Asset Pricing (Leopold S¨ogner) ,
Quantitative Finance Program
November 5, 2020 – due on November 20, 2020 (upload pdf via Learn WU)
Problem max. points points
Problem 1 25
Problem 2 25
Problem 3 25
Problem 4 25
1: Consider an economy where bid-ask spreads show up. A long spot position in asset D is the contract
(−Sa, D), Sa is called the ask price. A short spot position in asset D corresponds to (−Sb, D), Sb
is called the bid price. Sa ≥ Sb. In addition a risk-free discount bond is traded, where we get
(−ρb, 1K) and (−ρa, 1K). The lending rate is rl =
−1 while the borrowing rate is rb =
forward contracts the forward-ask and forward-bid prices are Fa and Fb, respectively The payoffs are
(0, D − Fa1K) for long forward position and (0, −D + Fb1K) for a short forward position; Fa ≤ Fb.
Show that in an arbitrage-free market,
rl ≤ rb, ρb ≤ ρa, Sb ≤ ρaFa and ρbFb ≤ Sa have to hold.
2: Construct your own example with J = 3 assets and K = 3 states, where the market is incomplete.
One of the three assets is a risk-free discount bond. Provide an argument or proof that the market
is incomplete in your example.
Derive the set of the state-prices and state price densities.
3: We defined a pricing cash flow as a cash-flow x
∗ ∈ X where there exist constants ρ and q such that
q 6= 0 and
x(0) + ρE(x(1)) + qCov(x
(1), x(1)) = 0 for all x ∈ X .
Consider the case where a risk-free asset is traded.
Suppose that somebody claims that the pricing cash-flow x
is equal to the cash-flow x
E. What happens (or what can go wrong), what are the implications? [Consider a case where x
E is proportional
Π and the case where this is not the case.]
4: Consider an economy where two risky assets are traded and a risk-free asset is traded. I.e. J = 3,
the number of states is finite.
Construct your own example: Obtain x
E, RΠ, a beta pricing returns R∗ and the highest attainable Share-ratio. Choose your own numbers and obtain these objects. Describe how the cash-flows
and returns are obtained (at least the formulas). Provide a figure where you also include/describe
the efficiency frontier.
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