Capital markets and portfolio theory (2000)

4 1.B Efficient portfolio in absence of a risk-free asset.. 1.B.ii Efficient portfolio and risk averse investorsDeÞne another optimization program P0, equivalent to P : P and P0 yield the

Roland Portait From the class notes taken by Peng Cheng

Novembre 2000

Table of Contents

PART I Standard (One Period) Portfolio Theory 1

1 Portfolio Choices 2

1.A Framework and notations 2

1.A.i No Risk-free Asset 2

1.A.ii With Risk-free Asset 4

1.B Efficient portfolio in absence of a risk-free asset 6

1.B.i Efficiency criteria 6

1.B.ii Efficient portfolio and risk averse investors 8

1.B.iii Efficient set 9

1.B.iv Two funds separation (Black) 10

1.C Efficient portfolio with a risk-free asset 11

1.D HARA preferences and Cass-Stiglitz 2 fund separation 14

1.D.i HARA (Hyperbolic Absolute Risk Aversion) 14

1.D.ii Cass and Stiglitz separation 15

2 Capital Market Equilibrium 17

2.A CAPM 17

2.A.i The Model 17

2.A.ii Geometry 19

2.A.iii CAPM as a Pricing and Equilibrium Model 19

2.A.iv Testing the CAPM 21

2.B Factor Models and APT 21

2.B.i K-factor models 21

2.B.ii APT 22

2.B.iii Arbitrage and Equilibrium 24

2.B.iv References 25

PART II Multiperiod Capital Market Theory : the Probabilistic Approach 26

3 Framework 27

3.A Probability Space and Information 27

3.B Asset Prices 28

3.B.i DeÞnitions and Notations 28

3.C Portfolio Strategies 29

3.C.i Notation: 29

3.C.ii Discrete Time 29

3.C.iii Continuous Time 30

4 AoA, Attainability and Completeness 32

4.A DeÞnitions 32

4.B Propositions on AoA and Completeness 35

4.B.i Correspondance between Q and Π : Main Results 35

4.B.ii Extensions 38

5 Alternative SpeciÞcations of Asset Prices 39

5.A Ito Process 39

5.B Diffusions 40

5.C Diffusion state variables 41

5.D Theory in the Ito-Diffusion Case 41

5.D.i Framework 41

5.D.ii Martingales 42

5.D.iii Redundancy and Completeness 42

5.D.iv Criteria for Recognizing a Complete Market 44

PART III State Variables Models: the PDE Approach 45

6 Framework 46

7 Discounting Under Uncertainty 48

7.A Ito’s lemma and the Dynkin Operator 48

7.B The Feynman-Kac Theorem 48

8 The PDE Approach 50

8.A Continuous Time APT 50

8.A.i Alternative decompositions of a return 50

8.A.ii The APT Model (continuous time version) 51

8.B One Factor Interest Rate Models 53

8.C Discounting Under Uncertainty 53

9 Links Between Probabilistic and PDE Approaches 55

9.A Probability Changes and the Radon-Nikodym Derivative 55

9.B Girsanov Theorem 56

9.C Risk Adjusted Drifts: Application of Girsanov Theorem 56

PART IV The Numeraire Approach 59

10 Introduction 60

11 Numeraire and Probability Changes 61

11.A Framework 61

11.A.i Assets 61

11.A.ii Numeraires 61

11.B Correspondence Between Numeraires and Martingale Probabilities 62 11.B.i Numeraire → Martingale Probabilities 62

