Absence of Arbitrage and Equilibrium
Econ 2100
Fall 2015
Lecture 26 (the last one), December 7
Outline
1
Asset Markets and Radner Equilibrium Recap
2
No Arbitrage
3
Martingale Pricing Theorem
4
Final Exam
Asset Markets
De…nition
An asset is a title to receive a certain amount of good 1 if and only if a particular
state s occurs at date 1.
Assets can be denominated in units of any good, we use good 1 by convention.
De…nition
The return matrix R is an S
asset k. That is:
K matrix
2
r11 ::
6 :: ::
6
R=6
6 r1s ::
4 :: ::
r1S ::
whose kth column is the return vector of
3
rk 1 :: rK 1
:: :: :: 7
7
rks :: rKs 7
7
:: :: :: 5
rkS :: rKS
De…nition
An asset structure R is complete if rank R = S, that is, if there are S assets with
linearly independent returns.
Budget Constraints
Assets are traded at date 0, while returns are realized at date 1.
At that time, agents decide how much to consume and trade on spot markets.
qk is the price of asset k, and q 2 RK are the asset prices.
zi 2 RK denote i’s asset holdings; we call zi a portfolio.
Given future spot prices p 2 RLS
+ , individual i’s budget constraints are
K
X
k =1
|
qk zki
{z
time 0
0
}
and
L
X
l =1
|
pls xlsi
L
X
l =1
pls ! lsi + p1s
{z
K
X
k =1
time 1
In equilibrium, future spot prices equal equilibrium prices.
zki rsk
for each s
}
Radner Equilibrium with Asset Markets
De…nition
A Radner equilibrium with asset markets is given by assets prices q 2 RK , spot
prices ps 2 RL for each s, portfolios zi 2 RK , and spot market plans xsi 2 RL for
each s such that:
1
for each i, zi and xi solve
max
zi 2RK ;xi 2RLS
+
Ui (x1i ; ::; xSi )
subject to
K
X
qk zki
0
and
ps xsi
ps ! si +
k =1
2
all markets clear; that is:
I
X
zki 0
and
i =1
K
X
p1s zki rsk
k =1
I
X
i =1
xsi
I
X
i =1
! si
for all s and k
Arbitrage
Arbitrage
An abitrage opportunity is the possibility to make strictly positive pro…ts at no risk.
Under what conditions on asset prices such opportunities do not exist?
De…nition
Asset prices q^ 2 RK satisfy absence of arbitrage if there does not exists a z 2 RK
such that
q^ z 0;
Rz 0
and
Rz 6= 0:
q^ are sometimes called no-arbitrage prices.
This can be interpreted as
q^ z 0
| {z }
a¤ordable portfolio
;
Rz
|
0
and
{z
Rz 6= 0
}
strictly positive pro…ts for sure
No arbitrage means there does not exists an a¤ordable portfolio that yields
non-negative returns in all states and strictly positive returns in some state.
Preferences are nowhere to be seen here.
Arbitrage and Asset Prices
No arbitrage imposes restrictions on prices.
No arbitrage and prices
Let
2
1
R=4 1
1
3
1 1
1 0 5
0 0
so that
2
3
z1 + z2 + z3
5
z1 + z2
Rz = 4
z1
Here, no arbitrage implies that if z1 > 0 then q1 > 0. Why?
Can always pick z2 and z3 equal to zero.
This result generalizes beyond the example.
Arbitrage and Asset Prices
No arbitrage and prices
Fact
If rk 0 and rk 6= 0 for all k (all assets have non-negative, non-zero, returns), then
no arbitrage implies qk > 0 for all k.
Think of the portfolio z de…ned as
zk > 0 and zk 0 = 0 for all k 0 6= k:
The portfolio returns are
Rz = rsk zk
this must be strictly positive in some state.
Therefore, the only way to avoid arbitrage is for the price of asset k to be
strictly positive.
Thus no arbitrage imposes
q z = qk zk
since we already have Rz
0 and Rz 6= 0:
0
No Arbitrage and Equilibrium Asset Prices
Theorem
Assume rk 0 and rk 6= 0 for all k (return vectors are nonnegative and non zero),
and suppose preferences are strictly monotone.
Then, any q 2 RK that is part of a Radner equilibrium satis…es no-arbitrage.
Under simple restrictions on assets and preferences, equilibrium implies
absence of arbitrage.
Intuitively, arbitrage implies some money is left on the table.
Given the e¢ ciency properties of equilibirum, this should not happen.
No Arbitrage and Equilibrium Asset Prices
We want to show that there cannot be arbitrage in equilibrium.
Proof.
Suppose not; then, q is an equilibrium and there exists a portfolio z such that
q
z
0;
Rz
0; and
Rz 6= 0
Suppose an individual buys some amount of this portfolio:
it doesn’t cost her anything, since q z 0;
given Rz 0 and Rz 6= 0, this action strictly increases her ‘…nancial’income
(all columns of R are positive and no entry is negative).
Hence, she can relax her budget constraint.
Because preferences are strictly monotone, she will increase her utility by using
this additional income to buy more of some goods on the spot markets at time
1.
