Sublinear and Locally Sublinear Prices

Universita Commerciale “Luigi Bocconi”

Facolta di Economia

Master of Science in Finance

Sublinear and Locally Sublinear Prices

Relatore

Prof. Erio Castagnoli

Controrelatore

Prof. Fabio Maccheroni

Candidato

Alessandro Plasmati1185198

Anno Accademico 2007/2008

Contents

1 Introduction 5

2 Linear Functionals 62.1 Criticism of Linear Prices . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Sublinear Functionals 73.1 Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3.1.1 Hahn-Banach Theorem . . . . . . . . . . . . . . . . . . . . . 73.1.2 General Representation of Sublinear Functionals . . . . . . . 93.1.3 Positivity of a Sublinear Functional . . . . . . . . . . . . . . 93.1.4 Monotonicity of a Sublinear Functional . . . . . . . . . . . . 10

4 A First (Sublinear) Extension of Linear Prices 114.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 No Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.3 Sublinear Extensions of Linear Functionals in Incomplete Markets . 124.4 The Financial Perspective: Super-replication . . . . . . . . . . . . . 134.5 A complete example . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

5 Sublinear Prices 155.1 The Concept of “Convenient Super-replication” . . . . . . . . . . . . 165.2 The properties of the pricing functional and their consequences on

the internal coherence of prices . . . . . . . . . . . . . . . . . . . . . 175.3 The Role of Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.4.1 Complete Market - Coherent Prices . . . . . . . . . . . . . . 195.4.2 Complete Market - Possible Super-replications . . . . . . . . 195.4.3 Complete Market - Arbitrages . . . . . . . . . . . . . . . . . 205.4.4 A further note . . . . . . . . . . . . . . . . . . . . . . . . . . 20

6 Frictions on the Riskless Asset 226.1 The probability vector and discount factor in a sublinear market . . 226.2 Dynamics of Super-replications and of Arbitrage . . . . . . . . . . . 24

6.2.1 Prices without Frictions . . . . . . . . . . . . . . . . . . . . . 246.2.2 Prices with Frictions on the Risky Assets - No Frictions on B 246.2.3 Prices with Frictions on the Risky Assets and on B . . . . . . 25

6.3 Possible Inversion of Discount Factors . . . . . . . . . . . . . . . . . 29

7 Beyond Sublinear Prices: Removal of Positive Homogeneity 337.1 A Graphical Analysis of Sublinear Prices . . . . . . . . . . . . . . . . 337.2 Prices Increasing in Trade Size . . . . . . . . . . . . . . . . . . . . . 347.3 A Formal Argument for Sublinear Price Increments . . . . . . . . . . 357.4 Deriving State Prices when Market Prices are Coherent . . . . . . . 377.5 Super-replications in this Setting . . . . . . . . . . . . . . . . . . . . 40

7.5.1 Deriving L+ in this Market . . . . . . . . . . . . . . . . . . . 457.6 The Pricing Functional with Multiple Price Changes . . . . . . . . . 47

8 Conclusions and Further Research 51

1 Introduction

The purpose of this work is to analyze extensions of linear pricing models whichincorporate a great deal of characteristics of real life prices, which can be generallyreferred to as transaction costs.

Linear prices can not account for the presence of frictions in the assets traded onthe market, and the presence of prices that allow for arbitrages creates the possibilityto set up trading strategies resulting in unbounded profits. The absence of any formof transaction costs and the possibility to scale the size of a trade without any effecton the price of the transaction are two assumptions that oversimplify the pricesystem and can be relaxed.

The properties of subadditivity and positive homogeneity that characterize sub-linear prices allow for the construction of a price system in which it is possible todifferentiate between bid and ask prices, leading to a more realistic model in whicharbitrages are more difficult because of frictions, and a weaker form of price incon-sistency arises, namely the possibility of convenient super-replications. We expandthe concept of internal consistency of a price system to the case where the risklessasset is affected by frictions and impose conditions on prices quoted on the marketso that arbitrages and super-replications are not allowed.

While a sublinear price system represents an improvement over the standard lin-ear case, it still suffers from the limitation that derives from the property of positivehomogeneity. The absence of any dependence of prices from the size of the trade,which is a corollary of positive homogeneity, automatically excludes any possiblemodeling of important issues such as liquidity of the securities and transparency ofprices.

In order to extend sublinear models we allow prices to vary at different thresholdsdetermined by portfolios of securities traded on the market. In this way, prices areincreasing with the size and direction of the trade, thus making the price system as awhole not positive homogeneous. We show that a model of this kind is characterizedby sublinear price increments and derive the pricing functional as the maximum ofa family of convex functionals, which in turn can be represented as maxima of affinefunctionals.

We conclude with interpretations of this model based mainly on the possibilityto accommodate the issues of liquidity, transparency, and market depth.

The analysis is organized as follows: in Section 2 we make a brief introductionabout linear prices, their usual representation, and the shortcomings of a linearpricing model. In Section 3 we show the mathematical properties of sublinearfunctionals. Section 4 contains a discussion about the possibility to extend a linearpricing model by introducing sublinear prices for non-replicable contingent claims.A fully sublinear model is presented in Section 5 and more insights into the problemsof internal coherence of prices in the sublinear case can be found in Section 6, wherethe riskless asset is affected by frictions. In Section 7 we develop a model for locallysublinear prices, where the removal of positive homogeneity allows us to limit thepossibility of arbitrages and convenient super-replications to a subset of the wholeprice system and also to introduce the issue of liquidity affecting the securitiestraded on the market. In the concluding remarks in Section 8 we provide somecomments about possible developments of future research on this topic.

5

2 Linear Functionals

Let X be a linear space over the field R of real numbers.

Definition 1. A map F : X 7→ R is called a functional. A functional F is linearif, for any x, y ∈X and α, β ∈ R,

F (αx+ βy) = αF (x) + βF (y) (2.1)

In financial mathematics, the usual assumption to build pricing models in dis-crete and continuous time is that linear functionals map payoff vectors into prices.We will confine ourselves to the finite case. The set Ω = ω1, ω2, . . . , ωm willrepresent the states of the world in a one-period model and the contingent claimy =

[y1 y2 . . . ym

]T can be written as

y =m∑

i=1

yiei ⇒ F (y) = F

(m∑

i=1

yiei

)=

m∑

i=1

yiF (ei) =m∑

i=1

yiϕi = ϕ · y (2.2)

where ei is the i-th canonical vector in Rm, i.e. its i-th component is equal to 1 andthe others are equal to 0. From the derivation in Equation 2.2, we deduce that anylinear functional is represented by a vector ϕ. Therefore, when X = Rm, the priceof any contingent claim y is given by

F (y) =m∑

i=1

yiF (ei) =[ϕ1 ϕ2 . . . ϕm

]

y1

y2...ym

= ϕ · y (2.3)

Remark 1. In general X is a subspace of Rm, instead of spanning the whole Rm.In fact, when X ⊂ Rm, there exists z /∈ X such that its price is not defined onthe market (and can not be defined by F ). This is a clear limitation of a linearpricing system.

Remark 2. It must be noted that linear prices have the following properties:

• Additivity: F (x + y) = F (x) + F (y) = ϕ · x + ϕ · y = ϕ · (x + y)

• Homogeneity: F (αx) = αF (x) = α(ϕ · x)

2.1 Criticism of Linear Prices

Linear prices can not account for the possibility of frictions in the prices of the secu-rities traded on the market. In case of market incompleteness and the introductionof non-replicable contingent claims (see Section 4.3), linear pricing systems do notdefine any price for them.

As a result, in a linear pricing system, only the following two cases are possible:

(a) y /∈X : the price of this contingent claim is not defined;

6

(b) x,y ∈X : the price to sell and to buy x (or y) is the same, and so is the priceto sell or to buy x and y together or separately, and to buy or sell one unitor a multiple of x (or y).

Point (b) can be summarized with the following table:

Linearity Meaning Reality

Fb(x) = Fa(x) No bid-ask spread Fb(x) 5 Fa(x)Fb(x + y) = F (x) + F (y) No multiple-sale advantage Fb(x + y) = Fb(x) + Fb(y)Fa(x + y) = F (x) + F (y) No multiple-purchase advantage Fa(x + y) 5 Fa(x) + Fa(y)

Fb(αx) = αF (x) Bid Quantity Invariance Fb(αx) T αFb(x)

Fa(αx) = αF (x) Ask Quantity Invariance Fa(αx) T αFa(x)

Remark 3. It is quite obvious to read the first three entries of the table as a directconsequence of the first property in Remark 2 (i.e. additivity). By substitutingthis property with subadditivity and constraining homogeneity to hold only onR+, the functional becomes sublinear.It is also natural to read the last two entries as a consequence of homogene-ity. However, in this case, it is not straightforward to determine the prevailingeffect on the price: in fact, there can be different factors pushing non-positivehomogeneous prices upward or downward at the same time.

3 Sublinear Functionals

Again, let X be a linear space over the field R of real numbers.

Definition 2. A functional F is sublinear if, for any x, y ∈X and α, β = 0,

F (αx+ βy) 5 αF (x) + βF (y) (3.1)

A sublinear functional has the following properties:

• Subadditivity: F (x + y) 5 F (x) + F (y)

• Positive Homogeneity: F (αx) = αF (x)

3.1 Characterization

In order to characterize a sublinear pricing system and outline the basic require-ments necessary for it to guarantee internal coherence of prices, we first devote ourattention to the mathematical tools that serve as background for the developmentof the pricing system.

3.1.1 Hahn-Banach Theorem

Theorem 1 (Hahn-Banach). Let p : Rm 7→ R be a sublinear functional. Let S bea vector subspace of R and f : S 7→ R a linear functional such that f(s) 5 p(s) forany s ∈ S . There always exists a linear extension F of f such that F (x) 5 p(x)for any x ∈ Rm

7

Proof. Let S ⊂ Rm, so that there exists y ∈ Rm\S . Be N the vector space gen-erated by S and y, i.e. the smallest vector space containing both S and y. Thereexists only one decomposition of x ∈ N such that x = s + λy, with s ∈ S , λ ∈ R.In fact, suppose we had, for instance, two possible decompositions, x = s1 + λ1yand x = s2 + λ2y. This would imply s1 + λ1y = s2 + λ2y⇒ s1 − s2 = (λ2 − λ1)y.Since y /∈ S it must be λ2 − λ1 = 0, which in turn would imply s1 = s2 (theirdifference must be 0).Whenever x = s + λy, a linear extension F of f must be such that

F (x) = f(s) + λF (y)

We need to show that it is possible to choose c = F (y) such that F 5 p on thewhole space N .

F (x) 5 p(x)⇒ F (s + λy) 5 p(s + λy)⇒ F (s)︸︷︷︸f(s)

+λF (y)︸︷︷︸c

5 p(s + λy)

for any s ∈ S and λ ∈ R. By positive homogeneity of p we obtain

c 5 p

(1λ

s + y)− f

(1λ

s)

if λ > 0

−c 5 p

(− 1λ

s + y)

+ f

(1λ

s)

if λ < 0

Based on the equivalence between 1λs ∈ S and s ∈ S , we need to choose c such

that:−p(−z− y)− f(z) 5 c 5 p(s + y)− f(s) for any z,s ∈ S

Notice that, (i) by linearity of f , we have f(s) − f(z) = f(s − z), that, (ii) byassumption, f(s− z) 5 p(s− z) and that, (iii) by sublinearity of p,

p(s− z + y − y) 5 p(s + y) + p(−z− y)

. Finally, putting (i), (ii) and (iii) together

f(s)− f(z) = f(s− z) 5 p(s− z) = p(s− z + y − y) 5 p(s + y) + p(−z− y)

Rearranging terms

−p(−z− y)− f(z) 5 p(s + y)− f(s) for any z,s ∈ S

Choose c so that

supz∈S

(− p(−z− y)− f(z)

)5 c 5 inf

s∈S

(p(s + y)− f(s)

)

We proved that there exists an extension of f on N . The Theorem is proven ifN = Rm. If not, we iterate our procedure by choosing y′ ∈ Rm\N and extendingthe functional again, this time to the vector space generated by N and y′, until,in a finite number of steps, f is extended to the whole Rm. lWe need to make an important point about Theorem 1.

