Asset Pricing Chapter XI. The Martingale Measure: Part I June 20, - - PowerPoint PPT Presentation

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing

Asset Pricing

Chapter XI. The Martingale Measure: Part I June 20, 2006

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing 1 (CAPM) E ˜ CF1 (1 + rf

1 + π)

; E ˜ CF 2 (1 + rf

2 + π)2 ;

E ˜ CF3 (1 + rf

3 + π)3 ; or

E ˜ CFτ − Πτ (1 + rf

τ )τ

. 2 (Risk Neutral) ˆ E ˜ CFτ (1 + rf

τ )τ ;

3 (Arrow-Debreu) X

θτ ∈Θτ

q(θτ )CF(θτ ), pj,t = E “ ˜ CF j,t+1 ” − cov( ˜ CFj,t+1, ˜ rM )[ E˜

rM −rf σ2 M

] 1 + rf , Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Existence of Risk Neutral Probabilities

The setting and the intuition

2 dates J possible states of nature at date 1 State 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), qe

i (θj, 1)

Securities market may or may not be complete S is the set of all fundamental securities, including bond and linear combination thereof

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Existence of Risk Neutral Probabilities

Existence of a set of numbers πRN

j

, ΣπRN

j

= 1 s.t qe

i (0) =

1 (1 + rf)EπRN(qe

i (θ, 1)) =

1 (1 + rf)

J

• j=1

πRN

j

qe

i (θj, 1) (1)

qe

i (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) = qe k (0) with

qe

k (θj, 1) ≥ qe s (θj, 1) for all j, and qe k (θˆ , 1) > qe s (θˆ , 1)

(3) = arbitrage opportunity

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

Consider a portfolio, P, composed of nb

P risk-free bonds and ni P

units of risky security i, i = 1, 2, ..., N. VP(0) = nb

Pqb(0) + N

• i=1

ni

Pqe i (0),

(4) VP(θj, 1) = nb

Pqb(1) + N

• i=1

ni

Pqe i (θj, 1).

(5)

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

Definition 11.1 A portfolio P in S constitutes an arbitrage opportunity provided the following conditions are satisfied: (i) VP(0) = 0, (6) (ii) VP(θj, 1) ≥ 0, for all j ∈ {1, 2, . . ., J}, (iii) VP(θˆ

, 1)

> 0, for at least one ˆ  ∈ {1, 2, . . ., J}.

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

Definition 11.2 A probability measure

• πRN

j

J

j=1 defined on the set of states (θj,

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

j

> 0, for all j = 1, 2, ..., J, and (7) (ii) qe

i (0)

= EπRN ˜ qe

i (θ, 1)

1 + rf

• ,

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

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

Table 11.1: Fundamental Securities for Example 11.1 Period t = 0 Prices Period t = 1 Payoffs θ1 θ2 qb(0): 1 qb(1): 1.1 1.1 qe(0): 4 qe(θj, 1): 3 7 complete markets no arbitrage opportunities "objective" state probabilities?

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

Table 11.2: Fundamental Securities for Example 11.2

Period t = 0 Prices Period t = 1 Payoffs θ1 θ2 θ3 qb(0): 1 qb(1): 1.1 1.1 1.1 qe

1(0): 2

qe

1(θj , 1):

3 2 1 qe

2(0): 3

qe

2(θj , 1):

1 4 6 2 = πRN

1

„ 3 1.1 « + πRN

2

„ 2 1.1 « + πRN

3

„ 1 1.1 « 3 = πRN

1

„ 1 1.1 « + πRN

2

„ 4 1.1 « + πRN

3

„ 6 1.1 « 1 = πRN

1

+ πRN

2

+ πRN

3

. The solution to this set of equations, πRN

1

= .3, πRN

2

= .6, πRN

3

= .1, Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

Table 11.3: Fundamental Securities for Example 11.3 Period t = 0 Prices Period t = 1 Payoffs θ1 θ2 θ3 qb(0): 1 qb(1): 1.1 1.1 1.1 qe

1(0): 2

qe

1(θj, 1):

1 2 3 2 = πRN

1

1 1.1

• + πRN

2

2 1.1

• + πRN

3

3 1.1

• 1

= πRN

1

+ πRN

2

+ πRN

3

System indeterminate; many solutions

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness 2.2 − πRN

1

= 2πRN

2

+ 3πRN

3

1 − πRN

1

= πRN

2

+ πRN

3

, πRN

1

> 0 πRN

2

= .8 − 2πRN

1

> 0 πRN

3

= .2 + πRN

1

> 0 0 < πRN

1

< .4, (πRN

1

, πRN

2

, πRN

3

) ∈ {(λ,8 − 2λ, .2 + λ) : 0 < λ < .4}

Risk Neutral probabilities are not uniquely defined!

