Time series analysis for financial market meltdowns

by Young Shin Kim, Svetlozar T. Rachev, MicheleLeonardo Bianchi, Ivan Mitov, Frank J. Fabozzi

No. 2 | AUGUST 2010

WORKING PAPER SERIES IN ECONOMICS

KIT – University of the State of Baden-Wuerttemberg andNational Laboratory of the Helmholtz Association econpapers.wiwi.kit.edu

Impressum

Karlsruher Institut für Technologie (KIT)

Fakultät für Wirtschaftswissenschaften

Institut für Wirtschaftspolitik und Wirtschaftsforschung (IWW)

Institut für Wirtschaftstheorie und Statistik (ETS)

Schlossbezirk 12

76131 Karlsruhe

KIT – Universität des Landes Baden-Württemberg und

nationales Forschungszentrum in der Helmholtz-Gemeinschaft

Working Paper Series in Economics

No. 2, August 2010

ISSN 2190-9806

econpapers.wiwi.kit.edu

Time Series Analysis for Financial Market Meltdowns

Young Shin Kima, Svetlozar T. Rachevb, Michele Leonardo Bianchic, IvanMitovd, Frank J. Fabozzie

aSchool of Economics and Business Engineering, University of Karlsruhe and KIT.bSchool of Economics and Business Engineering, University of Karlsruhe and KIT, and

Department of Statistics and Applied Probability, University of California, Santa Barbara, andFinAnalytica INC.

cSpecialized Intermediaries Supervision Department, Bank of Italy.dFinAnalytica INC.

eYale School of Management.

Abstract

There appears to be a consensus that the recent instability in global financial mar-kets may be attributable in part to the failure of financial modeling. More specif-ically, current risk models have failed to properly assess the risks associated withlarge adverse stock price behavior. In this paper, we first discuss the limitationsof classical time series models for forecasting financial market meltdowns. Thenwe set forth a framework capable of forecasting both extreme events and highlyvolatile markets. Based on the empirical evidence presented in this paper, ourframework offers an improvement over prevailing models for evaluating stockmarket risk exposure during distressed market periods.

Keywords: ARMA-GARCH model, α-stable distribution, tempered stabledistribution, value-at-risk (VaR), average value-at-risk (AVaR).2000 MSC: 60E07, 62M10, 91B28, 91B84

Email addresses: aaron.kim@kit.edu (Young Shin Kim), rachev@kit.edu(Svetlozar T. Rachev), micheleleonardo.bianchi@bancaditalia.it (MicheleLeonardo Bianchi), ivan.mitov@finanalytica.com (Ivan Mitov),frank.fabozzi@yale.edu (Frank J. Fabozzi)

Preprint submitted to Journal of Banking and Finance April 1, 2010

1. Introduction

The forecasting of the future behavior of the price of financial instruments isan essential activity in the implementation of risk management and portfolio allo-cation. The debate between the financial industry and regulators involves whetherthe sophisticated mathematical and statistical tools that have been employed inrisk management and valuation of complex financial instruments have played arole in the recent crisis. In particular, risk measures such as value-at-risk (VaR)and black-box models for assessing the risks that institutional investors and reg-ulated financial entities are exposed to have been singled out as the culprits (seeTurner (2009) and Sheedy (2009)). It is within this context that we discuss in thispaper a market model that is capable of explaining highly volatile periods. Wewill demonstrate that the proposed model together with a measure of risk knownas the average value-at-risk (AVaR) offers a more reliable risk assessment, particu-larly during financial crises. Furthermore, we will try to explain how “25 standarddeviation events” in the words of David Viniar, chief financial officer of GoldmanSachs, can occurring. We do so by measuring the probability of occurrence ofmarket crashes by looking at time-series data and showing that this probabilitystrictly depends on the distributional assumption. We then compare these prob-abilities to the “high standard deviation events” given by the normal probabilitydistribution that is typically assumed.

In order to obtain a good forecast for the distribution of returns, prediction offuture market volatility of the market is critical. Most of the recent empirical stud-ies have shown that the amplitude of daily returns varies across time. Moreover,there is ample empirical evidence that if volatility is high, it remains high, and ifit is low, it remains low. This means that volatility moves in clusters and for thisreason it is important to find a way to explain such observed patterns. This behav-ior, referred to as “volatility clustering”, refers to the tendency of large changesin asset prices (either positive or negative) to be followed by large changes, andsmall changes to be followed by small changes. The volatility clustering effectcan be captured by the autoregressive conditional heteroskedastic (ARCH) and thegeneralized ARCH (GARCH) models formulated by Engle (1982) and Bollerslev(1986), respectively. However, in this paper we provide empirical evidence thatsuggests that GARCH models based on the normal distribution would not haveperformed well in predicting real-world market crashes such as Black Monday(October 19, 1987) and, more recently, the global economic meltdown attributableto the subprime mortgage meltdown in 2007 and the Lehman Brothers failure inthe latter half year of 2008. One reason for the poor performance is due to the

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assumption that the innovation of the GARCH model is normally distributed.Asset management and pricing models require the proper modeling of the re-

turn distribution of financial assets. While the return distribution used in the tra-ditional theories of asset pricing such as the capital asset pricing model is thenormal distribution, numerous studies that have investigated the empirical behav-ior of asset returns in financial markets throughout the world reject the hypothesisthat asset return distributions are normally distributed. Returns from financial as-sets show well-defined patterns of leptokurtosis and skewness which cannot becaptured by the normality assumption.

Enhanced GARCH models with non-normal innovation distributions have beenproposed. For example, Menn and Rachev (2008) used GARCH models withα-stable innovations and the smoothly truncated α-stable innovations for optionpricing. A new class of distributions, the tempered stable distribution, has beenproposed recently to deal with the drawbacks of the α-stable distribution (see Kimet al. (2008) and Bianchi et al. (2010)).

Most importantly, a suitable measure has to be employed to evaluate marketrisk. The VaR measure has been adopted as a standard risk measure in the financialindustry, having been adopted by regulators to determine the capital requirementsfor both banking and trading books (see Kiff et al. (2007)). However, the limita-tions of the VaR measure have been well documented in the academic literature, aswell as among regulators and risk managers (see Bookstaber (2009)). Criticismsof this risk measure include (1) a short sample of historical observations is insuf-ficient to assess the risk one-day ahead, (2) the normal distributional assumptionis inadequate for forecasting extreme events, and (3) it is difficult to infer futurerisk from past observed patterns, particularly under stressed scenarios. In this pa-per, we address these three criticisms by (1) considering a ARMA-GARCH modelwith non-normal innovation, (2) estimating the model with a sample including 10years of daily data (including a more realistic measure of risk), principally fo-cused on the negative tail, and (3) backtesting the model during market shocks.By doing so, we hope to provide market participations with more reliable mathe-matical and statistical tools that can be used to try to understand complex financialmarket behavior. These tools cannot be used as black-boxes; market players haveto understand them to avoid financial debacles.

The risk measure we use in this study is AVaR, which is the average of VaRsless than the VaR for a given tail probability. AVaR, also called conditional value-

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at-risk (CVaR),1 is a superior risk measure to VaR because it satisfies all axioms ofa coherent risk measure and it is consistent with preference relations of risk-averseinvestors (see Rachev et al. (2007)). The closed-form solution for AVaR for theα-stable distribution, the skewed-t distribution, and the infinitely divisible distri-butions containing tempered stable distributions have been derived by Stoyanovet al. (2006), Dokov et al. (2008), and Kim et al. (to appear a), respectively.

Hence, in this paper, we discuss autoregressive moving average (ARMA)GARCH models with α-stable and tempered stable innovations and then assessthe forecasting performance of these models by comparing them to other time-series models that assume a normal innovation. We empirically test the perfor-mance of these models for the S&P 500 index (SPX) during stressed financialmarkets. The dataset includes the following stock market crashes: October 1987,October 1997, the turbulent period around the Asian Crisis in 1998 through 1999,the burst of the “dotcom bubble,” and the recent subprime mortgage crisis togetherwith the Lehman Brothers failure. We present VaR values for the two indexes forall of these periods. In our backtests of VaR, we evaluate the accuracy of the VaRmodels. Finally, we present a closed-form solution to the AVaR for the ARMA-GARCH model with tempered stable innovations, and compute AVaR values forthe two indexes.

The remainder of this paper is organized as follows. ARMA-GARCH modelswith the α-stable and tempered stable innovations are presented in Section 2. InSection 3, we discuss parameter estimation of the ARMA-GARCH models andforecasting return distributions for the two indexes for daily, weekly, and monthlyreturns. The VaR values and the backtesting of the ARMA-GARCH models withα-stable and tempered stable innovations are presented and the results then com-pared to the classical models such as the equally weighted moving average modeland ARMA-GARCH model with normal innovations. The closed-form solutionof the AVaR measure for the ARMA-GARCH model with tempered stable inno-vations is presented in Section 4, together with values of the AVaR for the twoindexes. In Section 5, we summarize our principal findings. We briefly reviewthree tempered stable distributions in the appendix.

2. ARMA-GARCH model with α-stable and tempered stable innovations

Let (St)t≥0 be the asset price process and (yt)t≥0 be the the return processof (St)t≥0 defined by yt = log St

St−1. We propose the ARMA(1,1)-GARCH(1,1)

1See Pflug (2000) and Rockafellar and Uryasev(2000, 2002).

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model: {yt = ayt−1 + bσt−1εt−1 + σtεt + c,

σ2t = α0 + α1σ

2t−1ε

2t−1 + β1σ

2t−1,

(1)

where ε0 = 0, and a sequence (εt)t∈N of independent and identically distributed(iid) real random variables. The innovation εt is assumed to be the standard nor-mal distribution. In this case, the ARMA(1,1)-GARCH(1,1) model is referred toas the normal-ARMA-GARCH model.

If εt is assumed to be α-stable2 and tempered stable innovations, then we ob-tain new ARMA(1,1)-GARCH(1,1) models. In this paper, we will consider threetempered stable distributions: the standard classical tempered stable (stdCTS),standard modified tempered stable (stdMTS), and standard rapidly decreasingtempered stable (stdRDTS) distributions. New ARMA(1,1)-GARCH(1,1) mod-els with the α-stable and three tempered stable innovations are defined as follows:

• Stable-ARMA-GARCH model3: εt ∼ Sα(σ, β, µ).

• CTS-ARMA-GARCH model: εt ∼ stdCTS(α, λ+, λ−).

• RDTS-ARMA-GARCH model: εt ∼ stdRDTS(α, λ+, λ−).

• MTS-ARMA-GARCH model: εt ∼ stdMTS(α, λ+, λ−).

Definitions and more details about the three tempered stable distributions are pre-sented in the appendix to this paper.

