Brainscape's Knowledge GenomeTM

Browse over 1 million classes created by top students, professors, publishers, and experts.


See full index

Asset Pricing > Stochastic Discount Factors and Arbitrage > Flashcards

Stochastic Discount Factors and Arbitrage Flashcards

1
Q

Motivation

A

Relationship between representation (∗) and absence of arbitrage

p = E[mx] leaves open the possibility that equation holds more generally
(ideally without any assumptions on investors and utility functions)

What minimal assumptions do we need to justify equation (∗)?

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
2
Q

Key result

A

there is a positive SDF m such that p = e(mx) holds for all assets if and only if there are no arbitrage opportunities

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
3
Q

State space

A

S = {1,…,S} (finite set of possible states of nature)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
4
Q

A real-valued random variable

A

is a function x : S → R
recall that a random variable is the same as an S-dimensional vector in RS

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
5
Q

X ⊂ RS

A

denotes the set of all payoffs the investor can invest in, the payoff space

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
6
Q

A price function

A

is a function p : X → R that assigns to each investable payoff x ∈ X its (ex-ante) market price p(x)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
7
Q

When Does p Have a SDF Representation?

A

Definition (SDF representation)

We say that the price function p has a SDF representation if there is a random variable m∈RS such that for all x∈X
p(x) = E[mx].

If p has a SDF representation, then our fundamental asset pricing equation holds for p

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
8
Q

Free Portfolio Formation – Motivation

A

Suppose an investor can buy two assets with payoffs x1, x2 ∈ X

Then the investor should also be able to buy both assets and achieve the payoff x1 + x2

More generally, let’s assume that the investor can also:

  • buy available assets in any fraction
  • short-sell all available assets
    Then the investor is able to form arbitrary portfolios of the two assets:
  • buy an arbitrary quantity a1 ∈ R of payoff x1
  • buy an arbitrary quantity a2 ∈ R of payoff x2

The total payoff of such a portfolio is
a1x1 + a2x2,

i.e. a linear combination of the payoffs x1 and x2

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
9
Q

Free Portfolio Formation – Formal Defintion

A

Assumption (free portfolio formation)

If x1,x2 ∈X and a1,a2 ∈R, then also a1x1+a2x2 ∈X

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
10
Q

Remarks: free portfolio formation

A

In the language of linear algebra: X is a linear space

For any set of basis payoffs x1, …, xn, all payoffs x ∈ X can be represented in the form

a1x1 +···+anxn

as a portfolio of the basis payoffs

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
11
Q

Law of one price motivation

A

The law of one price is the idea that any two asset portfolios that generate the same payoff should have the same price

Suppose you have three assets with the following payoffs available: x1, x2, x3 := a1x1 + a2x2

we can achieve payoff a1x1 + a2x2 by forming the following portfolio:

buy a1 units of asset 1, cost: a1p(x1)

buy a2 units of asset 2, cost: a2p(x2)

total cost of portfolio: a1p(x1) + a2p(x2)

we can achieve the same payoff by buying asset 3, cost: p(x3)

→ the law of one price suggests that p(x3) = a1p(x1) + a2p(x2)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
12
Q

Formal definition of Law of One Price

A

Definition (law of one price)

We say that a price function p : X → R satisfies the law of one price, if for all x1,x2 ∈ X and all a1,a2 ∈ R

p(a1x1 + a2x2) = a1p(x1) + a2p(x2)

In the language of linear algebra: the law of one price means that p is linear

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
13
Q

The Law of One Price and Arbitrage Opportunities

A

If the law of one price does not hold, an investor can make risk-free profits (“arbitrage profits”)

  • buy the cheaper version of a payoff, sell the more expensive
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
14
Q

Does the law of one price guarantee absence of arbitrage?

