1. Microfoundations of Financial Economics 2004-2005 1
Microfoundations of Financial Economics From Fisher to Arrow-Debreu
Professor André Farber
Solvay Business School
Université Libre de Bruxelles

2. Theory of asset pricing under certainty
1930
Fisher Theory of Interest
Williams Theory of Investment Value
1940
1950
Hirshleifer Theory of Optimal Investment Decisions
1960
PhD 01-2

3. Theory of asset pricing under uncertainty
1950
Markowitz Portfolio theory
Arrow State prices
1960
Arrow Debreu General equilibrium
Sharpe Lintner CAPM
1970
Black Scholes Merton OPM
Ross APT
Lucas Asset Prices
Ross Risk neutral pricing
Vasiceck Term structure
Cox Ross Rubinstein Binomial OPM
Harrison Kreps Martingales
1980
1990
Cochrane – Campbell: p = E(MX)
2000
PhD 01-2

4. Three views of asset pricing
General equilibrium
Mean variance efficiency
Beta pricing
Stochastic discount factors
Factor model + No arbitrage
Risk-neutral pricing
State prices linear pricing rule
Complete markets No arbitrage (NA) Law of one price (LOOP)
Adapted from Cochrane Figure 6.1
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5. Certainty: Irving Fisher
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
Gross interest rate: Rf = 1/m
Consider future payoff x
Price at time t = 0: p(x) = m x
Why?
otherwise, ARBITRAGE
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6. Where does m come from? Consumption over time:
Max utility function: U(c0, c1) U’i >0, concave
subject to budget constraint: c0 + m c1 = W
FOC:
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7. Technical details
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8. Using time separable utility
Suppose U(c0,c1) = u(c0) + β u(c1)
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9. Example As: Define: Three determinant of the interest rate: Impatience
Time preference
Growth rate of consumption
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10. Multiperiod model – certainty
Utility function:
Security: price = p future cash flows = {dt}
Optimum:
FOC:
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11. FOC: two periods version
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12. Introducing uncertainty
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
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13. Assumptions
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:
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14. Portfolio Composition S ×1 Payoff 1 × S h’x
Price h.p =h’p inner product
Example
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15. p.h ≤ 0 and h’x≥0 (with at least one positive payoff)
Arbitrage
General definition of an arbitrage:
p.h ≤ 0 and h’x≥0 (with at least one positive payoff)
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16. No arbitrage
Theorem: In complete markets, NA implies that exists a unique q>> 0 such that NA
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17. State price calculation
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18. Geometry
State 2
x(2) x q R
p(x) = cst
p(x) = 1
x(1)
State 1
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19. Stochastic discount factors
Define:
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20. Risk neutral probabilities
Define:
Note:
Looks like probabilities = risk neutral probabilities
New pricing formula:
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21. Geometry State 2 x(2) x q R p(x) = cst m p(x) = 1 x(1) State 1
x(1)
State 1
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22. Geometry (with rescaled values)
State 2
E(x) = cst m* 1* x*
State 1
p(x) = 0
E(x) = 0
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23. Beta pricing As: cov(m,x) = E(mx)-E(m)E(x) and E(m) = 1/Rf
Define gross return:
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24. Geometry State 2 p(x) = p[Projection of x on m] m* x* State 1 p(x) = 0
State 1
p(x) = 0
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25. Beta representation
Define :
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26. Tomorrow Where do the state prices, SDF, risk neutral proba come from?
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