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Fundamental Theorem of Asset Pricing under Transaction costs and Model uncertainty

Bayraktar, Erhan; Zhang, Yuchong
Tipo: Artigo de Revista Científica
Português
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16.74%
We prove the Fundamental Theorem of Asset Pricing for a discrete time financial market where trading is subject to proportional transaction cost and the asset price dynamic is modeled by a family of probability measures, possibly non-dominated. Using a backward-forward scheme, we show that when the market consists of a money market account and a single stock, no-arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of consistent price systems. We also show that when the market consists of multiple dynamically traded assets and satisfies \emph{efficient friction}, strict no-arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of strictly consistent price systems.; Comment: Final version. To appear in Mathematics of Operations Research

Trajectory based models. Evaluation of minmax pricing bounds

Degano, Ivan; Ferrando, Sebastian; Gonzalez, Alfredo
Tipo: Artigo de Revista Científica
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16.74%
The paper studies market models based on trajectory spaces, properties of such models are obtained without recourse to probabilistic assumptions. For a given European option, an interval of rational prices exists under a more general condition than the usual no-arbitrage requirement. The paper develops computational results in order to evaluate the option bounds; the global minmax optimization, defining the price interval, is reduced to a local minmax optimization via dynamic programming. A general class of trajectory sets is described for which the market model introduced by Britten Jones and Neuberger is nested as a particular case. We also develop a market model based on an operational setting constraining market movements and investor's portfolio rebalances. Numerical examples are presented, the effect of the presence of arbitrage on the price bounds is illustrated.; Comment: 45 pages, 15 figures

Karzand, Mina; Varshney, Lav R.
Tipo: Artigo de Revista Científica
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16.74%
The possibility of latency arbitrage in financial markets has led to the deployment of high-speed communication links between distant financial centers. These links are noisy and so there is a need for coding. In this paper, we develop a gametheoretic model of trading behavior where two traders compete to capture latency arbitrage opportunities using binary signalling. Different coding schemes are strategies that trade off between reliability and latency. When one trader has a better channel, the second trader should not compete. With statistically identical channels, we find there are two different regimes of channel noise for which: there is a unique Nash equilibrium yielding ties; and there are two Nash equilibria with different winners.; Comment: Will appear in IEEE International Symposium on Information Theory (ISIT), 2015

Financial Models with Defaultable Num\'eraires

Fisher, Travis; Pulido, Sergio; Ruf, Johannes
Tipo: Artigo de Revista Científica
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16.74%
Financial models are studied where each asset may potentially lose value relative to any other. To this end, the paradigm of a pre-determined num\'eraire is abandoned in favour of a symmetrical point of view where all assets have equal priority. This approach yields novel versions of the Fundamental Theorems of Asset Pricing, which clarify and extend non-classical pricing formulas used in the financial community. Furthermore, conditioning on non-devaluation, each asset can serve as proper num\'eraire and a classical no-arbitrage condition can be formulated. It is shown when and how these local conditions can be aggregated to a global no-arbitrage condition.

Dynamic Conic Finance via Backward Stochastic Difference Equations

Bielecki, Tomasz R.; Cialenco, Igor; Chen, Tao
Tipo: Artigo de Revista Científica
Português
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16.74%
We present an arbitrage free theoretical framework for modeling bid and ask prices of dividend paying securities in a discrete time setup using theory of dynamic acceptability indices. In the first part of the paper we develop the theory of dynamic subscale invariant performance measures, on a general probability space, and discrete time setup. We prove a representation theorem of such measures in terms of a family of dynamic convex risk measures, and provide a representation of dynamic risk measures in terms of g-expectations, and solutions of BS$\Delta$Es with convex drivers. We study the existence and uniqueness of the solutions, and derive a comparison theorem for corresponding BS$\Delta$Es. In the second part of the paper we discuss a market model for dividend paying securities by introducing the pricing operators that are defined in terms of dynamic acceptability indices, and find various properties of these operators. Using these pricing operators, we define the bid and ask prices for the underlying securities and then for derivatives in this market. We show that the obtained market model is arbitrage free, and we also prove a series of properties of these prices.; Comment: 65 pages

