A note on calculating expected shortfall for discrete time stochastic volatility models

Grabchak, Michael; Christou, Eliana Article A note on calculating expected shortfall for discrete time stochastic volatility models Financial Innovation Provided in Cooperation with: Springer Nature Suggested Citation: Grabchak, Michael; Christou, Eliana (2021) : A note on calculating expected shortfall for discrete time stochastic volatility models, Financial Innovation, ISSN 2199-4730, Springer, Heidelberg, Vol. 7, Iss. 1, pp. 1-16, https://doi.org/10.1186/s40854-021-00254-0 This Version is available at: https://hdl.handle.net/10419/237274 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ A note oncalculating expected shortfall fordiscrete time stochastic volatility models Michael Grabchak and Eliana Christou* Introduction Empirical studies consistently show that financial returns do not have a constant volatility and instead exhibit volatility clustering. This clustering is often modeled using GARCH or one of its variants. Perhaps the most prominent alternative to GARCH is the class of discrete time stochastic volatility (SV) models. These models are very flexible and capture additional stylized features of financial returns including skewness, excess kurtosis, and leverage effects (Cont and Tankov 2004). SV models are often credited to Taylor (1986) although they have a long prehistory. See Shephard (2005) or Taylor (1994) for a thorough review. A lot of research effort has focused on option pricing for SV models, much less attention has been paid to the problem of calculating risk. However, as we will see, this problem is not trivial even in the case when all of the parameters are explicitly known. In this paper, we introduce a Monte Carlo method for calculating expected shortfall (ES) for several important classes of SV models. ES is one of the best known and most commonly used measures of financial risk. It is, arguably, second in popularity only to Value-at-Risk (VaR). However, unlike VaR, ES is a coherent risk measure (Artzner etal. 1999) and it has been chosen to replace VaR as the measure determining a bank’s capital requirements in the Basel III regulatory framework (Basel Committee on Banking Supervision 2013). For more information on VaR, ES, and related risk measures, see e.g.McNeil etal. (2015) and the references therein. A well-known survey on the estimation of ES is given in Nadarajah etal. (2014). Among a long list of methodologies, that paper discusses the Abstract In this paper we consider the problem of estimating expected shortfall (ES) for discrete time stochastic volatility (SV) models. Specifically, we develop Monte Carlo methods to evaluate ES for a variety of commonly used SV models. This includes both models where the innovations are independent of the volatility and where there is dependence. This dependence aims to capture the well-known leverage effect. The performance of our Monte Carlo methods is analyzed through simulations and empirical analyses of four major US indices. Keywords: Expected shortfall, Stochastic volatility, Value-at-risk Open Access © The Author(s), 2021. 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RESEARCH Grabchakand Christou Financ Innov (2021) 7:43 https://doi.org/10.1186/s40854-021-00254-0 Financial Innovation *Correspondence: [email protected] Department of Mathematics and Statistics, University of North Carolina at Charlotte, 9201 University City Blvd, Charlotte, NC 28223, USA Page 2 of 16 Grabchakand Christou Financ Innov (2021) 7:43 estimation of ES under a GARCH model. However, we have not seen a discussion of ES estimation under an SV model in the literature. The difficulty in calculating ES for SV models lies in the fact that one needs to work with the product of two random variables and, even in the case where both terms in the product have simple distributions, the distribution of the product may be quite complicated. This is in contrast