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

Optimization and portfolio selection 175

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

Fx

er c x

()

(ln( ) )





1

22

f τ μ

σ

Note : erfc is the complementary error function (Ref. 3).

Analytic expression for the downside and upside statistics

Using the above formula for the lognormal density function, it is possible
to obtain analytic expressions for the integrals defining the basic statistics.
Following are fragments of Visual Basic code in the case Tau in a minimum
for analytic expressions of the basic statistics for a lognormal distribution (as
above, T is the DTR).
Upside probability or U can be written in terms of F ( x ), the distribution
function as:

UFx 1()

Upside potential or UP

I f T tau Then

b  T-tau

c  sqrt2 * sigma

UP  0.5 * Exp(mu  0.5 * sigma ^ 2) * (2-erfc((mu 

sigma ^ 2-Log(b)) /c))

Else

U P  tau  Exp(mu  sigma * sigma/2)

EndIf

Downside Variance or DV (this is the square of downside deviation)

I f T tau Then

b  T-tau

c  sqrt2 * sigma

a  (Log(b) -mu) /c

D V  0.5 * Exp(2 * mu  2 * sigma ^ 2) * (2-erfc(a-c))

-b * (2-erfc(a-c/2)) *Exp(mu  (sigma ^ 2) /2)  0.5 * b ^ 2 * (2-erfc(a))

Else

D V  0

End If