Forecasting volatility with a stacked model based on a hybridized Artificial Neural Network
FForecasting volatility with a stacked model based on ahybridized Artificial Neural Network
Eduardo Ramos-P´erez (1) ,Pablo J. Alonso-Gonz´alez (2) , Jos´e Javier N´u˜nez-Vel´azquez (2) (1) Ph D Student (Economics and Management Program). Universidad de Alcal´a.(2) Economics Department. Universidad de Alcal´a. ∗† Abstract
An appropriate calibration and forecasting of volatility and market risk are someof the main challenges faced by companies that have to manage the uncertaintyinherent to their investments or funding operations such as banks, pension fundsor insurance companies. This has become even more evident after the 2007-2008 Financial Crisis, when the forecasting models assessing the market risk andvolatility failed. Since then, a significant number of theoretical developmentsand methodologies have appeared to improve the accuracy of the volatility fore-casts and market risk assessments. Following this line of thinking, this paperintroduces a model based on using a set of Machine Learning techniques, such asGradient Descent Boosting, Random Forest, Support Vector Machine and Ar-tificial Neural Network, where those algorithms are stacked to predict S&P500volatility. The results suggest that our construction outperforms other habitualmodels on the ability to forecast the level of volatility, leading to a more accurateassessment of the market risk.
Keywords:
Machine learning, Stacking algorithms, Risk assessment, Volatility forecasting,Hybrid models
AMS Subject Classification:
During the Financial Crisis of 2007-2008, unexpected falls in stock prices resultedin significant losses for individual investors and financial institutions. Since then,new regulations have entered in force in order to ensure the correctness of the mar-ket risk assessment provided by financial institutions and to allow individual marketparticipants to be aware of the risk linked to financial products. As volatility is anindicator of the uncertainty associated with the asset profitability (Hull 2015 andRajashree and Ranjeeeta 2015), this variable tends to play a key role within the risk ∗ Authors’ address: (1)&(2) :Economics Department, Universidad de Alcal´a, Plaza de la Victoria2, 28802 Alcal´a de Henares, Spain. E–mails: P.J. Alonso-Gonz´alez, [email protected] , J.J.N´u˜nez, [email protected] , E. Ramos, [email protected] † Corresponding author: P. Alonso; Date: August 19, 2020. This manuscript version is made availableunder the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/ a r X i v : . [ q -f i n . R M ] A ug odels. In fact, events like the bankruptcy of LTCM in 1998 (Lowenstein 2000), thedotcom crash in 2001 (Aharon et al. 2010) or, more recently, the aforementionedFinancial Crisis of 2007-2008 were not foreseen by most of the risk models due toinaccurate estimates produced by the volatility forecasting models. It is worth men-tioning that, as volatility is not directly observed, before estimating any statisticalmodel it is necessary to select a volatility proxy (Poon and Granger 2003). In thefollowing paragraphs, the proposed methodology and main families of volatility fore-casting models (GARCH, Stochastic and Machine Learning) are presented.First of all, GARCH models are introduced as this family of models is probably themost widely used in the literature due to its ability to fit the volatility clustering(Mandelbrot 1963) empirically observed in financial time series. This auto-regressiveapproach and its generalization were developed by Engle (1982) and Bollerslev (1986)respectively. Classical GARCH models were discovered to be too rigid for fitting re-turns series, especially over a long time span, because the estimated persistence ofconditional variances is close to one (Bauwens et al. 2012). Therefore, more flexi-ble GARCH models were developed in order to overcome this problem. Engle andLee (1999) suggested a two equation model where each of them represents long-runand short-run components of volatility, respectively. Mixed-normal GARCH (Haaset al. 2004a) is a second way to deal with this problem. This kind of model allowsto choose amongst several regimes in each instant of time t. The drawback of thismethodology is that it assumes that the variables used to decide amongst regimes areall independent over time. To overcome this problem, Haas et al. (2004b) proposeda Markov-switching model where the parameters of a GARCH model change accord-ing to a Markov process. An extension of this kind of model can be found in Haasand Paolella (2012). Before concluding with the GARCH models, it is important tomention that volatility can behave differently depending on the trend of the market:bullish or bearish. To fit this behaviour, Nelson (1991) developed the EGARCHmodel that allows the sign and the volume of previous values to have separate im-pacts on the volatility forecasts. In addition to the EGARCH model, Glosten et al.