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From: Philip M on 20 Jul 2010 17:55
Hey guys,

I want to minimize the following function. I'm running a monte carlo simulation to calculate spreads of Collateralized Debt Obligations (CDOs). I want to minimize the difference of my model spread and the market spread by varying my model parameters "mu","a" and "b". To keep this model valid, a constraint has to be fulfilled for every point of valuating (r=1,...,20). I want to vary "mu" to keep the constraint fulfilled. In the end I would like to have a single parameter "a" and a single parameter "b" but a parameter vector "mu". I don't know how to set this up, which formula to use for multivariate minimization with constraints and how to tell matlab that it should vary only "mu" to match the constraints.

Thanks alot,

Philip!
-----------------------------------------------------------------------------------------------
Constraint:

expectedvalue=@(x) normcdf((mu+rho*sigma*x)/sqrt(1+sigma^2*(1-rho^2)))*normpdf(x);
(0.4)/p(r)=quad(expectedvalue,-999999,c(r));
------------------------------------------------------------------------------------------------
Function:

function f = funrr(mu,a,b)


q=100; % Number of runs
varomikron=0.04;


randZ=randn(q,12500);
randEps=randn(q,12500);
randOmik=normrnd(0,sqrt(varomikron),q,12500);

sigma=sqrt(b^2+varomikron);
rho=((a*b)/sigma);
l=zeros([20 125]);
Portfolioverlust=zeros(20,q);
Epsilon=zeros(20,1);
Omikron=zeros(20,1);
X=zeros(20,1);
Z=zeros(20,1);
v=zeros(q,1);
s=zeros(q,1);
pvprot=zeros(21,q);
pvdef=zeros(21,q);
LA=zeros(20,q);
LB=zeros(20,q);
LTranche=zeros(21,q);
spread=zeros(5,1);

spreadmodel=[0.0723309797736948 0.00474353239675 0.00125486203410625 0.0005703876529 0.00022735709814375];


r2007=xlsread('C:\Universitaet\Diplomarbeit\Daten\Spotrates','Tabelle1','A2:A21');
stetverzinsung=zeros(20,1);
for i = 1:20
stetverzinsung(i)=exp(-r2007(i)*(i/4));
end
k=1:20;
p=1-exp((-0.003514966*k)/4);
c = norminv(p);

Attachment=[0.00001 0.03 0.06 0.09 0.12];
Detachment=[0.03000 0.06 0.09 0.12 0.22];

for d=1:5
A=Attachment(d); %Attachment Punkt
B=Detachment(d); %Detachment Punkt
for i=1:q
Z(1)=randZ(i,(d-1)*20+1);
for j=1:125
Epsilon(j)=randEps(i,(d-1)*2500+j);
Omikron(j)=randOmik(i,(d-1)*2500+j);
X(j)=a*Z(1)+sqrt(1-a^2)*Epsilon(j);
if (X(j)<=c(1))
l(1,j)=1-normcdf(mu-b*Z(1)+Omikron(j));
else l(1,j)=0;
end
end
Portfolioverlust(1,i)=sum(l(1,:))/125;
for r=2:20
Z(r)=randZ(i,(d-1)*20+r);
for j=1:125
if (l(r-1,j)>0)
l(r,j)=l(r-1,j);
else
Epsilon(j)=randEps(i,(r-1)*125+(d-1)*2500+j);
Omikron(j)=randOmik(i,(r-1)*125+(d-1)*2500+j);
X(j)=a*Z(r)+sqrt(1-a^2)*Epsilon(j);
if (X(j)<=c(r))
l(r,j)=1-normcdf(mu-b*Z(r)+Omikron(j));
else l(r,j)=0;
end
end
end
Portfolioverlust(r,i)=sum(l(r,:))/125;
LA(r,i)=max(Portfolioverlust(r,i)-A,0);
LB(r,i)=max(Portfolioverlust(r,i)-B,0);
LTranche(r+1,i)=(1/(B-A))*(LA(r,i)-LB(r,i));
end
for h=2:21
pvprot(h-1,i)=stetverzinsung(h-1)*(LTranche(h,i)-LTranche(h-1,i));
pvdef(h-1,i)=0.25*stetverzinsung(h-1)*(1-LTranche(h,i));
end
if (d==1)
v(i)= -sum(pvdef(:,i))*0.05+sum(pvprot(:,i));
else s(i)=sum(pvprot(:,i))/sum(pvdef(:,i));
end
if (d==1)
spread(1)=sum(v)/q;
else spread(d)=sum(s)/q;
end
end
end

f = spread(1)-spreadmodel(1)+spread(2)-spreadmodel(2)+spread(3)-spreadmodel(3)+spread(4)-spreadmodel(4)+spread(5)-spreadmodel(5);
From: Alan Weiss on 21 Jul 2010 08:34
On 7/20/2010 5:55 PM, Philip M wrote:
> Hey guys,
>
> I want to minimize the following function. I'm running a monte carlo
> simulation to calculate spreads of Collateralized Debt Obligations
> (CDOs). I want to minimize the difference of my model spread and the
> market spread by varying my model parameters "mu","a" and "b". To keep
> this model valid, a constraint has to be fulfilled for every point of
> valuating (r=1,...,20). I want to vary "mu" to keep the constraint
> fulfilled. In the end I would like to have a single parameter "a" and a
> single parameter "b" but a parameter vector "mu". I don't know how to
> set this up, which formula to use for multivariate minimization with
> constraints and how to tell matlab that it should vary only "mu" to
> match the constraints.
>
> Thanks alot,
>
> Philip!
> -----------------------------------------------------------------------------------------------
>
> Constraint:
>
> expectedvalue=@(x)
> normcdf((mu+rho*sigma*x)/sqrt(1+sigma^2*(1-rho^2)))*normpdf(x);
> (0.4)/p(r)=quad(expectedvalue,-999999,c(r));
> ------------------------------------------------------------------------------------------------

Take a look at
http://www.mathworks.com/access/helpdesk/help/toolbox/optim/ug/brhkghv-7.html
to see how to pass extra parameters. In your case, if I understand, you
want a and b to be fixed, and you want mu to vary. So you would write
your objective as
function f = myobjective(a,b,mu)
....
and then call myobjective with the syntax
@(mu)myobjective(a,b,mu)

I recommend using the patternsearch solver for your problem, assuming
you have a Global Optimization Toolbox license (or Genetic Algorithm and
Direct Search Toolbox). If you have just Optimization Toolbox, I suggest
setting large values for the DiffMinChange and DiffMaxChange options,
since your objective function is going to be noisy. How large? It
depends on your problem, but large enough that your finite difference
estimates of gradients will be somewhat accurate.

If mu is one-dimensional, use the fminbnd solver, after you figure out
the bounds by experimentation.

Alan Weiss
MATLAB mathematical toolbox documentation
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