Fin 501:Asset Pricing I
Lecture 08: Multi
Multi--period Model
Prof. Markus K. Brunnermeier
Multi--period Model
Multi
Slide 08
08--1
Fin 501:Asset Pricing I
… the
th remaining
i i lectures
l t
•
•
•
•
•
•
Generalizing to multi-period setting
Ponzi Schemes,
Schemes Bubbles
Forward, Futures and Swaps
ICAPM and
d hhedging
d i ddemand
d
Multiple factor pricing models
Market efficiency
Multi--period Model
Multi
Slide 08
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Fin 501:Asset Pricing I
I t d ti
Introduction
• accommodate multiple and even infinitely many
periods.
• severall issues:
i
¾ how to define assets in an multi period model,
¾ how to model intertemporal preferences,
preferences
¾ what market completeness means in this environment,
¾ how the infinite horizon may the sensible definition of a
budget constraint (Ponzi schemes),
¾ and how the infinite horizon may affect pricing (bubbles).
•
This section is mostly based on Lengwiler (2004)
Multi--period Model
Multi
Slide 08
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Fin 501:Asset Pricing I
O
Overview
i
1. Generalizing the setting
¾ Trees, modeling information and learning
¾ Time
Time--preferences
2 Multi
2.
Multi--period SDF and event prices
¾ Static
Static--dynamic completeness
3 Dynamic
3.
D
i completion
l ti
4. Ponzi scheme and rational bubbles
5. Martingale process - EMM
Mean--Variance Analysis and CAPM
Mean
Slide 08
08--4
Fin 501:Asset Pricing I
0
1
2
3
many one period
i d models
d l
how to model information?
Multi--period Model
Multi
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Fin 501:Asset Pricing I
0
1
2
3
0
1
2
3
States s
Ω∪ ∅ ∪
F1
Multi--period Model
Multi
F2
Events Ai,t
Slide 08
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Fin 501:Asset Pricing I
M d li information
Modeling
i f
ti over time
ti
• Partition
• Field/Algebra
• Filtration
Multi--period Model
Multi
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Fin 501:Asset Pricing I
S
Some
probability
b bilit theory
th
• Measurability: A random variable y(s) is measure w.r.t.
algebra F if
¾ Pre-image of y(s) are events (elements of F)
• for each Ai ∈ F, y(s) = y(s’) for each s ∈ Ai and s’ ∈ Ai,
• yt(Ai):=
): yt(s),
(s) s ∈ Ai.
• Stochastic process: A collection of random variables yt(s)
for t = 0,…, T.
• Stochastic process is adapted to filtration F ={Ft}tT if each
yt((s)) is measurable w.r.t. Ft
¾ Cannot see in the future
Multi--period Model
Multi
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Fin 501:Asset Pricing I
f
from
static
t ti to
t dynamic…
d
i
asset holdings
Dynamic strategy (adapted process)
asset payoff x
Next period’s payoff xt+1+ pt+1
Pa off of a strategy
Payoff
strateg
span of assets
Marketed subspace of strategies
Market completeness
a) Static completeness (Debreu)
b) Dynamic completeness (Arrow)
No arbitrage w.r.t. holdings
No arbitrage w.r.t strategies
State prices q(s)
Event prices qt(At(s))
Multi--period Model
Multi
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Fin 501:Asset Pricing I
…from
f
static
t ti to
t dynamic
d
i
State prices q(s)
_
Risk free rate R
Risk neutral _prob.
π*(s) = q(s) R
Pricingg kernel
pj = E[kq x_j]
1 = E[kq] R
Event prices qt(A(s))
_
Risk free rate rt varies over time
Discount factor from t to 0 ρt(s)
Risk neutral prob.
π*(At(s)) = qt(At(s)) / ρt (At)
Pricingg kernel
kt ptj_= Et[kt+1(pjt+1+ xjt+1)]
kt = Rt+1 Et[kt+1]
Multi--period Model
Multi
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Fin 501:Asset Pricing I
… in more detail
A3 ‰ We recall the event tree that
captures the gradual resolution of
uncertainty.
ψ1(A4)
A1
ψ0(A4)
)
ψ2(A4) ‰ This tree has 7 events (A0 to A6).
