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, while asset 1 is a stock with prices
S

u
?




1 >>
>>
>>
1−p >
>
p

d for some constant rates of return d < u and a probability 0 < p < 1. Use the definition of arbitrage to show that there is no arbitrage if and only if d < 1 + r < u. When there is no arbitrage, find all equivalent martingale measures Q (relative to asset 0, as usual) and their densities dQ with respect to the objective measure P. dP Problem 3. Consider a two-asset model with prices given by
(B, S )

(3, 9)

; ww ww ww ww
/ (6, 8)
(4, 6)
GG 1/4
GG
GG
1/2 GG#
1/4

(6, 4)
Is there arbitrage in this market? If not, find all pricing kernels.
Problem 4. Consider a three-asset model with asset 0 cash B0 = B1 = 1 and assets 1 and
2 given by
(S (1) , S (2) )
(9, 8) w; ww ww w ww 1/3

(6, 7)

GG
GG
GG
2/3 GG#

(3, 5)
Is there arbitrage in this market? If not, find all equivalent martingale measures.
1

Problem 5. * Fix a probability space (Ω, F , P) and let Z be a random variable such that
Z > 0 a.s. and E(Z ) = 1. Define a set function Q on F by
Q(A) = E(Z 1A ).
Show that Q is a probability measure which is equivalent to P.
Problem 6. Let (Ω, F ,