Lecture 08: 08: Multi Multi--period Model period Model

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Lecture

Lecture 08: 08: Multi Multi--period Model period Model

Prof. Markus K. Brunnermeier Prof. Markus K. Brunnermeier

Slide Slide 0808--11 Multi

Multi--period Modelperiod Model

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th i i l t

… the remaining lectures

• Generalizing to multi-period setting

• Ponzi Schemes Bubbles Ponzi Schemes, Bubbles

• Forward, Futures and Swaps

ICAPM d h d i d d

• ICAPM and hedging demand

• Multiple factor pricing models

• Market efficiency

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I t d ti I t d ti

Introduction Introduction

• accommodate multiple and even infinitely many periods.

l i

• several issues:

¾ how to define assets in an multi period model,

¾ how to model intertemporal preferences

¾ how to model intertemporal 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)

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O i

Overview Overview

1. 1. Generalizing the setting Generalizing the setting

¾

¾ Trees, modeling information and learning Trees, modeling information and learning

¾

¾ Trees, modeling information and learning Trees, modeling information and learning

¾

¾ Time Time--preferences preferences

2 2 Multi Multi period SDF and event prices period SDF and event prices 2.

2. Multi Multi--period SDF and event prices period SDF and event prices

¾

¾ Static Static--dynamic completeness dynamic completeness

3 3 D D i i l ti l ti 3.

3. Dynamic completion Dynamic completion 4.

4. Ponzi Ponzi scheme and rational bubbles scheme and rational bubbles

Slide Slide 0808--44 Mean

Mean--Variance Analysis and CAPMVariance Analysis and CAPM

5. 5. Martingale process Martingale process -- EMM EMM

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0 1 2 3

i d d l many one period models

how to model information?

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States s

Multi--period Modelperiod Model F₁

F₂ Ω ∅∪ ∪

Events A_i,t

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M d li i f ti ti

Modeling information over time Modeling information over time

• Partition

• Field/Algebra

• Filtration

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S b bilit th S b bilit th

Some probability theory Some probability theory

•• Measurability: 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 A_i ∈ F, y(s) = y(s’) for each s ∈ A_i and s’ ∈ A_i,

• y (A ):= y (s) s ∈ A

• y_t(A_i): y_t(s), s ∈ A_i.

•• Stochastic process: Stochastic process: A collection of random variables y

_t

(s) for t = 0,…, T.

for t 0,…, T.

• Stochastic process is adapted to filtration adapted to filtration F F ={F

_t

}

_t^T

if each y

_t

(s) is measurable w.r.t. F

_t

Slide Slide 0808--88

y

_t

( )

_t

¾ Cannot see in the future

Multi

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f t ti t d i

from static to dynamic…

asset holdings Dynamic strategy (adapted process) asset holdings Dynamic strategy (adapted process) asset payoff x Next period’s payoff x_t+1+ p_t+1

Pa off of a strateg Payoff of a strategy

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 q_t(A_t(s))

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f t ti t d i

…from static to dynamic

State prices q(s) Event prices q

_t

(A(s))

Risk free rate R Risk free rate r

_t

varies over time

_ _

t

Discount factor from t to 0 ρ

_t

(s) Risk neutral prob. Risk neutral prob.

Risk neutral prob.

π

^*

(s) = q(s) R

Risk neutral prob.

π

^*

(A

_t

(s)) = q

_t

(A

_t

(s)) / ρ

_t

(A

_t

) Pricing kernel Pricing kernel

_

g

p

^j

= E[k

_q

x

^j

] 1 = E[k

_q

] R

g

k

_t

p

_t^j

= E

_t

[k

_t+1

(p

^j_t+1

+ x

^j_t+1

)]

k

_t

= R

_t+1

E

_t

[k

_t+1

]

_ _

1 E[k

_q

] R

^t ^t+1 ^t ^t+1

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in more detail in more detail

… in more detail

We recall the event tree that

A₁

A₃ We recall the event tree that

captures the gradual resolution of uncertainty.

