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Netscape Ipo Essay

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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