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Application of Fuzzy Asymmetric GARCH-Models to Forecasting of Volatility of Russian Stock Market Alexander Lepskiy, Artem Suevalov National Research University «Higher School of Economics» September 15, 2017

Purposes of the study • Testing of the methodology of forecasting volatility proposed in [Hung 2009] on the Russian stock market data; • Development and research of various modi cations of fuzzy asymmetric GARCH-models; • Comparative analysis of crisp and fuzzy asymmetric GARCH-models; • Research of the impact of some macroeconomic information on volatility. Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 2 / 20

GARCH(p,q)-model Generalized AutoRegressive Conditional Heteroscedasticity Bollerslev T. 1986 y ( t ) = u ( t ) + c , √ u ( t ) = σ ( t ) ε ( t ) , (1) q p ∑ ∑ σ 2 ( t ) = α 0 + α i u 2 ( t − i ) + β j σ 2 ( t − j ) , i =1 j =1 where y ( t ) is a random variable from stock market, ε ( t ) is a white noise process with zero mean and unit variance, σ ( t ) is a conditional variance of ε ( t ) , and α 0 , α i , β j , c are unknown parameters that needed to be estimated. Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 3 / 20

Asymmetric GARCH-models TGARCH: Zakoian J.M. 1994 GJR-GARCH: Glosten L.R. et al. 1993 y ( t ) = u ( t ) + c , √ u ( t ) = σ ( t ) ε ( t ) , q σ d ( t ) = α 0 + α (1) u d ( t − i ) I { u ( t − 1) > 0 } + ∑ (2) i i =1 q p α (2) u d ( t − k )(1 − I { u ( t − 1) > 0 } ) + β j σ d ( t − j ) , ∑ ∑ + i k =1 j =1 where d = 1 for TGARCH, and d = 2 for GJR-GARCH. Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 4 / 20

Asymmetric GARCH-models VSGARCH: Fornari A., Mele A. 1996 y ( t ) = u ( t ) + [ I { u ( t − 1) > 0 } c (1) + (1 − I { u ( t − 1) > 0 } ) c (2) ] , √ u ( t ) = σ ( t ) ε ( t ) , σ 2 ( t ) = I { u ( t − 1) > 0 } F (1) + (1 − I { u ( t − 1) > 0 } ) F (2) . (3) q p F ( k ) = α ( k ) α ( k ) β ( k ) ∑ u 2 ( t − i ) + ∑ σ 2 ( t − j ) . + i j 0 i =1 j =1 Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 5 / 20

Fuzzy asymmetric GARCH-model Hung J. 2009 y ( t ) = u ( t ) + [ I ( t ) c (1) + (1 − I ( t )) c (2) ] , (4) u ( t ) = √ σ ( t ) ε ( t ) , σ 2 ( t ) = I ( t ) F (1) + (1 − I ( t )) F (2) . where I(t) is switching function such that: { 1 , if y ( t − d ) ≥ r ( t ) , I ( t ) = (5) 0 , if y ( t − d ) < r ( t ) , where d is a lag, r ( t ) is a threshold. Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 6 / 20

Fuzzy logic system Consists of four components: • a fuzzifier • a fuzzy rule base • a fuzzy inference engine • a defuzzifier Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 7 / 20

Fuzzy Rule Base There are two types of rules: simple and complex. Simple rules use only one instument: • {if the MICEX index falls, then the probability of its fallen will increase} Complex rules use several instuments: • {if the MICEX index falls and USD/RUB currency rises, then the probability of its fallen will increase} Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 8 / 20

Modifications of an asymmetric fuzzy GARCH-model Let us introduce the following fuzzy asymmetric GARCH-models which will be used to predict the volatility of the Russian stock market. • Asymmetric GARCH-model with s-type switching function • Asymmetric GARCH-model with the characteristic function of comparing the fuzzy number-histogram and the fuzzy threshold • Asymmetric GARCH-model with a switching function of the index of fuzzy numbers pairwise comparison Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 9 / 20