This tree has 7 events (A to A ) ψ₁(A₄)

( )

A₀

A₁

A₄

This tree has 7 events (A₀ to A₆).

(Lengwiler uses e₀ to e₆)

3 time periods (0 to 2).

ψ₀(A₄)

ψ₂(A₄)

A₅ If A is some event, we denote the period it belongs to as τ(A).

So for instance τ(A )=1 τ(A )=2 A₂

A₆ t 1

t 0 t 2

So for instance, τ(A₂)=1, τ(A₄)=2.

We denote a path with ψ as follows…

t=1

t=0 t=2

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M lti l i d t i t M lti l i d t i t Multiple period uncertainty Multiple period uncertainty

A₃ Last period events have prob., π₃ - π₆.

A₁

A₄

The earlier events also have probabilities.

To be consistent the

A₀

A₅

To be consistent, the

probability of an event is equal to the sum of the

A₂

A₆

probabilities of its successor events.

So fo instance +

A₆ t=1

t=0 t=2 So for instance, π₁ = π₃+π₄.

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M lti l i d t M lti l i d t Multiple period assets Multiple period assets

A typical multiple period asset is a coupon bond:

⎪

⎪⎨

⎧

= +

<

= 1 coupon if ( ) ,

, )

( 0

if coupon

: ^*

*

t A

r_A τ

τ

⎪⎩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

finite maturity bond with no coupon.

It just pays 1 at expiration, and nothing otherwise.

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M lti l i d t M lti l i d t Multiple period assets Multiple period assets

• 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) ld b i d

could be striped.

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M lti l i d t M lti l i d t Multiple period assets Multiple period assets

Sh l i h f di id d f fi

• Shares are claims to the future dividends of a firm.

• Derivatives: A call option on a share, is an asset that pays either 0 or p k where p is the (random) price of pays either 0 or p - k, where p is the (random) price of the share and k is the strike (exercise) price.

• Options of this kind will be helpful when thinking p p g about how to make a market system complete.

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O i

Overview Overview

1. 1. Generalizing the setting Generalizing the setting

¾

¾ Trees, modeling information and learning Trees, modeling information and learning

¾

¾ Trees, modeling information and learning Trees, modeling information and learning

¾

¾ Time Time--preferences preferences

2 2 Multi Multi period SDF and event prices period SDF and event prices 2.

2. Multi Multi--period SDF and event prices period SDF and event prices

¾

¾ Static Static--dynamic completeness dynamic completeness

3 3 D D i i l ti l ti 3.

3. Dynamic completion Dynamic completion 4.

4. Ponzi Ponzi scheme and rational bubbles scheme and rational bubbles

5. 5. Martingale process Martingale process -- EMM EMM

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Ti f

Time preference Time preference

u(c ₀ ) + ∑ _t δ(t) E{u(c _t )}

• Discount factor δ(t) number between 0 and 1

• Assume δ(t)>δ(t+1) for all t.

S i i d 0 d k l f

• Suppose you are in period 0 and you make a plan of your present and future consumption: c

₀

, c

₁

, c

₂

, …

• The relation between consecutive consumption will 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)

• Time consistency δ(t) = δ

^t

(exponential discounting)

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Ti f

Time preference Time preference

• Canonical framework (exponential discounting) U(c) = E[ ∑ δ

^t

u(c

_t

)]

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(c ) + β ∑ δ

^t

u(c )]

U(c) = E[u(c

₀

) + β ∑ δ

^t

u(c

_t

)]

• Preference for the timing of uncertainty resolution

recursive utility formulation (Kreps Porteus 1978)

recursive utility formulation (Kreps-Porteus 1978)

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Digression:

Digression: ^g ^g Preference for the Preference for the timing of uncertainty resolution timing of uncertainty resolution

π $100 $150

$100

$100 π

$ 25 0 $150

$100

$ 25 Kreps-Porteus

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?

