SDF

Question 1

Complete markets definition of SDF

Answer 1

AD security price / probability of state

Question 2

Risk free is

Answer 2

1/E[M]

Question 3

Risk Neutral probability

Answer 3

M/E[M]

Question 4

Price of asset under rn

Answer 4

R_f ^-1 E^*[X]

Question 5

SDF under log utility

Answer 5

inverse of state-dependent growth optimal portfolio

Question 6

SDF assumptions in incomplete markets

Answer 6

A1) Portfolio formation: existence of linear combinations

A2) Law of one price: linearity

Question 7

Interpretation of incomplete market SDF

Answer 7

Portfolio of assets that best mimics the behavior of every agent’s SDF.
Note that the variance of incomplete markets SDF is greater than the SDF under complete markets (homework). Lowest variance since captures the HJ bounds (finds the most expensive asset by the standard of the market).

Question 8

A payoff space and pricing function have absence of arbitrage if

Answer 8

all X, s.t. X\geq0 always, have q(X)\geq0

and if all X that have X>0 with positive probability have q(X)>0.

Question 9

q=E(MX) and M_s>0 implies

Answer 9

absence of arb

Question 10

Absence of arb implies

Answer 10

exists M SDF such that M_s>0

Question 11

When is SDF not necessarily positive

Answer 11

When payoff is linear subspace but violates arbitrage! e.g. there is an ad security with 0 price.

Question 12

RP as function of SDF?

Answer 12

RP = -Cov(m_t+1,R_a,t+1-R_f,t+1)R_f

Question 13

RP as function of SDF when log normal

Answer 13

RP = -\sigma_amt

Question 14

HJ bound

Answer 14

inverse SR of SDF \geq SR of tangency

Question 15

log normal HF

Answer 15

\sigma_mt \geq RP / \sigma_at

Question 16

Define entropy

Answer 16

lnE(X)-Eln(X)

Question 17

Entropy bounds are

Answer 17

L(X)\geq RP

Question 18

What is the price of risk in a linear factor model?

Answer 18

b_kt * \sigma_kt^2

Question 19

Write down the heterogenous beliefs first order condition, what does it say?

Answer 19

q(s)/q(s’)=pi(s)v_c(c_s)/(pi(s’)v_c(c_s’) for all agents

People get rich since they think there’s a high probability of an event occurring.

Question 20

Write down Harrison and Kreps bellman

Answer 20

p(d_t)=\beta\max{D^i[d_t+1+p(d_t+1)|d_t]}

Question 21

Harrison Kreps assumptions

Answer 21

short sale constraint, heterogeneous beliefs, risk neutral, same preferences, dividend process.