Geodesic
Distributional uncertainty of the financial time series measured by $G$-expectation
Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 4, pp. 914-928 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Based on the law of large numbers and the central limit theorem under nonlinear expectation, we introduce a new method of using $G$-normal distribution to measure financial risks. Applying max-mean estimators and a small windows method, we establish autoregressive models to determine the parameters of $G$-normal distribution, i.e., the return, maximal, and minimal volatilities of the time series. Utilizing the value at risk (VaR) predictor model under $G$-normal distribution, we show that the $G$-VaR model gives an excellent performance in predicting the VaR for a benchmark dataset comparing to many well-known VaR predictors.
Keywords: autoregressive model, sublinear expectation, volatility uncertainty, $G$-VaR, $G$-normal distribution. 
@article{TVP_2021_66_4_a13,
     author = {Shige Peng and Shuzhen Yang},
     title = {Distributional uncertainty of the financial time series measured by $G$-expectation},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {914--928},
     year = {2021},
     volume = {66},
     number = {4},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a13/}
}
TY  - JOUR
AU  - Shige Peng
AU  - Shuzhen Yang
TI  - Distributional uncertainty of the financial time series measured by $G$-expectation
JO  - Teoriâ veroâtnostej i ee primeneniâ
PY  - 2021
SP  - 914
EP  - 928
VL  - 66
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a13/
LA  - ru
ID  - TVP_2021_66_4_a13
ER  - 
%0 Journal Article
%A Shige Peng
%A Shuzhen Yang
%T Distributional uncertainty of the financial time series measured by $G$-expectation
%J Teoriâ veroâtnostej i ee primeneniâ
%D 2021
%P 914-928
%V 66
%N 4
%U http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a13/
%G ru
%F TVP_2021_66_4_a13
Shige Peng; Shuzhen Yang. Distributional uncertainty of the financial time series measured by $G$-expectation. Teoriâ veroâtnostej i ee primeneniâ, Tome 66 (2021) no. 4, pp. 914-928. http://geodesic.mathdoc.fr/item/TVP_2021_66_4_a13/

[1] P. Artzner, F. Delbaen, J.-M. Eber, D. Heath, “Coherent measures of risk”, Math. Finance, 9:3 (1999), 203–228 | DOI | MR | Zbl

[2] M. Avellaneda, A. Levy, A. Parás, “Pricing and hedging derivative securities in markets with uncertain volatilities”, Appl. Math. Finance, 2:2 (1995), 73–88 | DOI | Zbl

[3] Zengjing Chen, L. Epstein, “Ambiguity, risk, and asset returns in continuous time”, Econometrica, 70:4 (2002), 1403–1443 | DOI | MR | Zbl

[4] R. Cont, “Model uncertainty and its impact on the pricing of derivative instruments”, Math. Finance, 16:3 (2006), 519–547 | DOI | MR | Zbl

[5] L. Denis, C. Martini, “A theoretical framework for the pricing of contingent claims in the presence of model uncertainty”, Ann. Appl. Probab., 16:2 (2006), 827–852 | DOI | MR | Zbl

[6] L. G. Epstein, Shaolin Ji, “Ambiguous volatility and asset pricing in continuous time”, Rev. Financ. Stud., 26:7 (2013), 1740–1786 | DOI

[7] Xiao Fang, Shige Peng, Qi-Man Shao, Yongsheng Song, “Limit theorems with rate of convergence under sublinear expectations”, Bernoulli, 25:4A (2019), 2564–2596 | DOI | MR | Zbl

[8] H. Föllmer, A. Schied, Stochastic finance. An introduction in discrete time, 3rd rev. ed., Walter de Gruyter Co., Berlin, 2011, xii+544 pp. | DOI | MR | Zbl

[9] Mingshang Hu, Shige Peng, Yongsheng Song, “Stein type characterization for $G$-normal distributions”, Electron. Commun. Probab., 22 (2017), 24, 12 pp. | DOI | MR | Zbl

[10] P. J. Huber, Robust statistics, Wiley Ser. Probab. Math. Statist., John Wiley Sons, Inc., New York, 1981, ix+308 pp. | DOI | MR | MR | Zbl | Zbl

[11] Hanqing Jin, Shige Peng, “Optimal unbiased estimation for maximal distribution”, Probab. Uncertain. Quant. Risk, 6:3 (2021), 189–198 | DOI

[12] J. Kerkhof, B. Melenberg, H. Schumacher, “Model risk and capital reserves”, J. Bank. Finance, 34:1 (2010), 267–279 | DOI

[13] N. V. Krylov, “On Shige Peng's central limit theorem”, Stochastic Process. Appl., 130:3 (2020), 1426–1434 | DOI | MR | Zbl

[14] K. Kuester, S. Mittnik, M. S. Paolella, “Value-at-risk prediction: a comparison of alternative strategies”, J. Financ. Econom., 4:1 (2006), 53–89 | DOI

[15] T. J. Lyons, “Uncertain volatility and the risk-free synthesis of derivatives”, Appl. Math. Finance, 2:2 (1995), 117–133 | DOI | Zbl

[16] S. Peng, “Backward SDE and related $g$-expectation”, Backward stochastic differential equations (Paris, 1995–1996), Pitman Res. Notes Math. Ser., 364, Longman, Harlow, 1997, 141–159 | MR | Zbl

[17] Shige Peng, “Filtration consistent nonlinear expectations and evaluations of contingent claims”, Acta Math. Appl. Sin. Engl. Ser., 20:2 (2004), 191–214 | DOI | MR | Zbl

[18] Shige Peng, “Nonlinear expectations and nonlinear Markov chains”, Chinese Ann. Math. Ser. B, 26:2 (2005), 159–184 | DOI | MR | Zbl

[19] Shige Peng, “$G$-expectation, $G$-Brownian motion and related stochastic calculus of {I}tô type”, Stochastic analysis and applications: The Abel Symposium 2005, Abel Symp., 2, Springer, Berlin, 2007, 541–567 | DOI | MR | Zbl

[20] Shige Peng, “Multi-dimensional $G$-Brownian motion and related stochastic calculus under $G$-expectation”, Stochastic Process. Appl., 118:12 (2008), 2223–2253 | DOI | MR | Zbl

[21] Shige Peng, “Theory, methods and meaning of nonlinear expectation theory”, in Chinese, Sci. Sin. Math., 47:10 (2017), 1223–1254 | DOI

[22] Shige Peng, Nonlinear expectations and stochastic calculus under uncertainty. With robust CLT and G-Brownian motion, Probab. Theory Stoch. Model., 95, Springer, Berlin, 2019, xiii+212 pp. | DOI | MR | Zbl

[23] Shige Peng, Shuzhen Yang, Jianfeng Yao, “Improving value-at-risk prediction under model uncertainty”, J. Financ. Econom., 2020, nbaa022, 32 pp., Publ. online | DOI

[24] Yongsheng Song, “Stein's method for law of large numbers under sublinear expectations”, Probab. Uncertain. Quant. Risk, 6:3 (2021), 199–212 | DOI

[25] Yongsheng Song, “Normal approximation by Stein's method under sublinear expectations”, Stochastic Process. Appl., 130:5 (2020), 2838–2850 | DOI | MR | Zbl

[26] P. Walley, Statistical reasoning with imprecise probabilities, Monogr. Statist. Appl. Probab., 42, Chapman and Hall, Ltd., London, 1991, xii+706 pp. | MR | Zbl