Substituting a = 0 and b = 0, α1 = 0, and β1 = 0 into (1), we obtain theconstant volatility (CV) market model. In the CV model, the conditional vari-ance becomes constant, i.e. σt =

√α0 for all t ≥ 0. Substituting a = 0 and

b = 0 into (1), we obtain the GARCH(1,1) model. If the innovation distributionsare normally distributed, the CV and GARCH(1,1) models are referred to as thenormal-CV model and the normal-GARCH model, respectively. Similarly, if in-novation distributions are α-stable distributed, the CV and GARCH(1,1) modelsare referred to as the stable-CV model and the stable-GARCH model, respec-tively. Moreover, if the innovation distributions are the three tempered-stable dis-tributions, the CV and GARCH(1,1) models are referred to as the tempered stableprocess (e.g., CTS-CV model and CTS-GARCH model).

2Extensive analysis of α-stable distributions and their properties can be found in Samorodnit-sky and Taqqu (1994), Rachev and Mittnik (2000), and Stoyanov and Racheva-Iotova (2004a,b).

3In subsequent discussions of the α-stable distributions in this paper, we restrict ourselves tothe non-Gaussian case in which 0 < α < 2.

5

For convenience, the normal time series model refers to the normal-CV, normal-GARCH, and normal-ARMA-GARCH models, the stable time series model refersto stable-CV, stable-GARCH, and stable-ARMA-GARCH models, and the tem-pered stable time series model refers to the CV, GARCH(1,1), and ARMA(1,1)-GARCH(1,1) models with the CTS, MTS, and RDTS innovations.

3. Parameter estimation and forecasting return distribution

In this section, we estimate the parameters for the CV model, the GARCH(1,1)model, and the ARMA(1,1)-GARCH(1,1) model. We use the historical data of theSPX. We then analyze the probability of market crashes.

Parameters of the CV, GARCH(1,1), and ARMA(1,1)-GARCH(1,1) modelsare estimated from daily, weekly, and monthly returns. For daily returns, we usethe closing price of those indexes. Weekly returns are calculated using closingprices on every Friday. If Friday is a holiday, then we use the trading day beforethat Friday. Monthly returns are calculated using the closing price of the lasttrading day of each month. In Figure 1, we show the daily, weekly, and monthlyreturns of the SPX.

Parameters of the normal-CV, normal-GARCH, and normal-ARMA-GARCHmodels are estimated using the maximum likelihood estimation method (MLE).For the other cases, the parameters are estimated as follows:

1. Estimate parameters α0, α1, β1, a, b, c with Student-t distributed innovationby the MLE. For estimating the parameters of the CV models, all param-eters except α0 are assumed to be zero. For the GARCH(1,1) model, theparameters a and b are assumed to be zero.

2. Extract residuals using the estimated parameters.

3. Fit the parameters of the innovation distribution (the α-stable, CTS, MTS,and RDTS distributions) to the extracted residuals using the MLE.4

3.1. Daily return of S&P 500 indexWe selected five days of market crashes for daily returns of the SPX: October

19, 1987 (Black-Monday), October 27, 1997 (Asian Turmoil), August 31, 1998(Russian Default), April 14, 2000 (Dotcom Collapse), and September 29, 2008

4See Bianchi et al. (to appear).

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(U.S. financial crisis). In order to investigate the forecasting performance of eachmodel, we estimate parameters using 10 years of historical data through the closesttrading day before the selected crash day. The returns and data for estimating theparameters are described in Table 1.

In Tables 3 to 7 we report estimation results for each dataset described in Ta-ble 1. These tables provide parameter estimation results for the CV, GARCH(1,1),and ARMA(1,1)-GARCH(1,1) models with the normal and Student-t distributedinnovations as well as estimated parameters for the α-stable, standard CTS, stan-dard MTS, and standard RDTS distributions. For the assessment of the goodness-of-fit, we use the Kolmogorov-Smirnov (KS) test and also calculate the Anderson-Darling (AD) statistic, the latter statistic providing a better test to evaluate the tailfit.

Based on the goodness-of-fit statistics reported in Tables 3 to 7, we concludethe following:

• For all six datasets, the three time series models based on the normal distri-bution are rejected by the KS test at the 1% significance level.

• The stable-CV model is rejected by the KS test at the 1% significance levelin the U.S. financial crisis case.

• The KS test applied to the data ending with September 26, 2008, the day be-fore the biggest loss during the U.S. financial crisis, shows that the CTS-CV,the MTS-CV, the RDTS-CV, the RDTS-GARCH, and the RDTS-ARMA-GARCH models are not rejected while all other models are rejected at the1% significance level.

• The AD statistic for the three time series models based on the normal dis-tribution are significantly larger than those of the other models. That meansthe normal distribution cannot describe the fat-tail behavior of the empiricalinnovation distribution.

Based on the estimated parameters, we can calculate the probabilities of occur-rence of those crashes. The average times of occurrence of crashes are calculatedby

1

250 · P [εt ≤ ε∗t ], (2)

where ε∗t is observed residual at time t. By taking into consideration the estimatesof the above selected models, in Table 3 we provide the residuals observed the

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day before the crash, and the probabilities that the Black Monday collapse willhappen, together with the average time of occurrence as defined in (2).

From the results reported in Table 3, we see that the probabilities that theBlack Monday decline will happen based on the estimated three normal time seriesmodels are considerably less than the other models. At the same time, averagetime of occurrence based on the three normal time series models are significantlylong. That is, the crash should happen only once every 7.966·10141 years under thenormal-CV model. If we use the normal-GARCH and normal-ARMA-GARCHmodels, the crash is expected to happen once every 2.554 · 1039, and 2.904 · 1039

years, respectively. Those time values are longer than the age of universe.5 Inreality, a similar fall should be expected once every 30–50 years, and hence thosethree models based on the normal distributional assumption are not realistic. Incontrast, the three models based on the α-stable distribution are more realistic thanthe three models based on the normal one. Average times of occurrence are 80years, 37.26 years, and 40.31 years for the stable-CV, stable-GARCH, and stable-ARMA-GARCH models, respectively. The nine models that consider temperedstable distributions have better performance than the corresponding three normaltime series models. The nine tempered stable time series models have longeraverage times of occurrence than the corresponding stable time series model, butsignificantly shorter than those of the corresponding normal time series models.

In Tables 4 to 6 we present the empirical analysis conducted for the AsianTurmoil (Table 4), the Russian Default (Table 5), and the Dotcom Collapse (Table6). In these investigations, we obtain similar results as in the Black Monday case.That is, the normal time series models are not realistic. Average times to occur-rence of the crash for the three stable time series models are shorter than the othermodels investigated.

We also investigate the probability that the crash on September 29, 2008 wouldoccur. On that day the SPX declined 9%. This crisis was caused by the defaultof Lehman Brothers on September 15, 2008, and the market at that time wascharacterized by increased volatility. We estimate parameters using the data untilSeptember 26, 2008 and forecast the probability that the crash on September 29,2008 will occur. Average times of occurrences are 1.99 · 1013, 502.1, and 732.3years for the normal-CV, normal-GARCH, and normal-ARMA-GARCH models,respectively. That is, we do not obtain any serious alerts from the three normal

5Astronomers estimated that the age of universe is between 12 and 14 billion years. Seehttp://map.gsfc.nasa.gov/.

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time series models even though the models do consider the volatility clusteringeffect. Constant volatility models such as the CTS-CV, MTS-CV, and RDTS-CVmodels do not perform satisfactorily either. In contrast, the α-stable and temperedstable time series models that account for volatility clustering have better fore-casting power. Those models provide very short times of occurrence for the crisis,and hence one can consider them to be superior early-warning models.

3.2. Weekly and monthly returns of S&P 500 indexNext, we selected three of the market crashes analyzed in Section 3.1. How-

ever, instead of considering daily returns, we analyze the time series of SPXweekly returns ending in the week October 19–23, 1987 (Black Monday), April10–14, 2000 (Dotcom Collapse), and October 6–10, 2008 (U.S. financial cri-sis). In order to investigate the forecasting performance of each model, we esti-mate parameters for historical weekly returns one week before the selected crashdays. Then, we also analyze the time series of SPX monthly returns, selecting themonths October 1987 and October 2008. The SPX dropped 24.54%, and 18.42%in October 1987 and October 2008, respectively. Table 2 describes returns anddata considered in the estimation.

Tables 8 to 12 report the estimated parameters of the CV, GARCH, and ARMA-GARCH models with normal and Student-t distributed innovations, the estimatesrelative to α-stable, standard CTS, standard MTS, and standard RDTS distribu-tions, the KS statistics with the corresponding p-values, and the AD statistics forall the models considered, for weekly and monthly returns. These results indicatethe following:

• For Black Monday and the U.S. financial crisis, the normal-CV and thestable-ARMA-GARCH models are rejected by the KS test at the 1% signif-icance level.

• In all cases, the stable-CV model is rejected by the KS test at the 1% signif-icance level.

• In the U.S. financial crisis case, the stable-GARCH model is rejected by theKS test at the 1% significance level.

• The three stable and nine tempered stable time series models have a betterAD statistic than those of the three normal time series models.

• The nine tempered stable time series models are not rejected by the KS testfor all datasets.

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Based on Tables 11 and 12, we conclude the following:

• The KS test does not reject any time series model at the 1% significancelevel.

• The three normal time series models have significantly larger AD statisticsthan the other models. That is, the normal time series models cannot explainthe tail properties of the monthly return distribution better than the othermodels investigated.

Based on the estimated parameters, we can calculate the probability that thosecrashes will happen. In Tables 8 to 10 we also provide observed residuals forweekly returns and probabilities that the collapse will happen in weekly analysis.In Tables 11 and 12 we provide the observed residuals for monthly returns, aswell as the probabilities that the collapse will happen in monthly analysis. In theweekly analysis, the average time of occurrence is calculated by

1

53 · P [εt ≤ ε∗t ],

where ε∗t is observed residual of weekly returns, while in the monthly analysis, itis calculated by

1

12 · P [εt ≤ ε∗t ],

where ε∗t is observed residual for monthly returns.Following are fair conclusions from the results reported in the tables:

• Average time of occurrences of the three normal time series models are verylarge and unrealistic in both the weekly analysis and the monthly analysis.

• The three stable time series models are more realistic than the three normaltime series models in view of average time of occurrences.

• In ARMA-GARCH models, average times of occurrence of the stable-AR-MA-GARCH models for both the weekly analysis and the monthly analysisare shorter than the other models investigated.

• Average times to occurrences of the CTS-ARMA-GARCH, MTS-ARMA-GARCH, and RDTS-ARMA-GARCH are longer than the three normal-ARMA-GARCH models and shorter than the three stable-ARMA-GARCHmodels for the weekly and monthly cases, respectively.

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• In the CV and GARCH models, we obtain the same conclusion as theARMA-GARCH model cases.

4. VaR and AVaR for ARMA-GARCH model

In this section we discuss the VaR and AVaR for the ARMA-GARCH modelwith stable and tempered stable innovations.

4.1. VaR and backtestingThe definition of VaR with the significance level η is

VaRη(X) = − inf{x ∈ R|P (X ≤ x) > η}.

