Searcharxiv⌕ Search

arXiv subjects

Constantinos Kardaras

Publications and source records attributed to Constantinos Kardaras.

At least 19 recordsLinked to original sources
arXiv2026

Log-optimality with small liability stream

In an incomplete financial market with general continuous semimartingale dynamics; we model an investor with log-utility preferences who, in addition to an initial capital, receives units of a non-traded endowment process. Using duality techniques, we derive the fourth-order expansion of the primal value function with respect to the units $ε$, held in the non-traded endowment. In turn, this lays the foundation for expanding the optimal wealth process, in this context, up to second order w.r.t. $ε$. The key processes underpinning the aforementioned results are given in terms of Kunita-Watanabe projections, mirroring the case of lower order expansions of similar nature. Both the case of finite and infinite horizons are treated in a unified manner.

q-fin.MF↗
arXiv2025

A dynamic competitive equilibrium model of irreversible capacity investment with stochastic demand and heterogeneous producers

We formulate a continuous-time competitive equilibrium model of irreversible capacity investment in which a continuum of heterogeneous producers supplies a single non-durable good subject to exogenous stochastic demand. Each producer optimally adjusts both output and capacity over time in response to endogenous price signals, while investment decisions are irreversible. Market clearing holds continuously, with prices evolving endogenously to balance aggregate supply and demand through a constant-elasticity demand function driven by a stochastic base component. The model admits a mean-field interpretation, as each producer's decisions both influence and are influenced by the aggregate behaviour of all others. We show that the equilibrium price process can be expressed as a nonlinear functional of the exogenous base demand, leading to a three-dimensional singular stochastic control problem for each producer. We derive an explicit solution to the associated Hamilton-Jacobi-Bellman equation, including a closed-form characterisation of the free-boundary surface separating investment and waiting regions.

math.PR↗
arXiv2023

Price Impact Under Heterogeneous Beliefs and Restricted Participation

We consider a financial market in which traders potentially face restrictions in trading some of the available securities. Traders are heterogeneous with respect to their beliefs and risk profiles, and the market is assumed thin: traders strategically trade against their price impacts. We prove existence and uniqueness of a corresponding equilibrium, and provide an efficient algorithm to numerically obtain the equilibrium prices and allocations given market's inputs. We find that restrictions may increase the market's welfare if traders have different views regarding the covariance matrix of securities returns. The latter heterogeneity regarding covariance matrix disagreement is essential in modelling; for instance, when traders agree on the covariance matrix, restricting participation in some securities for some traders leaves equilibrium prices unaltered in the unrestricted securities, a certainly undesirable model effect.

econ.GN↗
arXiv2022

Estimation of growth in fund models

Fund models are statistical descriptions of markets where all asset returns are spanned by the returns of a lower-dimensional collection of funds, modulo orthogonal noise. Equivalently, they may be characterised as models where the global growth-optimal portfolio only involves investment in the aforementioned funds. The loss of growth due to estimation error in fund models under local frequentist estimation is determined entirely by the number of funds. Furthermore, under a general filtering framework for Bayesian estimation, the loss of growth increases as the investment universe does. A shrinkage method that targets maximal growth with the least amount of deviation is proposed. Empirical evidence suggests that shrinkage gives a stable estimate that more closely follows growth potential than an unrestricted Bayesian estimate.

q-fin.PM↗
arXiv2020

Filtration shrinkage, the structure of deflators, and failure of market completeness

We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale part in the multiplicative Doob-Meyer decomposition of projected local martingale deflators are themselves local martingale deflators in the smaller information market. Via use of a Bayesian filtering approach, we demonstrate the exact mechanism of how updates on the possible class of models under less information result in the strict supermartingale property of projections of such deflators. Finally, we demonstrate that these projections are unable to span all possible local martingale deflators in the smaller information market, by investigating a situation where market completeness is not retained under filtration shrinkage.

math.PR↗
arXiv2019

Stochastic integration with respect to arbitrary collections of continuous semimartingales and applications to Mathematical Finance

