On Volatility Swaps for Stock Market Forecast: Application Example CAC 40 French Index

Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2014, Article ID 854578, 6 pages http://dx.doi.org/10.1155/2014/854578

Research Article On Volatility Swaps for Stock Market Forecast: Application Example CAC 40 French Index

Halim Zeghdoudi,1,2 Abdellah Lallouche,3 and Mohamed Riad Remita1

1 LaPSLaboratory,Badji-MokhtarUniversity,BP12,23000Annaba,Algeria 2 Department of Computing Mathematics and Physics, Waterford Institute of Technology, Waterford, Ireland 3 Universite´ 20 Aout, 1955 Skikda, Algeria

Correspondence should be addressed to Halim Zeghdoudi; [email protected]

Received 3 August 2014; Revised 21 September 2014; Accepted 29 September 2014; Published 9 November 2014

Academic Editor: Chin-Shang Li

Copyright © 2014 Halim Zeghdoudi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper focuses on the pricing of variance and volatility swaps under Heston model (1993). To this end, we apply this modelto the empirical financial data: CAC 40 French Index. More precisely, we make an application example for stock market forecast: CAC 40 French Index to price swap on the volatility using GARCH(1,1) model.

1. Introduction a price index. The underlying is usually a foreign exchange rate (very liquid market) but could be as well a single name Black and Scholes’ model [1]isoneofthemostsignificant equity or index. However, the variance swap is reliable in models discovered in finance in XXe century for valuing the index market because it can be replicated with a linear liquid European style vanilla option. Black-Scholes model combination of options and a dynamic position in futures. assumes that the volatility is constant but this assumption is Also, volatility swaps are not used only in finance and not always true. This model is not good for derivatives prices businesses but in energy markets and industry too. founded in finance and businesses market (see [2]). The variance swap contract contains two legs: fixed leg “The volatility of asset prices is an indispensable input in (variance strike) and floating leg (realized variance). There both pricing options and in risk management. Through the are several works which studied the variance swap portfolio introduction of volatility derivatives, volatility is now, in effect, theoryandoptimalportfolioofvarianceswapsbasedona atradablemarketinstrument”Broadie and Jain [3]. variance Gamma correlated (VGC) model (see Cao and Guo Volatility is one of the principal parameters employed to [5]). describe and measure the fluctuations of asset prices. It plays The goal of this paper is the valuation and hedging of a crucial role in the modern financial analysis concerning volatility swaps within the frame of a GARCH(1,1) stochastic risk management, option valuation, and asset allocation. volatility model under Heston model [6]. The Heston asset There are different types of volatilities: implied volatility, local process has a variance that follows a Cox et al. [7]process. volatility, and stochastic volatility (see Baili [4]). Also, we make an application by using CAC 40 French Index. To this end, the new financial products are variance Thestructureofthepaperisasfollows.Section2 andvolatilityswaps,whichplayadecisiveroleinvolatility considers representing the volatility swap and the variance hedging and speculation. Investment banks, currencies, stock swap. Section 3 describes the volatility swaps for Heston 2 indexes, finance, and businesses markets are useful for vari- model, gives explicit expression of 𝜎𝑡 , and discusses the ance and volatility swaps. relationship between GARCH and volatility swaps. Finally, Volatility swaps allow investors to trade and to control we make an application example for stock market forecast: the volatility of an asset directly. Moreover, they would trade CAC 40 French Index using GARCH/ARCH models. 2 Journal of Probability and Statistics

