Minimizes a function (variance over 1mln data-points)?

Questions
,
1

I have attached below one Mathematica notebook and a MATLAB file where I try to find the \theta which minimizes a function (variance over 1mln data-points).

The MATLAB code runs pretty fast, while the execution of the Mathematica code freezes my machine.

Any suggestions how to make the Mathematica code executable (at least)?

2

The problem is that:

$HistoryLength=1;
S=100;K=70;\[Sigma]=0.2;T=1;r=0.01;\[Alpha]=r-\[Sigma]^2/2;
n=100;
Z=RandomVariate[NormalDistribution[],n];
fVarPut[\[Theta]_]:=Variance[E^(-r T) Ramp[K-S E^(\[Alpha] T+\[Sigma] Sqrt[T](Z+\[Theta]))]E^(-\[Theta](Z+\[Theta])+\[Theta]^2/2)];

LogPlot[fVarPut[x],{x,-2,4},PlotRange->All]

looks like something like:

ScreenShot2016-10-24at14.04.50.png
ScreenShot2016-10-24at14.04.50.png752×486 12.2 KB

so

fVarPut[100]

will always give 0. it is ‘flat’ on the right (equal to 0, because of the Ramp function), so there is not one minimum…

3

The definition of pVarPut needs to signal that the function only works with a numeric argument.

In[1]:= S = 100; K = 70; [Sigma] = 0.2; T = 1; r = 0.01; [Alpha] =
r - [Sigma]^2/2;

In[2]:= n = 100;
Z = RandomVariate[NormalDistribution[], n];

In[4]:= fVarPut[\[Theta]_?NumberQ] := 
  Variance[E^(-r T)
     Ramp[K - 
      S E^(\[Alpha] T + \[Sigma] Sqrt[
         T] (Z + \[Theta]))] E^(-\[Theta] (Z + \[Theta]) + \[Theta]^2/
      2)];

In[5]:= (* freezes even with WorkingPrecision\[Rule]5 *)
NMinimize[fVarPut[x], x, WorkingPrecision -> 5]

Out[5]= {0, {x -> 3.3686}}

In[6]:= FindMinimum[fVarPut[x], {x, -2}, WorkingPrecision -> 5]

Out[6]= {1.0248*10^-382, {x -> -32.000}}
4

I don’t get any freeze with your code. However, it may run faster if you use the immediate assignment

fVarPut[\[Theta]_?NumberQ] = 
  Variance[E^(-r T) Ramp[
     K - S E^(\[Alpha] T + \[Sigma] Sqrt[
            T] (Z + \[Theta]))] E^(-\[Theta] (Z + \[Theta]) + \
\[Theta]^2/2)];

instead of the delayed assignment :=, because with delayed assignment Mathematica has to recalculate the Variance formula every time fVarPut is called.

5

Thank you very much all of you!

I was about to write that it freezes either way even if I ask for the minimum for the interval (-4,-1), when I read Frank’s paramount remark.

So much better now! :slight_smile:

6

This will calculate the variance in a symbolic way once; that might be or not be a smart idea depending on the number of points. Definitely not a smart move if n is large, it will do a lot of (comparatively) slow symbolic processing.

Frank Kampas’s solution is a good idea; to turn it into a numeric computation, which is very fast. It will avoid Mathematica trying to do any symbolic manipulation on it.

7

You will need to constrain it using something like:

FindMinimum[f,{x,x0,xmin,xmax}]

As this problem will have a minimum for x → -infinity and for x>a for some ‘a’ it is constant (kind of a minimum) (a depends on the random values).

8

I have a poor machine, Gianluca.. and a bit weird I guess, because it gives me that the implementation with “=” is slower than the one with “:=”