Autoregressive models of the time series under volatility uncertainty and application to VaR model
AAutoregressive models of the time series undervolatility uncertainty and application to VaR model
Shige Peng * Shuzhen Yang † Abstract
Financial time series admits inherent uncertainty and randomness that changesover time. To clearly describe volatility uncertainty of the time series, we assume thatthe volatility of risky assets holds value between the minimum volatility and maximumvolatility of the assets. This study establishes autoregressive models to determine themaximum and minimum volatilities, where the ratio of minimum volatility to maxi-mum volatility can measure volatility uncertainty. By utilizing the value at risk (VaR)predictor model under volatility uncertainty, we introduce the risk and uncertainty,and show that the autoregressive model of volatility uncertainty is a powerful tool inpredicting the VaR for a benchmark dataset.
KEYWORDS: Autoregressive model; sublinear expectation; volatility uncertainty; G-VaR;G-normal distribution.
To find an exact time series model is a basic goal in econometric theory. The time-varying models have achieved greatly in the application of time series analysis. The au-toregressive conditional heteroscedastic (ARCH) model is the seminal work among the * Institute of Mathematics, Shandong University, Jinan 250100, China, ([email protected]). † Shandong University-Zhong Tai Securities Institute for Financial Studies, Shandong University, PRChina, ([email protected]). a r X i v : . [ q -f i n . M F ] N ov evelopment of time-varying models . In Engle (1982), the conditional variance of assetsat time t was assumed to be the function of assets before time t , which deduced a time-varying, random variable model for the conditional variance. The ARCH model can cap-ture the local shifting of conditional variance of assets and be used to forecast one-periodvariance of assets.Di ff ering from the method developed in the ARCH model, for a given time series { Z t } nt = ,we originally propose a model to capture the volatility uncertainty of the time series toforecast one-period volatility uncertainty. Precisely, we assume the volatility of Z t holdsvalue in an interval, [ σ t , σ t ] , ≤ t ≤ n , where σ t is the minimum volatility and σ t isthe maximum volatility of Z t . To describe the distributions uncertainty of Z t , we assume Z t satisfies an infinite family of distributions, that is G-normal distribution N ( r t , [ σ t , σ t ])under a sublinear expectation . Recently, Peng et al. (2020) developed a VaR model underthe sublinear expectation , namely, G-VaR model. For a given risk level α ≤ .
5, theG-VaR model can be represented as:G-VaR α ( Z t ) = − r t − ¯ σ t Φ − (cid:32) σ t + σ t σ t α (cid:33) , (1.1)where r t is the mean return of the time series Z t , and Φ ( · ) is the cumulative distributionfunction of the standard normal distribution. By introducing an adaptive window W , whichsatisfies a convergence of results in the sample, Peng et al. (2020) compared the G-VaRmodel with some distributions under GARCH models to show the performances of the G-VaR model. However, the main contribution of Peng et al. (2020) is the explanation of theG-VaR model and the related empirical analysis. In particular, the empirical analysis ofPeng et al. (2020) showed that the G-VaR model can capture the long time average loss ofthe risky assets. In this study, we aim to establish a regressive model for forecasting thevolatility uncertainty of the time series in the local sense.It should be noted that the model developed in Peng et al. (2020) is a static model, inwhich we use the daily data to estimate the parameters and then calculate the risk for thefollowing period. On the contrary, we want to investigate a dynamic regressive model for ARCH model was first developed in Engle (1982), and then generalized to a GARCH model in Bollerslev(1986), which was widely used in financial time series analysis. See Section 2 of this paper, also see Peng (2019) for further details of sublinear expectation In particular, a worst-case distribution was derived to calculate the VaR under sublinear expectation inPeng et al. (2020). κ t = σ t σ t , and call this κ t the volatility uncertainty index of the risky asset. The G-VaR model inrepresentation (1.1) can be rewritten asG-VaR α ( Z t ) = − r t − ¯ σ t Φ − (cid:32) + κ t α (cid:33) . (1.2)From (1.2), we can distinguish the maximum volatility σ t and the volatility uncertaintyindex κ t in G-VaR α ( Z t ). The term ¯ σ t is the scale transformation of the standard normaldistribution Φ ( · ), and the term 1 + κ t α to 1 + κ t α , which issmaller than α . Hence, we can use the volatility uncertainty index κ t to modify the normaldistribution and capture the fat tail property of Z t .It is well known that financial time series have volatility clustering properties. Thus, weuse a p -th order autoregressive model that indicates the maximum volatility σ t and mini-mum volatility σ t to forecast one-step volatilities at a time t +
1. Furthermore, by assuming Z t satisfies an infinite family distribution, that is G-normal distribution under a sublinear ex-pectation, N ( r t , [ σ t , σ t ]), which can be used to describe the volatility uncertainty of Z t , wecan thus obtain a one-period forecast maximum volatility ˜ σ t + , minimum volatility ˜ σ t + andvolatility uncertainty index ˜ κ t + , and then forecast the VaR at the time t +