2. Volatility Swaps 3. Volatility Swaps for Heston Model

In this section we give some definitions and notations of swap, 3.1. Stochastic Volatility Model. Let (Ω, 𝐹,𝑡 𝐹 , P) be probability stock’s volatility, stock’s volatility swap, and variance swap. space with filtration 𝐹𝑡, 𝑡∈[0;𝑇]. We consider the risk- neutral Heston stochastic volatility model for the price 𝑆𝑡 and Definition 1. Swapswereintroducedinthe1980sandthere variance follows the following model: is an agreement between two parties to exchange cash flows at one or several future dates as defined by Bruce [8]. In 𝑑𝑆𝑡 =𝑟𝑡𝑆𝑡𝑑𝑡𝑡 +𝜎 𝑆𝑡𝑑𝑤1 (𝑡) , (𝑆1) this contract one party agrees to pay a fixed amount to a 2 2 2 counterpart which in turn honors the agreement by paying a 𝑑𝜎𝑡 =𝑘(𝜃−𝜎𝑡 )𝑑𝑡+𝜉𝜎𝑡𝑑𝑤2 (𝑡) , floating amount, which depends on the level of some specific underlying. By entering a swap, a market participant can where 𝑟𝑡 is deterministic interest rate, 𝜎0 >0and 𝜃>0are therefore exchange the exposure from the varying underlying short and long volatility, 𝑘>0is a reversion speed, 𝜉>0 by paying a fixed amount at certain future time points. is a volatility of volatility parameter, and 𝑤1(𝑡) and 𝑤2(𝑡) are independent standard Brownian motions. Definition 2. Astock’svolatilityisthesimplestmeasureofits We can rewrite the system (𝑆1) as follows: risk less or uncertainty. Formally, the volatility 𝜎𝑅(𝑆) is the annualized standard deviation of the stock’s returns during 𝑑𝑆𝑡 =𝑟𝑡𝑆𝑡𝑑𝑡𝑡 +𝜎 𝑆𝑡𝑑𝑤1 (𝑡) the period of interest, where the subscript 𝑅 denotes the 𝑑𝜎2 =𝑘(𝜃2 −𝜎2)𝑑𝑡+𝜌𝜉𝜎𝑑𝑤 (𝑡) 𝑆2 observed or “realized” volatility for the stock 𝑆. 𝑡 𝑡 𝑡 1 ( ) √ Definition 3 (see [9]). A stock volatility swap is a forward +𝜉 1−𝜌𝜎𝑡𝑑𝑤 (𝑡) , contract on the annualized volatility. Its payoff at expiration 𝑤(𝑡) is equal to where is standard Brownian motion which is independent of 𝑤1(𝑡) and the indicator economic 𝑋.Let 𝑁(𝜎 (𝑆) −𝐾 ), (𝑑𝑤 (𝑡), 𝑑𝑤 (𝑡)) = 𝜌𝑑𝑡 𝑅 vol (1) cov 1 2 ,andwecantransformthesystem (𝑆2) to (𝑆1) if we replace 𝜌𝑑𝑤1(𝑡) + √1−𝜌𝑑𝑤(𝑡)by 𝑑𝑤2(𝑡). 𝑇 𝜎 (𝑆) := √(1/𝑇) ∫ 𝜎2𝑑𝑠, 𝜎 where 𝑅 0 𝑠 𝑡 is a stochastic stock 𝐾 2 volatility, vol istheannualizedvolatilitydeliveryprice,and 3.2. Explicit Expression and Properties of 𝜎𝑡 . In this section we 𝑁 is the notional amount of the swap in Euro annualized reformulated the results obtained in [12], which are needed volatility point. for study of variance and volatility swaps, and price of pseudovariance, pseudovolatility, and the problems proposed Definition 4 (see [9]). A variance swap is a forward contract by He and Wang [13] for financial markets with deterministic on annualized variance, the square of the realized volatility. volatility as a function of time. This approach was first applied Its payoff at expiration is equal to to the study of stochastic stability of Cox-Ingersoll-Ross 2 processinSwishchukandKalemanova[14]. The Heston asset 𝑁(𝜎𝑅 (𝑆) −𝐾 ), (2) 2 var process has a variance 𝜎𝑡 that follows Cox et al. [7]process, (𝑆1) 𝐾 𝑁 describedbythesecondequationin .Ifthevolatility where var isthedeliverypriceforvarianceand is 𝜎𝑡 follows Ornstein-Uhlenbeck process (see, e.g., Oksendal thenotionalamountoftheswapinEurosperannualized 𝜎2 volatility point squared. [15]), then Ito’s lemma shows that the variance 𝑡 follows the process described exactly by the second equation in (𝑆1). 2 We start to define the following process and function: Notation 1. We note that 𝜎𝑅(𝑆) = 𝑉. V := 𝑒𝑘𝑡 (𝜎2 −𝜃2), Using the Brockhaus and Long [10]andJavaheri[11] 𝑡 𝑡 approximation which is used in the second order Taylor 𝑡 √𝑥 −2 𝑘Φ(𝑠) 2 2 2 2𝑘Φ(𝑠) −1 formula for ,wehave Φ (𝑡) := 𝜉 ∫ 𝑒 (𝜎0 −𝜃 + 𝑤̃2 (𝑠) +𝜃 𝑒 ) 𝑑𝑠. 0 (𝑉) 𝐸(√𝑉) ≈ √𝐸 (𝑉) − Var , (5) 8𝐸3/2 (𝑉) (3) −1 Definition 5. We define 𝐵(𝑡) := 𝑤̃2(Φ ),where𝑤̃2 is an F𝑡- (𝑉)/8𝐸3/2(𝑉) 𝑡 where Var is the convexity adjustment. Thus, to F̃ := F −1 measurable one-dimensional Wiener process, 𝑡 Φ𝑡 , calculate volatility swaps we need both 𝐸(𝑉) and Var(𝑉). −1 and 𝑡∧𝑠:=min(𝑡, 𝑠),whereΦ𝑡 is an inverse function of Φ𝑡. The realized discrete sampled variance is defined as 𝐵(𝑡) follows: The properties of are as follows: 𝑛 𝑆 F̃ 𝐸(𝐵(𝑡)) =0 𝑛 2 𝑡𝑖 (a) 𝑡-martingale and ; 𝑉𝑛 (𝑆) := ∑ln ( ), 𝑉:= lim 𝑉𝑛 (𝑆) , (𝑛−1) 𝑇 𝑆 𝑛→∞ 𝐸(𝐵2(𝑡)) = 𝜉2(((𝑒𝑘𝑡 −1)/𝑘)(𝜎2 −𝜃2)+((𝑒2𝑘𝑡 −1)/2𝑘)𝜃2) 𝑖=1 𝑡𝑖−1 (b) 0 ; (4) 2 𝑘(𝑡∧𝑠) 2 2 2𝑘(𝑡∧𝑠) (c) 𝐸(𝐵(𝑠)𝐵(𝑡)) =𝜉 (((𝑒 − 1)/𝑘)(𝜎0 −𝜃 ) + ((𝑒 − 2 where 𝑇 is the maturity (years or days). 1)/2𝑘)𝜃 ). Journal of Probability and Statistics 3

