Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-25T05:07:59.302Z Has data issue: false hasContentIssue false
  • English
  • Français

Comparison Results for GARCH Processes

Part of: Mathematical finance Distribution theory - Probability

Published online by Cambridge University Press:  30 January 2018

Fabio Bellini ,
Franco Pellerey ,
Carlo Sgarra  and
Salimeh Yasaei Sekeh
Fabio Bellini*
Affiliation:
Universitá di Milano Bicocca
Franco Pellerey*
Affiliation:
Politecnico di Torino
Carlo Sgarra*
Affiliation:
Politecnico di Milano
Salimeh Yasaei Sekeh*
Affiliation:
University of Bojnord
*
∗ Postal address: Dipartimento di Metodi Quantitativi, Universitá di Milano Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy. Email address: [email protected]
∗∗ Postal address: Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca Degli Abruzzi 24, 10129 Torino, Italy. Email address: [email protected]
∗∗∗ Postal address: Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy. Email address: [email protected]
∗∗∗∗ Postal address: Department of Mathematics, University of Bojnord, Bojnord, Iran. Email address: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider the problem of stochastic comparison of general GARCH-like processes for different parameters and different distributions of the innovations. We identify several stochastic orders that are propagated from the innovations to the GARCH process itself, and we discuss their interpretations. We focus on the convex order and show that in the case of symmetric innovations it is also propagated to the cumulated sums of the GARCH process. More generally, we discuss multivariate comparison results related to the multivariate convex and supermodular orders. Finally, we discuss ordering with respect to the parameters in the GARCH(1, 1) case.

Keywords

GARCHconvex orderpeakednesskurtosissupermodularity

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 91G70: Statistical methods, econometrics
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 3 , September 2014 , pp. 685 - 698
Copyright
© Applied Probability Trust 

References

Bellamy, N. and Jeanblanc, M. (2000). Incompleteness of markets driven by a mixed diffusion. Finance Stoch. 4, 209–222.CrossRefGoogle Scholar
Bergenthum, J. and Rüschendorf, L. (2006). Comparison of option prices in semimartingale models. Finance Stoch. 10, 222–249.Google Scholar
Birnbaum, Z. W. (1948). On random variables with comparable peakedness. Ann. Math. Statist. 19, 76–81.Google Scholar
Bollerslev, T. (1986). General autoregressive conditional heteroskedasticity. J. Econometrics 31, 307–327.Google Scholar
El Karoui, N., Jeanblanc-Picqué, M. and Shreve, S. E. (1998). Robustness of the Black and Scholes formula. Math. Finance 8, 93–126.Google Scholar
Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50, 987–1007.Google Scholar
Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Statist. 38, 1466–1474.Google Scholar
Finucan, H. M. (1964). A note on kurtosis. J. R. Statist. Soc. B 26, 111–112.Google Scholar
Gushchin, A. A. and Mordetski, È. (2002). Bounds on option prices for semimartingale market models. Proc. Steklov Inst. Math. 2002, 73–113.Google Scholar
Meester, L. E. and Shanthikumar, J. G. (1999). Stochastic convexity on general space. Math. Operat. Res. 24, 472–494.Google Scholar
Müller, A. and Stoyan, D. (2002). Comparison Methods for Stochastic Models and Risks. John Wiley, Chichester.Google Scholar
Møller, T. (2004). Stochastic orders in dynamic reinsurance markets. Finance Stoch. 8, 479–499.Google Scholar
Nelson, D. B. (1990). Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Shaked, M. and Shanthikumar, G. J. (2007). Stochastic Orders. Springer, New York.CrossRefGoogle Scholar