Hostname: page-component-8448b6f56d-qsmjn Total loading time: 0 Render date: 2024-04-23T14:51:41.578Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact simulation of Ornstein–Uhlenbeck tempered stable processes

Part of: Distribution theory Distribution theory - Probability Stochastic processes Probabilistic methods, simulation and stochastic differential equations Stochastic analysis

Published online by Cambridge University Press:  23 June 2021

Yan Qu ,
Angelos Dassios  and
Hongbiao Zhao Open the ORCID record for Hongbiao Zhao [Opens in a new window]
Yan Qu*
Affiliation:
Peking University
Angelos Dassios*
Affiliation:
London School of Economics and Political Science
Hongbiao Zhao*
Affiliation:
Shanghai University of Finance and Economics
*
*Postal address: School of Mathematical Sciences, Peking University, Beijing 100871, China.
**Postal address: Department of Statistics, London School of Economics, Houghton Street, LondonWC2A 2AE, UK.
***Postal address: School of Statistics and Management, Shanghai University of Finance and Economics, 777 Guoding Road, Shanghai 200433, China; Shanghai Institute of International Finance and Economics, 777 Guoding Road, Shanghai 200433, China. Email address: h.zhao1@lse.ac.uk.
Get access Rights & Permissions [Opens in a new window]

Abstract

There are two types of tempered stable (TS) based Ornstein–Uhlenbeck (OU) processes: (i) the OU-TS process, the OU process driven by a TS subordinator, and (ii) the TS-OU process, the OU process with TS marginal law. They have various applications in financial engineering and econometrics. In the literature, only the second type under the stationary assumption has an exact simulation algorithm. In this paper we develop a unified approach to exactly simulate both types without the stationary assumption. It is mainly based on the distributional decomposition of stochastic processes with the aid of an acceptance–rejection scheme. As the inverse Gaussian distribution is an important special case of TS distribution, we also provide tailored algorithms for the corresponding OU processes. Numerical experiments and tests are reported to demonstrate the accuracy and effectiveness of our algorithms, and some further extensions are also discussed.

Keywords

Monte Carlo simulationexact simulationnon-Gaussian Ornstein–Uhlenbeck processtempered stable subordinatortempered stable OU processOU tempered stable process

