No-arbitrage under a class of honest times
Tóm tắt
This paper quantifies the interplay between the no-arbitrage notion of no unbounded profit with bounded risk (NUPBR) and additional progressive information generated by a random time. This study complements the one of Aksamit et al. (Finance Stoch. 21:1103–1139, 2017) in which the authors have studied similar topics for the model stopped at the random time, while here we deal with the question of what happens after the random time. Given that the existing literature proves that NUPBR is always violated after honest times that avoid stopping times in a continuous filtration, we propose here a new class of honest times for which NUPBR can be preserved for some models. For these honest times, we obtain two principal results. The first result characterizes the pairs of initial market and honest time for which the resulting model preserves NUPBR, while the second result characterizes honest times that do not affect NUPBR of any quasi-left-continuous model (i.e., in which the asset price process has no predictable jump times). Furthermore, we construct explicitly local martingale deflators for a large class of models.
Tài liệu tham khảo
Acciaio, B., Fontana, C., Kardaras, C.: Arbitrage of the first kind and filtration enlargements in semimartingale financial models. Stoch. Process. Appl. 126, 1761–1784 (2016)
Aksamit, A.: Random Times, Enlargement of Filtration and Arbitrages. PhD Thesis, Evry-Val d’Essonne, University (2014). Available online at https://hal.archives-ouvertes.fr/tel-01016672/
Aksamit, A., Choulli, T., Deng, J., Jeanblanc, M.: Arbitrages in a progressive enlargement setting. In: Hillairet, C., et al. (eds.) Arbitrage, Credit and Informational Risks. Peking Univ. Ser. Math., vol. 5, pp. 53–86. World Scientific, Singapore (2014)
Aksamit, A., Choulli, T., Deng, J., Jeanblanc, M.: No-arbitrage up to random horizon for quasi-left-continuous models. Finance Stoch. 21, 1103–1139 (2017)
Aksamit, A., Choulli, T., Jeanblanc, M.: On an optional semimartingale decomposition and the existence of a deflator in an enlarged filtration. In: Donati-Martin, C., et al. (eds.) In Memoriam Marc Yor. Séminaire de Probabilités XLVII. Lecture Notes in Math., vol. 2137, pp. 187–218. Springer, Berlin (2015)
Aksamit, A., Choulli, T., Jeanblanc, M.: Classification of random time and applications. Working paper (2015). Available online at arXiv:1605.03905
Ankirchner, S., Dereich, S., Imkeller, P.: The Shannon information of filtrations and the additional logarithmic utility of insiders. Ann. Probab. 34, 743–778 (2006)
Ansel, J.-P., Stricker, C.: Couverture des actifs contingents. Ann. Inst. Henri Poincaré B, Probab. Stat. 30, 303–315 (1994)
Barlow, M.T.: Study of a filtration expanded to include an honest time. Probab. Theory Relat. Fields 44, 307–323 (1978)
Choulli, T., Deng, J., Ma, J.: How non-arbitrage, viability and numéraire portfolio are related. Finance Stoch. 19, 719–741 (2015)
Choulli, T., Stricker, C.: Deux applications de la décomposition de Galtchouk–Kunita–Watanabe. In: Azéma, J., et al. (eds.) Séminaire de Probabilités XXX. Lecture Notes in Math., vol. 1626, pp. 12–23. Springer, Berlin (1996)
Dellacherie, C.: Capacités et Processus Stochastiques. Springer, Berlin (1972)
Dellacherie, C., Meyer, P-A.: Probabilités et Potentiel, chapitres I–IV. Hermann, Paris (1980). English translation: Probabilities and Potentiel, chapters I–IV, North-Holland (1982)
Dellacherie, M., Maisonneuve, B., Meyer, P-A.: Probabilités et Potentiel, chapitres XVII–XXIV: Processus de Markov (fin), Compléments de Calcul Stochastique. Hermann, Paris (1992)
Deng, J.: Essays on Arbitrage Theory for a Class of Informational Markets. PhD Thesis, University of Alberta (2014). Available online at https://hdl.handle.net/10402/era.38797
Fontana, C., Jeanblanc, M., Song, S.: On arbitrages arising with honest times. Finance Stoch. 18, 515–543 (2014)
He, S.W., Wang, J.G., Yan, J.A.: Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton (1992)
Imkeller, P.: Random times at which insiders can have free lunches. Stoch. Stoch. Rep. 74, 465–487 (2002)
Jacod, J., Shiryaev, A.N.: Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin (2003)
Jeulin, T.: Semi-martingales et Grossissement d’une Filtration. Lecture Notes in Mathematics, vol. 833. Springer, Berlin (1980)
Kabanov, Y., Kardaras, C., Song, S.: No arbitrage of the first kind and local martingale numéraires. Finance Stoch. 20, 1097–1108 (2016)
Kardaras, C.: Market viability via absence of arbitrage of the first kind. Finance Stoch. 16, 651–667 (2012)
Loewenstein, M., Willard, G.A.: Local martingales, arbitrage, and viability. Free snacks and cheap thrills. Econ. Theory 16, 135–161 (2000)
Platen, E.: A benchmark approach to finance. Math. Finance 16, 131–151 (2006)
Takaoka, K., Schweizer, M.: A note on the condition of no unbounded profit with bounded risk. Finance Stoch. 18, 393–405 (2014)