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M lti

M lti i d i d P tf li Ch i P tf li Ch i

Digression:

Digression: Multi Multi--period period Portfolio Choice Portfolio Choice

Theorem 4.10 (Merton, 1971): Consider the above canonical ( ) multi-period consumption-saving-portfolio allocation problem.

Suppose U() displays CRRA, r_f is constant and {r} is i.i.d.

Then a/s is time invariant

Then a/s_t is time invariant.

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O i

Overview Overview

1. 1. Generalizing the setting Generalizing the setting

¾

¾ Trees, modeling information and learning Trees, modeling information and learning

¾

¾ Trees, modeling information and learning Trees, modeling information and learning

¾

¾ Time Time--preferences preferences

2 2 Multi Multi period SDF and event prices period SDF and event prices 2.

2. Multi Multi--period SDF and event prices period SDF and event prices

¾

¾ Pricing in a static dynamic model Pricing in a static dynamic model

3 3 D D i i l ti l ti 3.

3. Dynamic completion Dynamic completion 4.

4. Martingale process Martingale process -- EMM EMM

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A

A ii d d i i d l d l A

A static static dynamic model dynamic model

• We consider pricing in a model that contains many

• We consider pricing in a model that contains many periods (possibly infinitely many)…

• …and we assume that information is gradually and we assume that information is gradually revealed (this is the dynamic part)…

• …but we also assume that all assets are only traded …but we also assume that all assets are only traded

"at the beginning of time" (this is the static part).

• There is dynamics in the model because there is time, y , but the decision making is completely static.

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M i i ti i d

Maximization over many periods Maximization over many periods

• vNM exponential utility representative agent

• vNM exponential utility representative agent max{∑

_t=0

δ

^t

E{u(c

_t

)} | y-w ∈ B(p)}

• If all Arrow securities (conditional on each

event) are traded, we can express the first-order conditions as,

u'(c

₀

) = λ, δ

^τ(A)

π

_A

u'(w

_A

) = λq

_A

.

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M lti

M lti i d SDF i d SDF Multi

Multi--period SDF period SDF

• The equilibrium SDF is computed in the same fashion as

i h i d l b f

in the static model we saw before

) (

'

) (

'

0 )

(

w u

q _A ^A

A

A δτ

π_A ⁼ ⁽ ⁾

⎟⎟ ⎠

⎜⎜ ⎞

⎝

⎟⎟ ⎛

⎠

⎜⎜ ⎞

⎝

⋅ ⎛

⎟⎟ ⎠

⎜⎜ ⎞

⎝

= ⎛

−

)

( '

) (

' )

( '

) (

' )

( '

) (

'

) ( )

( ) ( 0

) (

1 ) 1 (

2 1

A A A

A A

w

A

u

w u

ψτ

ψ ψ

ψ

δ δ

δ _L

⎠

⎠ ⎝

⎝

⎠

⎝ u ( w ) u ( w ) u ( w )

)

1(A

m_ψ ₍ ₎

2 A

m_ψ ₍ ₎

1 )

(_A A

m_ψ_τ

× × L ×

= : M

_A

• We call m_A the "one-period ahead" SDF and M_A

)

1(A

ψ ψ₂(A) ψ_τ₍_A₎₋₁(A) A

the multi-period SDF (“state-price density”).

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Th f d t l i i f l

The fundamental pricing formula The fundamental pricing formula

• To price an arbitrary asset x, To price an arbitrary asset x,

portfolio of STRIPed cash flows, x

^j

= x

₁^j

+x

₂^j

+ L+x

∞j

, where x

_t^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

∑

=

t

j t t

j

E x

p {M }

t

• This is the fundamental pricing formula.

• Note that M_t= δ^t if the repr agent is risk neutral. The

fundamental pricing formula then just reduces to the present value of expected dividends, p_j = ∑ δ^t E{x_t^j}.

