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

Computing optimal mean/downside risk frontiers: the role of ellipticity 193

To check that this is a minimum, we compute:

u

 



β

γγ

θγ βθ

γθ

1

1

1

Δ()

()

−

⎛

⎝

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟⎟

(8.82)

u





βθγ βθγ

γγθ

()+−Δ()

()

1

(^1)
(8.83)
u
 

((βθγ ) (θγ ))
γθ
11
1
Δ
()
(8.84)
u


β
γγ
θγ
θγ
1 1
1
Δ()
()
(8.85)
u

β
γγθγ
1
1
Δ
()
(8.86)
u
β
γ
(8.87)
Considering now the second-order conditions, first:
uv 

 
β
γγθγ
β
γγ
θγ
1
1
1
1
Δ
Δ
()
()
(8.88)
uv


1
1
10
γθγ
θγ
Δ
Δ
()
()
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
(8.89)
Differentiating Equation (8.69),
∂
∂
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟
∂
∂
∂
∂
∂
∂
∂
∂
v
u
v
v
u
uv
v
u
v
u
v
u
(^22)
2
2
 2 () 10 γθ
(8.90)
Since ∂ v / ∂ u  0 at the minimum,
∂
∂
2
2
v 1
u uv



()
()
γθ
(8.91)