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Constantinos Kardaras

Publications and source records attributed to Constantinos Kardaras.

At least 37 records · Page 2Linked to original sources
arXiv2014

Valuation and parities for exchange options

Valuation and parity formulas for both European-style and American-style exchange options are presented in a general financial model allowing for jumps, possibility of default and "bubbles" in asset prices. The formulas are given via expectations of auxiliary probabilities using the change-of-numeraire technique. Extensive discussion is provided regarding the way that folklore results such as Merton's no-early-exercise theorem and traditional parity relations have to be altered in this more versatile framework.

q-fin.PR↗
arXiv2014

Maximality and numeraires in convex sets of nonnegative random variables

We introduce the concepts of max-closedness and numeraires of convex subsets in the nonnegative orthant of the topological vector space of all random variables built over a probability space, equipped with a topology consistent with convergence in probability. Max-closedness asks that maximal elements of the closure of a set already lie on the set. We discuss how numeraires arise naturally as strictly positive optimisers of certain concave monotone maximisation problems. It is further shown that the set of numeraires of a convex, max-closed and bounded set of of nonnegative random variables that contains at least one strictly positive element is dense in the set of its maximal elements.

math.FA↗
arXiv2013

On the closure in the Emery topology of semimartingale wealth-process sets

A wealth-process set is abstractly defined to consist of nonnegative càdlàg processes containing a strictly positive semimartingale and satisfying an intuitive re-balancing property. Under the condition of absence of arbitrage of the first kind, it is established that all wealth processes are semimartingales and that the closure of the wealth-process set in the Emery topology contains all "optimal" wealth processes.

q-fin.PM↗
arXiv2013

On the characterisation of honest times that avoid all stopping times

We present a short and self-contained proof of the following result: a random time is an honest time that avoids all stopping times if and only if it coincides with the (last) time of maximum of a nonnegative local martingale with zero terminal value and no jumps while at its running supremum, where the latter running supremum process is continuous. Illustrative examples involving local martingales with discontinuous paths are provided.

math.PR↗
arXiv2012

The numeraire property and long-term growth optimality for drawdown-constrained investments

We consider the portfolio choice problem for a long-run investor in a general continuous semimartingale model. We suggest to use path-wise growth optimality as the decision criterion and encode preferences through restrictions on the class of admissible wealth processes. Specifically, the investor is only interested in strategies which satisfy a given linear drawdown constraint. The paper introduces the numeraire property through the notion of expected relative return and shows that drawdown-constrained strategies with the numeraire property exist and are unique, but may depend on the financial planning horizon. However, when sampled at the times of its maximum and asymptotically as the time-horizon becomes distant, the drawdown-constrained numeraire portfolio is given explicitly through a model-independent transformation of the unconstrained numeraire portfolio. Further, it is established that the asymptotically growth-optimal strategy is obtained as limit of numeraire strategies on finite horizons.

q-fin.PM↗
arXiv2012

Uniform integrability and local convexity in $L^0$

Let $L^0$ be the vector space of all (equivalence classes of) real-valued random variables built over a probability space $(Ω, \mathcal{F}, P)$, equipped with a metric topology compatible with convergence in probability. In this work, we provide a necessary and sufficient structural condition that a set $X \subseteq L^0$ should satisfy in order to infer the existence of a probability $Q$ that is equivalent to $P$ and such that $X$ is uniformly $Q$-integrable. Furthermore, we connect the previous essentially measure-free version of uniform integrability with local convexity of the $L^0$-topology when restricted on convex, solid and bounded subsets of $L^0$.

math.FA↗
arXiv2012

Robust maximization of asymptotic growth

This paper addresses the question of how to invest in a robust growth-optimal way in a market where the instantaneous expected return of the underlying process is unknown. The optimal investment strategy is identified using a generalized version of the principal eigenfunction for an elliptic second-order differential operator, which depends on the covariance structure of the underlying process used for investing. The robust growth-optimal strategy can also be seen as a limit, as the terminal date goes to infinity, of optimal arbitrages in the terminology of Fernholz and Karatzas [Ann. Appl. Probab. 20 (2010) 1179-1204].

q-fin.PM↗
arXiv2012

Abstract, Classic, and Explicit Turnpikes

Portfolio turnpikes state that, as the investment horizon increases, optimal portfolios for generic utilities converge to those of isoelastic utilities. This paper proves three kinds of turnpikes. In a general semimartingale setting, the abstract turnpike states that optimal final payoffs and portfolios converge under their myopic probabilities. In diffusion models with several assets and a single state variable, the classic turnpike demonstrates that optimal portfolios converge under the physical probability; meanwhile the explicit turnpike identifies the limit of finite-horizon optimal portfolios as a long-run myopic portfolio defined in terms of the solution of an ergodic HJB equation.

