1 no-arbitrage pricing - paul kleinpaulklein.ca/newsite/teaching/809notes812.pdf · 2019-01-17 ·...
TRANSCRIPT
SIMON FRASER UNIVERSITY
BURNABY BRITISH COLUMBIA
Paul Klein
Office: WMC 3635
Phone: TBA
Email: paul klein [email protected]
URL: http://paulklein.ca/newsite/teaching/809.php
Economics 809
Advanced macroeconomic theory
Spring 2012
Lecture 8: Finance
1 No-arbitrage pricing
How far can we get in determining the “correct” price of assets by just as-
suming that there are no arbitrage opportunities? Notice that this is a weaker
assumption than (competitive) equilibrium. Surprisingly far, as it turns out. In
particular, we can work out how to price derivatives — claims that are defined
in terms of other securities, e.g. options. Arbitrage–free pricing is a lot like
expressing a vector as the linear combination of basis vectors.
1
1.1 A two–period model
Let t = 0, 1 (today and tomorrow). There areN securities. The price of security
n at time t is denoted by Snt . We write
St =
S1t
S2t...
SNt
.
S0 is deterministic, but S1 is stochastic. We write
S1(ω) =
S11(ω)
S21(ω)...
SN1 (ω)
where ω ∈ Ω = ω1, ω2, . . . , ωM.
Now define the matrix D via
DN×M
=
S11(ω1) S1
1(ω2) . . . S11(ωM)
S21(ω1)
. . . . . . ...... . . . . . . ...
SN1 (ω1) . . . . . . SN
1 (ωM)
=[d1 d2 . . . dM
].
Definition. A portfolio is a vector h ∈ RN . Interpretation: hn is the number
of securities of type n purchased at t = 0.
Remark: Fractional holdings as well as short positions (hn < 0) are allowed.
2
The value of a portfolio h at time t is given by
Vt(h) =N∑n=1
hnSnt = hTSt.
Definition. A vector h ∈ RN is called an arbitrage portfolio if
V0(h) < 0
and
V1(h) ≥ 0
for all ω ∈ Ω.
Remark. We can weaken the condition “for all ω ∈ Ω” to “with probability
one” if we want — but in this context that wouldn’t add much.
Theorem. Let securities prices S be as above. Then there exists no arbitrage
portfolio iff there exists a z ∈ RM+ such that
S0 = Dz.
Remark. This means that today’s (period 0’s) price vector has to lie in the
convex cone spanned by tomorrow’s (period 1’s) possible prices vectors. (A
convex cone is a subset C of a vector space X such that for any x, y ∈ C and
α ≥ 0 we have (αx) ∈ C and (x+ y) ∈ C. )
Proof. Absence of arbitrage opportunities means that the following system of
inequalities has no solution for h.hTS0 < 0
hTdj ≥ 0 for each j = 1, 2, . . . ,M.
Geometric interpretation: there is no hyperplane that separates S0 from the
columns of D. Such a hyperplane would have an arbitrage portfolio as a normal
3
vector. Now according to Farkas’ lemma, the non–existence of such a normal
vector is equivalent to the existence of non–negative numbers z1, z2, ..., zM such
that
S0 =M∑j=1
zjdj
or, equivalently,
S0 = Dz
where z ∈ RM+ .
We will prove Farkas’ lemma as a corollary of the Separating Hyperplane The-
orem.
Proposition (Farkas’ lemma). If Am×n
is a real matrix and if b ∈ Rm, then
exactly one of the following statements is true.
1. Ax = b for some x ∈ Rn+
2. (yTA) ∈ Rn+ and yT b < 0 for some y ∈ Rm.
Proof. In what follows we will sometimes say that x ≥ 0 when x is a vector.
This means that all the components are non–negative. To prove that (1) implies
that not (2), suppose that Ax = b for some x ≥ 0. Then yTAx = yT b. But
then (2) is not true. If it were, then yTA ≥ 0 and hence yTAx ≥ 0. But then
yT b ≥ 0. Hence (1) implies not (2). Next we show that not (1) implies (2). Let
X be the convex cone spanned by the columns a1, a2, ..., an of A, i.e.
