1. ECO568 - Uncertainty and Financial Markets
Arthur Charpentier and Alfred Galichon
Ecole polytechnique, January 6 2009
1. No-Arbitrage and the Law of One Price

2. General Introduction (1)
Features of Financial Economics
Abundance of …nancial data, empirical testability.
Interactions, collective behavior.
Equilibrium. Partial vs. General: how do you trust your model versus the
market?

3. General Introduction (2)
Rationality
Preferences + beliefs => choices
"Rational" preferences.
- “more” is always preferred to “less” - really?
- Risk-aversion: people buy insurance - but they also purchase lottery
tickets
Rational beliefs: agents’subjective probabilities = objective probabilities
(stationarity; no learning, no insider information, no overcon…dence)

4. General Introduction (3)
S. Ross’"three stylized facts" about …nancial markets:
E¢ cient Markets. Asset returns are unpredictable. Then what is this
course useful for?
Risk Premium. Risky assets have higher expected returns (ie. Lower
prices) than safe ones. By how much?
Correlation. Asset returns are cross-correlated. How to use it?
We shall come up with a consistent theory accounting for these facts.

5. Course outline
Lecture 1 (Jan 6): No Arbitrage and the Law of One Price
Lecture 2 (Jan 13): The Demand for Risk
Lecture 3 (Jan 20): Investment Theory (Static case): MVA, CAPM
Lecture 4 (Jan 21): Informational aspects
Lecture 5 (Feb 10) Investment Theory (Dynamic case, discrete time)
Lecture 6 (Feb 17): Investment Theory (Dynamic case, continuous time)
Lecture 7 (March 3): Term Structure Models
Lecture 8 (March 10): Empirical Puzzles in Asset Pricing
Lecture 9 (March 17): Risk management and Financial bubbles
Exam: March 24
Course webpage: http://groups.google.com/group/x-eco568.

6. Course organization
The class is organized as 1h45 lecture, then 1h45 section ("petite classe").
The lecture slides are distributed in class, and posted online after each lecture.
The "Polycopié de cours" by G. Demange and G. Laroque is avaibable from
the Scolarité. Students are advised to look at it, but the lecture slides are
self-contained.
Oral participation is strongly encouraged!

7. Today’ lecture outline
s
General introduction
No-Arbitrage and the Law of One Price
Section: Arbitrage Pricing Theory, Cox-Ross-Rubinstein

8. The setting
contingent states at time t = 1: ! 2 f1; :::; g.
Examples. ! 2 f0; 1g (no accident/accident); ! value the stock market
at time t = 1.
K assets available at time t = 0: k 2 f1; :::; Kg.
the tableau a! is the value of asset k in contingent state ! , pk price of
k
asset k at time t = 0. A market is the data a! ; pk .
k
Portfolio: combination of assets, z k quantity of asset k. Portfolio price
P k . Portfolio value (t = 1): P a! z k .
(t = 0): k pk z k k

9. Example. Consider the following tableau, where lines correspond to assets,
and columns to states
!
0 1 2 3
a= ;p =
1 1 0 1
what is the value of portfolio of 5 assets k = 1 and 2 asset k = 2 in the state
! = 2?

10. Arbitrage opportunity
De…nition. An arbitrage portfolio is a portfolio z k such that
P ! k P
k ak z 0 for all ! and k pk z k 0, at least one of these E + 1
inequalities being strict.
A market is arbitrage free if there is no arbitrage portfolio.
Proposition. A market a! ; pk without arbitrage opportunity satis…es the
k
Law of One Price:
if for two assets k and l, a! = a! for every state ! , then the assets have the
k l
same price: pk = pl .
0 1 0 1
2 0 1 2
B C B C
Example. Consider a = @1 2 4A ; and p = @1A. The portfolio z =
3 2 5 2
P ! k
( 1; 1; 1) is an arbitrage portfolio, as k ak z = 0 for every ! , and
P k
k pk z = 1.

11. State-price vectors and the
No-Arbitrage theorem
The No-Arbitrage Theorem. A market a! ; pk is arbitrage-free if and only
k
if there is a vector q! such that:
- q! > 0 for every state !
P
- pk = ! a! q! for every asset k
k
q! is then called a state-price vector.
Remark. 1. Although q! is in general not unique, this does not contradict
the law of one price, as every state-price vector q! yields the same set of asset
prices.
2. This notion does not involve the "statistical" probability per se (apart from
its support).

12. 0 1 0 1
2 0 1 1
B C B C
Example. Consider a = @1 2 4A ; and p = @1A. The market is arbitrage-
3 2 5 2
free, and q = (3=7; 0; 1=7) is a state-price vector.
P
Sketch of the proof of the No-Arbitrage theorem. ( is easy: k pk z k =
P P P P
k;! q! a! z k = k;! q! a! z k , thus k a! z k
k k k 0 implies k pk z k
0.
For the converse, we note that the investor is looking for the portfolio with
minimal cost which gets him a positive revenue with certainty in every state of
P P
the world. The investor solves V = inf z k z k pk s.t. k z k a! 0, and V
k
is to be interpreted as the sure gain of the investor. By positive homogeneity,
V = 0 or V = 1, and there is No arbitrage opportunity if and only if
V = 0.
P k P k !
Now, write the Lagrangian V = inf z supq 0 k z pk !;k z ak q! , where

13. q! are the Lagrange multipliers associated to the constraints. But by duality,
P P
one can invert the inf and the sup, and V = supq 0 inf z k z k pk !
! ak q! .
Therefore there is No arbitrage opportunity if and only if there is a vector q
such that
!
X X
inf z k pk a! q!
k =0
z
k !
P
hence ! q! a! = pk for all k. It remains to explain why one can choose
k
q! > 0. Remember, we have introduced the q! as Lagrange multipliers in
the investor’ constrained optimization problem. The null portfolio z = 0 is a
s
solution to the constrained optimization problem which has all the constraints
saturated. By a well-known result on linear programming, all the corresponding
Lagrange multipliers q! can be chosen strictly positive.
Remark: Game-theoretic interpretation. The proof has a game-theoretic
interpretation as a game between Investor and the Market, which sets the state

14. prices in order to minimize the Investor’ gain. The Market’ strategy is to
s s
choose a state-price vector q! , while the Investor’ strategy is to pick up his
s
investment portfolio z k .
The value of this game (for the Market) is then:
!
X X
V = inf sup z k pk a! q!
k
z q 0
k !
V is interpreted as the worse outcome for the Market facing a rational Investor.
This is a zero-sum game: the value of the game for the Investor is V , the
opposite og the value of the game for the Market. The duality principle is
equivalent to a min-max theorem, which precisely says that the value of this
game will be the same regardless whom (of the Investor or the Market) plays
…rst.

15. Consequence: Arrow-Debreu prices
De…nition. An Arrow-Debreu asset is an asset yielding 1 in state ! 0, and 0
otherwise. Denote a!0 = 1f!=!0g.
!
Proposition. If q is a state-price vector, then the price of a!0 is q!0 .
!
Moreover, if there exists a portfolio yielding 1 in state ! 0 and 0 otherwise, then
its price equals q!0 .
Example: Digital options, call options. Suppose a stock price a! can take
up to a …nite number of values v1; :::; vN . Then the digital option of strike vk
is the option d!k = 1fa! =vk g. This is precisely an Arrow-Debreu price.
v

16. Risk neutral probability
Denote k = 0 the riskless asset, ie. a! = 1 for every state ! , and call p0 the
0
price of that asset.
Introduce r the riskfree return r such that 1 + r = p . 1
P P 0
By the law of one price, p0 = ! q! , thus 1 = ! q! (1 + r).
P
Denoting ! = q! (1 + r), one has ! 0 and ! ! = 1, thus ! can be
interpreted as a probability: this is the risk h
neutral probability.
i
1 P ! z k = 1 E P a! z k , where P a! z k is the
One has p = 1+r !;k ! ak 1+r ! k k k k
portfolio contingent value at t = 1.
This probability has no reason to coincide with the "statistical" probability, the
di¤erence comes from the agents risk aversion. More on this soon!
! a!
k
One de…nes rk = pk the return of asset k in state ! . One has for each k,
P !
! ! rk = 1+ r , thus every return have the same risk-neutral probability,
which is the riskless return.

17. Example: Call option. The riskfree asset is a! = 1 + r with unit price, and
0
consider the stock a! with initial price S , where we suppose that the world has
1
only two states, ! = h in which case ah = S (1 + h), and ! = h in which
1
case al = S (1 + l). Suppose (1 + h) S > K > (1 + l) S , and consider the
1
+
call option c! on a1 with strike K . One has c! = a! K , so c! =
1
(1 + h) S K if ! = h, and c! = 0 if ! = l. One looks for the risk-neutral
probability ! . One has l (1 + l)+ h (1 + h) = 1+r, thus h = h l . Thus
r
l
h
the call price is C = 1+r ((1 + h) S r l
K ) = (1+r)(h l) ((1 + h) S K ).
Alternatively, one could also have looked for the value of a replicating portfolio
to hedge the value of the call. Call (z0; z1) such a portfolio; one has
z0 (1 + r) + z1 (1 + h) S = (1 + h) S K
z0 (1 + r ) + z1 (1 + l) S = 0

18. (1+l)[(1+h)S K] (1+h)S K
which solves into z0 = (1+r)(h l)
and z1 = (h l)S . By the law
of one price, the call value equals the initial value of the replicating portfolio,
r l
thus C = (1+r)(h l) ((1 + h) S K ).

