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

190 Optimizing Optimization

thus,

()^1 ()1

0

 r 1

P

P

p y

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟ exp

(8.59)

So that

() 11 rry ypi∑ωωiexp()()() 12  1 ωexp

(8.60)

and

ryp ωωexp() ( 1211 ) ( )expy

(8.61)

Therefore ,

sv rpppt r pdf r drp

t

()( ) ()



2

∫∞

(8.62)

If

rtp<<,expωωy 12  11 exp then y t ( ) ( ) ( )

(8.63)

The above transforms to a region ℜ in ( y (^) 1, y 2 ) space. Hence,
sv r()p ( 11 t ωωexp( ) (y 12 ) ( ))expy^2 pdf y y dy dy(12 1 2, )
∫ℜ
(8.64)
or
sv r() ()( )p  1 t pdf y y dy dy^2 12 1 2,ec 1 xp,()( )y pdf y y dy dy 1 12 1 2
∫∫ℜℜ∫
c (^112) ∫ℜ y y pdf y y dy dy
2
(( )) ( )exp12 1 2,
(8.65)
where c 1 and c 2 are some constants. None of the above integrals can be com-
puted in closed form, although they can be calculated by numerical methods.

8.4 Conic results

Using our definition of value at risk as VaR p  t σ (^) p  μ (^) p with t 0, and noting
from Proposition 8.1 that σp^2 must lie on the minimum variance frontier so that
σ
μγ βμ α
αγ β
p
2 pp
2
2
2



()
()
(8.66)