In this assignment you will practice matrix theory and linear algebra, and relate these concepts to models of financial markets.
1. Miscellaneous:
(a) Find the column space, row space, null space and left nullspace of the matrix
| 2 6 4 |
A = |1 3 2 |
(b) Consider the three vectors x1 = (1/√2, 0, -1/√2)T, x2 = (0, 1, 0)T , x3 = (1/√2, 0, 1/√2)T. Does the set A = {x1, x2, x3} form an orthonormal basis of R3?
2. Trading model: As in class, consider a model of a financial market with N assets and M possible outcomes. The payoff of asset n at time t = 1 if outcome m occurs is Dnm, where D ∈ RN×M, and its price at time zero is s0n , where s0 ∈ RN. A portfolio of stocks is represented by a vector h ∈ RN, where hn represents the number of shares owned of asset n. We also define the augmented matrix, D¯ = [-s0, D], the reachable payoff space R = {DTh : h ∈ RN }, and the augmented payoff space R¯ = {D¯ Th : h ∈ RN }. Consider the market with
|
- 2 |
1 |
1 |
| D¯ = |
- 3 |
1 |
2 |
|
- 16 |
8 |
9 |
(a) What is the t = 0 price of the portfolio h = (1, 1, 0)T in this market?
(b) Is this market complete?
(c) Does the law of one price (LOOP) hold in this market?
3. Law of one price: In class, we have discussed the law of one price (LOOP), which formalizes the very intuitive "no free lunch" idea that two portfolios that generate the same payoffs should have the same price in a well functioning market. We have also discussed the relation between this rule and the existence of a state price vector. Recall that a state-price vector, ψ ∈ RM , where M is the number of possible outcomes, is such that hTs0 = hT Dψ for all h ∈ RN.
The relation between the LOOP and the existence of a state price vector is one version of what is known as the fundamental theorem of asset pricing, and has two parts:
- Part 1: The LOOP holds if and only if there exists a state price vector.
- Part 2: In a market in which the LOOP holds, the state price vector is unique if and only if the market is complete.
A mathematical way of stating that the LOOP holds is that for all h ∈ RN :
hTD = 0 ⇒ hTs0 = 0.
(a) Use your knowledge from linear algebra to prove part 1 of the theorem.
(b) Use your knowledge from linear algebra to prove part 2 of the theorem.