Estimation in Financial Models - RiskLab

There are many possible weighting schemes. A popular one uses a Gaussian
kernel. With the notation z 0 j =(" j,1 ;" j,2 ;:::;" j,m )we have
w (s)
j =
where K() istheGaussian kernel
K(z s , z j )=
1
q
2jHj
exp
K(z s , z j )
P Tr=1
K(z r , z s ) ;
, 1
2 (z s , z j ) 0 H (z s , z j ) ;
with H = diag(h 1 ;:::;h m ) containing the bandwidths that were set to
^ k T ,1=(4+m) , where ^ k is the sample standard deviation of " s,k , the kth
component of z s , k =1;:::;m.
Another approach to nonparametric estimation of volatility is an approximation
using series expansion. The most frequently used series expansion in
economics is the Flexible Fourier Form (FFF) introduced by Gallant [28],
which leads to avolatility estimate of the form
( mX j
)
2X
^ t 2 = 0 + " t,j + j " 2 t,j + ( jk cos(k" t,j )+ jk sin(k" t,j )) ;
j=1
k=1
that means t 2 is estimated by a sum of a low-order polynomial and trigonometric
terms based on " t,j , j =1;:::;m. Note the disadvantage of the FFF
that the estimates of t
2 may benegative. We refer to [57] for more details.
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