Lemma 6. and taking (13) and variance formula we find

(a) Consider the following: Var (𝑉) 2 −𝑘𝑡 2 2 2 2 𝑘(𝑡∧𝑠) 𝜎𝑡 =𝑒 (𝜎0 −𝜃 +𝐵(𝑡)) +𝜃 , (6) 𝜉 𝑇 𝑒 −1 = ∬ [𝑒−𝑘(𝑡+𝑠) ( (𝜎2 −𝜃2) 𝑇2 𝑘 0 (b) 0 (14) 2𝑘(𝑡∧𝑠) 2 −𝑘𝑡 2 2 2 𝑒 −1 2 𝐸(𝜎𝑡 )=𝑒 (𝜎0 −𝜃 )+𝜃 , (7) + 𝜃 )] 𝑑𝑡𝑑𝑠; 2𝑘 (c) after calculations we obtain 𝑒𝑘(𝑡∧𝑠) −1 𝑒2𝑘(𝑡∧𝑠) −1 𝐸(𝜎2𝜎2)=𝜉2𝑒−𝑘(𝑡+𝑠) ( (𝜎2 −𝜃2)+ 𝜃2) 𝜉2𝑒−2𝑘𝑇 𝑠 𝑡 𝑘 0 2𝑘 (𝑉) = [(2𝑒2𝑘𝑇 −4𝑘𝑇𝑒𝑘𝑇 −2)(𝜎2 −𝜃2) Var 2𝑘3𝑇2 0 (15) 2 −𝑘(𝑡+𝑠) 2 2 −𝑘𝑡 2 2 2 2𝑘𝑇 2𝑘𝑇 𝑘𝑇 2 +𝑒 (𝜎0 −𝜃 ) +𝑒 (𝜎0 −𝜃 )𝜃 +(2𝑘𝑇𝑒 −3𝑒 +4𝑒 −1)𝜃 ]