Remark 4. Theorem 1 can be applied also when S = 0. It follows that, givena sublinear functional p, there is always an infinite number of linear functionalsin Rm that are smaller than or equal to p.

8

3.1.2 General Representation of Sublinear Functionals

There are three basic theorems useful for a no-arbitrage analysis of prices in asublinear setting.

Theorem 2. Any sublinear functional F : Rm 7→ R can be represented as themaximum of a family L of linear functionals ϕ · x, with L that can be always takenconvex and compact, that is,

F (x) = supϕ∈L

ϕ(x) for any x ∈ Rm

Proof. Take x,y ∈ Rm, α, β = 0. If F (x) = supϕ∈L ϕ(x), then,

F (αx + βy) = supϕ∈L

(ϕ(αx + βy)

)= sup

ϕ∈L

(αϕ(x) + βϕ(y)

)5

5 supϕ∈L

αϕ(x) + supϕ∈L

βϕ(y) = αF (x) + βF (y)

So, by Definition 2, the functional is sublinear.The viceversa is a consequence of the Hahn-Banach Theorem: F is sublinear, sothat the set L = linear ϕ : ϕ 5 F is non-empty. Therefore, for any x ∈ Rm thereexists at least one ϕ(x) = F (x). It follows that

F (x) = supϕ∈L

ϕ(x)

L is closed because if ϕn → ϕ with ϕ ∈ L, for ϕn(x) 5 F (x), ϕ(x) 5 F (x), whichimplies that ϕ ∈ L.L can always be taken convex because a convex combination of linear functionalsbounded from above is linear and bounded from above.L is bounded because −F (−x) 5 ϕ(x) 5 F (x) for all x ∈ Rm. l

3.1.3 Positivity of a Sublinear Functional

Definition 3. A sublinear functional F is positive if:

x = 0⇒ F (x) = 0

It is strictly positive if:x ≥ 0⇒ F (x) > 0

Theorem 3. Let F be a sublinear functional. The following are equivalent:

(a) there exists at least one (strictly) positive vector ϕ that belongs to the set Lsupporting F ;

(b) F is (strictly) positive, in the sense of Definition 3.

Proof. (a)⇒ (b).There exists ϕ = 0 such that

y = 0⇒ ϕ · y = 0⇒ F (y) = maxϕ · y = ϕ · y = 0

Proof. (b)⇒ (a).To show this, we need first to state the following (adapted):

9

Theorem 4. Let L be a closed and convex set in Rm disjoint from Rm+ . There

exists λ ∈ Rm\0 and c ∈ R such that

ϕ · λ < c < p · λ for any ϕ ∈ L,p ∈ Rm+

Back to our proof, we know that L is a compact and convex set of vectorsϕ ∈ Rm. Suppose no ϕ is positive, i.e. L ∩ Rm

+ = 0. By Theorem 4, there existsλ ∈ Rm and c ∈ R such that ϕ · λ < c < p · λ for any ϕ ∈ L,p ∈ Rm

+ . Since Rm+ is

a cone, and p · λ is bounded by c (from below), p · λ = 0 and c 5 0 for any p = 0.It follows that λ ≥ 0 and ϕ · λ < 0 for any ϕ ∈ L. Therefore F (λ) < 0, which isabsurd, because it violates the initial assumption that F is positive.

The strict version is proven analogously. l

3.1.4 Monotonicity of a Sublinear Functional

Definition 4. A sublinear functional F is monotone if:

x = y⇒ F (x) = F (y)

It is strictly monotone if:

x ≥ y⇒ F (x) > F (y)

Theorem 5. Let F be a sublinear functional. The following are equivalent:

(a) all vectors ϕ that belong to the set L supporting F are (strictly) positive;

(b) F is (strictly) monotone, in the sense of Definition 4.

Proof. (a)⇒ (b).Let us suppose that L contains only positive ϕ, and that x = y. It follows thatϕ · x = ϕ · y for any ϕ ∈ L, so that F (x) = F (y).Proof. (b)⇒ (a).Suppose now that L contains some non-positive ϕ. This implies that there existsx ≤ 0 such that ϕ ·x > 0. It follows that F (x) > 0. Finally, since F (0) = 0 we haveF (x) > 0 = F (0), but this is absurd since F is monotone by assumption.

The strict version is proven analogously. lRemark 5. If a sublinear functional F is monotone, then it is also positive. Infact, take x,y ∈ Rm, then, by monotonicity,

x = y⇒ F (x) = F (y)

It is sufficient to notice that positivity is a subcase of monotonicity where wechoose y = 0 and, obviously, x = y.

10

4 A First (Sublinear) Extension of Linear Prices

4.1 Setting

Let Ω be the set of states of the world in the finite case, Ω = ω1, ω2, . . . , ωi, . . . , ωm.The following summarizes our setting:

• there are n financial assets y1,y2, . . . ,yj , . . . ,yn traded on the market;

• yij represents the payoff of a given asset j if state i occurs at time 1;

• Y = [yij ] is the m× n matrix collecting all the yij ’s;

• any investment strategy in the n traded assets involves buying or short sellinga given number of units of a given asset. A vector a ∈ Rn will represent thenumber of units bought or sold for each of the n assets.

• π is the row vector containing the prices of the traded securities, i.e.

π = [π(y1), π(y2), . . . , π(yj), . . . , π(yn)]

4.2 No Arbitrage

Definition 5 (nai). A given market does not allow for arbitrage opportunities ofthe 1st kind if there exists no investment strategy a such that

Y a = 0 and π · a < 0

nai implies that the market can not be put in a condition of sure loss.

Definition 6 (naii). A given market does not allow for arbitrage opportunities ofthe 2nd kind if there exists no investment strategy a such that

Y a ≥ 0 and π · a 5 0

naii implies that the market can not be put in a condition where it never gains.

Remark 6. The concept of arbitrage is independent of the linearity of the pricesystem: some minor adjustments to the definition have to be made if the pricesystem is not linear, but only because the pricing functional (and its properties)change. For example, suppose that the price system is not linear: in this case,we have

π

n∑

j=1

ajyj

= π (Y a)

Therefore, in the non-linear case, we can define nai as the condition by which itis impossible to come up with an investment strategy a such that

Y a = 0 and π (Y a) < 0

This remark is meant to stress the point that the concept of arbitrage and itsrealization in a given setting (linear, sublinear, etc.) do not belong to the samelogical category. In fact, any coherent pricing system is dependent on no-arbitragerestrictions, therefore those restrictions, from the purely conceptual point of view,are logical priors to the construction of the models themselves.

11

Theorem 6 (Fundamental Theorem of Finance; de Finetti). Prices π 6= 0 do notallow for arbitrages of the first and of the second kind if and only if there exists (atleast) one (row) vector ϕ ∈ Rm

++ that solves the linear system ϕY = π.

4.3 Sublinear Extensions of Linear Functionals in Incomplete Mar-kets

Definition 7 (Incompleteness). A given market is incomplete if the rank of thematrix Y is strictly smaller than the number of states of the world (in our setting,rk(Y ) < m).

If the market is incomplete, there always exists x ∈ Rm such that the linearfunctional f : M 7→ R is undefined. In other words, M ⊂ Rm.

Suppose z ∈M . We obtain a replicating strategy of the payoff, call it a, solvingthe linear system

Y a = z

We then obtain f(z) as f(z) = π · a.Suppose now that we introduce in the market x /∈M , i.e. such that the linear

systemY a = x

is impossible. Obviously, when there exists no linear combination that replicates agiven payoff, a linear pricing functional is not able to define a price for it.

In this case, we need to extend the functional f to the whole Rm to be able toprice any x. By the Hahn-Banach Theorem (1), each linear functional on M can beextended in infinite ways to Rm. If we take the maximum of all the linear extensionsof the linear functionals on M at each point, we obtain a sublinear functional onRm. This functional represents the maximum finite sublinear extension of f .

In other words, pricing a non-replicable contingent claim requires to extend thefunctional to a sublinear one. By Theorem 2, any sublinear functional is of the form

F (x) = supϕ∈L

ϕ · x for any x ∈ Rm

with L non-empty, convex and compact.Therefore, F (x) can be interpreted as the price to buy x. If we want to sell x,

we can regard this sale as the purchase of −x: the selling price for −x is thereforegiven by −F (−x). We obtain:

−F (−x) = − supϕ∈L

ϕ · (−x) = infϕ∈L


To sum up:

• the price to buy (or ask price) is given by Fa(x) = supϕ∈L ϕ · x;

• the price to sell (or bid price) is given by Fb(x) = infϕ∈L ϕ · x.

The next step is to impose restrictions on the ϕ’s belonging to the set L so that theprice system is coherent and arbitrage free.

12

We need Theorem 6 to assess whether this market allows for arbitrage opportu-nities. The infinitely many solutions of the linear system ϕY = π all belong to theclosed set L. By Theorem 6, when a strictly positive ϕ > 0 solution exists, prices πare coherent: both nai and naii hold.When there exists at least one solution such that ϕ = 0, only nai holds. Let ussummarize:

(i) L+ = L ∩ Rm+ 6= 0 ⇔ only nai;

(ii) L++ = L ∩ Rm++ 6= 0 ⇔ naii and, a fortiori, nai;

Therefore, prices do not allow for arbitrages if:

F (x) = supϕ∈L++

ϕ · x = maxϕ∈L+


As we have already observed, ask prices can be immediately deduced from bidprices, and viceversa. Therefore, the analysis does not change if we speak of theformer (or of the latter), without explaining in detail what are the implications forthe other category.

4.4 The Financial Perspective: Super-replication

In the previous section we argued that in an incomplete market there always exists,by definition, some given x ∈ Rm such that the system Y a = x is impossible. Then,by exploiting the Hahn-Banach Theorem, we extended the linear functional to thewhole Rm: this extension leads to a sublinear functional expressing the upper boundof the set of coherent linear prices for x.

There is, however, another possible route that can be followed once we acknowl-edge that the linear system Y a = x is impossible. In fact, instead of approachingthe upper bound of the set of prices from below, by taking the linear convex com-bination of state prices ϕ1, ϕ2, . . . , ϕm in the set L+ that maximizes the prices of agiven set of payoffs x1, x2, . . . , xm, we can take the lower bound of the set of pricesobtained super-replicating x. We look for all the payoff vectors z ∈ M such thatzi = yi for all i = 1, 2, . . . ,m, and then take the cheapest one as the price of x, thatis,

F (x) = minz∈Mπ(z) : z = x for any x ∈ Rm

Remark 7. We can safely use the above formula for replicable contingent claimsas well. In fact, to avoid arbitrages, the minimum cost super-replication portfoliowhen y ∈M is the portfolio where just y is bought. Its cost is exactly π(y).Moreover, as will be shown in the example, no-arbitrage requires that the priceobtained with the two methods (super-replication and sublinear extension of thelinear functional) coincide.

4.5 A complete example

Let Ω = ω1, ω2, ω3 and

Y =

0 55 1015 7

π = [8 6.8]

13

The linear system ϕ · Y = π is given by:

5ϕ2 + 15ϕ3 = 85ϕ1 + 10ϕ2 + 7ϕ3 = 6.8

We set ϕ3 = α with α ∈ R so thatϕ1 = 4.6α− 1.84ϕ2 = 1.6− 3α

To exclude arbitrages, it must be ϕ > 0, which leads us toα < 8

15

α > 25

Let us take the non replicable random variable z = [13 11 22]T . Its linear priceis undefined: in fact we have that

π(z) = ϕ · z = 13(4.6α− 1.84) + 11(1.6− 3α) + 22α = 48.8α− 6.32

The lower bound of the range of prices consistent with no arbitrage is

Fb(z) = infαϕ · z = 48.8

(25

)− 6.32 = 13.2

and, analogously, the upper bound is

Fa(z) = supαϕ · z = 48.8

(815

)− 6.32 = 19.70667

—äm¨—

Let us now change perspective on the problem, and adopt the super-replicationargument.

