Asset Pricing - Chapter XI. The Martingale Measure: Part I for... · 11.1 Introduction 11.2 The...

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Asset PricingChapter XI. The Martingale Measure: Part I

June 20, 2006

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

1 (CAPM)ECF1

(1 + r f1 + π)

;ECF2

(1 + r f2 + π)2

;ECF3

(1 + r f3 + π)3

; orECFτ − Πτ

(1 + r fτ )τ

.

2 (Risk Neutral)ECFτ

(1 + r fτ )τ

;

3 (Arrow-Debreu) Xθτ∈Θτ

q(θτ )CF (θτ ),

pj,t =

E“

CF j,t+1

”− cov(CF j,t+1, rM )[

ErM−rfσ2

M]

1 + rf,

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Existence of Risk Neutral Probabilities

The setting and the intuition

2 datesJ possible states of nature at date 1State j = θj with probability πj

Risk free security qb(0) = 1, qb(1) ≡ qb(θj , 1) = (1 + rf )

i=1,..., N fundamental securities with prices qe(0), qei (θj , 1)

Securities market may or may not be completeS is the set of all fundamental securities, including bondand linear combination thereof

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Existence of Risk Neutral Probabilities

Existence of a set of numbers πRNj , ΣπRN

j = 1 s.t

qei (0) =

1(1 + rf )

EπRN (qei (θ, 1)) =

1(1 + rf )

J∑j=1

πRNj qe

i (θj , 1) (1)

qei (0) = πRN

1

(qe

i (θ1, 1)

1 + rf

)+......+πRN

J

(qe

i (θJ , 1)

1 + rf

), i = 1, 2, ..., N,

(2)No solution if: qe

s (0) = qek (0) with

qek (θj , 1) ≥ qe

s (θj , 1) for all j , and qek (θ, 1) > qe

s (θ, 1) (3)

= arbitrage opportunity

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Consider a portfolio, P, composed of nbP risk-free bonds and ni

Punits of risky security i , i = 1, 2, ..., N.

VP(0) = nbPqb(0) +

N∑i=1

niPqe

i (0), (4)

VP(θj , 1) = nbPqb(1) +

N∑i=1

niPqe

i (θj , 1). (5)

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Definition 11.1A portfolio P in S constitutes an arbitrage opportunity providedthe following conditions are satisfied:

(i) VP(0) = 0, (6)(ii) VP(θj , 1) ≥ 0, for all j ∈ {1, 2, . . ., J},(iii) VP(θ, 1) > 0, for at least one ∈ {1, 2, . . ., J}.

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Definition 11.2A probability measure

{πRN

j

}J

j=1defined on the set of states (θj ,

j = 1, 2, ..., J), is said to be a risk-neutral probability measure if

(i) πRNj > 0, for all j = 1, 2, ..., J, and (7)

(ii) qei (0) = EπRN

{qe

i (θ, 1)

1 + rf

},

for all fundamental risky securities i = 1, 2, ..., N in S.

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Table 11.1: Fundamental Securities for Example 11.1

Period t = 0 Prices Period t = 1 Payoffsθ1 θ2

qb(0): 1 qb(1): 1.1 1.1qe(0): 4 qe(θj , 1): 3 7

complete marketsno arbitrage opportunities"objective" state probabilities?

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Table 11.2: Fundamental Securities for Example 11.2

Period t = 0 Prices Period t = 1 Payoffsθ1 θ2 θ3

qb(0): 1 qb(1): 1.1 1.1 1.1qe

1 (0): 2 qe1 (θj , 1): 3 2 1

qe2 (0): 3 qe

2 (θj , 1): 1 4 6

2 = πRN1

„ 3

1.1

«+ π

RN2

„ 2

1.1

«+ π

RN3

„ 1

1.1

«

3 = πRN1

„ 1

1.1

«+ π

RN2

„ 4

1.1

«+ π

RN3

„ 6

1.1

«

1 = πRN1 + π

RN2 + π

RN3 .

The solution to this set of equations,

πRN1 = .3, π

RN2 = .6, π

RN3 = .1,

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Table 11.3: Fundamental Securities for Example 11.3

Period t = 0 Prices Period t = 1 Payoffsθ1 θ2 θ3

qb(0): 1 qb(1): 1.1 1.1 1.1qe

1(0): 2 qe1(θj , 1): 1 2 3

2 = πRN1

(1

1.1

)+ πRN

2

(2

1.1

)+ πRN

3

(3

1.1

)1 = πRN

1 + πRN2 + πRN

3

System indeterminate; many solutions

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

2.2 − πRN1 = 2π

RN2 + 3π

RN3

1 − πRN1 = π

RN2 + π

RN3 ,

πRN1 > 0

πRN2 = .8 − 2π

RN1 > 0

πRN3 = .2 + π

RN1 > 0

0 < πRN1 < .4,

(πRN1 , π

RN2 , π

RN3 ) ∈ {(λ,8 − 2λ, .2 + λ) : 0 < λ < .4}

Risk Neutral probabilities are not uniquely defined!

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Table 11.4: Fundamental Securities for Example 11.4

Period t = 0 Prices Period t = 1 Payoffsθ1 θ2 θ3

qb(0): 1 qb(1): 1.1 1.1 1.1qe

1(0): 2 qe1(θj , 1): 2 3 1

qe2(0): 2.5 qe

2(θj , 1): 4 5 3

an arbitrage opportunityNo solution (or solution with πRN

i = 0 for some i)

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Proposition 11.1 Consider the two-period setting describedearlier in this chapter. Then there exists arisk-neutral probability measure on S, if and only ifthere are no arbitrage opportunities among thefundamental securities.May not be unique!Until now: Fundamental securities in SNow: Portfolio of fundamental securities.

