MS&E111
Lecture 9
Introduction to Optimization
May 13, 2006
Prof. Amin Saberi
Scribed by: Yusuf Ozuysal
1
Applications of Linear Programming in Finance
Linear programming is one of the most widely used tools in finance. Using LP’s for solving
optimization problems encountered in finance has been a common practice for many years
and a great percentage of the courses thought on optimization in finance are devoted to
LP’s. While dealing with LP’s in finance one mostly needs a well established background on
probability theory and statistics also. However in this lecture, we will be visiting a simplified
version of a general problem linear problem in finance and using this simple instance to get
an idea of the general properties of such problems in finance.
1.1
Portfolio Optimization
A portfolio can be defined as a collection of financial assets where the cost for the assets are
known at the beginning of a period, but there are only statistical data available about the
change in these costs over the period. In portfolio optimization we are trying to buy from a
certain set of stocks in such a way that the value of the portfolio at the end of this period
will be maximized.
Example:
Let us look at the matrix of relative percentage gains given below. In the matrix
each column corresponds to a different asset and each row corresponds to a different event or
a scenario for the period we are interested in. So the entry
P
ij
is the percentage gain we will
receive from asset
j
, if scenario
i
happens. We are trying to find the optimal distribution of
assets that maximizes the estimated overall gain.
P
=
0
.
2
1
1
.
2
8
1
.
05
1
.
1
1
.
2
1
.
1
1
1
.
05
1
1
1
.
1
0
.
6
1
.
05
1
.
3
0
.
9
0
.
7
0
.
2
1
.
05
1
.
2
1
.
5
0
.
8
0
.
2
1
.
05
As can be seen from the matrix, the assets offer a variety of percentage gains for different
scenarios. For example, the asset in the last column can be thought as a bank investment
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MS&E 111: Introduction to Optimization  Spring 06
where we are gaining 5 percent of our investment no matter what happens in the market.
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 Spring '06
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 Operations Research, Linear Programming, LP

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