns Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Figure 5.1: Realized Variance (top) and Squared Returns (bottom) of the S&P500 Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Realized Variance: Four Stylized Facts • The top panel of Figure 5.2 shows the autocorrelation function of the S&P 500 RV series from Figure 5.1 • The bottom panel shows the corresponding ACF computed from daily squared returns • Notice how much more striking the evidence of variance persistence is in the top panel • Figure 5.2 illustrates the second stylized fact of RV: • RV is extremely persistent, which suggests that volatility may be forecastable at horizons beyond a few months as long as the information in intraday returns is used Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Figure 5.2: Autocorrelation of Realized Variance and Autocorrelation of Squared Returns Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 10 Range-based Proxies for Volatility 65 • Figure 5.7 shows the autocorrelation of RPt in the top panel • The first-order autocorrelation in the range-based variance proxy is around 0.60 (top panel) whereas it is only half of that in the squared-return proxy (bottom panel) • Furthermore, the range-based autocorrelations are much smoother and thus give a much more reliable picture of the persistence in variance than the squared returns in the bottom panel Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 66 Figure 5.6: Range-Based Variance Proxy (top) and Squared Returns (bottom) Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 67 Range-based Proxies for Volatility • Range-based volatility proxy does not make use of the daily open and close prices • Assuming that the asset log returns are normally distributed with zero mean and variance, 2; a more accurate range-based proxy can be derived as Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 68 Figure 5.7:Autocorrelation of Range-Based Variance Proxy and Autocorrelation of Squared Returns Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Range-based Proxies for Volatility • In the more general case where the mean return is not assumed to be zero the following range-based volatility proxy is available • All of these proxies are derived assuming that true variance is constant • For example, 30 days of high, low, open, and close information can be used to estimate the (constant) volatility for that period Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 69 Forecasting Volatility Using the Range 70 • Now we are going to use RPt in the HAR model • Several studies show that the log range is close to normally distributed as: • The strong persistence of the range and log normal property suggest a log HAR model of the form Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Forecasting Volatility Using the Range • where we have that • The range-based proxy can also be used as a regressor in GARCH-X models, for example Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 71 Forecasting Volatility Using the Range • A purely range-based model can be defined as • Finally, a Realized-GARCH style model (RangeGARCH) can be defined via Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 72 Forecasting Volatility Using the Range 73 • The Range-GARCH model can be estimated using bivariate maximum likelihood techniques using historical data on return, Rt, and on range proxy, RPt • ES and VaR can be constructed in the RP-based models by assuming that zt+1 is i.i.d normal where zt+1 = Rt+1/ t+1 in the GARCH-style models or in the HAR model Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 74 Range Based vs Realized Variance • For very liquid securities the RV modeling approach is useful as the intraday returns gives a very reliable estimate of today’s variance, which in turn helps forecast tomorrow’s variance • The GARCH estimate of today’s variance is heavily model dependent, whereas the realized variance for today is calculated from today’s squared intraday returns Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Range Based vs Realized Variance 75 • Shortcomings of realized variance approach includes • It requires high-quality intraday returns to be feasible • It is easy to calculate daily realized volatilities from 5minute returns, but it is difficult to construct at 10-year data set of 5-minute returns • Realized variance measures based on intraday returns can be noisy • This is especially true for securities with wide bid–ask spreads and infrequent trading • However, range-based variance measure is relatively immune to the market microstructure noise Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Range Based vs Realized Variance 76 • The true maximum can easily be calculated as the observed maximum less one half of the bid–ask spread • The true minimum as the observed minimum plus one half of the bid–ask price • The range-based variance measure thus has clear advantages in less liquid markets • In the absence of trading imperfections, range-based variance proxies can be shown to be only about as useful as 4-hour intraday returns Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen GARCH Variance Forecast Evaluation Revisited 77 • The realized variance measure can be used for evaluating the forecasts from variance models • If only squared returns are available then we can run the regression • where 2t+1/t is the forecast from the GARCH model • With RV-based estimates we can run the regression • where we have used the Average RV estimator as an example Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen GARCH Variance Forecast Evaluation Revisited • The range-based proxy can be used for evaluating the forecasts from variance models Thus we could run the regression • where RPt+1 can be constructed for example using Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 78 GARCH Variance Forecast Evaluation Revisited 79 • Using R2t+1 on the left-hand side of the regression could convey that the volatility forecast is poor • The fit of the regression will be low but it does not mean that the volatility forecast is poor • It could also mean that the volatility proxy is poor • If regressions using RVAvrt+1 or RPt+1 yield a much better fit than the regression using R2t+1 then the volatility forecast is much better than suggested by the noisy squared-return proxy Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen ... Financial Risk Management Second Edition © 2012 by Peter Christoffersen Figure 5.2: Autocorrelation of Realized Variance and Autocorrelation of Squared Returns Elements of Financial Risk Management. .. one-day-ahead forecast of RV is then Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen 18 Simple ARMA Models of Realized Variance • Since the log of RV is close to... estimate Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen Realized Variance: Four Stylized Facts • The top panel of Figure 5.1 shows the time series of daily