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 is not determined by the
nations popular vote.
EXAMPLE
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 binomial
lattice model, and then learn how to price options (d
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 wins $1 or loses $1 independent of the
past with probabil
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 variables
(rvs) X0 , X1 , X2 , . . . denoted by X = cfw_Xn : n
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 independent and uniformly randomly distributed on the
conti
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 (v T Qv 0 for any vector v Rn
this is denoted Q 0). We
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 1)
(1)
(2)
(3)
Ax b, (other linear constraints).
0 x.
(
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 , for all i. The method we discussed in class consisted
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 (computed from historical data),
r is the nvector of r