Atlas models of equity markets

Banner, Adrian D.; Fernholz, Robert; Karatzas, Ioannis

doi:10.1214/105051605000000449

Cited by 135 publications

(255 citation statements)

References 14 publications

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“…To explain this fact, Fernholz considers a special case of (1) called the Atlas model where δ n > 0, and δ i = 0 for all i = n. He conjectured that the capital distribution curve obtained from the diffusion satisfying the SDE for the Atlas model converges to a stationary distribution under which it is roughly a straight line. It is interesting to note that the phase transition phenomenon described in Theorem 2 predicts some remarkable outcomes about models of stock market capitalizations as given in Banner, Fernholz, and Karatzas [6], [13], and [14].…”

Section: The Market Modelmentioning

confidence: 87%

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A phase transition behavior for Brownian motions interacting through their ranks

Chatterjee

Pal

2009

Probab. Theory Relat. Fields

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Abstract. Consider a time-varying collection of n points on the positive real axis, modeled as Exponentials of n Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. If at each time point we divide the points by their sum, under suitable assumptions the rescaled point process converges to a stationary distribution (depending on n and the vector of drifts) as time goes to infinity. This stationary distribution can be exactly computed using a recent result of Pal and Pitman. The model and the rescaled point process are both central objects of study in models of equity markets introduced by Banner, Fernholz, and Karatzas. In this paper, we look at the behavior of this point process under the stationary measure as n tends to infinity. Under a certain 'continuity at the edge' condition on the drifts, we show that one of the following must happen: either (i) all points converge to 0, or (ii) the maximum goes to 1 and the rest go to 0, or (iii) the processes converge in law to a non-trivial Poisson-Dirichlet distribution. The underlying idea of the proof is inspired by Talagrand's analysis of the low temperature phase of Derrida's Random Energy Model of spin glasses. The main result establishes a universality property for the BFK models and aids in explicit asymptotic computations using known results about the Poisson-Dirichlet law.

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Section: The Market Modelmentioning

confidence: 87%

“…Stochastic Portfolio Theory. A detailed study of the solutions of SDE (1) in general was taken up in a paper by Banner, Fernholz, and Karatzas (BFK) [6]. These authors actually consider a more general class of SDEs than (1) in which the drifts as well as the volatilities of the different Brownian motions depend on their ranks.…”

Section: Introductionmentioning

confidence: 99%

A phase transition behavior for Brownian motions interacting through their ranks

Chatterjee

Pal

2009

Probab. Theory Relat. Fields

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“…11 Note that the existence of the limits in equations (3.16)-(3.18) is a weaker assumption than the existence of a stationary distribution of bank assets (Banner, Fernholz, and Karatzas, 2005). 12 We refer the reader to Fernholz (2016b) for a proof of the theorem.…”

Section: The Distribution Of Bank Assetsmentioning

confidence: 99%

Why Are Big Banks Getting Bigger?

Fernholz

Koch

2016

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The U.S. banking sector has become substantially more concentrated since the 1990s, raising questions about both the causes and implications of this consolidation. We address these questions using nonparametric empirical methods that characterize dynamic power law distributions in terms of two shaping factors -the reversion rates (a measure of crosssectional mean reversion) and idiosyncratic volatilities of assets for different size-ranked banks. Using quarterly data for subsidiary commercial banks and thrifts and their parent bank-holding companies, we show that the greater concentration of U.S. bank-holding company assets is a result of lower mean reversion, a result consistent with policy changes such as interstate branching deregulation and the repeal of Glass-Steagall. In contrast, the greater concentration of both U.S. commercial bank and thrift assets is a result of higher idiosyncratic volatility, yet, idiosyncratic volatility of parent bank-holding company assets fell. This contrast suggests that diversification through non-banking activities has reduced the idiosyncratic asset volatilities of the largest bank-holding companies and affected systemic risk.

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“…Then, fixing a probability density ψ ∈ C ∞ c (R) supported on [0, 3] with (5.18) c ψ = inf s∈ [1,2] {ψ(s)} > 0 , we set ψ δ (s) = δ −1 ψ(s/δ) and consider…”

Section: Proof Of Proposition 25mentioning

confidence: 99%

“…Proof of Part (a). Recall from [2,Section 3] that in any solution of (1.1) the ordered particles X (1) (t) ≤ X (2) (t) ≤ · · · ≤ X (N ) (t) satisfy the sds…”

Section: Proof Of Proposition 27mentioning

confidence: 99%

Large Deviations for Diffusions Interacting Through Their Ranks

Dembo

Shkolnikov

Varadhan

et al. 2016

Comm Pure Appl Math

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Abstract. We prove a Large Deviations Principle (ldp) for systems of diffusions (particles) interacting through their ranks, when the number of particles tends to infinity. We show that the limiting particle density is given by the unique solution of the appropriate McKean-Vlasov equation and that the corresponding cumulative distribution function evolves according to a non-degenerate generalized porous medium equation with convection. The large deviations rate function is provided in explicit form. This is the first instance of a ldp for interacting diffusions, where the interaction occurs both through the drift and the diffusion coefficients and where the rate function can be given explicitly. In the course of the proof, we obtain new regularity results for tilted versions of such generalized porous medium equation.

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Atlas models of equity markets

Cited by 135 publications

References 14 publications

A phase transition behavior for Brownian motions interacting through their ranks

A phase transition behavior for Brownian motions interacting through their ranks

Why Are Big Banks Getting Bigger?

Large Deviations for Diffusions Interacting Through Their Ranks

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