Considering the ARMA-GARCH model defined, we can define the VaR for theinformation until time t with significance level η as

VaRt,η(yt+1) = − inf{x ∈ R|Pt(yt+1 ≤ x) > η},

where Pt(A) is the conditional probability of a given event A for the informationuntil time t.

We consider five models: the normal-CV, the exponentially weighted mov-ing average (EWMA),6 the normal-ARMA-GARCH, the stable-ARMA-GARCH,and the CTS-ARMA-GARCH. For all five models we estimate the parametersby considering the time series from December 14, 2004 to December 31, 2008.For each daily estimation, we used about 10 years of historical return data forthe SPX. We then computed VaRs for the five models. Figure 2 shows the SPXdaily returns and negative values of daily VaRs with 1% significance level (i.e.−VaRt,0.01(yt+1)) for all five models considered. Based on those figures, we ob-serve that the normal-CV model is not able to capture market crashes.

For evaluating the accuracy of VaR for the five models, we perform the back-testing by Kupiec’s proportion of failures test developed by Kupiec (1995). Forthis, the number of violations (violations occur when the actual loss exceeds theestimate) from the empirical data are compared to the accepted number of ex-ceedances at a given significant level. In Table 13 we report the number of vio-lations and p-values of Kupiec’s backtest for the SPX. We count the number of

6We follow the EWMA model of J.P. Morgan RiskMetrics. In the model, the daily volatilityformula is given by σ2

t = λσ2t−1 + (1− λ)y2

t−1 with λ = 0.94.

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violations and the corresponding p-values for 1%-VaRs for the five models byconsidering different time periods: 1, 2, and 4 years.

From Table 13, we conclude the following for the SPX:

• All five models are not rejected at the significance level of 5% in the periodfrom (1) December 14, 2004 to December 15, 2005 and from December 16,2005 to December 20, 2006 (1-year backtest), and (2) December 14, 2004to December 20, 2006 (2-year backtest).

• For all crash dates investigated, the stable-ARMA-GARCH model is not re-jected at the significance level of 5%, and the CTS-ARMA-GARCH modelis not rejected at the significance level of 1%.

• The normal-CV, EWMA, normal-ARMA-GARCH models are rejected, butthe stable-ARMA-GARCH and CTS-ARMA-GARCH models are not re-jected at the significance level of 5% for the period from December 21,2006 to December 31, 2008 (2-year backtest) and December 14, 2004 toDecember 31, 2008 (4-year backtest).

According to the 1-year backtest, the discrepancy between the normal-CV,EWMA, normal-ARMA-GARCH, and stable and CTS-ARMA-GARCH is ob-served one year prior the crash of 2008. Hence, we consider that discrepancy asearly warning indicators of a forthcoming market crash.

4.2. AVaR for the ARMA-GARCH model with tempered stable innovationsIn this section, we discuss AVaR for the ARMA-GARCH model with CTS in-

novation, and provide an empirical example of the AVaR under the CTS-ARMA-GARCH model for the SPX.

The definition of AVaR with the significance level η is

AVaRη(X) =1

η

∫ η

0

VaRε(X)dε

where VaRε(X) is the VaR of X with the significance level ε. If the distributionof X is continuous, then we have

AV aRη(X) = −E [X|X < −V aRη(X)] .

Consider the ARMA-GARCH model defined in (1). Since for every t > 0 thedistribution of yt is continuous, we define the conditional AVaR of yt+1 for the

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information until time t with the significance level η by the following form:

AV aRt,η(yt+1) = −Et [yt+1|yt+1 < −V aRt,η(yt+1)] ,

where V aRt,η(yt+1) is the VaR of yt+1 for the information until time t with thesignificance level η and Et is the conditional expectation for the information untiltime t. By equation (1), we have

AV aRt,η(yt+1)

= −Et [ayt + bσtεt + σt+1εt+1 + c|yt+1 < −V aRt,η(yt+1)] .

Since yt, σt, εt, and σt+1 are determined at time t, we have

AV aRt,η(yt+1)

= −(c + ayt + bσtεt + σt+1Et [εt+1|yt+1 < −V aRt,η(yt+1)]).

Moreover, since we can prove that

−V aRt,η(yt+1) = ayt + bσtεt + σt+1(−V aRt,η(εt+1)) + c,

we obtain

AV aRt,η(yt+1) = −(c + ayt + bσtεt + σt+1Et [εt+1|εt+1 < −V aRt,η(εt+1)]),

orAV aRt,η(yt+1) = −(c + ayt + bσtεt) + σt+1AV aRt,η(εt+1).

Since εt+1 is independent of the information until time t and it is continuouslydistributed, we have

AV aRt,η(εt+1) = −E [εt+1|εt+1 < −V aRη(εt+1)] = AV aRη(εt+1).

If εt+1 is tempered stable, then the formula AV aRt,η(εt+1) can be obtained by thefollowing proposition (see Kim et al. (to appear a)).7

Proposition Let Y be a random variable for the return of an asset or portfolio.

7AVaR is defined on the loss distribution in Kim et al. (to appear a). We changed the propositionbecause we are looking at the return distribution.

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Suppose Y is infinitely divisible and the distribution function of Y is continuous.If there is ρ > 0 such that |φY (−u + iρ)| < ∞ for all u ∈ R, then

AVaRη(Y ) (3)

= VaRη(Y )− e−VaRη(Y )ρ

πη<

(∫ ∞

0

e−iuVaRη(Y )φY (−u + iρ)

(−u + iρ)2du

).

In Figure 3, we report the SPX daily returns and daily values of (−AVaRt,0.01

(yt+1)) for the normal-ARMA-GARCH and CTS-ARMA-GARCH models. Com-paring Figure 3 to Figure 2, one can graphically check the differences betweenthese two risk measures. In particular, AVaR seems to be more conservative un-der stressed scenarios. This means that AVaR can be considered a good indicatorduring highly volatile markets.

In Figure 4, we present the daily differences for the S&P500 between AVaRt,0.01

(yt+1) for the CTS-ARMA-GARCH model and VaRt,0.01(yt+1) for the normal-ARMA-GARCH model. The spreads from July 2007 to September 2008 are typ-ically larger than the spreads until July 2007. Finally, the index declines sharplyin September 2008. The same phenomenon is observed for the the daily differ-ences of AVaRt,0.01(yt+1) between the normal-ARMA-GARCH model and CTS-ARMA-GARCH model presented in Figure 5. Hence, we consider the increasingspread between 1%-AVaR for the CTS-ARMA-GARCH model and 1%-VaR (or1%AVaR) for the normal-ARMA-GARCH model and as early warning indicatorsof a pending market crash.

5. Conclusion

In this paper, we discussed models with stable and tempered stable innova-tions, and provided an assessment of their forecasting power relative to other mod-els widely used in the industry. The proposed models are applied to the analysisof the S&P 500 index during highly volatile markets.

Our first finding is that the time series models based on the assumption of anormal innovation do not provide a reliable forecast of the future distribution ofreturns, even if they account for volatility clustering. In particular, we our em-pirical evidence indicates that time series models with stable and tempered stableinnovations have better predictable power in measuring market risk compared tostandard models based on the normal distribution assumption.

We also analyzed the behavior of VaR depending on different distributionalassumption. We backtested VaR by considering the last four years of log returns

14

for the SPX. The Kupiec’s proportion of failures test rejects the three normal timeseries models considered, but does not reject the stable and tempered stable timeseries models.

Finally, we derived a closed-form solution for the AVaR for the ARMA-GARCHmodels with tempered stable innovation and applied this formula to calculate dailyAVaRs with the CTS-ARMA-GARCH models with respect to the last four yearsof data.

The principal findings of the paper are twofold. First, in a low volatile market,the models proposed in this paper are practically identical to the correspondingGaussian models. Second, the proposed models can be used as early warningsystems for a forthcoming sharp market downturn.

AcknowledgementRachev gratefully acknowledges research support through grants from the Di-

vision of Mathematical, Life and Physical Sciences, College of Letters and Sci-ence, University of California, Santa Barbara, the Deutschen Forschungsgemein-schaft, and the Deutscher Akademischer Austausch Dienst. Bianchi acknowl-edges that the views expressed in this paper are his alone and should not be at-tributed to those of his employer.

References

Bianchi, M. L., Rachev, S. T., Kim, Y. S., Fabozzi, F. J., 2010. Tempered infinitelydivisible distributions and processes. Theory of Probability and Its Applications(TVP), Society for Industrial and Applied Mathematics (SIAM).

Bianchi, M. L., Rachev, S. T., Kim, Y. S., Fabozzi, F. J., to appear. Tempered stabledistributions and processes in finance: numerical analysis. In: Corazza, M.,Pizzi, C. (Eds.), Mathematical and Statistical Methods for Actuarial Sciencesand Finance - MAF 2008.

Bollerslev, T., 1986. Generalized autoregressive conditional heteroskedasticity.Journal of Econometrics 31, 307–327.

Bookstaber, R., September 2009. The risks of financial modeling: VaR and theeconomic meltdown. Committee on Science and Technology, United StatesHouse of Representatives.

15

Boyarchenko, S. I., Levendorskii, S. Z., 2000. Option pricing for truncated Levyprocesses. International Journal of Theoretical and Applied Finance 3, 549–552.

Carr, P., Geman, H., Madan, D., Yor, M., 2002. The fine structure of asset returns:An empirical investigation. Journal of Business 75 (2), 305–332.

Dokov, S., Stoyanov, S. V., Rachev, S. T., 2008. Computing VaR and AVaR ofskewed-t distribution. Journal of Applied Functional Analysis 3 (1), 189–208.

Engle, R., 1982. Autoregressive conditional heteroskedasticity with estimates ofthe variance of united kingdom inflation. Econometrica 50, 987–1007.

Kiff, J., Kodres, L., Klueh, U., Mills, P., 2007. Do market risk management tech-niques amplify systemic risks? In: Global Financial Stability Report, FinancialMarket Turbulence Causes, Consequences, and Policies. World Economic andFinancial Surveys. International Monetary Fund, pp. October, 52–76.

Kim, Y. S., 2005. The modified tempered stable processes with application tofinance, ph.D thesis, Sogang University.

Kim, Y. S., Rachev, S. T., , Bianchi, M. L., Fabozzi, F. J., to appear a. ComputingVaR and AVaR in infinitely divisible distributions. Probability and Mathemati-cal Statistics.

Kim, Y. S., Rachev, S. T., Bianchi, M. L., Fabozzi, F. J., 2008. Financial marketmodels with Levy processes and time-varying volatility. Journal of Banking andFinance 32, 1363–1378.

Kim, Y. S., Rachev, S. T., Bianchi, M. L., Fabozzi, F. J., to appear b. Temperedstable and tempered infinitely divisible GARCH models. Journal of Bankingand Finance.