−𝑘𝑠 2 2 2 4 +𝑒 (𝜎0 −𝜃 )𝜃 +𝜃 . which achieves the proof. (8) Corollary 8. If 𝑘 is large enough, we find

Proof. See [12]. 2 𝐸 (𝑉) =𝜃, Var (𝑉) =0. (16) Theorem 7. One has Proof. The idea is the limit passage 𝑘→∞. (a) Remark 9. In this case a swap maturity 𝑇 does not influence 1−𝑒−𝑘𝑇 𝐸(𝑉) (𝑉) 𝐸 (𝑉) = (𝜎2 −𝜃2)+𝜃2, (9) and Var . 𝑘𝑇 0 3.3. GARCH(1,1) and Volatility Swaps. GARCH model is (b) needed for both the variance swap and the volatility swap. 𝜉2𝑒−2𝑘𝑇 The model for the variance in a continuous version for Heston (𝑉) = [(2𝑒2𝑘𝑇 −4𝑘𝑇𝑒𝑘𝑇 −2)(𝜎2 −𝜃2) model is Var 2𝑘3𝑇2 0 𝑑𝜎2 =𝑘(𝜃2 −𝜎2)𝑑𝑡+𝜉𝜎𝑑𝑤 (𝑡) . + (2𝑘𝑇𝑒2𝑘𝑇 −3𝑒2𝑘𝑇 +4𝑒𝑘𝑇 −1)𝜃2]. 𝑡 𝑡 𝑡 2 (17) (10) The discrete version of the GARCH(1,1) process is described by Engle and Mezrich [16]: 𝑉 Proof. (a) We obtain mean value for 2 ]𝑛+1 =(1−𝛼−𝛽)𝑉+𝛼𝑢𝑛 +𝛽]𝑛, (18) 1 𝑇 𝐸 (𝑉) = ∫ 𝐸(𝜎2)𝑑𝑡 𝑡 (11) where 𝑉 is the long-term variance, 𝑢𝑛 is the drift-adjusted 𝑇 0 2 stock return at time 𝑛, 𝛼 is the weight assigned to 𝑢𝑛,and using Lemma 6,andwefind 𝛽 is the weight assigned to ]𝑛.Furtherweusethefollowing

