Asset Pricing - Chapter XI. The Martingale Measure: Part I for... · 11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Deﬁnitions and Basic Results Arrow-Debreu

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



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



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



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



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




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




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




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





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






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




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






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




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




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




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




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




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




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




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



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



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



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

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Asset Pricing - Chapter XI. The Martingale Measure: Part I for... · 11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Deﬁnitions and Basic Results Arrow-Debreu