Bilateral counterparty risk valuation with stochastic dynamical models and application to Credit Default Swaps

Brigo, Damiano; Capponi, Agostino
Tipo: Artigo de Revista Científica
Português
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16.74%
We introduce the general arbitrage-free valuation framework for counterparty risk adjustments in presence of bilateral default risk, including default of the investor. We illustrate the symmetry in the valuation and show that the adjustment involves a long position in a put option plus a short position in a call option, both with zero strike and written on the residual net value of the contract at the relevant default times. We allow for correlation between the default times of the investor, counterparty and underlying portfolio risk factors. We use arbitrage-free stochastic dynamical models. We then specialize our analysis to Credit Default Swaps (CDS) as underlying portfolio, generalizing the work of Brigo and Chourdakis (2008) [5] who deal with unilateral and asymmetric counterparty risk. We introduce stochastic intensity models and a trivariate copula function on the default times exponential variables to model default dependence. Similarly to [5], we find that both default correlation and credit spread volatilities have a relevant and structured impact on the adjustment. Differently from [5], the two parties will now agree on the credit valuation adjustment. We study a case involving British Airways, Lehman Brothers and Royal Dutch Shell...

A note on the theory of fast money flow dynamics

Sokolov, Andrey; Kieu, Tien; Melatos, Andrew
Tipo: Artigo de Revista Científica
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16.74%
The gauge theory of arbitrage was introduced by Ilinski in [arXiv:hep-th/9710148] and applied to fast money flows in [arXiv:cond-mat/9902044]. The theory of fast money flow dynamics attempts to model the evolution of currency exchange rates and stock prices on short, e.g.\ intra-day, time scales. It has been used to explain some of the heuristic trading rules, known as technical analysis, that are used by professional traders in the equity and foreign exchange markets. A critique of some of the underlying assumptions of the gauge theory of arbitrage was presented by Sornette in [arXiv:cond-mat/9804045]. In this paper, we present a critique of the theory of fast money flow dynamics, which was not examined by Sornette. We demonstrate that the choice of the input parameters used in [arXiv:cond-mat/9902044] results in sinusoidal oscillations of the exchange rate, in conflict with the results presented in [arXiv:cond-mat/9902044]. We also find that the dynamics predicted by the theory are generally unstable in most realistic situations, with the exchange rate tending to zero or infinity exponentially.; Comment: accepted for publication in The European Physical Journal B

Outperforming the market portfolio with a given probability

Bayraktar, Erhan; Huang, Yu-Jui; Song, Qingshuo
Tipo: Artigo de Revista Científica
Português
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16.74%
Our goal is to resolve a problem proposed by Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.]: to characterize the minimum amount of initial capital with which an investor can beat the market portfolio with a certain probability, as a function of the market configuration and time to maturity. We show that this value function is the smallest nonnegative viscosity supersolution of a nonlinear PDE. As in Fernholz and Karatzas [On optimal arbitrage (2008) Columbia Univ.], we do not assume the existence of an equivalent local martingale measure, but merely the existence of a local martingale deflator.; Comment: Published in at http://dx.doi.org/10.1214/11-AAP799 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

On the range of admissible term-structures

Cousin, Areski; Niang, Ibrahima
Tipo: Artigo de Revista Científica
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16.74%
In this paper, we analyze the diversity of term structure functions (e.g., yield curves, swap curves, credit curves) constructed in a process which complies with some admissible properties: arbitrage-freeness, ability to fit market quotes and a certain degree of smooth- ness. When present values of building instruments are expressed as linear combinations of some primary quantities such as zero-coupon bonds, discount factor, or survival probabilit- ies, arbitrage-free bounds can be derived for those quantities at the most liquid maturities. As a matter of example, we present an iterative procedure that allows to compute model-free bounds for OIS-implied discount rates and CDS-implied default probabilities. We then show how mean-reverting term structure models can be used as generators of admissible curves. This framework is based on a particular specification of the mean-reverting level which al- lows to perfectly reproduce market quotes of standard vanilla interest-rate and default-risky securities while preserving a certain degree of smoothness. The numerical results suggest that, for both OIS discounting curves and CDS credit curves, the operational task of term- structure construction may be associated with a significant degree of uncertainty.

