arXiv subjects

Constantinos Kardaras

Publications and source records attributed to Constantinos Kardaras.

50 records · Page 3Linked to original sources

arXiv2010

Stability of the utility maximization problem with random endowment in incomplete markets

We perform a stability analysis for the utility maximization problem in a general semimartingale model where both liquid and illiquid assets (random endowments) are present. Small misspecifications of preferences (as modeled via expected utility), as well as views of the world or the market model (as modeled via subjective probabilities) are considered. Simple sufficient conditions are given for the problem to be well-posed, in the sense the optimal wealth and the marginal utility-based prices are continuous functionals of preferences and probabilistic views.

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arXiv2010

On the Dybvig-Ingersoll-Ross Theorem

The Dybvig-Ingersoll-Ross (DIR) theorem states that, in arbitrage-free term structure models, long-term yields and forward rates can never fall. We present a refined version of the DIR theorem, where we identify the reciprocal of the maturity date as the maximal order that long-term rates at earlier dates can dominate long-term rates at later dates. The viability assumption imposed on the market model is weaker than those appearing previously in the literature.

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arXiv2010

Arbitrage strategy

An arbitrage strategy allows a financial agent to make certain profit out of nothing, i.e., out of zero initial investment. This has to be disallowed on economic basis if the market is in equilibrium state, as opportunities for riskless profit would result in an instantaneous movement of prices of certain financial instruments. The principle of not allowing for arbitrage opportunities in financial markets has far-reaching consequences, most notably the option-pricing and hedging formulas in complete markets.

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arXiv2010

Free Lunch

The concept of absence of opportunities for free lunches is one of the pillars in the economic theory of financial markets. This natural assumption has proved very fruitful and has lead to great mathematical, as well as economical, insights in Quantitative Finance. Formulating rigorously the exact definition of absence of opportunities for riskless profit turned out to be a highly non-trivial fact that troubled mathematicians and economists for at least two decades. The purpose of this note is to give a quick (and, necessarily, incomplete) account of the recent work aimed at providing a simple and intuitive no-free-lunch assumption that would suffice in formulating a version of the celebrated Fundamental Theorem of Asset Pricing.

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arXiv2010

Stochastic discount factors

The valuation process that economic agents undergo for investments with uncertain payoff typically depends on their statistical views on possible future outcomes, their attitudes toward risk, and, of course, the payoff structure itself. Yields vary across different investment opportunities and their interrelations are difficult to explain. For the same agent, a different discounting factor has to be used for every separate valuation occasion. If, however, one is ready to accept discounting that varies randomly with the possible outcomes, and therefore accepts the concept of a stochastic discount factor, then an economically consistent theory can be developed. Asset valuation becomes a matter of randomly discounting payoffs under different states of nature and weighing them according to the agent's probability structure. The advantages of this approach are obvious, since a single discounting mechanism suffices to describe how any asset is priced by the agent.

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arXiv2009

Strict Local Martingale Deflators and Pricing American Call-Type Options

We solve the problem of pricing and optimal exercise of American call-type options in markets which do not necessarily admit an equivalent local martingale measure. This resolves an open question proposed by Fernholz and Karatzas [Stochastic Portfolio Theory: A Survey, Handbook of Numerical Analysis, 15:89-168, 2009].

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arXiv2009

Finitely additive probabilities and the Fundamental Theorem of Asset Pricing

This work aims at a deeper understanding of the mathematical implications of the economically-sound condition of absence of arbitrages of the first kind in a financial market. In the spirit of the Fundamental Theorem of Asset Pricing (FTAP), it is shown here that absence of arbitrages of the first kind in the market is equivalent to the existence of a finitely additive probability, weakly equivalent to the original and only locally countably additive, under which the discounted wealth processes become "local martingales". The aforementioned result is then used to obtain an independent proof of the FTAP of Delbaen and Schachermayer. Finally, an elementary and short treatment of the previous discussion is presented for the case of continuous-path semimartingale asset-price processes.

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arXiv2009

The continuous behavior of the numeraire portfolio under small changes in information structure, probabilistic views and investment constraints

The numeraire portfolio in a financial market is the unique positive wealth process that makes all other nonnegative wealth processes, when deflated by it, supermartingales. The numeraire portfolio depends on market characteristics, which include: (a) the information flow available to acting agents, given by a filtration; (b) the statistical evolution of the asset prices and, more generally, the states of nature, given by a probability measure; and (c) possible restrictions that acting agents might be facing on available investment strategies, modeled by a constraints set. In a financial market with continuous-path asset prices, we establish the stable behavior of the numeraire portfolio when each of the aforementioned market parameters is changed in an infinitesimal way.

