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

Some properties of averaging simulated optimization methods 243

Next , letting

Q

ab

bc



⎡

⎣

⎢

⎢

⎢

⎤

⎦

⎥

⎥

⎥

we have:

Ψ^121  ^2

1

T(/)(/)abc bcπ Tc

and via a transformation, uTΩΩi

(^1212)
μαˆ,and we get:
Ta b c(/)^2 ∼χ(,( /))^ NTabc^2  1 2
and
bc/ ∼Nb c(/, / )^1 Tc
Therefore,
σπ^211223
1
 pQ p k k k 
Tc
ˆ ⎡()/
⎣
⎢
⎢
⎤
⎦
⎥
⎥
where the three random variables
kkN 11 ∼∼χ()^2 Tk,(/,/) 2 bcT 1 c
and k (^3) NTabc 1
∼χ^22
(,( /))
are mutually independent.
We now have a straightforward method to generate, via simulation, the
mean – variance frontier and confidence intervals, which does not require hav-
ing to simulate the full portfolio.
For specified abˆ,ˆ,andcˆ along with T and N , and taking 5,000 replications
as an example:

1. Generate 5,000 observations on the three independent random variables, k 1 , k 2 ,

and k 3 i.e., k (^1) j , k (^2) j , k (^3) j , j  1, ... ,5,000.