(1993) proposed the GJR-GARCH to replicate the aforementioned behaviour. Otherdevelopments within this family can be found in Engle and Kroner (1995) with theirBEKK model, the factor model (Engle et al. 1990), the Constant Conditional Cor-relation model (Bollerslev 1990), the time-varying correlation model (Tse and Tsui2002), the dynamic correlation model (Engle 2002) or the multivariate GARCH ap-proach proposed by Kraft and Engle (1982) and Engle et al. (1984) and its financialimplementation by Bollerslev et al. (1988). More recently, Zhang et al. (2018) haveproposed a first order zero drift GARCH (ZD-GARCH) to study heteroscedasticityand conditional heteroscedasticity together.The second family is composed of those models which assume that the volatility isdriven by its own stochastic process. This approach was introduced by Taylor (1982)as an Euler approximation of the underlying diffusion model. Assuming that stockprices follow a Brownian motion, Heston (1993) derived a model where the volatil-ity follows an Ornstein-Uhlenbeck process. To derive the parameters of the HestonModel, two different strategies have been adopted in the literature: moment or sim-ulation. For the first one, the Generalized Method of Moments was proposed by2elino and Turnbull (1990) and Andersen and Sorensen (1999), while the simulationapproach has been used by Danielsson (2004), Durbin and Koopman (1997), Brotoand Ruiz (2004) or Andersen (2009), amongst others.The last family presented is Machine Learning, which comprises a set of techniquesused to analyse the future evolution of stock prices and volatility. These algorithmstry to learn automatically and recognize patterns in a large amount of data (Krollneret al. 2010). It is worth mentioning that the fitting of these algorithms is quite sen-sitive to the forecasting time-frame and the selected input variables. Armano et al.(2005) and de Faria et al. (2009) suggest using one day as a time-frame and laggedor technical indicators as input variables for the Machine Learning algorithms. Stockprices, volatilities and portfolio selection have been analysed using different method-ologies based on Machine Learning, such as Support Vector Machine (Gestel et al.2001), hidden Markov models (Gupta and Dhinga 2012 and Dias et al. 2019) or Ar-tificial Neural Networks (ANN) (Hamid and Iqbid 2002). These last authors showedthat volatility forecasts made by an ANN outperform the implied volatility derivedfrom Barone-Adesi and Whaley options models. Additionally, ANNs have been ap-plied successfully to other financial series different from volatility and stock prices:bond rates (Surkan and Xingren 2001) and bank failures (Hutchinson et al. 1994).Deep learning (LeCun et al. 2015) is a framework closely related with ANN whichhas been employed for predicting the evolution of Korean stock market index (Changet al. 2017).Despite the high performance of ANN, predictions derived from the use of this al-gorithm could be inaccurate when stock prices move sharply (Patel and Yalamalle2014). To overcome this problem, ANN were combined with other statistical models(Kristjanpoller et al. 2014) creating the so called hybrid models. Hybridization canbe defined as an approach in which several models are merged to form a new enhancedmodel in order to produce better forecasting results. Therefore, a hybrid model isa combination of the artificial intelligence techniques with some components of thetraditional forecasting models (like the ones presented within the GARCH family).Examples of this approach are discussed in Roh (2006), Hajizadeh et al. (2012),Lu et al. (2016) Monfared and Enke (2014) or Kristjanpoller et al. (2014), wheredifferent outputs from a GARCH-based model are used as inputs in an ANN. A moregeneral picture of this type of hybrid models is provided by Bildirici and Ersin (2009),since they compared and combined an ANN with different types of GARCH models(GARCH, EGARCH, GJR-GARCH, TGARCH, NGARCH, SAGARCH, PGARCH,APGARCH and NPGARCH). In addition to the above-mentioned researches, thistype of hybrid models has been broadly used in other papers. Bildirici and Ersin(2014) proposed a MS-GARCH with an ANN to improve the forecasting accuracy,Bektipratiwi and Irawan (2011) combined a radial basis function with an EGARCHto model stocks returns of an Indonesian bank and Arneric and Poklepovic (2016)developed an ANN model as an extension of a GJR-GARCH to forecast the marketreturns of six European emerging markets. GARCH-based models have been alsocombined with ANNs to predict the volatility in commodity markets, such as gold(Kristjanpoller and Minutolo 2015) or oil (Kristjanpoller and Minutolo 2016). In thislast case, the hybrid model included financial variables to improve the forecasts. This3trategy can also be found in Kristjanpoller and Hern´andez (2017). Kim and Won(2018) propose a hybrid model that combines a LSTM with various GARCH-typemodels to forecast the volatility of KOSPI index. A refinement of this model can befound in Back and Kim (2018). It should be mentioned that these models can begenerated in both directions: some outputs of a GARCH model can be used as inputof an ANN and vice versa (Lu et al. 2016). Finally, it should be noted that hybridi-sation can not only be made with ANN. Peng et al. (2018) proposed a structurecombining traditional GARCH-models with Support Vector Machine (SVM) (Cortesand Vapnik 1995).The research carried out along this paper develops a volatility forecasting model thatconsists of two different levels which is based on stacking algorithms methodology(Hastie et al. 2009) and statistical models of the Machine Learning family. RandomForest (RF) (Breiman 2001), Gradient Boosting (GB) with regression trees (Fried-man 2000) and Support Vector Machine (SVM) (Cortes and Vapnik 1995) are usedin the first level, while an ANN (Mcculloch and Pitts 1943) is incorporated within thesecond level of the stacked model (Stacked-ANN) in order to generate the volatilityforecasts. A different two-level approach can be found in Kristjanpoller and Minutolo(2018). They use an ANN-GARCH model with a pre-processing based on principalcomponents analysis to reduce the number of inputs employed in their network. Incontrast to the hybrid models defined previously, the proposed model is merging theresults arising from other machine learning algorithms which are free of some theo-retical assumptions like the use of a predefined distribution for the underlying assetreturns or the constant level of unconditional variance. Because of this and with theaim to build a more flexible model, the GARCH-based models are not present in theStacked-ANN architecture. The proposed model relies completely on the predictionsmade by machine learning algorithms and market data. Additionally, in the case ofthe Stacked-ANN the final forecasts made by the first level algorithms are directlyused as inputs within the ANN while, in most of the hybrid models discussed in theprevious paragraphs, sections of the GARCH-based models are inserted separatelyin the ANN.The rest of the paper proceeds as follows: Section 2 presents the set of volatilityforecasting models used for comparison purposes. Furthermore, the risk measuresand tests used to validate the results are discussed. In Section 3 the theoreticalbackground and architecture of the volatility forecasting model based on stackingalgorithms (Stacked-ANN) are explained. The empirical results of the different fore-casting models are shown in Section 4, where the accuracy and the risk measuresarising from the proposed model are compared with results obtained by the method-ologies explained in Section 2. Finally, Section 5 presents the main conclusions of theresults and comparisons carried out along Section 4.4 Benchmark models, risk measurements and statisticaltests
As stated above, this section is focused on explaining the benchmark models and thetests used to back-test the risk measurements. Thus, the first paragraphs are ded-icated to ANN, ANN-GARCH, ANN-EGARCH and Heston Model, while the endof this section is focused on the risk measurements and tests performed to validateand compare the results of the benchmark models with the one proposed in Section 3.The first benchmark model is a feed-forward ANN. Following the notation providedby Bishop (2006) and assuming that the algorithm has two hidden layers, the modelwould be defined by the following expression:ˆ σ t +1 = h (3) T (cid:88) k =1 w (3) p,k h (2) M (cid:88) j =1 w (2) k,j h (1) (cid:32) D (cid:88) i =1 w (1) j,i x i + w (1) j, (cid:33) + w (2) k, + w (3) p, (1)Where h ( n ) is the activation function associated with the layer n , w ( n ) z,v is the v-th weight associated with the neuron z inside the layer n and x i refers to the i inputvariable of database comprised by the explicative variables selected by the analyst.The second benchmark model is an ANN-GARCH( p , q ). As briefly introduced inSection 1, the aim of this hybrid model is to combine the GARCH( p , q ) estimateswith other input variables by using an ANN, which is a more flexible model thanGARCH( p , q ). Therefore, before starting with the fitting of the ANN, the parametersof the GARCH( p , q ) model need to be estimated:ˆ σ t = ω + q (cid:88) i =1 α i r t − i + p (cid:88) i =1 β i σ t − i / ˆ r t = ˆ σ t (cid:15) t (2)In this formulation ω , α i and β i are the parameters to be estimated, while r t and σ t refer to the return and volatility respectively. The returns distribution is determinedby the distribution selected for (cid:15) t . If a standardize normal or standardize Student’st-distribution is selected, then the returns generated by the model follow a con-ditional normal (CND) or conditional t-distribution (CTD) respectively (Bauwenset al. 2012). Once the GARCH( p , q ) parameters are estimated, (cid:80) qi =1 α i r t − and (cid:80) pi =1 β i σ t − can be computed and used as input (together with the rest of explica-tive variables) within the ANN.The third benchmark model is an ANN-EGARCH. The architecture of this modeland the previous one can be considered the same with the unique difference that thefirst step consists of fitting an EGARCH( p , q ) instead of a GARCH( p , q ) model. TheEGARCH( p , q ) can be defined as follows (Nelson 1991):log ˆ σ t = ω + p (cid:88) i =1 α i log ˆ σ t − i + q (cid:88) i =1 ( β i (cid:15) t − i + γ i ( | (cid:15) t − i | − E | (cid:15) t − i | )) (3)5nce the EGARCH is fitted, the following terms can be calculated and used as inputwithin the ANN together with the rest of the explicative variables selected by theanalyst: p (cid:88) i =1 α i log ˆ σ t − i q (cid:88) i =1 β i (cid:15) t − i q (cid:88) i =1 γ i ( | (cid:15) t − i | − E | (cid:15) t − i | ) (4)The last benchmark is the Heston (1993) Model. Even though this approach belongsto the stochastic family and the proposed one to the Machine Learning one, thismodel is going to be used as benchmark during this paper as this process is the mostwidely used within the family of the stochastic volatility models. It assumes thatchanges in stock prices through the time ( dX t ) follow a Brownian diffusion process: dX t = µX t dt + (cid:113) σ t X t dB t (5)Where B t ∼ N (0 , σ t t ). Therefore, if volatility follows an Ornstein-Uhlenbeck process,the changes in this variable are defined by the following expression: dσ t = θ ( υ − σ t ) dt + δσ t dB ∗ t (6)where υ is the long term volatility, θ is the rate of return to υ , δ is the volatility of σ t and B ∗ t is a Wiener process that has a correlation of ρ with B t .Once the four benchmark models have been explained, the section focuses on the riskmeasurements. As stated before, volatility plays a key role in market risk assessment.Therefore, the models will not be only compared in terms of accuracy, but the riskmeasurements arising from every volatility model are going to be tested. For thispurpose, VaR and CVaR have been selected as risk measures. Even though VaR isprobably the most used metric due to its simplicity and easy interpretation, CVaRhas been also included as it is considered to be a coherent risk measure (Artzner et al.1999). Consequently, for every volatility model the aforementioned risk measures aregoing to be computed and validated by means of the following tests: • Kupiec (1995) introduced a test in order to check if the number of VaR excessesare align with the level of confidence selected. • An extension of the previous test was developed by Christoffersen et al. (1997).The aim of this test is to validate that VaR excesses are independent, identicallydistributed and in line with the selected level of confidence. • Acerbi and Szekely (2014) developed a test (AS1) to assess the appropriatenessof the CVaR based on the assumption that VaR has been already tested andconsidered to be correct from a statistical point of view. The test is inspiredby the following equation: E (cid:20) r t CV aR α,t + 1 (cid:12)(cid:12)(cid:12)(cid:12) r t + V aR α,t < (cid:21) = 0 (7)As VaR needs to be previously validated, the result of this test has to be assessedtogether with the two aforementioned tests.6 In addition to the previous test, Acerbi and Szekely (2014) introduced anothermethod (AS2) to validate the CVaR without making any assumption about theappropriateness of the VaR. To do so, this test tries to check a CVaR expressionthat is not conditioned by the correctness of a previous VaR estimate.Before beginning with the Stacked-ANN architecture, it is worth noticing that thetwo first tests are parametric while the two last are non-parametric so, for furtherdetails about how to compute the statistics and their distributions please refer toaforementioned papers.
This section has been divided in several sub-sections in order to explain sequentiallythe proposed volatility forecasting model. As the Stacked-ANN model is composedby two different levels, the two first sub-sections are dedicated to the input data andthe algorithms within the first level of the Stacked-ANN model, while the third andforth sub-sections are focused on the data required to generate the stacking procedureand the details of the ANN fitted with the aforementioned information. (Figure 1explains briefly the process followed to estimate and test the Stacked-ANN model)
The first step is concerned with the creation of the database containing the volatilityproxy to be used as a response and the explanatory variables selected to fit thealgorithms. As the aim of the study is to predict future volatilities, the True RealizedVolatility (hereinafter TRV) is going to be used as response variable (Roh 2006):
T RV t = (cid:118)(cid:117)(cid:117)(cid:116) n n (cid:88) i =1 ( r t + i − − (cid:98) r t ) (8)Where (cid:98) r t = (cid:80) ni = n ( r t + i − ) /n and n = 5. The window has been selected to be largeenough to compute a stable TRV and small enough to avoid, as much as possible,mixing different volatility regimes.The variables given to the first level algorithms to forecast the TRV are the last 30volatilities computed with returns already observed in the market: V t = (cid:118)(cid:117)(cid:117)(cid:116) n n − (cid:88) i =0 ( r t − n + i − (cid:98) r t ) (9)Where (cid:98) r t = (cid:80) n − i =0 ( r t − n + i ) /n and n = 5. Only the last 30 volatilities have been se-lected because the correlations between previous volatilities and the TRV are residualand therefore their explanatory power is considered to be non-significant. The his-torical data to compute all the aforementioned variables is obtained by using the quantmod (Ryan and Ulrich 2017) package from the R project (R Core Team 2017)and, as suggested by Hastie et al. (2009), they will be scaled to the range [0 ,
1] to7mprove the training of the algorithms.Before beginning with the section related with the algorithms included within thefirst level, it is important to mention that the first 25% of the data is used to fit thefirst level algorithms, the next 50% is dedicated to the ANN estimation and the last25% is the test set. The comparison of the benchmark models with the proposed onein terms of accuracy and risk measurement will be made with a different set of datacontaining the information of the following year (e.g. if data from 2000 to 2007 isused to train and test the Stacked-ANN model, the out of sample data selected forcomparison purposes would be market movements happened during 2008).Figure 1: Stacked-ANN model structure