with GARCH models, where the problem of evaluating ES, essentially, reduces to that of calculating ES for the distribution of the innovations. The rest of this paper is organized as follows. In Sect.2 we formally define the SV model and give a simple Monte Carlo method for evaluating ES in this case. In Sect.3 we give a more sophisticated Monte Carlo method in the commonly used case where the innovations for the returns and for the volatility are independent. In Sect.4 we give a similar method for an important case with dependence, which aims to model leverage. In Sect.5 we illustrate our methodology on four major US indices. Some conclusions are given in Sect.6. Stochastic volatility Discrete time stochastic volatility models commonly assume that the financial (log) return, at time t, is given by where is the log variance. Here σ>0 , |φ|<1 , and µ∈R are parameters, and {ǫt} and {ηt} are sequences of independent and identically distributed (iid) random variables representing the innovations for rt and ht , respectively. We do not, in general, assume that for a given t, ǫt and ηt are independent of each other. Note that the log variance is modeled by an AR(1) process. The assumption that |φ|<1 ensures that this process is weakly stationary, see Ruppert and Matteson (2015). Under general conditions on the distributions of the innovations, this model can be seen as a discretization of a continuous time SV model where the log variance is modeled by a process of Ornstein–Uhlenbeck type, see Taylor (1994) or Barndorff-Nielsen and Shephard (2001). We are interested in evaluating the ES for this model. We begin by establishing some notation. Let Ft−1 denote the information set available at time t−1 . For simplicity, we sometimes write Pt−1 to denote the conditional probability Pt−1(·)=P(·|Ft−1) and Et−1 to denote the conditional expectation Et−1(·)=E(·|Ft−1) . For τ∈(0, 1) , the τ th VaR at time t, denoted by VaRτ(t) , is the smallest number for which Pt−1{rt<−VaRτ(t)}≤τ . Note that −VaRτ(t) is the τ th conditional (given Ft−1 ) quantile of rt . For this reason, we sometimes write Qτ(rt|Ft−1) for −VaRτ(t) . The τ th ES at time t, denoted by ESτ(t) , is defined by (1) rt=eh t /2ǫt, (2) ht=µ+φ(ht−1−µ) +ση t ES τ(t) = 1 ττ 0 VaRs(t)ds , Page 3 of 16 Grabchakand Christou Financ Innov (2021) 7:43 when the integral exists, and is undefined otherwise. The parameter τ is typically chosen to be a small number such as 0.01, 0.025, or 0.05. Throughout, we assume 1. that the distribution of rt is continuous, and 2. that it satisfies The second assumption ensures that ESτ(t) is well defined, while the first allows us to use the more explicit formula Here and throughout, we write 1{·} to denote the indicator function. Using the fact that the innovations are independent over time, together with basic properties of quantiles and expectations, we can write where a= Q τ e σY/2 Z  is the τ th (unconditional) quantile of the random variable eσY/2Z , and the joint distribution of (Y,Z) is the same as the joint distribution of (ηt,ǫt) . The difficulty in evaluating M is that we must work with the distribution of X=eσY/2Z , which can be complicated even when the distributions of Y and Z are fairly simple. Little is known about the distribution of X even in the case where Y and Z are both standard normal random variables, see Yang (2008) and the references therein. For this reason, we develop Monte Carlo methods to approximate M(τ,σ) . We begin by approximating a=Qτ(eσY/2Z) . Toward this end, fix some large integer N1 and simulate an iid sequence of bivariate random variables { (Y i ,Z i ) }N 1 i=1 from the joint distribution of (ηt,ǫt) . Next, for i=1, 2, ...,N1 , set Xi=eσY i /2Zi . Now sort these from smallest to largest to get X(1)≤X(2)≤ ··· ≤ X(N1) . Finally, approximate a=Qτ(eσY/2Z) by where ⌊·⌋ is the floor function. One can also use a smooth approximation using kernel estimators, see e.g. Sheather and Marron (1990). However, we