Reconsider the contingent claim ±z = ± [13 11 22]T . Our problems are thefollowing:

min(8a1 + 6.8a2)sub

5a2 = 135a1 + 10a2 = 1115a1 + 7a2 = 22

min(8a1 + 6.8a2)sub

5a2 = −135a1 + 10a2 = −1115a1 + 7a2 = −22

These problems can be solved graphically by considering the (two) shaded areasin Figure 1, that represent the regions of all the vectors that super-replicate z and−z.

As far as the replication of z is concerned, the coordinates of point A are givenby the intersection of the lines a2 = 2.6 and a2 = 22

7 − 157 a1, therefore the problem

is solved in vertex A where a1 = 0.253a2 = 2.6

14

Figure 1: Minimization Problem

(a) Super-replication of z

1 2 3-1

1

2

3

-1

A

a1

a2

0

(b) Super-replication of −z

-1-2-3 1

-1

-2

-3

1

B

a1

a2

0

The cost of this strategy is

8a1

∣∣a1=0.25333

+ 6.8a2

∣∣a2=2.6

= 19.70667

Analogously, as far as the replication of −z is concerned, the coordinates of pointB are given by the intersection of the lines a2 = −1

2a1 − 1110 and a2 = −22

7 − 157 a1,

therefore the problem is solved in vertex B wherea1 = −1.24347a2 = −0.47826

The cost of this strategy is

8a1

∣∣a1=−1.24347

+ 6.8a2

∣∣a2=−0.47826

= −13.2

The price to sell Fb(z) is therefore Fb(z) = −F (−z)︸︷︷︸−13.2

= 13.2.

Remark 8. Notice that

Fa(z) = supϕ∈L++

ϕ · z = miny∈Mπ(y) : y = z

Fb(z) = −F (−z) = infϕ∈L++

ϕ · z = miny∈Mπ(y) : y = −z

5 Sublinear Prices

Instead of using the properties of sublinearity only in order to extend a linear pricingsystem, we can build a model that allows the existence of bid-ask spreads possiblyfor all the traded assets on the market.

In this case, different prices to buy and sell a given security yj are allowed. Ifwe denote with Fb(yj) the price to sell and Fa(yj) the price to buy, we impose

15

that Fb(yj) 5 Fa(yj) for any j = 1, 2, . . . , n. The following notation is adopted:Fa and Fb are functionals and they map payoffs into prices (ask and bid prices,respectively); πa and πb are vectors in Rn collecting ask and bid prices of the nassets traded on the market.

The first modification with respect to the linear case is that the system ϕ ·Y = πis no longer suitable, due to the non uniqueness of π. In fact, any vector π ∈ Rn

such that πb 5 π 5 πa is consistent with the prices quoted by the market.Therefore, we now need to solve the system of linear inequalities

πb 5 ϕ · Y 5 πa

In this way, we obtain the set L collecting all the ϕ’s such that

L = ϕ : πb 5 ϕ · Y 5 πa

Clearly, it is of major importance to discuss the connection between the nature ofthe solutions of our problem above and the coherence of the prices on the market.

5.1 The Concept of “Convenient Super-replication”

The symmetry between buying and selling prices in linear pricing models is notpreserved when prices are sublinear, as we have seen.

This asymmetry generates the possibility that some contingent claims mightbe conveniently super-replicated, where conveniently refers to the fact that thereplicating strategy might be cheaper than the quoted price of the contingent claim.

In this respect, arbitrages of the first kind represent a subcategory of super-replication where the contingent claim z = 0 is replicated (not strictly super-replicated) at a better cost than F (0) = 0.

On the other hand, arbitrages of the second kind represent a subcategory ofsuper-replication where the contingent claim z = 0 is strictly super-replicated witha cost not worse than the cost of the replicated random variable, i.e. F (0) = 0.

Formally, we can give the following definitions:

Definition 8 (nsri). A given market does not allow for convenient super-repli-cation opportunities of the 1st kind if there exists no investment strategy a suchthat

Y a = yj and F (Y a) < F (yj)

where yj is one of the n assets traded on the market.It is now clear that, by choosing yj = 0, we collapse in the definition of nai (see

Definition 5).

Definition 9 (nsrii). A given market does not allow for convenient super-repli-cation opportunities of the 2nd kind if there exists no investment strategy a suchthat

Y a ≥ yj and F (Y a) 5 F (yj)

where yj is one of the n assets traded on the market.Also in this case, by choosing yj = 0, we collapse in the definition of naii (see

Definition 6).

16

5.2 The properties of the pricing functional and their consequenceson the internal coherence of prices

Let us define

(i) L = ϕ : πb 5 ϕ · Y 5 πa;

(ii) L+ = L ∩ Rm+ ;

(iii) L++ = L ∩ Rm++;

First, we want to analyze the relationship between the properties of a sublinearfunctional F and nsri/nsrii.

We recall from Definition 4 that if F is monotone, then x = y ⇒ F (x) = F (y)and that if F is strictly monotone, x ≥ y⇒ F (x) > F (y).

This is very useful in our analysis: a monotone functional guarantees that nsriholds. Analogously, a strictly monotone functional guarantees that nsrii holds.

However, this is not totally satisfying, as there is still no connection with theelements of the set L. We can employ Theorem 5, which states the equivalencebetween monotonicity of F and positivity of the ϕ’s in L. In other words,

• monotonicity ⇒ nsri and

• monotonicity ⇔ L ⊆ Rm+ (i.e. L = L+)

implyL = L+ ⇒ nsri⇒ nai

Moreover, when F is strictly monotone,

• strict monotonicity ⇒ nsrii and

• strict monotonicity ⇔ L ⊆ Rm++ (i.e. L = L++)

implyL = L++ ⇒ nsrii⇒ naii

We conclude that, for nsri to hold, all the ϕ’s in L must be positive, and, for nsriito hold, all the ϕ’s in L must be strictly positive.

On the other hand, we recall from Definition 3 that if F is positive, then x =0⇒ F (x) = 0 and that if F is strictly positive, x ≥ 0⇒ F (x) > 0.

Therefore, a positive functional guarantees that nai holds. Analogously, astrictly positive functional guarantees that naii holds. We can employ Theorem5, which states the equivalence between positivity of F and positivity of at leastone ϕ in L. In other words:

• positivity ⇒ nai and

• positivity ⇔ L+ 6= 0

implyL+ 6= 0 ⇒ nai

Moreover, when F is strictly positive,

17

• strict positivity ⇒ naii and

• strict positivity ⇔ L++

implyL++ ⇒ naii

We conclude that, for nai to hold, at least one ϕ in L must be positive, and, fornaii to hold, at least one ϕ in L must be strictly positive.

5.3 The Role of Prices

bid and ask prices reveal the presence of incoherent prices on the market, but in adifferent way.

Consider the following theorem for bid prices:

Theorem 7. The sublinear functional F is:

• monotone if and only if y = 0⇒ −F (−y) = 0;

• strictly monotone if and only if y ≥ 0⇒ −F (−y) > 0

In our case, since nsri (nsrii) is equivalent to monotonicity (strict monotonic-ity), bid prices reveal nsri (nsrii) if they are always non-negative (positive) forany non-negative (semi-positive) random variable.

We can summarize these concepts in this way:

positivity y = 0⇒ F (y) = 0 ask naistrict positivity y ≥ 0⇒ F (y) > 0 ask naii

monotonicity y = 0⇒ −F (−y) = 0 bid nsristrict monotonicity y ≥ 0⇒ −F (−y) > 0 bid nsrii

Let us now take all the random variables y such that y = 0. The market quotes(possibly) a bid price Fb(y) to sell the security and an ask price Fa(y) to buy thesame security. If we take the interval of prices determined by Fb(y) and Fa(y), i.e.[Fb(y) Fa(y)], there are three possible scenarios:

• [Fb(y), Fa(y)] ⊆ R+: prices do not allow for convenient super-replications(included arbitrages);

• [Fb(y), Fa(y)] overlaps with R+ but it is not included in it: prices allow forconvenient super-replications, but arbitrages are not possible;

• [Fb(y), Fa(y)] * R+, prices allow for arbitrages and, a fortiori, convenientsuper-replications.

The following graphical representation can be given:

18

Figure 2: Graphical Representation (y = 0)

Fb Fa

na,nsr

Fb Fa

Arbitrage

Fb Fa

sr

0

5.4 Example

5.4.1 Complete Market - Coherent Prices

Let Ω = ω1, ω2, ω3 and

Y =

1 0 51 5 101 15 7

πa = [1 8 6.8]

πb = [1 7.6 6.4]

The market is complete and the determinant of Y is −65. The set of consistent πis given by

π = [1 7.6 + 0.4α 6.4 + 0.4β]

We solve the linear system

ϕ1 + ϕ2 + ϕ3 = 15ϕ2 + 15ϕ3 = 7.6 + 0.4α5ϕ1 + 10ϕ2 + 7ϕ3 = 6.4 + 0.4β

It admits solution

ϕ =[

28.2− 1.2α− 4β65

5.8 + 0.8α+ 6β65

31 + 2α− 2β65

]0 5 α, β 5 1

For all possible values of α, β ∈ [0, 1], the vector ϕ is strictly positive, thereforensrii and naii hold.

Figure 3: nsr, na (y = 0)

Fb Fa

na,nsr

Fb Fa

Arbitrage

Fb Fa

sr

0

5.4.2 Complete Market - Possible Super-replications

Let us keep the same Y as the previous example and have the following price vectors:

πa = [1 8 6.8]πb = [1 7 5.0]


ϕ1 + ϕ2 + ϕ3 = 15ϕ2 + 15ϕ3 = 7 + α

5ϕ1 + 10ϕ2 + 7ϕ3 = 5 + 1.8β

19

It admits solution

ϕ =[

44− 3α− 18β65

−14− 2α+ 27β65

35 + 5α− 9β65

]0 5 α, β 5 1

Notice that ϕ2 assumes a negative value for β = 0. This means this market allowsfor convenient super-replications, even though it does not allow arbitrages, as someϕ’s are positive.

Figure 4: sr, na (y = 0)

Fb Fa

na,nsr

Fb Fa

Arbitrage

Fb Fa

sr

0

5.4.3 Complete Market - Arbitrages

Let us keep the same Y as the previous example and have the following price vectors:

πa = [1 7.1 5.1]πb = [1 7 5.0]


ϕ1 + ϕ2 + ϕ3 = 15ϕ2 + 15ϕ3 = 7 + 0.1α5ϕ1 + 10ϕ2 + 7ϕ3 = 5 + 0.1β

It admits solution

ϕ =[

44− 0.3α− β65

−14− 0.2α+ 1.5β65

35 + 0.5α− 0.5β65

]0 5 α, β 5 1

Notice that ϕ2 assumes a negative value for any 0 5 α, β 5 1. This means thismarket allows for arbitrage opportunities (and, of course, super-replications).

Figure 5: Arbitrage (y = 0)

Fb Fa

na,nsr

Fb Fa

Arbitrage

Fb Fa

sr

0

5.4.4 A further note

Further insights into the dynamics of super-replication and no-arbitrage can begained by adopting a different, probably unusual, view of the market.

In fact, it can be interesting to analyze the mechanisms and dynamics of theprice-setting process for the point of view of the market itself, that is, by thinking

20

in terms of multiple markets in which market-makers are free to set the prices, butthis freedom of choosing prices comes with the risk of incurring sure losses, thatis, arbitrages, and of losing opportunities to trade because of convenient super-replications by the counterparties.

More specifically, we can think of the market-maker as being able to set bidand ask prices at which he or she is willing to enter a transaction whereby itscounterparty buys or sells at ask and bid prices, respectively.

Suppose we start from bid and ask prices that do not allow neither for conve-nient super-replications nor for arbitrages, as in Section 5.4.1. We remind that theask price for asset y2 was 8, whereas the bid was 7.6; for asset y3, the ask pricewas 6.8, and the bid was 6.4.

On a parallel trading venue, a financial intermediary is trying to increase itsprofits by charging more on the spreads. In a complete market with sublinearprices, this can be done, for example, by keeping ask prices to the same level, andincreasing the spreads by lowering bid prices.

