Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T11:39:05.149Z Has data issue: false hasContentIssue false

Efficient simulation of Lévy-driven point processes

Published online by Cambridge University Press:  15 November 2019

Yan Qu*
Affiliation:
London School of Economics and Political Science
Angelos Dassios*
Affiliation:
London School of Economics and Political Science
Hongbiao Zhao*
Affiliation:
Shanghai University of Finance and Economics
*
*Postal address: Department of Statistics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK.
*Postal address: Department of Statistics, London School of Economics and Political Science, Houghton Street, London WC2A 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.

Abstract

In this paper, we introduce a new large family of Lévy-driven point processes with (and without) contagion, by generalising the classical self-exciting Hawkes process and doubly stochastic Poisson processes with non-Gaussian Lévy-driven Ornstein–Uhlenbeck-type intensities. The resulting framework may possess many desirable features such as skewness, leptokurtosis, mean-reverting dynamics, and more importantly, the ‘contagion’ or feedback effects, which could be very useful for modelling event arrivals in finance, economics, insurance, and many other fields. We characterise the distributional properties of this new class of point processes and develop an efficient sampling method for generating sample paths exactly. Our simulation scheme is mainly based on the distributional decomposition of the point process and its intensity process. Extensive numerical implementations and tests are reported to demonstrate the accuracy and effectiveness of our scheme. Moreover, we use portfolio risk management as an example to show the applicability and flexibility of our algorithms.

