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

174 Optimizing Optimization

Some auxiliary parameters

μ

σ

ln(Dif)

2

α

πσ



1

() 2

β

σ



1

() 2 2

Formula for the lognormal curve f ( x )

If the extreme value is a minimum and x is greater than the extreme value,
then:

f( )x exp( (ln( ) ) )

x

 x



        

α

βτμ

τ

2

If the extreme value is a maximum and x is less than the extreme value, then:

f( )x exp( (ln( ) ) )

x

 x



        

α

βμ

τ

τ^2

Formula for the lognormal cumulative distribution function F ( x )

If the extreme value is a minimum and x is greater than the extreme value,
then:

Fx

er c x

()

(ln( ) )





1

22

f τ μ

σ

DifDif  MeanMean TauTau

σln

SD

Dif

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟

⎛

⎝

⎜⎜

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟⎟

⎟⎟

2

σln 1

SD

Dif

⎛

⎝

⎜⎜

⎜⎜

⎞

⎠

⎟⎟

⎟⎟

⎛

⎝

⎜⎜

⎜⎜

⎜

⎞

⎠

⎟⎟

⎟⎟

⎟⎟

2

1