did not find much of an improvement when using these. Next, fix another large integer N2 and simulate a new iid sequence { (Y i ,Z i ) }N 2 i=1 from the joint distribution of (ηt,ǫt) and approximate M(τ,σ) by We note that, in principle one can use the same dataset to evaluate a and M(τ,σ) although for smaller sample sizes this may create bias. Either way, the difficulty with this (3) Et−1(|rt|)<∞. ES τ(t) = Et − 1[ − rt |− rt>VaRτ(t)] =− 1 τ Et − 1[rt1 { rt< − VaRτ(t) } ] . (4) ESτ(t)=−e{µ(1−φ)+φh t − 1 }/2M(τ,σ), (5) M (τ,σ) =1 τ E  eσY/2Z1 { eσY/2Z<a }, (6)  a= X (⌊τN1⌋), (7)  M 1(τ,σ) =1 N2τ N 2  i=1 eσYi/2Zi1{eσYi/2Zi≤  a} . Page 4 of 16 Grabchakand Christou Financ Innov (2021) 7:43 approach is that approximately (1−τ)100% of the simulated values will not satisfy the condition in the indicator function in (7) and will thus be thrown out. As such, very few values will actually be used in the sum. For this reason, we may need N2 to be an extremely large number to get a reasonable approximation. One could try to implement an importance sampling or related modification, but the fact that we are working with the product of two random variables, makes it difficult to use such an approach. Instead, we use the specific structure of this problem to implement an approach that works better in several important situations. Independent case It is commonly assumed that the sequences {ǫt} and {ηt} are mutually independent. For simplicity and to ensure that the distribution of the returns is continuous, we assume that the distributions of ǫt and ηt are both continuous, having probability density functions (pdfs) fǫ and fη , respectively. In order to guarantee that (3) holds, we must assume that By a conditioning argument, we have where This can be used to develop a Monte Carlo method for approximating M(τ,σ) as follows. Fix some large integer N1 and simulate two mutually independent sequences of iid random variables { Y i}N 1 i=1 and { Z i}N 1 i=1 , where Yi∼fη and Zi∼fǫ . Use these to approximate a=Qτ(eσY/2Z) by  a as in (6). Now choose another large integer N2 and simulate Y1,...,YN2 iid from fη . We can then approximate M(τ,σ) by We now give explicit formulas for H in several important situations. Throughout we assume that a≤0 , which holds for all reasonable choices of τ . Perhaps the most common assumptions are that fǫ is the pdf of a standard normal distribution or a t-distribution. In the standard normal case we have and in the case of a t-distribution with ν>1 degrees of freedom we have (8) E(|ǫt|)<∞and E(e0.5ση t )<∞. M (τ,σ) = 1 τE  eσY/2Z1{eσY/2Z<a}  = 1 τE  eσY/2E  Z1{eσY/2Z<a}|Y  = 1 τ E  eσY/2H(Y,a,σ)  , H (y,a,σ) = E  Z1 { eσy/2Z<a }= ae −σy/2 −∞ xfǫ(x)dx .  M 2(τ,σ) = 1 N2τ N 2  i=1 eσYi/2H(Yi,  a,σ) . (9) H (y,a,σ) =−1 √2π e−1 2a2e−σ y Page 5 of 16 Grabchakand Christou Financ Innov (2021) 7:43 In the above, we need ν>1 as otherwise (8) will not hold. In practice, it is often assumed that the distributions of returns are skewed. To capture this, skewed modifications of normal and t-distributions are often used. While there are a number of ways to introduce such modifications, we follow the approach of Fernandez and Steel (1998). In general, this approach can be described as follows. If f1 is the pdf of a distribution that is unimodal and symmetric around zero, then for γ>0 is a skewed modification of f1 . The parameter γ determines the skew of the distribution. When γ=1 the distribution is symmetric, when γ<1 it has a negative skewness, and when γ>1 it has a positive skewness. Using change of variables, it is straightforward to show that, if H1 corresponds to f1 , then corresponds to fγ . We can easily apply this to get explicit formulas for H in the cases of skewed modifications of normal and t-distributions. We now give a small simulation study to compare the performance of  M1(τ,σ) and  M2(τ,σ) . For these simulations we assume that ηt has a standard normal distribution, while for ǫt we