This contradicts the assumption that q is an equilibrium.
Equilibrium and Linear Asset Prices
Theorem (Martingale Pricing)
Assume rk 0 and rk 6= 0 for all k (return vectors are nonnegative and non zero),
and suppose preferences are strictly monotone. Then, for any q 2 RK that is part
of a Radner equilibrium, we can …nd non-negative numbers s 0 which satisfy
[q ]T =
Equivalently:
qk =
S
X
s rsk
R
for all k
s =1
What does this equality imply?
Suppose there is an asset which pays a constant amount in all states: that is
rsk = c for all s.
Normalize the price of this asset to be c, so that
S
S
X
X
c=
c
)
1
=
s
s =1
s
s =1
Thus, equilibrium asset prices must equal an expected value of their returns.
The ‘probability’used to compute the expected value is the same for all assets.
No-Arbitrage Implies Linear Asset Prices
Since we have already shown that equilibrium implies no-arbitrage, to prove
the Martingale Pricing Theorem we only need to prove the following lemma.
Lemma
Suppose rk 0 and rk 6= 0 for all k (return vectors are nonnegative and non zero).
If asset prices q 2 RK satisfy the no-arbitrage condition, then there exists a vector
2 RS+ such that
qT =
R
The proof follows from convexity of the set of no arbitrage portfolios.
The main step uses the separating hyperplane theorem, and should remind you
of the proof of the welfare theorems.
Separate the positive orthant from the set of incomes that can be obtained
given a no-arbitrage (equilibirum) price vector.
Proof of Lemma: Convex Sets
Assume no row of R is zero (there is no state where all assets yield zero).
This is without loss of generality because if such a row existed, one can pick an
arbitrary s for that row.
Pick a no arbitrage price vector q^ 2 RK
Given q^, consider the following set
V = v 2 RS : v = Rz for some z 2 RK with q^ z = 0
This set is convex (prove it).
It is the set of ‘…nancial’incomes attainable with ‘zero cost’portfolios.
Note that if v 2 V , then v 2 V
one can always ‘reverse’all trades in z and q^ ( z ) =
(^
q z ) = 0.
Furthermore, 0 2 V .
Since q^ z = 0, no-arbitrage implies the …nancial income from z cannot be
strictly positive (Rz 0).
Thus, V can intersect the S-dimensional non-negative orthant only at 0:
Graph these sets.
V \ RS+ n f0g = ;:
Proof of Lemma: Separation
Since V and RS+ n f0g are convex, and the origin (f0g) belongs to V , we can
apply the separating hyperplane theorem at 0.
0
By separating hyperplane theorem, there exists a nonzero
0
v
0
and
0
0
Notice that
v
0
0;
for each v 2 V
for each v 2 RS+ :
if not,
< 0 and for some v 2 RS++ one would have
separation.
0
Because v and
Since
0
v < 0 contradicting
v are in V we have
0
0
2 RS such that
v
0
and
0
0, these two inequalities imply
( v) =
0
(
v = 0.
0
v)
0
Proof of Lemma: Linear Pricing
Consider the row vector
0
R 2 RK .
We need to show that q^T is proportional to
0
R.
Because R has all nonnegative entries (and no zero row) and
0
R
T
0
and
0
0, we have
T
0
R 6= 0 :
Suppose q^T is not proportional to 0 R (there exists no real number
such that q^T = ( 0 R)) and …nd a contradiction.
>0
If this is the case, there exists a z such that q^ z = 0 (z is ortogonal to q^) and
0
R z > 0.
Let v = R z; then v 2 V and 0 v > 0 contradicting separation ( 0 v = 0 for
each v 2 V ).
Thus, q^T is proportional to
> 0.
If we let
Lemma.
=
0
0
R and q^T =
we get the result q^T =
0
R for some real number
R concluding the proof of the
No-Arbitrage Implies Linear Asset Prices
We have proved the following
Lemma
Suppose rk 0 and rk 6= 0 for all k (return vectors are nonnegative and non zero).
If asset prices q 2 RK satisfy the no-arbitrage condition, then there exists a vector
2 RS+ such that
qT =
R
Notice that all we need for linear pricing is no-arbitrage.
There are no restrictions on preferences for the Lemma!
The connection with equilibrium is made by adding strictly monotone
preferences.
Linear Pricing at Work
Implications of Linear Pricing
Suppose there is an asset that delivers one unit of good 1 in all states:
r1 = (1; ::; 1)
Let q be a no-arbitrage price vector and q1 = 1.
If
sais…es q T =
=(
R, we have
1 ; ::;
S)
0
and
S
X
s
=
r1 = q1 = 1
s =1
Hence, for all assets di¤erent from asset 1,
S
X
and
qk =
min rsk
s rsk
s =1
s
qk =
S
X
s =1
s rsk
max rsk
s
Intuitively, prices must be between the lowest and highest dividend.
Final Exam
Wednesday, December 16, at 9am.
Three Hours Long.
Open Class Handouts.
Cumulative.
Focuses more on topics after midterm.
Exercises and proofs as usual.