ECO568 - Uncertainty and Financial Markets

Arthur Charpentier and Alfred GalichonEcole polytechnique, January 6 2009

1. No-Arbitrage and the Law of One Price

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 themarket?

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 lotterytickets

� Rational beliefs: agents�subjective probabilities = objective probabilities

(stationarity; no learning, no insider information, no overcon�dence)

General Introduction (3)

S. Ross�"three stylized facts" about �nancial markets:

� E¢ cient Markets. Asset returns are unpredictable. Then what is thiscourse useful for?

� Risk Premium. Risky assets have higher expected returns (ie. Lowerprices) 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.

Course outline

Lecture 1 (Jan 6): No Arbitrage and the Law of One PriceLecture 2 (Jan 13): The Demand for RiskLecture 3 (Jan 20): Investment Theory (Static case): MVA, CAPMLecture 4 (Jan 21): Informational aspectsLecture 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 ModelsLecture 8 (March 10): Empirical Puzzles in Asset PricingLecture 9 (March 17): Risk management and Financial bubblesExam: March 24

Course webpage: http://groups.google.com/group/x-eco568.

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 fromthe Scolarité. Students are advised to look at it, but the lecture slides areself-contained.

Oral participation is strongly encouraged!

Today�s lecture outline

� General introduction

� No-Arbitrage and the Law of One Price

� Section: Arbitrage Pricing Theory, Cox-Ross-Rubinstein

The setting

� contingent states at time t = 1: ! 2 f1; :::;g.Examples. ! 2 f0; 1g (no accident/accident); ! value the stock marketat time t = 1.

� K assets available at time t = 0: k 2 f1; :::;Kg.the tableau a!k is the value of asset k in contingent state !, pk price ofasset k at time t = 0. A market is the data

�a!k ; pk

�.

� Portfolio: combination of assets, zk quantity of asset k. Portfolio price(t = 0):

Pk pkz

k. Portfolio value (t = 1):Pk a!kzk.

Example. Consider the following tableau, where lines correspond to assets,and columns to states

a =

0 1 21 1 0

!; p =

�31

�what is the value of portfolio of 5 assets k = 1 and 2 asset k = 2 in the state! = 2?

Arbitrage opportunityDe�nition. An arbitrage portfolio is a portfolio zk such thatPk a!kzk � 0 for all ! and

Pk pkz

k � 0, at least one of these E + 1inequalities being strict.A market is arbitrage free if there is no arbitrage portfolio.

Proposition. A market�a!k ; pk

�without arbitrage opportunity satis�es the

Law of One Price:if for two assets k and l, a!k = a!l for every state !, then the assets have thesame price: pk = pl.�

Example. Consider a =

0B@2 0 11 2 43 2 5

1CA ; and p =0B@212

1CA. The portfolio z =(�1;�1; 1) is an arbitrage portfolio, as Pk a!kzk = 0 for every !, andPk pkz

k = �1.

State-price vectors and theNo-Arbitrage theorem

The No-Arbitrage Theorem. A market�a!k ; pk

�is arbitrage-free if and only

if there is a vector q! such that:- q! > 0 for every state !- pk =

P! a

!k q! for every asset k

q! is then called a state-price vector.�

Remark. 1. Although q! is in general not unique, this does not contradictthe law of one price, as every state-price vector q! yields the same set of assetprices.2. This notion does not involve the "statistical" probability per se (apart fromits support).

Example. Consider a =

0B@2 0 11 2 43 2 5

1CA ; and p =0B@112

1CA. The market is arbitrage-free, and q = (3=7; 0; 1=7) is a state-price vector.

Sketch of the proof of the No-Arbitrage theorem. ( is easy:Pk pkz

k =Pk;!

�q!a!k

�zk =

Pk;! q!

�a!kz

k�, thus

Pk a!kzk � 0 implies

Pk pkz

k �0.

For the converse, we note that the investor is looking for the portfolio withminimal cost which gets him a positive revenue with certainty in every state ofthe world. The investor solves V = infz

Pk zkpk s.t.

