Stochastic Discount Factors and Arbitrage Flashcards
Motivation
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 (∗)?
Key result
there is a positive SDF m such that p = e(mx) holds for all assets if and only if there are no arbitrage opportunities
State space
S = {1,…,S} (finite set of possible states of nature)
A real-valued random variable
is a function x : S → R
recall that a random variable is the same as an S-dimensional vector in RS
X ⊂ RS
denotes the set of all payoffs the investor can invest in, the payoff space
A price function
is a function p : X → R that assigns to each investable payoff x ∈ X its (ex-ante) market price p(x)
When Does p Have a SDF Representation?
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
Free Portfolio Formation – Motivation
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
Free Portfolio Formation – Formal Defintion
Assumption (free portfolio formation)
If x1,x2 ∈X and a1,a2 ∈R, then also a1x1+a2x2 ∈X
Remarks: free portfolio formation
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
Law of one price motivation
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)
Formal definition of Law of One Price
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
The Law of One Price and Arbitrage Opportunities
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
Does the law of one price guarantee absence of arbitrage?
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
Absence of arbitrage - Formal definition
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.
Remarks for absence of arbitrage
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)
Complete and Incomplete markets
Definition (market (in-)completeness)
The asset market is called complete if X = RS
The asset market is called incomplete if it is not complete
Interpretation of market completeness
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
Contingent claims
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
Contingent claims
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)
Contingent claims and market completeness proposition
The asset market is complete ⇔ all contingent claims are available, η1,…,ηS ∈ X.
Proof of this
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
Example: Synthesizing Contingent Claims
Representing the price function
Suppose that
the asset market is complete
p satisfies the law of one price (i.e. is linear)
Representation of Price Function – Interpretation
Economic idea behind representation of price function
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 ))
State Diagram
Alternative State Space Geometry
SDFs
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?
Stochastic discount factors equation
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
Conclusion from this
in complete markets, any p that satisfies the law of one price has a SDF representation
does a SDF representation imply the law of one price?
yes, if p(x) = E[mx] for all x, then p is a linear function
Is the SDF representation unique?
yes, if p(x) = E[mx] for all x, then in particular for all s = 1,…,S
Deriving SDF equation
Conclusion (SDF representation in complete markets)
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)
When Does p even Satisfy Absence of Arbitrage?
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
When Does p even Satisfy Absence of Arbitrage? - conclusions
No arbitrage and SDFs - absence of arbitrage in complete markets
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.
Illustration: Nonpositive SDFs Imply Arbitrage Opportunities
Risk-neutral Probabilities
Introducing an investor into contingent claims market using Lagrangians
Law of One Price and SDFs: Main Theorem
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.
Main Theorem remarks
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
Proof: SDF Representation ⇒ Law of one Price
Proof: Law of one Price ⇒ SDF Representation – Lazy Mathematician’s Proof
Proof: Law of one Price ⇒ SDF Representation – Geometric Proof
Proof: Law of one Price ⇒ SDF Representation – Algebraic Proof
(Non-)Uniqueness of the SDF
What this tells us
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})
Example: Non-Uniqueness of SDFs
No Arbitrage and Positive SDFs: Main Theorem
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.
Remarks from this main theorem
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
Proof: Positive SDF ⇒ No Arbitrage
What properties does the absence of arbitrage require
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:
Idea of proof: Direction no Arbitrage ⇒ Positive SDF
Non-Uniqueness of Positive SDFs: Example
Summary
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
What states would you expect a boom and a recession?
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.