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arXiv2009

On the semimartingale property of discounted asset-price processes

A financial market model where agents trade using realistic combinations of buy-and-hold strategies is considered. Minimal assumptions are made on the discounted asset-price process - in particular, the semimartingale property is not assumed. Via a natural market viability assumption, namely, absence of arbitrages of the first kind, we establish that discounted asset-prices have to be semimartingales. In a slightly more specialized case, we extend the previous result in a weakened version of the Fundamental Theorem of Asset Pricing that involves strictly positive supermartingale deflators rather than Equivalent Martingale Measures.

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arXiv2009

Minimizing the expected market time to reach a certain wealth level

In a financial market model, we consider variations of the problem of minimizing the expected time to upcross a certain wealth level. For exponential Levy markets, we show the asymptotic optimality of the growth-optimal portfolio for the above problem and obtain tight bounds for the value function for any wealth level. In an Ito market, we employ the concept of market time, which is a clock that runs according to the underlying market growth. We show the optimality of the growth-optimal portfolio for minimizing the expected market time to reach any wealth level. This reveals a general definition of market time which can be useful from an investor's point of view. We utilize this last definition to extend the previous results in a general semimartingale setting.

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arXiv2008

Diversity and relative arbitrage in equity markets

A financial market is called "diverse" if no single stock is ever allowed to dominate the entire market in terms of relative capitalization. In the context of the standard Ito-process model initiated by Samuelson (1965) we formulate this property (and the allied, successively weaker notions of "weak diversity" and "asymptotic weak diversity") in precise terms. We show that diversity is possible to achieve, but delicate. Several illustrative examples are provided, which demonstrate that weakly-diverse financial markets contain relative arbitrage opportunities: it is possible to outperform (or underperform) such markets over sufficiently long time-horizons, and to underperform them significantly over arbitrary time-horizons. The existence of such relative arbitrage does not interfere with the development of option pricing, and has interesting consequences for the pricing of long-term warrants and for put-call parity. Several open questions are suggested for further study.

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arXiv2008

No-Free-Lunch equivalences for exponential Levy models

We provide equivalence of numerous no-free-lunch type conditions for financial markets where the asset prices are modeled as exponential Levy processes, under possible convex constraints in the use of investment strategies. The general message is the following: if any kind of free lunch exists in these models it has to be of the most egregious type, generating an increasing ealth. Furthermore, we connect the previous to the existence of the numeraire portfolio, both for its particular expositional clarity in exponential Levy models and as a first step in obtaining analogues of the no-free-lunch equivalences in general semimartingale models.

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arXiv2008

The numeraire portfolio in semimartingale financial models

We study the existence of the numeraire portfolio under predictable convex constraints in a general semimartingale model of a financial market. The numeraire portfolio generates a wealth process, with respect to which the relative wealth processes of all other portfolios are supermartingales. Necessary and sufficient conditions for the existence of the numeraire portfolio are obtained in terms of the triplet of predictable characteristics of the asset price process. This characterization is then used to obtain further necessary and sufficient conditions, in terms of a no-free-lunch-type notion. In particular, the full strength of the "No Free Lunch with Vanishing Risk" (NFLVR) is not needed, only the weaker "No Unbounded Profit with Bounded Risk" (NUPBR) condition that involves the boundedness in probability of the terminal values of wealth processes. We show that this notion is the minimal a-priori assumption required in order to proceed with utility optimization. The fact that it is expressed entirely in terms of predictable characteristics makes it easy to check, something that the stronger NFLVR condition lacks.

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arXiv2008

Balance, growth and diversity of financial markets

A financial market comprising of a certain number of distinct companies is considered, and the following statement is proved: either a specific agent will surely beat the whole market unconditionally in the long run, or (and this "or" is not exclusive) all the capital of the market will accumulate in one company. Thus, absence of any "free unbounded lunches relative to the total capital" opportunities lead to the most dramatic failure of diversity in the market: one company takes over all other until the end of time. In order to prove this, we introduce the notion of perfectly balanced markets, which is an equilibrium state in which the relative capitalization of each company is a martingale under the physical probability. Then, the weaker notion of balanced markets is discussed where the martingale property of the relative capitalizations holds only approximately, we show how these concepts relate to growth-optimality and efficiency of the market, as well as how we can infer a shadow interest rate that is implied in the economy in the absence of a bank.

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