Abstract
We study arbitrage opportunities, market viability and utility maximization in market models with an insider. Assuming that an economic agent possesses an additional information in the form of an \(\mathscr {F}_T\)-measurable discrete random variable G, we give criteria for the no unbounded profits with bounded risk property to hold, characterize optimal arbitrage strategies, and prove duality results for the utility maximization problem faced by the insider. Examples of markets satisfying NUPBR yet admitting arbitrage opportunities are provided. For the case when G is a continuous random variable, we consider the notion of no asymptotic arbitrage of the first kind (NAA1) and give an explicit construction for unbounded profits if NAA1 fails.
Similar content being viewed by others
Notes
The modified Bessel functions of the first kind is defined by the series representation \(I_{\alpha }(x) = \sum _{m \ge 0}\frac{1}{m!\Gamma (m+\alpha +1)}\left( \frac{x}{2} \right) ^{2m+\alpha }\), for a real number \(\alpha \) which is not a negative integer, and satisfies \(I_{-n}(x) = I_n(x)\) for integer n.
References
Acciaio, B., Fontana, C., Kardaras, C.: Arbitrage of the first kind and filtration enlargements in semimartingale financial models. Stochast. Process. Appl. 126, 1761–1784 (2016)
Aksamit, A., Choulli, T., Jeanblanc, M.: On an optional semimartingale decomposition and the existence of a deflator in an enlarged filtration, in In Memoriam Marc Yor-Séminaire de Probabilités XLVII, pp. 187–218. Springer (2015)
Amendinger, J., Imkeller, P., Schweizer, M.: Additional logarithmic utility of an insider. Stochast. Process. Appl. 75, 263–286 (1998)
Amendinger, J., Becherer, D., Schweizer, M.: A monetary value for initial information in portfolio optimization. Finance Stochast. 7, 29–46 (2003)
Ankirchner, S., Zwierz, J.: Initial enlargement of filtrations and entropy of Poisson compensators. J. Theor. Probab. 24, 93–117 (2011)
Ankirchner, S., Dereich, S., Imkeller, P.: The Shannon information of filtrations and the additional logarithmic utility of insiders. Ann. Probab. 34, 743–778 (2006)
Baldeaux, J., Platen, E.: Liability driven investments under a benchmark based approach. Preprint (2013)
Bremaud, P.: Point Processes and Queues: Martingale Dynamics. Springer, Berlin (1981)
Chau, N.H.: A Study of Arbitrage Opportunities in Financial Markets without Martingale Measures. Ph.D. thesis, Università degli Studi di Padova / Université Paris Diderot, Paris 7 (2016)
Chau, H.N., Tankov, P.: Market models with optimal arbitrage. SIAM J. Financ. Math. 6, 66–85 (2015)
Choulli, T., Deng, J., Ma, J.: How non-arbitrage, viability and numéraire portfolio are related. Finance Stochast. 19(4), 719–741 (2015)
Danilova, A., Monoyios, M., Ng, A.: Optimal investment with inside information and parameter uncertainty. Math. Financ. Econ. 3, 13–38 (2010)
Delbaen, F., Schachermayer, W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300, 463–520 (1994)
Delbaen, F., Schachermayer, W.: The no-arbitrage property under a change of numéraire. Stochast. Stochast. Rep. 53(3–4), 213–226 (1995)
Fernholz, D., Karatzas, I.: On optimal arbitrage. Ann. Appl. Probab. 20, 1179–1204 (2010)
Fontana, C.: No-arbitrage conditions and absolutely continuous changes of measure. In: Hillairet, C., Jeanblanc, M., Jiao, Y. (eds.) Arbitrage, Credit and Informational Risks: Peking University Series in Mathematics, vol. 5, pp. 3–18. World Scientific, Singapore (2014)
Fontana, C.: Weak and strong no-arbitrage conditions for continuous financial markets. Int. J. Theor. Appl. Finance 18, 1550005 (2015)
Grorud, A., Pontier, M.: Insider trading in a continuous time market model. Int. J. Theor. Appl. Finance 1, 331–347 (1998)
Imkeller, P., Perkowski, N.: The existence of dominating local martingale measures. Finance Stochast. 19(4), 685–717 (2015)
Imkeller, P., Pontier, M., Weisz, F.: Free lunch and arbitrage possibilities in a financial market model with an insider. Stochast. Process. Appl. 92, 103–130 (2001)
Jacod, J.: Grossissement initial, hypothèse (H\(^{\prime }\)) et théorème de Girsanov, in Grossissements de filtrations: exemples et applications. Springer, pp. 15–35 (1985)
Jacod, J.: Intégrales stochastiques par rapport à une semimartingale vectorielle et changements de filtration, Séminaire de Probabilités XIV 1978/79, pp. 161–172. Springer, Berlin (1980)
Jeanblanc, M., Yor, M., Chesney, M.: Mathematical Methods for Financial Markets. Springer, Berlin (2009)
Jeulin, T.: Semi-Martingales et Grossissement d’une Filtration, vol. 833 of Lecture Notes in Mathematics. Springer (1980)
