nach oben

Finance and Stochastics

Erschienen in:

01.10.2015

A convergence result for the Emery topology and a variant of the proof of the fundamental theorem of asset pricing

verfasst von: Christa Cuchiero, Josef Teichmann

Erschienen in: Finance and Stochastics | Ausgabe 4/2015

Einloggen

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

search-config

KI-gestützte Suche

Aus

Abstract

We show that no unbounded profit with bounded risk (NUPBR) implies predictable uniform tightness (P-UT), a boundedness property in the Emery topology introduced by Stricker (Séminaire de Probabilités de Strasbourg XIX, pp. 209–217, 1985). Combining this insight with well-known results of Mémin and Słominski (Séminaire de Probabilités de Strasbourg XXV, pp. 162–177, 1991) leads to a short variant of the proof of the fundamental theorem of asset pricing initially proved by Delbaen and Schachermayer (Math. Ann. 300:463–520, 1994). The results are formulated in the general setting of admissible portfolio wealth processes as laid down by Kabanov (Statistics and Control of Stochastic Processes, pp. 191–203, World Sci. Publ., River Edge, 1997).

Vorheriger Artikel How non-arbitrage, viability and numéraire portfolio are related

Nächster Artikel Aggregation-robustness and model uncertainty of regulatory risk measures

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

In order to show that the separating measure is a local martingale measure, the local boundedness assumption cannot be dropped.

Black, F., Scholes, M.S.: The pricing of options and corporate liabilities. J. Polit. Econ. 81, 637–654 (1973) CrossRefMATH

Dalang, R.C., Morton, A., Willinger, W.: Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Rep. 29, 185–201 (1990) MathSciNetCrossRefMATH

Delbaen, F.: Representing martingale measures when asset prices are continuous and bounded. Math. Finance 2, 107–130 (1992) CrossRefMATH

Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994) MathSciNetCrossRefMATH

Delbaen, F., Schachermayer, W.: The no-arbitrage property under a change of numéraire. Stoch. Stoch. Rep. 53, 213–226 (1995) MathSciNetCrossRefMATH

Delbaen, F., Schachermayer, W.: The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312, 215–250 (1998) MathSciNetCrossRefMATH

Delbaen, F., Schachermayer, W.: The Mathematics of Arbitrage. Springer, Berlin (2006) MATH

Emery, M.: Une topologie sur l’espace des semimartingales. In: Emery, M., Yor, M. (eds.) Séminaire de Probabilités, XIII, Univ. Strasbourg, Strasbourg, 1977/78. Lecture Notes in Math., vol. 721, pp. 260–280. Springer, Berlin (1979) CrossRef

Fernholz, R., Karatzas, I.: Stochastic portfolio theory: an overview. In: Bensoussan, A., et al. (eds.) Handbook of Numerical Analysis, vol. XV, pp. 89–167. North-Holland, Amsterdam (2009). Special Volume: Mathematical Modeling and Numerical Methods in Finance

10.

Gushchin, A.A.: On taking limits under the compensator sign. In: Ibragimov, I.A., et al. (eds.) Probability Theory and Mathematical Statistics, St. Petersburg, 1993, pp. 185–192. Gordon and Breach, Amsterdam (1996)

11.

Harrison, J.M., Kreps, D.M.: Martingales and arbitrage in multiperiod securities markets. J. Econ. Theory 20, 381–408 (1979) MathSciNetCrossRefMATH

12.

Harrison, J.M., Pliska, S.R.: Martingales and stochastic integrals in the theory of continuous trading. Stoch. Process. Appl. 11, 215–260 (1981) MathSciNetCrossRefMATH

13.

Imkeller, P., Perkowski, N.: The existence of dominating local martingale measures. Finance Stoch. 685–717 (2015, this issue). doi:10.1007/s00780-015-0264-0

14.

Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 288. Springer, Berlin (2003) CrossRefMATH

15.

Jakubowski, A., Mémin, J., Pagès, G.: Convergence en loi des suites d’intégrales stochastiques sur l’espace \(\mathbf{D}^{1}\) de Skorokhod. Probab. Theory Relat. Fields 81, 111–137 (1989) CrossRefMATH

16.

Kabanov, Y.M.: On the FTAP of Kreps–Delbaen–Schachermayer. In: Kabanov, Y., Rozovski, B., Shiryaev, A. (eds.) Statistics and Control of Stochastic Processes, Moscow, 1995/1996, pp. 191–203. World Sci. Publ., River Edge (1997)

17.

