arXiv subjects

Adam J. Ostaszewski

Publications and source records attributed to Adam J. Ostaszewski.

8 recordsLinked to original sources

arXiv2024

Damages and Materiality: Effects on voluntary disclosure

How should a court resolve a shareholder--management dispute following a materially significant price decline when it is suspected that management, at an earlier point in time, failed to update the market by disclosing a privately observed material event? A foundational result in this literature (Dye, 2017) shows that if a court publicly commits to increasing damages awards in an effort to deter nondisclosure, the policy may have a perverse effect: management may rationally choose to disclose even less. Schantl and Wagenhofer (2024) attribute this outcome to the pure insurance effect, whereby shareholders benefit from higher damages payments. They show that this result may be mitigated if management also face a fixed, exogenous reputational cost of nondisclosure. However, these reputational costs are independent of the model's equilibrium; furthermore, they assume that the court eventually observes the true state of the world with certainty (delayed omniscience by the court), and do not account for standards of materiality, which differ across legal systems. In contrast, we develop a dynamic continuous-time model in which both damages and materiality standards are endogenous. We show that, as damages awards increase, a previously unrecognized dynamic effect emerges: management rationally switch to a candid (full) disclosure strategy. Moreover, raising the materiality threshold induces this switch earlier, thereby increasing the extent of voluntary disclosure. Our analysis therefore demonstrates that regulators should recognize the complementary effects of damages and materiality standards. We further characterize what we term the legal consistency zone, in which higher damages awards, coupled with an appropriately chosen materiality standard, endogenously increase voluntary disclosure.

q-fin.GN↗

arXiv2022

The Kind of Silence: Managing a Reputation for Voluntary Disclosure in Financial Markets

In a continuous-time setting we investigate how the management of a firm controls a dynamic choice between two generic voluntary disclosure decision rules: one with full and transparent disclosure termed $\it{candid}$, the other, termed $\it{sparing}$, under which values only above a dynamic threshold are disclosed. We show how management are rewarded with a reputational premium for candour. The candid strategy is costly because the sparing alternative shields the firm from potential downgrades following low value disclosures. We show how parameters of the model such as news intensity, pay-for-performance and time-to-mandatory-disclosure determine the optimal choice of candid versus sparing strategy and the optimal time for management to switch between the two. The private news updates received by management are modelled with a Poisson process, occurring between the fixed mandatory disclosure dates, such as fiscal years or quarters, with the news received generated by a background Black-Scholes model of economic activity and of its partial observation. The model presented develops a number of insights, based on a very simple ordinary differential equation (ODE) characterizing equilibrium in a piecewise-deterministic model, derivable from the background Black-Scholes model. It is shown that in equilibrium when news intensity is low a firm may employ a candid disclosure strategy throughout, but will otherwise switch (alternate) between periods of being candid and periods of being sparing with the truth (or the other way about); that is, we are able to characterize when in equilibrium a candid firm will switch to adopting a sparing strategy. The model illustrates how parameters such as time to mandatory disclosure, news intensity and pay-for-performance may drive such switching behaviour. $\it{With\,constant\,pay\,for\,performance\,parameters,\,at\,most\,one\, switching\,occurs.}$

math.OC↗

arXiv2022

Freezing in Space-time: A functional equation linked with a PDE system

We analyze the functional equation $$F(x+F(x))=-F(x)$$ and reveal its relationship with a system of partial differential equations arising as the hydrodynamic limit of a system of pinned billiard balls on the line. The system of balls must freeze at some time, i.e., no velocity may change after the freezing time. The terminal velocity and the freezing time profiles play the role of boundary conditions for the PDEs (qua terminal conditions, despite being initial conditions from the wave equation perspective pursued here). Solutions to the functional equation provide the link between the freezing time and terminal velocity profiles on the one hand, and the solution to the PDE in the entirety of the space-time domain on the other.

math.AP↗

arXiv2019

On subadditive functions upper bounded on a 'large' set

The notion of a shift-compact set in an abelian topological group $X$ plays a significant role in functional equations and inequalities, especially so since each Borel set that is not Haar-meagre, alternatively not Haar-null, is necessarily shift-compact for $X$ completely metrizable (see \cite{BJ} and \cite{BinO8}). Although in general boundedness of a subadditive function does not imply its continuity, here we prove that each subadditive function $f:X\rightarrow \mathbb{R}$ (i.e. with the function satisfying $f(x+y)\leq f(x)+f(y)$ for $x,y\in X$) bounded above on a~shift-compact (non-Haar-null, non-Haar-meagre) set is locally bounded at each point of the domain. Our results refer to \cite[Chapter~XVI]{Kuczma} and papers by N.H.~Bingham and A.J.~Ostaszewski \cite{BO,BinO1,BinO2,BinO6,BinO7}.

math.CA↗

arXiv2019

A Functional Equation of Tail-balance for Continuous Signals in the Condorcet Jury Theorem

Consider an odd-sized jury, which determines a majority verdict between two equiprobable states of Nature. If each juror independently receives a binary signal identifying the correct state with identical probability $p$, then the probability of a correct verdict tends to one as the jury size tends to infinity (Condorcet, 1785). Recently, the first two authors developed a model where jurors sequentially receive signals from an interval according to a distribution, which depends on the state of Nature and on the juror's "ability", and vote sequentially. This paper shows that to mimic Condorcet's binary signal, such a distribution must satisfy a functional equation related to tail-balance, that is, to the ratio $α(t)$ of the probability that a mean-zero random variable satisfies $X >t$ given that $|X|>t$. In particular, we show that under natural symmetry assumptions the tail-balances $α(t)$ uniquely determine the distribution.

math.PR↗

arXiv2016

Stable laws and Beurling kernels

We identify a close relation between stable distributions and the limiting homomorphisms central to the theory of regular variation. In so doing some simplifications are achieved in the direct analysis of these laws in Pitman and Pitman (2016); stable distributions are themselves linked to homomorphy.

math.PR↗

arXiv2016

The Sound of Silence: equilibrium filtering and optimal censoring in financial markets

Following the approach of standard filtering theory, we analyse investor-valuation of firms, when these are modelled as geometric-Brownian state processes that are privately and partially observed, at random (Poisson) times, by agents. Tasked with disclosing forecast values, agents are able purposefully to withhold their observations; explicit filtering formulas are derived for downgrading the valuations in the absence of disclosures. The analysis is conducted for both a solitary firm and m co-dependent firms.

q-fin.MF↗

arXiv2014

Homomorphisms from Functional Equations: The Goldie Equation

The theory of regular variation, in its Karamata and Bojanić-Karamata/de Haan forms, is long established and makes essential use of the Cauchy functional equation. Both forms are subsumed within the recent theory of Beurling regular variation, developed elsewhere. Various generalizations of the Cauchy equation, including the Gołąb-Schinzel functional equation (GS), are prominent there. Here we unify their treatment by `algebraicization': extensive use of group structures introduced by Popa and Javor in the 1960s turn all the various solutions into homomorphisms, and show that (GS) is present everywhere, even if in a thick disguise.

math.CA↗