1, G-VaR α ( Z t + ).There are several advantages and contributions of this study:(i). By introducing the volatility uncertainty for financial time series, we use autoregres-sive models to forecast one-period maximum volatility, minimum volatility, and a volatilityuncertainty index. This new model can capture the local changing of volatility uncertainty.In particular, two parameters are introduced: the maximum volatility and volatility uncer-tainty index can be used to deal with issues regarding risk measures.(ii). Based on the maximum volatility and volatility uncertainty index, a new repre-sentation of G-VaR is given, in which we can distinguish the risk and uncertainty. It isconvenient to use this G-VaR model to conduct data analysis. Furthermore, we develop aLocalization Windows method to calculate the sample variance and use the nonlinear leastsquare estimator to estimate the parameters in the autoregressive models.(iii). Based on the benchmark S&P500 Index dataset, we verify the rationality of theautoregressive models, and predict the VaR for the S&P500 Index by G-VaR model. Fur-thermore, we calculate the likelihood ratio test and Christo ff erson independent test for the3esults of G-VaR model, which shows the powerful of these autoregressive models andLocalization Windows.This paper is organized as follows: In Section 1, we show the motivation of this studyand review the literature. We introduce and explain the concept of sublinear expectationsand G-normal distribution in Section 2. In Section 3, we formulate autoregressive modelsto indicate the maximum and minimum volatilities. In Section 4, we reveal how to imple-ment the first-order autoregressive models for the maximum and minimum volatilities, anddevelop a Localization Windows method to calculate the sample variance. In particular,based on the maximum volatility and volatility uncertainty index, a new representation forG-VaR model is given. In Section 5, based on the results of Section 4, we use this autore-gressive model to analyze the benchmark S&P500 Index dataset, and predict the VaR byG-VaR model. Finally, we conclude this paper in Section 6. In monograph Knight (1921), Knight distinguished three types of distribution uncer-tainties: The first one is that the future distribution exists and is known; the second is thatthe future distribution exists but is not known in advance; the third type of uncertaintycorresponds to a non-existent distribution. In the financial market, the distribution of timeseries usually satisfies the second and third kinds of uncertainty.The mean and variance are two important characteristics of the time series. By al-lowing the volatility of a stock to range between two extreme values σ min and σ max , amodel of pricing and hedging derivative securities and option portfolios was developed inAvellaneda et al. (1995). Lyons (1995) introduced optimal and risk-free strategies for inter-mediaries to meet their obligations under volatility uncertainty which takes value in someconvex region that varies with the price process, (see also Denis and Martini (2006), andreferences therein). Peng (1997) investigated a nonlinear expectation called g -expectation,which is a new formal mathematical approach to model mean uncertainty. Continuously,Chen and Epstein (2002) used the g -expectation introduced in Peng (1997) to describe thecontinuous-time inter-temporal version of multiple-priors utility. Furthermore, a separatepremium for ambiguity on top of the traditional premium for risk was developed in Chenand Epstein (2002).The notion of upper expectations was first discussed by Huber (1981) in robust statis-4ics, (see also Walley (1991)). Focusing on the measurement of both market risks andnonmarket risks, the concept of coherent risk measures was introduced in Artzner et al.(1999), see also F¨ollmer and Schied (2011). From a mathematical point of view, a gen-eral notation of nonlinear expectation was originally introduced in Peng (2004, 2005). Akind of nonlinear Brownian motion and related stochastic calculus under G-expectation(a nonlinear expectation) was developed in Peng (2006, 2008). A quantitative frameworkfor defining model uncertainty in option pricing models was considered in Cont (2006).Through examples, the di ff erence between model uncertainty and the more common no-tion of ”market risk” was illustrated. Kerkhof et al. (2010) proposed a procedure to takemodel risk into account in the computation of capital reserves, which can be used to addressVaR and expected shortfall, and thus distinguish estimation risk and miss-specification risk.Furthermore, a model of utility in a continuous-time framework that captures aversion toambiguity about both the volatility and the mean of returns was formulated in Epstein and Ji(2013). Recently, the systematic theory of G-expectation and a useful sublinear expectationwere concluded in monograph Peng (2019).Di ff ering from the traditional models on a constant one-step forecast variance, Engle(1982) introduced seminal autoregressive conditional heteroscedastic (ARCH) processes.This ARCH model assumed that the processes with a nonconstant variances conditionalon the past, but constant unconditional variances, (see also Bollerslev (1986) for generalARCH model). As Engle (1982) pointed out, and Mcnees (1979) suggested ”the inherentuncertainty or randomness associated with di ff erent forecast periods seems to vary widelyover time”. Thus ARCH (GARCH) model can forecast the variance which may changewith time.On the contrary, based on the sublinear expectation in Peng (2019), we want to proposethe inherent uncertainty behind the time series in this study. We assume that there is aninfinite family of probabilities { P θ } θ ∈ Θ behind the time series { Z t } nt = , where Θ denotes thepossible set. Note that, each P θ denotes a model, and thus the potential models are indeedinfinite. Hence, we need to find an appropriate model to capture the property of time series { Z t } nt = . The sublinear expectation provides a potential method to develop the worst-casemodel which contains the risk and volatility uncertainty. When forecasting the one-stepvolatility, the time series { Z t } nt = admits the maximum and minimum volatilities, which cancapture the local volatility uncertainty of the time series. Based on the changing of the5aximum and minimum volatilities, the worst-case distribution will vary with time.In the following, we review the literature on VaR. In the 1990s, J.P. Morgan proposedthe VaR model, which is used to calculate the downside risk in financial markets. BaselAccords I, II, and III have recorded the VaR measures into their recommendations to thebanking industry, which has accelerated the spread of VaR. There are many related bookson the theory and applications of the VaR model, and the VaR models are important topicsfor financial econometrics and risk management. We refer the reader to the recent reviewpapers by Kuester et al. (2006), Jorion (2010), Abad et al. (2014), Nadarajah and Chan(2016), and Zhang and Nadarajah (2017), among others. In addition, Abad et al. (2014)concluded fourteen papers that surveyed and compared di ff erent VaR methods through em-pirical analysis. Focusing on the daily NASDAQ Composite Index, based on a larger num-ber of VaR methods, Kuester et al. (2006) calculated the related daily VaR and statisticstesting for those VaR measures in terms of their prediction power.As most studies point out, applying an AR-GARCH model to filter the residuals in-stead of the original series can improve the predictions of the VaR model, further see Abadet al. (2014). Based on empirical analyses on the NASDAQ Composite Index, Kuester et al.