Using fmincon to optimize montecarlo

Hi All! I've written a montecarlo function for option pricing. I've pasted the code below. Now I am trying to use fmincon to find what value of s would maximize the output (avg) (so minimize -avg). i've tried reading the online help but i'm still pretty confused. can anyone point me in the right direction on how to start? i want p0 and k to be 50 and 52 respectively.

function [avg] = montecarlo(p0,k,s)
%UNTITLED8 Summary of this function goes here
%   Detailed explanation goes here
shocks = zeros (1000,20);   
y = randn(1000,20);
shocks = y;
logps = zeros (1000,20);

logps(:,1)= log(p0);

for i = 2:20
    for j = 1:1000
        logps(j,i) = logps(j,i-1)+.09*shocks(j,i);
    end
end

prices = exp(logps);

profits = zeros(1000,20);
payoffs = zeros(1000,1);
for i = 1:1000
    for j = 1:20     
        if (prices(i,j) > s)
            payoffs(i) = prices(i,j) - k;
            break
        else
                payoffs(i) = 0; 

          end
      end

end

avg = mean(payoffs);

end

Answer 1

[Matt J]

Your montecarlo() function is a piecewise constant function of s, and therefore is not differentiable. fmincon is therefore the wrong tool.

Really, what you should do is pre-compute the "prices" matrix - it never changes throughout the problem - and just evaluate montecarlo() for all s in the set of values returned by

>> [unique(prices);-Inf]

and look for the minimum. Since there are at most 20000 such values, it should be an easy brute force search.

You should also simplify your computation of avg to this more efficient one-liner

 avg = -k*mean(any(prices>s,2))

[Matt J]

In fact, your function is also a monotonic function of s, with a trivial minimum at

s=min(prices(:))-c;  %c>0 is any positive constant

So, I wonder if there isn't a mistake in your formulation.

[Matt J]

Really, what you should do is pre-compute the "prices" matrix - it never changes throughout the problem

Another reason to remove "prices" from your function and to pre-compute it is because it is random. It is impossible to iteratively minimize a function whose definition changes randomly every time the function is evaluated.