Of course, there are problems that the intermediary should address before de-ciding the level of bid prices. In fact, as we can see from the example in Section5.4.2, excessively low bid prices expose the market to the possibility of convenientsuper-replications.

Let us analyze the problem in more detail, with some numbers. We assume thatask prices are kept as they are in the first scenario. In the first scenario we hadobtained the following vector of consistent ϕs:

ϕ =[

28.2− 1.2α− 4β65

5.8 + 0.8α+ 6β65

31 + 2α− 2β65

]0 5 α, β 5 1

Notice that the component of ϕ that is “weaker” is ϕ2. Therefore, if we are interestedin pushing bid prices to the lower bound, ϕ2 is probably going to tell us when tostop, that is, it is likely to be the first to become negative as prices decrease.

Let us redefine the problem in terms of the spreads: we define s1 as Fa(y1) −Fb(y1), and s2 analogously. We must impose that 0 5 s1 5 8 and that 0 5 s2 5 6.8.We can solve ϕ · Y = πb and we find:

ϕ2 =11 + 2s1 − 15s2

65As we see in the figure, spreads must be limited to avoid super-replications. Forexample, on the graph we have visual evidence that, if bid prices drop as in thecase of section 5.4.2, the state-price ϕ2 becomes negative.

Figure 6: ϕ2 as a function of the spreads

01

23

45

67

8

0123456

-1.4-1.2

-1-0.8-0.6-0.4-0.2

00.20.40.6

ϕ2

ϕ2 as a function of the spreads

f(x,y)

s1

s2

-1.4-1.2-1-0.8-0.6-0.4-0.200.20.40.6

21

Remark 9. We have focused in our examples on a case in which the market iscomplete. The other case, that of market incompleteness, is not as satisfactoryfor a no arbitrage analysis for two main reasons:

(i) it does not have peculiarities of particular interest for a discussion withrespect to the complete case;

(ii) in general, the set L is so wide that it can not contain only positive vectorsϕ, i.e. super-replications are always possible.

The intuition is that, in this setting, the indeterminacy due to bid-ask spreadscoexists with the indeterminacy due to market incompleteness. Therefore, theoverall indeterminacy of prices is much larger.In conclusion, only two cases are possible when the market is incomplete:

(i) L contains some (at least one) ϕ ≥ 0, and arbitrages are impossible;

(ii) L does not contain any ϕ ≥ 0, and arbitrages are possible.

6 Frictions on the Riskless Asset

There are additional ways to interpret what we have developed so far, especiallywith respect to the concept of state prices.

In fact, consider the random variable 1 = [1 1 . . . 1], that is, the random vari-able paying a sure amount of money in any state. Of course, its price can beinterpreted as the discount factor from time 1 to time 0. We can compute its priceB as:

B = F (1) = ϕ · 1 =m∑

i=1

ϕi

The interesting thing is that any vector of state-prices can be rewritten in thefollowing way:

ϕ =m∑

i=1

ϕi

︸︷︷︸B

ϕ∑mi=1 ϕi

= Bp

where p is a vector whose components sum to 1. It can be interpreted as a proba-bility vector (as long as its components are positive).

Therefore, the price of a generic random variable y ∈M is given by

F (y) = ϕ · y = Bp · y = Ep [By]

This is an important result because it tells us that linear prices are expectationsof discounted payoffs. In the non-linear case, however, things are to some extentdifferent.

6.1 The probability vector and discount factor in a sublinear mar-ket

It is straightforward to transform the vector of state prices ϕ into the product of adiscount factor B and a probability vector p in the linear case.

22

However, when the market is incomplete and 1 /∈ M , the risk-free asset isaffected by frictions.

Example 6.1.1 (from Example 4.5). We had the following situation:

Y =

0 55 1015 7

π = [8 6.8]

The solutions are:

ϕ1 = 4.6α− 1.84ϕ2 = 1.6− 3αϕ3 = α

with 25 < α < 8

15 . The price of 1 = [1 1 1] is

Ba(1) = 2.6 815 − 0.24 = 1.146667

Bb(1) = 2.625 − 0.24 = 0.8

These prices can be interpreted as discount factors for sure amounts of money,that is, to finance a borrowing of 0.8, or any multiple of it, the borrower is requiredto repay 1 at maturity, scaled of course by the amount. Analogous reasoning appliesto investments: an investment that guarantees at maturity a capital of 1 requiresan initial outlay of 1.14667. When we deal with the possibility of different discountfactors, there are two main points to address:

(i) first of all, the expression “the price of a contingent claim is the discountedexpectation of its future payoffs” has now become ambiguous. In fact, thepresence of (at least) two prices for the riskless asset implies that any dis-counting could take place in (at least) two ways;

(ii) secondly, is the intuition that expectations of buying prices at time 1 shouldbe discounted with the buying price of the discount factor correct?

Before we address these problems, we move from the world of state-prices to theworld of risk-neutral probabilities by defining:

(i) P+ as the set of all the non negative neutral probabilities:

P+ = p : Bp ∈ L+

(ii) P++ as the set of all the strictly positive neutral probabilities:

P++ = p : Bp ∈ L++

Whenever P+ 6= ∅, P++ = P+.Therefore, we will have that the sublinear pricing functional F will be given by:

F (y) = maxp∈P+

Bp · y = maxp∈P+

Ep[By]

In general, B depends on p, unless the riskless asset is replicable, i.e. 1 ∈M .Before we continue with our example, we need to derive no-arbitrage conditions

(and non-super-replication conditions) in this setting.

23

6.2 Dynamics of Super-replications and of Arbitrage

Our discussion of the problems related to the internal coherence of prices startsby deriving conditions in the linear case, and extends up to the most complicatedcase where prices are affected by frictions and the riskless asset is also affected byfrictions.

6.2.1 Prices without Frictions

In the case of linear prices, when the same buying and selling price applies to a givenrandom variable y and there are no frictions on the riskless asset, i.e. 1 ∈M , theprice of any given random variable π(y) must be included between its discountedmaximum and the minimum payoff, that is

Bminiyi 5 π(y) 5 Bmax

iyi

where we assume that also y ∈M .

6.2.2 Prices with Frictions on the Risky Assets - No Frictions on B

In the case of different prices to buy and to sell y, we have that πa(y) = πb(y),therefore with Ba = Bb = B the coherence requirement reads

Bminiyi 5 πb(y) 5 πa(y) 5 Bmax

iyi for any y ∈ Rm

which can be rewritten in a more meaningful way as

miniyi 5 πb(y)

B5 πa(y)

B5 max


We can interpret this relationship as follows.If πa(y) > 0, this amount of money could be invested in the risk-free asset,

repaying at maturity exactly πa(y)B ; alternatively, it could be invested in the random

variable y. At maturity, if πa(y)B > maxi yi, the option to invest in the risk-free asset

super-replicates the investment in the random variable. It would be as if betting forSuperenalotto were always a worse option than putting money in a bank account.It must be πa(y) 5 maxi yi.

On the other hand, if πa(y)B < mini yi, the option to invest in the risk-free asset

is super-replicated by the investment in the random variable. It would be as ifbetting for Superenalotto were always a better option than putting money in abank account. It must be πa(y) = mini yi.

Let us assume that also πb(y) > 0. We have shown that it must be πa(y) 5maxi yi. Since by assumption πb(y) 5 πa(y), then it follows that, a fortiori, πb(y) 5maxi yi.

On the other hand, if πb(y)B < mini yi, the alternative to sell πb(y) units of the

risk-free asset at time 0 and having to repay πb(y)B is better than selling y in any

state. Since this option would imply a super-replication, it must be πb(y) = mini yi.Similar reasonings apply when prices are not positive. We can therefore derive

three possible cases depending on whether or not the conditions we described beforeare satisfied:

24

(i) the following holds

miniyi 5 πb(y)

B5 πa(y)

B5 max


and nsri and nai hold;

(ii) only one of the two inequalities holds, and the other does not. Where theinequality is not satisfied, super-replications, in the way explained above, cantake place, but not arbitrages;

(iii) both inequalities do not hold. Arbitrages can take place because both super-replications are convenient.

6.2.3 Prices with Frictions on the Risky Assets and on B

In this case, there are three possibilities for both super-replications and arbitrages,which require separate discussions.

Super-replications. For super-replications, we regard investments in (or bor-rowings of) the risk-free assets or in the random variable as alternative options:we discuss coherence requirements conditional on the choice to buy or sell havingalready being made. In other words, the direction of the trade is assumed: fromthat assumption onwards, we investigate the rational choice to make, when the co-herence requirements are not satisfied. Our purpose is to impose requirements onprices that make no alternative better than the other. Technically, we do so byruling out the possibility of convenient super-replications.

Case 1 πa(y) = πb(y) = 0

πa(y) can either be invested in the risk-free security, such that at maturity theamount πa(y)

Baaccrues, or it can be used to buy the random variable y. The random

variable should never pay out a maximum payoff smaller than the accrued amount,because it does not make sense to buy a random payoff smaller than a sure amountfor the same price. Conversely, it would not make sense for an investment in theriskless asset to always pay out less than a given random variable, at the same price,that is, the randomness of y must not super-replicate the sure amount in all states.To sum up, we have that

miniyi 5 πa(y)

Ba5 max

iyi

As far as selling prices are concerned, we could sell an amount equal to πb(y) ofthe riskless asset (at Bb) or sell y. It must be

miniyi 5 πb(y)

Bb5 max

iyi

If any of the inequalities above is not verified, super-replications of the purchase orof the sale of y can take place.

25

Case 2 πa(y) = 0 = πb(y)

Nothing changes for the buying price:

miniyi 5 πa(y)

Ba5 max

iyi

For selling prices, we have a negative inflow if we sell y, therefore we need to paythe amount of money |πb(y)|. Alternatively, this amount of money can be investedinto the risk-free asset, which at maturity is worth −πb(y)

Ba, where −πb(y) > 0. Since

we are comparing alternatives in which we sell y, the signs of the payoff are reversed,leading to

mini

(−yi) 5 −πb(y)Ba

5 maxi

(−yi)⇒ −maxi

(yi) 5 −πb(y)Ba

5 −mini

(yi)

which leads tomini

(yi) 5 πb(y)Ba

5 maxi

(yi)

Case 3 0 = πa(y) = πb(y)

In this case, if we buy the random variable, we have an inflow equal to its price.Alternatively, we can generate the same inflow, which is equal to −πa(y), by sellingthe degenerate random variable −πa(y)

Bbsuch that, at maturity, the following must

holdmini

(yi) 5 πa(y)Bb

5 maxi

(yi)

The reasoning for selling prices is identical to that of Case 2, so that

mini

(yi) 5 πb(y)Ba

5 maxi

(yi)

The interval S. Following from the previous discussion of the different possi-bilities related to the prices of the random variable y, we can construct a closedinterval, call it S, whose extremes are given by

sa(y) =

πa(y)Ba

for πa(y) = 0πa(y)Bb

for πa(y) < 0sb(y) =

πb(y)Bb

for πb(y) = 0πb(y)Ba

for πb(y) < 0

Let us define I(y) = [mini yi,maxi yi]. For nsri to hold, the extremes of the intervalS(y) must belong to I(y), so that S(y) ⊆ I(y).

26

Remark 10. By definition, Ba > Bb which implies 1Ba

< 1Bb

. Therefore, movingfrom πa(y) and πb(y), where it is always πb(y) 5 πa(y), to sa(y) and sb(y), thedirection of the inequality is not always preserved. It is straightforward to verifythat it can be sb(y) S sa(y).

Example 6.2.1 (sa > sb). Let us consider the random variable y = [2 4]T andits prices πa(y) = 2.4 and πb(y) = 1.8. If Ba = 2

3 and Ba = 35 , we have

sa(y) =2.42/3

= 3.6; sb(y) =1.83/5

= 3

therefore sa(y) > sb(y). Notice also that mini yi = y1 < sb(y) < sa(y) < y2 =maxi yi, therefore S(y) ⊂ I(y) and nsri holds.