(2006) conclude that “conditionally heteroskedastic models yield acceptable forecasts” andthat the conditional skewed- t (AR-GARCH-St) together with the conditional skewed- t cou-pled with EVT (AR-GARCH-St-EVT) perform best in general.The ARCH(GARCH) model can capture the local changing of variance by developinga random one-period forecast variance, which introduces an autoregressive conditional het-eroscedastic process, which is successful in analyzing financial time series. In this study,based on the G-normal distribution under a sublinear expectation, we want to use autore-gressive models for the maximum and minimum volatilities to capture the local changing ofvariance. A VaR forecast predictor, the G-VaR model shows that the volatility uncertaintyis a significant index for measuring the financial time series. For a given time series { Z t } nt = , to describe the volatility uncertainty of Z t , we assumethat each time t ∈ { , , · · · , n } , the volatility of Z t takes value in an interval, i.e., [ σ t , σ t ],where σ t is the minimum volatility and σ t is the maximum volatility of Z at the time t .6ntuitively, the time series { Z t } nt = admits the volatility uncertainty, which means thatthere are an infinite family of probabilities { P θ ( · ) , θ ∈ Θ } behind { Z t } nt = . Under eachprobability P θ , we can obtain a volatility value of Z t from [ σ t , σ t ]. In the following, weuse the infinite family of probabilities { P θ ( · ) , θ ∈ Θ } to describe the volatility uncertaintyof the time series { Z t } nt = . In Peng (2004, 2005), a nonlinear expectation was introduced anda useful sublinear expectation E [ · ] was developed, where E [ · ] defined on a linear space H of real valued functions on Ω . A sublinear expectation E [ · ] : H → R is satisfied, for anygiven X , Y ∈ H ,(i). E [ X ] ≤ E [ Y ] , X ≤ Y ;(ii). E [ c ] = c , c ∈ R ;(iii). E [ X + Y ] ≤ E [ X ] + E [ Y ];(iv). E [ λ X ] = λ E [ X ] , λ ≥ E [ · ] on infinite family distributions is givenin Theorem 2.1. For further details see Page 6, Theorem 1.2.1 of monograph Peng (2019). Theorem 2.1 (Peng (2019)) . Let E [ · ] be a sublinear expectation on H . There exists aninfinite family of linear expectation { E θ , θ ∈ Θ } such that E [ X ] = max θ ∈ Θ E θ [ X ] , X ∈ H . (2.1)Based on the Theorem 2.1, we can consider the time series { Z t } nt = under sublinear ex-pectation E [ · ], which can represent as an infinite family of distributions. In Peng (2019),a nonlinear expectation theory has been established completely, including the definitionsof an identically distributed, independent, and central limit theorem under sublinear expec-tation E [ · ]. To represent the limit distribution of the central limit theorem under sublinearexpectation, a nonlinear normal distribution (G-normal distribution) is developed. Here,the G-normal distribution does not denote a distribution but an infinite family of distribu-tions. Precisely, we give the exact definitions of identically distributed and independentunder sublinear expectation E [ · ]. Definition 2.1 (Peng (2019)) . Let ξ and ξ be two random variables defined on sublinearexpectation spaces ( Ω , H , E [ · ]) . They are called identically distributed, denoted by ξ d = ξ ,if E [ φ ( ξ )] = E [ φ ( ξ )] , ∀ φ ( · ) ∈ C l . Lip ( R ) . efinition 2.2 (Peng (2019)) . A random variable ξ ∈ H is said to be independent of ξ ∈ H , if for each φ ∈ C l . Lip ( R × R ) , we have E [ φ ( ξ , ξ )] = E [ E [ φ ( z , ξ )] z = ξ ] . Theorem 2.1 shows that identically distributed can be represented asmax θ ∈ Θ E θ [ φ ( ξ )] = max θ ∈ Θ E θ [ φ ( ξ )] , ∀ φ ( · ) ∈ C l . Lip ( R ) . Thus, the random variables ξ and ξ have the same maximum expectation for any givencriterion function φ ( · ) ∈ C l . Lip ( R ) under the maximum expectation of { E θ [ · ] , θ ∈ Θ } . Similarto the explanation of identically distributed, applying Theorem 2.1, ξ is independent from ξ and can be understood asmax θ ∈ Θ E θ [ φ ( ξ , ξ )] = max θ ∈ Θ E θ [max θ ∈ Θ E θ [ φ ( z , ξ )] z = ξ ] . When Θ is the set of a single point, the definition of independent under sublinear ex-pectation is identically with that in linear expectation. Note that a sublinear expectation E [ · ] = max θ ∈ Θ E θ [ · ], the independence of ξ from ξ means that the uncertainty of distribu-tions of ξ does not change with each realization of ξ ( ω ) = z , z ∈ R . Now, we construct a random variable ξ which satisfies the G-normal distribution. For agiven canonical probability space ( Ω , F , P ), Ω = C ([0 , { B t } ≤ t ≤ ,we define the probability measures P θ as follows. For A ∈ F , P θ ( A ) = P ◦ η − θ ( A ) = P ( η θ ∈ A ) , where η θ ( · ) = (cid:90) (cid:5) θ s dB s , θ ∈ Θ = L F ( Ω × [0 , , [ σ, σ ]) , and Θ is the set of all progressively measurable processes taking value on [ σ, σ ]. The col-lection of P θ s is denoted as { P θ } θ ∈ Θ . In this study, we do not consider the mean uncertaintyof variable ξ , i.e., E [ ξ ] = − E [ − ξ ] = . E [ · ] = max θ ∈ Θ E θ [ · ], it follows thatmax θ ∈ Θ E θ [ ξ ] = min θ ∈ Θ E θ [ ξ ] , which means that the maximum and minimum mean of ξ on θ ∈ Θ are equal. The distri-bution of ξ under P θ is F θ ( z ) : = P θ ( ξ ≤ z ), and this infinite family of distributions { F θ } θ ∈ Θ is chose as the family governing the random variable ξ . In the following, we use the so-called G-normal distribution N (0 , [ σ , σ ]) to represent the infinite family of distributions { F θ } θ ∈ Θ .In general, it is di ffi cult to calculate sublinear expectation E [ φ ( ξ )]. Based on a givenfunction φ ( · ) ∈ C l . Lip ( R ), we will reveal how to construct a random variable that satisfies aG-normal distribution using a nonlinear partial di ff erential equation. The partial di ff erentialequation is a useful tool to find the related parameter θ φ such that E [ φ ( ξ )] = E θ φ [ φ ( ξ )] andto calculate E θ [ φ ( ξ )]. Assumption 2.1.