19. Complete markets
De…nition. An asset class a! is complete if for every contingent payo¤ c! ,
k
there exists a portfolio z k such that c! = P a! z k .
k k
Proposition. An asset class a! is complete if and only if the rank of the matrix
k
a! is equal to the number of states . In that case, the state-price vector q!
k P
is unique, and a contingent payo¤ c! has price p = ! q! c! .
Important consequences:
1. When the markets are complete, there are at least as many assets than there
are states: K.
2. One can therefore eliminate E redundant assets which are linear com-
binations of independent assets, and suppose in practice that = K .

20. 0 1 0 1
2 0 1 1
B C B C
Example 1. Consider a = @1 2 4A ; and p = @1A. The market is
3 2 5 2
arbitrage-free, but not complete as the …rst two columns sum up to the third
one.
Example 2: options render the market complete. Consider a single asset
a = 5 2 3 with price 2. Consider the market made of a! , and call options
of strike 2 and 3, of payo¤ respectively (a! 2)+ and (a! 3)+.1
0 1 0 The market
5 2 3 1
B C B C
tablean can be written @3 0 1A, and the asset prices are @ p A. Provided
2 0 0 p0
p0 < p < 1, the new market is complete and arbitrage free.

21. Arbitrage bounds*
Proposition. Given an arbitrage-free market M = a! ; pk , de…ne PM the
k
set of probabilities wich are risk-neutral probabilities for this market.
(i) PM is a convex set.
(ii) PM is empty if and only if M o¤ers arbitrage opportunities.
(iii) PM is reduced to a point if and only if the market M is complete.
02P , 0 ? M, P 0 a!
(iv) If and M then that is ! ! k = 0 for all
k.

22. We introduce the notion of replicable claim.
De…nition. Given a contingent claim c! , one says that c! is replicable in the
market M if there exists a portfolio z such that
X
c! = z k a!
k
k
for all ! .
We use the following result to introduce the notion of arbitrage price bounds:
De…nition. For a general contingent claim c! , not necessarily replicable in the
market M , de…ne the lower and upper arbitrage price bounds of the claim as
1 1
v # (c)= # (c) and v " (c) = " (c) , where
1+r 1+r
! ]g and " = 1
# (c) = inf fE ! [c sup fE ! [c! ]g :
2PM 1 + r 2PM

23. We conclude with a result relating the notion of arbitrage price bounds and the
notion of replicability.
Theorem. The claim c! is replicable in the market M if hand only if #i(c) =
" (c). The set of No-Arbitrage prices for c! is given by v # (c) ; v " (c) .
Remark. If the claim c! is not replicable, then adding it to the market with
a price within arbitrage bounds will reduce the set PM : its dimension will
decrease by one.

24. Dynamic arbitrage
In the dynamic case, one observes the evolution of the assets over time, and one
can rebalance one’ portfolio over time. We take the time steps to be discrete.
s
Call a!jt the value of asset a at time t and in state ! , where !jt = ! 0! 1:::! t
(each time step brings on a new piece of randomness). One writes !j (t + 1) =
(!jt; ! t), and we shall suppose there is no uncertainty at date t = 0, namely
! 0 has only one value.
A strategy is the description of a portfolio in the dynamic case, allowing re-
balancing over time: formally a strategy z k (!jt) is the composition of the
investor’ portfolio at time t in the state of the world ! . Note that a strategy
s
can depend on ! 0, ! 1,..., ! t, but not on ! t+k : investors can have memory,

25. but cannot predict the future. The portfolio (liquidation) value at time t is
P k !jt
k z (!jt) ak .
P k !
An arbitrage opportunity is a strategy z such that k z (! 0 ) a k 0 0
P k !jt P k !jt
and k z (!jt 1) ak k z (!jt) ak . The interpretation is that such
strategy has a negative price and allows positive consumption
!jt X !jt !jt
cz = z k (!jt 1) ak z k (!jt) ak 0
k
at each time step.
De…nition. Markets are said to be dynamically complete if for every contingent
!jt
claim c!jt, there exists a portfolio z such that cz = c!jt.

26. Theorem. The three following conditions are equivalent:
(i) There is no arbitrage
P ! P !jt
(ii) There is a vector q!jt > 0 such that k z k (! 0) ak 0 = !jt q!jtcz ,
!jt
where cz is de…ned as above
(iii) For every !jt, there exists a vector q!t+1 (!jt) > 0 such that for all k,
!jt P !jt;! t+1
ak = ! t+1 ak q!t+1 (!jt).
Remark. One sees that q!jt and q!t+1 (!jt) are related by
q!jt+1
q!jt = q!t+1 (!jt) q!t (!jt 1) :::q!1 (!j0), and q!t+1 (!jt) = q .
!jt

27. Reference for the course
Campbell, J., Viceira, L. Strategic Assets Allocation, Oxford.
Demange, G., Laroque, G., Finance et Economie de l’Incertain, Economica.
Ingersoll, J., Theory of Financial Decision Making, Rowman & Little…eld.
Mas-Colell, Whinston, Green, Microeconomic Theory, Oxford.
Ross, S., Neoclassical Finance, Princeton.
Thank you!