• Bernoulli
  • Volume 20, Number 3 (2014), 1126-1164.

Small-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias

José E. Figueroa-López and Peter Tankov

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We characterize the small-time asymptotic behavior of the exit probability of a Lévy process out of a two-sided interval and of the law of its overshoot, conditionally on the terminal value of the process. The asymptotic expansions are given in the form of a first-order term and a precise computable error bound. As an important application of these formulas, we develop a novel adaptive discretization scheme for the Monte Carlo computation of functionals of killed Lévy processes with controlled bias. The considered functionals appear in several domains of mathematical finance (e.g., structural credit risk models, pricing of barrier options, and contingent convertible bonds) as well as in natural sciences. The proposed algorithm works by adding discretization points sampled from the Lévy bridge density to the skeleton of the process until the overall error for a given trajectory becomes smaller than the maximum tolerance given by the user.

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Bernoulli Volume 20, Number 3 (2014), 1126-1164.

First available in Project Euclid: 11 June 2014

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adaptive discretization barrier options bridge Monte Carlo methods exit probability killed Lévy process Lévy bridge small-time asymptotics


Figueroa-López, José E.; Tankov, Peter. Small-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias. Bernoulli 20 (2014), no. 3, 1126--1164. doi:10.3150/13-BEJ517.