Let us assume { ξ t } ≤ t ≤ satisfies the following stochastic di ff erential equa-tion, d ξ t = θ t dB t , Z = , under P θ , θ ∈ Θ = L F ( Ω × [0 , , [ σ, σ ]) , where Θ is the set of all progressively measurableprocesses taking value on [ σ, σ ] . The stochastic process { ξ t } ≤ t ≤ in Assumption 2.1 admits a related time-varying vari-ance θ ∈ Θ for the given probability measure P θ . Therefore, there are infinite family of dis-tributions behind the process { ξ t } ≤ t ≤ in Assumption 2.1 and we call ξ satisfies G-normaldistribution N (0 , [ σ , σ ]) . Let Assumption 2.1 holds, for a given function φ ( · ) ∈ C l . Lip ( R ),it follows that E [ φ ( ξ t )] = max θ ∈∈ Θ E θ [ φ ( ξ t )] = max θ ∈ Θ E θ [ φ ( (cid:90) t θ s d B s )] . Proposition 2.2.10 of Peng (2019) showed that u ( t , x ) = E [ φ ( ξ t + x )] is the unique viscositysolution of the following partial di ff erential equation: ∂ t u ( t , x ) − G ( ∂ xx u ( t , x )) = , t > , x ∈ R , (2.2) Based on Assumption 2.1, we use N (0 , [ σ , σ ]) to represent the infinite family of distributions { F θ } θ ∈ Θ behind the random variable ξ . u (0 , x ) = φ ( x ) , x ∈ R , where the function G ( · ) is defined as G ( a ) = (cid:16) σ a + − σ a − (cid:17) , a + = max( a , , and a − = max( − a , . (2.3)It should be noted that u (1 , = E [ φ ( ξ )]. For the given time-series { ξ t } ≤ t ≤ in Assump-tion 2.1, we can calculate the characteristic of variable ξ with function φ ( · ) under theinfinite family of distributions { F θ } θ ∈ Θ . Now, we present the main assumption for the time series { Z t } nt = : Assumption 3.1.
For a given positive integer p, at time s ∈ { , , · · · , p } , we assume Z s sat-isfies G-normal distribution N ( r s , [ σ s , σ s ]) , and Z h is independent from { Z , Z , · · · , Z h − } , ≤ h ≤ p under sublinear expectation E [ · ] , where r s is the mean of Z s , i.e, E [ Z s ] = − E [ − Z s ] = r s . For t > p, we assume Z t satisfies the autoregressive model:Z t = γ + γ Z t − + · · · + γ p Z t − p + ε t , (3.1) where ε t is independent from { Z t − p , Z t − p + , · · · , Z t − } under sublinear expectation E [ · ] and E [ − ε t ] = E [ ε t ] = , Remark 3.1.
It is di ffi cult to estimate the coe ffi cients { γ , · · · , γ p } directly under the givenG-normal distribution, since G-normal distribution denotes infinite family of distributions.Thus, we want to derive the autoregressive relation of parameters { r s } ns = and variances { σ s } ns = , { σ s } ns = of G-normal distribution, based on which we can forecast the one-perioddistribution of a risky asset Z. In Assumption 3.1, Z h is independent from { Z , Z , · · · , Z h − } , ≤ h ≤ p . Thus, we canshow that Z t , t > p satisfies the G-normal distribution, denoted as N ( r t , [ σ t , σ t ]). Takingsublinear expectation E [ · ] on both sides of equation (3.1), for t > p , it follows that E [ Z t ] = γ + γ E [ Z t − ] + · · · + γ t − p E [ Z t − p ] + E [ ε t ] , and E [ − Z t ] = − γ + γ E [ − Z t − ] + · · · + γ t − p E [ − Z t − p ] + E [ − ε t ] .
10y Assumption 3.1, we can obtain the autoregressive model for r t , r t = γ + γ r t − + · · · + γ p r t − p . (3.2)Combining equations (3.1) and (3.2), we have Z t − r t = γ ( Z t − − r t − ) + · · · + γ p ( Z t − p − r t − p ) + ε t , which deduces that E (cid:104) ( Z t − r t ) (cid:105) = E (cid:20)(cid:16) γ ( Z t − − r t − ) + · · · + γ p ( Z t − p − r t − p ) (cid:17) (cid:21) + E (cid:104) ε t (cid:105) (3.3)and − E (cid:104) − ( Z t − r t ) (cid:105) = − E (cid:20) − (cid:16) γ ( Z t − − r t − ) + · · · + γ p ( Z t − p − r t − p ) (cid:17) (cid:21) − E (cid:104) − ε t (cid:105) . (3.4)Note that σ t = E (cid:104) ( Z t − r t ) (cid:105) , σ t = − E (cid:104) − ( Z t − r t ) (cid:105) , t > p . By a simple calculation, equation (3.3) becomes σ t = E (cid:104) ε t (cid:105) + p (cid:88) i = γ i σ t − i + (cid:88) ≤ i < j ≤ p γ i γ j E (cid:104) ( Z t − i − r t − i )( Z t − j − r t − j ) (cid:105) . Denoting α + H ( t ) : = E (cid:104) ε t (cid:105) + (cid:88) ≤ i < j ≤ p γ i γ j E (cid:104) ( Z t − i − r t − i )( Z t − j − r t − j ) (cid:105) . (3.5)When H ( t ) can be approximated by { σ t − , · · · , σ t − p } , we have the autoregressive model: σ t = α + α σ t − + · · · + α