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O i

Overview Overview

1. 1. Generalizing the setting Generalizing the setting

¾

¾ Trees, modeling information and learning Trees, modeling information and learning

¾

¾ Trees, modeling information and learning Trees, modeling information and learning

¾

¾ Time Time--preferences preferences

2 2 Multi Multi period SDF and event prices period SDF and event prices 2.

2. Multi Multi--period SDF and event prices period SDF and event prices

¾

¾ Pricing in a static dynamic model Pricing in a static dynamic model

3 3 D D i i l ti l ti 3.

3. Dynamic completion Dynamic completion 4.

4. Ponzi Ponzi scheme and rational bubbles scheme and rational bubbles

5. 5. Martingale process Martingale process -- EMM EMM

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D i t di

Dynamic trading Dynamic trading

• In the "static dynamic" model we assumed that there

• In the static dynamic model we assumed that there were many periods and information was gradually revealed (this is the dynamic part)…

b ll d d " h b i i f i "

• …but all assets are traded "at the beginning of time"

(this is the static part).

• Now consequences of re-opening financial markets. Now consequences of re opening financial markets.

Assets can be traded at each instant.

• This has deep implications.

¾ ll d h b f il bl h

¾ allows us to reduce the number of assets available at each instant through dynamic completion.

¾ It opens up some nasty possibilities (Ponzi schemes and b bbl )

bubbles),

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C l ti ith h t

C l ti ith h t li li d d t t Completion with short

Completion with short--lived assets lived assets

• If the horizon is infinite, the number of events is also infinite. Does that imply that we need an p y

infinite number of assets to make the market complete? p

• 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)

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C l ti ith h t

C l ti ith h t li li d d t t Completion with short

Completion with short--lived assets lived assets

• 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.

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C l ti ith h t

C l ti ith h t li li d d t t Completion with short

Completion with short--lived assets lived assets

• 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 p

_t

be the price of the bond that begins in period t-1

d i i d

and matures in period t.

• For the market to be complete we need to be able to

t f lth b t t i d t j t

transfer wealth between any two periods, not just between consecutive periods.

• This can be achie ed ith a t di t t

• This can be achieved with a trading strategy.

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C l i i h h

C l i i h h li li d d

• Example: Suppose we want to transfer wealth from

Completion with short

Completion with short--lived assets lived assets

period 1 to period 3.

• In period 1 we cannot buy a bond that matures in period

3 b h b d i t t d d th

3, because such a bond is not traded then.

• Instead buy a bond that matures in period 2, for price p

₂

.

• In period 2 use the payoff of the period 2 bond to buy

• In period 2, use the payoff of the period-2 bond to buy period-3 bonds.

• In period 3, collect the payoff. 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 p

₂

p

₃

.

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C l i i h h

C l i i h h li li d d

With uncertainty the process is only slightly more

Completion with short

Completion with short--lived assets lived assets

With uncertainty the process is only slightly more complicated. It is easily understood with a graph.

event 3 Let p_A be the price of the asset

h i i A

event 1 event 3 that pays one unit in event A.

This asset is traded only in the event immediately preceding A.

event 0

event 5

event 4 We want to transfer wealth from event 0 to event 4.

Go backwards: in event 1, buy event 5 one event 4 asset for a price p₄.

The cost of this today is p₁p₄. The

ff i i i 4 d

In event 0, buy p₄ event 1 assets.

event 2

event 6

payoff is one unit in event 4 and nothing otherwise.

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C l i i h l

C l i i h l li li d d Completion with long

Completion with long--lived assets lived 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 for

¾bond can be purchased and sold in each period, for price p

_t

, t=1,…,T.

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C l ti ith l t it b d

Completion with a long maturity bond Completion with a long maturity bond

• So there are T prices 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 t'

in period t and selling it in period t .

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A i l i f ti t A i l i f ti t

1 0

A simple information tree A simple information tree

This information tree has three non trivial events plus four final event 1

q_1,1 q

non-trivial events plus four final states, so seven events altogether.