(Lengwiler uses e0 to e6)
A4
A0
‰
3 time periods (0 to 2).
A5 ‰ If A is some event, we denote the
period it belongs to as τ(A).
A2
A6
t 0
t=0
t 1
t=1
‰
So for instance,
instance τ(A2)=1,
)=1 τ(A4)=2.
)=2
‰
We denote a path with ψ as follows…
t 2
t=2
Multi--period Model
Multi
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Fin 501:Asset Pricing I
M lti l period
Multiple
i d uncertainty
t i t
A3
A1
A4
A0
‰
Last period events have prob.,
π 3 - π 6.
‰
The earlier events also have
probabilities.
‰
To be consistent
consistent, the
probability of an event is
equal to the sum of the
probabilities of its successor
events.
‰
So for
fo instance,
instance π1 = π3+π
+ 4.
A5
A2
A6
t=0
t=1
t=2
Multi--period Model
Multi
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Fin 501:Asset Pricing I
M lti l period
Multiple
i d assets
t
A typical multiple period asset is a coupon bond:
⎧coupon
if 0 < τ ( A) < t * ,
⎪
rA := ⎨1 + coupon if τ ( A) = t * ,
*
⎪0
if
(
)
>
.
τ
A
t
⎩
The coupon bond pays the coupon in each period &
pays the coupon plus the principal at maturity t*.
A consol is a coupon bond with t* = ∞;
it pays a coupon forever.
A discount bond (or zero-coupon bond)
finite maturity bond with no coupon.
coupon
It just pays 1 at expiration, and nothing otherwise.
Multi--period Model
Multi
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Fin 501:Asset Pricing I
M lti l period
Multiple
i d assets
t
• create STRIPS by extracting only those
payments that occur in a particular period.
¾ STRIPS are the same as discount bonds.
• More generally, arbitrary assets (not just bonds)
could
ld be
b striped.
i d
Multi--period Model
Multi
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Fin 501:Asset Pricing I
M lti l period
Multiple
i d assets
t
• Sh
Shares are claims
l i to the
h future
f
dividends
di id d off a firm.
fi
• Derivatives: A call option on a share, is an asset that
pays either 0 or p - k,
k where p is the (random) price of
the share and k is the strike (exercise) price.
p
of this kind will be helpful
p when thinkingg
• Options
about how to make a market system complete.
Multi--period Model
Multi
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Fin 501:Asset Pricing I
O
Overview
i
1. Generalizing the setting
¾ Trees, modeling information and learning
¾ Time
Time--preferences
2 Multi
2.
Multi--period SDF and event prices
¾ Static
Static--dynamic completeness
3 Dynamic
3.
D
i completion
l ti
4. Ponzi scheme and rational bubbles
5. Martingale process - EMM
Mean--Variance Analysis and CAPM
Mean
Slide 08
08--16
Fin 501:Asset Pricing I
Ti preference
Time
f
u(c0) + ∑t δ(t) E{u(ct)}
• Discount factor δ(t) number between 0 and 1
• Assume δ(t)>δ(t+1) for all t.
• Suppose
S
you are iin period
i d 0 andd you make
k a plan
l off
your present and future consumption: c0, c1, c2, …
• The relation between consecutive consumption will
depend on the interpersonal rate of substitution, which is
δ(t).
• Time consistency δ(t) = δ t (exponential discounting)
Multi--period Model
Multi
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Fin 501:Asset Pricing I
Ti preference
Time
f
• Canonical framework (exponential discounting)
U(c) = E[ ∑ δt u(ct)]
¾ prefers earlier uncertainty resolution if it affect action
¾ indifferent, if it does not affect action
• Time-inconsistent (hyperbolic discounting)
Special case: β−δ formulation
U(c) = E[u(c0) + β ∑ δt u(ct)]
• Preference for the timing of uncertainty resolution
recursive utility formulation (Kreps-Porteus
(Kreps Porteus 1978)
Slide 08
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Fin 501:Asset Pricing I
Preference for the
timing of uncertainty resolution
Digression:
g
π
0
$100
$150
$100
$ 25
$150
$100
Kreps-Porteus
π
$ 25
Early (late) resolution if W(P1,…) is convex (concave)
Example: do you want to know whether you will get cancer at the age of 55?
