Figure 6.5: Skewness and Kurtosis in the Asymmetric t Distribution
Tải bản đầy đủ - 0trang
The Asymmetric t distribution
• Notice that the symmetric t distribution is nested in the
asymmetric t distribution and can be derived by setting
d1 = d, d2 = 0 which implies A = 0 and B = 1
• Therefore,
• which yields
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
45
Estimation of d1 and d2
• MLE can be used to estimate the parameters of the
asymmetric distribution, d1 and d2
• The only complication is that the shape of the
distribution on each day depends on zt
• As before the likelihood function for the shock can
be defined as,
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
46
Estimation of d1 and d2
• Where
• This assumes the estimation of
is done without estimation error.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
47
Estimation of d1 and d2
• Alternatively the joint estimation of volatility and
distribution parameters can be done via,
• We can also estimate d1 and d2 using the sample
moments.
• Equations below can be solved numerically
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
48
Calculating VaR and ES
• Having estimated d1 and d2 we can calculate the
VaR of the portfolio return
• as
• Where
is the pth percentile of the
asymmetric t distribution.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
49
Calculating VaR and ES
•
is given by
• The ES is given by
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
50
QQ Plots
51
• Knowing the CDF we can construct the QQ plot as,
• where zi denotes the ith sorted standardized return
• The asymmetric t distribution is cumbersome to estimate
and implement but it is capable of fitting GARCH shocks
from daily asset returns quite well
• The t distributions attempt to fit the entire range of
outcomes using all the data available
• Consequently, the estimated parameters in the distribution
(for example d1 and d2 ) may be influenced excessively by
data values close to zero
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Figure 6.6: QQ Plot of S&P 500 GARCH Shocks
against the Asymmetric t Distribution
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
52
Extreme Value Theory (EVT)
53
• Typically, the biggest risks to a portfolio is the sudden
occurrence of a single large negative return.
• Having an as precise as possible knowledge of the
probabilities of such extremes is therefore at the essence
of financial risk management.
• Consequently, risk managers should focus attention
explicitly on modeling the tails of the returns distribution.
• Fortunately, a branch of statistics is devoted exactly to the
modeling of these extreme values.
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen