Estimation in Financial Models - RiskLab

After these theoretical considerations we deal with the actual calculation of
l n;N () for large values of N.
In order to maximize l n;N () numerical algorithms usually require the value
of l n;N () in a nite numberofpoints . For N =1 l n;N () is explicitly given,
because p 1 has a closed expression, but for N 2 this is in general not the
case.
For calculating l n;N () for N 2, we have to know how to calculate
p N (s; x; t; y; ) for all 0 s < t, x; y 2 IR d and 2 . Considering again
equation (3.23)
p N (s; x; t; y; ) =E P;s;x
p1 ( N,1 ;Y (N)
N,1
;t;y; ) ;
the idea is to calculate p N (s; x; t; y; ) by nding a good approximation to
the right hand side. Denote by (Uk m ) N,1;M
k=1;m=1
an i.i.d. sample from the r-
dimensional standard normal distribution. Then (Y m ) M m=1
=(YN,1) m M m=1
given
by the Euler approximation
Y m
0
= x; m =1;:::;M
Yk m = Y m
k,1
+ t , s
N
b( k,1;Yk,1; m )+
s
t , s
N
( k,1;Y m
k,1; ) U m k
for k =1;:::;N, 1 and m =1;:::;M, has the same distribution as an i.i.d.
sample of Y (N)
N,1
under P ;s;x .Thus we are able to approximate the right hand
side of (3.23), while we calculate
1
M
MX
m=1
p 1 ( N,1 ;Y m ;t;y; ) (3.24)
by means of the sequence (U m k ) N,1;M
k=1;m=1, for M chosen suciently large, and
thereby we are able to calculate p N (s; x; t; y; ) with any given accuracy.
From a practical point of view it is convenient to simulate the sample
(Uk m ) N,1;M
k=1;m=1
once (see Kloeden and Platen [45]) and store it. Then it can
be used to calculate p N (s; x; t; y; ) for all values of 0 s