Figure 5.2: Autocorrelation of Realized Variance and Autocorrelation of Squared Returns
Tải bản đầy đủ - 0trang
Realized Variance: Four Stylized Facts
11
• The top panel of Figure 5.3 shows a histogram of the RVs
from Figure 5.1
• The bottom panel of Figure 5.3 shows the histogram of
the natural logarithm of RV
• Figure 5.3 shows that the logarithm of RV is very close to
normally distributed whereas the level of RV is strongly
positively skewed with a long right tail
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Figure 5.3. Histogram of Realized Variance and log
Realized Variance
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
12
13
Realized Variance: Four Stylized Facts
• The approximate log normal property of RV is the third
stylized fact. We can write
• The fourth stylized fact of RV is that daily returns
divided by the square root of RV is very close to
following an i.i.d. (independently and identically
distributed) standard normal distribution:
Elements of Financial Risk Management Second Edition â 2012 by Peter Christoffersen
14
Realized Variance: Four Stylized Facts
RVmt+1 can only be computed at the end of day t+1, So this
result is not immediately useful for forecasting purposes
• If a good forecast RVmt+1/t can be made using information
available at time t then a normal distribution assumption
of
will be a decent first modeling strategy.
Approximately:
• where we have now standardized the return with the RV
forecast
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Realized Variance: Four Stylized Facts
15
• When constructing a good forecast for RVmt+1 , we need to
keep in mind the four stylized facts of RV:
– RV is a more precise indicator of daily variance than is
the daily squared return.
– RV has large positive autocorrelations for many lags.
– The log of RV is approximately normally distributed.
– The daily return divided by the square root of RV is
close to i.i.d. standard normal.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Forecasting Realized Variance
• Realized variances are very persistent
• So we need to consider forecasting models that
allow for current RV to matter for future RV
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
16
Simple ARMA Models of Realized Variance
17
• AR(1) model allows for persistence in a time series
• If we treat the estimated RVmt as an observed time series,
then we can assume the AR(1) forecasting model
• where t+1 is assumed to be uncorrelated over time and
have zero mean
• The parameters 0 and 1 can easily be estimated using
OLS.
• The 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 normally distributed we
may be better off modeling the RV in logs rather than
levels. We can therefore assume
• The normal property of ln (RVmt+1) will make the OLS
estimates of 0 and 1 better than those in the AR(1)
model for RVmt+1
• The AR(1) errors, t+1, are likely to have fat tails, which
in turn yield noisy parameter estimates
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Simple ARMA Models of Realized Variance
19
• As we have estimated it from intraday squared returns, the
RVmt+1 is not truly an observed time series but it can be
viewed as the true RV observed with a measurement error.
• If the true RV is AR(1) but we observed true RV plus an
i.i.d. measurement error then an ARMA(1,1) model is
likely to provide a good fit to the observed RV. We can
write
• which due to the MA term must be estimated using
maximum likelihood techniques
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
20
Simple ARMA Models of Realized Variance
• As the exponential function is not linear, we have in the
log RV model that
• Assuming normality of the error term we can use:
• In the AR(1) model the forecast for tomorrow is
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Simple ARMA Models of Realized Variance
21
• And for the ARMA(1,1) model we get,
• More sophisticated models such as long-memory ARMA
models can be used to model realized variance
• These models may yield better longer horizon variance
forecasts than the short-memory ARMA models
considered here
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen