Arbitrage and Hedging in a non probabilistic framework

Mathematics and Financial Economics 03/2011; 7(1). DOI: 10.1007/s11579-012-0074-5
Source: arXiv


The paper studies the concepts of hedging and arbitrage in a non
probabilistic framework. It provides conditions for non probabilistic arbitrage
based on the topological structure of the trajectory space and makes
connections with the usual notion of arbitrage. Several examples illustrate the
non probabilistic arbitrage as well perfect replication of options under
continuous and discontinuous trajectories, the results can then be applied in
probabilistic models path by path. The approach is related to recent financial
models that go beyond semimartingales, we remark on some of these connections
and provide applications of our results to some of these models.

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    • "The paper concentrates entirely on discrete, non probabilistic, market models extending the model in [11]. The setting could be considered as a discrete version of the non probabilistic, trajectory based, continuoustime models recently introduced in [3] and further developed in [4]. An example is given in Section 4 illustrating a general approach to constructing trajectory sets without using a priori probabilistic assumptions. "
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    ABSTRACT: The paper develops general, discrete, non-probabilistic market models and minmax price bounds leading to price intervals for European options. The approach provides the trajectory based analogue of martingale-like properties as well as a generalization that allows a limited notion of arbitrage in the market while still providing coherent option prices. Several properties of the price bounds are obtained, in particular a connection with risk neutral pricing is established for trajectory markets associated to a continuous-time martingale model.
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    ABSTRACT: The paper introduces general, discrete, non probabilistic models and a natural global minmax pricing rule that, for a given option, leads to a pricing interval. Conditions are described for the absence of arbitrage and a dynamic programming local minmax optimization is defined that evaluates the pricing interval bounds. © 2014 Springer International Publishing Switzerland. All rights are reserved.
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    ABSTRACT: The paper develops no arbitrage results for trajectory based models by imposing general constraints on the trading portfolios. The main condition imposed, in order to avoid arbitrage opportunities, is a local continuity requirement on the final portfolio value considered as a functional on the trajectory space. The paper shows this to be a natural requirement by proving that a large class of practical trading strategies, defi?ned by means of trajectory based stopping times, give rise to locally continuous functionals. The theory is illustrated, with some detail, for two specific trajectory models of practical interest. The implications for stochastic models which are not semimartingales are described. The present paper extends some of the results in [1] by incorporating in the formalism a larger set of trading portfolios.
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