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

Czichowsky, Christoph; Peyre, Rémi; Schachermayer, Walter; Yang, Junjian

doi:10.1007/s00780-017-0351-5

Cited by 24 publications

(30 citation statements)

References 34 publications

(79 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore, it would be crucial and interesting to extend the results of this paper to JSDEs driven by general Lévy processes. We also note that various theoretical results with applications for SDEs driven by fractional Brownian motions (fBms) have been studied extensively in literature; for instance, we refer the reader to [2,4,6,16,34] and the references therein. Thus, it would be important to extend our results to JSDEs driven by fBms.…”

Section: Discussionmentioning

confidence: 99%

Strong solutions for jump-type stochastic differential equations with non-Lipschitz coefficients

2019

View full text Add to dashboard Cite

show abstract

Section: Discussionmentioning

confidence: 99%

Strong solutions for jump-type stochastic differential equations with non-Lipschitz coefficients

2019

View full text Add to dashboard Cite

show abstract

“…The shot noise processes can be used to model limit order books for very-or ultra-high-frequency data. On the other hand, FBM has been used in portfolio optimization with transaction costs (Czichowsky, Peyre, Schachermayer, & Yang, 2018;Schachermayer, 2017) and rough stochastic volatility (Bayer, Friz, & Gatheral, 2016). It would be interesting to see how one can use the generalized FBM in these studies.…”

Section: Literature Reviewmentioning

confidence: 99%

Nonstationary self‐similar Gaussian processes as scaling limits of power‐law shot noise processes and generalizations of fractional Brownian motion

Pang

Taqqu

2019

High Frequency

View full text Add to dashboard Cite

We study shot noise processes with Poisson arrivals and nonstationary noises. The noises are conditionally independent given the arrival times, but the distribution of each noise does depend on its arrival time. We establish scaling limits for such shot noise processes in two situations: (a) the conditional variance functions of the noises have a power law and (b) the conditional noise distributions are piecewise. In both cases, the limit processes are self‐similar Gaussian with nonstationary increments. Motivated by these processes, we introduce new classes of self‐similar Gaussian processes with nonstationary increments, via the time‐domain integral representation, which are natural generalizations of fractional Brownian motions.

show abstract

“…Trading strategies are modelled by R 2 -valued, predictable processes ϕ = (ϕ 0 t , ϕ 1 t ) 0≤t≤T of finite variation, where ϕ 0 t and ϕ 1 t describe the holdings in the riskless and the risky asset, 2 Note added in proof: This question has been answered in [19] in the meantime. If the indirect utility is finite, it is sufficient for the existence of a shadow price that the price process is continuous and satisfies the condition (T W C) of "two way crossing"; see [5,47].…”

Section: Formulation Of the Problemmentioning

confidence: 99%

Portfolio optimisation beyond semimartingales: Shadow prices and fractional Brownian motion

Czichowsky¹,

Schachermayer²

2017

Ann. Appl. Probab.

Self Cite

View full text Add to dashboard Cite

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) \ {

show abstract

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

Cited by 24 publications

References 34 publications

Strong solutions for jump-type stochastic differential equations with non-Lipschitz coefficients

Strong solutions for jump-type stochastic differential equations with non-Lipschitz coefficients

Nonstationary self‐similar Gaussian processes as scaling limits of power‐law shot noise processes and generalizations of fractional Brownian motion

Portfolio optimisation beyond semimartingales: Shadow prices and fractional Brownian motion

Contact Info

Product

Resources

About