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2017 | OriginalPaper | Chapter

4. General Equilibrium Theory and No-Arbitrage

Authors : Emilio Barucci, Claudio Fontana

Published in: Financial Markets Theory

Publisher: Springer London

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Abstract

This chapter deals with general equilibrium theory in a risky environment, where agents interact in a financial market. The chapter starts by presenting the notion of Pareto optimality and its implications in terms of risk sharing. The concept of rational expectations equilibrium is introduced and characterized in the context of a two-period economy. Different financial market structures are considered, with a particular attention to the important case of complete markets. The last part of the chapter is devoted to the fundamental theorem of asset pricing, which relates the absence of arbitrage opportunities to the existence of a strictly positive linear pricing functional. The relation of this important result to the existence of an equilibrium of an economy and its implications for the valuation of financial assets are also discussed.

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previous chapter Portfolio, Insurance and Saving Decisions
next chapter Factor Asset Pricing Models: CAPM and APT
Footnotes
1
Some elements of the economy are common knowledge among the agents if each agent knows them, knows that the other agents know them, knows that the other agents know that he knows and so on. In that context, agents cannot agree to disagree (see Fuydenberg & Tirole [746] for a rigorous definition of common knowledge).
 
2
Using the notation a ls n introduced before, this corresponds to \(a_{s}^{n} = (d_{sn}, 0,\ldots, 0) \in \mathbb{R}^{L}\), where d sn is the dividend paid by asset n in correspondence of the state of world ω s , for all n = 1, …, N and s = 1, …, S.
 
3
In the present context, the separating hyperplane can be stated as follows (see Duffie [593]). Let M and H be closed convex cones in \(\mathbb{R}^{S+1}\) such that M ∩ H = {0}. Then, if H does not contain a linear subspace other than {0}, than there is a non-zero linear functional \(F: \mathbb{R}^{S+1} \rightarrow \mathbb{R}\) such that F(x) < F( y), for all x ∈ M and y ∈ H∖{0}.
 
4
In the following, with some abuse of notation, we equivalently denote by \(\tilde{x}\) or \(x = (x_{1},\ldots,x_{S})\, \in \, \mathbb{R}^{S}\) the random variable \(\tilde{x}\) taking values (x 1, …, x S ) in the S states of the world.
 
5
In the absence of arbitrage opportunities, it can be shown that \(\inf \{\,p^{\top }z: z \in \mathbb{R}^{N}\text{ and }Dz \geq c\} =\min \{\, p^{\top }z: z \in \mathbb{R}^{N}\text{ and }Dz \geq c\}\) and, similarly, \(\sup \{\,p^{\top }z: z \in \mathbb{R}^{N}\text{ and }Dz \leq c\} =\max \{\, p^{\top }z: z \in \mathbb{R}^{N}\text{ and }Dz \leq c\}\), meaning that the infimum and the supremum are actually attained by some portfolios, see Föllmer & Schied [721, Theorem 1.32].
 
6
With some abuse of notation, we denote by \(u_{\tilde{x}}^{i}\) the derivative of the function u i with respect to its second argument, for i = 1, …, I.
 
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Metadata
Title
General Equilibrium Theory and No-Arbitrage
Authors
Emilio Barucci
Claudio Fontana
Copyright Year
2017
Publisher
Springer London
Book
Financial Markets Theory

Print ISBN: 978-1-4471-7321-2

Electronic ISBN: 978-1-4471-7322-9

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-1-4471-7322-9_4