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

Table 11.4: Fundamental Securities for Example 11.4 Period t = 0 Prices Period t = 1 Payoffs θ1 θ2 θ3 qb(0): 1 qb(1): 1.1 1.1 1.1 qe

1(0): 2

qe

1(θj, 1):

2 3 1 qe

2(0): 2.5

qe

2(θj, 1):

4 5 3 an arbitrage opportunity No solution (or solution with πRN

i

= 0 for some i)

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

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

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

Proposition 11.2 Suppose the set of securities S is free of arbitrage opportunities. Then for any portfolio ˆ P in S Vˆ

P(0) =

1 (1 + rf)EπRN ˜ Vˆ

P(θ, 1),

(8) for any risk-neutral probability measure πRN on S.

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

Proof of Proposition 11.2

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

ˆ P

bonds and ni

ˆ P shares of fundamental risky asset i. In the

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

Vˆ

P(0) = nb ˆ Pqb(0) + N

• i=1

ni

ˆ P qe i (0) = nb ˆ P EπRN

• qb(1)

1+rf

• +

N

• i=1

ni

ˆ P EπRN

˜

qe

i (θ,1)

1+rf

• ,

for any risk neutral probability measure πRN, = EπRN   

nb

ˆ P qb(1)+ N

P

i=1

ni

ˆ P ˜

qe

i (θ,1)

1+rf

   =

1 (1+rf )EπRN

• ˜

Vˆ

P(θ, 1)

• .

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

What if risk neutral measure is not unique?

Proposition 11.2 remains valid: each of the multiple of risk neutral measures assign the same value to the fundamental securities an thus to the portfolio itself!

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

Proposition 11.3: Consider an arbitrary period t = 1 payoff ˜ x(θ, 1) and let M represent the set of all risk-neutral probability measures on the set S. Assume S contains no arbitrage

• pportunities. If

1 (1 + rf)EπRN ˜ x(θ, 1) = 1 (1 + rf)Eˆ

πRN ˜

x(θ, 1) for any πRN, ˆ πRN ∈ M, then there exists a portfolio in S with the same t = 1 payoff as ˜ x(θ, 1).

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

Proposition 11.4: Consider a set of securities S without arbitrage opportunities. Then S is complete if and only if there exists exactly one risk-neutral probability measure. Proof Suppose S is complete and there were two risk-neutral probability measures, {πRN

j

: j = 1, 2, . . . , J} and { πRN

j

: j = 1, 2, ..., J}. Then there must be at least one state ˆ  for which πRN

ˆ 

= πRN

ˆ 

. 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 = ˆ

j VP(θj, 1) = 1 j = ˆ j .

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing Definition 11.1 Definition 11.2 Proposition 11.1 Proposition 11.2 Proof of Proposition 11.2 Uniqueness

This is simply the statement of the existence of an Arrow-Debreu security associated with θˆ

.

But then {πRN

j

:j = 1, 2, ..., J} and { πRN

j

:j = 1, 2, ..., J} cannot both be risk-neutral measures as, by Proposition 11.2, VP(0) = 1 (1 + rf)EπRN ˜ VP(θ, 1) = πRN

ˆ j

(1 + rf) =

• πRN

ˆ j

(1 + rf) = 1 (1 + rf)E

πRN ˜

VP(θ, 1) = VP(0), a contradiction. Thus, there cannot be more than one risk-neutral probability measure in a complete market economy.

Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing

Arrow-Debreu Pricing:

qj (0) =

πRN j (1+rf )

Back to example 11.2. πRN

1

= .3, πRN

2

= .6, πRN

3

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

J

X

j=1

qj (0), and thus (1 + rf ) = 1 prf = 1

J

P

j=1

qj (0) We define the risk-neutral probabilities {πRN (θ)} according to πRN

j

= qj (0)

J

P

j=1

qj (0) (9) Asset Pricing

11.1 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing

Table 11.6 The Exchange Economy of Section 8.3 – Endowments and Preferences Endowments Preferences t = 0 t = 1 Agent 1 10 1 2 U1(c0, c1) = 1

2c1 0 + .9(1 3 ln(c1 1) + 2 3 ln(c1 2))

Agent 2 5 4 6 U2(c0, c1) = 1

2c2 0 + .9(1 3 ln(c2 1) + 2 3 ln(c2 2))

πRN

1

= .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 Introduction 11.2 The setting and the intuition 11.3 Notation, Definitions and Basic Results Arrow-Debreu Pricing

Table 11.7 Initial Holdings of Equity and Debt Achieving Equivalence with Arrow-Debreu Equilibrium Endowments t = 0 Consumption ˆ ni

e

ˆ ni

b

Agent 1: 10

1/ 2 1/ 2

Agent 2: 5 1 3 max(10 + 1q1(0) + 2q2(0) − c1

1q1(0) − c1 2q2(0)) + .9(1 3c1 1 + 2 3c1 2)

s.t. c1

1q1(0) + c1 2q2(0) ≤ 10 + q1(0) + 2q2(0)

The first order conditions are c1

1 :

q1(0) = 1

3.0.9

c1

2 :

q2(0) = 2

3.0.9

from which it follows that πRN

1

=

1 30.9

0.9 = 1 3 while πRN 2

=

2 30.9

0.9 = 2 3

Asset Pricing