The methods applied to optimize the hyper-parameters of the algorithms within thefirst level of the Stacked-ANN architecture are introduced below: • Minimization of the Mean Square Error (hereinafter, MMSE) for the wholedatabase to train the first level algorithms.8
Circular Block Bootstrap (CBB). This method (Politis and Romano 1991) gen-erates new samples by selecting random blocks from the original database. Thelength of these blocks is fixed and the procedure to calculate it was introducedby Politis and White (2004) and Patton et al. (2009). CBB can only be appliedto stationary time series. • Stationary Bootstrap (hereinafter, SB) (Politis and Romano 1994). Similar tothe case of CBB, this method can only be used with stationary time series.However, the difference with the former method is that the length of the blocksinstead of being fixed, it is randomly selected with a certain average that can becalculated using different approaches (see Politis and White 2004 and Pattonet al. 2009). • Maximum Entropy Bootstrap (hereinafter, MEB) (Vinod 2006 and Vinod andde Lacalle 2009). Unlike the two previous approaches, stationarity is not re-quired as the new samples are obtained from the maximum entropy distributionof the original time series. • H Cross-Validation (HCV). This method introduced by Chu and Marron (1991)tries to avoid the effect of the correlation that can exist between the responseand the explanatory variables while dealing with time series by eliminating hdata points between them. The bandwidth selection is obtained minimizing theabsolute autocorrelation between the response and explanatory variables, witha maximum width of 100 days.The optimum hyper-parameters combination of each one of the five previous meth-ods is obtained by applying grid search. Then, these combinations are tested againstdata out of sample (the following 50% of the database) to choose the most accurateoption for fitting the algorithm.As stated before, the first level of the stacked model architecture is composed by threealgorithms: Random Forest (RF) (Breiman 2001), Gradient Boosting with regressiontrees (GB) (Friedman 2000) and Support Vector Machine (SVM) (Cortes and Vapnik1995).
As explained in Section 3.1, the first 25% percent of data is dedicated to fit the firstlevel algorithms while the following 50% and 25% are used for fitting the ANN andtesting the results respectively. The explanatory variables given to the ANN are: • As with the first level algorithms, the last 30 volatilities ( V t , V t − , ..., V t − )scaled to the range [0 , • The True Realized Volatility forecasts made by the first level algorithms: Ran-dom forest ( (cid:91)
T RV t,RF ), Gradient boosting ( (cid:91)
T RV t,GB ) and Support Vector Ma-chine ( (cid:91)
T RV t,SV M ).The response variable is the
T RV t as defined in Section 3.1.9 .4 Second level: Stacking algorithm As stated previously, the last step of the Stacked-ANN model is the fitting of theANN, which is the algorithm stacking the forecasts made by the RF, GB and SVM.Before starting with the details of the ANN architecture, notice that the methodsand procedures related to the hyper-parameters optimization are the same as the firstlevel algorithms: Grid search in combination with the methods explained in Section3.2 and final hyper-parameters decision based on the out of sample error (last 25%of the database).Below, the main characteristics and details of the stacking algorithm are presented: • The feed-forward ANN has two hidden layers with 20 and 10 neurons respec-tively. The sigmoid activation function has been selected for all the neuronswithin the hidden layers while the linear activation function has been used inthe output layer, which is comprised by one neuron. • The optimization algorithm selected is Adaptive Moment Estimation (ADAM),which was created by Kingma and Ba (2014). This method consists in a pro-gressive adaptation of the initial learning rate, taking into consideration currentand previous gradients. The default calibration proposed by the authors is ap-plied: β = 0 . β = 0 . • The number of epochs are 10,000 and the batch size is equal to the length ofthe data used for training the ANN. • The backward pass calculations are done according to the selection of root meansquared error as a loss function. • As indicated in Section 3.1, the 50% of the information is selected for trainingthe ANN while the following 25% of the data is the test set. Note that the first25% of the data is used to fit the first level algorithms. • The parameter adjusting the level of L2 regularization ( φ ) and the initial learn-ing rate λ used within ADAM are the hyper-parameters to be optimized duringthe estimation process.Taking into consideration all the above-mentioned details, the T RV t forecasted bythe Stacked-ANN model (S-ANN) is obtained by means of the following expression: (cid:91) T RV t,S − ANN = (cid:98) f ( (cid:91) T RV t,RF , (cid:91) T RV t,GB , (cid:91) T RV t,SV M , V t , V t − , ..., V t − ) == h (3) (cid:88) k =1 w (3)1 ,k h (2) (cid:88) j =1 w (2) k,j h (1) (cid:32) (cid:88) i =1 w (1) j,i x i + w (1) j, (cid:33) + w (2) k, + w (3)1 , (10)As explained in Section 3.3, x i are the last 30 volatilities scaled to the range [0 , Results
During this section, the data used in the empirical analysis, the fitting process and thefinal comparison between the Stacked-ANN and the benchmark models are shown.