11.B.ii Probability → Numeraire 63

11.C Summary 63

12 The Numeraire (Growth Optimal) Portfolio 65

12.A DeÞnition and Characterization 65

12.A.i DeÞnition of the Numeraire (h, H) 65

12.A.ii Characterization and Composition of (h, H) 65

12.A.iii The Numeraire Portfolio and Radon-Nikodym Derivatives 69

12.B First Applications 69

12.B.i CAPM 70

12.B.ii Valuation 70

PART V Continuous Time Portfolio Optimization 72

13 Dynamic Consumption and Portfolio Choices (The Merton Model) 73

13.A Framework 73

13.A.i The Capital Market 73

13.A.ii The Investors (Consumers)’ Problem 74

13.B The Solution 74

13.B.i Sketch of the Method 74

13.B.ii Optimal portfolios and L + 2 funds separation 77

13.B.iii Intertemporal CAPM 78

14 THE ”EQUIVALENT” STATIC PROBLEM (Cox-Huang, Karatzas approach) 80

14.A Transforming the dynamic into a static problem 80

14.A.i The pure portfolio problem 80

14.A.ii The consumption-portfolio problem 82

14.B The solution in the case of complete markets 83

14.B.i Solution of the pure portfolio problem 83

14.B.ii Examples of speciÞc utility functions 85

14.B.iii Solution of the consumption-portfolio problem 86

14.B.iv General method for obtaining the optimal strategy x ∗∗ 87

14.C Equilibrium: the consumption based CAPM 88

PART VI STRATEGIC ASSET ALLOCATION 90

15 The problems 91

16 The optimal terminal wealth in the CRRA, mean-variance

and HARA cases 92

16.A Optimal wealth and strong 2 fund separation 92

16.B The minimum norm return 92

17 Optimal dynamic strategies for HARA utilities in two cases 93

17.A The GBM case 93

17.B Vasicek stochastic rates with stock trading 93

18 Assessing the theoretical grounds of the popular advice 94

18.A The bond/stock allocation puzzle 94

18.B The conventional wisdom 94

PART I Standard (One Period)

Portfolio Theory

Chapter 1

Portfolio Choices

In all the following we consider a single period or time interval (0 1), hence twoinstants t = 0 and t = 1

Consider an asset whose price is S(t) (no dividends or dividends reinvested).The return of this asset between two points in time (t = 0, 1) is:

R = S (1)− S (0)

S (0)

We now consider the case of a portfolio and distinguish the case where ariskless asset does not exist from the case where a risk free asset is traded

1.A.i No Risk-free Asset

There are N tradable risky assets noted i = 1, , N :

• The price of asset i is Si(t), t = 0, 1

• The return of asset i is

Ri = Si(1)− Si(0)

Si(0)

• The number of units of asset i in the portfolio is ni The portfolio is described

by the vector n(t); ni can be >0 (long position) or <0 (short position)

• Then the value of the portfolio, denoted by X (t), is

where x= (x1, x2, , xN)0 and 1 is the unit vector

• The return of the portfolio is the weighted average of the returns of itscomponents:

RX = x0R

• DeÞne µi = E [Ri] and µ= (µ1, µ2, , µN)0, then:

1.A.ii With Risk-free Asset

We now have N +1 assets, with asset 0 being the risk-free asset, and the remaining

N assets being the risky assets

• S0(1) = S0(0)· (1 + r) with r a deterministic interest rate.

• Again we can deÞne the portfolio in units, with n= (n0, n1, n2, , nN)0

• The portfolio can be similarly deÞned in weights:

xi = niS (0)

X (0)for the N risky assets (i = 1, 2, , N ), and

• The return of the portfolio is:

µX = E (RX) = r + x0πwhere π is the risk premium vector of the E (Ri− r)

• Also denote ΓN ×N as the variance-covariance matrix of the risky assets, then:

i=1 λiRi= k Q.E.D.

Remark 1 In the following sections we will assume that the assets are non-redundant (it is always possible to “drop” redundant assets if any).