Example 6.2.2 (sa < sb). For the same random variable and the same prices, ifBa = 4

5 and Ba = 12 , we have

sa(y) =2.44/5

= 3; sb(y) =1.81/2

= 3.6

therefore sa(y) < sb(y). Notice also that mini yi = y1 < sa(y) < sb(y) < y2 =maxi yi, therefore S(y) ⊂ I(y) and nsri holds also in this case.

Example 6.2.3 (sa = sb). If we want sa(y) = 2.4Ba

and sb(y) = 1.8Bb

to be equal, itmust be

2.4Ba

=1.8Bb⇒ Ba

Bb=

2.41.8

=43

Therefore if Bb = 35 and Ba = 4

5 , we have

sa(y) =2.44/5

= 3; sb(y) =1.83/5

= 3

In this case S(y) is a singleton and S(y) ⊂ I(y) and nsri holds.

—äm¨—

Arbitrages. As opposed to the case of super-replications, this time we regardtransactions in the risk-free assets not as alternatives to transactions in the randomvariable, but as a result of sales/purchases of the random variable y itself. In otherwords, we impose that the profile of the aggregated cash flows be always zero atinception.

Case 1 πa(y) = πb(y) = 0

We can buy y and pay its price πa(y), but our purchase must be financed by theshort sale of the degenerate random variable πa(y)

Bb. At maturity, the random amount

yi will be received. If this amount is greater than the cost of financing in any state,we have set up an arbitrage (cash flow is 0 at inception, positive at maturity).

27

Hence, it must be

πa(y)Bb

− yi 5 0 for some ωi ∈ Ω⇒ miniyi 5 πa(y)

Bb(6.1)

On the other hand, we can sell y and receive its price πb(y), but our sale must beinvested in the degenerate random variable πb(y)

Ba. At maturity, the random amount

yi will be paid. If this amount is smaller than the proceeds from investing in anystate, we have set up an arbitrage (cash flow is 0 at inception, positive at maturity).Hence, it must be

yi −πb(y)Ba

= 0 for some ωi ∈ Ω⇒ πb(y)Ba

5 maxiyi (6.2)

Case 2 πa(y) = 0 = πb(y)

Nothing changes for the purchase of y: the same condition as in Equation (6.1)applies.

On the other hand, we can sell y and pay its price πb(y), but our sale must befinanced by the short sale of the degenerate random variable −πb(y)

Bb. At maturity,

the random amount yi will be paid. If this amount is greater than the cost offinancing, we have set up an arbitrage (cash flow is 0 at inception, positive atmaturity). Hence, it must be

−yi +πb(y)Bb

5 0 for some ωi ∈ Ω⇒ πb(y)Bb

5 maxiyi (6.3)

Case 3 0 = πa(y) = πb(y)

We can buy y and receive its price πa(y), but the proceedings from the purchasemust be invested in the degenerate random variable −πa(y)

Ba. At maturity, the ran-

dom amount yi will be received. If this amount is greater than the proceeds fromthe riskless investment in any state, we have set up an arbitrage (cash flow is 0 atinception, positive at maturity). Hence, it must be

yi −πa(y)Ba

5 0 for some ωi ∈ Ω⇒ πa(y)Ba

= miniyi (6.4)

The same condition as in Equation (6.3) applies in case of a sale.

The interval S∗. In the same way as in the previous paragraph, we can con-struct a closed interval, call it S∗, whose extremes are given by

s∗a(y) =

πa(y)Bb

for πa(y) = 0πa(y)Ba

for πa(y) < 0s∗b(y) =

πb(y)Ba

for πb(y) = 0πb(y)Bb

for πb(y) < 0

28

The interval I is the same, i.e., I(y) = [mini yi,maxi yi]. For nai to hold, theinterval S∗(y) must not be disjoint from I(y), so that nai⇐⇒ S∗(y) ∩ I(y) 6= ∅.

Notice that, for any y ∈ Rm, S(y) ⊆ S∗(y). In fact, for any y ∈ Rm:

sa(y) 5 s∗a(y) sb(y) 5 s∗b(y)sb(y) 5 s∗a(y) sa(y) 5 s∗b(y)

therefore we conclude that:

(i) if sa(y) = sb(y) , then

s∗a(y) = sa(y) = sb(y) = s∗b(y)

(ii) if sb(y) = sa(y) , then

s∗a(y) = sb(y) = sa(y) = s∗b(y)

Remark 11. The fact that the interval S(y) is included in S∗(y) is rather com-forting. In fact, it points to the same principle we have defined in Definitions 8and 9: an arbitrage is a convenient super-replication, where the super-replicatedrandom variable is constrained to be the vector which is 0 in all of its components.When we find that the set S(y) is included in S∗(y), we conclude that wheneverthere are arbitrages opportunities on the market, there are necessarily super-replication opportunities. In other words, the fact that the no-arbitrage require-ment is less stringent than the absence of super-replication opportunities, as wehave already stated in the previous definitions, is confirmed by the fact that theset of prices consistent with no-arbitrage is wider than the set of prices consistentwith nsr, which entails S(y) ⊆ S∗(y).

Conclusion. We can conclude this long discussion with the following concise sum-mary of our findings:

(i) S(y) ⊆ I(y) nsri and nai hold;

(ii) S(y) * I(y) but S∗(y) ∩ I(y) 6= ∅ : only nai holds;

(iii) S∗(y) ∩ I(y) = ∅ : neither nsri nor nai hold. l

6.3 Possible Inversion of Discount Factors

We introduced prices at maturity s∗a because we are going to compare them withundiscounted expected values at maturity, obtained with the probability vector pderived above. In other words, we can add the following:

Definition 10. Undiscounted expected values are denoted by µ and defined as

µa(y) = maxp

Ep[y] µb(y) = minp

Ep[y]

29

Let us now continue with the previous example, so as to have an initial understand-ing of the relationship between µ and s∗.

Example 6.3.1 (follows from Example 6.1.1). The probability vector is given by

p =[

4.6α− 1.842.6α− 0.24

1.6− 3α2.6α− 0.24

α

2.6α− 0.24

]

The random variable z = [13 11 22] has prices

Fa(z) = 19.70667 Fb(z) = 13.2

which are obtained for α = 815 and α = 2

5 , respectively. They translate into pricesat maturity

s∗a(z) =Fa(z)Bb

=19.70667

0.8= 24.63334 s∗b(z) =

Fb(z)Ba

=13.2

1.146667= 11.51162

Let us now calculate µa(y) and µb(y)

µ = p · z =4.6α− 1.842.6α− 0.24

13 +1.6− 3α

2.6α− 0.2411 +

α

2.6α− 0.2422 =

48.8α− 6.322.6α− 0.24

with 25 < α < 8

15 . This implies that:

µa =48.8α− 6.322.6α− 0.24

∣∣∣α=8/15

= 17.18605

andµb =

48.8α− 6.322.6α− 0.24

∣∣∣α=2/5

= 16.5

Therefore both maximum values for Fa(z) and µa(z) are achieved for α = 815 and

both minimum values for Fb(z) and µa(z) are achieved for α = 25 .

Let us now price the random variable z′ = [13 11 45]. Its prices are

Fa(z′) = maxα

(71.8α− 6.32) = 31.97333 for α = 815

Fb(z′) = minα

(71.8α− 6.32) = 22.4 for α = 25

and its undiscounted expected values are given by

µa(z′) = maxα

(71.8α− 6.322.6α− 0.24

)= 28 for α = 2

5

µb(z′) = minα

(71.8α− 6.322.6α− 0.24

)= 27.88369 for α = 8

15

In this case, the maximum price Fa(z′) = 31.97333 is achieved by setting α = 815 .

The corresponding discount factor B(α) is therefore B( 815) = Ba = 1.14667. On

the other hand, µa(z′) is maximum for α = 25 and therefore its associated discount

factor is B(25) = 0.8. We can conclude that discount factors are not the same, and,

more specifically, they are reversed.

30

Before drawing some more generic conclusions after this example, let us try tofigure out in a more detailed way when the inversion of the discount factors takesplace, as it might be helpful for a more thorough understanding of the problem.

Let us assume that we want to know how, all other things being equal, themaximum undiscounted expectation µa(z′) changes as we let the payoff in state 3z′3 vary. In general, we derive that

µa(z′) = maxα

(26.8α− 6.32 + αz′3

2.6α− 0.24

)

This max is always achieved either for α = 815 or for α = 2

5 , therefore we can definetwo lines having the following coordinates:

26.8α− 6.32 + αz′32.6α− 0.24

=

26.8α−6.32+αz′32.6α−0.24

∣∣∣α=2/5

= 4.4+0.4z′30.8 = 5.5 + 0.5z′3

26.8α−6.32+αz′32.6α−0.24

∣∣∣α=8/15

= 7.973+0.533z′31.1467 = 6.95 + 0.465z′3

We can plot both lines on the same graph and verify that they cross at some point,thus making the “max” different depending on the value of z′3.

Figure 7: Inversion of Discount Factors. One line is dashed and the other is solidand they cross at point A.

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

605.5 + 0.5z′

3

6.953468 + 0.46511z′3

A

z′3

31

Figure 8: Inversion of Discount Factors. The solid line is the maximum undiscountedexpectation of the security µa(z′) and its slope changes at A.

0 10 20 30 40 50 60 70 80 900

10

20

30

40

50

60 maxα

(26.8α−6.32+αz′

32.6α−0.24

)=

6.953468 + 0.46511z′3 for z′3 ≦ 41.658585.5 + 0.5z′3 for z′3 > 41.65858

A

z′3

You can see from the figures that for any z′3 > 41.65858 the maximum is achievedby setting α = 2

15 , whose respective discount factor is Bb = 0.8.Therefore, for anyz′3 > 41.65858, discount factors are reversed.

We can now provide answers to the questions we had at the beginning of this section.First, the expression “the price of a contingent claim is the discounted expecta-

tion of its future payoffs” remains true, even though it is not precise, and not veryinformative either. In fact, while it is still true that prices are expected values offuture payoffs discounted to the present time, we have shown that it might well bethe case that the discount factor depends on the random variable that we want todiscount.

More specifically, the market acts as if it assigns preemptively the discountfactor which applies to a given random variable. When the undiscounted maximumexpected value of a random variable µa is discounted with the discount factor Ba,the market is associating this random variable with the category of sure amounts.In other words, µa expresses an opportunity cost with respect to Fa: the amountof money which will be obtained at time 1 from investing in the risk-free asset attime 0, without taking on any borrowings.

When the undiscounted maximum expected value of a random variable µa isdiscounted with the discount factor Bb, the market is not associating this randomvariable with the category of sure amounts. In this case, µa expresses the price atmaturity of Fa: the amount of money which does not allow the buyer to set up aposition such that, by means of borrowings, he is able to arbitrage the market.

32

7 Beyond Sublinear Prices: Removal of Positive Ho-mogeneity

We have seen that sublinear prices can accommodate the presence of bid-askspreads for all the traded securities. In a sublinear pricing system, the purchaseof a combination of two random variables has a smaller cost than the purchaseof the two random variables separately. In the literature that analyzes the com-ponents of the spreads, see for example [10] and [13], the following distinction isusually made:

(i) a component of the bid-ask spread is a decreasing function of the order size:it is the order processing cost. In fact, transaction costs per unit, such astrading commissions, are usually decreasing.

(ii) Let us assume that we want to buy a given quantity of a security. If we are thebuying at market prices, our order is matched with the lowest ask price: if theorder is large enough, the quantity supplied at a given moment at that askprice does not completely match our quantity. For the outstanding quantity,our order is matched to the current lowest ask price, which is higher thanthe price at which we bought the first units of the security. In this respect,spreads are increasing functions of the order size.

In mathematical terms, a model that would fulfill both requirements would belocally sublinear, so that each time our order is matched to another order on themarket the processing costs relative to that order are decreasing, but should also becharacterized by a pricing functional monotonically increasing in the trade size.

7.1 A Graphical Analysis of Sublinear Prices

This section provides an analysis of a sublinear pricing model which does not differmuch from our previous analysis of sublinear prices. It is however propaedeutic tothe development of a model in which the analysis of price increments can be tracedback to this discussion, i.e. price increments will be shown to be sublinear.Let:

Y =[2 86 4

]

The price vector is πb = [4 5] up to K = [0 0]T and, beyond K, let the pricevector be πa = [4.2 5.5]. In other words, the origin determines a price increment.Moreover, the market is complete.

