% This is material illustrating the methods from the book
% Financial Modelling - Theory, Implementation and Practice with Matlab
% source
% Wiley Finance Series
% ISBN 978-0-470-74489-5
%
% 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 file-exchange
% 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 [price, se, low, high] ...
= LongstaffSchwartz_M(S, M, Ns, g, df, B, Nr, NSim, level)
% Longstaff Schwartz method for pricing american options
v = g(:,end); % start for backward induction
% backward induction and regression from t_{Nr-1} up to t_1
for i = Nr-1:-1:1
index = find(g(:,i+1) > 0); % all ITM paths
s = S(index,i+1); % values of S at given time points
m = M(index,i+1); % values of M at given time points
v = v * df(i+1); % option value at t_i
Acell = B(s,m); % evaluate basis function in cell array B
A = cell2mat(Acell{:,:}); % convert to matrix
if i == 1 % for this period A is singluar therefore we only take S
A=[A(:,1:Ns) A(:,7)];
end
f = (A'*A)\(A'*v(index)); % determine coefficients
c = A*f; % continuation value
exercise = g(index,i+1) >= c; % early exercise
v(index(exercise)) = g(index(exercise),i+1);
end
price = mean(v * df(1)); % final option value
% standard error and confidence interval
sv = sqrt(1/(NSim-1)*sum((v - price * ones(NSim,1)).^2));
se = sv/sqrt(NSim);
low = price - norminv(level) * sv/sqrt(NSim);
high = price + norminv(level) * sv/sqrt(NSim);
end