consider two distributions: standard normal and student-t. The values of the parameters, σ and (in the case of the student-t distribution) ν , were calibrated according to the daily returns from January 2014 to December 2019 of the S&P 500 Index. This was done using the stochvol package for the statistical software R, see Kastner (2016). We also performed similar simulations where the parameters were calibrated to the daily returns over the same period from the Russell 2000 Index, the Dow Jones Industrial Average, and the NASDAQ Composite Index. However, the results were similar and are not presented in the interest of space. Since our goal is to compare the two methods for evaluating M(τ,σ) , we do not want issues with calculating a to interfere with the comparison. For this reason we choose N1=3∗107 to be a large value and use the same value of  a for all simulations with the same distribution. For N2 we consider a range of values from 100 to 5000 in increments of 100. For each value of N2 , we estimate  M1(τ,σ) and  M2(τ,σ) 1000 times and report the standard deviations and the means over these trials. For τ=0.01 , the results are presented in Fig.1. From the plots, we can see that the second method has significantly less variance and that the mean gets close to the true value much quicker. We also repeated the procedure for τ=0.025 and 0.05, but the results were similar and are thus omitted. We note that we cannot allow ηt to have a t-distribution, as this would violate assumption (8). Model withleverage In the literature of financial returns, the leverage effect is the empirically observed phenomenon that volatility tends to be negatively correlated with returns, see e.g. Cont and Tankov (2004). In the important case where ηt and ǫt are jointly Gaussian, leverage is (10) H (y,a,σ) = − √ νŴ{(ν +1)/2} √πŴ(ν/ 2 )(ν − 1 ) (1 + a2e−σy/ν)−(ν−1)/2 . fγ(x)= 2 γ + 1 γ [f1(x/γ )1{0≤x<∞} + f1(x/γ )1{−∞ <x<0} ] H γ(y,a,σ) =2 γ 3 +γ H1(y,γa, σ) Page 6 of 16 Grabchakand Christou Financ Innov (2021) 7:43 often modeled by assuming that the joint distribution of the random vector (ηt,ǫt) follows a bivariate normal distribution N(0, �) where the covariance matrix is for some ρ∈(−1, 1) , see Omori etal. (2007). When ρ<0 , the volatility is negatively correlated with the return, which captures the leverage effect. From properties of multivariate normal distributions, it follows that, in this case, where W1,W2 are iid N(0,1) random variables and d = denotes equality in distribution. There does not seem to be a standard way to model leverage in the non-Gaussian case. However, one approach is suggested, in a continuous time setting, by Eq.(8) in (11) �= 1ρ ρ1 , ǫ t d =ρW1+  1−ρ2W 2 ηt d =W1, 010002000300040005000 0.00.5 1.01.5 2.02.5 3.0 Standard Deviations For Two Methods sample size standard deviation 010002000300040005000 −2.96−2.92 −2.88−2.84 Means For Two Methods sample size mean a 010002000300040005000 0.00.5 1.01.5 2.02.5 3.0 Standard Deviations For Two Methods sample size standard deviation 010002000300040005000 −3.15−3.10 −3.05−3.00 −2.95 Means For Two Methods sample size mean b Fig. 1 Results for τ=0.01 . Results for  M1(τ,σ) are in dashed (red) line and the ones for  M2(τ,σ) are in solid (black) line. The dotted (blue) line corresponds to an approximation of  M2(τ,σ) based on a sample of size N2 = 108 . a Results for ǫ ∼ N(0, 1) with σ=0.3430 . Here, H is evaluated using (9). b Results for ǫ ∼ t37.9762 with σ=0.3228 . Here, H is evaluated using (10) Page 7 of 16 Grabchakand Christou Financ Innov (2021) 7:43 Barndorff-Nielsen and Shephard (2001). The idea is to add a constant times the innovation of the log volatility to the model for rt . A variant of this idea, which is consistent with how the Gaussian case is treated, is to consider the model where ht is as in (2) and {δt} and {ηt} are mutually independent sequences of iid random variables. The new parameter ρ∈(−1, 1) determines the dependence between the