Stochastic integrals are defined with respect to a collection $P = (P_i; \, i \in I)$ of continuous semimartingales, imposing no assumptions on the index set $I$ and the subspace of $\mathbb{R}^I$ where $P$ takes values. The integrals are constructed though finite-dimensional approximation, identifying the appropriate local geometry that allows extension to infinite dimensions. For local martingale integrators, the resulting space $\mathsf{S} (P)$ of stochastic integrals has an operational characterisation via a corresponding set of integrands $\mathsf{R} (C)$, constructed with only reference the covariation structure $C$ of $P$. This bijection between $\mathsf{R} (C)$ and the (closed in the semimartingale topology) set $\mathsf{S} (P)$ extends to families of continuous semimartingale integrators for which the drift process of $P$ belongs to $\mathsf{R} (C)$. In the context of infinite-asset models in Mathematical Finance, the latter structural condition is equivalent to a certain natural form of market viability. The enriched class of wealth processes via extended stochastic integrals leads to exact analogues of optional decomposition and hedging duality as the finite-asset case. A corresponding characterisation of market completeness in this setting is provided.

math.PR↗
arXiv2019

Projections of scaled Bessel processes

Let $X$ and $Y$ denote two independent squared Bessel processes of dimension $m$ and $n-m$, respectively, with $n\geq 2$ and $m \in [0, n)$, making $X+Y$ a squared Bessel process of dimension $n$. For appropriately chosen function $s$, the process $s (X+Y)$ is a local martingale. We study the representation and the dynamics of $s(X+Y)$, projected on the filtration generated by $X$. This projection is a strict supermartingale if, and only if, $m<2$. The finite-variation term in its Doob-Meyer decomposition only charges the support of the Markov local time of $X$ at zero.

math.PR↗
arXiv2018

Effective risk aversion in thin risk-sharing markets

We consider thin incomplete financial markets, where traders with heterogeneous preferences and risk exposures have motive to behave strategically regarding the demand schedules they submit, thereby impacting prices and allocations. We argue that traders relatively more exposed to market risk tend to submit more elastic demand functions. Noncompetitive equilibrium prices and allocations result as an outcome of a game among traders. General sufficient conditions for existence and uniqueness of such equilibrium are provided, with an extensive analysis of two-trader transactions. Even though strategic behaviour causes inefficient social allocations, traders with sufficiently high risk tolerance and/or large initial exposure to market risk obtain more utility gain in the noncompetitive equilibrium, when compared to the competitive one.

q-fin.MF↗
arXiv2018

Ergodic robust maximization of asymptotic growth

We consider the problem of robustly maximizing the growth rate of investor wealth in the presence of model uncertainty. Possible models are all those under which the assets' region $E$ and instantaneous covariation $c$ are known, and where additionally the assets are stable in that their occupancy time measures converge to a law with density $p$. This latter assumption is motivated by the observed stability of ranked relative market capitalizations for equity markets. We seek to identify the robust optimal growth rate, as well as a trading strategy which achieves this rate in all models. Under minimal assumptions upon $(E,c,p)$, we identify the robust growth rate with the Donsker-Varadhan rate function from occupancy time Large Deviations theory. We also prove existence of, and explicitly identify, the optimal trading strategy. We then apply our results in the case of drift uncertainty for ranked relative market capitalizations. Assuming regularity under symmetrization for the covariance and limiting density of the ranked capitalizations, we explicitly identify the robust optimal trading strategy in this setting.

q-fin.PM↗
arXiv2017

Incomplete stochastic equilibria for dynamic monetary utility

We study existence and uniqueness of continuous-time stochastic Radner equilibria in an incomplete market model among a group of agents whose preference is characterized by cash invariant time-consistent monetary utilities. An assumption of "smallness" type is shown to be sufficient for existence and uniqueness. In particular, this assumption encapsulates settings with small endowments, small time-horizon, or a large population of weakly heterogeneous agents. Central role in our analysis is played by a fully-coupled nonlinear system of quadratic BSDEs.

math.PR↗
arXiv2016

Equilibrium in risk-sharing games

The large majority of risk-sharing transactions involve few agents, each of whom can heavily influence the structure and the prices of securities. This paper proposes a game where agents' strategic sets consist of all possible sharing securities and pricing kernels that are consistent with Arrow-Debreu sharing rules. First, it is shown that agents' best response problems have unique solutions. The risk-sharing Nash equilibrium admits a finite-dimensional characterisation and it is proved to exist for arbitrary number of agents and be unique in the two-agent game. In equilibrium, agents declare beliefs on future random outcomes different than their actual probability assessments, and the risk-sharing securities are endogenously bounded, implying (among other things) loss of efficiency. In addition, an analysis regarding extremely risk tolerant agents indicates that they profit more from the Nash risk-sharing equilibrium as compared to the Arrow-Debreu one.

