The Annals of Applied Probability

Asymptotically optimal discretization of hedging strategies with jumps

Mathieu Rosenbaum and Peter Tankov

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In this work, we consider the hedging error due to discrete trading in models with jumps. Extending an approach developed by Fukasawa [In Stochastic Analysis with Financial Applications (2011) 331–346 Birkhäuser/Springer Basel AG] for continuous processes, we propose a framework enabling us to (asymptotically) optimize the discretization times. More precisely, a discretization rule is said to be optimal if for a given cost function, no strategy has (asymptotically, for large cost) a lower mean square discretization error for a smaller cost. We focus on discretization rules based on hitting times and give explicit expressions for the optimal rules within this class.

Article information

Ann. Appl. Probab. Volume 24, Number 3 (2014), 1002-1048.

First available in Project Euclid: 23 April 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H05: Stochastic integrals 91G20: Derivative securities

Discretization of stochastic integrals asymptotic optimality hitting times option hedging semimartingales with jumps Blumenthal–Getoor index


Rosenbaum, Mathieu; Tankov, Peter. Asymptotically optimal discretization of hedging strategies with jumps. Ann. Appl. Probab. 24 (2014), no. 3, 1002--1048. doi:10.1214/13-AAP940.

Export citation


  • [1] Aït-Sahalia, Y. and Jacod, J. (2009). Estimating the degree of activity of jumps in high frequency data. Ann. Statist. 37 2202–2244.
  • [2] Belomestny, D. (2010). Spectral estimation of the fractional order of a Lévy process. Ann. Statist. 38 317–351.
  • [3] Bertsimas, D., Kogan, L. and Lo, A. W. (2000). When is time continuous. J. Financ. Econ. 55 173–204.
  • [4] Blumenthal, R. M. and Getoor, R. K. (1961). Sample functions of stochastic processes with stationary independent increments. J. Math. Mech. 10 493–516.
  • [5] Blumenthal, R. M., Getoor, R. K. and Ray, D. B. (1961). On the distribution of first hits for the symmetric stable processes. Trans. Amer. Math. Soc. 99 540–554.
  • [6] Brodén, M. and Tankov, P. (2011). Tracking errors from discrete hedging in exponential Lévy models. Int. J. Theor. Appl. Finance 14 803–837.
  • [7] Carr, P., Geman, H., Madan, D. and Yor, M. (2002). The fine structure of asset returns: An empirical investigation. J. Bus. 75 305–332.
  • [8] Cont, R. and Mancini, C. (2011). Nonparametric tests for pathwise properties of semimartingales. Bernoulli 17 781–813.
  • [9] Cont, R., Tankov, P. and Voltchkova, E. (2007). Hedging with options in models with jumps. In Stochastic Analysis and Applications. Abel Symp. 2 197–217. Springer, Berlin.
  • [10] Černý, A. and Kallsen, J. (2007). On the structure of general mean-variance hedging strategies. Ann. Probab. 35 1479–1531.
  • [11] Figueroa-López, J. E. (2009). Nonparametric estimation of time-changed Lévy models under high-frequency data. Adv. in Appl. Probab. 41 1161–1188.
  • [12] Figueroa-López, J. E. (2012). Statistical estimation of Lévy-type stochastic volatility models. Ann. Finance 8 309–335.
  • [13] Föllmer, H. and Schweizer, M. (1991). Hedging of contingent claims under incomplete information. In Applied Stochastic Analysis (London, 1989). Stochastics Monogr. 5 389–414. Gordon and Breach, New York.
  • [14] Föllmer, H. and Sondermann, D. (1986). Hedging of nonredundant contingent claims. In Contributions to Mathematical Economics 205–223. North-Holland, Amsterdam.
  • [15] Fukasawa, M. (2011). Asymptotically efficient discrete hedging. In Stochastic Analysis with Financial Applications. Progress in Probability 65 331–346. Birkhäuser/Springer Basel AG, Basel.
  • [16] Getoor, R. K. (1961). First passage times for symmetric stable processes in space. Trans. Amer. Math. Soc. 101 75–90.
  • [17] Gradshetyn, I. and Ryzhik, I. (1995). Table of Integrals, Series and Products. Academic Press, San Diego.
  • [18] Hayashi, T. and Mykland, P. A. (2005). Evaluating hedging errors: An asymptotic approach. Math. Finance 15 309–343.
  • [19] Hubalek, F., Kallsen, J. and Krawczyk, L. (2006). Variance-optimal hedging for processes with stationary independent increments. Ann. Appl. Probab. 16 853–885.
  • [20] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften 288. Springer, Berlin.
  • [21] Merton, R. C. (1971). Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3 373–413.
  • [22] Rosenbaum, M. and Tankov, P. (2011). Asymptotic results for time-changed Lévy processes sampled at hitting times. Stochastic Process. Appl. 121 1607–1632.
  • [23] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge Univ. Press, Cambridge.
  • [24] Schweizer, M. (2001). A guided tour through quadratic hedging approaches. In Option Pricing, Interest Rates and Risk Management. Handb. Math. Finance 538–574. Cambridge Univ. Press, Cambridge.
  • [25] Tankov, P. and Voltchkova, E. (2009). Asymptotic analysis of hedging errors in models with jumps. Stochastic Process. Appl. 119 2004–2027.
  • [26] Woerner, J. H. C. (2007). Inference in Lévy-type stochastic volatility models. Adv. in Appl. Probab. 39 531–549.
  • [27] Zhang, R. (1999). Couverture approchée des options Européennes. Ph.D. thesis, Ecole Nationale des Ponts et Chaussées.