Estimation in Financial Models - RiskLab

and
~ n =
~ n
1 , e ~ n
Pni=1 X (i,1)
2
i,1
e ~ n
,
Pni=1 X i
2
i,1
Pni=1
1
2
i,1
: (3.58)
As regarding the martingale estimating function G n we are able to nd a
closed expression for only in a few cases where the diusion coecient is
rather simple. For instance, if (x) = p x (as in the square root diusion
model (1.5), the Cox Ingersoll Ross model) we have with a similar argument
as for F (that is, the conditional second moment solves an ordinary
dierential equation)
(x; ; ) = 2
2 2
( +2x)e
2
, 2( + x)e + :
As an extension we consider the mean-reverting process where > 0 enters
the diusion coecient
q
dX t = ,X t dt + + Xt 2 dW t :
With the same argument as above we obtain the conditional variance
(x; ; ) =x 2 e ,2 (e , 1) +
2 , 1 (1 , e(1,2) ):
We remark that in practical applications the estimating equations corresponding
to G n can be solved using a generalization of Newton's method.
Furthermore, if no explicit expressions for the conditional mean F and the
conditional variance are known, then F and can be approximated by the
sample mean and sample variance of a large number of simulated realizations
of the diusion process at the relevant time point.
(4) As an extension we shall combine martingale estimating functions
in an 'optimal' way.
The following approach is proposed by Bibby [5].
If the unknown parameter is multi-dimensional and a part of is only found
in the diusion coecient, then using the martingale estimating functions
generated by the conditional mean, ~G n and G n, leads to fewer estimation
equations than parameters. That is in this case neither G nor ~G can be
used to estimate the part of that only enters the diusion coecient. This
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