MSC classification

Primary: 65C05: Monte Carlo methods 62E15: Exact distribution theory 60G51: Processes with independent increments; L'evy processes
Secondary: 60E07: Infinitely divisible distributions; stable distributions 68U20: Simulation 60H35: Computational methods for stochastic equations
Type
Original Article
Information
Journal of Applied Probability , Volume 58 , Issue 2 , June 2021 , pp. 347 - 371
Check for updates
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Andrieu, C., Doucet, A. and Holenstein, R. (2010). Particle Markov chain Monte Carlo methods. J. R. Statist. Soc. B [Statist. Methodology] 72, 269–342.10.1111/j.1467-9868.2009.00736.xCrossRefGoogle Scholar
Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.10.1007/978-0-387-69033-9CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 41–68.10.1007/s007800050032CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Modelling by Lévy processes for financial econometrics. In Lévy Processes, eds O. Barndorff-Nielsen, S. Resnick and T. Mikosch, pp. 283–318. Birkhäuser, Boston.Google Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Statist. Soc. B [Statist. Methodology] 63, 167–241.10.1111/1467-9868.00282CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Normal modified stable processes. Discussion Paper Series no. 72, Department of Economics, University of Oxford.Google Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2002). Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. R. Statist. Soc. B [Statist. Methodology] 64, 253–280.10.1111/1467-9868.00336CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2003). Integrated OU processes and non-Gaussian OU-based stochastic volatility models. Scand. J. Statist. 30, 277–295.10.1111/1467-9469.00331CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2003). Realized power variation and stochastic volatility models. Bernoulli 9, 243–265.10.3150/bj/1068128977CrossRefGoogle Scholar
Barndorff-Nielsen, O. E., Jensen, J. L. and SØrensen, M. (1998). Some stationary processes in discrete and continuous time. Adv. Appl. Prob. 30, 989–1007.10.1239/aap/1035228204CrossRefGoogle Scholar
Barndorff-Nielsen, O. E., Nicolato, E. and Shephard, N. (2002). Some recent developments in stochastic volatility modelling. Quant. Finance 2, 11–23.10.1088/1469-7688/2/1/301CrossRefGoogle Scholar
Bertoin, J. (1998). Lévy Processes. Cambridge University Press, Cambridge.Google Scholar
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2002). The fine structure of asset returns: an empirical investigation. J. Business 75, 305–333.10.1086/338705CrossRefGoogle Scholar
Chen, Z., Feng, L. and Lin, X. (2012). Simulating Lévy processes from their characteristic functions and financial applications. ACM Trans. Model. Comput. Simul. 22, 14.Google Scholar
Chhikara, R. and Folks, L. (1989). The Inverse Gaussian Distribution: Theory, Methodology, and Applications. Marcel Dekker, New York.Google Scholar
Dassios, A. and Embrechts, P. (1989). Martingales and insurance risk. Stoch. Models 5, 181–217.CrossRefGoogle Scholar
Dassios, A. and Jang, J. (2003). Pricing of catastrophe reinsurance and derivatives using the Cox process with shot noise intensity. Finance Stoch. 7, 73–95.10.1007/s007800200079CrossRefGoogle Scholar
Dassios, A. and Zhao, H. (2011). A dynamic contagion process. Adv. Appl. Prob. 43, 814–846.10.1239/aap/1316792671CrossRefGoogle Scholar
Dassios, A. and Zhao, H. (2013). Exact simulation of Hawkes process with exponentially decaying intensity. Electron. Commun. Prob. 18, 1–13.10.1214/ECP.v18-2717CrossRefGoogle Scholar
Dassios, A. and Zhao, H. (2017). Efficient simulation of clustering jumps with CIR intensity. Operat. Res. 65, 1494–1515.10.1287/opre.2017.1640CrossRefGoogle Scholar
Dassios, A., Lim, J. W. and Qu, Y. (2020). Exact simulation of truncated Lévy subordinator. ACM Trans. Model. Comput. Simul. 30, 17.10.1145/3368088CrossRefGoogle Scholar
Dassios, A., Qu, Y. and Zhao, H. (2018). Exact simulation for a class of tempered stable and related distributions. ACM Trans. Model. Comput. Simul. 28, 20.10.1145/3184453CrossRefGoogle Scholar
Davis, M. H. A. (1984). Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J. R. Statist. Soc. B [Statist. Methodology] 46, 353–388.Google Scholar
Devroye, L. (2009). Random variate generation for exponentially and polynomially tilted stable distributions. ACM Trans. Model. Comput. Simul. 19, 1–20.10.1145/1596519.1596523CrossRefGoogle Scholar
Duffie, D. and Glynn, P. (1995). Efficient Monte Carlo simulation of security prices. Ann. Appl. Prob. 5, 897–905.CrossRefGoogle Scholar
Easley, D. and O’Hara, M. (1987). Price, trade size, and information in securities markets. J. Financial Economics 19, 69–90.10.1016/0304-405X(87)90029-8CrossRefGoogle Scholar
Engle, R. F. and Russell, J. R. (1998). Autoregressive conditional duration: a new model for irregularly spaced transaction data. Econometrica 66, 1127–1162.10.2307/2999632CrossRefGoogle Scholar
Gander, M. P. S. and Stephens, D. A. (2007). Simulation and inference for stochastic volatility models driven by Lévy processes. Biometrika 94, 627–646.10.1093/biomet/asm048CrossRefGoogle Scholar
