% 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 [lower, upper] = ...
UpperBound1(S, g, df, B, beta, lower, Nr, NSim, NSSim, ...
getpaths, payoff)
% method from Broadie see Chapter 8
v = g(:, end); % option value at maturity
c = zeros(NSim,Nr-1); % continuation value
stoppingtime = Nr * ones(NSim,1);
for i=Nr-1:-1:1
index = find(g(:,i)>0);
s = S(index, i+1);
v = v * df(i+1);
Acell = B(s);
A = cell2mat(Acell{:,:});
c(index,i) = A*beta(:,i); % cont value itm paths
earlyexercise = g(index,i) >= c(index,i);% exercise
stoppingtime(index(earlyexercise)) = i;
v(index(earlyexercise)) = g(index(earlyexercise),i);
end
% Computing the martingale pi
L = zeros(NSim,1);
expectation = zeros(NSim,Nr);
expectation(:,1) = lower * ones(NSim,1);
pi = zeros(NSim,Nr+1); % martingale
pi(:,1) = lower * ones(NSim,1); % set pi(0)
% perform the subsimulation for each value of the path set
for i=1:1:Nr-1
for j=1:NSim
expectation(j,i+1) = subsimulation(S(j,i+1), df, B, NSSim, Nr-i,...
beta(:,i:end), getpaths, payoff) * prod(df(1:i));
end
end
for i=1:1:Nr
i_exercise = stoppingtime == i; % check if early exercised
if i < Nr
L(i_exercise) = g(i_exercise, i) * prod(df(1:i));
L(~i_exercise) = expectation(~i_exercise,i+1);
else
L(i_exercise) = g(i_exercise, i) * prod(df(1:i));
end
pi(:,i+1) = pi(:,i) + L - expectation(:,i);
end
maximum = zeros(NSim,1);
for j=1:1:NSim
maximum(j) = L(j) + max(g(j,:) - pi(j,2:end)); % compute the max
end
upper = mean(maximum);
end
function y = subsimulation(S0, df, B, NSim, Nr, beta, gp,payoff)
S2 = gp(S0,NSim,Nr); S2 = S2(:,2:end); % paths
g2 = payoff(S2); % payoff
iVec = 1:NSim;
exercise = Nr * ones(NSim,1); % exercise per path
% determine exercise strategy for path set S2
for i=1:1:Nr-1
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