Asset Pricing Chapter VIII. Arrow-Debreu Pricing June 22, 2006 - - PowerPoint PPT Presentation

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Asset Pricing Chapter VIII. Arrow-Debreu Pricing June 22, 2006 - - PowerPoint PPT Presentation

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8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing:

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SLIDE 1

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks

Asset Pricing

Chapter VIII. Arrow-Debreu Pricing June 22, 2006

Asset Pricing

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SLIDE 2

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks

Setting: An Arrow Debreu Economy

General Equilibrium Supply = Demand With production (if desired) Static but Multi-period No restrictions on preferences The most general of the theories we shall consider

Asset Pricing

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SLIDE 3

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks

  • 1. Two dates: 0, 1 . This set-up, however, is fully

generalizable to multiple periods.

  • 2. N possible states of nature at date 1, which we index by

θ = 1, 2, ..., N with probabilities πθ

  • 3. One perishable (non storable) consumption good
  • 4. K agents, indexed k = 1,...,K, with preferences:

Uk

  • ck
  • + δk

N

  • θ=1

πθUk ck

θ

  • ;
  • 5. In addition, agent k’s endowment is described by the

vector {ek

0,

  • ek

θ

  • θ=1,2,...,N } Possibly:

uk(ck

0, ck θ1, ck θ2, ..., ck θN).

Asset Pricing

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SLIDE 4

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks

Traded Securities

Exclusively Arrow-Debreu securities (contingent claims): Security θ priced qθ promises delivery of one unit of commodity tomorrow if state θ is realized and nothing

  • therwise

How to secure one unit of consumption tomorrow?

Asset Pricing

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SLIDE 5

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks

max

(ck

0,ck 1,....,ck N)

Uk

0 (ck 0) + δk N

  • θ=1

πθUk(ck

θ )

s.t. ck

0 + N

  • θ=1

qθck

θ ≤ ek 0 + N

  • θ=1

qθek

θ

ck

0, ck 1, ...., ck N ≥ 0

(P) Equilibrium is a set of prices (q1, ....qN) such that:

  • 1. at those prices
  • ck

0, ..., ck N

  • solve problem (P), for all k,

and 2.

K

  • k=1

ck

0 = K

  • k=1

ek

0, K

  • k=1

Ck

θ = K

  • k=1

ek

θ, for every θ

Asset Pricing

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SLIDE 6

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Checking Pareto Optimality Interior vs. Corner Solution

Table 8.1. Endowments and Preferences in Our Reference Example Agents Endowments Preferences t = 0 t = 1 θ1 θ2 Agent 1 10 1 2

1 2c1 0 + 0.9

1

3 ln

  • c1

1

  • + 2

3 ln

  • c1

2

  • Agent 2

5 4 6

1 2c2 0 + 0.9

1

3 ln

  • c2

1

  • + 2

3 ln

  • c2

2

  • Asset Pricing
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SLIDE 7

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Checking Pareto Optimality Interior vs. Corner Solution

The respective agent problems are: Agent 1: max 1

2

  • 10 + 1q1 + 2q2 − c1

1q1 − c1 2q2

  • + 0.9

1

3 ln

  • c1

1

  • + 2

3 ln

  • c1

2

  • S.t. c1

1q1 + c1 2q2 ≤ 10 + q1 + 2q2, and c1 1, c1 2 ≥ 0

Agent 2: max 1

2

  • 5 + 4q1 + 6q2 − c2

1q1 − c2 2q2

  • + 0.9

1

3 ln

  • c2

1

  • + 2

3 ln

  • c2

2

  • S.t. c2

1q1 + c2 2q2 ≤ 5 + 4q1 + 6q2 and c2 1, c2 2 ≥ 0

Note: q1,q2 solve «key finance problem»!

Agent 1 : 8 > < > : c1

1 : q1 2

= 0.9 (Πθ)

1 c1 1

c1

2 : q2 2

= 0.9 “

2 3

”

1 c1 2

Agent 2 : 8 > < > : c2

1 : q1 2

= 0.9 “

1 3

”

1 c2 1

c2

2 : q2 2

= 0.9 “

2 3

”

1 c2 2

c1

2 + c2 2 = 8

Asset Pricing

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SLIDE 8

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Checking Pareto Optimality Interior vs. Corner Solution

qθ = δπθ ∂Uk

∂ck

θ

∂Uk ∂ck

, k, θ = 1, 2 (1) price of gold if state θ is realized price of gold today = MUk

θ

MUk , c1

1 = c2 1 = 2.5

c1

2 = c2 2 = 4

q1 = 2 (0.9) 1

3 1 c1

1

  • = 2 (0.9)

1

3

1

2.5

  • = (0.9)

1

3

4

5

  • = 0.24

q2 = 2 (0.9) 2

3 1 c1

2

  • = 2 (0.9)

2

3

1

4

  • = (0.9)

2

3

4

8

  • = 0.3

Asset Pricing

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SLIDE 9

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Checking Pareto Optimality Interior vs. Corner Solution

Table 8.2: Post-Trade Equilibrium Consumptions t = 0 t = 1 θ1 θ2 Agent 1: 9.04 2.5 4 Agent 2: 5.96 2.5 4 Total 15.00 5.0 8 It is Pareto optimal? H1 and H2 are satisfied!