p σ t − p . (3.6)Similarly, when H ( t ) can be approximated by { σ t − , · · · , σ t − p } , we have the autoregressivemodel for σ t : σ t = β + β σ t − + · · · + β p σ t − p , (3.7)where H ( t ) satisfies β + H ( t ) = − E (cid:104) − ε t (cid:105) − (cid:88) ≤ i < j ≤ p γ i γ j E (cid:104) − ( Z t − i − r t − i )( Z t − j − r t − j ) (cid:105) . xample 3.1. Now, we consider a simple case where p = . Based on equation (3.5), Wehave H ( t ) as follows: α + H ( t ) = E (cid:104) ε t (cid:105) + γ γ E [( Z t − − r t − )( Z t − − r t − )] . Putting Z t − − r t − = γ ( Z t − − r t − ) + γ ( Z t − − r t − ) , into the formula of H ( t ) , we have α + H ( t ) = E (cid:104) ε t (cid:105) + γ γ σ t − + γ γ E [( Z t − − r t − )( Z t − − r t − )] . When γ is su ffi ciently small, it follows that α + H ( t ) = E (cid:104) ε t (cid:105) + γ γ σ t − + o ( γ ) , where o ( γ ) is the infinitesimal of higher order of γ . Thus, we can assume σ t = α + α σ t − + α σ t − . From Assumption 3.1, we have that Z t − r t satisfies the G-normal distribution N (0 , [ σ t , σ t ]),where the minimum volatility and maximum volatility change with time t . We construct aindex which is used to measure the volatility uncertainty of Z t , κ t = σ t σ t ∈ [0 , . From the definition of volatility uncertainty index κ t , we can see that when κ t = κ t =
0, this is the largest volatility uncertainty that we canmeasure. Note that, Z t − r t satisfies G-normal distribution N (0 , [ σ t , σ t ]) which denotes aninfinite family distributions { P θ , θ ∈ Θ } , thus, we havemin θ ∈ Θ E θ [( Z t − r t ) ] = σ t , max θ ∈ Θ E θ [( Z t − r t ) ] = σ t . (3.8)The volatility uncertainty index κ t is denoted as κ t = (cid:115) min θ ∈ Θ E θ [( Z t − r t ) ]max θ ∈ Θ E θ [( Z t − r t ) ] , (3.9)12here E θ [ · ] is the expectation under probability P θ ( · ). Note that − E [ −· ] = min θ ∈ Θ E θ [ · ],applying Theorem 2.1, we have the following equivalence representations of (3.8) and (3.9)under sublinear expectation E [ · ], − E [ − ( Z t − r t ) ] = σ t , E [( Z t − r t ) ] = σ t . (3.10)and κ t = (cid:115) − E [ − ( Z t − r t ) ] E [( Z t − r t ) ] . (3.11)Based on the p -th autoregressive models (3.2), (3.6), and (3.7) with coe ffi cients ( γ , γ , · · · , γ p ),( α , α , · · · , α p ) and ( β , β , · · · , β p ), and the least square estimators of the coe ffi cients( r t − p , · · · , r p ), ( σ t − p , · · · , σ t ) and ( σ t − p , · · · , σ t ), we can obtain the one period forecast of r , σ and σ at time t +
1: ˜ r t + = γ + γ r t + · · · + γ p r t − p + , ˜ σ t + = α + α σ t + · · · + α p σ t − p + , and ˜ σ t + = β + β σ t + · · · + β p σ t − p + . Thus, the estimator for κ t + is, ˜ κ t + = ˜ σ t + ˜ σ t + . We consider a simple first-order autoregressive model which is used to conduct a dataanalysis. The maximum volatility σ t , minimum volatility σ t , and the return of financialtime series { Z t } nt = satisfy a first-order autoregressive model.Note that, we assume the volatility of Z t takes values in a interval [ σ t , σ t ]. To calculateparameters σ t , σ t , we consider the daily observation dataset { X t } nt = which comes from timeseries Z . At time t , we need to use the data { X s } s ≤ t to estimate the minimum and maximumvolatility. We can observe the daily data at time s ≤ t , X s . We denote the volatility estimatorat time s as ˆ σ s , s ≤ t , andˆ σ t − j = (cid:114) (cid:80) Li = ( X t − L + i − j − r t − j ) L − , ≤ j ≤ K − , L and K are two given positive integers, andˆ r t − j = (cid:80) Li = X t − L + i − j L . The estimators of the minimum and maximum volatilities are given as follows:ˆ σ t = min ≤ j ≤ K − ˆ σ t − j , ˆ σ t = max ≤ j ≤ K − ˆ σ t − j , and the volatility uncertainty index isˆ κ t = ˆ σ t ˆ σ t = min ≤ j ≤ K − ˆ σ t − j max ≤ j ≤ K − ˆ σ t − j . When the samples { X t − i } L + K − i = are independent and identical distribution under sublinearexpectation E [ · ], based on Section 5 of Peng et al. (2020), and Theorem 24 of Jin and Peng(2016), we can obtain that ˆ σ t is an unbiased estimator for the maximum volatility σ t , and ˆ σ t is an unbiased estimator for the minimum volatility σ t . We conclude the above estimationresults as follows: Lemma 4.1.
