A multidimensional bipolar theorem in L0(Rd;Ω,F,P)

Bouchard, Bruno; Mazliak, Laurent

doi:10.1016/s0304-4149(03)00073-5

Cited by 9 publications

(7 citation statements)

References 11 publications

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“…The understanding of the duality, sometimes called the "martingale method", in the context of portfolio optimisation goes back to [38,34,39] in the frictionless case. Under transaction costs, our work complements the dynamic duality results [16,17,20,22] for utility functions on the positive half-line as well as the static duality results [24,8,9,11,4] for (possibly) multi-variate utility functions.…”

Section: Introductionsupporting

confidence: 54%

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Portfolio optimisation beyond semimartingales: Shadow prices and fractional Brownian motion

Czichowsky¹,

Schachermayer²

2017

Ann. Appl. Probab.

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London School of Economics and Political Science and Universität WienWhile 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 Itô process. As a consequence, we obtain a rather surprising result on the pathwise behaviour of fractional Brownian motion: the trajectories may touch an Itô process in a one-sided manner without reflection. Introduction.Most of the literature in mathematical finance assumes that discounted prices S = (S t ) 0≤t≤T of risky assets are modelled by semimartingales. In frictionless financial markets, where arbitrary amounts of stock can be bought and sold at the same price S t , the semimartingale assumption is necessary. Otherwise, there would exist "arbitrage opportunities" (see [26], Theorem 7.2 for a precise statement) and optimal strategies for utility maximisation problems would fail to exist or yield infinite expected utility (see [2,40,42]).For non-semimartingale models based on fractional Brownian motion (B H t ) t≥0 such as the fractional Black-Scholes model S t = exp(μt + σ B H t ), where μ ∈ R, σ > 0 and Hurst parameter H ∈ (0, 1) \ {

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Section: Introductionsupporting

confidence: 54%

“…(A.9)Combining (A 8). and (A 9). gives lim n→∞ E U ′ g(x) g n − g(x) = 0 and therefore thatU ′ g(x) g n − g(x) + L 1 (P ) − −− → 0.The convergence h y(x) g n L 1 (P ) − −− → h y(x) g(x) then follows, since U ′ g(x) = y(x) h y(x) .…”

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confidence: 99%

Portfolio optimisation beyond semimartingales: Shadow prices and fractional Brownian motion

Czichowsky¹,

Schachermayer²

2017

Ann. Appl. Probab.

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“…For 0 ≤ t ≤ σ we may then explicitly calculate the primal optimizer 4) and the dual optimizer 5) so that…”

Section: Two Counterexamplesmentioning

confidence: 99%

Shadow Prices for Continuous Processes

Czichowsky

Schachermayer

Yang

2015

Mathematical Finance

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In a financial market with a continuous price process and proportional transaction costs, we investigate the problem of utility maximization of terminal wealth. We give sufficient conditions for the existence of a shadow price process, i.e., a least favorable frictionless market leading to the same optimal strategy and utility as in the original market under transaction costs. The crucial ingredients are the continuity of the price process and the hypothesis of "no unbounded profit with bounded risk". A counterexample reveals that these hypotheses cannot be relaxed. MSC 2010 Subject Classification: 91G10, 93E20, 60G48JEL Classification Codes: G11, C61

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“…It is well known that the existence of a shadow price is related to the solution of a suitable dual problem; see [21,10,12,13,11]. Under transaction costs, this duality goes back to the seminal work [8] of Cvitanić and Karatzas and has been subsequently extended to dynamic duality results [8,9,12,13,11] for utility functions on the positive half-line as well as static duality results [14,4,5,6,2] for (possibly) multi-variate utility functions.…”

Section: Introductionmentioning

confidence: 99%

Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs

et al. 2017

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We continue the analysis of our previous paper [13] pertaining to the existence of a shadow price process for portfolio optimisation under proportional transaction costs. There, we established a positive answer for a continuous price process S = (S t ) 0≤t≤T satisfying the condition (N U P BR) of "no unbounded profit with bounded risk". This condition requires that S is a semimartingale and therefore is too restrictive for applications to models driven by fractional Brownian motion. In the present paper, we derive the same conclusion under the weaker condition (T W C) of "two way crossing", which does not require S to be a semimartingale. Using a recent result of R. Peyre, this allows us to show the existence of a shadow price for exponential fractional Brownian motion and all utility functions defined on the positive half-line having reasonable asymptotic elasticity. Prime examples of such utilities are logarithmic or power utility. MSC 2010 Subject Classification: 91G10, 93E20, 60G48 JEL Classification Codes: G11, C61

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A multidimensional bipolar theorem in L0(Rd;Ω,F,P)

Cited by 9 publications

References 11 publications

Portfolio optimisation beyond semimartingales: Shadow prices and fractional Brownian motion

Portfolio optimisation beyond semimartingales: Shadow prices and fractional Brownian motion

Shadow Prices for Continuous Processes

Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs

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