Abstract
Extreme return financial time series are often challenging to model due to the presence of heavy temporal clustering of extremes and strong bursts of return volatility. One approach to model both these phenomena in extreme financial returns is the marked Hawkes self-exciting process. However, the Hawkes process restricts the arrival times of exogenously driven returns to follow a Poisson process and may fail to provide an adequate fit to data. In this work, we introduce a model for extreme financial returns, which provides added flexibility in the specification of the background arrival rate. Our model is a marked version of the recently proposed renewal Hawkes process, in which exogenously driven extreme returns arrive according to a renewal process rather than a Poisson process. We develop a procedure to evaluate the likelihood of the model, which can be optimized to obtain estimates of model parameters and their standard errors. We provide a method to assess the goodness-of-fit of the model based on the Rosenblatt residuals, as well as a procedure to simulate the model. We apply the proposed model to extreme negative returns for five stocks traded on the Australian Stock Exchange. The models identified for the stocks using in-sample data were found to be able to successfully forecast the out-of-sample risk measures such as the value at risk and provide a better quality of fit than the competing Hawkes model.
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References
Chavez-Demoulin, V., Davison, A.C., McNeil, A.J.: Estimating value-at-risk: A point process approach. Quantit. Finan 5(2), 227–234 (2005)
Chavez-Demoulin, V., McGill, J.: High-frequency financial data modeling using Hawkes processes. J. Bank. Financ. 36(12), 3415–3426 (2012)
Chen, F., Stindl, T.: Direct likelihood evaluation for the renewal Hawkes process. J. Comput. Graph. Stat. 27(1), 119–131 (2018)
Cline, D, Pu, H: A note on a simple Markov bilinear stochastic process. Statist. Probab. Lett. 56(3), 283–288 (2002)
Cui, Y., Lund, R.: A new look at time series of counts. Biometrika 96(4), 781–792 (2009)
Daley, D.J., Vere-Jones, D.: An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods, 2nd edn. Springer, New York (2003)
Embrechts, P., Mikosch, T., Klüppelberg, C.: Modelling Extremal Events for Insurance and Finance. Springer, New York (1997)
Embrechts, P., Liniger, T., Lin, L.: Multivariate Hawkes processes: An application to financial data. J. Appl. Probab. 48(A), 367–378 (2011)
Hawkes, A.G.: Spectra of some self-exciting and mutually exciting point processes. Biometrika 58(1), 83–90 (1971)
Herrera, R., Schipp, B.: Self-exciting extreme value models for stock market crashes. In: Statistical Inference, Econometric Analysis and Matrix Algebra, pp 209–231. Physica, Heidelberg (2009)
Kirchner, M.: Hawkes and INAR(\(\infty \)) processes. Stoch. Processes Appl. 126(8), 2494–2525 (2016)
McNeil, A.J., Frey, R.: Estimation of tail-related risk measures for heteroscedastic financial time series: an extreme value approach. J. Emp. Fin. 7(3), 271–300 (2000)
McNeil, A., Frey, R., Embrechts, P.: Quantitative Risk Management: Concepts, Techniques, and Tools. Princeton Series in Finance. Princeton University Press (2005)
Mina, J., Xiao, J.Y.: Return to riskmetrics: The evolution of a standard. RiskMetrics Group (2001)
R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna (2017)
Rosenblatt, M.: Remarks on a multivariate transformation. Ann. Math. Stat. 23(3), 470–472 (1952)
Stindl, T., Chen, F.: Likelihood based inference for the multivariate renewal Hawkes process. Comput. Stat. Data Anal. 123, 131–145 (2018)
Wheatley, S., Filimonov, V., Sornette, D.: The Hawkes process with renewal immigration & its estimation with an EM algorithm. Comput. Stat. Data Anal. 94(C), 120–135 (2016)
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The authors appreciate and thank the editor, the associate editor, and two anonymous reviewers for their very insightful comments that have led to a more consistent, improved and well presented paper.
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This research includes computations using the Linux computational cluster Katana supported by the Faculty of Science, UNSW Sydney. Stindl was supported by an Australian Government Research Training Program Scholarship. Chen was partly supported by a UNSW SFRGP grant.
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Stindl, T., Chen, F. Modeling extreme negative returns using marked renewal Hawkes processes. Extremes 22, 705–728 (2019). https://doi.org/10.1007/s10687-019-00352-4
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DOI: https://doi.org/10.1007/s10687-019-00352-4