A

Anwer: no, consider the following example
suppose S = 1 (the state is known), X = R, and p(x) = 0 for all x
this does not violate the law of one price (all prices are zero)
but investors can make risk-free profits
e.g. buy a claim to x = 1, pay zero, but get the positive payoff 1
→ To rule out risk-free profit opportunities, we also need to require that positive payoffs have positive prices

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
15
Q

Absence of arbitrage - Formal definition

A

Definition (absence of arbitrage)
We say that a price function p : X → R leaves no arbitrage opportunities if the following two conditions are satisfied:

  • p satisfies the law of one price;
  • every payoff x ∈ X that is nonnegative x ≥ 0 and positive in some states

(∃s ∈ S : x(s) > 0) has positive price, p(x) > 0.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
16
Q

Remarks for absence of arbitrage

A

The second property is equivalent to the following form of strict monotonicity:
if x,y ∈ X satisfy x ≤ y and x(s) < y(s) for some state, then p(x) < p(y)

(in words: if payoff y is at least as good as x in all states and better in some, it should have a higher price)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
17
Q

Complete and Incomplete markets

A

Definition (market (in-)completeness)

The asset market is called complete if X = RS

The asset market is called incomplete if it is not complete

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
18
Q

Interpretation of market completeness

A

In a complete market, the investor can purchase any potential payoff x
(i.e. any random variable)

In an incomplete market, there are some potential payoffs x that cannot be purchased

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
19
Q

Contingent claims

A

A contingent claim pays 1 in precisely one state, 0 otherwise
this means there are S different types of contingent claims, one for each state

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
20
Q

Contingent claims

A

A contingent claim pays 1 in precisely one state, 0 otherwise
this means there are S different types of contingent claims, one for each state

Mathematicalremark:thecontingentclaimpayoffsη1,…,ηS arepreciselythestandardbasis vectors e1, …, eS ∈ RS (the columns of the identity matrix)

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
21
Q

Contingent claims and market completeness proposition

A

The asset market is complete ⇔ all contingent claims are available, η1,…,ηS ∈ X.

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
22
Q

Proof of this

A

Very simply proof:

“⇒” if X = RS, then clearly η1,…,ηS ∈ X

“⇐” all x ∈ RS can be represented as portfolios (i.e. linear combinations)

x =x(1)η1 +x(2)η2 +···+x(S)ηS
of contingent claims

Contingent claims are useful conceptually because they have such a simple structure
When we work with them, we do not need to assume that they actually exist, just that they can be synthesized with available assets

How well did you know this?
1
Not at all
2
3
4
5
Perfectly
23
Q

Example: Synthesizing Contingent Claims

A
How well did you know this?
1
Not at all
2
3
4
5
Perfectly
24
Q

Representing the price function

A

Suppose that
the asset market is complete
p satisfies the law of one price (i.e. is linear)

25
Q

Representation of Price Function – Interpretation

A
26
Q

Economic idea behind representation of price function

A

complete markets: any payoff can be replicated by a portfolio of contingent claims

if the law of one price holds, the price of the payoff (p(x)) must equal the cost of the replicating portfolio (pcT x)

Remark: mathematical idea behind result

as a linear function, p has the matrix representation pcT = (p(η1), …, p(ηS ))

27
Q

State Diagram

A
28
Q

Alternative State Space Geometry

A
29
Q

SDFs

A

State prices pc(s) are difficult to interpret
does low pc(s) mean it is cheap to deliver payoffs in state s? or just that the probability π(s) is small?

30
Q

Stochastic discount factors equation

A

Scaling by probabilities isolates the first aspect:

m(s) is the cost of a claim with expected value of 1 that only pays off in state s

31
Q

Conclusion from this

A

in complete markets, any p that satisfies the law of one price has a SDF representation

32
Q

does a SDF representation imply the law of one price?

A

yes, if p(x) = E[mx] for all x, then p is a linear function

33
Q

Is the SDF representation unique?

A

yes, if p(x) = E[mx] for all x, then in particular for all s = 1,…,S

34
Q

Deriving SDF equation

A
35
Q

Conclusion (SDF representation in complete markets)

A

Suppose the asset market is complete. Then a price function p has a SDF representation if and only if the law of one price holds.

Furthermore, if these conditions are satisfied, the SDF representation is unique and satisfies
m(s) = pc(s)/π(s)

36
Q

When Does p even Satisfy Absence of Arbitrage?