Let a = [a1 a2] be the vector representing the portfolio of quantities boughtor sold of security y1 and y2, respectively. In Figure 9, we can see that K definesfour areas in the a1, a2 plane.

The vector of consistent prices was indicated by π and in our example it wouldbe defined as π = [4 + 0.2α 5 + 0.5β]. We solve πb 5 π 5 πa with π = ϕ · Y ,so that we can build the sets L,L+, L++ containing, respectively, all the solutionsof the system of linear inequalities, the positive solutions, and the strictly positiveones.

33

Figure 9: Areas determined by K

a1 ≦ 0a2 ≧ 0

a1 ≧ 0a2 ≦ 0

a1 ≦ 0a2 ≦ 0

a1 ≧ 0a2 ≧ 0

A1 A2

A3 A4

a1

a2

0

The fact that the market is complete allows us to price (sublinearly) any randomvariable, and, specifically, we can use state prices derived from quoted prices tocalculate these prices.

We underline once again that prices obtained in this way are positive homoge-neous. We are now going to discuss an example in which prices are not positivehomogeneous, therefore they are not sublinear. However, we can already hint at thefact that it is not necessary to impose that the point in which prices change mustbe the origin. In other words, we are not forced to set K at the origin, becausewe can always translate the axes so that K has coordinates 0, 0: when we do this,our point of view changes from prices to changes in prices. The intuition is that,consistently with the previous discussion, price changes be sublinear.

7.2 Prices Increasing in Trade Size

Let again

Y =[2 86 4

]

and let us focus only on buying prices, so that the example is less complicated.Let us assume that in this case the price vector is given by π′ = [4 5] until pointK = [10 20] and, beyond K, and by π′′ = [4.2 5.5]. Notice that π′ corresponds toπb in the previous example, as well as π′′ corresponds to πa. We can give a graphicalinterpretation to the problem as in Figure 10.

There are a few things worth mentioning that can help us identify the similarities(and the diffences) between this case (especially Panel (b) in Figure 10) and thecase of globally sublinear prices:

(i) prices in areas A1 in Figure 10 and A1 in Figure 9 are conceptually different.In fact, the former identifies an area in which buying prices are smallest. Inthe other three areas we are buying more of a security (or of both), thereforewe are charged a higher price for the incremental units, because we jump

34

Figure 10: Translation of K

(a) K in the a1, a2 plane

10

20 K

A1

A3

A2

A4

a1

a2

0

(b) K in the translated plane

A1

A3 A4

A2

a1

a2

0

in a more expensive price system. The latter, instead, separates the area ofselling prices from the area of buying prices. The important analogy is that,regardless of the sign of the transaction, in both cases there is a change in theprice system.

From the mathematical point of view the two situations are equivalent: theyare identical in the subcase in whichK coincides with the origin. In fact, noticethat the case in which K = [k1 k2] is a general case of the more restrictivecase in which prices change only at K = [0 0], which is the case we havedeveloped so far for sublinear prices, just because, by definition, sublinearprices change only we the sign of the transaction is different.

(ii) Variables on the axes in Panel (b) of Figure 10 are not quantities, but quantityincrements. If we denote with a the vector of quantity increments, then itscomponents are ai = ai − ki where i = 1, 2 in our simple example.

This is why, intuitively, price changes should be sublinear and not prices. Onlyin the particular case in which K coincides with the origin the whole pricesystem becomes sublinear, because this forces buying prices to be constantfor purchases (positive a), which is nothing else but the definition of positivehomogeneity. In this case it is easily verified that positive homogeneity doesnot hold.

7.3 A Formal Argument for Sublinear Price Increments

Let us indicate with a a portfolio and with P (a) its price: P (a) = π∗ · a, where π∗

is the price vector applicable to portfolio a.We have assumed that π∗ is increasing with a, that is, for any a1 = a2, π∗(a1) =

π∗(a2). Consequently, P (a) is also increasing with a:

a1 = a2 ⇐⇒ P (a1) = P (a2)

35

Intuitively, a given portfolio a2 is preferred or equivalent to another portfolio a1 ifand only if it results in final payments which are greater than payments resultingfrom a1. If we indicate preferences with %, where is the asymmetric part and ∼the symmetric one, we have

a2 % a1 ⇐⇒ Y · a2 = Y · a1 ⇐⇒ Y (a2 − a1) = 0

The set B = b : Y b = 0 is the cone collecting all and only those portfolios whichdeliver positive payments: b ∈ B ⇐⇒ a + b % a. For any a ∈ Rn,b ∈ B the priceP (a + b) must be equal to or greater than P (a), i.e. P (a + b) = P (a).

For the case in which prices change from π′ to π′′ at pointK, such that π′′ ≥ π′, let usset a = K. The price of portfolio a is P (a) = π′·a = π′·K and P (a+b) = π′·a+π·b.The vector π is defined as

πi =

π′i if bi 5 0π′′i if bi > 0

for i = 1, 2

and it must beπ · b = 0 if Y · b = 0

This condition is equivalent to state that there exists ϕ = 0 such that ϕ · y = π(ftf - Theorem 6).

Monotonicity of prices is guaranteed by ϕ = 0: state prices reveal the possibilityof convenient super-replications.

Let us indicate with y = Y · a = Y · K. If we call ϕ′ the vector collecting stateprices relative to the price vector π′, then the price of portfolio a is given by

P (a) = π′ ·K = ϕ′ · Y ·K = ϕ′ · y

and the price of portfolio a + b is given by

P (a + b) = π′ ·K + π · b = ϕ′ · y + ϕ · Y · b = ϕ′ · y + ϕ · y′

where y′ is a generic non-negative vector.Define the price increment ∆π(y′) as

∆π(y′) = ϕ · y′

∆π is clearly positive homogeneous:

∆π(αy′) = ϕ · αy′ = α∆π(y′)

For ∆π to be subadditive, it must be

∆π(y′ + x′) 5 ∆π(y′) + ∆π(x′)

Let us indicate with b and c the portfolios replicating the random variables y′ andx′, respectively (Y · b = y′ and Y · c = x′). The price vectors for b, c and b + cwill be π · b, π · c, and π · (b + c).

The following cases are possible:

36

(i) bi, ci > 0: πi = πi = πi = π′′i . This implies πibi + πici = πi(bi + ci)

(ii) bi, ci < 0: πi = πi = πi = π′i. This implies πibi + πici = πi(bi + ci)

(iii) bi > 0 and ci < 0. It can be a) bi + ci > 0 or b) bi + ci < 0.

In both cases, from bi > 0 we derive πi = π′′i and from ci < 0 we deriveπi = π′i.

(a) bi + ci > 0:

πibi + πici = π′′i bi + π′ci and πi(bi + ci) = π′′i (bi + ci)

which leads to

π′′i (bi + ci) = π′′i bi + π′′i ci <

π′′i bi + π′ici

where the inequality holds because π′′ = π′ with ci < 0 by assumption.

(b) bi + ci < 0:

πibi + πici = π′′i bi + π′ci and πi(bi + ci) = π′i(bi + ci)

which leads to

π′i(bi + ci) = π′ibi +π′ici < π′′i bi +

π′ici

where the inequality holds because π′′ = π′ with bi < 0 by assumption.

(iv) bi > 0 and ci < 0: the same reasoning as in point (iii) applies.

If we indicate with L the convex hull generated by the different ϕ, we obtain:

∆π(y′) = maxϕ∈L

ϕ · y′

or∆π(y′) = max

ϕ∈L+

ϕ · y′

if some ϕ are negative. Finally, we obtain the pricing functional as:

π(y) = π(y + y′) = ϕ′ · y + ∆π(y′) = ϕ′ · y + maxϕ∈L+

ϕ · (y − y)

7.4 Deriving State Prices when Market Prices are Coherent

Let us start with an example to connect the concept of sublinearity of price changeswe have discussed before with the concept of state prices, which we have alwaysindicated with ϕ.

Let again

Y =[2 86 4

]; π′ = [4 5] up to K = [10 20]T ; π′′ = [4.2 5.5] otherwise.

37

Let us take a random variable z whose replication portfolio is given by a = [11 15].In other words, this random variable is located in area A2 of Figure 11:

Figure 11: The replicating portfolio of z in area A2

10

20 Kba(11, 15)

A1

A3

A2

A4

a1

a2

0

The price of z is given by

π(z) = π′ ·K + π′ · [a−K]−︸︷︷︸decrements

+π′′ · [a−K]+︸︷︷︸increments

that is

π(z) = [4 5] ·[

1020

]+ [4 5] ·

[0−5

]+ [4.2 5.5] ·

[10

]= 119.2

Let us calculate the payoffs of the random variable z, of the random variable z whichis replicated exactly by portfolio K, of the random variable z+ obtained from theportfolio replicating price increments with respect to K, and, finally, of the randomvariable z− obtained from the portfolio replicating price decrements with respectto K. The first two are given by

z = Y · a =[2 86 4

]·[1115

]=[22 + 12066 + 60

]=[142126

]

z = Y ·K =[2 86 4

]·[1020

]=[20 + 16060 + 80

]=[180140

]

and the others by

z+ = Y · [a−K]+ =[2 86 4

]·[10

]=[26

]

z− = Y · [a−K]− =[2 86 4

]·[

0−5

]=[−40−20

]

Let us now calculate state prices:

38

in A1 the price vector is given by π = π′ = [4 5], therefore

[ϕ11 ϕ1

2]︸︷︷︸ϕ1

·[2 86 4

]= [4 5]⇒ ϕ1 = [0.35 0.55]

in A2 the price vector is given by π = [4.2 5], therefore

[ϕ21 ϕ2

2]︸︷︷︸ϕ2

·[2 86 4

]= [4.2 5]⇒ ϕ2 = [0.33 0.59]

in A3 the price vector is given by π = [4 5.5], therefore

[ϕ31 ϕ3

2]︸︷︷︸ϕ3

·[2 86 4

]= [4 5.5]⇒ ϕ3 = [0.425 0.525]

in A4 the price vector is given by π = π′′ = [4.2 5.5], therefore

[ϕ41 ϕ4

2]︸︷︷︸ϕ4

·[2 86 4

]= [4.2 5.5]⇒ ϕ4 = [0.405 0.565]

Let us now calculate the price of the random variable step by step.The price of z is calculated with the vector of state prices which applies in A1:

π(z) = ϕ1 · z = [0.35 0.55] ·[180140

]= 140

In order to check that our discussion about increments being sublinear (thereforemaxima of linear functionals) is correct, we calculate the price of increments usingall the possible state prices:

ϕ1 · z+ = [0.35 0.55] ·[26

]= 4 ϕ1 · z− = [0.35 0.55] ·

[−40−20

]= −25

ϕ2 · z+ = [0.33 0.59] ·[26

]= 4.2 ϕ2 · z− = [0.33 0.59] ·

[−40−20

]= −25

ϕ3 · z+ = [0.425 0.525] ·[26

]= 4 ϕ3 · z− = [0.425 0.525] ·

[−40−20

]= −27.5

ϕ4 · z+ = [0.405 0.565] ·[26

]= 4.2 ϕ4 · z− = [0.405 0.565] ·

[−40−20

]= −27.5

Now that we have all the numbers, we can calculate

π(z) = π(z) + maxϕ∈L

ϕ · z+

︸︷︷︸ϕ2·z+

+ maxϕ∈L

ϕ · z−︸︷︷︸

ϕ2·z−

= 140 + 4.2− 25 = 119.2

Our calculations show that prices obtained replicating the random variable andusing state prices (with sublinear increments) return the same price.

We have not yet discussed the problem of possible super-replications, since wehave excluded them a priori : for the moment, notice that state-prices reveal super-replications area by area. We will investigate how to deal with non-positive state-prices in the following section.

39

7.5 Super-replications in this Setting

The purpose of this section is to investigate intuitively and analytically the conse-quences of the presence of convenient super-replication opportunities in this setting.

