Simple Solver-type function

I am new to MATLAB and was wondering if someone could help me with a simple function. The idea is as follows:

I have two equations:

r_u = r0 + theta*delta + sigma*sqrt(delta)

r_d = r0 + theta*delta - sigma*sqrt(delta)

I need to find theta such that the above equations are satisfied as well as the equation below:

P = exp(-r0*delta)[0.5*exp(-r_u*delta) + 0.5*exp(-r_d*delta)]*100.

r0, delta, sigma and P are known. I need the function to calculate theta, r_u and r_d such that the above equations are true. I can do it in Excel using Solver but MATLAB is new to me and I would really appreciate any help I can get.

Answer 1

[Matt Tearle]

If you have Optimization Toolbox, you can use fsolve. Rewrite the three equations in the form F(x) = 0, where x is a vector of your three unknowns.

% set values
r0 = rand;
delta = rand;
sigma = rand;
P = rand;

% make function
f = @(x) [r0 + x(3)*delta + sigma*sqrt(delta) - x(1);
r0 + x(3)*delta - sigma*sqrt(delta) - x(2);
exp(-r0*delta)*(0.5*exp(-x(1)*delta) + 0.5*exp(-x(2)*delta))*100 - P];

% initial guess
x0 = rand(3,1);
% find solution
x0 = fsolve(f,x0)

Here, x(1) is r_u, x(2) is r_d, x(3) is theta.

EDIT TO ADD Having looked more closely at the math, I realized that you don't even need fsolve. The problem can be reduced to a single equation in one variable (theta):

% set values
r0 = rand;
delta = rand;
sigma = rand;
P = rand;

% make functions
r_u = @(theta) r0 + theta*delta + sigma*sqrt(delta);
r_d = @(theta) r0 + theta*delta - sigma*sqrt(delta);
f = @(theta) exp(-r0*delta)*(0.5*exp(-r_u(theta)*delta) + 0.5*exp(-r_u(theta)*delta))*100 - P;

% initial guess
theta0 = rand;
% find solution
theta0 = fzero(f,theta0)
r_u(theta0)
r_d(theta0)

[Julia]

My God, it's rediculously simple. Thank you very much for your help.

[Matt Tearle]

Wait until you see what happens when I actually think for a moment. See the above edit...