nach oben

Finance and Stochastics

Erschienen in:

17.11.2017

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

verfasst von: Christoph Czichowsky, Rémi Peyre, Walter Schachermayer, Junjian Yang

Erschienen in: Finance and Stochastics | Ausgabe 1/2018

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

The present paper accomplishes a major step towards a reconciliation of two conflicting approaches in mathematical finance: on the one hand, the mainstream approach based on the notion of no arbitrage (Black, Merton & Scholes), and on the other hand, the consideration of non-semimartingale price processes, the archetype of which being fractional Brownian motion (Mandelbrot). Imposing (arbitrarily small) proportional transaction costs and considering logarithmic utility optimisers, we are able to show the existence of a semimartingale, frictionless shadow price process for an exponential fractional Brownian financial market.

Vorheriger Artikel No-arbitrage under a class of honest times

Nächster Artikel Replicating portfolio approach to capital calculation

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Nur mit Berechtigung zugänglich

Here, we say that a property holds locally for the process \(S\) if there exists a localising sequence of stopping times \((\tau_{n})_{n=1}^{\infty}\) such that the stopped process \(S^{\tau_{n}}\) has this property for each \(n\).

Here, we mean that there exists one localised version of \(S\), not depending on \(\mu\), which admits a \(\mu \)-consistent price system for all \(\mu\in(0, 1)\).

The set \(\mathcal{B}(y)\) of all \(\lambda \) -consistent supermartingale deflators consists of all pairs of nonnegative càdlàg supermartingales \(Y = (Y^{0}_{t}, Y^{1}_{t})_{0 \leq t \leq T}\) such that \(\mathbb{\operatorname{E}}[Y^{0}_{0}] = y\), \(Y^{1} = Y^{0} \tilde{S}\) for some \([(1 - \lambda) S, S]\)-valued process \(\tilde{S} = (\tilde{S}_{t})_{0 \leq t \leq T}\), and \(Y^{0} (\phi^{0} + \phi^{1} \tilde{S}) = Y^{0} \phi^{0} + Y^{1} \phi ^{1}\) is a nonnegative càdlàg supermartingale for all \(\phi\in \mathcal{A} (1)\). Note that \(y \mathcal{Z} \subseteq\mathcal{B} (y)\) for \(y > 0\) by Proposition 2.6 of [11].

Equation (A.3) is actually not used in this article, but this result seemed to us worth being written.

Exact translation and scale invariance of fractional Brownian motion is actually not needed here: more precisely, exact invariance shortens the proof by allowing the use of Fernique’s theorem, but a slight refinement of that theorem would make the result work as soon as one has a bound of the type \(\operatorname{Var}[B^{H}_{t} - B^{H}_{s}] \leq C \left\lvert t - s\right\rvert ^{2 H}\); see e.g. [28, Lemma 4.2].

Ankirchner, S., Imkeller, P.: Finite utility on financial markets with asymmetric information and structure properties of the price dynamics. Ann. Inst. Henri Poincaré Probab. Stat. 41, 479–503 (2005) MathSciNetCrossRefMATH

Bayraktar, E., Yu, X.: On the market viability under proportional transaction costs. Mathematical Finance (2013). To appear. Available online at: arXiv:1312.3917

Bender, C.: Simple arbitrage. Ann. Appl. Probab. 22, 2067–2085 (2012) MathSciNetCrossRefMATH

Bouchard, B.: Utility maximization on the real line under proportional transaction costs. Finance Stoch. 6, 495–516 (2002) MathSciNetCrossRefMATH

Bouchard, B., Mazliak, L.: A multidimensional bipolar theorem in \(L^{0}(\mathbb{R}^{d};\Omega ,\mathcal {F},P)\). Stoch. Process. Appl. 107, 213–231 (2003) MathSciNetCrossRefMATH

Campi, L., Owen, M.P.: Multivariate utility maximization with proportional transaction costs. Finance Stoch. 15, 461–499 (2011) MathSciNetCrossRefMATH

Cvitanić, J., Karatzas, I.: Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Finance 6, 113–165 (1996) MathSciNetMATH

Cvitanić, J., Wang, H.: On optimal terminal wealth under transaction costs. J. Math. Econ. 35, 223–231 (2001) MathSciNetCrossRefMATH

Czichowsky, C., Muhle-Karbe, J., Schachermayer, W.: Transaction costs, shadow prices, and duality in discrete time. SIAM J. Financ. Math. 5, 258–277 (2014) MathSciNetCrossRefMATH

10.