Kim, Y. S., Rachev, S. T., Chung, D. M., Bianchi, M. L., 2009. The modifiedtempered stable distribution, GARCH-models and option pricing. Probabilityand Mathematical Statistics 29 (1), 91–117.

Koponen, I., 1995. Analytic approach to the problem of convergence of truncatedLevy flights towards the Gaussian stochastic process. Physical Review E 52,1197–1199.

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Kupiec, P., 1995. Techniques for verifying the accuracy of risk measurement mod-els. Journal of Derivatives 6, 6–24.

Menn, C., Rachev, S. T., 2008. Smoothly truncated stable distributions, GARCH-models, and option pricing. Mathematical Methods in Operation ResearchToappear.

Pflug, G., 2000. Some remarks on the value-at-risk and the conditional value-at-risk. In: Uryasev, S. (Ed.), Probabilistic Constrained Optimization: Methodol-ogy and Applications. Kluwer Academic Publishers, pp. 272–281.

Rachev, S. T., Mittnik, S., 2000. Stable Paretian Models in Finance. John Wiley& Sons, New York.

Rachev, S. T., Stoyanov, S., Fabozzi, F. J., 2007. Advanced Stochastic Models,Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, andPerformance Measures. John Wiley&Sons, Hoboken, New Jersey.

Rockafellar, R. T., Uryasev, S., 2000. Optimization of conditional value-at-risk.The Journal of Risk 2 (3), 21–41.

Rockafellar, R. T., Uryasev, S., 2002. Conditional value-at-risk for general lossdistributions. Journal of Banking & Finance 26, 1443–1471.

Samorodnitsky, G., Taqqu, M. S., 1994. Stable non-Gaussian random processes.Chapman & Hall/CRC.

Sheedy, E., 2009. It’s NOT the econometrics, stupid. Risk Magazine, September.

Stoyanov, S., Racheva-Iotova, B., 2004a. Univariate stable laws in the field of fi-nance. approximation of density and distribution functions. Journal of Concreteand Applicable Mathematics 2 (1), 37–58.

Stoyanov, S., Racheva-Iotova, B., 2004b. Univariate stable laws in the field offinance. parameter estimation. Journal of Concrete and Applicable Mathematics2 (4), 24–49.

Stoyanov, S., Samorodnitsky, G., Rachev, S. T., Ortobelli, S., 2006. Comput-ing the portfolio conditional value-at-risk in the α-stable case. Probability andMathematical Statistics 26 (1), 1–22.

17

Turner, A., March 2009. The Turner Review: A regulatory response to the globalbanking crisis. FSA.

Appendix

In this section we review three classes of tempered stable distributions formodeling a return distribution. Generally, these distributions do not have closed-form solutions for their probability density function. Instead, they are defined bytheir characteristic function.

Let α ∈ (0, 2) \ {1}, C, λ+, λ− > 0, and m ∈ R.

1. A random variable X is said to follow the classical tempered stable (CTS)distribution if the characteristic function of X is given by

φX(u) = φCTS(u; α, C, λ+, λ−,m) (4)= exp(ium− iuCΓ(1− α)(λα−1

+ − λα−1− )

+ CΓ(−α)((λ+ − iu)α − λα+ + (λ− + iu)α − λα

−)),

and we denote X ∼ CTS(α, C, λ+, λ−, m).8

2. A random variable X is said to follow the modified tempered stable (MTS)distribution9 if the characteristic function of X is given by

φX(u) = φMTS(u; α, C, λ+, λ−,m) (5)= exp(ium + C(GR(u; α, λ+) + GR(u; α, λ−))

+ iuC(GI(u; α, λ+)−GI(u; α, λ−))),

where for u ∈ R,

GR(x; α, λ) = 2−α+3

2√

πΓ(−α

2

) ((λ2 + x2)

α2 − λα

)

8The CTS distribution has been introduced under different names including: truncatedLevy flight by Koponen (1995), the KoBoL distribution by Boyarchenko and Levendorskii (2000),and the CGMY distribution by Carr et al. (2002).

9See Kim (2005) and Kim et al. (2009).

18

and

GI(x; α, λ) =2−α+1

2 Γ

(1− α

2

)λα−1

[2F1

(1,

1− α

2;3

2;−x2

λ2

)− 1

],

where 2F1 is the hypergeometric function. We denote an MTS distributedrandom variable X by X ∼ MTS( α, C, λ+, λ−, m).

3. A random variable X is said to follow the rapidly decreasing tempered sta-ble (RDTS) distribution 10 if the characteristic function of X is given by

φX(u) = φRDTS(u; α,C, λ+, λ−,m) (6)= exp (ium + C(G(iu; α, λ+) + G(−iu; α, λ−))) ,

where

G(x; α, λ) = 2−α2−1λαΓ

(−α

2

) (M

(−α

2,1

2;

x2

2λ2

)− 1

)

+ 2−α2− 1

2 λα−1xΓ

(1− α

2

)(M

(1− α

2,3

2;

x2

2λ2

)− 1

),

and M is the confluent hypergeometric function. In this case, we denoteX ∼ RDTS(α, C, λ+, λ−, m).

The cumulants of X are defined by

cn(X) =∂n

∂unlog E[eiuX ]|u=0, n = 1, 2, 3, · · · .

For the three tempered stable distributions, we have E[X] = c1(X) = m. Thecumulants of the three tempered stable distributions for n = 2, 3, · · · are presentedin the following table:

Distribution of X cn(X) for n = 2, 3, · · ·CTS CΓ(n− α)(λα−n

+ + (−1)nλα−n− )

MTS 2n−α+32 CΓ

(n+1

2

)Γ

(n−α

2

)(λα−n

+ + (−1)nλα−n− )

RDTS 2n−α−2

2 CΓ(

n−α2

) (λα−n

+ + (−1)nλα−n−

)

10See Kim et al. (to appear b).

19

By substituting the appropriate value for the two parameters m and C into thethree tempered stable distributions, we can obtain tempered stable distributionswith zero mean and unit variance. That is,

1. X ∼ CTS(α, C, λ+, λ−, 0) has zero mean and unit variance by substituting

C =(Γ(2− α)(λα−2

+ + λα−2− )

)−1, (7)

The random variable X is referred to as the standard CTS distribution withparameters (α,λ+,λ−) and denoted by X ∼ stdCTS(α, λ+, λ−).

2. X ∼ MTS(α, C, λ+, λ−, 0) has zero mean and unit variance by substituting

C = 2α+1

2

(√πΓ

(1− α

2

) (λα−2

+ + λα−2−

))−1

(8)

The random variable X is referred to as the standard MTS distribution anddenoted by X ∼ stdMTS(α, λ+, λ−).

3. X ∼ RDTS(α, C, λ+, λ−, 0) has zero mean and unit variance by substitut-ing

C = 2α2

(Γ

(1− α

2

) (λα−2

+ + λα−2−

))−1

(9)

The random variable X is referred to as the standard RDTS distribution anddenoted by X ∼ stdRDTS(α, λ+, λ−).

20

Table 1: Data for analyzing daily return of the SPX

Crash Date Return Data for forecasting Number ofobservations

Black Monday Oct. 19,1987 −23% daily return ending 2,490with Oct. 16, 1987

Asian Turmoil Oct. 27,1997 −7% daily return ending 2,504with Oct. 24, 1997

Russian Default Aug. 31,1998 −7% daily return ending 2,503with Aug. 28, 1998

Dotcom Collapse Apr. 14,2000 −6% daily return ending 2,504with Apr. 13, 2000

U.S. Financial Crisis Sept. 29,2008 −9% daily return ending 2,505with Sept. 26, 2008

Table 2: Data for analyzing weekly and monthly returns of the SPX

Data for weekly returns

Crash Date Return Data for forecasting Number ofobservations

Black Monday Oct. 19 − 23, 1987 −13.01% weekly return ending 1,461with Oct. 16, 1987

Dotcom Collapse Apr. 10 − 14, 2000 −11.14% weekly return ending 1,577with Apr. 7, 2000

U.S. Financial Oct. 6 − 10, 2008 −20.08% weekly return ending 1,578Crisis with Oct. 3, 2008

Data for monthly returns

Crash Date Return Data for forecasting Number ofobservations

Black Monday Oct. 1987 −24.54% monthly return ending 332with Sept. 1987

U.S. Financial Oct. 2008 −18.42% monthly return ending 359Crisis with Sept. 2008

21

Tabl

e3:

[Dai

lyan

alys

is-B

lack

Mon

day]

On

Oct

ober

19(M

onda

y),1

987

the

SPX

drop

ped

by23

%.

Fitti

ngth

em

odel

sto

ada

tase

ries

of2,

490

daily

obse

rvat

ions

endi

ngw

ithO

ctob

er16

(Fri

day)

,19

87yi

elds

the

follo

win

gre

sults

.

Mod

elIn

nova

tion

α0

α1

β1

ab

cK

SA

DR

esid

ual

Prob

abili

tyA

vera

getim

e

CV

Nor

mal

8.0

606·1

0−

54.4

448·1

0−

40.0

369

(0.0

022

)11.0

2−

25.5

62.3

7·1

0−

144

7.9

66·1

0141

t(d=

5.5

)8.1

175·1

0−

54.2

420·1

0−

4

CT

Sα

=1.8

781

λ+

=0.0

776

λ−

=0.0

826

0.0

172

(0.4

522

)0.0

693

−25.4

63.0

1·1

0−

61320

MT

Sα

=1.8

781

λ+

=0.1

865

λ−

=0.1

856

0.0

177

(0.4

162

)0.0

647

−25.4

68.0

7·1

0−

74918

RD

TS

α=

1.8

781

λ+

=0.0

972

λ−

=0.0

938

0.0

183

(0.3

765

)0.0

617

−25.4

62.9

5·1

0−

81.3

47·1

05

stab

leα

=1.8

781

β=

0.1

091

σ=

0.6

297

µ=

0.0

069

0.0

209

(0.2

258

)0.0

638

−25.4

64.9

6·1

0−

580

GA

RC

HN

orm

al1.3

480·1

0−

60.0

436

0.9

408

4.7

470·1

0−

40.0

332

(0.0

081

)4.7

78

−13.5

07.3

9·1

0−

42

2.5

54·1

039

t(d=

7.2

)1.0

912·1

0−

60.0

357

0.9

515

4.2

530·1

0−

4

CT

Sα

=1.9

135

λ+

=0.0

852

λ−

=0.0

861

0.0

194

(0.2

992

)0.0

585

−14.3

81.8

8·1

0−

5210.7

MT

Sα

=1.9

135

λ+

=0.2

169

λ−

=0.1

954

0.0

185

(0.3

603

)0.0

439

−14.3

81.1

2·1

0−

5352.9

RD

TS

α=

1.9

135

λ+

=0.1

179

λ−

=0.0

972

0.0

183

(0.3

677

)0.0

645

−14.3

85.0

4·1

0−

6787

stab

leα

=1.9

135

β=

0.0

640

σ=

0.6

531

µ=

0.0

005

0.0

189

(0.3

309

)0.0

600

−14.3

81.0

7·1

0−

437.2

6

AR

MA

-GA

RC

HN

orm

al1.2

721·1

0−

60.0

437

0.9

415

0.1

111

0.0

038

4.1

739·1

0−

40.0

320

(0.0

118

)8.3

12

−13.5

16.5

0·1

0−

42

2.9

04·1

039

t(d=

7.4

)9.8

761·1

0−

70.0

356

0.9

528

0.0

227

0.0

863

4.0

424·1

0−

4

CT

Sα

=1.9

163

λ+

=0.0

793

λ−

=0.0

837

0.0

170

(0.4

612

)0.0

579

−14.3

71.9

0·1

0−

5209.1

MT

Sα

=1.9

163

λ+

=0.2

045

λ−

=0.1

899

0.0

176

(0.4

200

)0.0

536

−14.3

71.1

7·1

0−

5339.2

RD

TS

α=

1.9

163

λ+

=0.1

109

λ−

=0.0

947

0.0

181

(0.3

853

)0.0

521

−14.3

75.5

2·1

0−

6719.4

stab

leα

=1.9

163

β=

0.1

034

σ=

0.6

545

µ=

0.0

027

0.0

190

(0.3

269

)0.0

453

−14.3

79.8

4·1

0−

540.3

1

22

Tabl

e4:

[Dai

lyan

alys

is-A

sian

Turm

oil]

On

the

Oct

ober

27(M

onda

y),1

997

the

SPX

drop

ped

by7%

.Fi

tting

the

mod

els

toa

data

seri

esof

2,50

4da

ilyob

serv

atio

nsen

ding

with

the

Oct

ober

24(F

rida

y),1

997

yiel

dsth

efo

llow

ing

resu

lts.

Mod

elIn

nova

tion

α0

α1

β1

ab

cK

SA

DR

esid

ual

Prob

abili

tyA

vera

getim

e

CV

Nor

mal

7.4

037·1

0−

55.3

247·1

0−

40.0

708

(0.0

000

)2.4

74·1

08

−8.3

34.1

0·1

0−

17

4.5

99·1

014

t(d=

3.5

)7.8

200·1

0−

56.4

055·1

0−

4

CT

Sα

=1.3

190

λ+

=0.3

681

λ−

=0.3

201

0.0

198

(0.2

790

)0.0

455

−8.1

22.2

5·1

0−

417.6

6

MT

Sα

=1.3

190

λ+

=0.5

721

λ−

=0.4

918

0.0

194

(0.3

031

)0.0

541

−8.1

21.5

2·1

0−

426.0

3

RD

TS

α=

1.3

291

λ+

=0.2

905

λ−

=0.2

271

0.0

202

(0.2

557

)0.1

164

−8.1

26.8

8·1

0−

557.7

stab

leα

=1.7

586

β=−

0.0

727

σ=

0.5

434

µ=−

0.0

089

0.0

295

(0.0

255

)0.0

644

−8.1

21.0

4·1

0−

33.8

21

GA

RC

HN

orm

al3.1

255·1

0−

70.0

240

0.9

710

5.1

106·1

0−

40.0

523

(0.0

000

)2.4

9·1

08

−7.2

52.1

2·1

0−

13

8.8

92·1

010

t(d=

5.1

)4.0

150·1

0−

70.0

344

0.9

605

6.4

544·1

0−

4

CT

Sα

=1.8

533

λ+

=0.2

211

λ−

=0.0

428

0.0

299

(0.0

222

)0.0

933

−7.1

01.8

0·1

0−

422.0

1

MT

Sα

=1.8

533

λ+

=0.4

718

λ−

=0.0

969

0.0

303

(0.0

196

)0.0

903

−7.1

01.6

8·1

0−

423.6

RD

TS

α=

1.8

533

λ+

=0.2

515

λ−

=0.0

460

0.0

283

(0.0

352

)0.0

840

−7.1

02.7

3·1

0−

414.5

6

stab

leα

=1.8

533

β=−

0.0

076

σ=

0.6

166

µ=

0.0

044

0.0

248

(0.0

915

)0.0

904

−7.1

07.8

6·1

0−

45.0

49

AR

MA

-GA

RC

HN

orm

al3.1

544·1

0−

70.0

243

0.9

706

−0.0

482

0.0

805

5.3

348·1

0−

40.0

505

(0.0

000

)2.4

88·1

08

−7.2

32.3

3·1

0−

13

8.1·1

010

t(d=

4.8

)3.4

659·1

0−

70.0

332

0.9

627

0.8

810

−0.9

139

7.6

174·1

0−

5

CT

Sα

=1.8

428

λ+

=0.2

469

λ−

=0.0

431

0.0

319

(0.0

122

)0.0

949

−7.1

91.9

8·1

0−

420.0

7

MT

Sα

=1.8

428

λ+

=0.5

138

λ−

=0.0

973

0.0

323

(0.0

105

)0.0

921

−7.1

91.3

8·1

0−

428.7

3

RD

TS

α=

1.8

428

λ+

=0.2

724

λ−

=0.0

463

0.0

299

(0.0

220

)0.0

844

−7.1

92.1

0·1

0−

418.8

7

stab

leα

=1.8

428

β=−

0.0

305

σ=

0.6

105

µ=

0.0

040

0.0

245

(0.0

976

)0.0

872

−7.1

98.4

3·1

0−

44.7

07

23

Tabl

e5:

[Dai

lyan

alys

is-R

ussi

anD

efau

lt]th

eA

ugus

t31

(Mon

day)

,199

8th

eSP

Xdr

oppe

dby

7%.

Fitti

ngth

em

odel

sto

ada

tase

ries

of2,

503

daily

obse

rvat

ions

endi

ngw

ithth

eA

ugus

t28

(Fri

day)

,199

7yi

elds

the

follo

win

gre

sults

.

Mod

elIn

nova

tion

α0

α1

β1

ab

cK

SA

DR

esid

ual

Prob

abili

tyA

vera

getim

e

CV

Nor

mal

6.5

602·1

0−

55.4

532·1

0−

40.0

623

(0.0

000

)6.0

22·1

05

−8.7

69.4

4·1

0−

19

2·1

016

t(d=

4.2

)6.7

229·1

0−

56.5

505·1

0−

4

CT

Sα

=1.3

576

λ+

=0.4

639

λ−

=0.3

815

0.0

214

(0.2

017

)0.0

491

−8.6

79.4

0·1

0−

542.2

2

MT

Sα

=1.3

576

λ+

=0.7

084

λ−

=0.5

823

0.0

207

(0.2

331

)0.0

671

−8.6

75.3

3·1

0−

574.4

5

RD

TS

α=

1.3

576

λ+

=0.3

481

λ−

=0.2

704

0.0

197

(0.2

821

)0.1

255

−8.6

71.2

4·1

0−

5320.4

stab

leα

=1.8

101

β=−

0.0

524

σ=

0.5

841

µ=−

0.0

016

0.0

259

(0.0

684

)0.0

776

−8.6

77.1

9·1

0−

45.5

15

GA

RC

HN

orm

al2.5

536·1

0−

70.0

251

0.9

719

5.3

767·1

0−

40.0

530

(0.0

000

)5.8

13·1

06

−5.7

15.6

7·1

0−

93.3

3·1

06

t(d=

5.2

)2.2

528·1

0−

70.0

295

0.9

683

6.6

287·1

0−

4

CT

Sα

=1.8

597

λ+

=0.2

397

λ−

=0.0

407

0.0

295

(0.0

254

)0.1

023

−5.5

16.0

6·1

0−

46.5

43

MT

Sα

=1.8

597

λ+

=0.4

992

λ−

=0.0

954

0.0

298

(0.0

227

)0.0

997

−5.5

15.5

5·1

0−

47.1

49

RD

TS

α=

1.8

597

λ+

=0.2

629

λ−

=0.0

458

0.0

277

(0.0

422

)0.0

931

−5.5

17.3

2·1

0−

45.4

2

stab

leα

=1.8

597

β=−

0.0

470

σ=

0.6

198

µ=

0.0

024

0.0

222

(0.1

662

)0.0

958

−5.5

11.2

9·1

0−

33.0

7

AR

MA

-GA

RC

HN

orm

al2.3

942·1

0−

70.0

250

0.9

723

0.0

606

−0.0

248

5.1

065·1

0−

40.0

500

(0.0

000

)5.6

64·1

06

−5.6

96.4

6·1

0−

92.9

22·1

06

t(d=

5.2

)2.3

890·1

0−

70.0

304

0.9

673

−0.0

278

0.0

391

6.7

898·1

0−

4

CT

Sα

=1.8

609

λ+

=0.2

367

λ−

=0.0

400

0.0

285

(0.0

332

)0.1

054

−5.4

76.1

7·1

0−

46.4

29

MT

Sα

=1.8

609

λ+

=0.4

953

λ−

=0.0

939

0.0

289

(0.0

295

)0.1

028

−5.4

75.8

0·1

0−

46.8

39

RD

TS

α=

1.8

609

λ+

=0.2

612

λ−

=0.0

450

0.0

269

(0.0

531

)0.0

961

−5.4

77.6

6·1

0−

45.1

8

stab

leα

=1.8

609

β=−

0.0

377

σ=

0.6

200

µ=

0.0

031

0.0

219

(0.1

782

)0.1

003

−5.4

71.2

8·1

0−

33.0

93

24

Tabl

e6:

[Dai

lyan

alys

is-D

otco

mC

olla

pse]

On

the

Apr

il14

,200

0th

eSP

Xdr

oppe

dby

6%.

Fitti

ngth

em

odel

sto

ada

tase

ries

of2,

504

daily

obse

rvat

ions

endi

ngw

ithth

eA

pril

13,2

000

yiel

dsth

efo

llow

ing

resu

lts.