Let

Y =[6 42 8

]and Y −1 =

[0.2 −0.1−0.05 0.15

]

If we want to translate the representation from the a1a2 plane to the plane wherethe payoffs y1 and y2 are on the axes we need to derive and then plot the followingtwo lines:[1020

]=[

0.2 −0.1−0.05 0.15

]·[y1

y2

]⇒

0.2y1 − 0.1y2 = 10−0.05y1 + 0.15y2 = 20

⇒y2 = 2y1 − 100y2 = 400

3 + 13y1

Figure 12: The two lines in the y1y2 plane

A1

A3

A2

A4

y1b

−400b

50b

−100

4003

b y(140, 180)b

y2

Let us calculate state prices when price vectors are given by

π′ = [1 2]π′′ = [7 5]

⇒ π =

[1 2] for a1 5 10, a2 5 20[7 2] for a1 > 10, a2 5 20[1 5] for a1 5 10, a2 > 20[7 5] for a1 > 10, a2 > 20

⇒ Π =

1 27 21 57 5

State prices for all the four areas are given by[ϕ1

1 ϕ12

][ϕ2

1 ϕ22

][ϕ3

1 ϕ32

][ϕ4

1 ϕ42

]= Π · Y −1 =

[0.1 0.2][1.3 −0.4][−0.05 0.65][1.15 0.05]

40

In areas A2 and A3 state prices are negative for some states, which hints at thefact that super-replications are possible. We will show two examples of super-replications, one for each area.

The random variable y = Y ·K =[140 180

]T has price

π(y) = π′ ·K = 10 · 1 + 20 · 2 = 50

Let us take a random variable z in A2 such that its payoff is[140 100

]T . In thisarea, the price ϕ2

2 relative to payoff y2 is negative. The portfolio replicating z isgiven by

a = Y −1 · z =[188

]

and its price by

π(z) = 50 + 7 · (18− 10) + 2 · (8− 20) = 82 (7.1)

The random variable z′ in A2 with payoff[140 120

]T super-replicates z, since it isz′ ≥ z. It is replicated by portfolio a′ = [16 11]T and its price is

π(z′) = 50 + 7 · (16− 10) + 2 · (11− 20) = 74

therefore, since π(z′) < π(z), z′ conveniently super-replicates z. Analogously, therandom variable z′′ in A2 with payoff

[140 160

]T super-replicates z′, since it isz′′ ≥ z′. It is replicated by portfolio a′′ = [12 17]T and its price is

π(z′′) = 50 + 7 · (12− 10) + 2 · (17− 20) = 58

therefore, since π(z′′) < π(z′), z′′ conveniently super-replicates z′ and z.However, we can still do better: we already know that y conveniently super-

replicates z′′: its price is 50 and it has a greater payoff in state 2. The questionis: can the random variable y be super-replicated as we have done so far, that is,keeping constant payoff in state 1 and increasing payoff in state 2? We can try withthe random variable z = [140 200]T : it super-replicates y, since it is z ≥ y. It isreplicated by portfolio a = [8 23]T and its price is

π(z) = 50 + 5 · (23− 20) + 1 · (8− 10) = 63

therefore, since π(z) > π(y), z does not conveniently super-replicate y.Figure 13 contains a graphical interpretation of the problem we have analyzed

so far. The explanation for the fact that the arrows in area A2 point towardsthe lines that separate the area itself from the others is the following: area A2 ischaracterized by a negative price for state 2, therefore pushing the payoff in state 2 ofa given random variable results in the price of the random variable decreasing. Thepossibility to push this payoff is limited by the existence of price systems where stateprices are coherent (i.e. positive) and do not allow for convenient super-replications.

Let us take a random variable x in A3 such that its payoff is[100 180

]T . Inthis area, the price ϕ3

1 relative to payoff y1 is negative. The portfolio replicating xis given by

a = Y −1 · x =[

222

]

41

Figure 13: Direction of super-replications in A2

(a) Super-replication of z by y

b

z

A1

A3

A2

A4

y1b

−400b

50b

−100

4003

b y(140, 180)b

y2

(b) Super-replications increase payoffs of negativestate prices.

A1

A3

A2

A4

y1b

−400b

50b

−100

4003

b

y2

and its price by

π(x) = 50 + 5 · (22− 20) + 1 · (2− 10) = 52

The random variable x′ in A3 with payoff[120 180

]T super-replicates x, since itis x′ ≥ x. It is replicated by portfolio a′ = [6 21]T and its price is

π(z′) = 50 + 5 · (21− 20) + 2 · (6− 10) = 51

therefore, since π(x′) < π(x), x′ conveniently super-replicates x.Again, we can still do better: we already know that y conveniently super-

replicates x′: its price is 50 and it has a greater payoff in state 1. Just to besure that we have to stop at y, let us take the random variable x = [160 180]T : itsuper-replicates y, since it is x ≥ y. It is replicated by portfolio a = [14 19]T andits price is

π(x) = 50 + 7 · (14− 10) + 2 · (19− 20) = 76

therefore, since π(x) > π(y), x does not conveniently super-replicate y.Figure 14 contains another graphical interpretation of the problem.

—äm¨—

Now let us focus on the case in which state prices. We could be tempted to extendthis interpretation about the “direction” of super-replications so that, in the oppo-site case to what we have analyzed so far, we could end up with a situation similarto Figure 15, where in Area 2 we have a negative state price for state 1 and in Area3 we have a negative state price for state 2. In analogy with what we have arguedbefore, we could say that pushing the payoff to the maximum possible value for a

42

Figure 14: Direction of super-replications in A3

(a) Super-replication of x by y

bx

A1

A3

A2

A4

y1b

−400b

50b

−100

4003

b y(140, 180)b

y2

(b) Super-replications increase payoffs of negativestate prices.

A1

A3

A2

A4

y1b

−400b

50b

−100

4003

b

y2

given random variable could result in the price of the random variable decreasing.In this case, where there is no line (and no change of price system) forcing us tostop from increasing a given payoff, we could conclude that the random variablewith the smallest price would be the random variable with a payoff correspondingto the negative state price equal to +∞, and with price −∞.

However, we will show that this argument is not correct.

Figure 15: Is this situation possible?

A1

A3

A2

A4

y1b

−400b

50b

−100

4003

b

y2

43

Let

Y =[2 86 4

]and Y −1 =

[−0.1 0.20.15 −0.05

]

We move from the representation in the a1a2 plane to the plane where the payoffsy1 and y2 are on the axes in the following way[1020

]=[−0.1 0.20.15 −0.05

]·[y1

y2

]⇒

0.2y2 − 0.1y1 = 10−0.05y2 + 0.15y1 = 20

⇒y2 = 50 + 0.5y1

y2 = −400 + 3y1

Let us calculate state prices when price vectors are given by

π′ = [1 2]π′′ = [7 5]

⇒ π =

[1 2] for a1 5 10, a2 5 20[7 2] for a1 > 10, a2 5 20[1 5] for a1 5 10, a2 > 20[7 5] for a1 > 10, a2 > 20

⇒ Π =

1 27 21 57 5

State prices for all the four areas are given by[ϕ1

1 ϕ12

][ϕ2

1 ϕ22

][ϕ3

1 ϕ32

][ϕ4

1 ϕ42

]= Π · Y −1 =

[0.1 0.2][−0.4 1.3][0.65 −0.05][0.05 1.15]

Also in this case, in areas A2 and A3 state prices are negative for some states,therefore super-replications are possible.

At this point, we might still think that Figure 15 represents a possible situation.However, let us first take four portfolios in the four different areas:

A1 : a1 =[24

]y1 = Y · a1 =

[2 86 4

]·[24

]=

[3628

]

A2 : a2 =[1118

]y2 = Y · a2 =

[2 86 4

]·[1118

]=[166138

]

A3 : a3 =[

921

]y3 = Y · a3 =

[2 86 4

]·[

921

]=[186138

]

A4 : a4 =[1121

]y4 = Y · a4 =

[2 86 4

]·[1121

]=[190150

]

We can plot the points along with the lines we have found before, and the result isshown in Figure 16.It is now clear that what actually happens is that areas A2 and A3 are reversed,with area A2 moving from the bottom right corner of the plane to the top left, andwith area A3 moving in the opposite direction.

The conclusion we can draw from this discussion is that in our setting super-replications occur and are limited to areas in which some state prices are negative.The minimum price which can not be conveniently super-replicated is attained atthe edge of a given area, that is, where a new price system is triggered by increasingthe payoff in the negative state, provided that the new price system is characterizedby coherent prices.

44

Figure 16: Inversion between Areas A2 and A3

A1

A2

A3

A4

b y1

by2

b y3

b y4

y1

b −400

b50

b−100

4003

b

y2

In this setting, super-replications are like viruses that spread over an entire area,prices are drugs giving immunity (health) to some areas and they “quarantine” thesuper-replication virus to areas where prices are “infected”, and state prices aretests revealing which areas are infected. In this respect, the arrows pointing tohealthy areas are improvements in the condition of ill random variables.

7.5.1 Deriving L+ in this Market

In this market, we have been pricing random variables only by replication, thatis, by obtaining the replicating portfolio and computing its price. Then, we havepointed out that there are some clear opportunities to conveniently super-replicatethose random variables which are located in areas of the plane where state pricesare not all positive. In order to find the price at which it is not possible to furtherimprove the payoff corresponding to the negative state price, i.e. the “clean” priceof the random variable, we have pushed the random variable in the y1y2 plane tothe edge of the area.

However, it is possible to move from replicating portfolios to state prices withouttoo much effort. In order to price with the functional we have derived before, thatis

π(y) = ϕ′ · y + maxϕ∈L+

ϕ · (y − y)

we need to build the convex hull L generated by the state prices in the four areas:

45

in the first example we analyzed before, where

ϕ1 =ϕ2 =ϕ3 =ϕ4 =

[ϕ1

1 ϕ12

][ϕ2

1 ϕ22

][ϕ3

1 ϕ32

][ϕ4

1 ϕ42

]=

[0.1 0.2][1.3 −0.4][−0.05 0.65][1.15 0.05]

L is obtained as the set of all convex combinations of state prices (see representationin Figure 17): Since some state prices are negative, in order to obtain the price such

Figure 17: L

Lϕ1

ϕ2

b ϕ2(1.3,−0.4)

b

ϕ1(0.1, 0.2)

bϕ3(−0.05, 0.65)

bϕ4(1.15, 0.05)

that super-replications are not allowed, we need to derive L+ from L (see Figure18). In order to verify our claim that super-replications are excluded by imposing

Figure 18: L+

L+ϕ1

ϕ2

b ϕ2(1.3,−0.4)

b

ϕ1(0.1, 0.2)

bϕ3′

(0, 0.625)

b

ϕ3′′(0, 0.5)

b

ϕ2′(0.5, 0)

b

ϕ2′′(1.16667, 0)

bϕ3(−0.05, 0.65)

bϕ4(1.15, 0.05)

that random variables be priced with state prices belonging to L+, let us go back

46

to our previous example, where random variable z =[140 100

]T was replicated by

a =[18 8

]T . Also, z − y =[0 −80

]T . We can price z when super-replicationsare allowed as

πR(z) = ϕ′ · y + maxϕ∈L

ϕ · (z− y) =[0.1 0.2

]·[140180

]+[1.3 −0.4

]·[

0−80

]= 82

which coincides with the price obtained from the replicating portfolio (see page 41,Equation 7.1). On the other hand, if we want to know the price at which super-replications are not possible, we need to calculate

π(z) = ϕ′ · y + maxϕ∈L+

ϕ · (z− y) =[0.1 0.2

]·[140180

]+[0 0.5

]·[

0−80

]= 50

There are two things which are important to remark: first, the price at which therandom variable can not be conveniently super-replicated is the same we found by“pushing” the payoff up to the point in which it reaches the edge of its region.Second, notice that the random variable z =

[140 100

]T belongs to area A3, and

it is conveniently super-replicated by random variable y =[140 180

]T in area A1.The state-price in L+ which is such that ϕ · (z − y) is maximum is exactly thestate-price we obtain from the intersection of the segment linking state price ϕ3

and state price ϕ1 and of the ϕ2 axis in Figure 18, which we have called ϕ3′′ .