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

18.

Kardaras, C.: A structural characterization of numéraires of convex sets of nonnegative random variables. Positivity 16, 245–253 (2012) MathSciNetCrossRefMATH

19.

Kardaras, C.: Generalized supermartingale deflators under limited information. Math. Finance 23, 186–197 (2013) MathSciNetCrossRefMATH

20.

Kardaras, C.: On the closure in the Emery topology of semimartingale wealth-process sets. Ann. Appl. Probab. 23, 1355–1376 (2013) MathSciNetCrossRefMATH

21.

Kreps, D.M.: Arbitrage and equilibrium in economies with infinitely many commodities. J. Math. Econ. 8, 15–35 (1981) MathSciNetCrossRefMATH

22.

Kurtz, T.G., Protter, P.: Weak limit theorems for stochastic integrals and stochastic differential equations. Ann. Probab. 19, 1035–1070 (1991) MathSciNetCrossRefMATH

23.

Kurtz, T.G., Protter, P.E.: Weak convergence of stochastic integrals and differential equations. In: Talay, D., Tubaro, L. (eds.) Probabilistic Models for Nonlinear Partial Differential Equations, Montecatini Terme, 1995. Lecture Notes in Math., vol. 1627, pp. 1–41. Springer, Berlin (1996) CrossRef

24.

Mémin, J.: Espaces de semi martingales et changements de probabilité. Z. Wahrscheinlichkeitstheor. Verw. Geb. 52, 9–39 (1980) CrossRefMATH

25.

Mémin, J., Słominski, L.: Condition UT et stabilité en loi des solutions d’équations différentielles stochastiques. In: Azéma, J., Yor, M., Meyer, P.-A. (eds.) Séminaire de Probabilités de Strasbourg XXV. Lecture Notes in Math., vol. 1485, pp. 162–177 (1991) CrossRef

26.

Merton, R.C.: Theory of rational option pricing. Bell J. Econ. Manag. Sci. 4(1), 141–183 (1973) MathSciNetCrossRefMATH

27.

Meyer, P.-A.: Martingales and stochastic integrals. I. In: Dold, A., Eckmann, B. (eds.) Lecture Notes in Math., vol. 284. Springer, Berlin (1972)

28.

Rokhlin, D.B.: On the existence of an equivalent supermartingale density for a fork-convex family of random processes. Mat. Zametki 87, 594–603 (2010) MathSciNetCrossRefMATH

29.

Ross, S.A.: A simple approach to the valuation of risky streams. J. Bus. 51, 453–475 (1978) CrossRef

30.

Schachermayer, W.: Martingale measures for discrete-time processes with infinite horizon. Math. Finance 4, 25–55 (1994) MathSciNetCrossRefMATH

31.

Schachermayer, W.: Fundamental theorem of asset pricing. In: Cont, R. (ed.) Encyclopedia of Quantitative Finance, pp. 792–801. Wiley, New York (2010)

32.

Stricker, C.: Lois de semimartingales et critères de compacité. In: Azéma, J., Yor, M. (eds.) Séminaire de Probabilités de Strasbourg XIX. Lecture Notes in Math., vol. 1123, pp. 209–217 (1985)

33.

Stricker, C.: Arbitrage et lois de martingale. Ann. Inst. Henri Poincaré Probab. Stat. 26, 451–460 (1990) MathSciNetMATH

34.

Yan, J.A.: Caractérisation d’une classe d’ensembles convexes de \(L^{1}\) ou \(H^{1}\). In: Azéma, J., Yor, M. (eds.) Séminaire de Probabilités XIV, Paris, 1978/1979. Lecture Notes in Math., vol. 784, pp. 220–222. Springer, Berlin (1980)

Titel: A convergence result for the Emery topology and a variant of the proof of the fundamental theorem of asset pricing
verfasst von: Christa Cuchiero
Josef Teichmann
Publikationsdatum: 01.10.2015
Verlag: Springer Berlin Heidelberg
Erschienen in: Finance and Stochastics / Ausgabe 4/2015
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-015-0276-9

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 4/2015

Pricing and hedging Asian-style options on energy

Discretely monitored first passage problems and barrier options: an eigenfunction expansion approach

RETRACTED ARTICLE: The distribution of the maximum of a variance gamma process and path-dependent option pricing

How non-arbitrage, viability and numéraire portfolio are related

Dynamic credit investment in partially observed markets

An optimal consumption problem in finite time with a constraint on the ruin probability