Type
Original Article
Copyright
© Applied Probability Trust 2019 

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

Acharya, V. V., DeMarzo, P. and Kremer, I. (2011). Endogenous information flows and the clustering of announcements. Amer. Econom. Rev. 101, 2955–79.10.1257/aer.101.7.2955CrossRefGoogle Scholar
Ahnert, T. and Kakhbod, A. (2017). Information choice and amplification of financial crises. Rev. Financial Studies 30, 21302178.CrossRefGoogle Scholar
At-Sahalia, Y. and Jacod, J. (2009). Estimating the degree of activity of jumps in high frequency data. Ann. Statist. 37, 22022244.CrossRefGoogle Scholar
At-Sahalia, Y. and Jacod, J. (2011). Testing whether jumps have finite or infinite activity. Ann. Statist. 39, 16891719.CrossRefGoogle Scholar
At-Sahalia, Y. and Jacod, J. (2014). High-Frequency Financial Econometrics. Princeton University Press, NJ.Google Scholar
At-Sahalia, Y., Cacho-Diaz, J. and Laeven, R. J. (2015). Modeling financial contagion using mutually exciting jump processes. J. Financial Econometrics 117, 585606.CrossRefGoogle Scholar
At-Sahalia, Y., Laeven, R. J. and Pelizzon, L. (2014). Mutual excitation in Eurozone sovereign CDS. J. Econometrics 183, 151167.CrossRefGoogle Scholar
Asmussen, S. and Glynn, P. W. (2007). Stochastic Simulation: Algorithms and Analysis. Springer, New York.CrossRefGoogle Scholar
Azizpour, S., Giesecke, K. and Schwenkler, G. (2018). Exploring the sources of default clustering. J. Financial Econometrics 129, 154183.10.1016/j.jfineco.2018.04.008CrossRefGoogle Scholar
Bacry, E., Delattre, S., Hoffmann, M. and Muzy, J.-F. (2013). Modelling microstructure noise with mutually exciting point processes. Quant. Finance 13, 6577.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. (1997). Normal inverse Gaussian distributions and stochastic volatility modelling. Scand. J. Statist. 24, 113.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. (1998). Processes of normal inverse Gaussian type. Finance Stoch. 2, 4168.10.1007/s007800050032CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. (2001). Superposition of Ornstein–Uhlenbeck type processes. Theory Prob. Appl. 45, 175194.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2001). Modelling by Lévy processess for financial econometrics. In Lévy Processes, eds Barndorff-Nielsen, O. E., Resnick, S. I., and Mikosch, T., pp. 283318. Birkhäuser, Boston.CrossRefGoogle 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 63, 167241.CrossRefGoogle 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 64, 253280.10.1111/1467-9868.00336CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2003). Integrated OU processes and non-Gaussian OU-based stochastic volatility models. Scand. J. Statist. 30, 277295.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E. and Shephard, N. (2003). Realized power variation and stochastic volatility models. Bernoulli 9, 243265.CrossRefGoogle Scholar
Barndorff-Nielsen, O. E., Jensen, J. L. and Sørensen, M. (1998). Some stationary processes in discrete and continuous time. Adv. Appl. Prob. 30, 9891007.10.1239/aap/1035228204CrossRefGoogle Scholar
Bowsher, C. G. (2007). Modelling security market events in continuous time: intensity based, multivariate point process models. J. Econometrics 141, 876912.CrossRefGoogle Scholar
Brémaud, P. and Massoulié, L. (1996). Stability of nonlinear Hawkes processes. Ann. Prob. 24, 15631588.Google Scholar
Brémaud, P. and Massoulié, L. (2002). Power spectra of general shot noises and Hawkes point processes with a random excitation. Adv. Appl. Prob. 34, 205222.CrossRefGoogle Scholar
Brix, A. (1999). Generalized gamma measures and shot-noise Cox processes. Adv. Appl. Prob. 31, 929953.CrossRefGoogle Scholar
Broadie, M. and Kaya, Ö. (2006). Exact simulation of stochastic volatility and other affine jump diffusion processes. Operat. Res. 54, 217231.CrossRefGoogle Scholar
Brunnermeier, M. K. (2009). Deciphering the liquidity and credit crunch 2007–2008. J. Econom. Perspectives 23, 77100.CrossRefGoogle Scholar
Brunnermeier, M. K. and Pedersen, L. H. (2009). Market liquidity and funding liquidity. Rev. Financial Studies 22, 22012238.10.1093/rfs/hhn098CrossRefGoogle Scholar
Caccioli, F., Shrestha, M., Moore, C. and Farmer, J. D. (2014). Stability analysis of financial contagion due to overlapping portfolios. J. Bank. Finance 46, 233245.CrossRefGoogle Scholar
Cai, N., Song, Y. and Chen, N. (2017). Exact simulation of the SABR model. Operat. Res. 65, 931951.10.1287/opre.2017.1617CrossRefGoogle Scholar
Carr, P., Geman, H., Madan, D. B. and Yor, M. (2003). Stochastic volatility for Lévy processes. Math. Finance 13, 345382.CrossRefGoogle Scholar
Centanni, S. and Minozzo, M. (2006). A Monte Carlo approach to filtering for a class of marked doubly stochastic Poisson processes. J. Amer. Statist. Assoc. 101, 15821597.CrossRefGoogle Scholar
Chen, N. and Huang, Z. (2013). Localization and exact simulation of Brownian motion-driven stochastic differential equations. Math. Operat. Res. 38, 591616.CrossRefGoogle 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, 14:114:26.CrossRefGoogle Scholar
Chhikara, R. and Folks, L. (1989). The Inverse Gaussian Distribution: Theory, Methodology, and Applications. Marcel Dekker, New York.Google Scholar
Cont, R. (2001). Empirical properties of asset returns: stylized facts and statistical issues. Quant. Finance 1, 223236.CrossRefGoogle Scholar
Cont, R. and Tankov, P. (2004). Financial Modelling with Jump Processes. CRC Press, Boca Raton.Google Scholar
Cont, R. and Wagalath, L. (2013). Running for the exit: distressed selling and endogenous correlation in financial markets. Math. Finance 23, 718741.CrossRefGoogle Scholar
Cont, R. and Wagalath, L. (2016). Fire sales forensics: measuring endogenous risk. Math. Finance 26, 835866.CrossRefGoogle Scholar
Corsi, F., Marmi, S. and Lillo, F. (2016). When micro prudence increases macro risk: the destabilizing effects of financial innovation, leverage, and diversification. Operat. Res. 64, 10731088.CrossRefGoogle Scholar
Cox, D. R. (1955). Some statistical methods connected with series of events. J. R. Statist. Soc. B 17, 129164.Google Scholar
Cox, D. R. (1972). Regression models and life-tables. J. R. Statist. Soc. B 34, 187220.Google Scholar
Crane, R. and Sornette, D. (2008). Robust dynamic classes revealed by measuring the response function of a social system. Proc. Nat. Acad. Sci. USA 105, 1564915653.CrossRefGoogle ScholarPubMed
CreditRisk+ (1997). CreditRisk+: A Credit Risk Management Framework. Credit Suisse First Boston International, New York.Google Scholar
Daley, D. J. and Vere-Jones, D. (2003). An Introduction to the Theory of Point Processes, vol. I: Elementary Theory and Methods. Springer, New York.Google Scholar
Das, S. R., Duffie, D., Kapadia, N. and Saita, L. (2007). Common failings: how corporate defaults are correlated. J. Finance 62, 93117.CrossRefGoogle Scholar
Dassios, A. and Embrechts, P. (1989). Martingales and insurance risk. Stoch. Models 5, 181217.CrossRefGoogle Scholar
Dassios, A. and Zhao, H. (2011). A dynamic contagion process. Adv. Appl. Prob. 43, 814846.CrossRefGoogle Scholar
Dassios, A. and Zhao, H. (2013). Exact simulation of Hawkes process with exponentially decaying intensity. Electron. Commun. Prob. 18, 113.CrossRefGoogle Scholar
Dassios, A. and Zhao, H. (2017). A generalised contagion process with an application to credit risk. Int. J. Theor. Appl. Finance 20, 133.CrossRefGoogle Scholar
Dassios, A. and Zhao, H. (2017). Efficient simulation of clustering jumps with CIR intensity. Operat. Res. 65, 14941515.CrossRefGoogle 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, 20:120:21.CrossRefGoogle Scholar
Davis, M. H. (1984). Piecewise-deterministic Markov processes: a general class of non-diffusion stochastic models. J. R. Statist. Soc. B 46, 353388.Google Scholar
Davis, M. H. (1993). Markov Models and Optimization. Chapman & Hall/CRC, London.CrossRefGoogle Scholar
Devroye, L. (2009). Random variate generation for exponentially and polynomially tilted stable distributions. ACM Trans. Model. Comput. Simul. 19, 120.CrossRefGoogle Scholar
Duffie, D. and Gârleanu, N. (2001). Risk and valuation of collateralized debt obligations. Financial Analysts Journal 57, 4159.CrossRefGoogle Scholar
Duffie, D., Eckner, A., Horel, G. and Saita, L. (2009). Frailty correlated default. J. Finance 64, 20892123.CrossRefGoogle Scholar
Duffie, D., Filipovic, D. and Schachermayer, W. (2003). Affine processes and applications in finance. Ann. Appl. Prob. 13, 9841053.Google Scholar
Eberlein, E., Madan, D., Pistorius, M. and Yor, M. (2013). A simple stochastic rate model for rate equity hybrid products. Appl. Math. Finance 20, 461488.CrossRefGoogle Scholar
Eisenberg, L. and Noe, T. H. (2001). Systemic risk in financial systems. Manag. Sci. 47, 236249.CrossRefGoogle Scholar
Elsinger, H., Lehar, A. and Summer, M. (2006). Risk assessment for banking systems. Manag. Sci. 52, 13011314.CrossRefGoogle Scholar
Embrechts, P., Liniger, T. and Lin, L. (2011). Multivariate Hawkes processes: an application to financial data. J. Appl. Prob. 48A, 367378.CrossRefGoogle Scholar
Errais, E., Giesecke, K. and Goldberg, L. R. (2010). Affine point processes and portfolio credit risk. SIAM J. Financ. Math. 1, 642665.CrossRefGoogle Scholar
Gençay, R., Dacorogna, M., Muller, U. A., Pictet, O. and Olsen, R. (2001). An Introduction to High-Frequency Finance. Academic Press, San Diego.Google Scholar
Giesecke, K., Kakavand, H. and Mousavi, M. (2011). Exact simulation of point processes with stochastic intensities. Operat. Res. 59, 12331245.CrossRefGoogle Scholar