X =
a ∈ Rn : a =
n∑i=1
λiai; λi ≥ 0, i = 1, 2, . . . , n
.
Suppose there is no x ≥ 0 such that Ax = b. Then b /∈ X. Since X is closed
and convex, the Separating Hyperplane Theorem says that there is a y ∈ Rm
4
such that yTa > yT b for all a ∈ X. Since 0 ∈ X, yT0 > yT b and consequently
yT b < 0. Now suppose (to yield a contradiction) that not yTA ≥ 0. Then there
is a column in A, say ak, such that yTak < 0. Since X is a convex cone and
ak ∈ X, we have (αak) ∈ X for all α ≥ 0. But by the (absurd) supposition,
for sufficiently large α, we have yT (αak) = α(yTak) < yT b, and this contradicts
separation so the supposition cannot be true. Thus yTai ≥ 0 for all columns ai
of A, i.e. yTA ≥ 0.
Separating Hyperplane Theorem. Let X ⊂ Rn be closed and convex, and
suppose y /∈ X. Then there is an a ∈ R and an h ∈ Rn such that hTx > a > hTy
for all x ∈ X.
Proof. Omitted.
A popular interpretation of the result S0 = Dz is the following (but beware of
over–interpretation). Define
qi =ziβ
where
β =M∑i=1
zi.
Then q can be thought of as a probability distribution on Ω and we may conclude
the following.
Theorem. The market S is arbitrage–free iff there is a scalar β > 0 and a
probability measure Q such that
S0 = βEQ[S1] = β[q1S1(ω1) + q2S1(ω2) + . . .+ qMS1(ωM)]
and we call this a martingale measure (for reasons that are not immediately
obvious in this context). Economically speaking, the qis are “state prices”.
Relative prices of $1 in state i. Arrow–Debreu prices.
5
Yet another approach is to define a so–called pricing kernel. For an arbitrage–
free market S, there exists a (scalar) random variable m : Ω → R such that
such that
S0 = E[m · S1]
where the expectation is now taken under the “objective” measure P . Provided
the outcomes ωi all obtain with strictly positive probabilities, we have
m(ωi) = βQ(ωi)/P (ωi) = βqi/pi.
where the pis are the objective probabilities. So the pricing kernel takes care
both of discounting and the change of measure.
1.1.1 Pricing contingent claims
Definition. A contingent claim is a mapping X : Ω → R. We represent it as a
vector x ∈ RM . Interpretation: the contract x entitles the owner to $xi in state
ωi.
Theorem. Let S be an arbitrage–free market. Then there exist β, q such that
if each contingent claim x is priced according to
π0[X] = βqTx = βEQ[X]
then the market consisting of all contingent claims is arbitrage free.
Example. Let Ω = ω1, ω2. Let
S10 = b S2
0 = s
S11(ω1) = (1 + r)b S1
1(ω2) = (1 + r)b
S21(ω1) = x S2
1(ω2) = y
where r > 0 and, without loss of generality, y ≤ x. Then
D =
[(1 + r)b (1 + r)b
x y
].
6
Abusing the notation somewhat, define q = q1. No arbitrage impliesb = β[q(1 + r)b+ (1− q)(1 + r)b]
s = β[qx+ (1− q)y]
0 ≤ q ≤ 1.
We may conclude that
β =1
1 + rand
q =s(1 + r)− y
x− y, 1− q =
x− s(1 + r)
x− y.
The condition 0 ≤ q ≤ 1 then becomes
0 ≤ s(1 + r)− y
x− y≤ 1.
That is to say, x = y and
y ≤ s(1 + r) ≤ x.
This is not a surprising no–arbitrage condition. In this case, it turns out that
there is a unique martingale measure, so all contingent claims can be uniquely
priced. Below we give a more general treatment of this phenomenon, and it
turns out that this uniqueness is intimately related to the notion of market
completeness.
1.1.2 Completeness and uniqueness of martingale measure
Can contingent claims be prices in a unique way? One class of contingent claims
certainly can: the hedgeable ones.
