Temi d'esame svolti Finanza matematica

D+

PaDu pbDU el

wap, we

+ Exe 32103

-0.02x 0.02.3

PaDV 3 e

1 887.486

= =

x 1

= Exe-002x+ 52103 -0.02.5

E, Trxne-gin

PbDUB 3 e =

=

=

2414.1636

=

PcDU 4635.5085

=

pdDU 9175. 1622

= 191.7096

3301.64962,

2

es inversa

matrice 2x2

aw (2)

UCIS=-e- d

1

=

1

2

e x b

A A +

=

= = -

Dett C

C

-

if 1.05

=

, exx DEX

=

/2x x

e 2

1.1 V v - -

- = = zx

4 a

1.2 -

-

PEX X

x-2xp2

V

Det 2

=

2x(1 32

-

=

(a) ptf diversificato

stime

x 10.5

t C per =

rendiwer]

u(.) exP,

V 105

(e-

= pg.

* f1)

r

-

* -0.150/2x

DEx 0.85 0.05x

I

x

-

2xx- =

=

= 0.058/2x

p12

0.15

p zx 4x(1 0.15a

a

- -

- +

-

=10.05x-0.150/2x][4x(1 125 impongs entrambe

de

->

- - pifdiv)

0.05812x

2 0.159354x21 23 (0,1)

le we

- -

+

senzel=" =

E 30.05x-0.150/2x][4x(1- -

125

0- 1 I

-

-0.05012x

[ 0.159][4 pi))

0 x21- 2

+

= = -

30.05x-0.150/2x][4x(1- - -0.05p(2x

[ 0.15a][4 57))

125 0

-

10 1 x21

2 -

+

=

10.05x 0.15p(x) 0.3-0.05952x21

0 =

- 1

=

0 32)

4x(1

4x(1 p2) -

- 0.3-8.058/2x 482

4

0 = -

=

-0.1512x24

0.05x 42

0 X

-

= 0.3-0.050/2x>0

·

0.05x-0.15p/2x 0

>

· 0.050/2x 0.3

=

0.055x 0.1595230

- 0.32

0.3

5x = x

5x2,0.15852 =

0.0585210.858122

3528

=

0.85 0.5

4

= 72

1

x

x,1802 = 0.858/2x 14 482

0.3 0

=

x

. -

- -

+

0.158/2x 402

x(4

0.05x- 20

· -

- [...]

5x(4

0.15152 482

0.055x 0

- -

- =

avoia

0.15852

4825x

(0.05 4

- + =

0.150/2

5x2, 0.85 4p20

4

- +

0.15222

x =

in

2005

E x 4.5

=>> * 4.5

=>

, 0.001293

*

(b) otims

3, ptf

x = 125-

x= -0.150/2x][4x(1- 0.003746

10.05x

0.5:

20p = -

· =

1

2 0.15a354x21 11 -

0.0502x 0.02653

- -

+

W* 0.0371

-0.5

rep =

· = 0.0401

varianza

Differenziale

(2) di

3e 10.5

x = =

p

sifen?

3

es 1.01

rf = 1

0.83

1.02 0.16 - 6.63-1.184

0.04 0.552

V

V

e =

= = -

0.22 0.04

0.04 4.851

1.86 0.574

1.104

- -

0.28

0.04

1.09 0.5743.713

0.552

0.83 - -

Bl C

A wP

frontiera: hETP]

al +

g

= B

meet

A met

B

meet A

met

e) e) 1)

2) A(V

h

A(v c(v

B(V -

- - -

g =

- -

= D D

'e

A DV - 11.2517

matD.matc. A

mat =

= = A)t.

eTV e

B met 11.8135

(mat c. A

= mat

= =

DS

1

1Tv

C 10.7293

(met

(metD) met c. =

= =

A2

BC

D 0.1498

=

= - =

=

R x,1)

2 (!)