−𝑘𝑇 relationship (19) to calculate the discrete GARCH(1,1) param- 1−𝑒 2 2 2 eters: 𝐸 (𝑉) = (𝜎0 −𝜃 )+𝜃 . (12) 𝑘𝑇 𝐶 𝑉= 2 2 (b) Variance for 𝑉 equals Var(𝑉) = 𝐸(𝑉 )−𝐸(𝑉),and 1−𝛼−𝛽 the second moment may be found as follows: using formula 𝑉 𝑉 2 2 𝑇 2 2 𝜃= ,𝜎= (8) of Lemma 6: 𝐸(𝑉 )=(1/𝑇)∬ 𝐸(𝜎𝑡 𝜎𝑠 )𝑑𝑡𝑑𝑠, 0 0 Δ𝑡𝐿 Δ𝑡𝑆 (19) 𝐸(𝑉2) 1−𝛼−𝛽 𝑘= Δ𝑡 𝜉2 𝑇 𝑒𝑘(𝑡∧𝑠) −1 = ∬ [𝑒−𝑘(𝑡+𝑠) ( (𝜎2 −𝜃2) 2 2 0 2 𝛼 (𝐾−1) 𝑇 0 𝑘 𝜉 = , (13) Δ𝑡 2𝑘(𝑡∧𝑠) 𝑒 −1 2 + 𝜃 )] 𝑑𝑡𝑑𝑠 where Δ𝑡𝐿 = 1/252, 252 trading days in any given year, and 2𝑘 Δ𝑡𝑆 =1/63,63tradingdaysinanygiventhreemonths. Now, we will briefly discuss the validity of the assumption +𝐸2 (𝑉) that the risk-neutral process for the instantaneous variance is 4 Journal of Probability and Statistics a continuous time limit of a GARCH(1,1) process. It is well Table1:Unitroottest. known that this limit has the property that the increment Test ADF PP in instantaneous variance is conditionally uncorrelated with 𝑆 −34.16458 −35.01017 the return of the underlying asset. This unfortunately implies cac that, at each maturity 𝑇, the implied volatility is symmetric. Hence, for assets whose options are priced consistently with a symmetric smile, these observations can be used either 0.4 to initially calibrate the model or as a test of the model’s 0.3 0.2 validity. It is worth mentioning that it is not suitable to use at- 0.1 the-money implied volatilities in general to price a seasoned 0.0 volatility swap. However, our GARCH(1,1) approximation −0.1 shouldstillbeprettyrobust. −0.2 −0.3 −0.4 4. Application 250 500 750 1000 In this section, we apply the analytical solutions from Sec- Y YF tion 3 topriceaswaponthevolatilityoftheCAC40French Index for five years (October 2009–April 2013). Figure 1: GARCH(1,1) CAC 40 French Index forecasting. The first step of this application is to study the stationarity of the series. To this end, we used the unit root test of Dickey- Fuller (ADF) and Philips Peron´ test (PP). 0.23456 4.1.UnitRootTestsandDescriptiveAnalysis. In this section, Remark 10. If the nonadjusted strike is equal to ,then 0.23456 − 0.13376 = 0.1008 we summarized unit root tests and descriptive analysis results theadjustedstrikeisequalto . 𝑆 of cac (see Table 1). 𝐸(𝑉) Unit root test confirms the stationarity of the series. According to Figure 3 is increasing exponentially 𝑇→∞ × −6 In Table 2 all statistic parameters of CAC 40 French Index and converges when towards 3.3140 10 . (𝑉) areshown.Fortheanalysis1155 observations were taken. But Var is increasing linearly during the first year [1, ∞[ Mean of time series is 0.0000528,median0,andstandard and is decreasing exponentially during years when (𝑉) → 0 𝑇→∞ deviation 0.014589. Skewness of CAC 40 French Index is Var ,if . −0.078899,soitisnegativeandthemeanislargerthanthe median, and there is left-skewed distribution. Kurtosis is 7.255109, large than 3, so we called leptokurtic, indicating 4.2. Conclusions. According to results founded, the higher peak and fatter tails than the normal distribution. GARCH(1,1) is a very good model for modeling the volatility Jarque-Bera is 809.0892.Sowecanforecastanuptrend. swaps for stock market. Also, we remark the influence of the GARCH(1,1) models are clearly the best performing French financial crisis (2009) on CAC 40 French Index. models as they receive the lowest score on fitting metrics Moreover, we presented a probabilistic approach, based whilst representing the lowest MAE, RMSE, MAPE, SEE, on changing of time method, to study variance and volatility and BIC among all models. They are closely followed by swaps for stock market with underlying asset and variance GARCH(2,1) which is placed comfortably lower than both that follow the Heston model. We obtained the formulas ARCH(2) and ARCH(4). However the GARCH(1,1) model for variance and volatility swaps but with another structure is simple and easy to handle. The results also show that and another application to those in the papers by Brockhaus GARCH(1,1) model improves the forecasting performance and Long [10]andSwishchuk[12]. As an application of (see Table 3). our analytical solutions, we provided a numerical example using CAC 40 French Index to price swap on the volatility Numerical Applications.WehaveusedEviewssoftware,and (Figure 1). −7 we found 𝐶 =2.03× 10 , 𝛼 = −0.008411, 𝛽 = 0.980310, Also, we compared the forecasting performance of sev- and 𝐾 = 7.255109. To this end, we find the following: 𝑉 = eral GARCH models using different distributions for CAC −7 72.23942208 × 10 ; 𝜃 = 0.00182043; 𝜎0 = 0.0004551; 𝑘= 40 French Index. We found that the GARCH(1,1) skewed 2 𝑡 7.081452; 𝜉 = 0.111 51. Student model is the most promising for characterizing We use the relations (9) and (10) for a swap maturity 𝑇= thedynamicbehaviourofthesereturnsasitreflectstheir 0.9 years, and we find underlying process in terms of serial correlation, asymmetric volatility clustering, and leptokurtic innovation. The results −6 𝐸 (𝑉) = 2.8273 × 10 , also show that GARCH(1,1) model improves the forecast- (20) ing performance. This result later further implies that the −9 Var (𝑉) = 5.0873 × 10 . GARCH(1,1) model might be more useful than the other (2) (4) 3/2 three models (ARCH ,ARCH ,andGARCH(2,1))when The convexity adjustment is Var(𝑉)/8𝐸 (𝑉) = 0.13376 and implementing risk management strategies for CAC 40 French 𝐸(√𝑉) ≈ −0.13208. Index (Figure 2). Journal of Probability and Statistics 5