Slide 08
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Fin 501:Asset Pricing I
Digression: MultiM lti-period
Multi
i d
P tf li Choice
Portfolio
Ch i
Theorem 4.10 ((Merton, 1971):
) Consider the above canonical
multi-period consumption-saving-portfolio allocation problem.
Suppose U() displays CRRA, rf is constant and {r} is i.i.d.
Then a/st is time invariant.
invariant
Slide 08
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Fin 501:Asset Pricing I
O
Overview
i
1. Generalizing the setting
¾ Trees, modeling information and learning
¾ Time
Time--preferences
2 Multi
2.
Multi--period SDF and event prices
¾ Pricing in a static dynamic model
3 Dynamic
3.
D
i completion
l ti
4. Martingale process - EMM
Mean--Variance Analysis and CAPM
Mean
Slide 08
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Fin 501:Asset Pricing I
A static
i dynamic
d
i model
d l
• We consider pricing in a model that contains many
periods (possibly infinitely many)…
• …and
and we assume that information is gradually
revealed (this is the dynamic part)…
• …but we also assume that all assets are only traded
"at the beginning of time" (this is the static part).
• There is dynamics
y
in the model because there is time,,
but the decision making is completely static.
Multi--period Model
Multi
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Fin 501:Asset Pricing I
M i i ti over many periods
Maximization
i d
• vNM exponential utility representative agent
max{∑t=0 δt E{u(ct)} | y-w ∈ B(p)}
• If all Arrow securities (conditional on each
event) are traded, we can express the first-order
conditions as,
u'(c0) = λ, δτ(A)πAu'(wA) = λqA.
Multi--period Model
Multi
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Fin 501:Asset Pricing I
MultiM lti-period
Multi
i d SDF
• The equilibrium SDF is computed in the same fashion as
i the
in
h static
i model
d Al we saw before
b f
qA
πA
= δ τ ( A)
u' (w )
u ' ( w0 )
⎛ u ' ( wψ 1 ( A) ) ⎞ ⎛ u ' ( wψ 2 ( A) ) ⎞ ⎛
u' (w A ) ⎞
⎟
⎟⎟ ⋅ ⎜⎜ δ
⎟L⎜ δ
= ⎜⎜ δ
ψ
(
A
)
ψ 1 ( A) ⎟
0
⎜ u ' ( w τ ( A )−1 ) ⎟
u
'
(
w
)
u
'
(
w
)
⎝
⎠ ⎝
⎠ ⎝
⎠
mψ 1 ( A)
•
×
mψ 2 ( A)
×L×
mψ τ ( A )−1 ( A)
=: M A
We call mA the "one-period ahead" SDF and MA
the multi-period SDF (“state-price density”).
Multi--period Model
Multi
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Fin 501:Asset Pricing I
Th fundamental
The
f d
t l pricing
i i formula
f
l
• To price an arbitrary asset x,
portfolio of STRIPed cash flows, x j = x1j +x2j +L+x∞j,
where xt j denotes the cash-flows in period t.
• The price of asset x j is simply the sum of the prices of
its STRIPed payoffs, so
p j = ∑ E{M t xt }
j
t
•
This is the fundamental pricing formula.
•
Note that Mt = δt if the repr agent is risk neutral. The
fundamental pricing formula then just reduces to the present
value of expected dividends, pj = ∑ δt E{xtj}.
Multi--period Model
Multi
Slide 08
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Fin 501:Asset Pricing I
O
Overview
i
1. Generalizing the setting
¾ Trees, modeling information and learning
¾ Time
Time--preferences
2 Multi
2.
Multi--period SDF and event prices
¾ Pricing in a static dynamic model
3 Dynamic
3.
D
i completion
l ti
4. Ponzi scheme and rational bubbles
5. Martingale process - EMM
Slide 08
08--26
Fin 501:Asset Pricing I
D
Dynamic
i ttrading
di
• In the "static
static dynamic
dynamic" model we assumed that there
were many periods and information was gradually
revealed (this is the dynamic part)…
• …but
b all
ll assets are traded
d d "at
" the
h bbeginning
i i off time"
i "
(this is the static part).
