On the stickiness property

Bayraktar, Erhan; Sayit, Hasanjan

doi:10.1080/14697680903341806

Cited by 9 publications

(15 citation statements)

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“…In the frictionless case one typically assumes the existence of an equivalent local martingale measure for the price process having suitable integrability properties to achieve this. In contrast, our sufficient conditions under transaction costs are more robust and hold in a wide variety of models; see [3,14,28,32,31,35,46,45] 1 . They apply in particular to the fractional Black-Scholes model and exponential utility.…”

Section: Introductionmentioning

confidence: 95%

“…exists in the sense of [50]. Here, A U (x; S) denotes the set of all S-integrable (in the sense of Itô), predictable processes ϑ = (ϑ t ) 0≤t≤T such that there exists a sequence (ϑ n ) ∞ n=1 of self-financing and admissible trading strategies ϑ n = (ϑ n t ) 0≤t≤T without transaction costs 3…”

Section: Formulation Of the Problemmentioning

confidence: 99%

“…> 0 by the duality relation (3. 3) and therefore that the dual optimiser Z = ( Z 0 , Z 1 ) is in Z λ e . It then follows along the same arguments as in the proof of Proposition 3.2 after replacing ϕ n = (ϕ 0,n , ϕ 1,n ) by ϕ n = ( ϕ 0,n , ϕ 1,n ) andZ = (Z 0 ,Z 1…”

Section: )mentioning

confidence: 99%

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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: Introductionmentioning

confidence: 95%

Section: Formulation Of the Problemmentioning

confidence: 99%

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Portfolio optimisation beyond semimartingales: Shadow prices and fractional Brownian motion

Czichowsky¹,

Schachermayer²

2017

Ann. Appl. Probab.

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“…One well‐known example is the conditional full support condition proposed by Guasoni, Rásonyi, and Schachermayer (). Other related sufficient conditions are discussed in Bayraktar and Sayit (), Maris, Mbakop, and Sayit (), and Sayit and Viens (). Recently, for continuous price processes, Rásonyi and Schachermayer () built the equivalence between the absence of arbitrage with general strategies for any small constant transaction cost

λ > 0

and the existence of a CPS for any small transaction cost

λ > 0

.…”

Section: Introductionmentioning

confidence: 99%

On the market viability under proportional transaction costs

Bayraktar

2017

Mathematical Finance

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Abstract. This paper studies the market viability with proportional transaction costs. Instead of requiring the existence of strictly consistent price systems (SCPS) as in the literature, we show that strictly consistent local martingale systems (SCLMS) can successfully serve as the dual elements such that the market viability can be verified. We introduce two weaker notions of no arbitrage conditions on market models named no unbounded profit with bounded risk (NUPBR) and no local arbitrage with bounded portfolios (NLABP). In particular, we show that the NUPBR and NLABP conditions in the robust sense for the smaller bid-ask spreads is the equivalent characterization of the existence of SCLMS for general market models. We also discuss the implications for the utility maximization problem.

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“…and the results on stickiness for a single stock in Guasoni [10] and Bayraktar and Sayit [1] are relevant to this model. As demonstrated in Guasoni [10], the presence of liquidation costs kY t |θ t | in 2 gives more flexibility on the price processes: all the regular strong Markov processes and processes with full support do not admit arbitrage in this model.…”

mentioning

confidence: 99%