In order to analyse the models under different market conditions, the algorithms havebeen trained and tested five different times with the S&P 500 volatilities observedin the following periods: 2000-2007, 2001-2008, 2002-2009, 2009-2016 and 2010-2017.As stated in Section 3.1, during the training and testing of the models the first 25%of the periods selected is dedicated to fit the first level algorithms, the next 50% isused to optimize the ANN while the last 25% is reserved for testing purposes. Theyear after the aforementioned periods (2008, 2009, 2010, 2017 and 2018 respectivelyfor each period) has been used to compare the out of sample results of the Stacked-ANN with the benchmark models. The first three data-sets have been selected inorder to analyse the performance of the models during the years after the financialcrisis, when the markets where dominated by a high volatile regime. Although theyears influenced by the financial crisis are valuable to test the accuracy of the volatil-ity forecasting models, the two last data-sets have been selected in order to analysethe models performance with the most recent data. Additionally, the lower level ofvolatility during the last periods, especially in 2017, allows to assess the robustnessof the models by analysing them in different market conditions. In order to supportthe explanations given during this paragraph, Table 1 summarizes the moments ofthe TRV during the different periods selected to compare the models:Table 1: True Realised Volatility statisticsPeriod Mean STD Skewness KurtosisYear 2008 0.022 0.016 1.510 4.519Year 2009 0.015 0.008 0.853 3.248Year 2010 0.010 0.006 0.854 3.736Year 2017 0.004 0.002 0.911 3.369Year 2018 0.009 0.006 1.406 4.702
Source : own elaborationIn addition, the Kolmogorov-Smirnov test has been applied sequentially to the TRVin order to assess statistically if the behaviour of the volatility changes over the dif-ferent periods. As 2008 is the year when the most extreme events related with crisishappened and the market changed from a low to a high volatile regime, the skewnessand mean of that year volatility is higher than the one related with 2009. Becauseof that, the aforementioned test reveals that the volatility of 2008 and 2009 do notbelong to the same distribution ( KS p − value = 0 . KS p − value = 0 . KS p − value = 0 . Source : own elaborationAs the critical values are − .
63 and − .
43 with a probability of 5% and 1% respec-tively, it can be concluded that the data meet the requirements imposed by CBB andSB methods.Previously to the fitting of the algorithms, the parameters needed for the differentbootstrap and cross validation methods are obtained by means of the methodologiespresented in Section 3.2. As the Stacked-ANN architecture is comprised by two dif-ferent levels, the length of blocks for CBB, the average of the blocks for SB and thedistance, h , to be used within the HCV method are obtained for both, the data-setto fit first level algorithms and the one dedicated to the second level. Table 3 sum-marizes the former parameters and it shows non-significant changes over time for thedifferent periods and levels: 12able 3: Calibration of the elements for bootstrap and CVData for training Data for trainingMethod Period 1st level algorithms 2nd level algorithmCBB Block (2000-2007) 28 63CBB Block (2001-2008) 36 58CBB Block (2002-2009) 40 56CBB Block (2009-2016) 39 58CBB Block (2010-2017) 38 30SB Block average (2000-2007) 25 55SB Block average (2001-2008) 32 51SB Block average (2002-2009) 35 49SB Block average (2009-2016) 34 51SB Block average (2010-2017) 33 27HCV length (2000-2007) 26 31HCV length (2001-2008) 31 51HCV length (2002-2009) 31 40HCV length (2009-2016) 32 55HCV length (2010-2017) 35 27 Source : own elaboration
As explained in Section 3.2, different approaches have been followed to find theoptimum hyper-parameter combination. Table 4 shows the methods that minimizethe out of sample error per each algorithm and period:Table 4: Methods optimizing OOS errorStacking Gradient SupportPeriod Algorithm (ANN) Random Forest Boosting Vector Machine(2000-2007) ME SB CBB SB(2001-2008) CBB CBB CBB SB(2002-2009) CBB CBB CBB CBB(2009-2016) HCV HCV HCV SB(2010-2017) SB CBB SB SB