Kabanov, Y., Kardaras, C., Song, S.: No arbitrage and local martingale deflators. ArXiv preprint arXiv:1501.04363 (2015)
Kabanov, Y.M.: On the FTAP of Kreps–Delbaen–Schachermayer, in Statistics and Control of Random Processes, pp. 191–203. World Scientific, The Liptser Festschrift (1997)
Kabanov, Y.M., Kramkov, D.O.: Large financial markets: asymptotic arbitrage and contiguity. Theory Probab. Appl. 39, 182–187 (1994)
Karatzas, I., Kardaras, C.: The numéraire portfolio in semimartingale financial models. Finance Stochast. 11, 447–493 (2007)
Kardaras, C.: Market viability via absence of arbitrage of the first kind. Finance Stochast. 16, 651–667 (2012)
Kardaras, C., Kreher, D., Nikeghbali, A.: Strict local martingales and bubbles. Ann. Appl. Probab. 25, 1827–1867 (2015)
Kohatsu-Higa, A., Yamazato, M.: Insider models with finite utility in markets with jumps. Appl. Math. Optim. 64, 217–255 (2011)
Kramkov, D., Schachermayer, W.: The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Ann. Appl. Probab. 9, 904–950 (1999)
Levental, S., Skorohod, A.V.: A necessary and sufficient condition for absence of arbitrage with tame portfolios. Ann. Appl. Probab. 5, 906–925 (1995)
Meyer, P.-A., Jacod, J.: Sur un théoreme de. Séminaire de probabilités de Strasbourg 12, 57–60 (1978)
Mostovyi, O.: Necessary and sufficient conditions in the problem of optimal investment with intermediate consumption. Finance Stochast. 19, 135–159 (2015)
Pikovsky, I., Karatzas, I.: Anticipative portfolio optimization. Adv. Appl. Probab. 28, 1095–1122 (1996)
Protter, P.: Stochastic Integration and Differential Equations, 2nd edn. Springer, Berlin (2003)
Ruf, J., Runggaldier, W.: A systematic approach to constructing market models with arbitrage. In: Hillairet, C., Jeanblanc, M., Jiao, Y. (eds.) Arbitrage, Credit and Informational Risks: Peking University Series in Mathematics, vol. 5, pp. 19–28. World Scientific, Singapore (2014)
Song, S.: An alternative proof of a result of Takaoka. arXiv:1306.1062 (2013)
Takaoka, K., Schweizer, M.: A note on the condition of no unbounded profit with bounded risk. Finance Stochast. 18, 393–405 (2014)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
The research of Chau Ngoc Huy was supported by Natixis Foundation for Quantitative Research and the “Lendület” Grant LP2015-6 of the Hungarian Academy of Sciences. The research of Peter Tankov was supported by the chair “Financial Risks” sponsored by Société Générale. We thank the referees for helpful comments on the previous version of this paper.
Appendix
Appendix
Lemma 4.11
Assume that X, Y are two independent exponential random variables with parameters \(\alpha , \beta \), respectively. Then the random variable \(Z = \frac{\alpha X}{\beta Y}\) has density \(1/(1+ z)^2.\)
Proof
For \(z > 0\). we compute the cumulative distribution of Z
The density of Z is obtained by taking derivative of the cumulative distribution of Z with respect to z. \(\square \)
Definition 4.12
(Optional projection—Definition 5.2.1 of [23]) Let X be a bounded (or positive) process, and \(\mathbb {F}\) a given filtration. The optional projection of X is the unique optional process \({}^{o}X\) which satisfies
almost surely for any \(\mathbb {F}\)-stopping time \(\tau .\)
The following result helps us to find the compensator of a process when passing to smaller filtrations.
Lemma 4.13
Let \(\mathbb {G}, \mathbb {H}\) be filtrations such that \(\mathscr {G}_t \subset \mathscr {H}_t,\) for all \(t \in [0,T]\). Let X be a \(\mathbb {G}\)-adapted process. Suppose that the process \(M_t := X_t - \int \nolimits _0^t {\lambda _udu}\) is a \(\mathbb {H}\)-martingale, where \(\lambda \ge 0\). Then the process \(M^G_t :=X_t - \int \nolimits _0^t {^{o}\lambda _u du}\) is a \(\mathbb {G}\)-martingale, where \(^{o}\lambda \) is the optional projection of \(\lambda \) onto \(\mathbb {G}\).
Proof
Since \(\lambda _u \ge 0,\) the optional projection \(^{o}\lambda \) exists and for fixed u, it holds that \(^{o}\lambda _u = \mathbb {E}[\lambda _u| \mathscr {G}_u]\) almost surely. If \(0\le s <t\) and H is bounded and \(\mathscr {G}_s\)-measurable, then, by Fubini’s Theorem
Hence \(M^G\) is a \(\mathbb {G}\)-martingale.\(\square \)
Rights and permissions
About this article
Cite this article
Chau, H.N., Runggaldier, W.J. & Tankov, P. Arbitrage and utility maximization in market models with an insider. Math Finan Econ 12, 589–614 (2018). https://doi.org/10.1007/s11579-018-0217-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11579-018-0217-4
Keywords
- Initial enlargement of filtration
- Optimal arbitrage
- No unbounded profits with bounded risk
- Incomplete markets
- Hedging
- Utility maximization