1.B Efficient portfolio in absence of a risk-free asset

1.B.i Efficiency criteria

DeÞnition 2 Portfolio (x∗, X∗) is efficient if ∀y, σ Y < σ X ∗ ⇒ µ Y < µX∗ and σ Y =

σ X ∗ ⇒ µ Y ≤ µ X ∗

Consider any efficient portfolio (x∗, X∗) and let variance(RX) = k

x∗ solves the optimization program (P ) :

max

x E [RX] s.t x0Γx = k ; x01= 1The Lagrangian is:

Remark that these Þrst order conditions are necessary and also sufficient for thesolution being a maximum since the second order conditions hold (L(x) is strictlyconcave -Γ positive deÞnite).

Remark 2 The second term can be considered as the additional required rate of return (risk premium), proportional to cov (R i , R X ).

Remark 3 If cov (R i , R X ) = 0, then µi= λ.

Remark 4 Also note:

of risk for any asset i embedded in the portfolio X.

1.B.ii Efficient portfolio and risk averse investors

DeÞne another optimization program (P0), equivalent to (P ) :

( (P ) and (P0) yield the same solutions since they have the same Lagrangian)(P0) writes, equivalently:

The Þrst order conditions of (P’) write as for (P): µ − θΓx∗−λ1 = 0 (with onlyone multiplier for (P0)

• Consider the case of minimum variance portfolio where θ = ∞, i.e

Γk1 − λ1 = 0Together with the constraint k011= 1 gives:

10Γ−11Thus:

k1 = λΓ−11

10Γ−11· Γ−11

1.B.iii Efficient set

DeÞnition 3 The Efficient Set is the set of all x ∗ that obey the Þrst order condition alently, it is the set of all x ∗ that solve the optimization program (P 0 ) ∀θ ≥ 0.

Equiv-Recall that the Þrst order condition for (P0) is:

µ− θΓx∗−λ1 = 0DeÞne risk tolerance bθ as the inverse of risk aversion, i.e

bθ = 1θThen x∗ can be solved as:

bθ10Γ−1µ− bθλ10Γ−11= 1or:

bθ10Γ−1µ− bθλ10Γ−11= bθθThis solves for λ:

λ = 1

0Γ−1µ−θ

10Γ−11Then:

We recognize in the Þrst term the minimum variance portfolio (k1) and we call

k2 the second term:

21= 0 Any efficient portfolio is thusthe sum of k1 (the minimum variance portfolio) and k2 which is a zero weight(zero investment) portfolio As it could be expected, an investor with a zero risktolerance will hold only k1; If he has a positive risk tolerance bθ he will add a risktaking the form bθk2 in order to increase the expected return The efficient set cannow be caracterized as:

ES = n

x∗|x∗ = k1+ bθk2 ∀bθ > 0oSince the expected return x∗0µis linear in bθ and the variance is quadratic in bθ, inthe (σ2, R) space the efficient portfolios are represented by the efficient frontier,which is a parabola Each point on the efficient frontier corresponds to a given θ,the slope of the parabola at this point being equal to θ2 (the shadow price in (P )

of the constraint on variance)

In the (σ, R) space the efficient frontier is an hyperbola

1.B.iv Two funds separation (Black)

Theorem 2

Consider any two efficient portfolio x and y:

1 Any convex combination of x and y is efficient, i.e.∀ u ∈ [0, 1] , ux+ (1 − u) y ∈ES

2 Any efficient portfolio is a combination of x and y (not necessarily a convexcombination)

3 The whole parabola (efficient and inefficient frontier) is generated by (all)combinations of x and y

• Since x∈ ES and y∈ ES, for some positive bθX and bθY , we have:

x = k1+ bθXk2

y = k1+ bθYk2Let z = ux + (1 − u)y, then:

• Let z∈ ES, then z = k1 + bθZk2 for some bθZ > 0 For any x∈ ES and

1.C Efficient portfolio with a risk-free asset

Consider Þgure 1 where the upper branch of the hyperbola EFR represents, in the(σ, E) space, the efficient portfolios in absence of a riskless asset Assume now thatexists a risk free asset 0 yielding the certain return r M stands for the tangencypoint of the hyperbola EFR with a straight line drown from r representing asset 0.Point M represents a portfolio composed only of risky assets, called the tangentportfolio