Table 2 Mean Median Std.Dev. Skewness Kurtosis Jarque-B 𝑆 5.28𝐸 − 5 − cac 0.0000 0.014589 0.078899 7.255109 809.0892

Table 3 2 Models Adju 𝑅 SEE BIC RMSE MAE MAPE ARCH(2) 0.989953 0.007369 −2.620676 0.013674 0.009786 3.612218 ARCH(4) 0.989971 0.007062 −2.801014 0.010689 0.007441 3.469134 GARCH(2,1) 0.992352 0.003072 −7.893673 0.002668 0.002835 2.946543 GARCH(1,1) 0.999122 0.002672 −8.993776 0.002668 0.001983 2.743416

×10−2 Line plot of Var(V) donneeś volatility 3v∗8c 0.30 3.0 0.28 2.5 0.26 2.0

0.24 ) 1.5 V 0.22 1.0 0.20 Var( 0.5 0.18 0.0 250 500 750 1000 −0.5 0123456789 Conditional standard deviation Maturity (years)

Figure 2: CAC 40 French Index conditional variance. (a) Line plot of E(V) donneeś volatility 3v ∗8c 0.20 Appendix 0.18 0.16

We give a reminder for each parameter. )

V 0.14 (1) Std. Dev. (standard deviation) is a measure of disper- ( sion or spread in the series. The standard deviation is given E 0.12 by 0.10 0.08 0 123456789 1 𝑁 √ 2 𝑠= ∑ (𝑦𝑖 − 𝑦) , (A.1) Maturity (years) 𝑁−1𝑖=1 (b) where 𝑁 is the number of observations in the current sample Figure 3: CAC 40 French Index 𝐸(𝑉) and Var(𝑉). and 𝑦 isthemeanoftheseries. (2) Skewness is a measure of asymmetry of the distribution of the series around its mean. Skewness is computed as (4) Jarque-Bera is a test statistic for testing whether the series is normally distributed. The statistic is computed as 1 𝑁 𝑦 − 𝑦 3 𝑆= ∑ ( 𝑖 ) , 𝑁 𝜎̂ (A.2) 𝑁 (𝐾−3)2 𝑖=1 = (𝑆2 + ), Jarque-Bera 6 4 (A.4) where 𝜎̂ is an estimator for the standard deviation that isbasedonthebiasedestimatorforthevariance(𝜎=̂ where 𝑆 is the skewness and 𝐾 is the kurtosis. 𝑠√(𝑁 − 1)/𝑁). (5) = (3) Kurtosis measures the peakedness or flatness of the Mean󵄨 absolute󵄨 error (MAE) is as follows: MAE (1/𝑁) ∑𝑁 󵄨𝑦 − 𝑦̂ 󵄨 distribution of the series. Kurtosis is computed as 𝑖=1 󵄨 𝑖 𝑖󵄨. (6) Mean absolute percentage error (MAPE) is as follows: 𝑁 󵄨 󵄨 𝑁 4 󵄨 󵄨 1 𝑦 − 𝑦 MAPE =∑𝑖=1 󵄨(𝑦𝑖 − 𝑦̂𝑖)/𝑦𝑖󵄨. 𝐾= ∑ ( 𝑖 ) , 𝑁 𝜎̂ (A.3) (7) Root mean squared error (RMSE) is as follows: 𝑖=1 √ 𝑁 2 RMSE = (1/𝑁) ∑𝑖=1 (𝑦𝑖 − 𝑦̂𝑖) . 2 where 𝜎̂ isagainbasedonthebiasedestimatorforthevari- (8) Adjusted R-squared (adjust 𝑅 ) is considered. ance. (9) Sum error of regression (SEE) is considered. 6 Journal of Probability and Statistics

(10) Schwartz criterion (BIC) is measured by 𝑛 ln (SEE) + 𝑘 ln (𝑛).

Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment This work was given ATRST (ex: ANDRU) financing within the framework of the PNR Project (Number 8/u23/1050) and Averroes` Program.

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