It seems as if we would need six Arrow securities (for events 1 and 2

0 1

event 0 q_1,0

q_{2 0}

q_2,1 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

1 0

q_2,0

q_1,2 q

market cannot be complete, right?

Wrong! Dynamic trading provides a way to fully insure each event

0 1

event 2

q_2,2 separately.

Note that there are six prices because each asset is traded in

0 1

asset 1 asset 2

three events.

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O

O i d h ldi i d h ldi One

One--period holding period holding

• Call “trading strategy [j,A]” the cash flow of asset j that g gy [j ] j is purchased in event A and is sold one period later.

• How many such assets exist? What are their cash flows?

p_1,1 1 0 p_2,1

There are six such strategies:

[1,0], [1,1], [1,2], [2,0], [2,1], [2,2].

(N h hi i i ll ffi i 0

1 1 0 p_1,0

p_2,0

(Note that this is potentially sufficient to span the complete space.)

“Strategy [1,1]" costs p_1,1 and pays t 1 i th fi t fi l t t d

p_1,2 0 1 p

out 1 in the first final state and zero in all other events.

“Strategy [1,0]" costs p_1,0 and pays o t p in e ent 1 p in e ent 2

p_2,2 0 1 out p_1,1 in event 1, p_1,2 in event 2, and zero in all the final states.

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Th t d d ff t i

The extended payoff matrix The extended payoff matrix

The trading strategies [1,0] … [2,2] give rise to a new 6x6 payoff matrix

asset [1,0] [2,0] [1,1] [2,1] [1,2] [2,2]

event 0 –p_{1 0} –p_{2 0} 0 0 0 0

payoff matrix.

p_1,0 p_2,0

event 1 p_1,1 p_2,1 –p_1,1 –p_2,1 0 0

event 2 p_1,2_1,2 p_2,2_2,2 0 0 –p_1,2_1,2 –p_2,2_2,2

state 1 0 0 1 0 0 0

state 3 0 0 0 1 0 0

state 3 0 0 0 0 1 0

state 4 0 0 0 0 0 1

This matrix is regular (and hence the market complete) if the grey submatrix is regular (= of rank 2).

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Th t d d ff t i

The extended payoff matrix The extended payoff matrix

• Is the gray submatrix regular?

• Components of submatrix are prices of the two assets, conditional on period 1 events.

Th i hi h ( ) d ( )

• There are cases in which (p

₁₁

, p

₂₁

) and (p

₁₂

, p

₂₂

) are collinear in equilibrium.

• If per capita endowment is the same in event 1 and 2 in If per capita endowment is the same in event 1 and 2, in state 1 and 3, and in state 2 and 4, respectively, and if the probability of reaching state 1 after event 1 is the

h b bili f hi 3 f 2

same as the probability of reaching state 3 after event 2 Æ submatrix is singular

(only of rank 1)

.

• But then events 1 and 2 are effectively identical and we

• But then events 1 and 2 are effectively identical, and we

may collapse them into a single event.

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Th t d d ff t i

The extended payoff matrix The extended payoff matrix

• A random square matrix is regular So outside of special

• A random square matrix is regular. So outside of special cases, the gray submatrix is regular (“almost surely”).

• The 2x2 submatrix may still be singular by accident The 2x2 submatrix may still be singular by accident.

• In that case it can be made regular again by applying a small perturbation of the returns of the long-lived assets, small perturbation of the returns of the long lived assets, by perturbing aggregate endowment, the probabilities, or the utility function.

• Generically, the market is dynamically complete.

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H l k ?

How many assets to complete market?

b hi b Th i b f b h

• branching number = The maximum number of branches fanning out from any event in the uncertainty tree.

• This is also the number of assets necessary to achieve

• 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

continuously trading few assets, thereby generating a

y g y g g

continuity of trading strategies and of prices.