• Efficient frontier in presence of a riskless asset

Proposition 2

1 Asset 0 is efficient

2 Consider any portfolio X Any combination of 0 and X yielding

R = uRX+ (1− u) r, lies on the straight line connecting 0 and X in the (σ, E)space

3 Any feasible portfolio which representative point is not on r − M (such as X)

is dominated by portfolios in r − M The straight line r − M is the efficientfrontier and is called the Capital Market Line

4 (Tobin’s Two-fund Separation) Any efficient portfolio is a combination of anytwo efficient portfolios, for instance 0 and M

5 Any efficient portfolio writes:

µ − r1 = θΓx∗Then:

µ − r1 ¢

Also: m 0 1 = 1, then:

1 0 Γ −1 ¡

µ − r1 ¢

Q.E.D.

Remark 5 Given a risk tolerance b θ:

• bθ < bθM, the portfolio is long in 0 and m

• bθ > bθM, the portfolio shorts 0

Remark 6 We deÞne later the market portfolio as a portfolio containing all the risky assets present in the market (and only risky assets) In absence of riskless asset the market portfolio is efficient iif its representative point belongs to the hyperbola EFR In presence of a risk free asset the necessary and sufficient condition for the market portfolio to be efficient is that it coincides with the tangent portfolio m (which is the only efficient portfolio of EFR, in presence of a risk free asset) Would all investors face the same efficient frontier (it would be the case under homogeneous expectations and horizon) and would they all follow the mean-variance criteria, they would all hold combinations of 0 and M and the tangent portfolio M would necessarily coincide with the market portfolio.

1.D HARA preferences and Cass-Stiglitz 2 fund separation

A rational agent (in the sense of Von Neumann-Morgenstern) should maximizethe expected utility of wealth E [U (W )]

1.D.i HARA (Hyperbolic Absolute Risk Aversion)

A utility function U (W ) belongs to HARA class if it writes:

U (W ) = γ

1− γ

·

bθ + Wγ

and the relative risk tolerance (RRT) is:

RRT = bθ

1γ

We obtain CRRA, i.e constant relative risk aversion

A limit case of CRRA is obtained for γ = 1 which can be showed to beequivalent to the Log utility

U (W ) = W −W

2

2bθi.e the quadratic utility function

3 Using a quadratic utility function implies a mean-variance criteria; Indeed:

min var (RX) s.t E [RX] = bE (and x01= 1)

4 Three undesirable features of the quadratic utility:

— Saturation at W = b θ (for that wealth U (W ) = W −W2bθ2 is maximum; U (W ) decreases for W > b θ!)

— ARA increasing with wealth (it is commonly admitted that ARA decreases for most agents).

— Indifference to skewness (only the two Þrst moments of W matter), whereas most investors actually like skewness.

1.D.ii Cass and Stiglitz separation

Cass and Stiglitz showed that all HARA investors sharing the same exponential

parameter γ can build their optimal portfolios by mixing the two same funds.When a risk free asset exists it can be chosen as one of the two funds Since allquadratic (mean-variance) investors exhibit the same γ(= −1) Tobin and Black

2 fund separation are particular cases of Cass and Stiglitz separation Cass andStiglitz conditions on the utility functions for separation to hold for investorssharing the same exponential parameter are summarized in the following table

@r (under complete markets ∃ r) quadratic or CRRA2

2

in the particular case of CRRA one fund suffices (for a given γ the portfolio is the same for all W

Chapter 2

Capital Market Equilibrium

2.A.i The Model

Consider again N risky assets (a risk free asset may exist or not) The marketvalue of asset i is Vi, then (by deÞnition of the market portfolio) it’s weight in themarket portfolio is:

mi = PNVi

i=1Vi

The return of the market portfolio is:

RM = m0R

Hypothesis 1 (H) : The market portfolio M is efficient.

