Geodesic
Turbulence on financial markets and multiplicative cascade model of volatility
Matematičeskoe modelirovanie, Tome 32 (2020) no. 12, pp. 43-54.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we develop a volatility model on multiple time horizons taking into account a distribution of frequencies of price oscillations. The main point of the model lies in the ability to analyze and exploit the «carrier frequencies» of market prices to gain more precise estimate of the current volatility. Our focus is on the determination of market structure, implied in price dynamics and assuming different market agents, which work on different time-scales. Finally, in order to examine proposed model, we compare volatility estimations calculated for S 500 index with VIX index, as the main objective indicator of market volatility. Comparison of historical volatility, MCM model and the proposed model showed the advantage of the last one in terms of mean absolute percentage error.
Keywords: volatility model, financial market, G.14.
Mots-clés : turbulence, J.E.L. Classification: G.10 
@article{MM_2020_32_12_a3,
     author = {E. E. Nikulin and A. A. Pekhterev},
     title = {Turbulence on financial markets and multiplicative cascade model of volatility},
     journal = {Matemati\v{c}eskoe modelirovanie},
     pages = {43--54},
     publisher = {mathdoc},
     volume = {32},
     number = {12},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MM_2020_32_12_a3/}
}
TY  - JOUR
AU  - E. E. Nikulin
AU  - A. A. Pekhterev
TI  - Turbulence on financial markets and multiplicative cascade model of volatility
JO  - Matematičeskoe modelirovanie
PY  - 2020
SP  - 43
EP  - 54
VL  - 32
IS  - 12
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MM_2020_32_12_a3/
LA  - ru
ID  - MM_2020_32_12_a3
ER  - 
%0 Journal Article
%A E. E. Nikulin
%A A. A. Pekhterev
%T Turbulence on financial markets and multiplicative cascade model of volatility
%J Matematičeskoe modelirovanie
%D 2020
%P 43-54
%V 32
%N 12
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MM_2020_32_12_a3/
%G ru
%F MM_2020_32_12_a3
E. E. Nikulin; A. A. Pekhterev. Turbulence on financial markets and multiplicative cascade model of volatility. Matematičeskoe modelirovanie, Tome 32 (2020) no. 12, pp. 43-54. http://geodesic.mathdoc.fr/item/MM_2020_32_12_a3/

[1] B. Mandelbrot, “The variation of certain speculative prices”, Journal of Business, 36:4 (1963), 394–419 | DOI

[2] R. F. Engle, “Autoregressive conditional heteroscedasticity with estimates of variance of United Kingdom inflation”, Econometrica, 50:4, July (1982), 987–1008 | DOI | MR

[3] T. Bollerslev, “Generalized autoregressive conditional heteroscedasticity”, Journal of Econometrics, 31:3, April (1986), 307–327 | DOI | MR | Zbl

[4] D. Nelson, “Conditional heteroskedasticity in asset returns: A new approach”, Econometrica, 59:2, March (1991), 347–370 | DOI | MR | Zbl

[5] R. F. Engle, “Autoregressive conditional heteroscedasticity with estimates of the stock volatility and the crash of 87: Discussion”, The Review of Financial Studies, 3:1, March (1990), 103–106 | DOI

[6] R. F. Engle, V. K. Ng, “Measuring and testing the impact of news on volatility”, Journal of Finance, 48:5, December (1993), 1749–1778 | DOI

[7] P. Hansen, A. Lunde, “A forecast comparison of volatility models: Does anything beat a garch(1,1)”, Journal of Applied Econometrics, 20:7, March (2005), 873–889 | DOI | MR

[8] F. Black, M. Scholes, “Pricing of options and corporate liabilities”, Journal of Political Economy, 81:3, May–June (1973), 637–654 | DOI | MR | Zbl

[9] R. Merton, “Theory of rational option pricing”, Bell Journal of Economics and Management Sci., 4:1, Spring (1973), 141–183 | DOI | MR | Zbl

[10] S. Taylor, Financial Returns Modelled by the Product of two Stochastic Processes a Study of the Daily Sugar Prices 1961–1979, v. 1, North-Holland, Amsterdam, 1982