return and the volatility. When ρ=0 , the model reduces to the independent case. Note that this is equivalent to taking in (1). For simplicity we assume that the distribution of δ1 has pdf fδ . We now give an approach for evaluating M(τ,σ) . In this case, we can write Zd =ρW1+ 1 −ρ 2 W2 and Yd =W1 , where W1,W2 are independent random variables with W1∼fη and W2∼fδ . It follows that where and Here is the cumulative distribution function (cdf) of the distribution of δt and r t = eht/2  1 − ρ2δt + ρηt , ǫt=1−ρ 2 δt+ρηt E eσY/2Z1{eσY/2Z<a}  =EeσW1/2(ρW1+1−ρ2W2)1{eσW1/2(ρW1+1−ρ2W2)<a}  =ρEeσW1/2W11{eσW1/2(ρW1+1−ρ2W2)<a} +1−ρ2EeσW1/2W21{eσW1/2(ρW1+1−ρ2W2)<a} =ρEeσW1/2W1E1{eσW1/2(ρW1+1−ρ2W2)<a}|W1 +1−ρ2EeσW1/2EW21{eσW1/2(ρW1+1−ρ2W2)<a}|W1  =ρEeσW1/2W1H1(W1,a,σ,ρ) +  1−ρ2E  eσW1/2H2(W1,a,σ,ρ)  , H 1(y,a,σ,ρ) =Fδ  e−σy/2a−ρy  1 − ρ2  H 2(y,a,σ,ρ) =Gδ  e−σy/2a−ρy  1 − ρ2 . F δ(b) =b −∞ fδ(x)dx,b ∈R Page 8 of 16 Grabchakand Christou Financ Innov (2021) 7:43 In the case where the distribution of δt is standard normal and b≤0 , Gδ(b)=−ϕ(b) where ϕ(x)=e− x 2 /2 /√2π is the pdf of the standard normal distribution. The above suggests the following Monte Carlo method. First, we approximate a by  a as in (6). Next, choose a large integer N2 and simulate W1,...,WN2 iid from fη . We can then approximate M(τ,σ) by We again perform a small simulation study to compare the performance of  M1(τ,σ) and  M2(τ,σ) . For these simulations we assume that δt and ηt are independent standard normal random variables, or equivalently that (ηt,ǫt)∼N(0, �) , where  is given by (11). The values of σ and ρ are calibrated according to the daily returns from January 2014 to December 2019 of the S&P 500 Index. This was again done using the stochvol package for R. For simulations we again took N1=3∗107 and N2 from 100 to 5000 in increments of 100 and repeated each simulation over 1000 iterations. The results for τ=0.01 are given in Fig.2. We again see that the second method has less variance and that the mean gets close to the true value quicker. However, the differences are not as strong in this case. As before, we also considered the case where the parameters were calibrated to the other three indices and when τ=0.025 and 0.05. Those results were similar and are not presented here in the interest of space. Data analysis In this section we perform a data analysis, where we estimate ES using SV models for four major US indices: the S&P 500 Index, the Russell 2000 Index, the Dow Jones Industrial Average, and the NASDAQ Composite Index. In all cases we use daily returns from G δ(b) =b −∞ xfδ(x)dx,b ∈ R .  M 2(τ,σ) =1 N2τ N 2  i=1 eσWi/2  ρWiH1(Wi,  a,σ,ρ) +  1−ρ2H2(Wi,  a,σ,ρ) . 01000 2000 3000 4000 5000 1234 Standard Deviations For Two Methods sample size standard deviation 01000 2000 3000 4000 5000 −4.25−4.20 −4.15−4.10 −4.05 Means For Two Methods sample size mean Fig. 2 Results for τ = 0.01 . Results for  M1(τ , σ) are in dashed (red) line and the ones for  M2(τ , σ) are in solid (black) line. Here the calibrated parameter values are σ = 0.4011 and ρ =− 0.7596 . The dotted (blue) line corresponds to an approximation of  M2(τ , σ) based on a sample of size N2 =10 8 Page 15 of 16 Grabchakand Christou Financ Innov (2021) 7:43 Abbreviations SV: Stochastic volatility; ES: Expected shortfall; VaR: Value-at-risk; iid: Independent and identically distributed. Acknowledgements The authors would like to thank the anonymous referees, whose comments lead to improvements in the presentation of this paper. Authors’ contributions All the authors contributed equally to this work. All authors read and approved the final manuscript. Funding This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors. Availability of data and materials The datasets analyzed in this study are available from https:// finan ce. yahoo. com. Declaration Competing interests The authors declare that they have no competing interests. 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