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

Computing optimal mean/downside risk frontiers: the role of ellipticity 187

So , examine the integral in brackets:

Jqy

yt

dy

p

p

p





exp() exp

1 ()

22

2

(^0) σπ^2
μ
σ
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
∞
∫
(8.39)
and let t  μ (^) p   μ (^) t , so
J
yy qy
dy
p
tp t
p

1  
2
22
2
222
(^0) σπ^2
μσ μ
σ
exp
()
⎛
⎝
⎜⎜
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
⎟
∞
∫
(8.40)
J
t q
p
tp
p


exp exp
μ
σ
μσ
σ
2
2
222
222
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
()
⎛
⎝
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎞
⎠
⎟⎟⎟
⎟⎟
⎟⎟
⎟⎟
()()−
⎛
⎝
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟
1
22
2 2
σπ^2
σμ
p σ
pt
p
yq
exp d

yy
0
∞
∫
(8.41)
Transform yz
yq
yz q
pt
p
  ppt


σμ
σ
σσμ
2
()()⇒ (^2) , so
J
qq z
dz
tp




exp exp
2
2
1
2 2
μσ^222
π
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
qq ⎟⎟⎟
()σμσpt p^2 
∞
∫ /
(8.42)
J
qqtp qpt
p




exp
2
2
1
μσ σμ^222
σ
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
Φ ⎟⎟
⎟⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
(8.43)
That is,
It
q
qq qtppt
p
()
∂  
∂
⎛
⎝
⎜⎜
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
⎛
⎝
⎜⎜
⎜⎜
⎜
2 ⎞
2
2 22 2
2
exp
μσσμ
σ
Φ
⎠⎠
⎟⎟
⎟⎟
⎟⎟
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
q (^0)
(8.44)
It
q
q
qq q
pt
tppt
p
()
∂  
∂
()
⎛
⎝
⎜⎜
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
σμ
μσσμ
σ
2
2 22 2
2
exp Φ
⎛⎛
⎝
⎜⎜
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
⎡
⎣
⎢
⎢
⎢
⎛
⎝
⎜⎜
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
 ′

σ
μσ
p
qqtp
exp
2
2
22
Φ
qq pt
p q
σμ
σ
2
0


⎛
⎝
⎜⎜
⎜⎜
⎜⎜
⎞
⎠
⎟⎟
⎟⎟
⎟⎟
⎤
⎦
⎥
⎥
⎥
(8.45)