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  • [1] Avramidis, A.N. and L’Ecuyer, P. (2006). Efficient Monte Carlo and quasi-Monte Carlo option pricing under the variance-gamma model. Management Science 52 1930–1944.
  • [2] Baldi, P. (1995). Exact asymptotics for the probability of exit from a domain and applications to simulation. Ann. Probab. 23 1644–1670.
  • [3] Barthelemy, P., Bertolotti, J. and Wiersma, D.S. (2008). A Lévy flight for light. Nature 453 495–498.
  • [4] Becker, M. (2010). Comment on “Correcting for simulation bias in Monte Carlo methods to value exotic options in models driven by Lévy processes” by C. Ribeiro and N. Webber [Appl. Math. Finance 13 (2006) 333–352]. Appl. Math. Finance 17 133–146.
  • [5] Benhamou, S. (2007). How many animals really do the Lévy walk? Ecology 88 1962–1969.
  • [6] Bertoin, J. (1996). Lévy Processes. Cambridge Tracts in Mathematics 121. Cambridge: Cambridge Univ. Press.
  • [7] Boyarchenko, S. and Levendorskiĭ, S. (2002). Barrier options and touch-and-out options under regular Lévy processes of exponential type. Ann. Appl. Probab. 12 1261–1298.
  • [8] Buldyrev, S.V., Gitterman, M., Havlin, S., Kazakov, A.Y., da Luz, M.G.E., Raposo, E.P., Stanley, H.E. and Viswanathan, G.M. (2001). Properties of Lévy flights on an interval with absorbing boundaries. Physica A: Statistical Mechanics and Its Applications 302 148–161.
  • [9] Chambers, J.M., Mallows, C.L. and Stuck, B.W. (1976). A method for simulating stable random variables. J. Amer. Statist. Assoc. 71 340–344.
  • [10] Chechkin, A.V., Gonchar, V.Yu., Klafter, J. and Metzler, R. (2005). Barrier crossing of a Lévy flight. Europhys. Lett. 72 348–354.
  • [11] Comte, F. and Genon-Catalot, V. (2011). Estimation for Lévy processes from high frequency data within a long time interval. Ann. Statist. 39 803–837.
  • [12] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC Financial Mathematics Series. Boca Raton, FL: Chapman & Hall/CRC.
  • [13] Corcuera, J., De Spiegeleer, J., Ferreiro-Castilla, A., Kyprianou, A.E., Madan, D. and Schoutens, W. (2011) Efficient pricing of contingent convertibles under smile conform models. Preprint. Available at
  • [14] Devroye, L. (1986). Non-Uniform Random Variate Generation. New York: Springer.
  • [15] Dzougoutov, A., Moon, K.-S., von Schwerin, E., Szepessy, A. and Tempone, R. (2005). Adaptive Monte Carlo algorithms for stopped diffusion. In Multiscale Methods in Science and Engineering (B. Enguist, P. Lötstedt and O. Runborg, eds.). Lecture Notes in Computational Science and Engineering 44 59–88. Berlin: Springer.
  • [16] Fang, F., Jonsson, H., Schoutens, W. and Oosterlee, C. (2010). Fast valuation and calibration of credit default swaps under Lévy dynamics. J. Comput. Finance 14 57–86.
  • [17] Figueroa-López, J.E. (2011). Sieve-based confidence intervals and bands for Lévy densities. Bernoulli 17 643–670.
  • [18] Figueroa-López, J.E. and Forde, M. (2012). The small-maturity smile for exponential Lévy models. SIAM J. Financial Math. 3 33–65.
  • [19] Figueroa-López, J.E., Gong, R. and Houdré, C. (2012). Small-time expansions of the distributions, densities, and option prices of stochastic volatility models with Lévy jumps. Stochastic Process. Appl. 122 1808–1839.
  • [20] Figueroa-López, J.E. and Houdré, C. (2009). Small-time expansions for the transition distributions of Lévy processes. Stochastic Process. Appl. 119 3862–3889.
  • [21] Figueroa-López, J.E. and Ouyang, C. (2011) Small-time expansions for local jump-diffusion models with infinite jump activity. Preprint. Available at arXiv:1108.3386v2 [math.PR].
  • [22] Garbaczewski, P. and Stephanovich, V. (2009). Lévy flights in confining potentials. Phys. Rev. E (3) 80 031113.
  • [23] Górska, K. and Penson, K.A. (2011). Lévy stable two-sided distributions: Exact and explicit densities for asymmetric case. Phys. Rev. E (3) 83 061125.
  • [24] Gradshetyn, I. and Ryzhik, I. (1995). Table of Integrals, Series and Products. San Diego: Academic Press.
  • [25] Houdré, C. (2002). Remarks on deviation inequalities for functions of infinitely divisible random vectors. Ann. Probab. 30 1223–1237.
  • [26] Johnson, N.L. and Rogers, C.A. (1951). The moment problem for unimodal distributions. Ann. Math. Statistics 22 433–439.
  • [27] Kou, S.G. and Wang, H. (2003). First passage times of a jump diffusion process. Adv. in Appl. Probab. 35 504–531.
  • [28] Kuznetsov, A., Kyprianou, A.E., Pardo, J.C. and van Schaik, K. (2011). A Wiener–Hopf Monte Carlo simulation technique for Lévy processes. Ann. Appl. Probab. 21 2171–2190.
  • [29] Kyprianou, A.E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Universitext. Berlin: Springer.
  • [30] Léandre, R. (1987) Densité en temps petit d’un processus de sauts. In Séminaire de Probabilités XXI (J. Azéma, M. Yor and P.A. Meyer, eds.). Lecture Notes in Math. 1247 81–99. Berlin: Springer.
  • [31] Metwally, S. and Atiya, A. (2002). Using Brownian bridge for fast simulation of jump-diffusion processes and barrier options. Journal of Derivatives Fall 143–154.
  • [32] Metzler, R. and Klafter, J. (2000). The random walk’s guide to anomalous diffusion: A fractional dynamics approach. Phys. Rep. 339 1–77.
  • [33] Moon, K.-S. (2008). Efficient Monte Carlo algorithm for pricing barrier options. Commun. Korean Math. Soc. 23 285–294.
  • [34] Mordecki, E., Szepessy, A., Tempone, R. and Zouraris, G.E. (2008). Adaptive weak approximation of diffusions with jumps. SIAM J. Numer. Anal. 46 1732–1768.
  • [35] Protter, P.E. (2004). Stochastic Integration and Differential Equations, 2nd ed. Applications of Mathematics (New York) 21. Berlin: Springer.
  • [36] Ribeiro, C. and Webber, N. (2006). Correcting for simulation bias in Monte Carlo methods to value exotic options in models driven by Lévy processes. Appl. Math. Finance 13 333–352.
  • [37] Rosenbaum, M. and Tankov, P. (2011). Asymptotic results for time-changed Lévy processes sampled at hitting times. Stochastic Process. Appl. 121 1607–1632.
  • [38] Rüschendorf, L. and Woerner, J.H.C. (2002). Expansion of transition distributions of Lévy processes in small time. Bernoulli 8 81–96.
  • [39] Samorodnitsky, G. and Taqqu, M.S. (1994). Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance. Stochastic Modeling. New York: Chapman & Hall.
  • [40] Sato, K.-i. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics 68. Cambridge: Cambridge Univ. Press.
  • [41] Shlesinger, M.F., Zaslavsky, G.M. and Frisch, U., eds. (1995). Lévy Flights and Related Topics in Physics. Lecture Notes in Physics 450. Berlin: Springer.
  • [42] Szepessy, A., Tempone, R. and Zouraris, G.E. (2001). Adaptive weak approximation of stochastic differential equations. Comm. Pure Appl. Math. 54 1169–1214.
  • [43] Tankov, P. (2010). Pricing and hedging in exponential Lévy models: Review of recent results. In Paris–Princeton Lectures on Mathematical Finance 2010. Lecture Notes in Math. 2003 319–359. Berlin: Springer.
  • [44] Viswanathan, G.M., Afanasyev, V., Buldyrev, S.V., Murphy, E.J., Prince, P.A. and Stanley, H.E. (1996). Lévy flight search patterns of wandering albatrosses. Nature 381 413–415.
  • [45] Webber, N. (2005). Simulation methods with Lévy processes. In Exotic Option Pricing and Advanced Lévy Models (A. Kyprianou, W. Schoutens and P. Wilmott, eds.) 29–49. Chichester: Wiley.