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E l Bl k

E l Bl k S h l S h l f f l l Example: Black

Example: Black--Scholes formula Scholes formula

• Cox, Ross, Rubinstein binominal tree model of B-S

• Stock price goes up or down

(follows binominal tree)

i i

interest rate is constant

• Market is dynamically complete with 2 assets

¾ St k

¾ Stock

¾ Risk-free asset (bond)

• Replicate payoff of a call option with Replicate payoff of a call option with (dynamic Δ-hedging)

• (later more)

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O i

Overview Overview

1. 1. Generalizing the setting Generalizing the setting

¾

¾ Trees, modeling information and learning Trees, modeling information and learning

¾

¾ Trees, modeling information and learning Trees, modeling information and learning

¾

¾ Time Time--preferences preferences

2 2 Multi Multi period SDF and event prices period SDF and event prices 2.

2. Multi Multi--period SDF and event prices period SDF and event prices

¾

¾ Pricing in a static dynamic model Pricing in a static dynamic model

3 3 D D i i l ti l ti 3.

3. Dynamic completion Dynamic completion 4.

4. Ponzi Ponzi schemes and rational bubbles schemes and rational bubbles

5. 5. Martingale Martingale -- EMM EMM

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P i h

Ponzi schemes:

Ponzi schemes: infinite horizon max. problem infinite horizon max. problem

I fi it h i ll t t b bit il

• Infinite horizon allows agents to borrow an arbitrarily large amount without effectively ever repaying, by

rolling over debt forever rolling over debt forever.

¾ Ponzi scheme - allows infinite consumption.

• Consider an infinite horizon model, no uncertainty, and Consider an infinite horizon model, no uncertainty, and a complete set of short-lived bonds.

[z^t is the amount of bonds maturing in period t in the portfolio, β_tt is the price of this bond as of period t-1]

)

( ⁰ ⁰ ¹ ¹ ⎪

⎬⎫

⎪⎨

⎧ ^∞ − ≤ −

∑ ^δ

^t ^c ^w

^β

^z

0 . for

) (

max

1 1 1

1

1 1 0

0

0 ⎪⎭⎬

⎪⎩⎨

>

−

≤

− ₊ ₊

∑

=

t z

z w

c c u

t t t t

t t

β

δ β

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P i h

P i h lli lli d b f d b f

Ponzi schemes:

Ponzi schemes: rolling over debt forever rolling over debt forever

• The following consumption path is possible: g p p p c

_t

= w

_t

+1 for all t.

• Note that agent consumes more than his endowment in each period, forever.

• This can be financed with ever increasing debt:

z

₁

=-1/β

₁

, z

₂

=(-1+z

₁

)/β

₂

, z

₃

=(-1+z

₂

)/β

₃

…

• 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

b d d b W d ddi i l i

unbounded above. We need an additional constraint.

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P i h t lit

Ponzi schemes: transversality Ponzi schemes: transversality

• The constraint that is typically imposed on top of the budget constraint is the transversality

condition,

lim

_t→∞_t→∞

β β

_t_t

z

_t_t

≥ 0.

• This constraint implies that the value of debt cannot diverge to infinity.

cannot diverge to infinity.

¾More precisely, it requires that all debt must be redeemed eventually (i.e. in the limit).

y ( )

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R ti l B bbl

R ti l B bbl Th Th i i f f l l

Rational Bubbles:

Rational Bubbles: The price of a consol The price of a consol

• In an infinite dynamic model in which assets are In an infinite dynamic model, in which assets are traded repeatedly, there are additional solutions besides the "fundamental pricing formula "

besides the fundamental pricing formula.

¾New solutions have “bubble component”.

¾No market clearing at infinity

¾No market clearing at infinity.

• Consider

¾

¾model without uncertainty

¾Consol bond delivering $1 in each period forever

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Th i f l

The price of a consol The price of a consol

• Our formula

p

_t

= E

_t

[M

_t+1

(p

_t+1

+ d

_t+1

)]

• in the case of certainty with d=1 p

_t

= M

_t+1

p

_t+1

+ M

_t+1

• solve forward –

(many solutions)

∑

∞

^lim ^.

component bubble

value l

fundamenta 0 1

4 4 3 4

4 2 4 3 1

42

1

^t ^t ^T ^T

p

^T

p =

^∞₌

M +

_→_∞

M

⇒ ∑

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Th i f l

The price of a consol The price of a consol

• Consider case without uncertainty. (event tree is line)

• According to the static dynamic model (the

• According to the static-dynamic model (the

fundamental pricing formula), the price of the consol is p

₀

= ∑

_t

M

_t

.