[11] Steven L. Heston, “A closed-form solution for options with stochastic volatility with applications to bond and currency options”, Review of Financial Studies, 6:2 (1993), 327–343 | DOI | MR | Zbl

[12] J. Hull, A. White, “The pricing of options on assets with stochastic volatilities”, Journal of Finance, 42:2 (1987), 281–300 | DOI

[13] L. Scott, “Option pricing when the variance changes randomly: Theory, estimation, and an application”, Journal of Financial and Quantitative Analysis, 22:4 (1987), 419–438 | DOI

[14] E. Stein, J. Stein, “Stock price distributions with stochastic volatility: an analytic approach”, Review of Financial Studies, 4:4 (1991), 727–752 | DOI | Zbl

[15] T. Andersen, T. Bollerslev, F. Diebold, P. Labys, “The distribution of exchange rate volatility”, Journal of the American Statistical Association, 96:453 (2001), 42–55 | DOI | MR | Zbl

[16] T. Andersen, T. Bollerslev, F. Diebold, P. Labys, “Modeling and forecasting realized volatility”, Econometrica, 71:2 (2003), 579–625 | DOI | MR | Zbl

[17] O. Barndorff-Nielsen, N. Shephard, “Econometric analysis of realized volatility and its use in estimating stochastic volatility models”, J. of Royal Stat. Soc., 64:2 (2002), 253–280 | DOI | MR | Zbl

[18] M. Dacorogna, U. Muller, R. Dave, R. Olsen, O. Pictet, Modelling Short-term Volatility with GARCH and HARCH models, John Wiley, N.Y., 1998

[19] G. Zumbach, “Volatility processes and volatility forecast with long memory”, Quantitative Finance, 4:1 (2004), 70–86 | MR | Zbl

[20] V. M. Isakov, S. I. Kabanikhin, A. A. Shanin, M. A. Shishlenin, S. Zhang, “Algorithm for Determining the Volatility Function in the Black-Scholes Model”, Computational Mathematics and Mathematical Physics, 59:10 (2019), 1753–1758 | DOI | MR | Zbl

[21] A. N. Shiryaev, Osnovy stokhasticheskoi finansovoi matematiki, Fazis, M., 2004

[22] L. Onsager, “Statistical hydrodynamics”, Il Nuovo Cimento, 6, supp. (1949), 279–287 | DOI | MR

[23] G. Gauthier, M. T. Reeves, X. Yu, A. S. Bradley, M. A. Baker, T. A. Bell, H. Rubinsztein-Dunlop, M. J. Davis, T. W. Neely, “Giant vortex clusters in a two-dimensional quantum fluid”, Science, 364:6447 (2019), 1264–1267 | DOI | MR | Zbl

[24] S. P. Johnstone, A. J. Groszek, P. T. Starkey, C. J. Billington, T. P. Simula, K. Helmerson, “Evolution of large-scale flow from turbulence in a two-dimensional superfluid”, Science, 364:6447 (2019), 1267–1271 | DOI | MR | Zbl

[25] S. Ghashghaie, W. Breymann, J. Peinke, P. Talkner, Y. Dodge, “Turbulent cascades in foreign exchange markets”, Nature, 381:6585 (1996), 767–770 | DOI

[26] U. Frisch, Turbulence, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[27] W. Breymann, S. Ghashghaie, P. Talkner, “A stochastic cascade model for FX dynamics”, Intern. J. of Theoretical Applied Finance, 2000, no. 03, 357–360, arXiv: cond-mat/0004179 | DOI | Zbl

[28] A. Subbotin, A multi-horizon scale for volatility, CES Working Paper 20, 2008 https://halshs.archives-ouvertes.fr/halshs-00261514

[29] A. Subbotin, “Modelirovanie volatilnosti: ot uslovnoi geteroskedastichnosti k kaskadam na mnozhestvennykh gorizontakh”, Prikladnaia ekonometrika, 3:15 (2009), 94–138

[30] J. Muzy, J. Delour, E. Bacry, “Modelling fluctuations of financial time series: from cascade process to stochastic volatility model”, The Europ. Phys. J. B, 17:3 (2000), 537–548 | DOI