Mock esame Metodi quantitativi per la finanza II

M

e

u

egy # that super-replicates X (1) is such that:

u u u u

5#

A) # = 5# ; B) # = ;

0 1 0 1

u u u u

3#

C) # = 3# ; D) # = ; E) none of the other answers is correct.

0 1 0 1

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2 Given the one-period market (with riskfree rate r = 0)

2 3

1 0:5 1 question 2

answer

to

info useful

6 7 realy

not

This is

-

1+0 4 4 1 1 1

6 7

0 6 7

M = with objective probability masses P (! ) = , P (! ) = and P (! ) = ,

1 2 3

4 5 4 4 2

1+0 0 4

1+0 0 0   2

e e e e e

 

the no-arbitrage price of the payo Y (1) = 2B (1) + 2 S (1) S (1) S (1) + S (1) is such that:

1 1 2 2

A) Y (0) = 2; B) Y (0) = 3;

C) Y (0) = 4; D) Y (0) > 4; E) none of the other answers is correct.

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3 Start from the setup of Question 2 and consider the initial wealth W = 1. Given the risk

h i h i

0

e e

 P P



aversion parameter  > 0, the strategy # solves the problem max E V (1) V ar V (1) . The

# #

# ;#

1 2

 12

strategy # has the same wealth variance of a strategy #, with # = # = , when:

1 2

q p 11

3

A)  = ; B)  = ;

11 3

q p 11

3

C)  = ; D)  = ; E) none of the other answers is correct.

22 22

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4 Start from the setup of Question 2 and consider the initial wealth W . Among the strategies

0

made of risky securities only, the strategy , expressed in terms of portfolio shares, that minimizes the

wealth variance is such that:

A)  = 1; B)  = 0;

1 2

C)  = 1; D)  = 1; E) none of the other answers is correct.

0 2

Alessandro Sbuelz, Andrea Tarelli - SBFA, UCSC - QUANTITATIVE METHODS FOR FINANCE 1

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5 Under the objective probability measure P , the underlying stock value dynamics is

dS P

dt



= (r q + ) + dz ;

S

where r is the riskfree rate, q is the dividend yield,  is the volatility parameter,  is the risk premium

 

P

( is the market price of risk), and z is a Wiener process under P . S is the positive current stock

value (t is the current date). The European call, the European put, and the futures contract have the same

maturity/delivery date T > t (K is the common strike price of the two options). Their current no-arbitrage

values are c (S; t), p (S; t), and F (S; t), respectively. They are such that:

r(T t) r(T t)



A) c (S; t) = p (S; t) + e F (S; t) e K;

r(T t) r(T t)



B) c (S; t) = p (S; t) + e F (S; t) e K;

r(T t) r(T t)

 

C) c (S; t) = p (S; t) e F (S; t) e K;

r(T t) r(T t)

 

D) c (S; t) = p (S; t) e F (S; t) e K;

E) none of the other answers is correct.

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6 Start from the setup of Question 5. The digital option with the terminal payo 1 (for

fSKg

!

t T ) has current no-arbitrage value D (S; t), which is such that:

 

12 P

2 2

dD dt



A) =  + S (r q + ) + S  + Dqdt + Sdz ;

 

12 P

2 2

dD dt



B) =  + S (r q) + S  + Sdz ;

P

dD

C) = E [dD] + Sdt + Sdz ;

t P

dD 

D) E [dD] = (=D) Sdz ;

t

E) none of the other answers is correct.

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P

dr dt dz

 

7 Assume the Vasicek model (the short rate r has dynamics = (r r) +  under the

r r 6

objective probability measure P with  > 0). The market price of interest rate risk is  (with  = 0) The

r r

zero-coupon bond value B (r; t) with maturity date T > t is such that:

 



1e



A) B = B ;

rr  

 

1e

B) B = B ;

rr r 

 



e



C) B = B ;

rr r 

 



1e



D) B = B ;

rr r 

E) none of the other answers is correct.

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8 Start from the setup of Question 7. The zero-coupon bond price B (r; t) with maturity date

T > t is such that: 12 2 2

 

A) B B  (r r) + B  = Br ;

t r rr r

12 2

B  

B) B  (r r) + B  = Br;

t r rr r

12 2 2

B  

C) B  (r r) + B r  = Br;

t r rr r

12 2



D) B + B  (r r) + B  = Br;

t r rr r

E) none of the other answers is correct.

Alessandro Sbuelz, Andrea Tarelli - SBFA, UCSC - QUANTITATIVE METHODS FOR FINANCE 2

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