Source : own elaborationRegardless of the period, the empirical results suggest that CBB and SB outperformthe rest of the methods. These outcomes are expected as these two methods basedon re-sampling blocks from the original database are specifically prepared to workwith stationary time series. Table 5 summarizes the hyper-parameters suggested bythe methods shown in Table 4: 13able 5: Final hyper-parametersStacking Gradient SupportPeriod Algorithm (ANN) Random Forest Boosting Vector Machine(2000-2007) φ = 0 N = 10 B = 1479 γ = 0 . λ = 0 . Obs = 24 λ = 0 . (cid:15) = 0 . φ = 0 . N = 10 B = 3000 γ = 0 . λ = 0 . Obs = 107 λ = 0 . (cid:15) = 0 . φ = 0 N = 1 B = 3583 γ = 0 . λ = 0 . Obs = 37 λ = 0 . (cid:15) = 0 . φ = 0 . N = 30 B = 1000 γ = 0 . λ = 0 . Obs = 118 λ = 0 . (cid:15) = 0 . φ = 0 . N = 7 B = 1000 γ = 0 . λ = 0 . Obs = 175 λ = 0 . (cid:15) = 0 . Source : own elaborationWhere λ is the learning rate of the ANN and GB, φ the parameter adjusting thelevel of L2 regularization of the ANN, B the number of iterations performed whilefitting the GB, N the number of variables randomly selected by the RF and Obs theminimum number of observations to be kept in the terminal nodes of every fitted treewithin the RF architecture. Finally, γ refers to the parameter included within theradial basis function kernel (the lower the parameter, the higher the non-linearity)and (cid:15) defines the threshold where the error begins to be penalized by the SVM. Once the Stacked-ANN is fitted, its performance is compared with the benchmarkmodels explained in Section 2 (ANN, ANN-GARCH(1,1), ANN-EGARCH(1,1) andHeston Model). Before beginning with the comparisons, the three following remarksabout the benchmark models have to be done: • Due to the nature of the Heston Model, 20,000 simulations per each day havebeen computed and the daily average of them has been taken to assess itsaccuracy. • The GARCH(1,1) and EGARCH(1,1) (included in the ANN-GARCH(1,1) andANN-EGARCH(1,1) architecture respectively) have been estimated assumingStudent-t innovations. • The fitting procedure and architecture of the ANNs included within ANN-GARCH(1,1), ANN-EGARCH(1,1) and ANN models are the same as the onesexplained for the Stacked-ANN (see Section 3.4).Table 6 shows the out of sample error of the different periods selected to compare theperformance and robustness of the Stacked-ANN with the benchmark models. Theresults shown in this table suggest the following conclusions:14able 6: Accuracy analysisRMSE: RMSE: RMSE: RMSE: RMSE:Model 2008 2009 2010 2017 2018Stacked-ANN 0.01192 0.00534 0.00494 0.00254 0.00544ANN-EGARCH 0.01332 0.00588 0.00537 0.00276 0.00571ANN-GARCH 0.01335 0.00584 0.00539 0.00263 0.00575Heston 0.02066 0.00714 0.00547 0.00359 0.00610ANN 0.01526 0.00615 0.00541 0.00274 0.00590
Source : own elaboration • Regardless of the period, the Stacked-ANN outperforms other hybrid modelsbased on auto-regressive methodologies like ANN-GARCH and ANN-EGARCH.In relative terms, minor deviations are observed between the different periods. • All the hybridized models tend to outperform the pure ANN model. • As expected due to the extremely high volatilities observed during the financialcrisis, the results show that, regardless of the model, 2008 forecasts are lessaccurate. All the models minimize their error rate in the year with the lowestlevel volatility, 2017. • The forecasts made by the Heston Model tend to be the less accurate due tothe non-predictive nature of this model.In addition to the above-mentioned analysis, the risk measures obtained by usingeach one of the volatility models are tested. In order to do so, a returns distributionis selected for each one of the forecasting volatility methods. As described in Section2, Heston Model requires the changes in stock prices to follow a Brownian diffusionprocess. Nevertheless, for the rest of the benchmark models and the Stacked-ANN(which are free of assumptions about the returns) a Student t-distribution has beencombined with the different volatility forecasts. This assumption about Student t-distribution has been selected when possible as returns tend to be leptokurtic andheavier-tailed than Normal distribution (McNeil et al. 2015).Before analysing the results of the tests presented in Section 2, it is worth mentioningthat the level of confidence (99%) and number of days (10) selected are based on theones set by Basel Directive, whose aim is to monitor, amongst others, the marketrisk. Table 7 shows the p-value of the tests dedicated to VaR (Kupiec and Christof-fersen) and CVaR (AS1 and AS2). If a 95% is set as confidence level, Stacked-ANNin combination with Student t-distribution is the only model that produces an ap-propriate p-value for Kupiec, AS1 and AS2 tests in every period under analysis. Allthe models show difficulties to produce a p-value higher or equal than 0 .