Definition. A contingent claim X is said to be hedgeable if there is an h ∈ RN
such that
V1(h) ≡ X
7
i.e.
hTS1(ω) = X(ω)
for all ω ∈ Ω or
hTdi = X(ωi)
for all i = 1, 2, . . . ,M or
hTD = xT
i.e. hedgeability of X boils down to the corresponding x being in the row space
of D.
Proposition. Suppose the contingent claim X is hedgeable via h. Then the
only price of X at 0 that is consistent with no arbitrage is
π0[X] = hTS0.
Proof. Exercise.
Proposition. Suppose X is a contingent claim that is hedgeable via h and that
it is also hedgeable via g. Suppose also that there are no arbitrage opportunities.
Then
hTS0 = gTS0.
Proof. Exercise.
Thus any hedgeable contingent claim can be uniquely priced. Under what
circumstances can all contingent claims be uniquely priced?
Definition. A market S is said to be complete if all contingent claims are
hedgeable.
Proposition. A market S is complete iff rank(D) ≥ M . A necessary condition
is N ≥ M .
Proof. Exercise.
8
Meta–theorem. A market S is generically arbitrage–free and complete if
N = M . If N > M it is generically not arbitrage–free. If N < M it is
incomplete.
Proposition. Suppose a market is arbitrage–free and complete. Then there
is a unique probability measure Q such that every contingent claim is priced
according to
π0[X] = βEQ[X]
where
β = π[1]
where 1 is the “unit” contingent claim that delivers $1 no matter what.
Proof.
π0[X] = hTS0 = hTβEQ[S1] =
= βEQ[hTS1] = βEQ[X].
Meanwhile,
π0[1] = βEQ[1] = β.
2 Equilibrium pricing
In general equilibrium, relative prices are given by marginal rates of substitu-
tion. This gives a way of determining prices even if they are not given exoge-
nously. Lucas (1978) pioneered the pricing of assets in dynamic equilibrium.
We will assume here that preferences are represented by
E
[ ∞∑t=0
βt c1−αt
1− α
].
9
Now let’s suppose that a single asset with stochastic return rt is available, so
that the consumer’s period-by-period budget constraint is
at+1 = wt + rtat − ct
Suppose that the information flow is given by the filtration Ft∞t=0 generated
by the processes rt and wt. Applying the principles of stochastic dynamic
optimization, we form the Hamiltonian:
H = βt c1−αt
1− α+ λt+1[wt + rtat − ct]
and the optimality conditions are
E[λt+1|Ft]rt = λt
and
βtc−αt = E[λt+1|Ft].
Apparently—though it is not immediately obvious why this is relevant—
E[λt+2|Ft+1]rt+1 = λt+1
and
βt+1c−αt+1 = E[λt+2|Ft+1]
Thus
βt+1c−αt+1rt+1 = λt+1 (1)
and you may recall that
βtc−αt = E[λt+1|Ft].
Thus, taking expectations with respect to Ft on each side of Equation (1), we
get
βt+1E[c−αt+1rt+1|Ft
]= βtc−α
t
10
or, simplifying and rearranging,
1 = βE[c−α
t+1rt+1|Ft]
c−αt
establishing that
mt+1 = βc−αt+1rt+1
c−αt
is a pricing kernel for this economy.
2.1 What kind of assets yield high average returns?
It is tempting to conclude that, in general, if people are risk averse, then risky
assets have to yield a high return to compensate for risk. That would be wrong,
however. What matters is how good an asset is in providing insurance against
consumption variance. An risky asset that pays more when consumption is low
and less when it is high is actually better than a safe asset and will still be held
even if the average return is lower than that of a safe asset.
To establish this result, we will assume that consumption growth ln ct+1 − ln ct
is i.i.d. normally distributed with mean µ and variance σ2. The normal distri-
bution is convenient because the linear combination of two random variables is
also normally distributed.
We also need the following result. If X is normally distributed with mean µX
and variance σ2X , then
E[eX ] = exp
µX +
1
2σ2X
.
It follows that if X and Y are joint normal with means µX and µX , variances
σ2X and σ2
Y and covariance σXY then
E[eX+Y ] = exp
µX + µY +
1
2σ2X +
1
2σ2Y + σXY
.