7.6 The Pricing Functional with Multiple Price Changes

Let K1,K2, . . . ,Kt the t portfolios triggering changes in the price system. If thereare m states of the world, 2m (or less for areas on the border of the plane) is thenumber of verteces enclosing areas in which a different pricing vector π determinesstate prices, the possibility of convenient super-replications, and the possibility ofarbitrages.

In a generic area s, if the vector ϕs > 0, then convenient super-replications arenot possible. On the other hand, if ϕs 6> 0, area t is entirely characterized by thepossibility of convenient super-replications.

In order to remove convenient super-replications from the market, non-negativeconvex combinations between ϕs and the other ϕ’s in areas surrounding ϕs shouldbe used in place of ϕs.

In each area, the price of a random variable y can be obtained in two ways:

(i) by taking one of the verteces Kr of generic area s to which the portfolio a,which replicates y, belongs (notice that in this case the plane is defined byquantities, i.e. a1 and a2), and computing

P (a) = Pr(Kr) + Ps(a−Kr)

(ii) by taking the price of random variable yr = Y · Kr (which is the randomvariable corresponding to a vertex of the area and the plane is defined bypayoffs, i.e. y1 and y2) and adding the price of y − yr, such that

π(y) = π(yr) + π(y − yr) = π(yr) + ϕs · (y − yr)

47

Let us now fix yr. The pricing functional in the 2m areas of which yr is a vertex is

π(y) = π(yr) + maxϕ · (y − yr)

where the maximum refers to all the areas of which yr is a vertex.Moreover, if we consider also all the functionals obtained from the other ys, they

are all smaller than or equal to π. In fact, in order to calculate those functionals weuse price vectors surrounding ys smaller with respect to the positive components ofy − ys and greater with respect to the negative components of y − ys.

Therefore, we can conclude that

π(y) = maxr

[π(yr) + max

ϕ∈Lr

ϕ · (y − yr)]

We can manipulate the functional in the following way:

π(y) = maxr

[π(yr) + max

ϕ∈Lr

ϕ · (y − yr)]

= maxr

maxϕ∈Lr

[π(yr) + ϕ · (y − yr)

]

= maxr

maxϕ∈Lr

[π(yr) + ϕ · y − ϕ · yr

]

= maxr

maxϕ∈Lr

(ϕ · y + cϕ

)

= maxϕ∈L

(ϕ · y + cϕ

)

where L =⋃Lr and cϕ = π(yr)− ϕ · yr. The component of the functional cϕ can

be interpreted as the difference between the market price of yr, that is, the price ofthe portfolio replicating yr computed with market prices increasing in trade size,and ϕ ·yr, which is the price of yr consistent with the price system to which randomvariable y belongs.

In this respect, the price assigned to yr in the current price system by ϕ is moreexpensive than its real price π(yr). Therefore, since it must be π(yr) 5 ϕ · yr ifnsr holds, cϕ must be negative, i.e. cϕ 5 0.

Before we move on to show an example, let us focus for a moment on possibleinterpretations of cϕ.

First of all, notice that the pricing functional π(y) = maxϕ∈L(ϕ·y+cϕ

)extends

the sublinear pricing functional π(y) = maxϕ∈L ϕ ·y: in fact, in a sublinear marketthe only price change occurs at K = [0 0]T , which makes π(y) = 0 and cϕ = 0.

In the sublinear case, the market acts as if it sets its price as the most expensivepossible one among the different prices proposed by the intermediaries, where stateprices ϕ in the set L+ represent the different prices set by the intermediaries.

Also in this case the market sets the most expensive price as the the marketprice. However, the way in which the different intermediaries set their prices isinfluenced now by the incremental quantities they are willing to supply: in fact, ifthey supply securities when the buyer has reached their level of incremental supply,thereby eroding the liquidity of the security, the price per unit is higher. Thehigher expense per unit is partially subsidized by the intermediary by discountinga fixed amount cϕ from the total price. However, since market prices are set in a

48

conservative way (i.e. the market price is a maximum), a large cϕ will be appliedonly in a situation where quantities are also large.

Example 7.6.1. Let

Y =[2 86 4

] π′ = [ 4 5 ] up to K = [10 20]T ;π′′ = [ 4.2 5.5 ] up to K ′ = [50 80]T ;π′′′ = [4.4 6 ] above K ′;

The plane is divided by point K and K ′ as Figure 19 indicates.

Figure 19: The plane with multiple price changes

10

20 K

50

80 K ′

b

b

A1 A2 A5

A3 A4 A6

A7 A8 A9

a1

a2

0

We can calculate cϕ as follows:

cϕ1 = 0cϕ2 = −0.2 · 10 = −2cϕ3 = −0.5 · 20 = −10cϕ4 = cϕ2 + cϕ3 = −12cϕ5 = cϕ2 − 0.2 · 50 = −12cϕ6 = cϕ4 − 0.2 · 50 = −22cϕ7 = cϕ3 − 0.5 · 80 = −50cϕ8 = cϕ4 − 0.5 · 80 = −52cϕ9 = cϕ5 + cϕ7 = −62

49

The price of the random variable z =[142 126

]T , which is replicated by portfolio

a =[11 15

]T , belongs to area A2 and has replication price 119.2 (see Example inSection 7.4, page 37). In this example, the vectors of consistent state prices are:

ϕ1 = [ϕ11 ϕ1

2]ϕ2 = [ϕ2

1 ϕ22]

ϕ3 = [ϕ31 ϕ3

2]ϕ4 = [ϕ4

1 ϕ42]

ϕ5 = [ϕ51 ϕ5

2]ϕ6 = [ϕ6

1 ϕ62]

ϕ7 = [ϕ71 ϕ7

2]ϕ8 = [ϕ8

1 ϕ82]

ϕ9 = [ϕ91 ϕ9

2]

= [0.35 0.55]= [0.33 0.59]= [0.425 0.525]= [0.405 0.565]= [0.31 0.63]= [0.385 0.605]= [0.5 0.5]= [0.48 0.54]= [0.46 0.58]

We can calculate the price of z as

π(z) = ϕ2 · z + cϕ2 = [0.33 0.59] ·[142126

]− 2 = 121.2− 2 = 119.2

and we find the same price as with the replicating portfolio. Moreorever, let uscalculate

ϕ1 · zϕ2 · zϕ3 · zϕ4 · zϕ5 · zϕ6 · zϕ7 · zϕ8 · zϕ9 · z

= [0.35 0.55] · [ 142 126]T = 119= [0.33 0.59] · [ 142 126]T = 121.2= [0.425 0.525] · [ 142 126]T = 126.5= [0.405 0.565] · [ 142 126]T = 128.7= [0.31 0.63] · [ 142 126]T = 123.4= [0.385 0.605] · [ 142 126]T = 130.9= [0.5 0.5] · [ 142 126]T = 134= [0.48 0.54] · [ 142 126]T = 136.2= [0.46 0.58] · [ 142 126]T = 138.4

which lead us to the price of the random variable if we consider the maximum valueof ϕs · z added to the respective cϕs

ϕ1 · z + cϕ1

ϕ2 · z + cϕ2

ϕ3 · z + cϕ3

ϕ4 · z + cϕ4

ϕ5 · z + cϕ5

ϕ6 · z + cϕ6

ϕ7 · z + cϕ7

ϕ8 · z + cϕ8

ϕ9 · z + cϕ9

= 119 + 0 = 119= 121.2− 2 = 119.2 ⇐ max= 126.5− 10 = 116.5= 128.7− 12 = 116.7= 123.4− 12 = 111.4= 130.9− 22 = 108.9= 134− 50 = 84= 136.2− 52 = 84.2= 138.4− 62 = 76.4

and again, we find the price to be 119.2.

50

8 Conclusions and Further Research

Consistently with the never-ending research for increasingly more general modelswhich is peculiar to every branch of mathematical science, our analysis of mod-els that make use of the property of sublinearity for prices and price incrementsallows us to give a better representation of reality without making assumptionswhich constrast with the model they start from (i.e. the linear pricing model), and,as such, these models represent generalizations which can, given some additionalassumptions, revert to the standard linear model.

Sublinear models, which have been analyzed in the first part of this work, canaccommodate limits to arbitrage arising from transaction costs, taxes and bid-askspreads. Frictions can be incorporated in all the assets traded on the market, inreplicable contingent claims as well as in the non-replicable ones, and, finally, in theriskless asset. We have also shown that the distinction and the derivation of therisk-neutral probabilities from state-prices is not as straightforward as in the linearcase, and there are cases in which the discounted expectation of future payoffs isnot equal to the price of the contingent claim.

Locally sublinear pricing models, on the other hand, are even more versatile andpowerful, since, at the lowest level of sophistication (with only one price change),they allow for the possibility of different price systems not only between buyingand selling prices, but between the price of any random variable depending onthe direction and size of a given trade. At a higher level of complexity, locallysublinear prices can accommodate a representation of a market in which assets arenot infinitely liquid, thereby removing positive homogeneity from prices, which isa property of sublinear prices. The removal of this property is also very useful inrepresenting the possibility, often present on real markets, that arbitrages be feasibleonly for limited amounts.

Especially with respect to locally sublinear prices, there are still very promisingresearch opportunities that deserve to be explored. First of all, it could be inter-esting to analyze the properties and behavior of risk-neutral probabilities, discountfactors, borrowing and investing rates. Moreover, the dimensions of the payoff ma-trix could be increased, and state prices could be derived for the case in which themarket is not complete. Finally, the fact that the functional appears to be themaximum of affine functionals suggests the possibility that the constant cϕ and theambiguity index in the variational representation of preferences in [11] be linked.

In addition to this, a more in-depth analysis of cϕ and of the composition of thegrid of points where the price system changes could provide some more interestinginsights, which we can anticipate and summarize qualitatively here.

Transparency A more transparent price system allows for broad disseminationof information and, as such, promotes what is generally referred to as pricediscovery. In other words, a market where prices increase gradually with thesize of the trades enables participants who want to buy or sell to find out, or,better, to have a reasonably clear idea, of the price at which a given trade- where the word trade is meant to specify simultaneously the size of theorder and its price - can be agreed. In our setting, a transparent price system

51

would be structured with a sufficiently large number of grid points such thatcϕ’s variations increase proportionally with size increments. Notice that thisis a different concept from liquidity: in fact, we have willingly ignored thedistance between two points in the grid. As long as the grid gives a realisticrepresentation of the prices at which market participants can expect to trade,transparency is satisfied, even though we have ignored that it might well bepossible that even for minimal quantities prices could experience substantialvariations.

Liquidity As we have already mentioned, the liquidity of a given market is anindication of the ease with which it allows to buy and sell the securities, andmultiples thereof, that can be traded. In this respect, liquidity is one of thecomponents of bid-ask spreads, and lack of liquidity for a given security in oursetting would be signaled by prices increasing more than linearly with respectto quantities. Moreover, the liquidity of the security would be increased anddecreased by moving the grid on the right and on the left (notice that isdifferent from increasing and decreasing cϕ, which refer to the market as awhole, as we argue below).

Depth If the most appropriate concept relating the variation of prices with in-creases in the size of the trade a given security is that of liquidity, when weanalyze the market as a whole we generally refer to the depth of the market.The depth of a market can be assessed by analyzing the behavior of priceswhen large trades (or block trades) hit the market. In other words, a marketlacks depth if trading is difficult in any large quantities, which results in bid-ask spreads increasing with the size of the trades. In our setting, the differentcϕ’s indicate if and how variations in prices make large trades difficult: forinstance, we could compare two markets based on values of the different cϕ,provided that they have the same K. It is obvious that the three concepts oftransparency, liquidity and depth are not mutually exclusive and often over-lap, but it is nonetheless important to understand that a model for locallysublinear prices is much more realistic in that it relaxes the assumption ofpositive homogeneity, and all the three concepts we have explained are key tothe financial interpretation of what the removal of a mathematical propertyactually entails.

52

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Sublinear and Locally Sublinear Prices