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Proposition 11.2 Suppose the set of securities S is free ofarbitrage opportunities. Then for any portfolio P inS

VP(0) =1

(1 + rf )EπRN VP(θ, 1), (8)

for any risk-neutral probability measure πRN on S.

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Proof of Proposition 11.2

Let P be an arbitrary portfolio in S, and let it be composed of nbP

bonds and niP

shares of fundamental risky asset i . In the

absence of arbitrage, P must be priced equal to the value of itsconstituent securities, in other words,

VP(0) = nbP

qb(0) +N∑

i=1ni

Pqe

i (0) = nbP

EπRN

(qb(1)1+rf

)+

N∑i=1

niP

EπRN

(qe

i (θ,1)1+rf

),

for any risk neutral probability measure πRN ,

= EπRN

nbP

qb(1)+NP

i=1ni

Pqe

i (θ,1)

1+rf

= 1(1+rf )

EπRN

(VP(θ, 1)

).

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

What if risk neutral measure is not unique?

Proposition 11.2 remains valid: each of the multiple of riskneutral measures assign the same value to the fundamentalsecurities an thus to the portfolio itself!

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Proposition 11.3: Consider an arbitrary period t = 1 payoffx(θ, 1) and let M represent the set of all risk-neutral probabilitymeasures on the set S. Assume S contains no arbitrageopportunities. If

1(1 + rf )

EπRN x(θ, 1) =1

(1 + rf )EπRN x(θ, 1) for any πRN , πRN ∈ M,

then there exists a portfolio in S with the same t = 1 payoff asx(θ, 1).

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

Proposition 11.4: Consider a set of securities S withoutarbitrage opportunities. Then S is complete if and only if thereexists exactly one risk-neutral probability measure.Proof Suppose S is complete and there were two risk-neutralprobability measures, {πRN

j : j = 1, 2, . . . , J} and {~πRNj :

j = 1, 2, ..., J}. Then there must be at least one state for whichπRN

6= ~πRN . Since the market is complete, one must be able to

construct a portfolio P in S such that

VP(0) > 0, and

{VP(θj , 1) = 0 j 6= jVP(θj , 1) = 1 j = j

.

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Definition 11.1Definition 11.2Proposition 11.1Proposition 11.2Proof of Proposition 11.2Uniqueness

This is simply the statement of the existence of anArrow-Debreu security associated with θ.But then {πRN

j :j = 1, 2, ..., J} and {~πRNj :j = 1, 2, ..., J} cannot

both be risk-neutral measures as, by Proposition 11.2,

VP(0) =1

(1 + rf )EπRN VP(θ, 1) =

πRNj

(1 + rf )

6=~πRN

j

(1 + rf )=

1(1 + rf )

E~πRN VP(θ, 1)

= VP(0), a contradiction.

Thus, there cannot be more than one risk-neutral probabilitymeasure in a complete market economy.

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Arrow-Debreu Pricing:

qj (0) =πRN

j(1+rf )

Back to example 11.2.π

RN1 = .3, π

RN2 = .6, π

RN3 = .1,

q1(0) = .3/1.1 = .27; q2(0) = .6/1.1 = .55; q3(0) = .1/1.1 = .09.Conversely:

prf =JX

j=1

qj (0),

and thus

(1 + rf ) =1

prf

=1

JPj=1

qj (0)

We define the risk-neutral probabilities {πRN (θ)} according to

πRNj =

qj (0)

JPj=1

qj (0)

(9)

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Table 11.6 The Exchange Economy of Section 8.3 –Endowments and Preferences

Endowments Preferencest = 0 t = 1

Agent 1 10 1 2 U1(c0, c1) = 12c1

0 + .9(13 ln(c1

1) + 23 ln(c1

2))

Agent 2 5 4 6 U2(c0, c1) = 12c2

0 + .9(13 ln(c2

1) + 23 ln(c2

2))

πRN1 =

.24

.54, and πRN

2 =.30.54

.

Suppose a stock were traded where qe(θ1, 1) = 1, and qe(θ2, 1) = 3.By risk-neutral valuation (or equivalently, using Arrow-Debreu prices),its period t = 0 price must be

qe(0) = .54[.24.54

(1) +.30.54

(3)

]= 1.14;

the price of the risk-free security is qb(0) = .54.

Asset Pricing

11.1 Introduction11.2 The setting and the intuition

11.3 Notation, Definitions and Basic ResultsArrow-Debreu Pricing

Table 11.7 Initial Holdings of Equity and Debt AchievingEquivalence with Arrow-Debreu Equilibrium Endowments

t = 0Consumption ni

e nib

Agent 1: 10 1/2 1/2

Agent 2: 5 1 3

max(10 + 1q1(0) + 2q2(0)− c11q1(0)− c1

2q2(0)) + .9(13c1

1 + 23c1

2)s.t. c1

1q1(0) + c12q2(0) ≤ 10 + q1(0) + 2q2(0)

The first order conditions arec1

1 : q1(0) = 13 .0.9

c12 : q2(0) = 2

3 .0.9

from which it follows that πRN1 =

13 0.90.9 = 1

3 while πRN2 =

23 0.90.9 = 2

3Asset Pricing