Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Finance and Stochastics
Erschienen in: Finance and Stochastics 2/2018

21.02.2018

Stability of Radner equilibria with respect to small frictions

Erschienen in: Finance and Stochastics | Ausgabe 2/2018

Einloggen

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

search-config
loading …

Abstract

We study risk-sharing equilibria with trading subject to small proportional transaction costs. We show that the frictionless equilibrium prices also form an “asymptotic equilibrium” in the small-cost limit. More precisely, there exist asymptotically optimal policies for all agents and a split of the trading cost according to their risk aversions for which the frictionless equilibrium prices still clear the market. Starting from a frictionless equilibrium, this allows studying the interplay of volatility, liquidity and trading volume.
Vorheriger Artikel Perfect hedging under endogenous permanent market impacts
Nächster Artikel Correction to: Yield curve shapes and the asymptotic short rate distribution in affine one-factor models

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
Anhänge
Nur mit Berechtigung zugänglich
Fußnoten
1
The numerical results of [48, 30] also confirm the accuracy of the small-cost asymptotics for reasonable parameter values. For this reason, these expansions allow one to “reveal the salient features of the problem while remaining a good approximation to the full but more complicated model” [70].
 
2
This use of approximate optimality is similar to the notion of \(\varepsilon\)-equilibria in game theory; cf. [59] and many more recent studies.
 
3
If agents have identical risk aversions as in [48], each of them pays the same cost. In general, more risk-averse agents have a stronger motive to trade and are therefore willing to bear a larger share. This assumption is made for tractability; it is relaxed in [7] in a simpler model with quadratic trading costs and preferences, where only expected returns, but not volatilities and interest rates, are determined in equilibrium.
 
4
This suggests that relaxing individual optimality to an asymptotic version does not affect the equilibrium implications at the leading order. At higher orders, other variations of the model such as initial positions, liquidation conventions, dividend specifications, finite time horizons, etc., also play a non-negligible role. A rigorous proof of such a result in a general setting is an important direction for future research.
 
5
Note that the liquidation times \(\widehat {\tau}^{1}_{\varepsilon}\) and \(\widehat {\tau} ^{2}_{\varepsilon}\) of Agents 1 and 2, defined in (A.14), coincide.
 
6
To obtain a formula for leading-order trading volume in their model, multiply the inverse of the approximate waiting time [48, Eq. (30)] between successive trades with the corresponding trade size, which is given by the half-width of the no-trade region from [48, Theorem 4].
 
7
As \(\varphi\) is làdlàg (because it is of finite variation), integration by parts is well defined, and as \(\Delta S^{i,\varepsilon}\) is continuous, we need not consider left and right limits.
 
