UpperBound3(S, g, df, B, f, Nr, NSim, NSSim, ...

% This is material illustrating the methods from the book
% Financial Modelling  Theory, Implementation and Practice with Matlab
% source
% Wiley Finance Series
% ISBN 9780470744895
%
% Date: 02.05.2012
%
% Authors: Joerg Kienitz
% Daniel Wetterau
%
% Please send comments, suggestions, bugs, code etc. to
% kienitzwetterau_FinModelling@gmx.de
%
% (C) Joerg Kienitz, Daniel Wetterau
%
% Since this piece of code is distributed via the mathworks fileexchange
% it is covered by the BSD license
%
% This code is being provided solely for information and general
% illustrative purposes. The authors will not be responsible for the
% consequences of reliance upon using the code or for numbers produced
% from using the code.
function [lower, upper] = UpperBound3(S, g, df, B, f, Nr, NSim, NSSim, ...
getpaths, payoff)
% method from Broadie for computing upper bounds, see Chapter 8
iVec = 1:NSSim;
v = g(:,end); % start for backward induction
c = zeros(NSim,Nr1); % continuation value
% backward induction and regression from t_{Nr1} up to t_1
for i = Nr1:1:1
index = find(g(:,i) > 0); % all ITM paths
s = S(index,i+1); % values of S at given time point
v = v * df(i+1); % option value at t_i
Acell = B(s); % evaluate basis function in cell array B
A = cell2mat(Acell{:,:}); % convert to matrix
c(index,i) = A*f(:,i); % continuation value
exercise = g(index,i) >= c(index,i); % early exercise
v(index(exercise)) = g(index(exercise),i);
end
lower = mean(v * df(1)); % final option value
% Computing the martingale numerically, martingale = pi
L = zeros(NSim,1); %erster Teil von delta_i vgl. Formel (6.6)
expectation = zeros(NSim,Nr); %zweiter Teil von delta_i vgl. Formel (6.6)
expectation(:,1) = lower * ones(NSim,1);
for i=1:1:Nr1
for j=1:NSim
expectation(j,i+1) = subsimulation(S(j,i+1), df, B, NSSim, Nri,...
f(:,i:end), iVec(1:NSSim), getpaths, payoff) * prod(df(1:i));
end
end
pi = zeros(NSim,Nr+1); % stores the values of the constructed martingale
% first time step and last are different
i_exercise = g(:,1) >= c(:,1) & c(:,1) > 0; % exercise in this case
L(i_exercise) = g(i_exercise, 1) * prod(df(1:1));
L(~i_exercise) = expectation(~i_exercise,2);
pi(:,2) = pi(:,1) + L  expectation(:,1);
for i=2:1:Nr1
% exercise in this case if not already exercised
i_exercise = g(:,i) >= c(:,i) & c(:,i) > 0 & ~i_exercise;
L(i_exercise) = g(i_exercise, i) * prod(df(1:i));
L(~i_exercise) = expectation(~i_exercise,i+1);
pi(:,i+1) = pi(:,i) + L  expectation(:,i);
end
% finally exercise if in the money and not already exercised
i_exercise = g(:,Nr) > 0 & ~i_exercise;
pi(i_exercise,Nr+1) = pi(i_exercise,Nr) + g(i_exercise,Nr)* prod(df(1:Nr));
pi(~i_exercise,Nr+1) = pi(~i_exercise,Nr);
% upper bound using the martingale pi
maximum = zeros(NSim,1);
for j=1:1:NSim
maximum(j) = max(g(j,:)  pi(j,2:end)); %vgl. Formel (6.5)
end
upper = mean(maximum);
end
function y = subsimulation(S0, df, B, NSim, Nr, beta, iVec, gp,payoff)
S2 = gp(S0,NSim,Nr); S2 = S2(:,2:end); % paths
g2 = payoff(S2); % payoff
exercise = Nr * ones(NSim,1); % exercise per path
for i=1:1:Nr1
i_nexercised = exercise == Nr;
I_nexercise = iVec(i_nexercised);
s = S2(i_nexercised,i);
Acell = B(s);
A = cell2mat(Acell{:,:});
c = A * beta(:,i);
i_exercise = g2(i_nexercised,i) >= c & g2(i_nexercised,i) > 0;
exercise(I_nexercise(i_exercise)) = i;
end
summe=0;
for j=1:1:NSim
summe = summe + g2(j,exercise(j)) * prod(df(1:exercise(j)));
end
y = summe / NSim; % MC value from subsimulation
end