For a given time t and a su ffi ciently small δ > , there exists positive integersK and L such that, E (cid:104)(cid:12)(cid:12)(cid:12) ˆ σ t − σ t (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) ˆ σ t − σ t (cid:12)(cid:12)(cid:12) + | ˆ r t − r t | (cid:105) < δ. Based on Lemma 4.1, for the given time t , we can use data { X t − s } L + K − s = to estimatethe maximum volatility σ t and volatility uncertainty index κ t , and use the data { X t − s } L − s = toestimate the mean of the return r t . The question is how to find ”good” integers K and L .Based on classical statistic results, we can reduce the variance of the sample variance byincreasing the numbers of K and L . In practice, the financial time series does not stabilize,and it does with volatility clustering characteristics. We comment that the data X s cancontains more information on parameters ( σ t , σ t , r t ), when s is nearly with time t . Thisis because the data generation mechanisms of X t will vary with the parameters ( σ t , σ t , r t )with time t . Consequently, we conclude that using small value of K and L can obtain betterestimators for ( σ t , σ t , r t ). For the given small values of positive integers K and L , K and L are called Localization Windows. For the statistical method under sublinear expectation E [ · ], see Jin and Peng (2016). This estimationmethod is called ”Max-Mean calculation” in Peng (2017)
14n this study, we identify the small values of K and L as Localization Windows, whichmeans that K and L can reflect the local changing of risky asset. Based on the LocalizationWindows K and L , we can obtain the least square estimators for ( α , α ), ( β , β ), and( γ , γ ). Furthermore, we find that the Localization Windows K and L can take values inthe following sets, K ∈ A = { , , · · · , } , L ∈ B = { , , · · · , } , which means that we only need to consider two to six weeks of daily data. By combiningdi ff erent values of K and L from sets A and B , we show that the G-VaR model given inSubsection 4.1 can achieve a better performance at least for the stock index dataset inSection 5.Note that one year and four years of daily data were used to estimate the parameters ofthe G-VaR model in Peng et al. (2020), which can describe the long time average loss ofthe risky asset, but not the local changing of risky assets. In this study, based on the au-toregressive model for volatility uncertainty and the Localization Windows, we can obtainthe one-period forecast of the parameters of the G-VaR model, and capture the volatilityclustering properties of risky assets, as well as the long time average loss of the risky asset.See Section 5 for more details. Peng et al. (2020) considered a new VaR model- G-VaR under sublinear expectation E [ · ]for the time series { Z t } nt = that follows a G-normal distribution N ( r t , [ σ t , σ t ]) depending onthree positive parameters ( r t , σ t , σ t ). Here, G-VaR is defined by the sublinear expectation E [1 Z t ≤ x ] but not a distribution function F ( x ), that is,G-VaR α ( Z t ) : = − inf { x ∈ R : E [1( Z t ≤ x )] > α } , (4.1)where 1( Z t ≤ x ) is an indicator function, when Z t ≤ x , 1( Z t ≤ x ) =
1, when Z t > x ,1( Z t ≤ x ) =
0. A worst case distribution ˆ F ( · ) was introduced,ˆ F ( x ) = E [1 Z t ≤ x ] = u (1 , x ) , where u (1 , x ) is the solution of partial di ff erential equation (2.2) at a time t =
1, with theinitial condition u (0 , x ) = ≤ x ). Based on the explicit solution of (2.2), ˆ F ( · ) has the15ollowing closed-form expressions,ˆ F ( x ) = σ t σ t + σ t Φ (cid:32) x − r t σ t (cid:33) I ( x ≤ + (cid:40) − σ t σ t + σ t Φ (cid:32) − x − r t σ t (cid:33)(cid:41) I ( x > , (4.2)where Φ ( · ) is the distribution function of the standard normal distribution. The G-VaRmodel in (4.1) is equal to G-VaR α ( Z t ) = − ˆ F − ( α ) . (4.3)Note that G-VaR model is used to measure the risk of asset { Z t } nt = , thus we set α ≤ . α ( Z t ) = − r t − ¯ σ t Φ − (cid:32) σ t + σ t σ t α (cid:33) . (4.4)By introducing an adaptive window W that satisfies the convergence results in the sample ,Peng et al. (2020) compares this G-VaR model with some distributions under GARCHmodels and presents the performance of G-VaR models.It should be noted that the parameters σ t and κ t denote the risk and volatility uncertaintyof asset Z t , which motivates us to develop autoregressive models for the maximum volatility σ t , minimum volatility σ t and obtain the volatility uncertainty index κ . Hence, we rewritethe representations of (4.2) and (4.4) as follows:ˆ F ( x ) = + κ t Φ (cid:32) x − r t σ t (cid:33) I ( x ≤ + (cid:40) − κ t + κ t Φ (cid:32) − x − r t σ t (cid:33)(cid:41) I ( x > , (4.5)and G-VaR α ( Z t ) = − r t − σ t Φ − (cid:32) + κ t α (cid:33) , (4.6)where κ t = σ t σ t . In the formula (4.6), we can distinguish the influences of σ t and κ t , wherethe role of maximum volatility σ t is same as the volatility in the VaR model with classicalnormal distribution. When κ t =
0, the volatility uncertainty index κ t can move the risk level α from α to α κ t = σ t = σ t , there is no volatility uncertainty, and G-VaR α ( Z t ) = The Condition 5.1 in Peng et al. (2020) assumed the time series { Z t } nt = has the property that for given awindow W and risk level α , there exists a window W such that I ( W ) = α , where 1 ≤ W ≤ W , and I ( W ) = lim n →∞ n − W n (cid:88) ¯ t = W I ( Z ¯ t + < − G-VaR W α, ¯ t ( Z ¯ t + )) . In the representation of I ( W ), we can see that we need to use partial samples to find an adaptive window W . r t − ¯ σ t Φ − ( α ), which is equal to the VaR under normal distribution N ( r t , σ t ). In particular,we can use G-VaR α ( Z t ) = − r t − ¯ σ t Φ − (0 . α ) to calculate the VaR of dataset { Z t } nt = with thelargest volatility uncertainty . Now, we use the first-order autoregressive models to study the stock index S&P500Index . We denote the 100 times log-returns daily data of S&P500 Index as Z , and oneobservation sequence { X t } nt = is from January 4, 2010 to July 