Self Study Questions Topic 6

Question 1

m

Answer 1

states

Question 2

n

Answer 2

basis assets

Question 3

complete market

Answer 3

• all possible payoff profiles can be created (replicated in a market) - otherwise = incomplete
• m states need n greater than or equal to m basis assets

-m states n=m provide complete market with fewest possible basis assets (no redundant)

Question 4

redundant assets

Answer 4

• asset created by combination amongst other basis assets = redundant
• incomplete & complete = have redundant
• basis assets exceeds number of states = redundant assets

n>m there are n-m redundant basis assets

Question 5

linear independence

Answer 5

• cant create any of the basis assets from any combination of the others

columns linearly independent of eachother
rows are linearly independent of eachother

Question 6

4 hedging cases

Answer 6

1. complete market without redundant basis assets
2. complete market with redundant basis assets
3. incomplete market without redundant basis assets
4. incomplete market with redundant basis assets

Question 7

complete market without redundant basis assets

Answer 7

x = inverse matrix A multiplied by b

Question 8

complete market with redundant basis assets

Answer 8

same as without redundant
but first have to remove (n-m) = the number of redundant basis assets

Question 9

incomplete market without redundant basis assets

Answer 9

• transpose matrix A
• A transposed A is a square matrix and then is inverted

everything before the x on the left just becomes an identity matrix so therefore = 1

Question 10

incomplete market with redundant basis assets

Answer 10

same as without redundant but have to remove redundant to produce a reduced for A which is not square but has full rank

Question 11

type 1 arbitrage

Answer 11

1. no initial inv cost
2. payoff not negative in any state
3. payoff positive in at least 1 state

Question 12

type 2 arbitrage

Answer 12

1. upfront investment releases cash
2. payoff is zero

exploits mispricing of a redundant asset (over or under)

Question 13

Arbitrage Theorem

Answer 13

S = prices in the market
A transpose t = transpose of payoff matrix
fi = column vector of prices for AD securities

if find set of positive state prices which this is satisfied = NO ARBITRAGE

Question 14

asset prices using risk-neutral probabilities

Answer 14

Question 15

risky assets

Answer 15

Question 16

risk neutral probabilities

Answer 16

Question 17

Rf in the binomial case

Answer 17

Question 18

A

Answer 18

asset payoffs

Question 19

x

Answer 19

weights

Question 20

Ax

Answer 20

portfolio payoff

Ax = b

Question 21

b

Answer 21

target focus asset payoffs

payoff vector want to replicate/hedge

Question 22

A⁻¹

Answer 22

inverse matrix

adj (B) divided by determinant B

1. cofactor matrix
2. transpose cofactor matrix = adj
3. determinant
4. adj/determinant

Question 23

Im

Answer 23

identity matrix

Question 24

(A^{T}

Answer 24

transpose matrix

flip along the diagonal
rows become columns columns become rows

Question 25

S

Answer 25

asset prices S = fi transposed x b

Question 26

A hat (^)

Answer 26

asset returns

Question 27

e

Answer 27

arrow debreu securities elementary securities

Question 28

φ

Answer 28

fi state prices

Question 29

adj(B)

Answer 29

adjugate matrix transpose of the cofactor matrix

Question 30

|B|

Answer 30

determinant of a matrix remember to swap the signs

Question 31

(B^{C}\)

Answer 31

cofactor matrix replacing each element with its cofactor

Question 32

invertibility of a matrix

Answer 32

find its inverse if - square matrix - full rank - non-zero determinant

Question 33

full rank

Answer 33

- rows are linearly independent - columns are linearly independent - reduction to row-echelon form

Question 34

how do we know if market is complete

Answer 34

1. rows/columns linearly independent 2. payoff matrix has a zero determinant 3. can be inverted 4. can be put in row-echelon form

Question 35

standard row echelon form

Answer 35

main diagonal of 1s lower left anything below that main diagonal composed of zeros