q-fin.PM↗
arXiv2011

Valuation equations for stochastic volatility models

We analyze the valuation partial differential equation for European contingent claims in a general framework of stochastic volatility models where the diffusion coefficients may grow faster than linearly and degenerate on the boundaries of the state space. We allow for various types of model behavior: the volatility process in our model can potentially reach zero and either stay there or instantaneously reflect, and the asset-price process may be a strict local martingale. Our main result is a necessary and sufficient condition on the uniqueness of classical solutions to the valuation equation: the value function is the unique nonnegative classical solution to the valuation equation among functions with at most linear growth if and only if the asset-price is a martingale.

math.PR↗
arXiv2011

Maximum penalized quasi-likelihood estimation of the diffusion function

We develop a maximum penalized quasi-likelihood estimator for estimating in a nonparametric way the diffusion function of a diffusion process, as an alternative to more traditional kernel-based estimators. After developing a numerical scheme for computing the maximizer of the penalized maximum quasi-likelihood function, we study the asymptotic properties of our estimator by way of simulation. Under the assumption that overnight London Interbank Offered Rates (LIBOR); the USD/EUR, USD/GBP, JPY/USD, and EUR/USD nominal exchange rates; and 1-month, 3-month, and 30-year Treasury bond yields are generated by diffusion processes, we use our numerical scheme to estimate the diffusion function.

q-fin.ST↗
arXiv2010

A time before which insiders would not undertake risk

A continuous-path semimartingale market model with wealth processes discounted by a riskless asset is considered. The numeraire portfolio is the unique strictly positive wealth process that, when used as a benchmark to denominate all other wealth, makes all wealth processes local martingales. It is assumed that the numeraire portfolio exists and that its wealth increases to infinity as time goes to infinity. Under this setting, an initial enlargement of the filtration is performed, by including the overall minimum of the numeraire portfolio. It is established that all nonnegative wealth processes, when stopped at the time of the overall minimum of the numeraire portfolio, become local martingales in the enlarged filtration. This implies that risk-averse insider traders would refrain from investing in the risky assets before that time. A partial converse to the previous result is also established in the case of complete markets, showing that the time of the overall minimum of the numeraire portfolio is in a certain sense unique in rendering undesirable the act of undertaking risky positions before it. The aforementioned results shed light to the importance of the numeraire portfolio as an indicator of overall market performance.

q-fin.PM↗
arXiv2010

Numéraire-invariant preferences in financial modeling

We provide an axiomatic foundation for the representation of numéraire-invariant preferences of economic agents acting in a financial market. In a static environment, the simple axioms turn out to be equivalent to the following choice rule: the agent prefers one outcome over another if and only if the expected (under the agent's subjective probability) relative rate of return of the latter outcome with respect to the former is nonpositive. With the addition of a transitivity requirement, this last preference relation has an extension that can be numerically represented by expected logarithmic utility. We also treat the case of a dynamic environment where consumption streams are the objects of choice. There, a novel result concerning a canonical representation of unit-mass optional measures enables us to explicitly solve the investment--consumption problem by separating the two aspects of investment and consumption. Finally, we give an application to the problem of optimal numéraire investment with a random time-horizon.

q-fin.GN↗
arXiv2010

Generalized supermartingale deflators under limited information

We undertake a study of markets from the perspective of a financial agent with limited access to information. The set of wealth processes available to the agent is structured with reasonable economic properties, instead of the usual practice of taking it to consist of stochastic integrals against a semimartingale integrator. We obtain the equivalence of the boundedness in probability of the set of terminal wealth outcomes (which in turn is equivalent to the weak market viability condition of absence of arbitrage of the first kind) with the existence of at least one strictly positive deflator that makes the deflated wealth processes have a generalized supermartingale property.

q-fin.GN↗
arXiv2010

Efficient estimation of one-dimensional diffusion first passage time densities via Monte Carlo simulation

We propose a method for estimating first passage time densities of one-dimensional diffusions via Monte Carlo simulation. Our approach involves a representation of the first passage time density as expectation of a functional of the three-dimensional Brownian bridge. As the latter process can be simulated exactly, our method leads to almost unbiased estimators. Furthermore, since the density is estimated directly, a convergence of order $1 / \sqrt{N}$, where $N$ is the sample size, is achieved, the last being in sharp contrast to the slower non-parametric rates achieved by kernel smoothing of cumulative distribution functions.

math.PR↗
arXiv2010

Market viability via absence of arbitrage of the first kind

In a semimartingale financial market model, it is shown that there is equivalence between absence of arbitrage of the first kind (a weak viability condition) and the existence of a strictly positive process that acts as a local martingale deflator on nonnegative wealth processes.

q-fin.PR↗
arXiv2010

Multiplicative approximation of wealth processes involving no-short-sale strategies via simple trading

A financial market model with general semimartingale asset-price processes and where agents can only trade using no-short-sales strategies is considered. We show that wealth processes using continuous trading can be approximated very closely by wealth processes using simple combinations of buy-and-hold trading. This approximation is based on controlling the proportions of wealth invested in the assets. As an application, the utility maximization problem is considered and it is shown that optimal expected utilities and wealth processes resulting from continuous trading can be approximated arbitrarily well by the use of simple combinations of buy-and-hold strategies.

q-fin.PM↗