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

Part of: Probabilistic methods, simulation and stochastic differential equations Mathematical finance

Published online by Cambridge University Press:  30 March 2016

Aleksandar Mijatović ,
Martijn R. Pistorius  and
Johannes Stolte
Aleksandar Mijatović*
Affiliation:
Imperial College London
Martijn R. Pistorius*
Affiliation:
Imperial College London
Johannes Stolte*
Affiliation:
Imperial College London
*
∗ Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
∗ Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
∗ Postal address: Department of Mathematics, Imperial College London, South Kensington Campus, London SW7 2AZ, UK.
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Abstract

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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.

Keywords

Lévy bridge processmixed-exponential jump-diffusionfirst-passage timeoccupation timeMarkov bridge sampling.

MSC classification

Primary: 65C05: Monte Carlo methods
Secondary: 91G60: Numerical methods (including Monte Carlo methods)

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 52 , Issue 4 , December 2015 , pp. 1076 - 1096
Copyright
Copyright © Applied Probability Trust 2015 

References

[1] Asmussen, S. and Albrecher, H. (2010). Ruin Probabilities , 2nd edn. World Scientific, Hackensack, NJ.Google Scholar
[2] Asmussen, S. and Rosiński, J. (2001). Approximations of small jumps of Lévy processes with a view towards simulation. J. Appl. Prob. 38, 482–493.Google Scholar
[3] Asmussen, S., Avram, F. and Usabel, M. (2002). Erlangian approximations of finite-horizon ruin probabilities. ASTIN Bull. 32, 267–281.Google Scholar
[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, 75–107.Google Scholar
[5] Bartholomew, D. J. (1969). Sufficient conditions for a mixture of exponentials to be a probability density function. Ann. Math. Statist. 40, 2183–2188.CrossRefGoogle Scholar
[6] Bates, D. S. (1996). Jumps and stochastic volatility: exchange rate processes implicit in Deutsche Mark options. Rev. Financial Studies 9, 69–107.Google Scholar
[7] Bertoin, J. (1996). Lévy Processes. Cambridge University Press.Google Scholar
[8] Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2006). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12, 1077–1098.CrossRefGoogle Scholar
[9] Botta, R. F. and Harris, C. M. (1986). Approximation with generalized hyperexponential distributions: weak convergence results. Queueing Systems 1, 169–190.Google Scholar
[10] Boyarchenko, M. and Levendorskii, S. (2012). Valuation of continuously monitored double barrier options and related securities. Math. Finance 22, 419–444.Google Scholar
[11] Boyle, P., Broadie, M. and Glasserman, P. (1997). Monte Carlo methods for security pricing. J. Econom. Dynam. Control 21, 1267–1321.Google Scholar
[12] Cai, N. and Kou, S. G. (2011). Option pricing under a mixed-exponential jump diffusion model. Manag. Sci. 57, 2067–2081.Google Scholar
[13] Carr, P. (1998). Randomization and the American put. Rev. Financial Studies 11, 597–626.Google Scholar
[14] Chaumont, L. and Uribe Bravo, G. (2011). Markovian bridges: weak continuity and pathwise constructions. Ann. Prob. 39, 609–647.CrossRefGoogle Scholar
[15] Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
[16] Dereich, S. (2011). Multilevel Monte Carlo algorithms for Lévy-driven SDEs with Gaussian correction. Ann. Appl. Prob. 21, 283–311.Google Scholar
[17] Feller, W. (1966). An Introduction to Probability Theory and Its Applications , Vol. II. John Wiley, New York.Google Scholar
[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, 985–1010.Google Scholar
[19] Figueroa-López, J. E. and Tankov, P. (2014). Small-time asymptotics of stopped Lévy bridges and simulation schemes with controlled bias. Bernoulli 20, 1126–1164.Google Scholar
[20] Gatheral, J. (2006). The Volatility Surface: A Practitioner's Guide. John Wiley, Hoboken, NJ.Google Scholar
[21] Gobet, E. (2000). Weak approximation of killed diffusion using Euler schemes. Stoch. Process. Appl. 87, 167–197.CrossRefGoogle Scholar
[22] Haug, E. G. (2007). The Complete Guide to Option Pricing Formulas, 2nd edn. McGraw-Hill, New York.Google Scholar
[23] Higham, D. J. and Mao, X. (2005). Convergence of Monte Carlo simulations involving the mean-reverting square root process. J. Comp. Finance 8, 35–62.Google Scholar
[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, 629–644.Google Scholar
[25] Kallenberg, O. (2002). Foundations of Modern Probability , 2nd edn. Springer, New York.Google Scholar
[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, 3234–3254.Google Scholar
[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. 49–80.Google Scholar
[28] Kloeden, P. E. and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Berlin.Google Scholar
[29] Kou, S. G. and Wang, H. (2004). Option pricing under a double exponential jump diffusion model. Manag. Sci. 50, 1178–1192.Google Scholar
[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, 2171–2190.Google Scholar
[31] Kyprianou, A. E. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
[32] Kyprianou, A. E. and Pistorius, M. R. (2003). Perpetual options and Canadization through fluctuation theory. Ann. Appl. Prob. 13, 1077–1098.Google Scholar
[33] Levendorskii, S. (2011). Convergence of price and sensitivities in Carr's randomization approximation globally and near barrier. SIAM J. Financial Math. 2, 79–111.CrossRefGoogle Scholar
[34] Lewis, A. L. and Mordecki, E. (2008). Wiener-Hopf factorization for Lévy processes having positive jumps with rational transforms. J. Appl. Prob. 45, 118–134.Google Scholar
[35] Marchuk, G. I. and Shaidurov, V. V. (1983). Difference Methods and Their Extrapolations. Springer, New York.Google Scholar
[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, 43–54.CrossRefGoogle Scholar
[37] Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (2002). Numerical Recipes in C++ , 2nd edn. Cambridge University Press.Google Scholar
[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, 127–145.CrossRefGoogle Scholar
[39] Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
[40] Sidi, A. (2003). Practical Extrapolation Methods: Theory and Applications. Cambridge University Press.Google Scholar
[41] Stolte, J. (2013). On accurate and efficient valuation of financial contracts under models with jumps. Doctoral Thesis, Imperial College London.Google Scholar
[42] Uribe Bravo, G. (2014). Bridges of Lévy processes conditioned to stay positive. Bernoulli 20, 190–206.Google Scholar