Solving non linear system of equations

Hello!I have some troubles while solving the non-linear system.

http://saveimg.ru/show-image.php?id=19f0894034eb37dc9fef4a7a088c2a55 this is a link to my system, which I need to solve against V and sigma_v

I wrote the following code in mm-file

function y = mm(x,D,r,delta_t,S,sigma_s)
d1=(log(x(2)/D)+(r-0.5*x(1)*x(1)*delta_t))/(x(1)*sqrt(delta_t));
d2=d1-x(1)*sqrt(delta_t);
y=((sigma_s*S-normcdf(d1,0,1)*x(1)*x(2))^2+(S-x(2)*normcdf(d1,0,1)-D*exp(-r*delta_t)*normcdf(d2,0,1))^2);

and I use the following command:

x=fminsearch('mm(x,5000000,0.0071,2548,74513269760,0.000665992)',[10 0.0001])

but the answer is 10 0.00001 which is not correct

So I'd be glad if someone shows me a mistake, which I made

Thx in advance!

Answer 1

[Teja Muppirala]

Wow. Now this is really like finding a needle in a haystack.

But FMINSEARCH can find it if you just give it a better starting condition, and enough function evaluations:

opts = optimset('tolfun',0,'tolx',0,'maxfun',Inf,'display','iter');
xmin = fminsearch(@(x)mm(x,5000000,0.0071,2548,74513269760,0.000665992),[0.0001 1e10],opts)

This gives you the answer:

xmin(1) = 6.659920000005961e-004
xmin(2) = 7.451326975993047e+010

The difficulty was in finding a good initial condition. How did I do it? By making a giant plot over logarthmic space:

jvec = linspace(-10,20,201);
kvec = linspace(-10,20,201);

A = zeros(numel(jvec),numel(kvec));

for jj = 1:numel(jvec)
    for kk = 1:numel(kvec)
        A(jj,kk) = mm(10.^[jvec(jj) kvec(kk)],5000000,0.0071,2548,74513269760,0.000665992);
    end
end

figure
imagesc(kvec,jvec,log10(A)); colorbar;

Answer 2

[terance]

That's cool!!!Thx for your help!But I have a silly questuin: where did you find on a plot the initial condition?Am I right, that if there is a red light then it is max and vise verse for blue one.

PS when I use this script on my Mac I dont receive the same results as you((( I receive the following

xmin =

   1.0e+10 *

    0.0000    7.4513

[Teja Muppirala]

Your results are the same as mine. This is just a difference in how many digits are displayed. If you do

format long

and then view the result, it is the same. Also, because the numbers are very differently scaled, it helps to view them one at a time

xmin(1)

xmin(2)

On the plot, red means high and blue means low, just as the colorbar indicates. I just looked for where there were blue areas (and actually I replotted that graph several times, changing the limits and zooming in a little each time until I was pretty certain where a good initial condition was).