Matematica per l’Economia e la Finanza
Part I: Matlab Programming and Optimization Methods in
Portfolio Management.
M. Dolfin
Univ. of Messina - Italy
https://www.marinadolfin.it
UNIME, April 2017
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Outline
Part I
Matlab programming and optimization methods in portfolio
management.
Part II
Monte Carlo simulations - Option Pricing with Binomial
lattices and Black-Scholes formula
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Motivation
Why Matlab
Both basic and sophisticated functions;
built-in functions or personally designed M-files;
rich set of graphical capabilities;
useful toolboxes;
widely diffused;
efficiently used in computational finance.
Many functionalities within a single integrated framework.
www.mathworks.com
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Motivation
Need for theory
Developing your own solutions;
choose the most appropriate algorithm;
use methods properly;
understand what is going wrong.
Computational mathematics may be quite different from
’pencil-and-paper’ mathematics.
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Readings
P. Brandimarte,
Numerical Methods in Finance and Economics. A
Matlab-based introduction, Wiley-Interscience (2006).
W. J. Palm III,
Introduction to Matlab for Engineers (3rd ed.), McGraw-Hill
(2011).
www.mathworks.com
Further readings
M. J. Miranda, P.L. Fackler, Applied computational Economics
and Finance, MIT Press, Cambridge (2002)
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Start Matlab
Standard desktop configuration
Current Directory
(top left).
Command Window (center).
Workspace
(top right).
Detail
(bottom left).
Command History
(bottom right).
Symbols of basic operations:
+
-
/
*
ˆ
\
= to register a value into a specific variable.
==
Equality operator.
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Exercises
Compute
6
10 13
+
18
+ 5(92 )
5(7)
(ans = 410.1297)
6(351/4 ) + 140.35
(ans = 17.1123)
Ex.1
A cylinder tank has height h = 15m and radius r = 8m.
Compute the radius of a tank with the same height and a
volume 20% greater (V = πr 2 h). (ans = 8.7636)
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Numerical arrays
Ordered collection of numbers
x = [0, 4, 3, 6]
y = [2, 7, 1, 6]
z = x + y to sum two arrays
Equally spaced numbers: an example
u = 0 : 0.1 : 10;
w = 5 ∗ sin(u);
plot(u, w)
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Ex.2
Plot the function
s = 2 sin(3t + 2) +
√
5t + 1
over the interval 0 ≤ t ≤ 5, in the same window of the plot of
the previous function.
Put a title and properly label the axis.
Edit the plot and change the thickness of the line and the
fonts.
Save the figure in the desktop.
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Matrices
You may think matrices as a collection of numbers
z columns
}|
{
(1 3 5 2
rows 8 4 2 0
7 0 2 8
in a square format
Example: index the rows by companies (stocks) and the
columns by dates. Sensible data: stock price on a day.
dates
z
}|
{
(400 500 5000
companies 200 100 4500
100 100 5500
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M. Dolfin - Univ. of Messina - ITALY
Matrix dimension
Number of rows first followed by the number of columns.
1 3 5 2
A = 8 4 2 0
7 0 2 8
A3×4 ;
A(3 × 4)
One may use the notation (days × #assets) to identify not only
the size of the matrix but also the nature of the information.
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Tables
Information are appended on the left and top borders
Mon
Tue
Wed
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Bond
54
55
56
Stock
21
18
27
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Example: Prices of two assets (a bond and a stock)
One bond costs $54 and one stock costs $21; a two-element
row vector describes the data
p = 54 21
whose entries are the prices (expressed in an appropriate unit
of measure).
Numbers of shares of the two securities
Two-element column vector
1
n=
,
D(days × #securities)
2
One can evaluate the portfolio with the matrix product.
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Rules of multiplication
In symbols
P = p1 p2 ;
q1
Q=
q2
V = p1 · q1 + p2 · q2
Matrix multiplication - Rule
The numbers of columns in the first matrix must be equal to the
number of rows in the second → the inner dimensions of the
two matrix must be equal.