11
With the assumptions we have made so far we are able to derive an explicit
expression for the price of a bond—a riskless asset that yields precisely 1 unit
of consumption in period 1. Denote the period 0 price of a bond by q. Then
q = E[m · 1] = βE
[c−αt+1
c−αt
]=
= βE[exp−α(ln ct+1 − ln ct)] =
β exp
−αµ+
1
2α2σ2
.
On the other hand, the risk-free rate r∗ is
r∗ =1
q= β−1 exp
αµ− 1
2α2σ2
.
(The risk free rate might have depended on time. But it doesn’t in this case,
so we omit the time subscript.)
Now let’s introduce an arbitrary risky asset with stochastic rate of return rt.
Let’s assume ln rt+1 and ln ct+1 − ln ct are joint normal. Then Aiyagari (1993)
claims (rightly of course) that
E[rt+1]
r∗= exp αCOV(ln rt+1, ln ct+1 − ln ct) .
Let’s try to verify this! First of all, we have the following beautiful and very
general result which holds for all rates of return processes rt:
E[mt+1 · rt+1] = 1.
for some non-negative stochastic process mt (adapted to Ft∞t=0) With our
specification of preferences, as we have seen,
mt+1 = β exp−α(ln ct+1 − ln ct)
12
and the equation E[mt+1 · rt+1] = 1 becomes
βE[expln rt+1 − α[ln ct+1 − ln ct]] = 1
which implies
β exp
µln rt+1
+1
2σ2ln rt+1
− αµ+1
2α2σ2 − αCOV(ln rt+1, ln ct+1 − ln ct)
= 1.
This can be factorized to become
β exp
µln rt+1
+1
2σ2ln rt+1
exp
−αµ+
1
2σ2
exp −αCOV(ln rt+1, ln ct+1 − ln ct) = 1
and the result follows.
We take away from this that an asset earns a premium on average when its re-
turn is positively correlated with consumption growth—or negatively correlated
with future marginal utility.
2.2 The risky asset is a claim to consumption
Here we will assume, in the spirit of Mehra and Prescott (1985), that the risky
asset return is proportional to per-capita consumption growth, i.e.
rt+1 = A · ct+1
ct.
for some A > 0. But then the covariance between ln r and ln ct+1 − ln ct is just
the variance of ln ct+1 − ln ct! Plugging this into Aiyagari’s formula, we get
E[rt+1]
r∗= expασ2.
This is a very satisfying result, because it means that the ratio of returns
depends on the variance of consumption growth and α only.
13
According to my calculations, based on data from the U.S. Bureau of Economic
Analysis, the growth rate of annual per capita consumption in the United States
has averaged about 2.0 percent between 1929 and 2005, with a standard de-
viation of about 0.02. If we believe that rE = 1.06 and rB = 1.01 then we
get
α ≈ 0.05
0.022= 125.
Now if α = 125 and µ = 0.02 we can compute β from the formula
r∗ = β−1 exp
αµ− 1
2α2σ2
.
The result is
β ≈ 0.53.
So we can make the model fit the data. Is there really an equity premium
puzzle?
2.3 The more general case
Consider the approximate version of Aiyagari’s formula:
E[r]− r∗ ≈ αCOV(ln r, ln ct+1 − ln ct).
Now apply Cauchy-Schwarz’ inequality which says that, for any random vari-
ables X and Y
COV(X,Y ) ≤√
V(X) · V(Y ) = STD(X) · STD(Y ).
We get (approximately at least)
E[r]− r∗ ≤ αSTD(ln r) · STD(ln ct+1 − ln ct).
14
According to Aiyagari (1993), the standard deviation of the return on equity
is about 0.07. Thus the predicted equity premium is about 0.0014α. If the
equity premium is 0.05, we must have α ≥ 36. So do we really have an equity
premium puzzle?
Most economists believe there is. Indeed, the conventional wisdom is that none
of the proposed solutions have been successful. Kocherlakota (1996) concludes
that it is still a puzzle. This in spite of, for example,
• Constantinides (1990). Habit formation.
