%% price testing thing
N = 3;
T = 1;
dT = T/N;
u = exp(sigma*sqrt(dT)); %p85
d = exp(-sigma*sqrt(dT)); %p85
q = (exp_r^dT - d)/(u-d);
pricemat = pricematfun(S0, u, d, N)
derivpricematfun(pricemat, K, q)

%% price testing thing
N = 4;
T = 1;
dT = T/N;
K = 100;
S0 = 90;
exp_r = 1.05;
sigma = 0.2;

u = exp(sigma*sqrt(dT)); %p85
d = exp(-sigma*sqrt(dT)); %p85
q = (exp_r^dT - d)/(u-d);
pricemat = pricematfun(S0, u, d, N);
derivpricemat = derivpricematfun(pricemat, K, q);
derivpricemat(1,1)


%%
% 3.1
% call = right to buy at price K
% put = right to sell at price K

K = 100;
S0 = 90;
exp_r = 1.05;
sigma = 0.2;
N = 4;
T = 1;

% dT = T/N;
% u = exp(sigma*sqrt(dT)); %p85
% d = exp(-sigma*sqrt(dT)); %p85
% qu = (exp_r^dT - d)/(u-d);
% qd = (u-exp_r^dT)/(u-d);
pi0_eu = zeros(1,N);
%pi0_eu = pi0_value_europecall(T, N, sigma, exp_r, S0, K)
for n = 1:N
pi0_eu(n) = pi0_value_europecall(T, n, sigma, exp_r, S0, K);
end
plot(pi0_eu)
pi0_eu(end)
%(St-k)+, eu call, St > k




%% 3.2
% K = 100;
% S0 = 90;
% exp_r = 1.05;
% sigma = 0.2;
% N = 3;
% T = 1;

pi0_us = zeros(1,N);
for n = 1:N
pi0_us(n) = pi0_value_americanput(T, n, sigma, exp_r, S0, K);
end

plot(pi0_us)
pi0_us(end)

%% 3.3

%% forward contract at t=0
% K = 100;
% S0 = 90;
% exp_r = 1.05;
% sigma = 0.2;
% N = 3;
% T = 1;

pi0_forward = zeros(1,N);
for n = 1:N
pi0_forward(n) = pi0_value_forward(T, n, sigma, exp_r, S0, K);
end

plot(pi0_forward)
pi0_forward(end)

lower = S0-K
mid = pi0_eu-pi0_us(end)
upper = pi0_forward(end)



%%
function y = europe_call(K, price)
y = max(price - K, 0);
end

function y = american_put(K, price_current)
y = max(K - price_current, 0);
end

function y = forward(S0, price_current)
y = max(S0 - price_current, 0);
end


function pi0 = pi0_value_europecall(T, N, sigma, exp_r, S0, K)
dT = T/N;
u = exp(sigma*sqrt(dT)); %p85
d = exp(-sigma*sqrt(dT)); %p85

qu = (exp_r^dT - d)/(u-d);
qd = (u-exp_r^dT)/(u-d);

sum = 0;
for k = 0:N
    price = S0*u^k*d^(N-k);
    sum = sum + nchoosek(N,k)*qu^k*(1-qu)^(N-k)*europe_call(K, price);
end
pi0 = exp_r^T*sum;

end

function pi0 = pi0_value_americanput(T, N, sigma, exp_r, S0, K)
dT = T/N;
u = exp(sigma*sqrt(dT)); %p85
d = exp(-sigma*sqrt(dT)); %p85

qu = (exp_r^dT - d)/(u-d);
qd = (u-exp_r^dT)/(u-d);

sum = 0;
for k = 0:N
    price = S0*u^k*d^(N-k);
    sum = sum + nchoosek(N,k)*qu^k*(1-qu)^(N-k)*american_put(K, price);
end
pi0 = exp_r^T*sum;

end

function pi0 = pi0_value_forward(T, N, sigma, exp_r, S0, K)
dT = T/N;
u = exp(sigma*sqrt(dT)); %p85
d = exp(-sigma*sqrt(dT)); %p85

qu = (exp_r^dT - d)/(u-d);
qd = (u-exp_r^dT)/(u-d);

sum = 0;
for k = 0:N
    price = S0*u^k*d^(N-k);
    sum = sum + nchoosek(N,k)*qu^k*(1-qu)^(N-k)*forward(S0, price);
end
pi0 = exp_r^T*sum;

end



function pricemat = pricematfun(S0, u, d, N)
    pricemat = zeros(N+1,N+1);
    for n = 1:N+1
        for i = 1:n
            pricemat(i,n) = S0*u^(i-1)*d^(n-i);
            %S0*u^k*d^(N-k)
        end
    end
end

function derivpricemat = derivpricematfun(mat, K, q)
    derivpricemat = zeros(size(mat));
    N = length(mat);
    for n = 1:N
        for i = 1:N-n+1
            if n == 1
                derivpricemat(i,N-n+1) = max(mat(i,N-n+1)-K,0);
            else
                derivpricemat(i,N-n+1) = derivpricemat(i,N-n+2)*(1-q)+derivpricemat(i+1,N-n+2)*(q);
            end

        end
    end

%     derivpricemat = derivpricemat(1:end-1,1:end-1);
%     derivpricemat = flip(derivpricemat,2)';
end