Gander, M. P. S. and Stephens, D. A. (2007). Stochastic volatility modelling in continuous time with general marginal distributions: inference, prediction and model selection. J. Statist. Planning Infer. 137, 3068–3081.10.1016/j.jspi.2006.07.015CrossRefGoogle Scholar
Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. Springer, New York.10.1007/978-0-387-21617-1CrossRefGoogle Scholar
Glasserman, P. and Liu, Z. (2010). Sensitivity estimates from characteristic functions. Operat. Res. 58, 1611–1623.10.1287/opre.1100.0837CrossRefGoogle Scholar
Hofert, M. (2011). Sampling exponentially tilted stable distributions. ACM Trans. Model. Comput. Simul. 22, 1–11.CrossRefGoogle Scholar
Jongbloed, G., van der Meulen, F. H. and van der Vaart, A. W. (2005). Nonparametric inference for Lévy-driven Ornstein–Uhlenbeck processes. Bernoulli 11, 759–791.10.3150/bj/1130077593CrossRefGoogle Scholar
Kallsen, J., Muhle-Karbe, J. and Voß, M. (2011). Pricing options on variance in affine stochastic volatility models. Math. Finance 21, 627–641.Google Scholar
Li, L. and Linetsky, V. (2013). Optimal stopping and early exercise: an eigenfunction expansion approach. Operat. Res. 61, 625–643.10.1287/opre.2013.1167CrossRefGoogle Scholar
Li, L. and Linetsky, V. (2014). Time-changed Ornstein–Uhlenbeck processes and their applications in commodity derivative models. Math. Finance 24, 289–330.10.1111/mafi.12003CrossRefGoogle Scholar
Li, L. and Linetsky, V. (2015). Discretely monitored first passage problems and barrier options: an eigenfunction expansion approach. Finance Stoch. 19, 941–977.CrossRefGoogle Scholar
Lukacs, E. (1969). A characterization of stable processes. J. Appl. Prob. 6, 409–418.CrossRefGoogle Scholar
Mendoza-Arriaga, R. and Linetsky, V. (2014). Time-changed CIR default intensities with two-sided mean-reverting jumps. Ann. Appl. Prob. 24, 811–856.10.1214/13-AAP936CrossRefGoogle Scholar
Mendoza-Arriaga, R. and Linetsky, V. (2016). Multivariate subordination of Markov processes with financial applications. Math. Finance 26, 699–747.10.1111/mafi.12061CrossRefGoogle Scholar
Michael, J. R., Schucany, W. R. and Haas, R. W. (1976). Generating random variates using transformations with multiple roots. Amer. Statist. 30, 88–90.Google Scholar
Nicolato, E. and Venardos, E. (2003). Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type. Math. Finance 13, 445–466.10.1111/1467-9965.t01-1-00175CrossRefGoogle Scholar
Norberg, R. (2004). Vasiček beyond the normal. Math. Finance 14, 585–604.10.1111/j.0960-1627.2004.00206.xCrossRefGoogle Scholar
Qu, Y., Dassios, A. and Zhao, H. (2019). Efficient simulation of Lévy-driven point processes. Adv. Appl. Prob. 51, 927–966.10.1017/apr.2019.44CrossRefGoogle Scholar
Qu, Y., Dassios, A. and Zhao, H. (2019). Random variate generation for exponential and gamma tilted stable distributions. To appear in ACM Trans. Model. Comput. Simul. Google Scholar
Qu, Y., Dassios, A. and Zhao, H. (2020). Exact simulation of gamma-driven Ornstein–Uhlenbeck processes with finite and infinite activity jumps. J. Operat. Res. Soc. Available at https://doi.org/10.1080/01605682.2019.1657368.Google Scholar
Rosiński, J. (2001). Series representations of Lévy processes from the perspective of point processes. In Lévy Processes, eds O. Barndorff-Nielsen, S. Resnick and T. Mikosch, pp. 401–415. Birkhäuser, Boston.10.1007/978-1-4612-0197-7_18CrossRefGoogle Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge.Google Scholar
Sato, K.-I. and Yamazato, M. (1984). Operator-selfdecomposable distributions as limit distributions of processes of Ornstein–Uhlenbeck type. Stoch. Process. Appl. 17, 73–100.10.1016/0304-4149(84)90312-0CrossRefGoogle Scholar
Schoutens, W. (2003). Lévy Processes in Finance: Pricing Financial Derivatives. Wiley, New York.10.1002/0470870230CrossRefGoogle Scholar
Todorov, V. (2015). Jump activity estimation for pure-jump semimartingales via self-normalized statistics. Ann. Statist. 43, 1831–1864.10.1214/15-AOS1327CrossRefGoogle Scholar
Todorov, V. and Tauchen, G. (2006). Simulation methods for Lévy-driven continuous-time autoregressive moving average (CARMA) stochastic volatility models. J. Business Econom. Statist. 24, 455–469.CrossRefGoogle Scholar
Uhlenbeck, G. E. and Ornstein, L. S. (1930). On the theory of the Brownian motion. Phys. Rev. 36, 823–841.10.1103/PhysRev.36.823CrossRefGoogle Scholar
Wolfe, S. J. (1982). On a continuous analogue of the stochastic difference equation $x_n=\rho x_{n-1}+b_n$ . Stoch. Process. Appl. 12, 301–312.10.1016/0304-4149(82)90050-3CrossRefGoogle Scholar
Zhang, S. (2011). Exact simulation of tempered stable Ornstein–Uhlenbeck processes. J. Statist. Comput. Simul. 81, 1533–1544.10.1080/00949655.2010.494247CrossRefGoogle Scholar
Zhang, S. and Zhang, X. (2009). On the transition law of tempered stable Ornstein–Uhlenbeck processes. J. Appl. Prob. 46, 721–731.10.1017/S0021900200005842CrossRefGoogle Scholar
Zolotarev, V. M. (1966). On representation of stable laws by integrals. In Selected Translations in Mathematical Statistics and Probability, vol. 6, pp. 84–88. American Mathematical Society.Google Scholar