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Finance and Stochastics
Erschienen in: Finance and Stochastics 3/2018

25.05.2018

Equilibrium returns with transaction costs

verfasst von: Bruno Bouchard, Masaaki Fukasawa, Martin Herdegen, Johannes Muhle-Karbe

Erschienen in: Finance and Stochastics | Ausgabe 3/2018

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Abstract

We study how trading costs are reflected in equilibrium returns. To this end, we develop a tractable continuous-time risk-sharing model, where heterogeneous mean–variance investors trade subject to a quadratic transaction cost. The corresponding equilibrium is characterized as the unique solution of a system of coupled but linear forward–backward stochastic differential equations. Explicit solutions are obtained in a number of concrete settings. The sluggishness of the frictional portfolios makes the corresponding equilibrium returns mean-reverting. Compared to the frictionless case, expected returns are higher if the more risk-averse agents are net sellers or if the asset supply expands over time.
Vorheriger Artikel Robust pricing–hedging dualities in continuous time
Nächster Artikel Non-implementability of Arrow–Debreu equilibria by continuous trading under volatility uncertainty

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Fußnoten
1
Liquidity premia with exogenous asset prices are studied by [13, 26, 33, 19, 14], for example.
 
2
Mean-reverting fundamentals also drive the mean-reverting dynamics in the overlapping-generations model with linear costs studied in [44], for example.
 
3
This will be the time-discount rate below; for infinite-horizon models, it needs to be strictly positive.
 
4
If one instead assumes that the volatility follows some (sufficiently integrable) stochastic process https://static-content.springer.com/image/art%3A10.1007%2Fs00780-018-0366-6/MediaObjects/780_2018_366_IEq31_HTML.gif , the subsequent characterization of individually optimal strategies and equilibrium returns in terms of coupled but linear FBSDEs as in (A.1) and (A.2) still applies. However, the stochastic volatility then appears in the coefficients of this equation, so that the solution can no longer be characterized (semi-)explicitly in terms of matrix power series. Instead, a “backward stochastic Riccati differential equation” appears as a crucial new ingredient already in the one-dimensional models with exogenous price dynamics studied by [31, 6].
 
5
The assumption of quadratic rather than proportional costs is made for tractability. However, buoyed by the results from the partial equilibrium literature, we expect the qualitative properties of our results to be robust across different small transaction costs; compare with the discussion in [36].
 
6
More general specifications do no seem natural for the tax interpretation of the model. Note, however, that the mathematical analysis below only uses that \(\varLambda\) is symmetric and positive definite.
 
7
This means that agents stop trading near maturity when there is not enough time left to recuperate the costs of further transactions. If \(T=\infty\), this terminal condition is replaced by the transversality conditions implicit in https://static-content.springer.com/image/art%3A10.1007%2Fs00780-018-0366-6/MediaObjects/780_2018_366_IEq75_HTML.gif for \(\delta>0\).
 
8
Several groups of noise traders with different mean positions as considered in [18, Sect. 4] can be treated analogously.
 
9
Due to the degeneracy of the forward component (A.1), general FBSDE theory as in [15] only yields local existence. However, the direct arguments developed below allow us to establish global existence and also lead to explicit representations of the solution in terms of matrix power series.
 
10
Note that \(\int_{0}^{\infty}e^{-\sqrt{\Delta} s} B \tilde{\xi}_{s} \,ds\) is square-integrable because https://static-content.springer.com/image/art%3A10.1007%2Fs00780-018-0366-6/MediaObjects/780_2018_366_IEq290_HTML.gif and all eigenvalues of \(\sqrt{\Delta}\) are at least \(\delta/2\).
 
11
Note that the inverses are well defined by Lemma A.3.
 
12
If \(A \in\mathbb{R}^{\ell\times\ell}\) and all eigenvalues of \(A\) are real, \(O\) can be taken as an open neighborhood of \(\lambda_{1}, \dots, \lambda_{m}\) in ℝ, provided that \(f\) is also real-valued.
 
13
If \(A\), \(\lambda_{1}, \ldots, \lambda _{m}\), \(O\) and \(f\) are all real-valued and \(f\) is defined on the spectrum of \(A\), the Hermite interpolating polynomial is also real-valued.
 
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Metadaten
Titel
Equilibrium returns with transaction costs
verfasst von
Bruno Bouchard
Masaaki Fukasawa
Martin Herdegen
Johannes Muhle-Karbe
Publikationsdatum
25.05.2018
Verlag
Springer Berlin Heidelberg
Erschienen in
Finance and Stochastics / Ausgabe 3/2018
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI
https://doi.org/10.1007/s00780-018-0366-6

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