05 for theChristoffersen test because VaR exceedances tend to happen in a short period of timeinstead of spread over the period analysed. It is worth mentioning that the hybridmodels taken as benchmark (ANN-EGARCH and ANN-GARCH) also fail in produc-ing an appropriate value for the Kupiec test in several periods while, as stated before,15he proposed hybrid model (Stacked-ANN) pass the test for every period. Finally,Heston Model tends to produce less appropriate risk measures due to the distributionconstrain mentioned previously.Table 7: P-value of the VaR and CVaR testsPeriod: Period: Period: Period: Period:Model Test 2008 2009 2010 2017 2018Stacked-ANN Kupiec 0.85 0.84 0.65 0.85 0.85Christ. 0.01 0.79 0.02 0.01 0.01AS1 0.66 0.85 0.61 0.90 0.91AS2 0.56 0.63 0.36 0.67 0.69ANN-EGARCH Kupiec 0.12 0.12 0.84 0.03 0.03Christ. 0.00 0.00 0.01 0.03 0.03AS1 0.52 0.85 0.61 1.00 1.00AS2 0.07 0.19 0.62 0.91 0.91ANN-GARCH Kupiec 0.12 0.03 0.01 0.03 0.03Christ. 0.00 0.03 0.00 0.03 0.03AS1 0.51 1.00 0.77 1.00 1.00AS2 0.08 0.92 0.05 0.85 0.89Heston Model Kupiec 0.00 0.00 0.65 0.03 0.00Christ. 0.00 0.00 0.59 0.03 0.00AS1 0.00 0.01 0.83 1.00 0.06AS2 0.00 0.00 0.36 0.92 0.00ANN Kupiec 0.65 0.04 0.65 0.30 0.29Christ. 0.02 0.00 0.00 0.00 0.00AS1 0.24 0.86 0.59 0.81 0.00AS2 0.29 0.11 0.35 0.24 0.00
Source : own elaboration 16
Conclusions
This paper introduces a Stacked-ANN model based only on Machine Learning tech-niques with the aim to improve the accuracy of the volatility forecasts made by otherhybrid models based on a combination of GARCH or EGARCH with ANNs. Its pre-dictive power and performance has been tested in terms of RMSE, VaR and CVaR.Two main results have to be pointed out. Firstly, the Stacked-ANN has been able togenerate more accurate volatility forecasts than other models in a high volatile regimeperiod like the one occurred after the Financial Crisis of 2007-2008. The models out-performed by the Stacked-ANN during that time lapse are other hybrid models likeANN-GARCH and ANN-EGARCH, the most widely used stochastic volatility the-ory (Heston Model) and a feed-forward ANN without any combination with otheralgorithms or statistical models. Notwithstanding the Stacked-ANN performance, itis observed for every model that the higher the volatility the lower the accuracy. Inaddition to this analysis, the Stacked-ANN has been tested with the most recent data(2017 and 2018) in order to check its performance in the current market conditions.As it occurred with the tests carried out during the financial crisis, the proposedarchitecture outperforms the benchmark models in terms of accuracy. The superiorperformance shown by the Stacked-ANN in periods with different levels of volatilityare due to the model flexibility. In contrast with ANN-GARCH or ANN-EGARCH,the inputs introduced in the ANN stacked model do not follow any theoretical as-sumption about the returns distribution or volatility. As explained throughout Sec-tion 3, the architecture proposed uses previous volatilities and forecasts made by arandom forest, gradient boosting with regression trees and support vector machineas inputs. Before beginning with the second point of the conclusion, it is worthmentioning that it has been empirically demonstrated that block bootstrap methodsare of special interest when fitting algorithms to volatility as these procedures areespecially prepared to work with stationary time series.Secondly, the forecasts made by the volatility models have been combined with a cer-tain distribution in order to compute the VaR and CVaR for all the different periodsanalysed. The distribution selected has been the Student’s t-distribution for everymodel with the exception of the Heston Model which requires changes in asset pricesto follow a Brownian diffusion process. The empirical results demonstrated that onlythe Stacked-ANN model is able to produce an appropriate p-value for Kupiec, AS1and AS2 tests in every period under analysis, including those ones related with thefinancial crisis.The aforementioned flexibility and predictive power of the Stacked-ANN comparedwith other volatility models suggest to develop further investigations about the im-plications of using this model for derivative valuation purposes. As the price of theseinstruments is closely related to the volatility of the underlying assets, further re-searches should be done in order to compare the implied volatilities observed in themarket with the ones arising from the proposed model. If the volatility measured bythe Stacked-ANN is more accurate than market expectations, it would be possible toidentify under and overvalued derivatives.17 eferences
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