The Annals of Applied Probability

Strict local martingales and bubbles

Constantinos Kardaras, Dörte Kreher, and Ashkan Nikeghbali

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This paper deals with asset price bubbles modeled by strict local martingales. With any strict local martingale, one can associate a new measure, which is studied in detail in the first part of the paper. In the second part, we determine the “default term” apparent in risk-neutral option prices if the underlying stock exhibits a bubble modeled by a strict local martingale. Results for certain path dependent options and last passage time formulas are given.

Article information

Ann. Appl. Probab., Volume 25, Number 4 (2015), 1827-1867.

Received: December 2013
First available in Project Euclid: 21 May 2015

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Zentralblatt MATH identifier

Primary: 91G99: None of the above, but in this section 60G30: Continuity and singularity of induced measures 60G44: Martingales with continuous parameter 91G20: Derivative securities

Strict local martingales bubbles


Kardaras, Constantinos; Kreher, Dörte; Nikeghbali, Ashkan. Strict local martingales and bubbles. Ann. Appl. Probab. 25 (2015), no. 4, 1827--1867. doi:10.1214/14-AAP1037.

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  • [1] Bayraktar, E., Kardaras, C. and Xing, H. (2012). Strict local martingale deflators and valuing American call-type options. Finance Stoch. 16 275–291.
  • [2] Carr, P., Fisher, T. and Ruf, J. (2014). On the hedging of options on exploding exchange rates. Finance Stoch. 18 115–144.
  • [3] Cheridito, P., Nikeghbali, A. and Platen, E. (2012). Processes of class sigma, last passage times, and drawdowns. SIAM J. Financial Math. 3 280–303.
  • [4] Chybiryakov, O. (2007). Itô’s integrated formula for strict local martingales with jumps. In Séminaire de Probabilités XL. Lecture Notes in Math. 1899 375–388. Springer, Berlin.
  • [5] Cox, A. M. G. and Hobson, D. G. (2005). Local martingales, bubbles and option prices. Finance Stoch. 9 477–492.
  • [6] Delbaen, F. and Schachermayer, W. (1995). Arbitrage possibilities in Bessel processes and their relations to local martingales. Probab. Theory Related Fields 102 357–366.
  • [7] Delbaen, F. and Schachermayer, W. (1998). The fundamental theorem of asset pricing for unbounded stochastic processes. Math. Ann. 312 215–250.
  • [8] Delbaen, F. and Shirakawa, H. (2002). No arbitrage condition for positive diffusion price processes. Asia-Pac. Financ. Mark. 9 159–168.
  • [9] Dellacherie, C. (1969). Ensembles aléatoires. I. In Séminaire de Probabilités, III (Univ. Strasbourg, 1967/68) 97–114. Springer, Berlin.
  • [10] Elworthy, K. D., Li, X. M. and Yor, M. (1997). On the tails of the supremum and the quadratic variation of strictly local martingales. In Séminaire de Probabilités, XXXI. Lecture Notes in Math. 1655 113–125. Springer, Berlin.
  • [11] Elworthy, K. D., Li, X.-M. and Yor, M. (1999). The importance of strictly local martingales; applications to radial Ornstein–Uhlenbeck processes. Probab. Theory Related Fields 115 325–355.
  • [12] Eršov, M. P. (1975). Extension of measures and stochastic equations. Theory Probab. Appl. 19 431–444.
  • [13] Fernholz, E. R. and Karatzas, I. (2009). Stochastic portfolio theory: An overview. In Special Volume: Mathematical Modeling and Numerical Methods in Finance (P. G. Ciarlet et al., eds). Handbook of Numerical Analysis XV 89–167. Elsevier, Oxford.
  • [14] Fernholz, D. and Karatzas, I. (2010). On optimal arbitrage. Ann. Appl. Probab. 20 1179–1204.
  • [15] Föllmer, H. (1972). The exit measure of a supermartingale. Z. Wahrsch. Verw. Gebiete 21 154–166.
  • [16] Föllmer, H. and Protter, P. (2011). Local martingales and filtration shrinkage. ESAIM Probab. Stat. 15 S25–S38.
  • [17] Hulley, H. (2010). The economic plausibility of strict local martingales in financial modelling. In Contemporary Quantitative Finance 53–75. Springer, Berlin.
  • [18] Jacod, J. (1979). Calcul Stochastique et Problèmes de Martingales. Lecture Notes in Math. 714. Springer, Berlin.
  • [19] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
  • [20] Jarrow, R. A. and Protter, P. (2009). Forward and futures prices with bubbles. Int. J. Theor. Appl. Finance 12 901–924.
  • [21] Jarrow, R. A., Protter, P. and Shimbo, K. (2007). Asset price bubbles in complete markets. In Advances in Mathematical Finance. Appl. Numer. Harmon. Anal. 97–121. Birkhäuser, Boston, MA.
  • [22] Jarrow, R. A., Protter, P. and Shimbo, K. (2010). Asset price bubbles in incomplete markets. Math. Finance 20 145–185.
  • [23] Kreher, D. and Nikeghbali, A. (2013). A new kind of augmentation of filtrations suitable for a change of probability measure by a strict local martingale. Preprint, available at arXiv:1108.4243v2.
  • [24] Madan, D., Roynette, B. and Yor, M. (2008a). From Black–Scholes formula, to local times and last passage times for certain submartingales. Available at
  • [25] Madan, D., Roynette, B. and Yor, M. (2008b). Option prices as probabilities. Finance Res. Lett. 5 79–87.
  • [26] Madan, D. B. and Yor, M. (2006). Ito’s integrated formula for strict local martingales. In In Memoriam Paul-André Meyer: Séminaire de Probabilités XXXIX. Lecture Notes in Math. 1874 157–170. Springer, Berlin.
  • [27] Meyer, P. A. (1972). La mesure de H. Föllmer en théorie des surmartingales. In Séminaire de Probabilités, VI (Univ. Strasbourg, Année Universitaire 19701971; Journées Probabilistes de Strasbourg, 1971) 118–129. Lecture Notes in Math. 258. Springer, Berlin.
  • [28] Mijatović, A. and Urusov, M. (2012). On the martingale property of certain local martingales. Probab. Theory Related Fields 152 1–30.
  • [29] Najnudel, J. and Nikeghbali, A. (2011). A new kind of augmentation of filtrations. ESAIM Probab. Stat. 15 S39–S57.
  • [30] Pal, S. and Protter, P. (2010). Analysis of continuous strict local martingales via $h$-transforms. Stochastic Process. Appl. 120 1424–1443.
  • [31] Parthasarathy, K. R. (1967). Probability Measures on Metric Spaces. Probability and Mathematical Statistics 3. Academic Press, New York.
  • [32] Platen, E. and Heath, D. (2006). A Benchmark Approach to Quantitative Finance. Springer, Berlin.
  • [33] Profeta, C., Roynette, B. and Yor, M. (2010). Option Prices as Probabilities. A New Look at Generalized Black–Scholes Formulae. Springer, Berlin.
  • [34] Protter, Ph. E. (2013). Strict local martingales with jumps. Preprint, available at arXiv:1307.2436v2.
  • [35] Revuz, D. and Yor, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [36] Ruf, J. (2013). Hedging under arbitrage. Math. Finance 23 297–317.
  • [37] Yen, J.-Y. and Yor, M. (2011). Call option prices based on Bessel processes. Methodol. Comput. Appl. Probab. 13 329–347.