• Now consequences of re
re-opening
opening financial markets.
Assets can be traded at each instant.
• This has deep implications.
¾ allows
ll
us to reduce
d
the
h number
b off assets available
il bl at eachh
instant through dynamic completion.
¾ It opens up some nasty possibilities (Ponzi schemes and
b bbl )
bubbles),
Multi--period Model
Multi
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Fin 501:Asset Pricing I
C
Completion
l ti with
ith shortshort
h t-lived
li d assets
t
• If the horizon is infinite, the number of events is
p y that we need an
also infinite. Does that imply
infinite number of assets to make the market
p
complete?
• Do we need assets with all possible times to
maturity and events to have a complete market?
• No. Dynamic completion.
Arrow (1953) and Guesnerie and Jaffray (1974)
Multi--period Model
Multi
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Fin 501:Asset Pricing I
C
Completion
l ti with
ith shortshort
h t-lived
li d assets
t
• Asset is `short lived´ if it pays out only in the
period immediately after the asset is issued.
• Suppose for each event A and each successor
event A' there is an asset that pays in A' and
nothing otherwise.
• It is possible to achieve arbitrary transfers
between all events in the event tree by trading
only these short-lived assets.
• This is straightforward if there is no uncertainty.
Multi--period Model
Multi
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Fin 501:Asset Pricing I
C
Completion
l ti with
ith shortshort
h t-lived
li d assets
t
• Without uncertainty, and T periods (T can be infinite),
there are T one period assets, from period 0 to period 1,
from period 1 to 2, etc.
• Let pt be the price of the bond that begins in period t-1
andd matures iin period
i d t.
• For the market to be complete we need to be able to
t
transfer
f wealth
lth between
b t
any two
t periods,
i d nott just
j t
between consecutive periods.
• This can be achie
achieved
ed with
ith a trading
t di strategy.
t t
Multi--period Model
Multi
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Fin 501:Asset Pricing I
C
Completion
l i with
i h shortshort
h -lived
li d assets
• Example: Suppose we want to transfer wealth from
period 1 to period 3.
• In period 1 we cannot buy a bond that matures in period
3 because
3,
b
suchh a bbond
d iis nott traded
t d d then.
th
• Instead buy a bond that matures in period 2, for price p2.
• In period 2,
2 use the payoff of the period-2
period 2 bond to buy
period-3 bonds.
• In period 3, collect the payoff.
• The result is a transfer of wealth from period 1 to period
3. The price, as of period 1, for one unit of purchasing
power in period 3, is p2p3.
Multi--period Model
Multi
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Fin 501:Asset Pricing I
C
Completion
l i with
i h shortshort
h -lived
li d assets
With uncertainty the process is only slightly more
complicated. It is easily understood with a graph.
event 1
event 3
event 4
event 0
event 5
event 2
event 6
‰
Let pA be the price of the asset
that
h pays one unit
i in
i event A.
A
This asset is traded only in the
event immediately preceding A.
We want to transfer wealth from
event 0 to event 4.
‰ Go backwards: in event 1, buy
one event 4 asset for a price p4.
‰ In event 0, buy p4 event 1 assets.
‰ The cost of this today is p1p4. The
payoff
ff iis one unit
i in
i event 4 and
d
nothing otherwise.
‰
Multi--period Model
Multi
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Fin 501:Asset Pricing I
C
Completion
l i with
i h longlong
l
-lived
li d assets
• dynamic completion with long-lived assets,
Kreps (1982)
• T-period model without uncertainty (T < ∞).
• assume there is a single asset:
¾a discount bond maturing in T.
¾bond can be purchased and sold in each period,
period for
price pt, t=1,…,T.
Multi--period Model
Multi
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Fin 501:Asset Pricing I
Completion
C
l ti with
ith a long
l
maturity
t it bond
b d
• So there are T prices
(not simultaneously, but sequentially).
• Purchasing power can be transferred from
period t to period t' > t by purchasing the bond
in period t and selling it in period tt'.