Mod

elIn

nova

tion

α0

α1

β1

ab

cK

SA

DR

esid

ual

Prob

abili

tyA

vera

getim

e

CV

Nor

mal

8.4

098·1

0−

55.7

153·1

0−

40.0

665

(0.0

000

)1.1

48·1

04

−6.6

11.9

2·1

0−

11

9.8

16·1

08

t(d=

3.7

)9.0

250·1

0−

56.5

042·1

0−

4

CT

Sα

=1.3

344

λ+

=0.3

800

λ−

=0.3

402

0.0

216

(0.1

895

)0.0

444

−6.3

95.4

6·1

0−

47.2

64

MT

Sα

=1.3

344

λ+

=0.5

868

λ−

=0.5

267

0.0

209

(0.2

238

)0.0

449

−6.3

94.2

6·1

0−

49.3

21

RD

TS

α=

1.3

344

λ+

=0.2

920

λ−

=0.2

546

0.0

204

(0.2

479

)0.0

858

−6.3

92.4

4·1

0−

416.2

7

stab

leα

=1.7

792

β=−

0.0

653

σ=

0.5

570

µ=−

0.0

046

0.0

321

(0.0

112

)0.0

721

−6.3

91.4

7·1

0−

32.7

04

GA

RC

HN

orm

al3.7

464·1

0−

70.0

462

0.9

509

5.8

652·1

0−

40.0

424

(0.0

002

)253.2

−4.3

18.2

3·1

0−

62292

t(d=

5.8

)2.0

000·1

0−

70.0

383

0.9

612

6.4

505·1

0−

4

CT

Sα

=1.8

826

λ+

=0.2

911

λ−

=0.0

382

0.0

290

(0.0

293

)0.0

923

−4.2

41.6

4·1

0−

32.4

24

MT

Sα

=1.8

826

λ+

=0.5

814

λ−

=0.0

951

0.0

293

(0.0

263

)0.0

907

−4.2

41.5

4·1

0−

32.5

83

RD

TS

α=

1.8

826

λ+

=0.2

934

λ−

=0.0

468

0.0

271

(0.0

503

)0.0

839

−4.2

41.7

7·1

0−

32.2

38

stab

leα

=1.8

826

β=−

0.0

684

σ=

0.6

334

µ=

0.0

009

0.0

211

(0.2

138

)0.0

866

−4.2

41.9

4·1

0−

32.0

42

AR

MA

-GA

RC

HN

orm

al4.0

049·1

0−

70.0

479

0.9

489

0.0

824

−0.0

307

5.4

464·1

0−

40.0

414

(0.0

004

)243.9

−5.6

96.4

6·1

0−

92.9

22·1

06

t(d=

5.8

)2.2

440·1

0−

70.0

389

0.9

601

−0.9

832

0.9

896

1.2

602·1

0−

3

CT

Sα

=1.8

844

λ+

=0.2

912

λ−

=0.0

382

0.0

277

(0.0

422

)0.0

941

−5.4

76.1

7·1

0−

46.4

29

MT

Sα

=1.8

844

λ+

=0.5

817

λ−

=0.0

954

0.0

280

(0.0

386

)0.0

925

−5.4

75.8

0·1

0−

46.8

39

RD

TS

α=

1.8

844

λ+

=0.2

921

λ−

=0.0

471

0.0

257

(0.0

720

)0.0

854

−5.4

77.6

6·1

0−

45.1

8

stab

leα

=1.8

844

β=−

0.0

612

σ=

0.6

347

µ=

0.0

023

0.0

208

(0.2

287

)0.0

880

−5.4

71.2

8·1

0−

33.0

93

25

Tabl

e7:

[Dai

lyan

alys

is-U

.S.F

inan

cial

Cri

sis]

On

the

Sept

embe

r29

(Mon

day)

,200

8th

eSP

Xdr

oppe

dby

9%.

Fitti

ngth

em

odel

sto

ada

tase

ries

of2,

505

daily

obse

rvat

ions

endi

ngw

ithth

eSe

ptem

ber2

6(F

rida

y),2

008

yiel

dsth

efo

llow

ing

resu

lts.

Mod

elIn

nova

tion

α0

α1

β1

ab

cK

SA

DR

esid

ual

Prob

abili

tyA

vera

getim

e

CV

Nor

mal

1.3

471·1

0−

45.9

697·1

0−

50.0

532

(0.0

000

)1.1

94

−7.9

59.4

8·1

0−

16

1.9

91·1

013

t(d=

4.2

)1.4

349·1

0−

42.0

184·1

0−

4

CT

Sα

=1.3

648

λ+

=0.4

466

λ−

=0.3

891

0.0

207

(0.2

332

)0.0

588

−7.7

11.5

9·1

0−

424.9

9

MT

Sα

=1.3

648

λ+

=0.6

802

λ−

=0.6

043

0.0

195

(0.2

933

)0.0

553

−7.7

19.4

8·1

0−

541.8

7

RD

TS

α=

1.3

648

λ+

=0.3

255

λ−

=0.2

971

0.0

189

(0.3

318

)0.0

488

−7.7

12.1

1·1

0−

5188.5

stab

leα

=1.8

198

β=−

0.4

876

σ=

0.5

870

µ=−

0.0

577

0.0

694

(0.0

000

)0.1

648

−7.7

11.2

3·1

0−

33.2

36

GA

RC

HN

orm

al8.5

044·1

0−

70.0

620

0.9

330

3.0

978·1

0−

40.0

356

(0.0

034

)131.6

−3.9

63.7

6·1

0−

5502.1

t(d=

9.8

)5.0

125·1

0−

70.0

631

0.9

358

4.0

891·1

0−

4

CT

Sα

=1.7

485

λ+

=1.1

223

λ−

=0.3

720

0.0

360

(0.0

029

)0.0

788

−3.8

91.2

7·1

0−

33.1

32

MT

Sα

=1.7

485

λ+

=1.7

016

λ−

=0.6

410

0.0

355

(0.0

035

)0.0

790

−3.8

91.2

2·1

0−

33.2

44

RD

TS

α=

1.7

485

λ+

=0.8

719

λ−

=0.2

751

0.0

274

(0.0

460

)0.0

829

−3.8

91.2

1·1

0−

33.2

81

stab

leα

=1.9

427

β=−

1.0

000

σ=

0.6

728

µ=−

0.0

443

0.0

428

(0.0

002

)0.0

875

−3.8

92.5

1·1

0−

31.5

83

AR

MA

-GA

RC

HN

orm

al8.4

382·1

0−

70.0

610

0.9

339

0.7

618

−0.8

109

7.6

710·1

0−

50.0

369

(0.0

021

)167.2

−4.0

52.5

8·1

0−

5732.3

t(d=

9.5

)4.8

918·1

0−

70.0

628

0.9

362

0.7

157

−0.7

725

1.2

236·1

0−

4

CT

Sα

=1.7

467

λ+

=1.7

836

λ−

=0.3

547

0.0

384

(0.0

012

)0.1

037

−3.9

71.2

9·1

0−

33.0

74

MT

Sα

=1.7

467

λ+

=2.5

175

λ−

=0.6

143

0.0

382

(0.0

013

)0.2

421

−3.9

71.2

6·1

0−

33.1

44

RD

TS

α=

1.7

467

λ+

=1.3

930

λ−

=0.2

706

0.0

230

(0.1

398

)0.0

814

−3.9

71.2

3·1

0−

33.2

32

stab

leα

=1.9

408

β=−

1.0

000

σ=

0.6

714

µ=−

0.0

318

0.0

419

(0.0

003

)0.0

832

−3.9

72.4

2·1

0−

31.6

39

26

Tabl

e8:

[Wee

kly

anal

ysis

-Bla

ckM

onda

y]O

nO

ctob

er23

(Fri

day)

,198

7th

eSP

Xdr

oppe

dby

13.0

1%.F

ittin

gth

em

odel

sto

ada

tase

ries

of1,

461

wee

kly

retu

rns

endi

ngw

ithO

ctob

er16

(Fri

day)

,198

7yi

elds

the

follo

win

gre

sults

.

Mod

elIn

nova

tion

α0

α1

β1

ab

cK

SA

DR

esid

ual

Prob

abili

tyA

vera

getim

e

CV

Nor

mal

3.8

743·1

0−

41.0

667·1

0−

30.0

457

(0.0

043

)1.1

55

−6.6

61.3

5·1

0−

11

6.6

59·1

09

t(d=

5.3

)3.8

743·1

0−

41.6

507·1

0−

3

CT

Sα

=1.6

732

λ+

=0.3

532

λ−

=0.1

783

0.0

247

(0.3

336

)0.0

727

−6.6

95.1

1·1

0−

436.9

1

MT

Sα

=1.6

732

λ+

=0.5

942

λ−

=0.3

481

0.0

260

(0.2

759

)0.0

670

−6.6

94.0

0·1

0−

447.1

5

RD

TS

α=

1.6

732

λ+

=0.2

649

λ−

=0.1

768

0.0

296

(0.1

537

)0.0

652

−6.6

92.7

5·1

0−

468.6

stab

leα

=1.8

591

β=−

0.8

588

σ=

0.6

197

µ=−

0.0

690

0.0

848

(0.0

000

)0.1

881

−6.6

91.6

5·1

0−

311.4

6

GA

RC

HN

orm

al1.3

360·1

0−

50.1

426

0.8

306

1.6

183·1

0−

30.0

362

(0.0

421

)0.2

784

−2.9

01.8

6·1

0−

348.2

5

t(d=

10.2

)1.3

147·1

0−

50.1

379

0.8

349

1.8

142·1

0−

3

CT

Sα

=1.7

489

λ+

=0.9

288

λ−

=0.2

687

0.0

280

(0.1

975

)0.0

855

−2.9

45.4

7·1

0−

33.4

5

MT

Sα

=1.7

489

λ+

=1.5

430

λ−

=0.5

177

0.0

279

(0.2

029

)0.0

877

−2.9

45.0

0·1

0−

33.7

72

RD

TS

α=

1.7

489

λ+

=0.9

916

λ−

=0.2

569

0.0

284

(0.1

848

)0.0

726

−2.9

46.4

4·1

0−

32.9

28

stab

leα

=1.9

433

β=−

1.0

000

σ=

0.6

719

µ=−

0.0

049

0.0

402

(0.0

175

)0.0

790

−2.9

45.9

4·1

0−

33.1

74

AR

MA

-GA

RC

HN

orm

al1.3

088·1

0−

50.1

441

0.8

301

0.1

698

−0.1

040

1.3

525·1

0−

30.0

380

(0.0

290

)0.2

775

−2.7

92.6

4·1

0−

334

t(d=

10.0

)1.2

923·1

0−

50.1

383

0.8

351

0.2

044

−0.1

377

1.4

415·1

0−

3

CT

Sα

=1.7

489

λ+

=0.8

297

λ−

=0.2

749

0.0

286

(0.1

800

)0.0

781

−2.8

36.4

0·1

0−

32.9

49

MT

Sα

=1.7

489

λ+

=1.3

740

λ−

=0.5

249

0.0

286

(0.1

783

)0.0

799

−2.8

36.2

6·1

0−

33.0

15

RD

TS

α=

1.7

489

λ+

=0.7

575

λ−

=0.2

516

0.0

247

(0.3

309

)0.0

701

−2.8

37.3

3·1

0−

32.5

75

stab

leα

=1.9

433

β=−

1.0

000

σ=

0.6

717

µ=−

0.0

151

0.0

426

(0.0

097

)0.0

838

−2.8

36.9

9·1

0−

32.6

99

27

Tabl

e9:

[Wee

kly

anal

ysis

-Dot

com

Col

laps

e]O

nth

eA

pril

14,

2000

the

SPX

drop

ped

by11

.14%

.Fi

tting

the

mod

els

toa

data

seri

esof

1,57

7w

eekl

yre

turn

sen

ding

with

the

Apr

il7,

2000

yiel

dsth

efo

llow

ing

resu

lts.