Diverse Market Models of Competing Brownian Particles with Splits and Mergers

Karatzas, Ioannis; Sarantsev, Andrey
Tipo: Artigo de Revista Científica
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16.74%
We study models of regulatory breakup, in the spirit of Strong and Fouque (2011) but with a fluctuating number of companies. An important class of market models is based on systems of competing Brownian particles: each company has a capitalization whose logarithm behaves as a Brownian motion with drift and diffusion coefficients depending on its current rank. We study such models with a fluctuating number of companies: If at some moment the share of the total market capitalization of a company reaches a fixed level, then the company is split into two parts of random size. Companies are also allowed to merge, when an exponential clock rings. We find conditions under which this system is non-explosive (that is, the number of companies remains finite at all times) and diverse, yet does not admit arbitrage opportunities.; Comment: 22 pages. Keywords: competing Brownian particles, splits, mergers, diverse markets, arbitrage opportunity, portfolio

Phenomenology of the Interest Rate Curve

Bouchaud, J. -P.; Sagna, N.; Cont, R.; El-Karoui, N.; Potters, M.
Tipo: Artigo de Revista Científica
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16.74%
This paper contains a phenomenological description of the whole U.S. forward rate curve (FRC), based on an data in the period 1990-1996. We find that the average FRC (measured from the spot rate) grows as the square-root of the maturity, with a prefactor which is comparable to the spot rate volatility. This suggests that forward rate market prices include a risk premium, comparable to the probable changes of the spot rate between now and maturity, which can be understood as a Value-at-Risk' type of pricing. The instantaneous FRC however departs form a simple square-root law. The distortion is maximum around one year, and reflects the market anticipation of a local trend on the spot rate. This anticipated trend is shown to be calibrated on the past behaviour of the spot itself. We show that this is consistent with the volatility hump' around one year found by several authors (and which we confirm). Finally, the number of independent components needed to interpret most of the FRC fluctuations is found to be small. We rationalize this by showing that the dynamical evolution of the FRC contains a stabilizing second derivative (line tension) term, which tends to suppress short scale distortions of the FRC. This shape dependent term could lead...

B-spline techniques for volatility modeling

Corlay, Sylvain
Tipo: Artigo de Revista Científica
Português
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16.74%
This paper is devoted to the application of B-splines to volatility modeling, specifically the calibration of the leverage function in stochastic local volatility models and the parameterization of an arbitrage-free implied volatility surface calibrated to sparse option data. We use an extension of classical B-splines obtained by including basis functions with infinite support. We first come back to the application of shape-constrained B-splines to the estimation of conditional expectations, not merely from a scatter plot but also from the given marginal distributions. An application is the Monte Carlo calibration of stochastic local volatility models by Markov projection. Then we present a new technique for the calibration of an implied volatility surface to sparse option data. We use a B-spline parameterization of the Radon-Nikodym derivative of the underlying's risk-neutral probability density with respect to a roughly calibrated base model. We show that this method provides smooth arbitrage-free implied volatility surfaces. Finally, we sketch a Galerkin method with B-spline finite elements to the solution of the partial differential equation satisfied by the Radon-Nikodym derivative.; Comment: 25 pages

Portfolio optimisation beyond semimartingales: shadow prices and fractional Brownian motion

Czichowsky, Christoph; Schachermayer, Walter
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.74%
While absence of arbitrage in frictionless financial markets requires price processes to be semimartingales, non-semimartingales can be used to model prices in an arbitrage-free way, if proportional transaction costs are taken into account. In this paper, we show, for a class of price processes which are not necessarily semimartingales, the existence of an optimal trading strategy for utility maximisation under transaction costs by establishing the existence of a so-called shadow price. This is a semimartingale price process, taking values in the bid ask spread, such that frictionless trading for that price process leads to the same optimal strategy and utility as the original problem under transaction costs. Our results combine arguments from convex duality with the stickiness condition introduced by P. Guasoni. They apply in particular to exponential utility and geometric fractional Brownian motion. In this case, the shadow price is an Ito process. As a consequence we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an Ito process in a one-sided manner without reflection.; Comment: 29 pages