M := m × m × × m

p

₀

∑

_t

M

_t

.

_M

t := m₁ × m₂ × L × mt

• Consider re-opening markets now. The price at time 0 is just the sum of all Arrow prices, so

time 0 is just the sum of all Arrow prices, so

• p₀ = ∑_t=1^∞ M_t.

• q_t is the marginal rate of substitution between

1 '( )

.

) ( '

0

= ∑

^∞= ⁰

t u w

w t u ^t

p δ

• q_t is the marginal rate of substitution between consumption in period t and in period 0,

• q_t = δ^t (u'(w^t)/u'(w⁰)), so

1 ( )

∑

t u w

q_t ( ( ) ( ))

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Th i f l t

Th i f l t 1 1

The price of a consol at

The price of a consol at tt=1 =1

• At time 1 the price of the consol is the sum of the At time 1, the price of the consol is the sum of the remaining Arrow securities,

2 '( )

.

) ( 1 '

1

∑

^∞= ¹

=

−

t u w

w t u ^t

p δ

2 ( )

1

∑

t= u w

• This can be reformulated,

) ( ' )

(

1 ' ⁰

∑

^∞

=

− ^u ^w ^t ^u ^w^t

p δ δ

• The second part (the sum from 2 to ∞) is almost

2 '( )

.

) (

1

=

' ¹

∑

= ⁰

t u w

w

p δ

u

δ

equal to p₀,

(

₁ _'^'₍⁽ ⁾₎

)

_'^'₍⁽ ⁾₎

2 '( )

) ( '

0 1 0

0 u w

w u

t u w

w u t

t u w

w u

t ^t δ ^t δ

δ =

∑

−

∑

^∞=

∞

=

.

) ( '

0 ⁰

1

w u

w

p − δ

u

=

(

₁ ₍ ₎

)

₍ ₎

2 ( ) t u w u w

t u w

∑

= = u ( w )

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Consol: Solving forward Consol: Solving forward Consol: Solving forward Consol: Solving forward

• More generally, we can express the price at time t+1 as a function of the price at time t,

(

^'⁽_'₍ ₎⁾

)

^,^thus

) ( '

) ( ' 1 1

1

1 t

t t

t

w u

w u w t

u w u

t q

p ₊ =δ⁻ + −δ ⁺

) 1 ( '

) ( ' )

( '

) (

' ¹ ¹

+

+ +

= _t

w u

w u w

u w u

t p

p δ ^t_t δ ^t_t

= m

_t₊₁

+ m

_t₊₁

p

_t₊₁

.

• We can solve this forward by substituting the t+1 version of this equation into the t version, ad

infinitum,

.

0 Mt limT MT pT

p = ^∞ +

⇒

∑

,

=L +

+

= +

= ₁ ₁ ₁ ₁ ₁[ ₂ ₂ ₂]

0 m m p m m m m p

p

. lim

component

bubble value

l fundamenta 0 1

4 4 3 4

4 2 43 1

42

1^t M^t ^T M^T p^T

p ₌ + _→_∞

⇒

∑

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M b bbl

Money as a bubble Money as a bubble

lim M p

M

p = ∑

^∞

+

• The fundamental value = price in the static-dynamic model.

. lim

component bubble

value l

fundamenta 0 1

4 4 3 4

4 2 4 3 1

42

1

^t

M

^t ^T

M

^T

p

^T

p = ∑

₌

+

_→_∞

p y

• 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 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 i li M 0

imposing lim_T→∞ M_T p_T = 0.

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