Fractional Brownian motion satisfies two-way crossing

Peyre, Rémi

doi:10.3150/16-bej858

Cited by 7 publications

(4 citation statements)

References 12 publications

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“…Bender [2] have given a necessary and sufficient condition for the market S being free of arbitrage in the class S(F) of all simple strategies when S is continuous. Moreover, using Bender [2]'s result, Peyre [41] has shown that there is no arbitrage in the class S(F) for the fractional Brownian motion market.…”

Section: Trading With Simple Strategiesmentioning

confidence: 99%

No arbitrage and lead–lag relationships

Hayashi

Koike

2019

Statistics & Probability Letters

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The existence of time-lagged cross-correlations between the returns of a pair of assets, which is known as the lead-lag relationship, is a well-known stylized fact in financial econometrics. Recently some continuous-time models have been proposed to take account of the lead-lag relationship. Such a model does not follow a semimartingale as long as the lead-lag relationship is present, so it admits an arbitrage without market frictions. In this paper we show that they are free of arbitrage if we take account of market frictions such as the presence of minimal waiting time on subsequent transactions or transaction costs.

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Section: Trading With Simple Strategiesmentioning

confidence: 99%

No arbitrage and lead–lag relationships

Hayashi

Koike

2019

Statistics & Probability Letters

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“…These include the stickiness condition [22] or the conditional full support property [21]. Additionally, conditions on the fine structure of the paths may be required to prevent arbitrage opportunities by trading on arbitrarily small time intervals including conditions on the pathwise quadratic variation [30,5] or the two-way crossing property [7,28]. While in these works a first shift of paradigms from the probabilistic martingale condition to paths properties may be visible, the paths properties are still formulated with respect to a reference probability measure.…”

Section: Introductionmentioning

confidence: 99%

Model-Free Finance and Non-Lattice Integration

Bender,

Ferrando,

Gonzalez

2021

Preprint

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Starting solely with a set of possible prices for a traded asset S (in infinite discrete time) expressed in units of a numeraire, we explain how to construct a Daniell type of integral representing prices of integrable functions depending on the asset. Such functions include the values of simple dynamic portfolios obtained by trading with S and the numeraire. The space of elementary integrable functions, i.e. the said portfolio values, is not a vector lattice. It then follows that the integral is not classical, i.e. it is not associated to a measure. The essential ingredient in constructing the integral is a weak version of the no-arbitrage condition but here expressed in terms of properties of the trajectory space. We also discuss the continuity conditions imposed by Leinert (Archiv der Mathematik, 1982) and König (Mathematische Annalen, 1982) in the abstract theory of non-lattice integration from a financial point of view and establish some connections between these continuity conditions and the existence of martingale measures.

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“…Remark 1.3. Let us remark that Assumptions 1.1-1.2 hold true for reasonable semi-martingale models and important non semi-martingale models such as the exponential fractional Brownian motion (see [10] for Assumption 1.1 and [3,13] for Assumption 1.2).…”

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confidence: 99%

A Note on Utility Maximization with Proportional Transaction Costs and Stability of Optimal Portfolios

Bayraktar¹,

Czichowsky²,

Dolinskyi³

et al. 2021

Preprint

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The aim of this short note is to establish a limit theorem for the optimal trading strategies in the setup of the utility maximization problem with proportional transaction costs. This limit theorem resolves the open question from [4]. The main idea of our proof is to establish a uniqueness result for the optimal strategy. Surprisingly, up to date, there are no results related to the uniqueness of the optimal trading strategy. The proof of the uniqueness is heavily based on the dual approach which was developed recently in [6,7,8].

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Fractional Brownian motion satisfies two-way crossing

Cited by 7 publications

References 12 publications

No arbitrage and lead–lag relationships

No arbitrage and lead–lag relationships

Model-Free Finance and Non-Lattice Integration

A Note on Utility Maximization with Proportional Transaction Costs and Stability of Optimal Portfolios

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