Giesecke, K., Longstaff, F. A., Schaefer, S. and Strebulaev, I. (2011). Corporate bond default risk: a 150-year perspective. J. Financial Econometrics 102, 233250.CrossRefGoogle Scholar
Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering. Springer, New York.CrossRefGoogle Scholar
Glasserman, P. and Liu, Z. (2010). Sensitivity estimates from characteristic functions. Operat. Res. 58, 16111623.CrossRefGoogle Scholar
Gordy, M. B. (2000). A comparative anatomy of credit risk models. J. Bank. Finance 24, 119149.CrossRefGoogle Scholar
Gordy, M. B. (2003). A risk-factor model foundation for ratings-based bank capital rules. J. Financial Intermediation 12, 199232.CrossRefGoogle Scholar
Hainaut, D. and Devolder, P. (2008). Mortality modelling with Lévy processes. Insurance Math. Econom. 42, 409418.CrossRefGoogle Scholar
Hawkes, A. G. (1971). Point spectra of some mutually exciting point processes. J. R. Statist. Soc. B 33, 438443.Google Scholar
Hawkes, A. G. (1971). Spectra of some self-exciting and mutually exciting point processes. Biometrika 58, 8390.CrossRefGoogle Scholar
Hawkes, A. G. and Oakes, D. (1974). A cluster process representation of a self-exciting process. J. Appl. Prob. 11, 493503.CrossRefGoogle Scholar
Hofert, M. (2011). Sampling exponentially tilted stable distributions. ACM Trans. Model. Comput. Simul. 22, 111.CrossRefGoogle Scholar
Kang, C., Kang, W. and Lee, J. M. (2017). Exact simulation of the Wishart multidimensional stochastic volatility model. Operat. Res. 65, 11901206.CrossRefGoogle Scholar
Krishnamurthy, A. (2010). How debt markets have malfunctioned in the crisis. J. Econom. Perspectives 24, 328.CrossRefGoogle Scholar
Kyprianou, A. (2006). Introductory Lectures on Fluctuations of Lévy Processes with Applications. Springer, Berlin.Google Scholar
Large, J. (2007). Measuring the resiliency of an electronic limit order book. J. Financial Markets 10, 125.CrossRefGoogle Scholar
Lee, S. S. and Hannig, J. (2010). Detecting jumps from Lévy jump diffusion processes. J. Financial Econometrics 96, 271290.CrossRefGoogle Scholar
Lewis, P. A. and Shedler, G. S. (1979). Simulation of nonhomogeneous Poisson processes by thinning. Naval Res. Logist. Quart. 26, 403413.CrossRefGoogle Scholar
Li, H., Wells, M. T. and Cindy, L. Y. (2008). A Bayesian analysis of return dynamics with Lévy jumps. Rev. Financial Studies 21, 23452378.CrossRefGoogle Scholar
Li, L. and Linetsky, V. (2014). Time-changed Ornstein–Uhlenbeck processes and their applications in commodity derivative models. Math. Finance 24, 289330.CrossRefGoogle Scholar
Longstaff, F. A. and Rajan, A. (2008). An empirical analysis of the pricing of collateralized debt obligations. J. Finance 63, 529563.CrossRefGoogle Scholar
Madan, D. B. and Seneta, E. (1990). The variance gamma (V.G.) model for share market returns. J. Business 63, 511524.CrossRefGoogle Scholar
Madan, D. B., Carr, P. P. and Chang, E. C. (1998). The variance gamma process and option pricing. Europ. Finance Rev. 2, 79105.CrossRefGoogle Scholar
Michael, J. R., Schucany, W. R. and Haas, R. W. (1976). Generating random variates using transformations with multiple roots. Amer. Statistician 30, 8890.Google Scholar
Morris, S. and Shin, H. S. (2004). Liquidity black holes. Rev. Finance 8, 118.CrossRefGoogle Scholar
Nicolato, E. and Venardos, E. (2003). Option pricing in stochastic volatility models of the Ornstein–Uhlenbeck type. Math. Finance 13, 445466.CrossRefGoogle Scholar
Poterba, J. M. and Summers, L. H. (1988). Mean reversion in stock prices: evidence and implications. J. Financial Econometrics 22, 2759.CrossRefGoogle Scholar
Qu, Y., Dassios, A. and Zhao, H. (2019). Exact simulation for tempered stable distributions. Working paper, London School of Economics.Google Scholar
Qu, Y., Dassios, A. and Zhao, H. (2019). Exact simulation of gamma-driven Ornstein–Uhlenbeck processes with finite and infinite activity jumps. Working paper, London School of Economics.Google Scholar
Rosiski, J. (2001). Series representations of Lévy processes from the perspective of point processes. In Lévy Processes, eds Barndorff-Nielsen, O. E., Resnick, S. I., and Mikosch, T., pp. 401415. Birkhäuser, Boston.CrossRefGoogle Scholar
Rosiski, J. (2007). Tempering stable processes. Stoch. Process. Appl. 117, 677707.CrossRefGoogle Scholar
Rydberg, T. H. and Shephard, N. (2000). A modelling framework for the prices and times made on the NYSE. In Nonlinear and Nonstationary Signal Processing (Isaac Newton Institute Series), eds Fitzgerald, W., Smith, R., Walden, A., and Young, P.. Cambridge University Press.Google Scholar
Sato, K.-I. (1999). Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Schoutens, W. and Cariboni, J. (2010). Lévy Processes in Credit Risk. John Wiley, Chichester.Google Scholar