q-fin.RM↗
arXiv2016

Continuous-time perpetuities and time reversal of diffusions

We consider the problem of estimating the joint distribution of a continuous-time perpetuity and the underlying factors which govern the cash flow rate, in an ergodic Markov model. Two approaches are used to obtain the distribution. The first identifies a partial differential equation for the conditional cumulative distribution function of the perpetuity given the initial factor value, which under certain conditions ensures the existence of a density for the perpetuity. The second (and more general) approach, identifies the joint law as the stationary distribution of an ergodic multi-dimensional diffusion using techniques of time reversal. This later approach allows for efficient use of Monte-Carlo simulation when estimating the distribution, as the distribution is obtained by sampling a single path of the reversed process.

math.PR↗
arXiv2015

No arbitrage and local martingale deflators

A supermartingale deflator (resp., local martingale deflator) multiplicatively transforms nonnegative wealth processes into supermartingales (resp., local martingales). The supermartingale numeraire (resp., local martingale numeraire) is the wealth processes whose reciprocal is a supermartingale deflator (resp., local martingale deflator). It has been established in previous literature that absence of arbitrage of the first kind (NA1) is equivalent to existence of the supermartingale numeraire, and further equivalent to existence of a strictly positive local martingale deflator; however, under NA1, the local martingale numeraire may fail to exist. In this work, we establish that, under NA1, any total-variation neighbourhood of the original probability has an equivalent probability under which the local martingale numeraire exists. This result, available previously only for single risky-asset models, is in striking resemblance with the fact that any total-variation neighbourhood of a separating measure contains an equivalent $σ$-martingale measure. The presentation of our main result is relatively self-contained, including a proof of existence of the supermartingale numeraire under NA1. We further show that, if the Levy measures of the asset-price process have finite support, NA1 is equivalent to existence of the local martingale numeraire with respect to the original probability.

math.PR↗
arXiv2015

Robust Fundamental Theorem for Continuous Processes

We study a continuous-time financial market with continuous price processes under model uncertainty, modeled via a family $\mathcal{P}$ of possible physical measures. A robust notion ${\rm NA}_{1}(\mathcal{P})$ of no-arbitrage of the first kind is introduced; it postulates that a nonnegative, nonvanishing claim cannot be superhedged for free by using simple trading strategies. Our first main result is a version of the fundamental theorem of asset pricing: ${\rm NA}_{1}(\mathcal{P})$ holds if and only if every $P\in\mathcal{P}$ admits a martingale measure which is equivalent up to a certain lifetime. The second main result provides the existence of optimal superhedging strategies for general contingent claims and a representation of the superhedging price in terms of martingale measures.

q-fin.MF↗
arXiv2015

Strict local martingales and bubbles

This paper deals with asset price bubbles modeled by strict local martingales. With any strict local martingale, one can associate a new measure, which is studied in detail in the first part of the paper. In the second part, we determine the "default term" apparent in risk-neutral option prices if the underlying stock exhibits a bubble modeled by a strict local martingale. Results for certain path dependent options and last passage time formulas are given.

math.PR↗
arXiv2015

Arbitrage of the first kind and filtration enlargements in semimartingale financial models

In a general semimartingale financial model, we study the stability of the No Arbitrage of the First Kind (NA1) (or, equivalently, No Unbounded Profit with Bounded Risk) condition under initial and under progressive filtration enlargements. In both cases, we provide a simple and general condition which is sufficient to ensure this stability for any fixed semimartingale model. Furthermore, we give a characterisation of the NA1 stability for all semimartingale models.

math.PR↗
arXiv2015

On the stochastic behaviour of optional processes up to random times

In this paper, a study of random times on filtered probability spaces is undertaken. The main message is that, as long as distributional properties of optional processes up to the random time are involved, there is no loss of generality in assuming that the random time is actually a randomised stopping time. This perspective has advantages in both the theoretical and practical study of optional processes up to random times. Applications are given to financial mathematics, as well as to the study of the stochastic behaviour of Brownian motion with drift up to its time of overall maximum as well as up to last-passage times over finite intervals. Furthermore, a novel proof of the Jeulin-Yor decomposition formula via Girsanov's theorem is provided.

math.PR↗