Literatur
1.
2.
Almgren, R.F., Chriss, N.: Optimal execution of portfolio transactions. J. Risk 3, 5–40 (2001) CrossRef
3.
Almgren, R.F., Li, T.M.: Option hedging with smooth market impact. Mark. Microstruct. Liq. 2, 1650002 (2016) CrossRef
4.
5.
Biagini, S., Frittelli, M.: Utility maximization in incomplete markets for unbounded processes. Finance Stoch. 9, 493–517 (2005) MathSciNetCrossRefMATH
6.
Biagini, S., Frittelli, M.: The supermartingale property of the optimal wealth process for general semimartingales. Finance Stoch. 11, 253–266 (2007) MathSciNetCrossRefMATH
7.
8.
9.
10.
Cai, J., Rosenbaum, M., Tankov, P.: Asymptotic lower bounds for optimal tracking: a linear programming approach. Ann. Appl. Probab. 27, 2455–2514 (2017) MathSciNetCrossRefMATH
11.
Cai, J., Rosenbaum, M., Tankov, P.: Asymptotic optimal tracking: feedback strategies. Stochastics 89, 943–966 (2017) MathSciNetCrossRef
12.
13.
Christensen, P.O., Larsen, K.: Incomplete continuous-time securities markets with stochastic income volatility. Rev. Asset Pricing Stud. 4, 247–285 (2014) CrossRef
14.
Christensen, P.O., Larsen, K., Munk, C.: Equilibrium in securities markets with heterogeneous investors and unspanned income risk. J. Econ. Theory 147, 1035–1063 (2012) MathSciNetCrossRefMATH
15.
Constantinides, G.M.: Capital market equilibrium with transaction costs. J. Polit. Econ. 94, 842–862 (1986) CrossRef
16.
Cvitanić, J., Karatzas, I.: Hedging and portfolio optimization under transaction costs: a martingale approach. Math. Finance 6, 133–165 (1996) MathSciNetCrossRefMATH
17.
18.
19.
Delbaen, F., Grandits, P., Rheinländer, T., Samperi, D., Schweizer, M., Stricker, C.: Exponential hedging and entropic penalties. Math. Finance 12, 99–123 (2002) MathSciNetCrossRefMATH
20.
Dellacherie, C., Meyer, P.-A.: Probabilities and Potential B: Theory of Martingales. North-Holland, Amsterdam (1982) MATH
21.
Dixit, A.: Analytical approximations in models of hysteresis. Rev. Econ. Stud. 58, 141–151 (1991) CrossRef
22.
Duffie, D.: Dynamic Asset Pricing Theory, 3rd edn. Princeton University Press, Princeton (2001) MATH
23.
24.
Dumas, B., Luciano, E.: An exact solution to a dynamic portfolio choice problem under transactions costs. J. Finance 46, 577–595 (1991) CrossRef
25.
Elworthy, K.D., Li, X.-M., Yor, M.: The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes. Probab. Theory Relat. Fields 115, 325–355 (1999) MathSciNetCrossRefMATH
26.
27.
Frittelli, M.: The minimal entropy martingale measure and the valuation problem in incomplete markets. Math. Finance 10, 39–52 (2000) MathSciNetCrossRefMATH
28.
Gârleanu, N., Pedersen, L.H.: Dynamic trading with predictable returns and transaction costs. J. Finance 68, 2309–2340 (2013) CrossRef
29.
30.
Gerhold, S., Guasoni, P., Muhle-Karbe, J., Schachermayer, W.: Transaction costs, trading volume, and the liquidity premium. Finance Stoch. 18, 1–37 (2014) MathSciNetCrossRefMATH
31.
32.
Heaton, J., Lucas, D.: Evaluating the effects of incomplete markets on risk sharing and asset pricing. J. Polit. Econ. 104, 443–487 (1993) CrossRef
33.
34.
35.
Janeček, K., Shreve, S.E.: Asymptotic analysis for optimal investment and consumption with transaction costs. Finance Stoch. 8, 181–206 (2004) MathSciNetCrossRefMATH
36.
Janeček, K., Shreve, S.E.: Futures trading with transaction costs. Ill. J. Math. 54, 1239–1284 (2010) MathSciNetMATH
37.
Kabanov, Y., Stricker, C.: On the optimal portfolio for the exponential utility maximization: remarks to the six-author paper. Math. Finance 12, 125–134 (2002) MathSciNetCrossRefMATH
38.
Kallenberg, O.: Foundations of Modern Probability, 2nd edn. Springer, New York (2002) CrossRefMATH
39.
40.
Kallsen, J., Muhle-Karbe, J.: On using shadow prices in portfolio optimization with transaction costs. Ann. Appl. Probab. 20, 1341–1358 (2010) MathSciNetCrossRefMATH
41.
42.
Kallsen, J., Muhle-Karbe, J.: The general structure of optimal investment and consumption with small transaction costs. Math. Finance 27, 659–703 (2017) MathSciNetCrossRefMATH
43.
Karatzas, I., Lehoczky, J.P., Shreve, S.E.: Equilibrium models with singular asset prices. Math. Finance 1(3), 11–29 (1991) CrossRefMATH