Mod

elIn

nova

tion

α0

α1

β1

ab

cK

SA

DR

esid

ual

Prob

abili

tyA

vera

getim

e

CV

Nor

mal

4.4

794·1

0−

41.8

034·1

0−

30.0

375

(0.0

232

)41.6

4−

5.3

54.4

8·1

0−

82.0

01·1

06

t(d=

6.0

)4.4

452·1

0−

42.2

638·1

0−

3

CT

Sα

=1.6

955

λ+

=0.3

916

λ−

=0.2

030

0.0

212

(0.4

695

)0.0

531

−5.3

98.5

7·1

0−

422.0

1

MT

Sα

=1.6

955

λ+

=0.6

707

λ−

=0.3

834

0.0

214

(0.4

602

)0.0

535

−5.3

98.0

5·1

0−

423.4

4

RD

TS

α=

1.6

955

λ+

=0.3

044

λ−

=0.1

849

0.0

242

(0.3

108

)0.0

497

−5.3

96.9

4·1

0−

427.2

1

stab

leα

=1.8

838

β=−

0.6

164

σ=

0.6

354

µ=−

0.0

385

0.0

571

(0.0

001

)0.1

131

−5.3

91.8

2·1

0−

310.3

9

GA

RC

HN

orm

al1.6

004·1

0−

50.1

139

0.8

526

2.0

853·1

0−

30.0

316

(0.0

842

)0.4

072

−3.8

65.6

9·1

0−

51577

t(d=

12.3

)1.4

605·1

0−

50.1

064

0.8

623

2.2

815·1

0−

3

CT

Sα

=1.7

621

λ+

=1.0

769

λ−

=0.3

246

0.0

226

(0.3

902

)0.0

602

−3.8

71.3

9·1

0−

313.5

5

MT

Sα

=1.7

621

λ+

=1.7

727

λ−

=0.6

007

0.0

228

(0.3

795

)0.0

598

−3.8

71.0

7·1

0−

317.6

5

RD

TS

α=

1.7

621

λ+

=0.8

615

λ−

=0.2

772

0.0

270

(0.1

963

)0.0

528

−3.8

71.5

0·1

0−

312.6

stab

leα

=1.9

579

β=−

1.0

000

σ=

0.6

818

µ=

0.0

011

0.0

340

(0.0

512

)0.0

667

−3.8

71.8

4·1

0−

310.2

5

AR

MA

-GA

RC

HN

orm

al1.6

031·1

0−

50.1

148

0.8

515

−0.1

471

0.0

957

2.3

911·1

0−

30.0

320

(0.0

774

)0.3

841

−3.8

65.7

4·1

0−

51563

t(d=

12.4

)1.4

679·1

0−

50.1

072

0.8

613

−0.1

981

0.1

485

2.7

310·1

0−

3

CT

Sα

=1.7

611

λ+

=1.1

165

λ−

=0.3

190

0.0

237

(0.3

322

)0.0

684

−3.8

71.4

0·1

0−

313.4

7

MT

Sα

=1.7

611

λ+

=1.8

492

λ−

=0.5

945

0.0

238

(0.3

261

)0.0

678

−3.8

71.0

2·1

0−

318.5

4

RD

TS

α=

1.7

611

λ+

=0.9

119

λ−

=0.2

766

0.0

270

(0.1

982

)0.0

527

−3.8

71.5

3·1

0−

312.3

1

stab

leα

=1.9

567

β=−

1.0

000

σ=

0.6

814

µ=

0.0

069

0.0

317

(0.0

825

)0.0

622

−3.8

71.8

9·1

0−

310.0

1

28

Tabl

e10

:[W

eekl

yan

alys

is-U

.S.F

inan

cial

Cri

sis]

On

the

Oct

ober

10(F

rida

y),2

008,

the

wee

kly

retu

rnof

the

SPX

drop

ped

by20

.08%

.Fitt

ing

the

mod

els

toa

data

seri

esof

1,57

8w

eekl

yre

turn

sen

ding

with

the

Oct

ober

3(F

rida

y),

2008

yiel

dsth

efo

llow

ing

resu

lts.

Mod

elIn

nova

tion

α0

α1

β1

ab

cK

SA

DR

esid

ual

Prob

abili

tyA

vera

getim

e

CV

Nor

mal

4.5

519·1

0−

41.4

972·1

0−

30.0

428

(0.0

060

)33.9

4−

9.4

81.2

3·1

0−

21

7.3

04·1

019

t(d=

5.5

)4.5

234·1

0−

42.1

609·1

0−

3

CT

Sα

=1.6

810

λ+

=0.4

180

λ−

=0.1

736

0.0

244

(0.3

018

)0.0

614

−9.5

41.4

1·1

0−

4133.8

MT

Sα

=1.6

810

λ+

=0.7

334

λ−

=0.3

246

0.0

244

(0.2

997

)0.0

616

−9.5

48.7

0·1

0−

5216.9

RD

TS

α=

1.6

810

λ+

=0.3

633

λ−

=0.1

531

0.0

220

(0.4

252

)0.0

557

−9.5

45.3

9·1

0−

5350

stab

leα

=1.8

678

β=−

0.5

103

σ=

0.6

246

µ=−

0.0

377

0.0

549

(0.0

001

)0.1

090

−9.5

46.0

8·1

0−

431.0

3

GA

RC

HN

orm

al9.5

757·1

0−

60.0

912

0.8

920

1.9

028·1

0−

30.0

331

(0.0

620

)1.2

54

−5.5

01.9

5·1

0−

84.5

98·1

06

t(d=

7.9

)6.9

895·1

0−

60.0

698

0.9

171

2.2

927·1

0−

3

CT

Sα

=1.7

251

λ+

=1.0

163

λ−

=0.2

051

0.0

264

(0.2

173

)0.0

644

−5.9

33.4

0·1

0−

455.5

2

MT

Sα

=1.7

251

λ+

=1.6

675

λ−

=0.3

938

0.0

273

(0.1

879

)0.0

664

−5.9

31.2

3·1

0−

4153.1

RD

TS

α=

1.7

251

λ+

=0.8

589

λ−

=0.1

916

0.0

199

(0.5

513

)0.0

482

−5.9

34.1

4·1

0−

445.5

3

stab

leα

=1.9

168

β=−

1.0

000

σ=

0.6

566

µ=−

0.0

301

0.0

513

(0.0

005

)0.1

015

−5.9

31.3

3·1

0−

314.1

8

AR

MA

-GA

RC

HN

orm

al1.1

370·1

0−

50.1

024

0.8

778

−0.3

992

0.3

205

2.7

223·1

0−

30.0

351

(0.0

403

)0.9

502

−5.3

73.8

4·1

0−

82.3

36·1

06

t(d=

7.8

)7.1

469·1

0−

60.0

720

0.9

146

−0.4

602

0.3

828

3.3

859·1

0−

3

CT

Sα

=1.7

229

λ+

=1.0

587

λ−

=0.1

978

0.0

262

(0.2

258

)0.0

671

−5.9

93.1

2·1

0−

460.5

3

MT

Sα

=1.7

229

λ+

=1.7

442

λ−

=0.3

827

0.0

273

(0.1

873

)0.0

749

−5.9

99.9

4·1

0−

5189.8

RD

TS

α=

1.7

229

λ+

=0.9

381

λ−

=0.1

887

0.0

200

(0.5

491

)0.0

504

−5.9

94.1

8·1

0−

445.0

9

stab

leα

=1.9

144

β=−

1.0

000

σ=

0.6

553

µ=−

0.0

256

0.0

495

(0.0

008

)0.0

979

−5.9

91.3

4·1

0−

314.0

8

29

Tabl

e11

:[M

onth

lyan

alys

is-B

lack

Mon

day]

On

Oct

ober

1987

the

SPX

drop

ped

by24

.54%

.Fitt

ing

the

mod

els

toa

data

seri

esof

332

mon

thly

retu

rns

endi

ngw

ithSe

ptem

ber1

987

yiel

dsth

efo

llow

ing

resu

lts:

Mod

elIn

nova

tion

α0

α1

β1

ab

cK

SA

DR

esid

ual

Prob

abili

tyA

vera

getim

e

CV

Nor

mal

1.7

083·1

0−

35.2

881·1

0−

30.0

456

(0.4

813

)0.1

453

−6.0

76.5

6·1

0−

10

6.0

44·1

08

t(d=

7.7

)1.7

267·1

0−

35.8

053·1

0−

3

CT

Sα

=1.7

296

λ+

=0.4

410

λ−

=0.3

131

0.0

336

(0.8

354

)0.1

025

−6.0

52.4

7·1

0−

4336.9

MT

Sα

=1.7

296

λ+

=0.7

528

λ−

=0.5

779

0.0

335

(0.8

369

)0.1

019

−6.0

51.7

0·1

0−

4490.4

RD

TS

α=

1.7

296

λ+

=0.3

463

λ−

=0.2

818

0.0

334

(0.8

398

)0.1

023

−6.0

58.1

2·1

0−

51026

stab

leα

=1.9

218

β=−

1.0

000

σ=

0.6

582

µ=−

0.0

385

0.0

537

(0.2

893

)0.1

628

−6.0

51.2

0·1

0−

369.5

GA

RC

HN

orm

al2.1

009·1

0−

40.0

970

0.7

814

5.7

438·1

0−

30.0

516

(0.3

275

)0.1

715

−6.2

22.4

4·1

0−

10

1.6

23·1

09

t(d=

10.1

)1.9

588·1

0−

40.0

957

0.7

930

6.0

702·1

0−

3

CT

Sα

=1.7

551

λ+

=1.9

364

λ−

=0.4

299

0.0

510

(0.3

429

)0.1

431

−6.1

73.4

0·1

0−

52452

MT

Sα

=1.7

551

λ+

=2.5

304

λ−

=0.7

333

0.0

506

(0.3

521

)0.1

431

−6.1

71.4

5·1

0−

55758

RD

TS

α=

1.7

551

λ+

=1.2

137

λ−

=0.3

391

0.0

415

(0.6

051

)0.1

243

−6.1

73.2

4·1

0−

52570

stab

leα

=1.9

501

β=−

1.0

000

σ=

0.6

724

µ=−

0.0

060

0.0

374

(0.7

370

)0.1

179

−6.1

77.1

4·1

0−

4116.7

AR

MA

-GA

RC

HN

orm

al2.0

619·1

0−

40.1

080

0.7

720

−0.8

308

0.8

932

1.0

849·1

0−

20.0

436

(0.5

398

)0.1

716

−6.2

81.7

2·1

0−

10

2.3

06·1

09

t(d=

11.3

)1.9

494·1

0−

40.1

038

0.7

843

−0.8

444

0.8

977

1.1

303·1

0−

2

CT

Sα

=1.7

568

λ+

=2.7

246

λ−

=0.5

849

0.0

483

(0.4

090

)0.1

424

−6.2

26.5

5·1

0−

61.2

73·1

04

MT

Sα

=1.7

568

λ+

=2.9

115

λ−

=0.9

105

0.0

474

(0.4

345

)0.1

415

−6.2

26.2

1·1

0−

61.3

42·1

04

RD

TS

α=

1.7

568

λ+

=1.3

247

λ−

=0.4

045

0.0

375

(0.7

268

)0.1

167

−6.2

21.1

6·1

0−

57168

stab

leα

=1.9

520

β=−

1.0

000

σ=

0.6

748

µ=−

0.0

072

0.0

337

(0.8

423

)0.1

349

−6.2

26.7

8·1

0−

4122.9

30

Tabl

e12

:[M

onth

lyan

alys

is-U

.S.F

inan

cial

Cri

sis]

On

the

Oct

ober

2008

,the

mon

thly

retu

rnof

the

SPX

drop

ped

by18

.42%

.Fi

tting

the

mod

els

toa

data

seri

esof

359

mon

thly

retu

rns

endi

ngw

ithth

eSe

ptem

ber

2008

yiel

dsth

efo

llow

ing

resu

lts.

