nmf 1.1 introduction

8/13/2019 NMF 1.1 Introduction

1/13

1

Numerical (and Statistical)Methods for Finance

Module 1Introductory Lecture

Contents

Course objectives and development

Recalls on the spreadsheet Introductory models


2/13

2

Course objectives

To introduce theoretical and practicalmodels on financial calculus

To make students aware of the appropriatePC use in the financial applications

To show the PC potential in a wide range of

financial applcations. To enhance the model building and

implementation process.

Course features

Lectures Supervised lab activities Home work required to reach the

proper background for good benefitfrom the course

Specific home work


3/13

3

Assessment rules

Written test on a paper sheet, using a pen(not a pencil !) and a pocket calculator

Completion of the models made available indraft, in .pdf format, on the course site, to bealso enriched for wider applications

Oral test on the background theory and themodels implementation details

Assessment criteria

The final evaluation of the module will based on the gradesachieved in the three tasks: Written test Colloquium

Models completion The student must prove his/her own intellectual property of

the presented models. Furthermore, these caracteristics will be appreciated:

Autonomy Models correctness Initiative Ability to access data sources


4/13

4

Non exhaustive models sample list

Accumulation functionShort term treasury bondsFinancial tables and representation of thecash flows of an annuity transaction

Debt redemption schedules (amortizationplan) at a fixed or variabile interest rateAccumulation plan (fixed and variable rate)

Models (cont.)

Value function vs interest rate, IRRAdjusted Present Value

Duration of a general portfolio and of asecurity paying fixed couponsIRR-COST vs IRR_VARPseudo-random numbers generation


5/13

5

Spreadsheet recalls Copy and Paste Relative and absolute references Native functions of a spreadsheet:

Mathematical Financial Date

Logical Statistical

Graphs

Introductory models

Sequence and sequence of partial sums Pithagorean Table

Drawing a real function of a real variable ina closed domain

Computation of a polynomial pn(x) Intersection between two functions Generation of a random time series


6/13

6

Sequences and partial sums

ProblemRepresent on a spreadsheet an arbitrary

sequence showing a finite number ofterms, randomly generated, and computethe partial sums sequence.

Then, deal with arithmetic and geometricprogressions

Sequence and partial sums

{an}n=1,2, Sequence {Sn}n=1,2, Partial sums sequence defined as

{F n} n=1,2 , Partial products sequence==

n

t t n aS

1

=

=n

t t n aF

1


7/13

7

Sequence and partial sums(example)

t a t S t

0 2 2

1 3 5

2 -1 4

3 1 5

4 2 7

5 -1 6

6 3 9

7 4 13

Recurrence equations and closed formof the partial sums for the aritmetic and

geometric progressions

( )[ ]( )

qq

aS qaqaa

nnd aa

S td ad aa

aS aaaS

n

nt

t t

nt t

t t t

t

ss

t

sst

===

+++

=+=+=

+=+==

+

==

11

12

1

001

0001

1

1

00

d: common difference

q: common ratio a 0 : scale factor


8/13

8

Arithmetic progressions

( )[ ]( )

( )Parabola

222

222

12

:nprogressioArithmetic

002

200

00

01

1

and adn

dndnana

nnd aa

S

td ad aa

d aa

n

t t

t t

+++=

=+++

=

=+++

=

+=+=

=

Geometric progressions

( )

1111

1lim

11

...1

:nprogressioGeometric

01

0

1

02

00

0

01

1


9/13

9

Drawing a function in a domain [a,b]

Intersection points between twofunctions

Example:

=

=

=

=

=++=++

+=++=

2

1

1

0

0221132

1132

2

2

1

1

22

2

y

x

y

x

x x x x x

x y x x y


10/13

10

Intersection points between twofunctions

Graphical representation:

Beyond the analytical solutions

What can we do if we are not able to find thesolution in closed form?

We proceed numerically

How is it performed automatically on aspreadsheet?


11/13

11

Computation of a polynomial for givenvalues of x

( )

( )

( )

( ) 64242821

2

5.44112121

1

4221

4,1,2,21

...

3

3

233

3210

01

110

=++=

=++=

++=

====

=++++= =

p

p

x x x x p

aaaa

xaa xa xa xa x pn

h

hnhnn

nnn

Numerical computation of apolynomial by recurrence

( )( )

( )

( )

( ) ( ) ( ) 001

12

11

0

11

10

12

11

01

11

10

,

:equationRecurrence

...

...

...

...

1

a x pa x x p x p

a xa xa xa

a xa xa xa

a xa xa x p

a xa xa xa x p

nnn

n

p

nnn

x p

nnnn

nnn

n

nnnn

n

n

n

=+=

++++=

=++++

+++=

++++=

4 4 4 4 34 4 4 4 21

4 4 4 4 4 84 4 4 4 4 76


12/13

12


( )

( )

( )

( )21

44121

1

21

1123

1

23

2121

1

21

1

:getwouldwe1forexamplepreviousIn the

3

2

1

0

=+=

==

=+=

=

=

p

p

p

p

x


( )

( )

( )

( ) 4221

1221

2212

1

:getwouldweanyforexamplepreviousIn the

233

22

1

0

++=

+=

+=

=

x x x x p

x x x p

x x p

x p

x


13/13

13

Numerical computation of a polynomialfor a set of values of x

p0(x) p1(x ) p2(x) p3(x)t 0 1 2 3at -0,5 2 -1 4

i x p0 0,00 -0,5 2 -1 4,0001 0,50 -0,5 1,75 -0,125 3,9382 1,00 -0,5 1,5 0,5 4,5003 1,50 -0,5 1,25 0,875 5,3134 2,00 -0,5 1 1 6,0005 2,50 -0,5 0,75 0,875 6,1886 3,00 -0,5 0,5 0,5 5,5007 3,50 -0,5 0,25 -0,125 3,5638 4,00 -0,5 0 -1 0,0009 4,50 -0,5 -0,3 -2,125 -5,563

10 5,00 -0,5 -0,5 -3,5 -13,500 -15. 00 0

-10. 00 0

-5 . 0 0 0

0 . 0 0 0

5 . 0 0 0

10. 00 0

0 . 0 0 0 . 5 0 1. 0 0 1.5 0 2 . 0 0 2 .5 0 3 . 0 0 3 . 5 0 4 . 0 0 4 . 5 0 5 . 0 0

x

nmf 1.1 introduction

Documents