Remark 7 The market portfolio would be efficient if all investors would hold efficient folios (since a combination of efficient portfolios is efficient).

2 Conversely, if there exist θ and λ such that, for i = 1, , N : µi =

λ + θcov (RM, Ri), then (H) is true

The proof comes directly from Theorem 1.

Q.E.D.

Remark 8 θ can be interpreted as the risk aversion of the average (representative) investor.

Remark 9 CAPM holds for any portfolio (x, X).

Indeed, call RX its return and consider the case where no risk free asset exists(x01= 1) :

Remark 11 λ and θ are the same for all assets or portfolios

Remark 12 For the market portfolio:

µM = λ + θcov (RM, RM)

= λ + θσ2MTherefore:

θ = µM− λ

σ 2 M

¸ cov (RM, Ri)

βi= cov (RM, Ri)

σ 2 M

Then we may write the CAPM equation in the alternative form:

Corollary 1 (0−beta CAPM) If M is efficient, for any zero beta portfolio or asset Z: E [R i ] =

µZ+ βi(µM− µ Z )

Corollary 2 (Standard CAPM) : If there exists a risk-free asset yielding r (which is a ticular zero beta asset)

par-E [R i ] = r + βi(µM− r) Note that µZ= r for any zero beta portfolio or asset.

2.A.ii Geometry

missing

2.A.iii CAPM as a Pricing and Equilibrium Model

• For a security delivering eV (1) at time 1(the pdf of eV (1) is given, thusE( eV (1)) and cov( eV (1) , RM) are known), what is its price V (0) at time 0?

Let’s assume that there exists a risk-free asset, then:

Ehe

V (1)i

V (0) = E [1 + R] = 1 + r + θcov

Ãe

Then:

Ehe

V (0) =

Ehe

at the risk-free rate

However this asset may be an element of the market portfolio M (unless thisclaim is in zero net supply ) and therefore the previous pricing formula isnot a closed form general equilibrium relation

• In fact CAPM is an equilibrium condition stemming from the demand side;The equilibrium price can only be otained by specifying the supply side (inthe previous example the supply was a right on an exogeneous cash ßow X).General equilibrium requires a speciÞcation of the supply of all securitiestraded in the market

— Consider the N risky assets together and we look for their equilibrium prices We assume Þrst an inelastic supply Assume that asset i delivers e V i (1), an exogenous cash ßow, at time 1, what is its price at time 0?

E h e

¸ cov

à e

¸ cov (R M , R i ) and

1 + RM =

P N i=1 Vi(0) · (1 + R i )

P N i=1 Vi(0)

2.A.iv Testing the CAPM

One remark about this important empirical topic

Testing the CAPM is equivalent to testing (H) However, how should we deÞnethe market portfolio and how to measure the market return?

Usually the market portfolio is proxied by stock (plus bond) indices But results

on stock indices do not include all assets in M (non tradable assets, art, ) Hence

we test the efficiency of the index and not that of M (Roll’s Critique)

2.B.i K-factor models

Hypothesis 2 There exist K factors, F k , k = 1, 2, K with

with β k being the kthrow of β.

• In practice, we should have large N and small K, so that in estimating thevariance-covariance matrix,

we only need to estimate K terms of σ2

k and run N regressions for estimatingthe βik

• In CAPM or in the Markowitz model, without the factor decomposition, weneed to estimate N (N − 1) /2 terms

• A Particular case: K = 1 boils down into the market model that writes:

We assume that the returns are generated by a K factors linear process previouslydeÞned that writes:

• λ0 is the required rate of return without systematic risk.

• λk is the market price of risk k

• λkβik is the risk premium imposed to security i because it has a risk k ofintensity βik

Consider any well-diversiÞed zero investment portfolio satisfying:

x01 = 0 or x⊥ 1

x0βk = 0 or x⊥βk for k = 1, , K hence:

x is any element of h

vect ³ 1,β

1 , β

2 , , β

K

´i⊥Also x 0 ² = 0 (since it is well diversiÞed); Then:

µ is orthogonal to any element x of [vect(1, β1, β2, , βK)] ⊥ , i.e.