Pk zka!k � 0, and �V

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 ifV = 0.Now, write the Lagrangian V = infz supq�0

Pk zkpk �

P!;k z

ka!k q!, where

q! are the Lagrange multipliers associated to the constraints. But by duality,one can invert the inf and the sup, and V = supq�0 infz

Pk zk�pk �

P! a

!k q!

�.

Therefore there is No arbitrage opportunity if and only if there is a vector qsuch that

infz

Xk

zk pk �

X!a!k q!

!= 0

henceP! q!a

!k = pk for all k. It remains to explain why one can choose

q! > 0. Remember, we have introduced the q! as Lagrange multipliers inthe investor�s constrained optimization problem. The null portfolio z = 0 is asolution to the constrained optimization problem which has all the constraintssaturated. By a well-known result on linear programming, all the correspondingLagrange multipliers q! can be chosen strictly positive. �

Remark: Game-theoretic interpretation. The proof has a game-theoreticinterpretation as a game between Investor and the Market, which sets the state

prices in order to minimize the Investor�s gain. The Market�s strategy is tochoose a state-price vector q!, while the Investor�s strategy is to pick up hisinvestment portfolio zk.The value of this game (for the Market) is then:

V = infzsupq�0

Xk

zk pk �

X!a!k q!

!

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 , theopposite og the value of the game for the Market. The duality principle isequivalent to a min-max theorem, which precisely says that the value of thisgame will be the same regardless whom (of the Investor or the Market) plays�rst.

Consequence: Arrow-Debreu prices

De�nition. An Arrow-Debreu asset is an asset yielding 1 in state !0, and 0otherwise. 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, thenits price equals q!0.�

Example: Digital options, call options. Suppose a stock price a! can takeup to a �nite number of values v1; :::; vN . Then the digital option of strike vkis the option d!vk = 1fa!=vkg. This is precisely an Arrow-Debreu price.

Risk neutral probabilityDenote k = 0 the riskless asset, ie. a!0 = 1 for every state !, and call p0 theprice of that asset.Introduce r the riskfree return r such that 1 + r = 1

p0.

By the law of one price, p0 =P! q!, thus 1 =

P! q! (1 + r).

Denoting ��! = q! (1 + r), one has ��! � 0 and

P! �

�! = 1, thus �

�! can be

interpreted as a probability: this is the risk neutral probability.One has p = 1

1+r

P!;k �

�!a!kzk = 1

1+rE�!

hPk a!kzki, where

Pk a!kzk is the

portfolio contingent value at t = 1.This probability has no reason to coincide with the "statistical" probability, thedi¤erence comes from the agents risk aversion. More on this soon!

One de�nes r!k =a!kpkthe return of asset k in state !. One has for each k,P

! �!r!k = 1+r, thus every return have the same risk-neutral probability,

which is the riskless return.

Example: Call option. The riskfree asset is a!0 = 1 + r with unit price, andconsider the stock a!1 with initial price S, where we suppose that the world hasonly two states, ! = h in which case ah1 = S (1 + h), and ! = h in whichcase al1 = S (1 + l). Suppose (1 + h)S > K > (1 + l)S, and consider the

call option c! on a1 with strike K. One has c! =�a!1 �K

�+, so c! =

(1 + h)S �K if ! = h, and c! = 0 if ! = l. One looks for the risk-neutralprobability ��!. One has �

�l (1 + l)+�

�h (1 + h) = 1+r, thus �

�h =

r�lh�l. Thus

the call price is C =��h1+r ((1 + h)S �K) =

r�l(1+r)(h�l) ((1 + h)S �K).

Alternatively, one could also have looked for the value of a replicating portfolioto hedge the value of the call. Call (z0; z1) such a portfolio; one has

z0 (1 + r) + z1 (1 + h)S = (1 + h)S �Kz0 (1 + r) + z1 (1 + l)S = 0

which solves into z0 = �(1+l)[(1+h)S�K](1+r)(h�l) and z1 =

(1+h)S�K(h�l)S . By the law

of one price, the call value equals the initial value of the replicating portfolio,thus C = r�l

(1+r)(h�l) ((1 + h)S �K).