Mod

elIn

nova

tion

α0

α1

β1

ab

cK

SA

DR

esid

ual

Prob

abili

tyA

vera

getim

e

CV

Nor

mal

1.8

356·1

0−

37.0

363·1

0−

30.0

620

(0.1

213

)63.9

−4.4

64.0

1·1

0−

69.8

81·1

04

t(d=

5.6

)1.8

217·1

0−

39.2

266·1

0−

3

CT

Sα

=1.6

968

λ+

=1.4

962

λ−

=0.1

570

0.0

463

(0.4

114

)0.2

044

−4.5

37.1

7·1

0−

4116.2

MT

Sα

=1.6

968

λ+

=2.2

526

λ−

=0.2

925

0.0

485

(0.3

562

)0.4

703

−4.5

32.4

9·1

0−

4334.3

RD

TS

α=

1.6

968

λ+

=1.1

662

λ−

=0.1

439

0.0

325

(0.8

301

)0.0

758

−4.5

32.4

5·1

0−

333.9

4

stab

leα

=1.8

853

β=−

1.0

000

σ=

0.6

333

µ=−

0.0

451

0.0

769

(0.0

278

)0.1

482

−4.5

33.2

7·1

0−

325.4

6

GA

RC

HN

orm

al4.5

429·1

0−

50.1

061

0.8

810

6.6

06·1

0−

30.0

667

(0.0

782

)23.2

9−

3.6

31.4

4·1

0−

42746

t(d=

5.9

)8.7

296·1

0−

50.1

064

0.8

549

9.1

459·1

0−

3

CT

Sα

=1.6

988

λ+

=2.1

451

λ−

=0.1

680

0.0

461

(0.4

200

)0.4

000

−3.5

81.1

8·1

0−

370.6

MT

Sα

=1.6

988

λ+

=3.1

288

λ−

=0.3

110

0.0

484

(0.3

595

)1.1

868

−3.5

84.0

9·1

0−

4203.8

RD

TS

α=

1.6

988

λ+

=1.5

484

λ−

=0.1

510

0.0

382

(0.6

584

)0.0

985

−3.5

85.3

5·1

0−

315.5

7

stab

leα

=1.8

876

β=−

1.0

000

σ=

0.6

351

µ=−

0.0

356

0.0

748

(0.0

350

)0.1

440

−3.5

85.6

3·1

0−

314.7

9

AR

MA

-GA

RC

HN

orm

al4.2

245·1

0−

50.1

179

0.8

727

0.9

511

−0.9

728

3.2

909·1

0−

40.0

637

(0.1

043

)10.7

2−

3.5

51.9

0·1

0−

42089

t(d=

5.7

)8.9

769·1

0−

50.1

087

0.8

521

0.3

140

−0.3

626

6.2

904·1

0−

3

CT

Sα

=1.6

953

λ+

=2.4

886

λ−

=0.1

665

0.0

447

(0.4

580

)0.5

968

−3.6

37.7

1·1

0−

4108.1

MT

Sα

=1.6

953

λ+

=3.5

328

λ−

=0.3

069

0.0

472

(0.3

894

)1.9

558

−3.6

32.3

1·1

0−

4361.3

RD

TS

α=

1.6

953

λ+

=1.6

659

λ−

=0.1

487

0.0

396

(0.6

144

)0.0

847

−3.6

35.2

8·1

0−

315.7

7

stab

leα

=1.8

836

β=−

1.0

000

σ=

0.6

322

µ=−

0.0

354

0.0

759

(0.0

311

)0.1

463

−3.6

35.5

9·1

0−

314.9

31

Tabl

e13

:N

umbe

rofv

iola

tions

(N)a

ndp

valu

esof

Kup

iec’

spr

opor

tion

offa

ilure

ste

stfo

rthe

S&P

500.

1yea

r(25

5da

ys)

2ye

ars

(510

days

)4

year

s(1

020

days

)

Mod

elD

ec.1

4,20

04D

ec.1

6,20

05D

ec.2

1,20

06D

ec.2

8,20

07D

ec.1

4,20

04D

ec.2

1,20

06D

ec.1

4,20

04∼

Dec

.15,

2005

∼D

ec.2

0,20

06∼

Dec

.27,

2007

∼D

ec.3

1,20

08∼

Dec

.20,

2006

∼D

ec.3

1,20

08∼

Dec

.31,

2008

N(p

-val

ue)

N(p

-val

ue)

N(p

-val

ue)

N(p

-val

ue)

N(p

-val

ue)

N(p

-val

ue)

N(p

-val

ue)

Nor

mal

-CV

2(0.

7190

)4(

0.39

95)

15(0

.000

0)25

(0.0

000)

6(0.

6968

)40

(0.0

000)

46(0

.000

0)E

WM

A3(

0.78

29)

5(0.

1729

)12

(0.0

000)

9(0.

0016

)8(

0.23

34)

21(0

.000

0)29

(0.0

000)

Nor

mal

-AR

MA

-GA

RC

H1(

0.26

60)

3(0.

7829

)9(

0.00

16)

10(0

.000

4)4(

0.61

09)

19(0

.000

0)23

(0.0

005)

Stab

le-A

RM

A-G

AR

CH

03(

0.78

29)

6(0.

0646

)4(

0.39

95)

3(0.

3113

)10

(0.0

539)

13(0

.398

0)C

TS-

AR

MA

-GA

RC

H0

3(0.

7829

)7(

0.02

11)

4(0.

3995

)3(

0.31

13)

11(0

.022

9)14

(0.2

577)

32

1987

, Blac

k Mon

day

1997

, Asia

n tu

rmoil

1998

, Rus

sian

defa

ult

2000

, Dot

com

colla

pse

2008

, Leh

man

Bro

ther

s

SPX daily returns

1987

, Blac

k Mon

day

1997

, Asia

n tu

rmoil

1998

, Rus

sian

defa

ult

2000

, Dot

com

colla

pse

2008

, Leh

man

Bro

ther

s

SPX weekly returns

1987

, Blac

k Mon

day

1997

, Asia

n tu

rmoil

1998

, Rus

sian

defa

ult

2000

, Dot

com

colla

pse

2008

, Leh

man

Bro

ther

s

SPX monthly returns

Figure 1: Daily, weekly, and monthly returns of the SPX from January 1987 toDecember 2008.

33

Dec2004

Mar2005

Jun2005

Sep2005

Dec2005

Mar2006

Jun2006

Sep2006

Dec2006

Mar2007

Jun2007

Sep2007

Dec2007

Mar2008

Jun2008

Sep2008

Dec2008

Normal−CV

Dec2004

Mar2005

Jun2005

Sep2005

Dec2005

Mar2006

Jun2006

Sep2006

Dec2006

Mar2007

Jun2007

Sep2007

Dec2007

Mar2008

Jun2008

Sep2008

Dec2008

EWMA

Dec2004

Mar2005

Jun2005

Sep2005

Dec2005

Mar2006

Jun2006

Sep2006

Dec2006

Mar2007

Jun2007

Sep2007

Dec2007

Mar2008

Jun2008

Sep2008

Dec2008

Normal−ARMA−GARCH

Dec2004

Mar2005

Jun2005

Sep2005

Dec2005

Mar2006

Jun2006

Sep2006

Dec2006

Mar2007

Jun2007

Sep2007

Dec2007

Mar2008

Jun2008

Sep2008

Dec2008

Stable−ARMA−GARCH

Dec2004

Mar2005

Jun2005

Sep2005

Dec2005

Mar2006

Jun2006

Sep2006

Dec2006

Mar2007

Jun2007

Sep2007

Dec2007

Mar2008

Jun2008

Sep2008

Dec2008

CTS−ARMA−GARCH

Figure 2: VaR (−VaRt,η(yt+1)) for the SPX returns for the normal-CV, the EWMA,the normal-ARMA-GARCH, the stable-ARMA-GARCH, and the CTS-ARMA-GARCHmodel. 34

Dec2004

Mar2005

Jun2005

Sep2005

Dec2005

Mar2006

Jun2006

Sep2006

Dec2006

Mar2007

Jun2007

Sep2007

Dec2007

Mar2008

Jun2008

Sep2008

Dec2008

SPX log returnsAVaR Normal−ARMA−GARCHAVaR CTS−ARMA−GARCH

Figure 3: The 1%-AVaRs (−AVaRt,0.01(yt+1)) of the SPX, for the normal-ARMA-GARCH and CTS-ARMA-GARCH models.

−0.02

−0.01

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07SPX

Dec2004Feb2005

May2005Jul2005

Sep2005

Dec2005Feb2006

May2006Jul2006

Sep2006

Dec2006Feb2007

May2007Jul2007

Sep2007

Dec2007Feb2008

May2008Jul2008

Sep2008

Dec2008

Figure 4: The differences between 1%-VaR for the normal-ARMA-GARCH and 1%-AVaR for the CTS-ARMA-GARCH models with respect to the SPX. The differences areincreasing in the solid rectangles.

−0.005

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045SPX

Dec2004Feb2005

May2005Jul2005

Sep2005

Dec2005Feb2006

May2006Jul2006

Sep2006

Dec2006Feb2007

May2007Jul2007

Sep2007

Dec2007Feb2008

May2008Jul2008

Sep2008

Dec2008

Figure 5: The differences between 1%-AVaR for the normal-ARMA-GARCH and 1%-AVaR for the CTS-ARMA-GARCH models with respect to the SPX. The differences areincreasing in the solid rectangles.

35

No. 2

No. 1

Young Shin Kim, Svetlozar T. Rachev, Michele Leonardo Bianchi, Ivan

Mitov, Frank J. Fabozzi: Time series analysis for financial market

meltdowns, August 2010

Biliana Güner, Svetlozar T. Rachev, Daniel Edelman, Frank J. Fabozzi:

Bayesian inference for hedge funds with stable distribution of returns,

August 2010

recent issues

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