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Mathematics and Financial Economics
Erschienen in: Mathematics and Financial Economics 4/2013

01.09.2013

Scale-invariant asset pricing and consumption/portfolio choice with general attitudes toward risk and uncertainty

verfasst von: Costis Skiadas

Erschienen in: Mathematics and Financial Economics | Ausgabe 4/2013

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Abstract

Motivated by notions of aversion to Knightian uncertainty, this paper develops the theory of competitive asset pricing and consumption/portfolio choice with homothetic recursive preferences that allow essentially any homothetic uncertainty averse certainty-equivalent form. The market structure is scale invariant but otherwise general, allowing any trading constraints that scale with wealth. Technicalities are minimized by assuming a finite information tree. Pricing restrictions in terms of consumption growth and market returns are derived and a simple recursive method for solving the corresponding optimal consumption/portfolio choice problem is established.
Vorheriger Artikel Financial market equilibria with heterogeneous agents: CAPM and market segmentation
Nächster Artikel Acceptability indexes via -expectations: an application to liquidity risk

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Fußnoten
1
The homothetic case of second-order expected utility of [21, 27] is also consistent with the present paper. Skiadas [36] argues that in high frequency with Brownian/Poisson information, recursive utility with smooth second order expected utility becomes quantitatively indistinguishable from expected utility, a fact that limits the appeal of this approach for asset pricing and portfolio choice.
 
2
This is the same as saying that we fix a positive definite matrix \(\Pi \in \mathbb{R }^{n\times n}\) and write \(\left( x\mid y\right) =x^{\prime }\Pi y\) for all column vectors \(x,y\in \left( 0,\infty \right) ^{n}\).
 
3
The term subdifferential is often used in place of superdifferential, following [28], who nevertheless suggests on p. 308 that the term “superdifferential” might be more appropriate. Our use of the term corresponds to Clarke’s notion of generalized gradient (see, for example, [6]) as well as the way the term is used in the literature of viscosity solutions of Hamilton-Jacobi-Bellman equations (see, for example, [2]). The terms Fréchet superdifferential (and \(D\)-superdifferential for the finite-dimensional case) also map to what we call superdifferential, given concavity (see [7, p. 142] ).
 
4
This definition is consistent with the use of \(u_{\gamma }\) in [39], where it is important that the range of \(u_{\gamma }\) is either \(\pm \left( 0,\infty \right) \) or \( \mathbb{R } .\) Nothing would change in the present paper, however, if we were to redefine \(u_{\gamma }\left( z\right) =\big ( z^{1-\gamma }-1\big ) /\left( 1-\gamma \right) ,\) which is more consistent in the sense that \(u_{1} =\lim _{\gamma \rightarrow 1}u_{\gamma }.\)
 
5
See [38] for an axiomatic characterization of recursive utility with a constant EIS, constant rate of impatience and constant source-dependent CRRA.
 
6
Note that the presentation in [39] is based on a filtration that is generated by two sources of uncertainty, an ambiguous one (horse-race uncertainty) and an unambiguous one (roulette risk). One could take the filtration \(\left\{ \mathcal{F }_{t}\right\} \) of the present paper to coincide with that in [39] (as, for example, would be required if one were to adopt the source-dependent CRRA specification). In Example 15, we take the alternative view that roulette risk is nontradeable and this paper’s filtration \(\left\{ \mathcal{F }_{t}\right\} \) is identified with the horse-race filtration in [39].
 
7
Clarke’s result can be applied at the CE \(\nu ={\bar{\upsilon }}_{F,t}\) at each nonterminal spot \(\left( F,t\right) \). A quick way to show the superdifferential expression is to use an envelope theorem for the directional derivatives of \(\nu \) at \(w,\) which characterize \(\partial \nu \left( w\right) \) via Theorem 23.2 of [28].
 
8
In fact, \(h_{t}\) need not even be monotone; the proportional aggregator \(g_{t}\left( x\right) =\sqrt{x}\exp \left( e^{-1}-e^{-x}\right) \) satisfies condition 17 but defines the elasticity function \(h_{t}\left( x\right) =\left( 1/2\right) +xe^{-x}\), which is hump-shaped.
 
9
To see why, fix any nonterminal spot \(\left( F,t-1\right) \) with immediate successor spots \(\left( F_{1},t\right) ,\dots ,\left( F_{n},t\right) .\) Let \({\bar{\upsilon }}_{F,t-1}:\left( 0,\infty \right) ^{n}\rightarrow \left( 0,\infty \right) \) be the CE defined by (13), and for each \(\alpha \in A\left( F,t-1\right) \) let \(\bar{R}_{F,t}\left( \alpha \right) \) be the vector in \( \mathbb{R } ^{n}\) whose \(i\)th component is the realization of \(R_{F,t}\left( \alpha \right) \) at spot \(\left( F_{i},t\right) .\) Lemma 1 applies with \(\nu ={\bar{\upsilon }}_{F,t-1},\) \(X=\left\{ x\in \mathbb{R } ^{n}:x\le \bar{R}_{F,t}\left( \alpha \right) -\bar{R}_{F,t}\left( \psi \left( F,t\right) \right) ,\ \alpha \in A\left( F,t-1\right) \right\} ,\) and \(\left( x\mid y\right) =\sum _{i=1}^{n}x_{i}y_{i}P\left[ F_{i}\mid F\right] .\)
 
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Metadaten
Titel
Scale-invariant asset pricing and consumption/portfolio choice with general attitudes toward risk and uncertainty
verfasst von
Costis Skiadas
Publikationsdatum
01.09.2013
Verlag
Springer Berlin Heidelberg
Erschienen in
Mathematics and Financial Economics / Ausgabe 4/2013
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI
https://doi.org/10.1007/s11579-013-0103-z

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