• Huggett (1993). Market incompleteness.
2.4 An infinitely-lived asset
Like Mehra and Prescott (1985), I now assume that the risky asset is a claim
to per-capita consumption in every period from the next one onwards. (Unlike
them, I assume log-normal and i.i.d. consumption growth.) The purpose of
this is to give an example of an asset whose rate of return is proportional to
consumption growth. Denote the price of the risky asset by pt. Might there be
a stationary equilibrium where pt = Act for some A > 0? If so,
Act = βEt
c−αt+1c
αt · [ct+1 + Act+1]
.
It follows that (since we assume ct to be known when expectations are evaluated)
A = β(1 + A)Et exp [(1− α)(ln ct+1 − ln ct] .
Hence
A = β(1 + A) exp
(1− α)µ+
1
2(1− α)2σ2
.
15
Rearrangement (for no obvious purpose right now) yields
1 + A
A= β−1 exp
−(1− α)µ− 1
2(1− α)2σ2.
Given what we know already,
Et[rt+1] =1 + A
Aexp
µ+
1
2σ2
.
Substituting in our expression for (1 +A)/A we get
E[rt+1] = Et[rt+1] = β−1 exp
αµ+ α
(1− 1
2α
)σ2
.
Summarizing our results so far, we haveE[rt+1] = β−1 exp
αµ+ α
(1− 1
2α)σ2
r∗ = β−1 expαµ− 1
2α2σ2
.
It follows from this thatE[rt+1]
r∗= expασ2.
2.5 The original Mehra-Prescott paper
Mehra and Prescott (1985) begin their paper by noting the following fact. Be-
tween 1889 and 1978, the average return on equity was 7 percent per year and
the average return on bonds was less than one percent. The difference (by def-
inition) is called the equity premium. Jagannathan et al. (2000) claim that this
premium has declined significantly, and that it was approximately zero 1970-
2000. But this new fact (if indeed it is a fact) was not known by Mehra and
Prescott in 1985.
Mehra and Prescott analyze an exchange (endowment) economy where a rep-
resentative consumer maximizes
E
[ ∞∑t=0
βtu(ct)
]16
where
u(c) =c1−σ − 1
1− σ.
Output follows an exogenous stochastic process satisfying
yt+1 = xt+1yt
and since all agents are alike and there is no trade, we have
ct = yt.
Meanwhile, xt is a finite–state Markov chain in discrete time. Specifically,
xt ∈ λ1, λ2, . . . , λn
and
P [xt+1 = λj|xt = λi] = φij.
It follows that the distribution of xt satisfies
µt+1 = ΦTµt.
Suppose there is a unique stationary distribution π such thatπ = ΦTπ
πT1 = 1.
and
limt→∞
µt = π
for all µ0 ∈ Rn such that µT0 1 = 1.
Definition. A share of equity at t is a claim to ys∞s=t+1. Denote the price of
this bundle of claims by pt.
The price of equity satisfies the following difference equation.
u′(yt)pt = βEt [u′(yt+1)(pt+1 + yt+1)] .
17
The presence of the conditional expectation is justified in my handout on
stochastic dynamic optimization.
Might there exist a pricing function
pt = p(y, i)
where y is current output and i is the current state of x? We can define it
implicitly via
p(y, i) = βn∑
j=1
φij(λjy)−σ[p(λjy, j) + λjy]y
σ.
It turns out that this function not only exists but is linear in y! So there exist
constants ω1, ω2, . . . , ωn such that
p(y, i) = ωi y.
A system of equations for these constants is given by
ωi = βn∑
j=1
φijλ1−σj (ωj + 1); i = 1, . . . , n.
We now want to derive the average return on equity. We begin by defining the
period return on equity when going from state i to state j via
Reij =
p(λjy, j) + λjy − p(y, i)
p(y, i)=
λj(ωj + 1)
ωi− 1.
The conditionally expected return on equity is
Rei =
n∑j=1
φijReij
and the unconditionally expected return (equal to the long–run average by the
LOLN) is
Re =n∑
i=1
πiRei .
