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http://aleplasmati.comuv.com/?page_id=2Alessandro Plasmati's final thesis for the Master of Science in Finance at Bocconi University, Milan, Italy.The purpose of this work is to analyze extensions of linear pricing models which incorporate a great deal of characteristics of real life prices, which can be generally referred to as transaction costs. Linear prices can not account for the presence of frictions in the assets traded on the market, and the presence of prices that allow for arbitrages creates the possibility to set up trading strategies resulting in unbounded proﬁts. The absence of any form of transaction costs and the possibility to scale the size of a trade without any eﬀect on the price of the transaction are two assumptions that oversimplify the price system 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 to diﬀerentiate between bid and ask prices, leading to a more realistic model in which arbitrages are more diﬃcult because of frictions, and a weaker form of price incon- sistency arises, namely the possibility of convenient super-replications. We expand the concept of internal consistency of a price system to the case where the riskless asset is aﬀected by frictions and impose conditions on prices quoted on the market so that arbitrages and super-replications are not allowed.

` Universita Commerciale Luigi Bocconi Facolt` di Economia a Master of Science in Finance

Sublinear and Locally Sublinear Prices

Relatore

Prof. Erio CastagnoliControrelatore

Prof. Fabio Maccheroni

Candidato

Alessandro Plasmati

1185198

Anno Accademico 2007/2008

Contents1 Introduction 2 Linear Functionals 2.1 Criticism of Linear Prices . . . . . . . . . . . . . . . . . . . . . . . . 3 Sublinear Functionals 3.1 Characterization . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Hahn-Banach Theorem . . . . . . . . . . . . . . 3.1.2 General Representation of Sublinear Functionals 3.1.3 Positivity of a Sublinear Functional . . . . . . . 3.1.4 Monotonicity of a Sublinear Functional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 6 6 7 7 7 9 9 10 11 11 11 12 13 13 15 16 17 18 19 19 19 20 20 22 22 24 24 24 25 29 33 33 34 35 37 40 45 47

4 A First (Sublinear) Extension of Linear Prices 4.1 Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 No Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Sublinear Extensions of Linear Functionals in Incomplete Markets 4.4 The Financial Perspective: Super-replication . . . . . . . . . . . . 4.5 A complete example . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Sublinear Prices 5.1 The Concept of Convenient Super-replication . . . . . . . . . . . . 5.2 The properties of the pricing functional and their consequences on the internal coherence of prices . . . . . . . . . . . . . . . . . . . . . 5.3 The Role of Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Complete Market - Coherent Prices . . . . . . . . . . . . . . 5.4.2 Complete Market - Possible Super-replications . . . . . . . . 5.4.3 Complete Market - Arbitrages . . . . . . . . . . . . . . . . . 5.4.4 A further note . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Frictions on the Riskless Asset 6.1 The probability vector and discount factor in a sublinear market . . 6.2 Dynamics of Super-replications and of Arbitrage . . . . . . . . . . . 6.2.1 Prices without Frictions . . . . . . . . . . . . . . . . . . . . . 6.2.2 Prices with Frictions on the Risky Assets - No Frictions on B 6.2.3 Prices with Frictions on the Risky Assets and on B . . . . . . 6.3 Possible Inversion of Discount Factors . . . . . . . . . . . . . . . . . 7 Beyond Sublinear Prices: Removal of Positive Homogeneity 7.1 A Graphical Analysis of Sublinear Prices . . . . . . . . . . . . . 7.2 Prices Increasing in Trade Size . . . . . . . . . . . . . . . . . . 7.3 A Formal Argument for Sublinear Price Increments . . . . . . . 7.4 Deriving State Prices when Market Prices are Coherent . . . . 7.5 Super-replications in this Setting . . . . . . . . . . . . . . . . . 7.5.1 Deriving L+ in this Market . . . . . . . . . . . . . . . . 7.6 The Pricing Functional with Multiple Price Changes . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 Conclusions and Further Research