Multi--period Model
Multi
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Fin 501:Asset Pricing I
A simple
i l information
i f
ti tree
t
1
0
event 1
q1,1
q2,1
q1,0
q2,0
20
0
1
1
0
event 0
q1,2
q2,2
event 2
0
asset 1
1
This information tree has three
non trivial events plus four final
non-trivial
states, so seven events altogether.
It seems as if we would need six
Arrow securities (for events 1 and 2
and for the four final states) to
have a complete market. Yet we
have only two assets. So the
market cannot be complete, right?
Wrong! Dynamic trading provides
a way to fully insure each event
separately.
Note that there are six prices
because each asset is traded in
three events.
asset 2
Slide 08
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Fin 501:Asset Pricing I
O -period
OneOne
i d holding
h ldi
• Call “tradingg strategy
gy [j,A]”
[j ] the cash flow of asset j that
is purchased in event A and is sold one period later.
• How many such assets exist? What are their cash flows?
p1,1
p2,1
p1,0
p2,0
1 0
0 1
1 0
p1,2
p2,2
0 1
There are six such strategies:
[1,0], [1,1], [1,2], [2,0], [2,1], [2,2].
(Note that
(N
h this
hi iis potentially
i ll sufficient
ffi i
to span the complete space.)
“Strategy [1,1]" costs p1,1 and pays
outt 1 iin th
the fi
firstt final
fi l state
t t and
d zero
in all other events.
“Strategy [1,0]" costs p1,0 and pays
o t p1,1 in e
out
event
ent 1,
1 p1,2 in event
e ent 2,
2
and zero in all the final states.
Multi--period Model
Multi
Slide 08
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Fin 501:Asset Pricing I
Th extended
The
t d d payoff
ff matrix
ti
The trading strategies [1,0] … [2,2] give rise to a new 6x6
payoff matrix.
matrix
asset
event 0
event 1
event 2
state 1
state 3
state 3
state 4
[1,0]
–p1,0
10
p1,1
p1,2
0
0
0
0
[2,0]
–p2,0
20
p2,1
p2,2
0
0
0
0
[1,1]
0
–p1,1
0
1
0
0
0
[2,1]
0
–p2,1
0
0
1
0
0
[1,2]
0
0
–p1,2
0
0
1
0
[2,2]
0
0
–p2,2
0
0
0
1
This matrix is regular (and hence the market complete) if the
grey submatrix is regular (= of rank 2).
Multi--period Model
Multi
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Fin 501:Asset Pricing I
Th extended
The
t d d payoff
ff matrix
ti
• Is the gray submatrix regular?
• Components of submatrix are prices of the two assets,
conditional on period 1 events.
• There
Th are cases in
i which
hi h (p
( 11, p21) and
d ((p12, p22) are
collinear in equilibrium.
2 in
• If per capita endowment is the same in event 1 and 2,
state 1 and 3, and in state 2 and 4, respectively, and if
the probability of reaching state 1 after event 1 is the
same as the
h probability
b bili off reaching
hi state 3 after
f event 2
Æ submatrix is singular (only of rank 1).
• But then events 1 and 2 are effectively identical
identical, and we
may collapse them into a single event.
Multi--period Model
Multi
Slide 08
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Fin 501:Asset Pricing I
Th extended
The
t d d payoff
ff matrix
ti
• A random square matrix is regular.
regular So outside of special
cases, the gray submatrix is regular (“almost surely”).
• The 2x2 submatrix may still be singular by accident
accident.
• In that case it can be made regular again by applying a
small perturbation of the returns of the long
long-lived
lived assets,
by perturbing aggregate endowment, the probabilities,
or the utility function.
• Generically, the market is dynamically complete.
Multi--period Model
Multi
Slide 08
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Fin 501:Asset Pricing I
H many assets to complete
How
l market?
k ?
• b
branching
hi number
b = The
Th maximum
i
number
b off branches
b
h
fanning out from any event in the uncertainty tree.
• This is also the number of assets necessary to achieve
dynamic completion.
• Generalization by Duffie and Huang (1985):
continuous time Æ continuity of events Æ but a small
number of assets is sufficient.
• The large power of the event space is matched by
continuouslyy tradingg few assets, thereby
y ggenerating
ga
continuity of trading strategies and of prices.