Modifications of an asymmetric fuzzy GARCH-model Asymmetric GARCH-model with s-type switching function The main formula is defined as Hung model, but characteristic function is defined as follows:  t > a + ∆ , 1 ,    t − ( a − ∆)  I ( t ) = (6) a − ∆ ≤ t ≤ a + ∆ , , 2∆   t < a − ∆ ,  0 ,  where a , ∆ are parameters that are estimated with the model coefficients by the MLE. Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 10 / 20

Modifications of an asymmetric fuzzy GARCH-model Asymmetric GARCH-model with s-type switching function 1.5 1 I(t) 0.5 0 -0.5 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 t Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 11 / 20

Modifications of an asymmetric fuzzy GARCH-model Asymmetric GARCH-model with the characteristic function of comparing the fuzzy number-histogram and the fuzzy threshold The main formula is defined as Hung model, but characteristic function is defined as follows: r d ( t ) ≻ h , { 1 , I ( t ) = (7) иначе , 0 , where r d ( t ) is the fuzzy number-histogram constructed on last d values of y ( t ) , h is the fuzzy threshold, ≻ is some operation of comparing fuzzy numbers. Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 12 / 20

Modifications of an asymmetric fuzzy GARCH-model Asymmetric GARCH-model with the characteristic function of comparing the fuzzy number-histogram and the fuzzy threshold Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 13 / 20

Modifications of an asymmetric fuzzy GARCH-model Asymmetric GARCH-model with a switching function of the index of fuzzy numbers pairwise comparison The main formula is defined as Hung model, but characteristic function is defined as follows: I ( t ) = R ( r d ( t ) , h ) , (8) where r d ( t ) and h are the fuzzy number-histogram and the fuzzy threshold defined in the previous section. As an index of pairwise comparison we used the popular Baas-Kwakernaak index [Baas S.M., Kwakernaak H. 1977] R ( r d ( t ) , h ) = sup min { µ r d ( t ) ( i ) , µ h ( j ) } , (9) i ≥ j where µ r d ( t ) and µ h are membership functions of the fuzzy number-histogram and the fuzzy threshold, respectively. Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 14 / 20

Data description The models were tested on MICEX and RTS indices, the dollar exchange rate for the period from January 1, 2015 to December 31, 2016. The dates for the indices and the dollar exchange rate were matched where there were gaps for any instrument. The models were trained on 95% of the data and the prediction was carried out for the next 5 days. Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 15 / 20

Testing results MICEX index • MSFE - the mean square forecast error • MAFE - the mean absolute forecast error • LAFE - the largest absolute forecast error MSFE MAFE LAFE GJR-GARCH 1.25E-7 3.10E-4 5.37E-4 Fuzzy GJR-GARCH 4.99E-9 5.27E-5 1.11E-4 Model 1 3.90E-8 1.58E-4 3.47E-4 Model 2 2.69E-9 4.87E-5 7.42E-5 Model 3 1.95E-6 1.22E-3 2.14E-3 Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 16 / 20

Testing results MICEX index Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 17 / 20

Testing results RTS index • MSFE - the mean square forecast error • MAFE - the mean absolute forecast error • LAFE - the largest absolute forecast error MSFE MAFE LAFE GJR-GARCH 7.20E-7 8.13E-4 1.21E-3 Fuzzy GJR-GARCH 7.35E-7 7.80E-4 1.12E-3 Model 1 3.84E-7 5.53E-4 8.13E-4 Model 2 6.03E-7 7.03E-4 9.63E-4 Model 3 4.70E-7 6.15E-4 8.75E-4 Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 18 / 20

Testing results RTS index Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 19 / 20

Results • Practically all considered fuzzy asymmetric GARCH-models have better prognostic ability than their crisp analogues • Using of expert information does not significantly improve the result • The predictive ability of various fuzzy models is significantly different on MICEX and RTS indices Alexander Lepskiy, Artem Suevalov Higher School of Economics September 15, 2017 20 / 20