51

1

Introduction

The purpose of this work is to analyze extensions of linear pricing models which incorporate a great deal of characteristics of real life prices, which can be generally referred to as transaction costs. Linear prices can not account for the presence of frictions in the assets traded on the market, and the presence of prices that allow for arbitrages creates the possibility to set up trading strategies resulting in unbounded prots. The absence of any form of transaction costs and the possibility to scale the size of a trade without any eect on the price of the transaction are two assumptions that oversimplify the price system and can be relaxed. The properties of subadditivity and positive homogeneity that characterize sublinear prices allow for the construction of a price system in which it is possible to dierentiate between bid and ask prices, leading to a more realistic model in which arbitrages are more dicult because of frictions, and a weaker form of price inconsistency arises, namely the possibility of convenient super-replications. We expand the concept of internal consistency of a price system to the case where the riskless asset is aected by frictions and impose conditions on prices quoted on the market so that arbitrages and super-replications are not allowed. While a sublinear price system represents an improvement over the standard linear case, it still suers from the limitation that derives from the property of positive homogeneity. The absence of any dependence of prices from the size of the trade, which is a corollary of positive homogeneity, automatically excludes any possible modeling of important issues such as liquidity of the securities and transparency of prices. In order to extend sublinear models we allow prices to vary at dierent thresholds determined by portfolios of securities traded on the market. In this way, prices are increasing with the size and direction of the trade, thus making the price system as a whole not positive homogeneous. We show that a model of this kind is characterized by sublinear price increments and derive the pricing functional as the maximum of a family of convex functionals, which in turn can be represented as maxima of ane functionals. We conclude with interpretations of this model based mainly on the possibility to accommodate the issues of liquidity, transparency, and market depth. The analysis is organized as follows: in Section 2 we make a brief introduction about linear prices, their usual representation, and the shortcomings of a linear pricing model. In Section 3 we show the mathematical properties of sublinear functionals. Section 4 contains a discussion about the possibility to extend a linear pricing model by introducing sublinear prices for non-replicable contingent claims. A fully sublinear model is presented in Section 5 and more insights into the problems of internal coherence of prices in the sublinear case can be found in Section 6, where the riskless asset is aected by frictions. In Section 7 we develop a model for locally sublinear prices, where the removal of positive homogeneity allows us to limit the possibility of arbitrages and convenient super-replications to a subset of the whole price system and also to introduce the issue of liquidity aecting the securities traded on the market. In the concluding remarks in Section 8 we provide some comments about possible developments of future research on this topic. 5

2

Linear Functionals

Let X be a linear space over the eld R of real numbers. Denition 1. A map F : X R is called a functional. A functional F is linear if, for any x, y X and , R, F (x + y) = F (x) + F (y) (2.1)

In nancial mathematics, the usual assumption to build pricing models in discrete and continuous time is that linear functionals map payo vectors into prices. We will conne ourselves to the nite case. The set = {1 , 2 , . . . , m } will represent the states of the world in a one-period model and the contingent claim T can be written as y = y1 y2 . . . ymm m m m

y=i=1

yi ei F (y) = F

yi e ii=1

=i=1

yi F (ei ) =i=1

yi i = y

(2.2)

where ei is the i-th canonical vector in Rm , i.e. its i-th component is equal to 1 and the others are equal to 0. From the derivation in Equation 2.2, we deduce that any linear functional is represented by a vector . Therefore, when X = Rm , the price of any contingent claim y is given by y1 m y2 F (y) = yi F (ei ) = 1 2 . . . m . = y (2.3) . .i=1

ym

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 dened on / the market (and can not be dened by F ). This is a clear limitation of a linear pricing system. Remark 2. It must be noted that linear prices have the following properties: Additivity: F (x + y) = F (x) + F (y) = x + y = (x + y) Homogeneity: F (x) = F (x) = ( x)

2.1

Criticism of Linear Prices

Linear prices can not account for the possibility of frictions in the prices of the securities traded on the market. In case of market incompleteness and the introduction of non-replicable contingent claims (see Section 4.3), linear pricing systems do not dene 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 dened; / 6

(b) x, y X : the price to sell and to buy x (or y) is the same, and so is the price to sell or to buy x and y together or separately, and to buy or sell one unit or a multiple of x (or y). Point (b) can be summarized with the following table:Linearity Fb (x) = Fa (x) Fb (x + y) = F (x) + F (y) Fa (x + y) = F (x) + F (y) Fb (x) = F (x) Fa (x) = F (x) Meaning No bid-ask spread No multiple-sale advantage No multiple-purchase advantage Bid Quantity Invariance Ask Quantity Invariance Reality Fb (x) Fa (x) Fb (x + y) Fb (x) + Fb (y) Fa (x + y) Fa (x) + Fa (y) Fb (x) Fb (x) Fa (x) Fa (x)

Remark 3. It is quite obvious to read the rst three entries of the table as a direct consequence of the rst property in Remark 2 (i.e. additivity). By substituting this property with subadditivity and constraining homogeneity to hold only on R+ , the functional becomes sublinear. It is also natural to read the last two entries as a consequence of homogeneity. However, in this case, it is not straightforward to determine the prevailing eect on the price: in fact, there can be dierent factors pushing non-positive homogeneous prices upward or downward at the same time.

3

Sublinear Functionals0, (3.1)

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