A

Recall definition of absence of arbitrage: two requirements on p

  • law of one price
  • nonnegative payoffs that are positive in some state have positive price
37
Q

When Does p even Satisfy Absence of Arbitrage? - conclusions

A
38
Q

No arbitrage and SDFs - absence of arbitrage in complete markets

A

We have seen previously: under law of one price, the unique SDF is m(s) = pc(s)/π(s)

this is positive, if and only if pc(s) is positive

Combining with result from previous slide yields:

Conclusion (absence of arbitrage in complete markets)
Suppose the asset market is complete.

Then the following are equivalent for a price function p:

p leaves no arbitrage opportunities;
p satisfies the law of one price and contingent claim prices are positive,

pc(1), …, pc(S) > 0;

p has a SDF representation with a a positive SDF m.

39
Q

Illustration: Nonpositive SDFs Imply Arbitrage Opportunities

A
40
Q

Risk-neutral Probabilities

A
41
Q

Introducing an investor into contingent claims market using Lagrangians

A
42
Q

Law of One Price and SDFs: Main Theorem

A

Theorem (SDF representation theorem)

A price function p has a SDF representation if and only if the law of one price holds.

In this case, there is a unique payoff x∗ ∈ X such that
p(x) = E[x∗x]

for all x ∈ X.

43
Q

Main Theorem remarks

A

m = x∗ is of course a possible SDF that represents p

The theorem does not say that the SDF is unique, just that there is a unique one that happens to be also a payoff

44
Q

Proof: SDF Representation ⇒ Law of one Price

A
45
Q

Proof: Law of one Price ⇒ SDF Representation – Lazy Mathematician’s Proof

A
46
Q

Proof: Law of one Price ⇒ SDF Representation – Geometric Proof

A
47
Q

Proof: Law of one Price ⇒ SDF Representation – Algebraic Proof

A
48
Q

(Non-)Uniqueness of the SDF

A
49
Q

What this tells us

A

This tells us four things:

x∗ = proj(m | X) for any SDF m that represents p

any m of the form m=x∗+ε with ε∈X⊥ is also a SDF that represents p

any SDF that represents p is as in (b)

there is a unique SDF if and only if markets are complete (as only then X ⊥ = {0})

50
Q

Example: Non-Uniqueness of SDFs

A
51
Q

No Arbitrage and Positive SDFs: Main Theorem

A

Theorem (fundamental theorem of asset pricing)

A price function p has a SDF representation with a positive SDF m > 0 if and only if it leaves no arbitrage opportunities.

52
Q

Remarks from this main theorem

A

This theorem does not say anything about uniqueness

It also does not say that all SDFs that represent m must be positive, just that there is one

In particular, x∗, the unique SDF contained in X, may not be positive

53
Q

Proof: Positive SDF ⇒ No Arbitrage

A
54
Q

What properties does the absence of arbitrage require

A

the law of one price: this already follows from the previous theorem if x ≥ 0 and x(s) > 0 for some states s, then p(x) > 0:

55
Q

Idea of proof: Direction no Arbitrage ⇒ Positive SDF

A
56
Q

Non-Uniqueness of Positive SDFs: Example

A
57
Q

Summary

A

Main questions of this lecture:
what assumptions are needed to write a formula p = E[mx]?

in which cases is this possible with m > 0?

We have provided a complete answer:

SDF representation ⇔ law of one price holds

positive SDF representation ⇔ market leaves no arbitrage opportunities

Equivalence results hold in both complete and incomplete markets

58
Q

What states would you expect a boom and a recession?

A

For the reasoning behind this answer, we can think beyond the arbitrage-free market model and bring in some insights from the consumption-based model. For any investor in this asset market, the consumption-based model tells us that m is proportional to marginal utility and marginal utility is high when consumption is low.

After trading has happened, m must align with the marginal utility of all investors, so states of high m are states in which all investors consume little and states of low m are states in which all investors consume much.

The former likely coincide with recessions, the latter with booms. Because m(s) is largest for s = 2, this is plausibly the recession state. Because m(s) is smallest for s = 4, this is plausibly the boom state.

Asset Pricing (9 decks)

Brainscape helps you reach your goals faster, through stronger study habits.
© 2024 Bold Learning Solutions. Terms and Conditions