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

232 Optimizing Optimization

and

E

k

(( ) )k k k

()

χ() ,

ν

ν ν

ν

2 2

22

  2



Γ

Γ

()

/

Eg

ekk

F

N

k

N

k T

ThN

[] N () ,;

 





 



/

/

/

(^21)
22
1
2
11
2
1
2
1
2
ΓΓ
ΓΓ
()()
()
ν
ν
22 h
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟
Therefore,
EEg
ekk
F
kkk
Th N
k N
[( ) ]αλ( )
λν
ν




 

/
/
(^21)
22
1
2
11
2
ΓΓ
ΓΓ
()()
()()
NN
k
N
Th


1 
2
1
2
,;/ 2
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟
(10.8)
ETE Eg
e
kkk
ThN kk
k N
(( ) ( )
()




 

λ
λν
ν
/
/
/
2
(^21)
2222
1
2 2
ΓΓ
ΓΓ
()()
()
111
1
22
1
2
F 2
NkN
Th



,;/
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟
(10.9)
EIR Eg
(( ) )kk ( /2)
(10.10)
In particular, if we consider the means of the three quantities, we have:
Ea e F
N
N
N
()ˆ  Th






ΓΓ
ΓΓ
1
2
1
22
2
11
1
2
1
1
2
1
()
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟
()()
ν
λ ν
/ ,,;
()
()
,;
N
Th
N
F
N
Th







1
2
2
1
2
1
1
2
11 2
/
/
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
⎟
⎛
⎝
⎜⎜
⎜
⎞
⎠
⎟⎟
λν ⎟⎟
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥










 
NT
hN
NT
h
TN
1
2
1
1
1
22
1
λν
λν λ ν
ν
() ( )
() ()
;
Since the true α  1/ λ h , we can readily develop an unbiased estimator of α via
a simple transformation:
E
T
a
N
T
()ν
λ
α




21
ˆ
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