17, 2020, with a total of n = Z t obeys the G-normal distribution N ( r t , [ σ t , σ t ]).The maximum volatility σ t , minimum volatility σ t , and return r t satisfy σ t = α + α σ t − ; (5.1) σ t = β + β σ t − ; (5.2) r t = γ + γ r t − . (5.3)For the given time s ≤ t , by Lemma 4.1, for the given Localization Windows K and L ,we assume the samples { X s − i } L + K − i = are independent under sublinear expectation E [ · ] , andwe can use data { X s − i } L + K − i = to obtain a sample estimator ˆ σ s for the maximum volatility σ s , estimator ˆ σ s for volatility uncertainty index σ s , and the data { X s − i } L − i = to obtain a sam-ple estimator ˆ r s for return r s . Based on the least square estimator, and sample estima-tors { ˆ σ s , ˆ σ s , ˆ r s } ts = t − N + , we can obtain the estimators ( ˆ α , ˆ α ), ( ˆ β , ˆ β ) and ( ˆ γ , ˆ γ ) for coe ffi -cients ( α , α ), ( β , β ) and ( γ , γ ), and one-period forecast parameters ( ˜ σ t + , ˜ σ t + , ˜ r t + ) for( σ t + , σ t + , r t + ).We first plot the sample estimators of the maximum and minimum volatilities ( ˆ σ t , ˆ σ t ),where time t is between July 2010 to July 2020. The dataset are downloaded from https: // finance.yahoo.com / lookup. Here, we assume the samples { X s − i } L + K − i = to satisfy the G-normal distribution N (0 , [ σ s , σ s ]). Note that,a G-normal distribution denotes an infinite family of distributions, thus for a given simple X s , which can begenerated from N (0 , [ σ i , σ i ]) , i = s , · · · , s + L + K −
500 1000 1500 2000 25000102030405060 From July 2010 to July 2020 Maximum varianceMinimum variance0 50 100 150 200 2500102030405060 From July 2019 to July 2020 Maximum varianceMinimum variance
Figure 1: Dynamic value of maximum and minimum volatilities for the log-return ofS&P500 Index.In the first picture of Figure 1, we can see that the dynamic value of the maximumvolatility is similar manner to the value of the minimum volatility. Hence, the maximumand minimum volatilities can reflect the same changing properties of the dataset X t . Inthe second picture of Figure 1, we show the details of maximum and minimum volatilitiesfrom July 2019 to July 2020. Furthermore, we can see that there is a little change inthe distance between the maximum and minimum volatilities, ˆ σ t − ˆ σ t with time t , whichmay be a useful property to study the volatility uncertainty. In the following, we take K = , L = , N = { ˆ σ s , ˆ σ s , ˆ r s } ts = t − N + .18
10 20 30 40 50 600204060 Maximum variance at t−1 M a x i m u m v a r i an c e a t t Autoregressive relation0 5 10 15 20 25 30 35 40 45 500204060 Minimum variance at t−1 M i n i m u m v a r i an c e a t t Autoregressive relation−2.5 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2−4−202 Return at t−1 R e t u r n a t t Autoregressive relation
Figure 2: Scatter points of ( ˆ σ t − , ˆ σ t ), ( ˆ σ t − , ˆ σ t ), and (ˆ r t − , ˆ r t ) for log-returns of S&P500Index.Based on the nonlinear least square estimator in Section 4, we can obtain the coe ffi -cients, ( ˆ α , ˆ α ) = (0 . , . β , ˆ β ) = (0 . , . γ , ˆ γ ) = (0 . , . σ t + , ˜ σ t + , ˜ r t + ) is given as follows:˜ σ t + = . + . σ t ;˜ σ t + = . + . σ t ;˜ r t + = . + . r t . In Figure 2, the red lines are the corresponding fitting linear functions for sequences( ˆ σ t − , ˆ σ t ), ( ˆ σ t − , ˆ σ t ), and (ˆ r t − , ˆ r t ). The correlation coe ffi cient and R is given as follows:corr( ˆ σ t − , ˆ σ t ) = . , corr( ˆ σ t − , ˆ σ t ) = . , corr(ˆ r t − , ˆ r t ) = . ( ˆ σ t − , ˆ σ t ) = . , R ( ˆ σ t − , ˆ σ t ) = . , R (ˆ r t − , ˆ r t ) = . , which show that the liner relations of sequences ( ˆ σ t − , ˆ σ t ), ( ˆ σ t − , ˆ σ t ), and (ˆ r t − , ˆ r t ) arereasonable. 19
500 1000 1500 2000 2500−20−15−10−50510 From July 2010 to July 2020 log−return−G−VaR0 50 100 150 200 250−20−15−10−50510 From July 2019 to July 2020 log−return−G−VaR
Figure 3: G-VaR value for log-return of S&P500 Index and the dynamic value of log-returns of S&P500 Index with risk level α = . − G-VaR α ( Z t + ) model, whereG-VaR α ( Z t + ) = − ˜ r t + − ˜ σ t + Φ − (cid:32) + ˜ κ t + α (cid:33) , and the time t + α ( Z t + ) can capture the local changes of the log-returns ofS&P500 Index.We consider Localization Windows 5 ≤ K , L ≤
15, which are used to estimate the max-imum volatility ˜ σ t + and the volatility uncertainty index ˜ κ t + . The excellent performanceof the G-VaR model in Figure 3 indicates that the worst case distribution can capture thelocal changing of a log-return of S&P500 Index. Therefore, we comment that the Local-ization Windows K and L are important for estimating the local maximum volatility andvolatility uncertainty index. Furthermore, the one-period forecast maximum volatility ˜ σ t + and volatility uncertainty index ˜ κ t + can be used to capture the quantile of the distributionof the log-returns of S&P500 Index. 20
500 1000 1500 2000 250000.020.040.060.08 From July 2010 to July 2020 The violations0 50 100 150 200 2500.0490.050.0510.0520.0530.054 From July 2019 to July 2020 The violations
Figure 4: The number of sample violations of the log-returns of S&P500 Index.The count numbers of the violations of the log-returns of S&P500 Index are given asfollows: m ( h ) = {− G-VaR α ( Z t + ) < X t + , − G-VaR α ( Z t + ) < X t + , t < t ≤ h } ; m ( h ) = {− G-VaR α ( Z t + ) < X t + , − G-VaR α ( Z t + ) > X t + , t < t ≤ h } ; m ( h ) = {− G-VaR α ( Z t + ) > X t + , − G-VaR α ( Z t + ) < X t + , t < t ≤ h } ; m ( h ) = {− G-VaR α ( Z t + ) > X t + , − G-VaR α ( Z t + ) > X t + , t < t ≤ h } , where t is the initial date which is used to forecast G-VaR α ( Z t + ), and {·} denotes thecount numbers which satisfies the violation conditions. Furthermore, we set m ( h ) = m ( h ) + m ( h ) , m ( h ) = m ( h ) + m ( h ) , and ˆ α ( h ) = m ( h ) m ( h ) + m ( h ) , − ˆ α ( h ) = m ( h ) m ( h ) + m ( h ) ; π ( h ) = m ( h ) m ( h ) + m ( h ) , π ( h ) = m ( h ) m ( h ) + m ( h ) ; π ( h ) = m ( h ) + m ( h ) m ( h ) + m ( h ) + m ( h ) + m ( h ) .
