Question 1 [25 marks]
Answer the following questions, justifying your answers:
a. Let B be a Brownian motion. For each of the following processes, cal
culate the conditional expectation at s < t and decide whether it is a
i. Xt = tBt
ii. Zt = Bt3 − tBt.
b. A tennis match between Roger Federer and Rafael Nadal is played with
the best-of-five rule, that is, the two opponents play successive sets
until one of them (the match winner) has won three sets. The match is
then over. Consider the following random variables:
• X = number of sets won by Federer
• Y = number of sets won by Nadal
• Z = total number of sets played in the match
• B = payoff offered by bookmakers, paying $100 if Federer wins the
match, nothing otherwise
Answer the following questions:
i. Describe a sample space where these random variables are de
ii. Write out explicitly σ (B).
iii. Is it true that σ (X, Y ) = σ (X, Z)? Briefly justify your answer.
iv. Is it true that σ (X) = σ (B)? Briefly justify your answer.
Question 2 [15 marks]
Suppose that there are three market scenarios, and there are three basis
assets with payoff matrix and price vector as follows:
a. Perform Gaussian elimination (showing all steps) to determine how
many redundant asset there are in this market. Is the market complete?
b. Find all state prices which are consistent with the Law of One Price,
and based on this finding determine whether there are any arbitrage
opportunities. If there is an arbitrage opportunity, state what type of
arbitrage it is.
c. Calculate the risk-free return and the risk-neutral probabilities.
d. A focus asset with payoff 1 2 0′ is introduced. What are the possible
no-arbitrage prices of the focus asset?