Multi--period Model
Multi
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Fin 501:Asset Pricing I
E
Example:
l BlackBlack
Bl k-Scholes
S h l formula
f
l
• Cox, Ross, Rubinstein binominal tree model of B-S
• Stock price goes up or down (follows binominal tree)
i
interest
rate is
i constant
• Market is dynamically complete with 2 assets
¾ Stock
St k
¾ Risk-free asset (bond)
• Replicate payoff of a call option with
(dynamic Δ-hedging)
• (later more)
Multi--period Model
Multi
Slide 08
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Fin 501:Asset Pricing I
O
Overview
i
1. Generalizing the setting
¾ Trees, modeling information and learning
¾ Time
Time--preferences
2 Multi
2.
Multi--period SDF and event prices
¾ Pricing in a static dynamic model
3 Dynamic
3.
D
i completion
l ti
4. Ponzi schemes and rational bubbles
5. Martingale - EMM
Slide 08
08--42
Fin 501:Asset Pricing I
P i schemes:
Ponzi
h
infinite horizon max. problem
• IInfinite
fi it horizon
h i
allows
ll
agents
t to
t borrow
b
an arbitrarily
bit il
large amount without effectively ever repaying, by
rolling over debt forever
forever.
¾ Ponzi scheme - allows infinite consumption.
• Consider an infinite horizon model, no uncertainty, and
a complete set of short-lived bonds.
[zt is the amount of bonds maturing in period t in the portfolio, βt
is the price of this bond as of period t-1]
⎧⎪ ∞ t
⎫⎪
c0 − w0 ≤ − β1 z1
max ⎨∑ δ u (ct )
⎬.
c1 − w1 ≤ zt − β t +1 zt +1 for t > 0⎪⎭
⎪⎩ t =0
Multi--period Model
Multi
Slide 08
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Fin 501:Asset Pricing I
P i schemes:
Ponzi
h
rolling
lli over debt
d b forever
f
• The followingg consumption
p
path
p is possible:
p
ct = wt+1 for all t.
• Note that agent consumes more than his endowment in
each period, forever.
• This can be financed with ever increasing debt:
z1=-1/β1, z2=(-1+z1)/β2, z3=(-1+z2)/β3 …
• Ponzi schemes can never be part of an equilibrium. In
fact, such a scheme even destroys the existence of a
utility maximum because the choice set of an agent is
unbounded
b
d d above.
b
W
We needd an additional
ddi i l constraint.
i
Multi--period Model
Multi
Slide 08
08--44
Fin 501:Asset Pricing I
P i schemes:
Ponzi
h
transversality
t
lit
• The constraint that is typically imposed on top
of the budget constraint is the transversality
condition,
limt→∞ βt zt ≥ 0.
• This constraint implies that the value of debt
cannot diverge to infinity.
¾More precisely, it requires that all debt must be
redeemed eventuallyy (i.e.
( in the limit).
)
Multi--period Model
Multi
Slide 08
08--45
Fin 501:Asset Pricing I
R ti l Bubbles:
Rational
B bbl The
Th price
i off a consoll
• In an infinite dynamic model,
model in which assets are
traded repeatedly, there are additional solutions
besides the "fundamental
fundamental pricing formula
formula."
¾New solutions have “bubble component”.
¾No market clearing at infinity.
infinity
• Consider
¾model without uncertainty
¾Consol bond delivering $1 in each period forever
Multi--period Model
Multi
Slide 08
08--46
Fin 501:Asset Pricing I
Th price
The
i off a consoll
• Our formula
pt = Et[Mt+1 (pt+1+ dt+1)]
• in the case of certainty with d=1
pt = Mt+1 pt+1 + Mt+1
• solve forward – (many solutions)
⇒ p0 =
M
∑
1
424
3
∞
t =1
t
fundamental value
+ lim T →∞ M T pT .
144244
3
bubble component
Multi--period Model
Multi
Slide 08
08--47
Fin 501:Asset Pricing I
Th price
The
i off a consoll
• Consider case without uncertainty. (event tree is line)
• According to the static-dynamic
static dynamic model (the
fundamental pricing formula), the price of the consol is
p0 = ∑t Mt.