21n Figure 4, we show the dynamic value of ˆ α ( h ), where h is from July 2010 to July 2020in the first picture, and from July 2016 to July 2020 in the second picture. We can see thatˆ α ( h ) is stable from h = t +
250 with standard deviation 0 . t + t + . t +
251 to t + Z t + ) model, we use the test of alikelihood ratio for a Bernoulli trial and the test of a Christo ff erson independent to verifyit. Let ˆ α ( h ) be the sample violations rate, and denote the likelihood ratio test statistics, T ( h ) = m ( h ) ln ˆ α ( h ) α + m ( h ) ln 1 − ˆ α ( h )1 − α , and the Christo ff erson independent test statistics, T ( h ) = (cid:34) (1 − π ( h )) m ( h ) ( π ( h )) m ( h ) (1 − π ( h )) m ( h ) ( π ( h )) m ( h ) (1 − π ( h )) m ( h ) + m ( h ) ( π ( h )) m ( h ) + m ( h ) (cid:35) . Applying the well-known asymptotic χ (1) distribution, the p -value of the test is,LR huc = P (cid:16) χ (1) > T ( h ) (cid:17) , and independent test of violations point,LR hind = P (cid:16) χ (1) > T ( h ) (cid:17) . In the following, we conclude the testing results of S&P500 Index with α = . α = . h − t ˆ α ( h ) LR huc LR hind α ( h ), LR huc , LR hind , and 100VaR, respectively, where 100VaR is 100times VaR under G-VaR. It should be noted thatLR huc > . , LR hind > . , ff erson independent test statistics arethrough under the confidential level 95%.Similarly, we take K = , L = , N = α = .
01 in Table 2.
Figure 5: G-VaR value for log-return of S&P500 Index and the dynamic value of log-returns of S&P500 Index with risk level α = .
500 1000 1500 2000 250000.0050.010.0150.020.025 From July 2010 to July 2020 The violations0 50 100 150 200 2500.01260.01280.0130.01320.01340.0136 From July 2019 to July 2020 The violations
Figure 6: The number of sample violations of the log-returns of S&P500 Index.Table 2: Testing results of S&P500 Index with α = . h − t ˆ α ( h ) LR huc LR hind As Kuester et al. (2006) pointed out that the best VaR predictions for the benchmarkare obtained by AR-GARCH filtered modeling such as the recommended AR-GARCH-Skewed-t or AR-GARCH-Skewed-t-EVT models. In Peng et al. (2020), the G-VaR pre-dictor is compared with these two models and a more traditional AR-GARCH-Normalpredictor. Hence, we use the dataset S&P500 Index from January 3, 2000 to February 7,2018 in Peng et al. (2020), to compare the models developed in this study with the above24our models. We denote G-VaR ∗ as the VaR model developed in Peng et al. (2020). Partialresults of Table 3 were downloaded from Table 7 of Peng et al. (2020) with historical datawindow W = K , L , N ) = (5 , , α = .
05 and ( K , L , N ) = (6 , , α = . α ˆ α ( h ) LR huc LR hind ∗ : 200101-201802 0.05 0.050 ∗ , GARCH(1,1)-N, GARCH(1,1)-St, and GARCH(1,1)-St-EVT. We denote the val-ues of the likelihood ratio test LR huc and the Christo ff erson independent test LR hind as bold-face type which are through the tests under the confidential level 95%. We can see that theG-VaR ∗ model in Peng et al. (2020) is the best VaR model among the above VaR modelsbased on the value LR huc . Hence, G-VaR ∗ model can capture the long time average lossof S&P500 Index. In fact, It should be noted that LR huc can test the accuracy of the VaRmodel and LR hind can test the independence of the violation points. Combining the values ofLR huc and LR hind , we can see that models G-VaR and GARCH(1,1)-St-EVT are the best for α = .
05, and G-VaR is the best for α = .
01. In particular, we also use the G-VaR modelto predict VaR for NASDAQ Composite Index and CSI300, and we have a performance25imilar to that of G-VaR model as in Table 3. Hence, we can see that the autoregressivemodels developed in this study are powerful tools to investigate the excellent performanceof the G-VaR model.
To capture the inherent uncertainty of time series, the parameters of maximum volatil-ity, minimum volatility, and the volatility uncertainty index are introduced, and then au-toregressive models are developed, indicating the maximum and minimum volatilities toforecast the one-period value, where the maximum volatility and the ratio of minimumvolatility to maximum volatility represents the risk and uncertainty of the time series. Inthe autoregressive models, we assume the maximum and minimum volatilities have the p -th autoregressive e ff ect. Hence, we can obtain the one-period forecast maximum volatility,minimum volatility and volatility uncertainty index.Based on the assumption that the time series satisfies a G-normal distribution under sub-linear expectation, we can use a worst-case distribution to capture the distribution propertyof the time series, where the G-normal distribution denotes an infinite family of distribu-tions. In Peng et al. (2020), the empirical analysis is given based on an adaptive window W , which can capture the long time loss of the time series, but can not follow the lo-cal changing of the time series. Di ff ering from Peng et al. (2020), we develop a conceptof Localization Windows to estimate the local sample variance. Then, combining the au-toregressive models, we can obtain a one-period forecast maximum volatility, minimumvolatility and volatility uncertainty index. In Section 5, we apply this new mechanism topredict the VaR of S&P500 Index by the G-VaR α ( · ) model, which shows the rationality ofthe autoregressive models and the Localization Windows.As shown in this study, we find that the volatility uncertainty index κ t = σ t σ t is a powerfultool to represent the inherent uncertainty of the time series. Thus, when studying the relateddistribution property of time series, the index κ t should be considered. Furthermore, wecomment that it is di ffi cult to distinguish the mean uncertainty and volatility uncertaintyof time series. Hence, it is interesting to develop a mean uncertainty index, and a relatedautoregressive model. However, we consider the volatility uncertainty but not the meanuncertainty in this study. Future work on the relationship between the volatility uncertainty26nd the mean uncertainty should be considered and investigated. References
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