Microfoundations of Financial Economics 2004-2005 1.2 From Fisher to Arrow-Debreu

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Microfoundations of Financial Economics 2004-2005 1.2 From Fisher to Arrow-Debreu

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Microfoundations of Financial Economics 2004-2005 1.2 From Fisher to Arrow-Debreu

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Microfoundations of Financial Economics2004-20051.2 From Fisher to Arrow-Debreu

Professor André Farber

Solvay Business School

Université Libre de Bruxelles

1930

FisherTheory of Interest

WilliamsTheory of Investment Value

1940

1950

HirshleiferTheory of Optimal Investment Decisions

1960

PhD 01-2

1950

ArrowState prices

MarkowitzPortfolio theory

1960

Arrow DebreuGeneral equilibrium

Sharpe LintnerCAPM

1970

Black Scholes MertonOPM

RossAPT

LucasAsset Prices

RossRisk neutral pricing

VasiceckTerm structure

Harrison KrepsMartingales

Cox Ross RubinsteinBinomial OPM

1980

1990

Cochrane – Campbell: p = E(MX)

2000

PhD 01-2

General equilibrium

Mean variance efficiency

Beta pricing

Stochastic discount factors

Factor model+No arbitrage

Risk-neutral pricing

State priceslinear pricing rule

Complete marketsNo arbitrage (NA)Law of one price (LOOP)

Adapted from Cochrane Figure 6.1

PhD 01-2

- Setting:
- 1 good – price at time 0 = 1 (numeraire)
- Constant price (no inflation)
- 1 security: zero-coupon, face value = 1, price at time t=0: m

- otherwise, ARBITRAGE

PhD 01-2

- Consumption over time:
- Max utility function:U(c0, c1)U’i >0, concave
- subject to budget constraint:c0 + mc1 = W
- FOC:

PhD 01-2

PhD 01-2

Suppose U(c0,c1) = u(c0) + β u(c1)

PhD 01-2

As:

Define:

Three determinant of the interest rate:

Impatience

Time preference

Growth rate of consumption

PhD 01-2

Utility function:

Security:price = pfuture cash flows = {dt}

Optimum:

FOC:

PhD 01-2

PhD 01-2

- Setting
- 1 period
- S states of nature :S = {s} finite
- Probabilities: π(s)
- S traded securities: price p(S×1 vector)
- Future payoffs conditional on state of nature: x(s)
- xi =[xi(1), xi(2), …, xi(S)] 1 ×S vector
- Matrix of payoffs: S×S matrix

PhD 01-2

- Payoff space: X: set of all the payoffs that investor can purchase
- Complete markets: X = RS
- Portfolio formation:
- Law of one price, linear pricing rule:

PhD 01-2

- Composition S ×1

- Payoff 1×Sh’x

- Priceh.p =h’p inner product

Example

PhD 01-2

- General definition of an arbitrage:
p.h ≤ 0 and h’x≥0 (with at least one positive payoff)

PhD 01-2

- Theorem: In complete markets, NA implies that exists a unique q>> 0 such that

NA

PhD 01-2

PhD 01-2

State 2

x(2)

x

q

R

p(x) = cst

p(x) = 1

0

x(1)

State 1

PhD 01-2

Define:

PhD 01-2

Define:

Note:

Looks like probabilities = risk neutral probabilities

New pricing formula:

PhD 01-2

State 2

x(2)

x

q

R

p(x) = cst

m

p(x) = 1

0

x(1)

State 1

PhD 01-2

State 2

E(x) = cst

m*

x*

1*

0

State 1

p(x) = 0

E(x) = 0

PhD 01-2

As: cov(m,x) = E(mx)-E(m)E(x) and E(m) = 1/Rf

Define gross return:

PhD 01-2

State 2

p(x) = p[Projection of x on m]

m*

x*

0

State 1

p(x) = 0

PhD 01-2

Define :

PhD 01-2

- Where do the state prices, SDF, risk neutral proba come from?

PhD 01-2