Optimizing Optimization: The Next Generation of Optimization Applications and Theory (Quantitative Finance)

128 Optimizing Optimization

to estimate the parameters (ˆv,,μσγiiiiˆ ,ˆ) of each component. Then, an estimator
of matrix Σ is given by

ˆ cov( )

()()

∑ ˆˆ

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

 ⎟⎟⎟





V

2

24

2

2

2

2

v

vv

v

γγ

where γˆ()γγ 114 ,..., and cov( V ) is the variance – covariance matrix of V.
Table 5.4 reports the estimated parameters of the multivariate skewed Student’s
t -distribution for the 14 factors.
Since we have estimated all the parameters of Y and Z , we can generate
S scenarios for Y and, independently, S scenarios for Z , and using Equation
(5.5) we obtain S scenarios for the vector of standardized innovations
uuˆ [ˆˆ 114 ,...,u] that is asymmetric t -distributed. Denote these scenarios by
()VV 114 ()ss,,... () for s  1, ... , S and denote the marginal distributions FxVj() for
1  j  14 of the estimated 14-dimensional asymmetric t -distribution by
F V ( x 1 , ... , x 14 )  P ( V 1  x 1 , ... , V 14  x 14 ). Then, considering UFV()js  Vj()j()s ,
1  j  14; 1  s  S , we can generate S scenarios ()UU 114 ()ss,, ,... () s  1, ... , S of
the uniform random vector ( U 1 , ... , U 14 ) (with support on the 14-dimensional
unit cube) and whose distribution is given by the copula:

Ct()(()()) 114 ,,......t F F tVV 11 1 ,,F tV 141 14 ; 01114 ti ; i.

Considering the stable distributed marginal sample distribution function of
the j -th standardized innovation FjuˆjT, 1 ; ,..., 114 (see Equation (5.4)) and
the scenarios Uj()s for 1  j  14; 1  s  S , then we can generate S scenarios
of the vector of standardized innovations (taking into account the dependence
structure of the vector) uuus S()Ts 11 (,, ), ,,T(, )^1 s ......T( , )^141 s 1 valued at time
T  1 assuming:

uFU jT(, )js 1 ()ujT, 1 ^1 ()j()s ;;. 1141 sS

Once we have described the multivariate behavior of the standardized inno-
vation at time T  1 using relation (5.3), we can generate S scenarios of the
vector of innovation:

εεε σ σT()s 11 ()(, )T^1 s,,......T( , )^141 s () 11 ,TuT(, )^1 s 1 ,, 14 ,T 1 uus ST(,)^14  1 s ,,, 1 ...,

where σ (^) j (^) , (^) T (^)  1 are still defined by Equation (5.3). Thus, using relation (5.3),
we can generate S scenarios of the vector of factors fffT
s
T
s
T
s
 111141 
() ()
,
()
, ,...,
⎡
⎣⎢
⎤
⎦⎥^
valued at time T  1. Observe that this procedure can always be used