Mt := m1 × m2 × L × mt
•
•
•
•
Consider re-opening markets now. The price at
time 0 is just the sum of all Arrow prices, so
p0 = ∑t=1∞ Mt.
qt is the marginal rate of substitution between
consumption in period t and in period 0,
qt = δt ((u'(w
( t))/u'(w
( 0))
)),
Multi--period Model
Multi
so
p0 = ∑t =1 δ
∞
t u '( w t )
u '( w 0 )
Slide 08
08--48
.
Fin 501:Asset Pricing I
Th price
The
i off a consoll att t=1
1
• At time 1,
1 the price of the consol is the sum of the
remaining Arrow securities,
∞
p1 = ∑t = 2 δ t −1 uu ''(( ww )) .
t
1
•
This can be reformulated,
p1 = δ
•
−1 u '( w0 )
u '( w1 )
∑
∞
δ
t =2
t u '( w t )
u '( w 0 )
.
The second part (the sum from 2 to ∞) is almost
equal to p0,
∑
∞
t =2
δ
t u '( w t )
u '( w 0 )
=
(∑
∞
t =1
δ
t u '( w t )
u '( w 0 )
)− δ
Multi--period Model
Multi
u '( w1 )
u '( w 0 )
= p0 − δ
u '( w1 )
u '(w
( w0 )
.
Slide 08
08--49
Fin 501:Asset Pricing I
Consol: Solving forward
• More generally, we can express the price at time t+1
as a function of the price at time t,
(
pt +1 = δ −1 uu'('(wwt +1)) qt − δ
t
pt = δ
•
u '( wt +1 )
u '( w t )
+δ
),
u '( wt +1 )
u '( wt )
u '( wt +1 )
u '( w t )
thus
pt +1 = mt +1 + mt +1 pt +1.
We can solve this forward by substituting the t+1
version of this equation into the t version, ad
infinitum,,
p0 = m1 + m1 p1 = m1 + m1[m2 + m2 p2 ] = L
⇒ p0 =
M
∑
1
424
3
∞
t =1
t
fundamental value
+ limT →∞ MT pT .
14
4244
3
Multi--period Model
Multi
bubble component
Slide 08
08--50
Fin 501:Asset Pricing I
M
Money
as a bbubble
bbl
p0 =
M
∑
1
424
3
∞
t =1
t
fundamental value
+ lim T →∞ M T pT .
144244
3
bubble component
• The fundamental value = p
price in the static-dynamic
y
model.
• Repeated trading gives rise to the possibility of a bubble.
• Fiat money can be understood as an asset with no dividends.
In the static-dynamic model, such an asset would have no
value (the present value of zero is zero). But if there is a
bubble on the price of fiat money, then it can have positive
value (Bewley, 1980).
• In asset pricing theory, we often rule out bubbles simply by
i
imposing
i lim
li T→∞ MT pT = 0.
0
Multi--period Model
Multi
Slide 08
08--51
Fin 501:Asset Pricing I
O
Overview
i
1. Generalizing the setting
¾ Trees, modeling information and learning
¾ Time
Time--preferences
2 Multi
2.
Multi--period SDF and event prices
¾ Pricing in a static dynamic model
3 Dynamic
3.
D
i completion
l ti
4. Ponzi schemes and rational bubbles
5. Martingale - EMM
Mean--Variance Analysis and CAPM
Mean
Slide 08
08--52
Fin 501:Asset Pricing I
M ti l
Martingales
• Let X1 be a random variable and
let x1 be the realization of this random variable.
• Let X2 be another random variable and assume that the
distribution of X2 depends on x1.
• Let X3 be a third random variable and assume that the
di ib i off X3 depends
distribution
d
d on x1, x2.
• Such a sequence of random variables, (X1,X2,X3,…), is
called
ll d a stochastic
t h ti process.
• A stochastic process is a martingale if
E[ t+1 | xt , … , x1] = xt .
E[x
Multi--period Model
Multi
Slide 08
08--53
Fin 501:Asset Pricing I
Hi t
History
off the
th wordd martingale
ti l
•
•
•
•
“martingale” originally refers to a sort of pants worn by “Martigaux” people
living in Martigues located in Provence in the south of France. By analogy, it is
used to refer to a strap in equestrian. This strap is tied at one end to the girth of
the saddle and at the other end to the head of the horse. It has the shape of a
fork and divides in two.
