Figure 9.4: Simulated Threshold Correlations from the Symmetric t Distribution with Various Parameters
Tải bản đầy đủ - 0trang
Multivariate Standardized t
Distribution
17
• In the case of n assets we have the multivariate t distribution
• Where
• Using the density definition we can construct the
likelihood function
• which can be maximized to estimate d
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Multivariate Standardized t
Distribution
18
• The correlation matrix can be preestimated using
• The correlation matrix can also be made dynamic, and
can be estimated using the DCC approach
• An easier estimate of d can be obtained by computing the
kurtosis, 2, of each of the n variables
• The relationship between excess kurtosis and d is
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Multivariate Standardized t
Distribution
19
• Using all the information in the n variables we can estimate d using
• where 2,i is sample excess kurtosis of ith variable
• The standardized symmetric n dimensional t variable can
be simulated as follows
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Multivariate Standardized t
Distribution
20
• where W is a univariate inverse gamma random variable,
• where U is a vector of multivariate standard normal variables,
• where U and W are independent
• The simulated z will have a mean of zero, a standard deviation
of one, and a correlation matrix
• Once we have simulated MC realizations of vector z we can
simulate MC realizations of the vector of asset returns and
from this the portfolio VaR and ES can be computed by
simulation as well
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Multivariate Asymmetric t
Distribution
Let be an nì1 vector of asymmetry parameters
The asymmetric t distribution is then defined by
where
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
21
Multivariate Asymmetric t
Distribution
22
• where
is the so-called modified Bessel function of
the third kind, which can be evaluated in Excel using the
formula besselk(x, (d+n)/2)
• Note that the vector and matrix are constructed so that
the vector of random shocks z will have a mean of zero, a
standard deviation of one, and the correlation matrix
• Note also that if = 0 then
• Note that the asymmetric t distribution will converge to the
symmetric t distribution as the asymmetry parameter
vector goes to a vector of zeros
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
Figure 9.5: Simulated Threshold Correlations from the
Asymmetric t Distribution with Various Linear
Correlations
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen
23
24
Multivariate Asymmetric t Distribution
• From the density
function
we can construct the likelihood
• which can be maximized to estimate the scalar d and
vector
• The correlation matrix can be preestimated using
• The correlation matrix can be made dynamic, t, and
can be estimated using the DCC approach
Elements of Financial Risk Management Second Edition © 2012 by Peter Christoffersen