Asset Pricing

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SLIDE 10

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Checking Pareto Optimality Interior vs. Corner Solution

Checking Pareto Optimality

max {c1

0,c1 1,c1 2}

u1(c1

0, c1 1, c1 2) + λu2(c2 0, c2 1, c2 2)

s.t. c1

0 + c2 0 = 15; c1 1 + c2 1 = 5; c1 2 + c2 2 = 8,

c1

0, c1 1, c1 2, c2 0, c2 1, c2 2 ≥ 0

u1 u2 = u1

1

u2

1

= u1

2

u2

2

= λ (2) 1/2 1/2 = (0.9)1

3 1 c1

1

(0.9)1

3 1 c2

1

= (0.9)2

3 1 c1

2

(0.9)2

3 1 c2

2 Asset Pricing

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SLIDE 11

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Checking Pareto Optimality Interior vs. Corner Solution

Box 8-1 Interior vs. Corner Solution q0 ∂Uk ∂ck ≤ δπθ ∂Uk ∂ck

θ

, if ck

0 > 0, and k, θ = 1, 2

(3)

Asset Pricing

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SLIDE 12

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Checking Pareto Optimality Interior vs. Corner Solution

If one security (q1) only (incomplete markets): Table 8.3 : The Post-Trade Allocation t = 0 t = 1 θ1 θ2 Agent 1: 9.64 2.5 2 Agent 2: 5.36 2.5 6 Total 15.00 5.0 8 Agent 1: 5.51 instead of 5.62 Agent 2: 4.03 instead of 4.09

Asset Pricing

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SLIDE 13

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks

Table 8.4: The New Endowment Matrix t = 0 t = 1 θ1 θ2 Agent 1 4 1 5 Agent 2 4 5 1

Asset Pricing

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SLIDE 14

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Table 8.5: Agents’ Utility In The Absence of Trade State-Contingent Utility Expected Utility in Period 1 θ1 θ2 Agent 1 ln(1) = 0 ln(5) = 1.609 1/2 ln(1) + 1/2 ln(5) = 8047 Agent 2 ln(5) = 1.609 ln(1) = 0 1/2 ln(1) + 1/2 ln(5) = 8047 Table 8.6: The Desirable Trades And Post-Trade Consumptions Date 1 Endowments Pre-Trade Consumption Post-Trade θ1 θ2 θ1 θ2 Agent 1 1 5 [⇓2] 3 3 Agent 2 5 [⇑2] 1 3 3

Post-trade allocation is Pareto Optimal After trade, EU in period 1 is approx. 1.1 for both agents.

Asset Pricing

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SLIDE 15

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks

Table 8.7 : Another Set of Initial Allocations

t = 0 t = 1 θ1 θ2 Agent 1 4 1 3 Agent 2 4 5 3

Table 8.8 : Plausible Trades And Post-Trade Consumptions

Date 1 Endowments Pre-trade Consumption Post-trade θ1 θ2 θ1 θ2 Agent 1 1 3 [⇓1] 2 2 Agent 2 5 [⇑1] 3 4 4

Post-trade allocation is PO Features perfect risk sharing or full mutual insurance (no aggregate risk)

Asset Pricing

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SLIDE 16

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks

u1 u2 = u1

1

u2

1

= u1

2

u2

2

= λ (4) u1

1

u2

1

= u1

2

u2

2

= u1

1

u1

2

= u2

1

u2

2

(5)

1 If one of the two agents is fully insured, the other must be as well. 2 More generally, if the MRS are to differ from q, given that they must be equal between them, the low consumption - high MU state must be the same for both agents and similarly for the high consumption - low MU state. 3 If there is aggregate risk, however, the above reasoning also implies that, at a Pareto optimum, it is shared "proportionately" among agents with same risk tolerance. 4 Finally, if agents are differentially risk averse, in a Pareto optimal allocation the less risk averse will typically provide some insurance services to the more risk averse. 5 More generally, optimal risk sharing dictates that the agent most tolerant of risk bears a disproportionate share of it. Asset Pricing

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SLIDE 17

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks

Implementing Pareto Optimal Allocations: On the Possibility of Market failure

(i) Agent 1 solves: max (4 − qQz1

Q) + [1/2ln(1 + z1 Q) + 1/2ln(5)]

s.t. qQz1

Q ≤ 4

(ii) Agent 2 solves: max (4 − qQz2

Q) + [1/2ln(5 + z2 Q) + 1/2ln(1)]

s.t. qQz2

Q ≤ 4

Assuming an interior solution, the FOCs are (i)’ : −qQ + 1

2

  • 1

1+z1

Q

  • = 0;

(ii)′ : −qQ + 1

2

  • 1

5+z2

Q

  • = 0 ⇒

1 1+z1

Q =

1 5+z2

Q ; Asset Pricing

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SLIDE 18

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks Table 8.9: Market Allocation When Both Securities Are Traded t = 0 t = 1 θ1 θ2 Agent 1 4 3 3 Agent 2 4 3 3 Table 8.10: The Net Gains From Trade Expected Utility Levels and Net Trading Gains (Gain to issuer in bold) No Trade Trade Only Q Trade Both Qand R EU EU ∆EU(i) EU ∆EU(ii) Agent 1 4.8047 5.0206 0.2159 5.0986 0.0726 Agent 2 4.8047 4.883 0.0783 5.0986 0.2156 Total 0.2942 0.2882 Asset Pricing

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SLIDE 19

8.1 Setting: An Arrow Debreu Economy 8.2 Competitive Equilibrium and Pareto Optimality Illustrated 8.3 Pareto Optimality and risk Sharing 8.4 Implementing Pareto Optimal Allocations: On the Possibility of Market failure 8.5 Arrow-Debreu pricing: concluding remarks

The «father» of all asset pricing relationships A reference Conceptual import Hard to identify pure state of nature Static

Asset Pricing