µ ∈ vect(1, β 1 , β2, , βK) implying that exist K + 1 scalars such that : µ = λ 0 1 + λ 1 β1+ + λ K βK

Q.E.D.

• In the particular case where there is a risk-free asset, then:

µ0 = λ0 = rand

µi = r + λ1βi1+ + λKβiK

2.B.iii Arbitrage and Equilibrium

• Equilibrium implies AoA, but the inverse is not true

• AoA conditions do not involve utility functions

2.B.iv References

Dumas-Allaz, 1995 ; Demange-Rochet, 1992

PART II

Multiperiod Capital Market Theory : the Probabilistic Approach

Chapter 3

Framework

3.A Probability Space and Information

We consider the usual probability triplet (Ω, F, P ), where F is a σ-algebra on Ωrepresenting the observable events at time T

Information in the period [0, T ] is represented by a Þltration {Ft}t∈[0,T ], where Ft

is the set of observable events at time t (represented by a σ−algebra), and thesequence {Ft}t∈[0,T ] satisÞes the ”usual” conditions:

F0 = {null events and a.s event}

In the discrete time setting, all transactions take place at discrete points, i.e.,

t = 1, 2, , T In the continuous time setting, transactions take place continuously,i.e., t ∈ [0, T ]

We assume a frictionless market, continuously open in the continuous time work

frame-3.B Asset Prices

3.B.i DeÞnitions and Notations

There are N + 1 assets traded in the market, one being the locally risk-free set, denoted by 0, and the remaining N being the risky assets The prices ofthose assets are noted Si(t) ( for i = 0, 1, , N ); S(t) = (S1(t), , SN(t))0 or(S0(t), S1(t), , SN(t))0 (depending on the context) is the N (or N + 1) dimen-sional column vector of asset prices Without loss of generality it will generally

2 In the continuous time context:

• r(t) is stochastic but Ft-adapted

For a risk-free asset:

dS0 = S0rdtor

S0(t) = eR0tr(u)du

with S0(0) = 1

• For a risky asset we will usually assume that prices follow Ito processes:

dSi = Siµidt + Siσi0dwwith risk induced by w, the vector of standard Brownian Motions

Technical conditions (e.g., the integrability conditions) apply

If Si follows Ito process, we preclude jumps If jumps are involved, however, then

a rather general assumtion is that Si follows a semi-martingale process A slightlymore speciÞc assumption is that asset prices follow processes that yield a.s RightContinuous and Left Limited (RCLL) paths When considering the possibility of

jumps we will assume RCLL processes for the asset prices to avoid the soÞstication

of semi martingales3

• It is worthwhile to note that Ito processes ⊂ RCLL ⊂ Semi − martingales

• Most of the results of the next chapter (On AOA and completeness) hold inthe semi-martingale case

3.C Portfolio Strategies

3.C.i Notation:

weights on risky assets

• S(N +1)×1 the vector of the N+1 asset prices

• X (t) = n0(t)S(t) the value of the portfolio at t

• (n,X) or (x,X) a strategy

3.C.ii Discrete Time

[t− 1, t[ is period t − 1;at time t S(t) is set and, just after, n(t) is choosen

• During period t − 1, the value of the portfolio will evolve:

Consider the integral: R

φ (u) dS In a regular integral of this form dS is inÞnitesmal, while in a jump process it can assume some Þnite value somewhere.