Complete markets

De�nition. An asset class a!k is complete if for every contingent payo¤ c!,

there exists a portfolio zk such that c! =Pk a!kzk.

Proposition. An asset class a!k is complete if and only if the rank of the matrixa!k is equal to the number of states . In that case, the state-price vector q!is unique, and a contingent payo¤ c! has price p =

P! q!c

!.�Important consequences:1. When the markets are complete, there are at least as many assets than thereare states: � K.2. One can therefore eliminate � E redundant assets which are linear com-binations of independent assets, and suppose in practice that = K.

Example 1. Consider a =

0B@2 0 11 2 43 2 5

1CA ; and p =

0B@112

1CA. The market is

arbitrage-free, but not complete as the �rst two columns sum up to the thirdone.

Example 2: options render the market complete. Consider a single asseta =

�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)+. The market

tablean can be written

0B@5 2 33 0 12 0 0

1CA, and the asset prices are0B@1pp0

1CA. Providedp0 < p < 1, the new market is complete and arbitrage free.

Arbitrage bounds*

Proposition. Given an arbitrage-free market M =�a!k ; pk

�, de�ne PM the

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.

(iv) If � and �0 2 PM , then �� �0 ?M , that isP��! � �0!

�a!k = 0 for all

k.�

We introduce the notion of replicable claim.De�nition. Given a contingent claim c!, one says that c! is replicable in themarket M if there exists a portfolio z such that

c! =Xk

zka!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 themarket M , de�ne the lower and upper arbitrage price bounds of the claim as

v# (c) =1

1 + r�# (c) and v" (c) =

1

1 + r�" (c) , where

�# (c) = inf�2PM

fE�! [c!]g and �" =1

1 + rsup�2PM

fE�! [c!]g :

We conclude with a result relating the notion of arbitrage price bounds and thenotion of replicability.Theorem. The claim c! is replicable in the market M if and only if �# (c) =�" (c). The set of No-Arbitrage prices for c! is given by

hv# (c) ; v" (c)

i.�

Remark. If the claim c! is not replicable, then adding it to the market witha price within arbitrage bounds will reduce the set PM : its dimension willdecrease by one.

Dynamic arbitrage

In the dynamic case, one observes the evolution of the assets over time, and onecan rebalance one�s portfolio over time. We take the time steps to be discrete.

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 zk (!jt) is the composition of theinvestor�s portfolio at time t in the state of the world !. Note that a strategycan depend on !0, !1,..., !t, but not on !t+k: investors can have memory,

but cannot predict the future. The portfolio (liquidation) value at time t isPk zk (!jt) a!jtk .

An arbitrage opportunity is a strategy z such thatPk zk (!0) a

!0k � 0

andPk zk (!jt� 1) a!jtk � P

k zk (!jt) a!jtk . The interpretation is that such

strategy has a negative price and allows positive consumption

c!jtz =

Xk

�zk (!jt� 1) a!jtk � zk (!jt) a!jtk

�� 0

at each time step.

De�nition. Markets are said to be dynamically complete if for every contingentclaim c!jt, there exists a portfolio z such that c!jtz = c!jt.

Theorem. The three following conditions are equivalent:(i) There is no arbitrage

(ii) There is a vector q!jt > 0 such thatPk zk (!0) a

!0k =

P!jt q!jtc

!jtz ,

where c!jtz is de�ned as above(iii) For every !jt, there exists a vector q!t+1 (!jt) > 0 such that for all k,

a!jtk =

P!t+1 a

!jt;!t+1k q!t+1 (!jt).�

Remark. One sees that q!jt and q!t+1 (!jt) are related byq!jt = q!t+1 (!jt) q!t (!jt� 1) :::q!1 (!j0), and q!t+1 (!jt) =

q!jt+1q!jt

.

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!