Skip to main content Accessibility help

Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications

  • Aleksandar Mijatović (a1), Martijn R. Pistorius (a1) and Johannes Stolte (a1)


We develop a new Monte Carlo variance reduction method to estimate the expectation of two commonly encountered path-dependent functionals: first-passage times and occupation times of sets. The method is based on a recursive approximation of the first-passage time probability and expected occupation time of sets of a Lévy bridge process that relies in part on a randomisation of the time parameter. We establish this recursion for general Lévy processes and derive its explicit form for mixed-exponential jump-diffusions, a dense subclass (in the sense of weak approximation) of Lévy processes, which includes Brownian motion with drift, Kou's double-exponential model, and hyperexponential jump-diffusion models. We present a highly accurate numerical realisation and derive error estimates. By way of illustration the method is applied to the valuation of range accruals and barrier options under exponential Lévy models and Bates-type stochastic volatility models with exponential jumps. Compared with standard Monte Carlo methods, we find that the method is significantly more efficient.


Corresponding author

Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
∗∗ Email address:
∗∗∗ Email address:
∗∗∗∗ Email address:


Hide All
[1] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities , 2nd edn. World Scientific, Hackensack, NJ.
[2] Asmussen, S. and Rosiński, J. (2001). Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Prob. 38, 482493.
[3] Asmussen, S., Avram, F. and Usabel, M. (2002). Erlangian approximations of finite-horizon ruin probabilities. ASTIN Bull. 32, 267281.
[4] Avram, F., Chan, T. and Usabel, M. (2002). On the valuation of constant barrier options under spectrally one-sided exponential Lévy models and Carr's approximation for American puts. Stoch. Process. Appl. 100, 75107.
[5] Bartholomew, D. J. (1969). Sufficient conditions for a mixture of exponentials to be a probability density function. Ann. Math. Statist. 40, 21832188.
[6] Bates, D. S. (1996). Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options. Rev. Financial Studies 9, 69107.
[7] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.
[8] Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12, 10771098.
[9] Botta, R. F. and Harris, C. M. (1986). Approximation with generalized hyperexponential distributions: weak convergence results. Queueing Systems 1, 169190.
[10] Boyarchenko, M. and Levendorskii, S. (2012). Valuation of continuously monitored double barrier options and related securities. Math. Finance 22, 419444.
[11] Boyle, P., Broadie, M. and Glasserman, P. (1997). Monte Carlo methods for security pricing. J. Econom. Dynam. Control 21, 12671321.
[12] Cai, N. and Kou, S. G. (2011). Option pricing under a mixed-exponential jump diffusion model. Manag. Sci. 57, 20672081.
[13] Carr, P. (1998). Randomization and the American put. Rev. Financial Studies 11, 597626.
[14] Chaumont, L. and Uribe Bravo, G. (2011). Markovian bridges: weak continuity and pathwise constructions. Ann. Prob. 39, 609647.
[15] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL.
[16] Dereich, S. (2011). Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction. Ann. Appl. Prob. 21, 283311.
[17] Feller, W. (1966). An Introduction to Probability Theory and Its Applications , Vol. II. John Wiley, New York.
[18] Ferreiro-Castilla, A., Kyprianou, A. E., Scheichl, R. and Suryanarayana, G. (2014). Multilevel Monte Carlo simulation for Lévy processes based on the Wiener-Hopf factorisation. Stoch. Process. Appl. 124, 9851010.
[19] Figueroa-López, J. E. and Tankov, P. (2014). Small-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias. Bernoulli 20, 11261164.
[20] Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. John Wiley, Hoboken, NJ.
[21] Gobet, E. (2000). Weak approximation of killed diffusion using Euler schemes. Stoch. Process. Appl. 87, 167197.
[22] Haug, E. G. (2007). The Complete Guide to Option Pricing Formulas, 2nd edn. McGraw-Hill, New York.
[23] Higham, D. J. and Mao, X. (2005). Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comp. Finance 8, 3562.
[24] Jeannin, M. and Pistorius, M. (2010). A transform approach to compute prices and Greeks of barrier options driven by a class of Lévy processes. Quant. Finance 10, 629644.
[25] Kallenberg, O. (2002). Foundations of Modern Probability , 2nd edn. Springer, New York.
[26] Kleinert, F. and Van Schaik, K. (2015). A variation of the Canadisation algorithm for the pricing of American options driven by Lévy processes. Stoch. Process. Appl. 125, 32343254.
[27] Kloeden, P. and Neuenkirch, A. (2013). Convergence of numerical methods for stochastic differential equations in mathematical finance. In Recent Developments in Computational Finance , World Scientific, Hackensack, NJ, pp. 4980.
[28] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.
[29] Kou, S. G. and Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Manag. Sci. 50, 11781192.
[30] 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. Prob. 21, 21712190.
[31] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.
[32] Kyprianou, A. E. and Pistorius, M. R. (2003). Perpetual options and Canadization through fluctuation theory. Ann. Appl. Prob. 13, 10771098.
[33] Levendorskii, S. (2011). Convergence of price and sensitivities in Carr's randomization approximation globally and near barrier. SIAM J. Financial Math. 2, 79111.
[34] Lewis, A. L. and Mordecki, E. (2008). Wiener-Hopf factorization for Lévy processes having positive jumps with rational transforms. J. Appl. Prob. 45, 118134.
[35] Marchuk, G. I. and Shaidurov, V. V. (1983). Difference Methods and Their Extrapolations. Springer, New York.
[36] Metwally, S. A. K. and Atiya, A. F. (2002). Using Brownian bridge for fast simulation of jump-diffusion processes and barrier options. J. Derivatives 10, 4354.
[37] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (2002). Numerical Recipes in C++ , 2nd edn. Cambridge University Press.
[38] Ruf, J. and Scherer, M. (2011). Pricing corporate bonds in an arbitrary jump-diffusion model based on an improved Brownian-bridge algorithm. J. Comp. Finance 14, 127145.
[39] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.
[40] Sidi, A. (2003). Practical Extrapolation Methods: Theory and Applications. Cambridge University Press.
[41] Stolte, J. (2013). On accurate and efficient valuation of financial contracts under models with jumps. Doctoral Thesis, Imperial College London.
[42] Uribe Bravo, G. (2014). Bridges of Lévy processes conditioned to stay positive. Bernoulli 20, 190206.


MSC classification

Randomisation and recursion methods for mixed-exponential Lévy models, with financial applications

  • Aleksandar Mijatović (a1), Martijn R. Pistorius (a1) and Johannes Stolte (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed