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Assignment 1

Asset Pricing (Leopold S¨ogner) ,

Quantitative Finance Program

WU Vienna

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

Points 100

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

ρa

−1 while the borrowing rate is rb =

1

ρb

−1. For

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.

1

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

to x

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

Π, 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.

2

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