% Define the parameters of the model
dTm = 0.1; % Mean-reversion speed parameter
a0 = 0.5; % Fourier series coefficient
a1 = 0.2; % Fourier series coefficient
ac = [0.3, 0.4, 0.1]; % Fourier series coefficients
as = [0.1, 0.2, 0.3]; % Fourier series coefficients
K = 3; % Number of terms in the Fourier series
Tm = 25.0; % Terminal condition parameter
% Define the grid parameters
tStart = 0; % Start time
tEnd = 1; % End time
tStep = 0.01; % Time step size
TStart = 20; % Start temperature
TEnd = 30; % End temperature
TStep = 0.1; % Temperature step size
% Create the grid
t = tStart:tStep:tEnd;
T = TStart:TStep:TEnd;
% Initialize the solution grid
U = zeros(length(T), length(t)); % Initialize the grid with zeros
% Loop over time steps
for n = 2:length(t)
% Update the temperature values based on the PDE using finite differences
% Compute the fuzzy coefficients based on Tm and the Fourier series
alpha = a0 + a1 * t(n) + sum(ac .* cos(K*t(n)*Tm) + as .* sin(K*t(n)*Tm));
sigma = abs(sin(t(n))); % Just an example for fuzzy volatility, modify as needed
% Compute the finite difference approximation
for i = 2:length(T)-1
U(i, n) = U(i, n-1) + tStep * (dTm * (Tm - T(i)) + alpha * (Tm - T(i)) + sigma * randn);
end
end
% Inference of Risk-Neutral Probability and Optimization Problem
Fm = [10, 12, 14, 16, 18]; % Observed market prices of weather futures
tm = 1; % Expiration time of weather futures
delta = 0.95; % Risk-free discount factor
r = 0.03; % Risk-free interest rate
% Define the Risk-Neutral Expectation operator E_Q
% Modify this part based on your specific requirements
E_Q = @(I_Htm, Ft) mean(Ft(I_Htm)); % Define your E_Q function here
% Define the condition for weather futures I_Htm
% Modify this part based on your specific requirements
I_Htm = T > 25; % Define your I_Htm condition here
% Define additional variables for the E_Q calculation
% Modify this part based on your specific requirements
Ft = U(:, end); % Define your Ft variable here
% Set the optimization problem
objective = @(fQ) sum((delta * exp(-r * (tm - t)) .* E_Q(I_Htm, Ft, fQ) - Fm).^2);
x0 = [0.1, 0.2, 0.3]; % Initial guess for the risk-neutral probabilities
% Define the optimization solver options
options = optimoptions('fmincon');
options.Algorithm = 'interior-point'; % Choose the algorithm (e.g., 'interior-point', 'sqp', 'active-set')
options.MaxIterations = 100; % Maximum number of iterations
options.Display = 'iter'; % Display the iterative output
% Define constraints for fmincon
% Define the constraints for fmincon
A = [1, -1, 1; -1, 2, 3]; % Coefficients of the linear inequality constraints
b = [5; 10]; % Right-hand side values of the linear inequality constraints
Aeq = [2, 3, -1]; % Coefficients of the linear equality constraints
beq = [4]; % Right-hand side values of the linear equality constraints
lb = [0; 0; 0]; % Lower bounds on the variables
ub = [1; 1; 1]; % Upper bounds on the variables
% Solve the optimization problem
[fQ_opt, fval] = fmincon(objective, x0, A, b, Aeq, beq, LB, UB, [], options);
% Display the optimal risk-neutral probabilities and the minimum objective value
disp('Optimal Risk-Neutral Probabilities:');
disp(fQ_opt);
disp('Minimum Objective Value:');
disp(fval);
% Plot the solution
surf(t, T, U);
xlabel('Time');
ylabel('Temperature');
zlabel('Solution');