= D 4.9723 vie:

D) 4.988

mat C.

v-1= mat A

(mat

matc. =

=

3.1712 3.388

2.5856 2.874

B A

e

e n

g 16.1531

= =

- 4.5002

11.6551 16.1531

i!+- ETIP]

wi 4.5002

=

=> 11.6551

-11.9816 A

meet

000

b) tangente

port, met 2 met

rf2)/2

I =

1(e rf1) e-

e-

- - v

v - 0.8798

-

we = => -

(e-rf1)

V 0.4469

I - =

(metD A)

A

met met

metc. c. mat 0.6329

- 0.25

VarIFPS

pAf

(C) con = 12

5(ETEP) t

32SP1 = - +

⑧

- I]"

[SFP)

⑧ 1

ETEPS 1.09548

- +

=> = F

10.09625 7 wP 16.1531

1,43!+- 0.2923

=> -

= 4.50021.09548 0.3062

=

11.6551

-11.9816 0.7863

frontiera?

wo=

(d) di

é

3

3

3 3.1.06 3.

43.1.02

d3 1.09 1.0567

ET e frontiera

vede su

se

+ =

+ -

= i

16.1531

+ 0.0516

- -

d3

. Et

4.5002 0.2889 I

- = 0.4892

11.6551

D

C wol?

varianza ce

tra

la

cambia

come wel metB A

(rd) wd met 0.09778

v

var matA

=

= =

var(*) DY metB

WC

NCTV 0.07111

metD

mat

= = =

(F%) var(rd) 0.02667

var - =

ETFe]

(e) 1.05

= differente)

Mup (tha

del

var senzen rf

e

con

uf

senza

· We ETe]

16.1531

!+

= 0.4423

- =

4.5002 0.3013

11.6551

-11.9816 0.2563

B

A C

D

(E) metc

we'V 0.1414

matD mat c

var We= =

=

con f

· 141

(con

ptf rf) pg

FP

ver rf1

(ETFP) -

(re)= 0.0533

ver =

H 2H B-2Arf (r=2 0.03

=

=

var(e,) -var(rei) 0.0881

Ivar= =

es4 3 2 2 = 1 2

4

Il

D 3

- -

D 5

↳ 9

↳ =

= 4

13 4

5

- 2.85

2 -

I 3 6 I 7

- X

D

arbitraggio?

(a) to $

x

arbitr DTm mco

soluzione

ammette

I s q=

=> q0

(D) -

m = 3 bix

P

6

-4

11 -

I 2.85

m =

= 4 5

-

13 I

-2 47

- 13

-

met

C 0.19

0.17

co 0.77

1. -

I >O 0.2768(13)

<

* 0 (2.483;3,59841

0.1731.113) ->

x

* - d

>8 0.1911(13)

x a

13 A

2.95 arbitraggio

(b) per x = implicito

rendimento of prices

lettore state

,bx (i)

a

(D) -

m = =

=

R3<0

m =

=> ms

io 0.9498 1.0529

if me = r f

- =

= =

⑳mice?

(4,5,2)

(x 2.95c

= =

DNC

vale:

c = 6.5

c

w D -

=

=> = 0.3846

-

es5 ,

rischiosi

2

N. -

1.02 0.18 0.02 0.05

rf =

rin I

ETFm) 1.15 0.03

0.02 0.1

=

+ = 0.03 0.88

0.05

ICAPM]

10.6,0.4)

WP = ↑ 0.4.0.1

0.62.0.18

(PP) 0.0904

(a) ver +2.0.02.0.6.0.4

= =

+

FM)

COVCF,

Bim 0.03

0.05 0.62592m

(b) 0.375

= = =

= = 0.08

S"(EM) 0.08

(C) EtEn]-rf

il rischio:

premio per Bim(E[m]

ET] rf) 0.625(1.15-1.02) 0.08125

-rf= = =

-

-

premis per

rischio

il

titolo I

Ett) 0.375(1.15-1.02) 0.04875

Vf =

=

- 0.4(ETR)

ETEP] (ETY) rf) 0.06825

rf)

0.6.

