Optimization and Simulation for USA Presidential
Elections
Professor Karl Sigman
Columbia University
New York City
USA
1/20
USA Presidential Election Modeling
The winner of a USA Presidential Election
c 2016 by Karl Sigman
Copyright
1
Modeling risky assets and pricing options: The Binomial Lattice Model (BLM)
In these notes we will present a Markov chain model for risky assets known as the binomia
c 2015 by Karl Sigman
Copyright
1
Gamblers Ruin Problem
Let N 2 be an integer and let 1 i N 1. Consider a gambler who starts with an initial
fortune of $i and then on each successive gamble either wi
c 2016 by Karl Sigman
Copyright
1
1.1
Discretetime Markov chains
Stochastic processes in discrete time
A stochastic process in discrete time n IN = cfw_0, 1, 2, . . . is a sequence of random variabl
Random Number Generators
Professor Karl Sigman
Columbia University
Department of IEOR
New York City
USA
1/14
Introduction
Your computer generates" numbers U1 , U2 , U3 , . . . that are
considered inde
Applications Programming for Financial Engineering
INDUSTRIAL 633

Spring 2012
IEOR 4500
Quick Review of the Principal Components Method
Suppose Q is the covariance matrix for the returns of n assets. Then Q is symmetric
(qij = qji for all indices i, j ) and positivesemidenite
Applications Programming for Financial Engineering
INDUSTRIAL 633

Spring 2012
IEOR 4500
Maximizing the Sharpe ratio
Suppose we have the setting for a meanvariance portfolio optimization problem:
,
the vector of mean returns
Q,
the covariance matrix
xj = 1, (proportions add to
Applications Programming for Financial Engineering
INDUSTRIAL 633

Spring 2012
How to solve simple QPs The problem we discussed in class had the following general structure: Minimize
i 2 i x2 + 2 i i<j
ij xi xj

i
i xi , (*)
Subject to the constraints:
i
xi = 1, and
li xi ui ,
Applications Programming for Financial Engineering
INDUSTRIAL 633

Spring 2012
IEOR 4500
Factor models
Suppose we have a collection of n assets. A factor model for the asset returns is a statistical
model of the form:
r = +
+ V Tf
(1)
where
is the nvector of expected returns (