(m × p) · (p × n) → (m × n)
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Rules of multiplication
In symbols
p11 q1 + p12 q2
p11 p12 p21 p22 q1 = p21 q1 + p22 q2
q2
p31 q1 + p32 q2
p31 p32
(m × p) · (p × n) → (m × n)
(3 × 2) (2 × 1) → (3 × 1)
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Ex.3
Portfolio evaluation: multiply the number of shares of each
security by the corresponding price of shares of each security
and then sum the results
Matrix of Prices
Matrix of Quantities
P(1 × #assets),
Q(#assets × 1),
Value = product of price and quantity
Master in Economia Bancaria e Finanziaria
P = 54 21 .
1
Q=
.
2
V =P∗Q.
M. Dolfin - Univ. of Messina - ITALY
Ex.3 (with different values) - Value of a portfolio on 3
different days
Prices
Mon
Tue
Wed
Bond
54
55
56
Stock
21
18
27
Quantities
Bond
Stock
1
2
Table of results
Master in Economia Bancaria e Finanziaria
Mon
Tue
96
91
M. Dolfin - Univ. of Messina - ITALY
Ex.3 - Value of two portfolios on 3 different days.
Prices
Mon
Tue
Wed
Bond
54
55
56
Stock
21
18
27
Bond
Stock
PortA PortB
1
5
2
2
Mon
Tue
Wed
PortA PortB
96
312
91
311
110
334
Quantities
Table of results
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Addition
Two matrices with the same dimension may be added together.
Result: a new matrix in which each element is the sum of the
corresponding elements of the two matrices.
Exercise n.5
port A =
Bond
Stock
1
;
2
port all = port A + port B;
port B =
Bond
Stock
port all =
Bond
Stock
5
2
6
4
One may also add a constant to every element in a matrix.
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Subtraction
Each element of the second matrix is subtracted from the
corresponding element of the first one.
Addition and and subtraction operate on an element-by element
basis.
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Element-by-element operations
The relevant operator is preceded with a dot. The dimensions
must be the same. It’s also called array operation.
port A =
port A.∗port B =
Bond
Stock
Bond
Stock
Master in Economia Bancaria e Finanziaria
1
;
2
5
;
4
port B =
Bond
Stock
port A./port B =
5
2
Bond
Stock
0.2
1
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Element-by-element operations - Ex.4 (Yields)
Values of stock dividends
Dividends = 1.90 0.40 1.56 4.50
Closing prices
Prices = 25.626 17.75 26.125 60.50
The array division of dividends by prices, gives the yield
Yields = Dividends./Prices
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Transpose
A matrix can be mirrored with respect to the diagonal; i.e. the
rows becomes columns and vice versa.
A=
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1 4 9
;
6 8 10
1 6
A0 = 4 8
9 10
M. Dolfin - Univ. of Messina - ITALY
Multiple operations
Can be performed by using an expression indicating the
desired computations which may include multiple matrix and/or
array operations (the dimensions must be appropriate for the
operations).
Example
B = inv (A) ∗ (b ∗ c)
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Matrix division
Primarily used for solving equations.
For instance, given a set of (solvable) linear equations
A X = B,
An×n ,
Xn×1 ,
Bn×1
Matlab uses two matrix divisions symbols:
left division (backslash \) → solves for A X = B.
right division (slash /) → X A = B.
A must be a nonsingular square matrix.
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Ex.5: left division operator
Linear algebraic system of equations
6x + 12y + 4z = 70
7x − 2y + 3z = 5
2x + 8y − 9z = 64
Create a bidimensional array (matrix)
6 12 4
A = 7 −2 3
2 8 −9
70
B=5
64
solution = A\B
Master in Economia Bancaria e Finanziaria
M. Dolfin - Univ. of Messina - ITALY
Inversion ( nonsingular square matrices)
Inverse of a matrix
When multiplied by the matrix, gives an identity matrix.
A−1 ;
in Matlab
inv (A)
A−1 A = I
The inverse of a scalar, i.e. a 1 × 1 matrix, is the reciprocal.
M = (4); I = (1);
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M−1 =
1
;
4
1
4=1
4
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