Informatics and Applications
2020, Volume 14, Issue 1, pp 4855
RISKNEUTRAL DYNAMICS FOR THE ARIMAGARCH RANDOM PROCESS WITH ERRORS DISTRIBUTED ACCORDING TO THE JOHNSON'S Su LAW
 A. R. Danilishin
 D. Yu. Golembiovsky
Abstract
Riskneutral world is one of the fundamental principles of financial mathematics, for definition of a fair value of derivative financial instruments. The article deals with the construction of riskneutral dynamics for the ARIMAGARCH (Autoregressive Integrated Moving Average, Generalized AutoRegressive Conditional Heteroskedasticity) random process with errors distributed according to the Johnson's SU law. Methods for finding riskneutral coefficients require the existence of a generating function of moments (examples of such transformations are the Escher transformation, the extended Girsanov principle). A generating function of moments is not known for Student and Johnson's SU distributions. The authors form a generating function of moments for the Johnson's SU distribution and prove that a modification of the extended Girsanov principle may obtain a riskneutral measure with respect to the chosen distribution.
[+] References (14)
 Hull, J. 2018. Options, futures, and other derivatives. 10th ed.Pearson.896 p.
 Patton, A. 2015. Quantitative finance. London:University
of London Press Publisher. 65p.
 Akgiray, V. 1989. Conditional heteroscedasticity in time
series of stock returns: Evidence and forecasts. J. Bus.
62(1):55–80. doi:10.1086/296451
 Terasvirta, T. 2009. An introduction to univariate
GARCH models. Handbook of financial time series. Eds. T.G. Andersen, R.A. Davis, J.P. Kreiss, and
Th.V. Mikosch. Berlin–Heidelberg: Springer. 10:17–42.
doi:10.1007/9783540712978 1.
 Follmer, H., and A. Schied. 2002. Stochastic finance: An introduction in discrete time. Berlin: Walter de Gruyter. 422 p.
 Bollerslev, T. 1987. A conditionally heteroskedastic time series model for speculative prices and rates of return. Rev. Econ. Stat. 69(3):542547. doi: 10.2307/1925546.
 Simonato, J. G. 2012. GARCH processes with skewed and leptokurtic innovations: Revisiting the Johnson SU case. Available at: https://ssrn.com/abstract=2060994 (accessed May 18, 2012).
 Elliott, R. J., and D. B. Madan. 1998. A Discrete time equivalent martingale measure. Math. Financ. 8(2):127 152. doi: 10.1111/14679965.00048.
 Yi, X. 2013. Comparison of option pricing between ARMAGARCH and GARCHM models. London, Ontario, Canada: University ofWestern Ontario. MoS Thesis. 73 p.
 Enrique, R., and L. Escobar. 2006. Using moment generating functions to derive mixture distributions. Am. Stat. 60(1):7580. doi: 10.1198/000313006X90819.
 Simonato, J. G., and L. Stentoft. 2015. Which pricing approach for options under GARCH with nonnormal innovations? Available at: https://www.degroote. mcmaster.ca/files/2015/11/SimonatoStentoft.pdf (accessed November 2015).
 Williams, D. 1991. Probability with martingales. Cambridge: Cambridge University Press. 251 p.
 Cameron, R. H., and W T. Martin. 1945. Transformation of Wiener integrals under a general class of linear transformations translations. T. Am. Math. Soc. 58:184219. doi: 10.1090/S00029947194500132401.
 Bell, D. 1991. Transformations of measures on an infinitedimensional vector space. Seminar on stochastic processes, 1990. Eds. E. Qinlar, P. J. Fitzsimmons, and R. J. Williams. Progress in probability book ser. Birkhauser Boston. 24:1525. doi: 10.1007/978l468405620_3.
[+] About this article
Title
RISKNEUTRAL DYNAMICS FOR THE ARIMAGARCH RANDOM PROCESS WITH ERRORS DISTRIBUTED ACCORDING TO THE JOHNSON'S SU LAW
Journal
Informatics and Applications
2020, Volume 14, Issue 1, pp 4855
Cover Date
20200330
DOI
10.14357/19922264200107
Print ISSN
19922264
Publisher
Institute of Informatics Problems, Russian Academy of Sciences
Additional Links
Key words
ARIMA; GARCH; riskneutral measure; Girsanov extended principle; Johnson's SU ; option pricing
Authors
A. R. Danilishin and D. Yu. Golembiovsky ,
Author Affiliations
Department of Operations Research, Faculty of Computational Mathematics and Cybernetics, M. V. Lomonosov Moscow State University, 152 Leninskiye Gory, Moscow 119991, GSP1, Russian Federation
Department of Banking, Sinergy University, 80G Leningradskiy Prospect, Moscow 125190, Russian Federation