-rf -

+

-

= =

es6 P(Xk) Fx(k)

=

e

E 0.03

con p 1

= ⑧

-

I 0.02

= p =

0.95

p 0.85

= ⑧ O

-

0.03- ⑧ O i

I

4 ↳K

VERO.05, E50.05

a) . -2

-18

I I min 0.03

e

↳0 p

I =

0.02

p =

0.95

p =

vaR0.05)

([< 0.05

P =

veR0.05, VerR0.83)

P(I)

Y,

P( 0.03

= =

- -

VerR0.85)

) 0.03

F5 =

-

=> VerR0.83=-2

-

=> VerR0.85=

2

vaR"

I

ESP= du

· EvaRi 0.03 (VaR0.83 +VaR0.03]

jos 1863

Esos 240

= = =

=

Eg0.05 Yb

Vapors Y

I+

(b) iid

per

e =

E 0.03

con p =

I 0.02

= p =

0.95

p = 4

0.832 9.18

18-10

E -

p =

=

- 1.2.18-3

210.03110.82)

10-2 =

p

- =

b 2(0.03)(0.95)

1 0.057

10 p

- + = =

= 4

-2 2 0.822 4.18 -

p

- =

= 0.038

2(0.02710.95)

- p =

=

0.952 0.9025

p = = P(Ypsk) Fy(k)

=

1

4 ⑧

-

I 9.10 -

-20mIn p = 8.8973- ⑧ I

3

1.2.10

amminente -

p

b = 0.8595- ⑧ I

=> 0.057

I p = 0.8391

0.85 ⑧ I

-

4

4.18 - X

p 1.18-3-

= 2. I

⑧

0.038

p 4

= 9.10 - 0

⑧

-

0.9823

p = I

is I

-i

-20 a >I

2

-I

-

Ver0.051

Fy) 0.03

=

-

-Ver0.83=-9 VerR0.05=9

=>

0

Eg0.83 20)

(9 820

12

=

· =

+

+

06/23

1

es -10k cedola p(0, T

T V. H.

"2 99.6

ZCB 108 2(sem.)

I 103.5

CB 188 98.8

2 ann

100

CB 3 56

58 ↳ ann

CB bootstrapping? ,)

tassi composte

(cap

curve con i

(1+

=

2

-

i)0,x))

100(1

99.6 = +

0.808

i(0,42) =

· 2 1

ic0,11)

-

103.3 -

2(1 i(0,")) 102(1

+

= + +

2 1

I =0.005V

S -

-

3

i(0,1) -2(1 8-10

183.5 -

=

· +

102 2

1 i(0,2))

1(1 i(0,1))

98.8 1011

- -

= +

+ + 2

1 -

[

i(0,2) I V

98.8 -

i20,1)

(1 0.0162

· -

= =

-1

+

101 3

+ 2 -

-

4(1 10,21

i(0,1 10,3))

54(1

4(1 -

56 +

= + +

+ 0.0390X

3

- 2 -

L (4(1 I

i10,21)

10,1 4(1+ -

i10,3) 56 +

+

· - 1

= =

-

54 intermedi!!

tassi

per

i (0,2.25)

· W

0.25.0.039

i(0,1.25) 0.022

0.75.0.0162

8.839- =

= +

ais NO!

0.8162 =

2 giucts

es tults

+

rf 1.015

= 1.02 0.02 0.04

0.12

- - 0.03

0,18

1.85 0.02

4, V

2

1

re, =

=

+ 1.08 0.830.24

0.04

-ur D

e ax

-

u(x) 1

x

= =

- 0.3)

a) (a

otimo

ptf = mat

v (+1) ametD)

-

x 1 0.2416

-

= = W

-

= 0.5288

A 0.8769

b) B

ptf tang

1(e rf1) =

mat mat

- A)

(metC

v metD-

-

we A

=(metD) 0.2075

= -

e-rf1 0.4542

-

I 8.7532

~

IV

(2) 0.31

dieng,

oximo a

ptf =

10.1390 ~

0.6481

x = 0.9330

C VW

w

(W) 0.2735

var = =

wTVw

var(N) 0.2477

= = v

(4)

1 0.0258

(W)

ver

var

= - =

da

3 correggere

es +

I

re 1.02 0.02 0.04

0.12 0.03

0,18

1.85 0.02

V

2 =

= 1.08 0.83

0.04 0.24

Al D

=

(a) FP: hE[]

g

w +

Bl Al

met

met e) e), 2)

2) A(v

h

A(v c(v

B(V -

- - -

g =

-

= D

Bl

met

R- 'e D-mat

A matB. A 14.2038

mat

= =