In comparison to this division, martingale refers to a strategy which consists in
playing twice the amount you lost at the previous round. Now, it refers to any
strategy used to increase one's probability to win by respecting the rules.
The notion of martingale appears in 1718 (The Doctrine of Chance by
Abraham de Moivre) referring to a strategy that makes you sure to win in a fair
game.
See also www.math.harvard.edu/~ctm/sem/martingales.pdf
Multi--period Model
Multi
Slide 08
08--54
Fin 501:Asset Pricing I
Pi
Prices
are martingales…
ti l
•
•
Samuelson (1965) has argues that prices have
to be martingales in equilibrium.
one has to assume that
1. no discounting
2 no dividend payments (intermediate cash flows)
2.
3. representative agent is risk-neutral
1 With
1.
Wi h discounting:
di
i
¾ Discounted price process should follow
martingale
i l
E[δ t+1| pt] = pt .
E[δp
Multi--period Model
Multi
Slide 08
08--55
Fin 501:Asset Pricing I
R i
Reinvesting
ti dividends
di id d
2 With dividend payments
2.
¾ because pt depends on the dividend of the asset in
period t+1,
t+1 but pt+1 does not
(these are ex-dividend prices).
¾ Consider,
Consider value of a fund that keeps reinvesting
the dividends follows a martingale
LeRoyy ((1989).
)
Multi--period Model
Multi
Slide 08
08--56
Fin 501:Asset Pricing I
Reinvesting dividends
• Consider a fund owning nothing but one units of asset j.
• The value of this fund at time 0 is f0 = p0 = E[∑t=1∞ δt xtj]
= δ E[x1j+p1]. (if representative agent is risk-neutral)
• After receiving dividends x1j (which are state contingent)
it buys more of asset j at the then current price p1, so the
f d then
fund
h owns 1 + x1j/p
/ 1 units
i off the
h asset.
• The discounted value of the fund is then
f1 = δ p1 (1+x1j/p1) = δ (p1+x1j) = p0 = f0,
so the discounted value of the fund is indeed a
martingale.
i l
Multi--period Model
Multi
Slide 08
08--57
Fin 501:Asset Pricing I
…and
and with risk aversion?
• A similar statement is true if the representative agent is
risk averse.
averse
• The difference is that
¾ we must discount with the risk-free interest rate,, not with the
discount factor,
¾ we must use the risk-neutral probabilities (also called
equivalent martingale measure for obvious reasons) instead
of the objective probabilities.
• Just as in the 2-period model, we define the riskneutrall probabilities
b bili i as
π∗A = πA MA / ρτ(A),
where ρτ(A) is the discount-factor from event A to 0.
(state dependent)
Multi--period Model
Multi
Slide 08
08--58
Fin 501:Asset Pricing I
…and with risk aversion?
• The initial value of the fund is
f0 = p0 = ∑tt=11∞ E[Mt xt j].
• Let us elaborate on this a bit,
f0 = E[∑t=1∞ Mt xt j] = E{M1 x1 j + ∑t=2∞ Mt xt j]
= E[M1 (x1 j + ∑t=2∞ ∏t'=2t Mt' xt j )]
= E[M1 (x1 j + p1)].
• E* = expectations under the risk-neutral distribution
π∗, this can be rewritten as
f0 = ρ1 E*[x
E*[ 1 j + p1] = ρ1 E*[f1].
]
• The properly discounted (ρ instead of δ) and properly
expected (π∗ instead of π) value of the fund is indeed a
martingale.
Multi--period Model
Multi
Slide 08
08--59
Fin 501:Asset Pricing I
O
Overview
i
1. Generalizing the setting
¾ Trees, modeling information and learning
¾ Time
Time--preferences
2 Multi
2.
Multi--period SDF and event prices
¾ Static
Static--dynamic completeness
3 Dynamic
3.
D
i completion
l ti
4. Ponzi scheme and rational bubbles
5. Martingale process - EMM
Slide 08
08--60