Czichowsky, C., Schachermayer, W.: Duality theory for portfolio optimisation under transaction costs. Ann. Appl. Probab. 26, 1888–1941 (2016) MathSciNetCrossRefMATH

11.

Czichowsky, C., Schachermayer, W., Yang, J.: Shadow prices for continuous processes. Math. Finance 27, 623–658 (2017) MathSciNetCrossRefMATH

12.

Deelstra, G., Pham, H., Touzi, N.: Dual formulation of the utility maximization problem under transaction costs. Ann. Appl. Probab. 11, 1353–1383 (2001) MathSciNetCrossRefMATH

13.

Dudley, R.M.: Uniform Central Limit Theorems, 2nd edn. Cambridge Studies in Advanced Mathematics, vol. 142. Cambridge University Press, New York (2014) MATH

14.

Guasoni, P.: Optimal investment with transaction costs and without semimartingales. Ann. Appl. Probab. 12, 1227–1246 (2002) MathSciNetCrossRefMATH

15.

Guasoni, P.: No arbitrage under transaction costs, with fractional Brownian motion and beyond. Math. Finance 16, 569–582 (2006) MathSciNetCrossRefMATH

16.

Guasoni, P., Rásonyi, M., Schachermayer, W.: The fundamental theorem of asset pricing for continuous processes under small transaction costs. Ann. Finance 6, 157–191 (2010) CrossRefMATH

17.

He, H., Pearson, N.D.: Consumption and portfolio policies with incomplete markets and short-sale constraints: the infinite-dimensional case. J. Econ. Theory 54, 259–304 (1991) MathSciNetCrossRefMATH

18.

Kabanov, Y.M., Last, G.: Hedging under transaction costs in currency markets: a continuous-time model. Math. Finance 12, 63–70 (2002) MathSciNetCrossRefMATH

19.

Kallsen, J., Muhle-Karbe, J.: Existence of shadow prices in finite probability spaces. Math. Methods Oper. Res. 73, 251–262 (2011) MathSciNetCrossRefMATH

20.

Karatzas, I., Kardaras, C.: The numéraire portfolio in semimartingale financial models. Finance Stoch. 11, 447–493 (2007) MathSciNetCrossRefMATH

21.

Karatzas, I., Lehoczky, J.P., Shreve, S.E.: Optimal portfolio and consumption decisions for a “small investor” on a finite horizon. SIAM J. Control Optim. 25, 1557–1586 (1987) MathSciNetCrossRefMATH

22.

Karatzas, I., Lehoczky, J.P., Shreve, S.E., Xu, G.-L.: Martingale and duality methods for utility maximization in an incomplete market. SIAM J. Control Optim. 29, 702–730 (1991) MathSciNetCrossRefMATH

23.

Kardaras, C., Platen, E.: On the semimartingale property of discounted asset-price processes. Stoch. Process. Appl. 121, 2678–2691 (2011) MathSciNetCrossRefMATH

24.

Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999) MathSciNetCrossRefMATH

25.

Kramkov, D., Schachermayer, W.: Necessary and sufficient conditions in the problem of optimal investment in incomplete markets. Ann. Appl. Probab. 13, 1504–1516 (2003) MathSciNetCrossRefMATH

26.

Larsen, K., Žitković, G.: On the semimartingale property via bounded logarithmic utility. Ann. Finance 4, 255–268 (2008) CrossRefMATH

27.

Mandelbrot, B.: The variation of some other speculative prices. J. Bus. 40, 393–413 (1967) CrossRef

28.

Peyre, R.: Fractional Brownian motion satisfies two-way crossing. Bernoulli 23, 3571–3597 (2017) MathSciNetCrossRefMATH

29.

Schachermayer, W.: The Asymptotic Theory of Transaction Costs. Zürich Lecture Notes in Advanced Mathematics. EMS Publishing House, Zürich (2017) CrossRefMATH

Titel: Shadow prices, fractional Brownian motion, and portfolio optimisation under transaction costs
verfasst von: Christoph Czichowsky
Rémi Peyre
Walter Schachermayer
Junjian Yang
Publikationsdatum: 17.11.2017
Verlag: Springer Berlin Heidelberg
Erschienen in: Finance and Stochastics / Ausgabe 1/2018
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-017-0351-5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Weitere Artikel der Ausgabe 1/2018

Dynamic programming approach to principal–agent problems

No-arbitrage under a class of honest times

Replicating portfolio approach to capital calculation

Time-consistent stopping under decreasing impatience

Financial equilibrium with asymmetric information and random horizon

Optimal liquidation under stochastic liquidity