Arbitrages in a Progressive Enlargement Setting

Aksamit, Anna; Choulli, Tahir; Deng, Jun; Jeanblanc, Monique
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.74%
This paper completes the analysis of Choulli et al. Non-Arbitrage up to Random Horizons and after Honest Times for Semimartingale Models and contains two principal contributions. The first contribution consists in providing and analysing many practical examples of market models that admit classical arbitrages while they preserve the No Unbounded Profit with Bounded Risk (NUPBR hereafter) under random horizon and when an honest time is incorporated for particular cases of models. For these markets, we calculate explicitly the arbitrage opportunities. The second contribution lies in providing simple proofs for the stability of the No Unbounded Profit with Bounded Risk under random horizon and after honest time satisfying additional important condition for particular cases of models.

Self-Financing, Replicating Hedging Strategies, an incomplete thermodynamic analogy

McCauley, Joesph L.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
16.74%
In the theory of riskfree hedges in continuous time finance, one can start with the delta-hedge and derive the option pricing equation, or one can start with the replicating, self-financing hedging strategy and derive both the delta-hedge and the option pricing partial differential equation. Approximately reversible trading is implicitly assumed in both cases. The option pricing equation is not restricted to the standard Black-Scholes equation when nontrivial volatility is assumed, but produces option pricing in agreement with the empirical distribution for the right choice of volatility in a stochastic description of fluctuations. The replicating, self-financing hedging strategy provides us with an incomplete analogy with thermodynamics where liquidity plays the role of the heat bath, the absence of arbitrage is analgous to thermal equilibrium, but there is no role played by the entropy of the returns distribution, which cannot reach a maximum/equilibrium. We emphasize strongly that the no-arbitrage assumption is not an equilibrium assumption, as is taught in economics, but provides only an incomplete, very limited analogy with the idea of thermal equilibrium.; Comment: 10 pages, no figures

A note on the Fundamental Theorem of Asset Pricing under model uncertainty

Bayraktar, Erhan; Zhang, Yuchong; Zhou, Zhou
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.74%
We show that the results of ArXiv:1305.6008 on the Fundamental Theorem of Asset Pricing and the super-hedging theorem can be extended to the case in which the options available for static hedging (\emph{hedging options}) are quoted with bid-ask spreads. In this set-up, we need to work with the notion of \emph{robust no-arbitrage} which turns out to be equivalent to no-arbitrage under the additional assumption that hedging options with non-zero spread are \emph{non-redundant}. A key result is the closedness of the set of attainable claims, which requires a new proof in our setting.; Comment: Final version. To appear in Risks

On pricing kernels, information and risk

Wilcox, D. L.; Gebbie, T. J.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.74%
We discuss the finding that cross-sectional characteristic based models have yielded portfolios with higher excess monthly returns but lower risk than their arbitrage pricing theory counterparts in an analysis of equity returns of stocks listed on the JSE. Under the assumption of general no-arbitrage conditions, we argue that evidence in favour of characteristic based pricing implies that information is more likely assimilated by means of nonlinear pricing kernels for the markets considered.; Comment: 20 pages, 3 figures, 1 table

Implicit transaction costs and the fundamental theorems of asset pricing

Allaj, Erindi
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.74%

Kinetic properties in inhomogeneous self-aware media

Morozovskiy, A.; Snarskii, A. A.; Bezsudnov, I. V.; Sevryukov, V. A.; Malinsky, J.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
16.74%
The new framework for finance is proposed. This framework based on three known approaches in econophysics. Assumptions of the framework are the following: 1. For the majority of situations market follows non-arbitrage condition. 2. For the small number of situations market influenced by the actions of big firms. 3. If actions of big players lead to the arbitrage opportunity, small players could self-organize to take advantage of this opportunity. The framework is an attempt to combine approaches of Bouchaud, Gabaix, Sornette, Stanley and coauthors. Suggested framework is applied for the analysis of market impact models, behavior of big players, self-organization of market firm and volatility description.; Comment: Chapter 23 of the book "Transport processes in macroscopically disordered media (From mean field theory to percolation)" accepted for publication by Springer [32 pages, 1 figure, 2 tables]

Consistent Long-Term Yield Curve Prediction

Teichmann, Josef; Wüthrich, Mario V.