18
Now consider a riskless one–period bond. How are we to price it, in terms of
current consumption? Since it is a claim to 1 unit of consumption in each state,
the formula is
q(y, i) = βn∑
j=1
φij
u′(λjy)
u′(y)· 1.
With our specification of preferences, we get
q(y, i) = βn∑
j=1
φijλ−σj .
Incidentally,βu′(ct+1)
u′(ct)
constitutes a pricing kernel for this economy. We can now define the period
return on a bond in state i via
Rbi =
1
q(y, i)− 1
and the unconditionally expected return is
Rb =n∑
i=1
πiRbi .
The equity premium is then defined as Re −Rb.
But how are we to judge whether theory is consistent with Re = 0.07 and
Rb = 0.01? We calibrate! Here are the parameters we need to determine.
(1) The states λ1, λ2, . . . , λn and the transition probabilities φij; i, j = 1, 2, . . . , n.
(2) The preference parameters β and σ.
We begin with (1).
There is data on consumption growth rates and they have an average µ.
19
Impose symmetry so that Φ = ΦT .
Set n = 2.λ1 = 1 + µ+ δ
λ2 = 1 + µ− δ.
Φ =
[φ 1− φ
1− φ φ
]
Apparently π1 = π2 =1
2.
Finally, use δ and φ to match variance and autocorrelation! The (unconditional)
variance of consumption growth is
1
2[δ2] +
1
2[δ2] = δ2.
Meanwhile, the autocovariance is
1
2[φ · δ2 − (1− φ)δ2] +
1
2[φδ2 − (1− φ)δ2]
and the autocorrelation is the autocovariance divided by the variance, i.e.
1
2[φ− 1 + φ]
1
2[φ− 1 + φ] =
1
2[4φ− 2] = 2φ− 1.
Mehra and Prescott (1985) find that, with annual data,
µ = 0.018
δ = 0.036
and
φ = 0.43
resulting from an autocorrelation of consumption growth equal to −0.14.
(2) Determining β and σ. It would, in principle, be feasible to choose β and
σ so as to match Re and Rb. Strangely, Mehra and Prescott (1985) do not
20
consider this alternative. If they did, they would find that σ must be be very
large. Apparently Mehra and Prescott (1985) take as an axiom that σ < 10.
So they conclude that the model cannot account for the facts.
Apparently most economists are still uncomfortable with σ ≫ 1. It is not
altogether clear why. But if one were to estimate β and σ in the way that I
suggested, the parameter estimates would vary wildly with the sampling period.
This suggests that there might be something fishy with the model.
Many attempts have been made since Mehra and Prescott (1985) to alter pref-
erences in such a way as to solve the equity puzzle. Have any of these attempts
been successful and persuasive? Kocherlakota (1996) doesn’t think so.
Exercise 1 Suppose consumption ct satisfies
ln ct+1 = ρ ln ct + εt+1
where εt is i.i.d. normal with mean 0 and variance σ2. Assume the same pref-
erences as in these lecture notes. Suppose 0 < ρ < 1. Find an expression for
the period t riskless rate of return.
21
References
Aiyagari, S. R. (1993). Explaining financial market facts: The importance of incomplete marketsand transactions. Federal Reserve Bank of Minneapolis Quarterly Review 17 (1), 17–31.
Constantinides, G. (1990). Habit formation: A resolution of the equity premium puzzle. Journalof Political Economy 98 (3).
Huggett, M. (1993). The risk free rate in heterogeneous-agents, incomplete insurance economies.Journal of Economic Dynamics and Control 17 (5/6), 953–970.
Jagannathan, R., E. R. McGrattan, and A. Scherbina (2000). The declining U.S. equity pre-mium. Federal Reserve Bank of Minneapolis Quarterly Review 24 (4), 3–19.
Kocherlakota, N. R. (1996, March). The equity premium: It’s still a puzzle. Journal ofEconomic Literature 34 (1), 42–71.
Lucas, R. E. (1978, November). Asset prices in an exchange economy. Econometrica 46 (6),1429–1445.
Mehra, R. and E. Prescott (1985). The equity premium: A puzzle. Journal of MonetaryEconomics 15, 145–61.
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