The second term can be deemed as the net cash inßow added to the portfolio attime t Indeed it can be decomposed into two terms: −S0(t)n(t− 1), the value ofassets sold at time t, and S0(t)n(t), the algebric value of assets purchased (may

DeÞnition 6 (Self-Þnancing Portfolio) When at each time t the net inßow is 0, the strategy

is said to be self-Þnancing, i.e., if (n,X) is self-Þnancing, then:

3.C.iii Continuous Time

• The gain process in [t, t + dt) is deÞned as:

dG(t) = n0(t)dS(t)and

G(t) =

Z t 0

dG(u) =

Z t 0

n0(u)dS(u)

• The change of the portfolio value is found to be:

• Again, in a self-Þnancing strategy: dX(t) = dG(t), and X(t) = X(0) + G(t);

As in the discrete time case, the self Þnancing property as well as theexpression of the gain do not depend on the choosen numeraire

Chapter 4

AoA, Attainability and

Completeness

DeÞnition 7 strategy (n,X) is admissible if:

1 n(t) is Ft adapted and satisÞes some technical conditions4

2 X(t)∈ L1,2

3 (This is an additional condition imposed sometimes) X(t) is bounded frombelow to avoid doubling Strategies5

DeÞnition 8 A is the set of admissible strategies

DeÞnition 9 A 0 = { Self-Þnancing and admissible strategies}

We now work with A0, i.e., ∀ (n,X) ∈ A0, dX = n0dS

5

In a Doubling Strategy the gambler bets 2 when losing 1 and bets 4 when losing 2

It is also possible to deÞne a strategy by a vector of weights xN ×1 The weight ofthe risk-free asset in the portfolio is then 1 − x01.

DeÞnition 10 (a,A) is an arbitrage if:

1 (a,A) ∈ A0

2 A(0) = n0(0)S(0) =0, (i.e., zero initial investment)

3 A(T )≥ 0 a.s (i.e., non-negative cash ßow at the end)

4 E [A(T )|F0] > 0

There is an arbitrage opportunity each time that a strategy (x, X) in A0 dominatesanother strategy (y, Y ) in A0 (i.e X(T ) ≥ Y (T ) a.s and E[X(T )] ≥ E[Y (T )] forthe same initial investment X(0) = Y (0); or X(T ) = Y (T ) a.s.with X(0) < Y (0)).Arbitrage is built by being long in (x, X) and short in (y, Y )

Example 1 X(T ) ≥ S 0 (T ) = eR0Tr(u)du a.s.; E(X(T ) − S 0 (T )) > 0 and X(0) = 1

Example 2 X(T ) = K, a constant, while X(0) < KB T (0) where B T (0) denotes the value

at time 0 of a zero-coupon bond yielding 1 at time T.

The previous considerations imply:

with X(T ) = Y (T ) a.s., then in AoA: X(0) = Y (0).

DeÞnition 11 C eT is a contingent claim if

1 CeT is FT measurable.

2 CeT ∈ L1,2 (Þnite mean and variance)

3 (goes with hypothesis on admissible strategies) eCT is bounded from below.DeÞnition 12 C , {the set of contingent claims}

Example 3 The terminal values of N + 1 primitive assets are contingent claims.

Example 4 ∀A ∈ F T , the indicator function 1A is a contingent claim.

DeÞnition 13 C e T ∈ C is attainable if ∃ (c,C) ∈ A 0 with C(T ) = e C T a.s We say e C T is attained by (c,C) or (c,C) yields e C T

DeÞnition 14 Ca= {attainable contingent claims}

DeÞnition 15 C n = {non-attainable contingent claims}

DeÞnition 16 The market is (dynamically) complete when all contingent claims are able, i.e., C a = C or C n = ∅.

attain-Remark 13 Market completeness is unrealistic in discrete time, but less unrealistic in tinuous time In continuous time the possibility of rebalancing at each point of time allows

con-a much lcon-arger spcon-anning When completeness is obtcon-ained through continuous rebcon-alcon-ancing, the market is said “dynamically” complete.

DeÞnition 17 A pricing formula π maps C onto R To be viable, π must satisfy:

1 π is linear, i.e., ∀λ1, λ2, eCT ∈ C, and eC0

CT

´

= X(0)DeÞnition 18 Π = {π|π viable}