44.
Karatzas, I., Shreve, S.E.: Methods of Mathematical Finance. Springer, New York (1998) CrossRefMATH
45.
46.
Korn, R.: Portfolio optimisation with strictly positive transaction costs and impulse control. Finance Stoch. 2, 85–114 (1998) CrossRefMATH
47.
Liu, H., Loewenstein, M.: Optimal portfolio selection with transaction costs and finite horizons. Rev. Financ. Stud. 15, 805–835 (2002) CrossRef
48.
Lo, A., Mamaysky, H., Wang, J.: Asset prices and trading volume under fixed transactions costs. J. Polit. Econ. 112, 1054–1090 (2004) CrossRef
49.
50.
Martin, R.: Optimal trading under proportional transaction costs. RISK August, 54–59 (2014)
51.
Meyer, P.-A.: Inégalités de normes pour les intégrales stochastiques. In: Dellacherie, C., et al. (eds.) Sém. Probab. Strasbourg XII. Lecture Notes in Mathematics, vol. 649, pp. 757–762 (1978)
52.
Moreau, L., Muhle-Karbe, J., Soner, H.M.: Trading with small price impact. Math. Finance 27, 350–400 (2017) MathSciNetCrossRef
53.
54.
Owen, M.P., Žitković, G.: Optimal investment with an unbounded random endowment and utility-based pricing. Math. Finance 19, 129–159 (2009) MathSciNetCrossRefMATH
55.
56.
Possamaï, D., Soner, H.M., Touzi, N.: Homogenization and asymptotics for small transaction costs: the multidimensional case. Commun. Partial Differ. Equ. 40, 2005–2046 (2015) MathSciNetCrossRefMATH
57.
Protter, P.E.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2005) CrossRef
58.
Radner, R.: Existence of equilibrium of plans, prices, and price expectations in a sequence of markets. Econometrica 2, 289–303 (1972) MathSciNetCrossRefMATH
59.
Radner, R.: Collusive behavior in noncooperative epsilon-equilibria of oligopolies with long but finite lives. J. Econ. Theory 22, 136–154 (1980) CrossRefMATH
60.
Rogers, L.C.G.: Why is the effect of proportional transaction costs \(O(\delta ^{2/3})\)? In: Yin, G., Zhang, Q. (eds.) Mathematics of Finance, pp. 303–308. Am. Math. Soc., Providence (2004) CrossRef
61.
Rosenbaum, M., Tankov, P.: Asymptotically optimal discretization of hedging strategies with jumps. Ann. Appl. Probab. 24, 1002–1048 (2014) MathSciNetCrossRefMATH
62.
Schachermayer, W.: Optimal investment in incomplete markets when wealth may become negative. Ann. Appl. Probab. 11, 694–734 (2001) MathSciNetCrossRefMATH
63.
64.
Shreve, S.E., Soner, H.M.: Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4, 609–692 (1994) MathSciNetCrossRefMATH
65.
Soner, H.M., Touzi, N.: Homogenization and asymptotics for small transaction costs. SIAM J. Control Optim. 51, 2893–2921 (2013) MathSciNetCrossRefMATH
66.
Vayanos, D.: Transaction costs and asset prices: a dynamic equilibrium model. Rev. Financ. Stud. 11, 1–58 (1998) CrossRef
67.
Vayanos, D., Vila, J.-L.: Equilibrium interest rate and liquidity premium with transaction costs. Econ. Theory 13, 509–539 (1999) CrossRefMATH
68.
Wang, J.: A model of intertemporal asset prices under asymmetric information. Rev. Econ. Stud. 60, 249–282 (1993) CrossRefMATH
69.
70.
Whalley, A.E., Wilmott, P.: An asymptotic analysis of an optimal hedging model for option pricing with transaction costs. Math. Finance 7, 307–324 (1997) MathSciNetCrossRefMATH
71.
Metadaten
Titel
Stability of Radner equilibria with respect to small frictions
Publikationsdatum
21.02.2018
Erschienen in
Finance and Stochastics / Ausgabe 2/2018
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI
https://doi.org/10.1007/s00780-018-0354-x

Weitere Artikel der Ausgabe 2/2018

Finance and Stochastics 2/2018 Zur Ausgabe

OriginalPaper

Perfect hedging under endogenous permanent market impacts

Correction

Correction to: Yield curve shapes and the asymptotic short rate distribution in affine one-factor models

OriginalPaper

Approximation of forward curve models in commodity markets with arbitrage-free finite-dimensional models

OriginalPaper

An expansion in the model space in the context of utility maximization

OriginalPaper

A risk-neutral equilibrium leading to uncertain volatility pricing

OriginalPaper

The microstructural foundations of leverage effect and rough volatility