1431 words - 6 pages

Advanced Financial Models

Example sheet 1 - Michaelmas 2010

Michael Tehranchi

Problem 1. In a one-period model, a num´raire asset is called risk-free if its time-1 price

e

i

i

is not random: if asset i is risk-free, then S1 = (1 + ri )S0 for a real constant ri > −1, called

the risk-free rate of return. Suppose that a market model has at least one risk-free asset.

Show that if there is no arbitrage, then the risk-free rate of return is unique, in the sense

that if both asset i and asset j are risk-free then ri = rj .

Problem 2. (Binomial model) Consider a market with two assets. Asset 0 is a risk-free

bond with

B0 = 1, and B1 = 1 + r

for some constant interest rate r > −1, ...view middle of the document...

Fix a

constant vector α ∈ Rd and deﬁne an equivalent measure Q on (Ω, F ) by the density

dQ

2

= eα·W −|α| /2 .

dP

ˆ

Prove that the random variable W = W − α has the Nd (0, I ) distribution under Q.

Problem 7. Consider a d + 1 asset model, where asset 0 is cash B0 = 1 = B1 while assets

1, . . . , d have time-1 prices S1 with the Nd (µ, V ) distribution, where µ ∈ Rd and V is a

positive semi-deﬁnite d × d matrix. Use the previous problem to show that there is no

arbitrage if V is non-singular. Show that there might be arbitrage (depending on the values

of µ and S0 ) if V is singular.

Problem 8. (Stiemke’s theorem) Let A be a m × n matrix. Prove that exactly one of the

following statements is true:

• There exists an x ∈ Rn with xi > 0 for all i = 1, . . . , n such that Ax = 0.

• There exists a y ∈ Rm with (AT y )i ≥ 0 for all i = 1, . . . , n such that AT y = 0.

What does this have to do with ﬁnance?

Problem 9. Let X1 , . . . , Xd be d random variables. We aim to show that if

a · X ≥ 0 a.s. for some a ∈ Rd ⇒ a · X = 0 a.s.

then there exists a random variable Z > 0 a.s. such that E(Z |Xi |) < ∞ and E(ZXi ) = 0 for

all i.

(1) Why does the above theorem imply the hard direction of the 1FTAP?

(2) Why is there no loss of generality in assuming that the random variables X1 , . . . , Xd

are linearly independent in the sense that if b · X = 0 a.s. then b = 0.

2

(3) Let F (a) = E(ea·X −|X | ). Show that F is ﬁnite-valued and smooth.

(4) We aim now to show that there exists a∗ ∈ Rd which minimizes F . Assuming this

∗

2

for the moment, show that Z = ea ·X −|X | satisﬁes the conclusions of the statement.

(5) Now suppose for the sake of ﬁnding a contradiction that F does not achieve its

minimum. Let (an )n be such that F (an ) ↓ inf a F (a). Why can we assume that (an )n

is unbounded?

(6) Let an = an /|an |. Why does the sequence (ˆn ) have a convergent subsequence?

ˆ

a

(7) Suppose |an | → ∞ and an → α. Show that P(α · X > 0) = 0.

ˆ

(8) Use the hypothesis of the statement to show α · X = 0 a.s. Why is this our desired

contradiction?

Problem 10. Let X1 , . . . , Xd and Y random variables, and let

Z = {Z > 0 a.s. : E(Z |Xi |) < ∞ and E(ZXi ) = 0 for all i} = ∅.

2

We aim to show that if E(ZY ) = 0 for all Z ∈ Z such that E(Z |Y |) < ∞, then

Y = a · X a.s. for some a ∈ Rd .

(1) Why does this statement imply the hard direction of the characterization of attainable

claims?

(2) How can we modify the argument in the previous problem to ﬁnd random variables

Zr ∈ Z of the form

2

2

Zr = ear ·X +rY −|X | −Y

for r ∈ {0, 1}.

(3) Show that Y = b · X + log(Z1 /Z0...

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