ON CONDITIONAL MOMENTS OF GARCH MODELS, WITH APPLICATIONS TO MULTIPLE PERIOD VALUE AT RISK ESTIMATION

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1 Statistica Sinica 13(23), ON CONDITIONAL MOMENTS OF GARCH MODELS, WITH APPLICATIONS TO MULTIPLE PERIOD VALUE AT RISK ESTIMATION Ci-Ming Wong and Mike K. P. So TeHongKongUniversityofScience& Tecnology Abstract: In tis article, te exact conditional second, tird and fourt moments of returns and teir temporal aggregates are derived under Quadratic GARCH models. Tree multiple period Value at Risk estimation metods are proposed. Two metods are based on te exact second to fourt moments and te oter adopts a Monte Carlo approac. Some simulations sow tat te multiple period Value at Risk calculated from an asymmetric t-distribution wit te variance, skewness parameter and te degrees of freedom cosen to matc te second to fourt moments of te aggregate returns is close to te one obtained by Monte Carlo simulations. Using some market indices for illustration, te proposed Value at Risk estimation metods are found to be superior to some standard approaces suc as RiskMetrics. Key words and prases: Aggregate returns, eteroskedastic models, kurtosis, Monte Carlo metods, skewness, square root of time rule, volatility. 1. Introduction Tis article studies te conditional moments of temporal aggregate returns under some GARCH specifications. Let r t be te return at time t and Ω t be te information up to time t. Te aggregate return R t, at time t for a orizon is given by R t, = r t r t+. Denote te conditional variance of r t given Ω t 1 by σt 2. It is well-known tat if te variances are constant, σt 2 = σ2, and te returns r t are uncorrelated, te variance of te aggregate returns R t, is simply σ 2. In oter words, under te random walk ypotesis, te standard deviation or volatility of R t, is obtained by scaling σ wit. Tis simple scaling metod is called te square root of time rule, or rule. For example, if σ is te constant standard deviation of daily returns, te annual standard deviation is usually referred to 252σ, under te assumption tat we ave approximately 252 trading days per year. Altoug tis square root of time rule is widely accepted by practitioners to do annualization and to measure te risk in different orizons, its restrictions and problems are

2 116 CHI-MING WONG AND MIKE K. P. SO well known. For example, J. P. Morgan (1996, p.87) stated tat Typically, te square root of time rule results from te assumption tat variances are constant. Also, Diebold, Hickman, Inoue and Scuermann (1998) stated tat Te common practice of converting 1-day volatility estimates to -day estimates by scaling by is inappropriate and produces overestimates of te variability of long-orizon volatility. In ligt of te restrictions and problems of te rule pointed out in te literature, it is important to furter examine te rule in various scenarios. We focus our study on te GARCH framework. Te appropriateness of te rule depends on te conditional second moment properties of te aggregate returns. Recently, many studies ave investigated te moment properties of GARCH processes. See, for example, He and Teräsvirta (1999a, 1999b) and Duan, Gautier and Simonato (1999). Wile existing results on GARCH moments involve mainly te unconditional moments, Capters 3 and 7 of Tsay (22) studied te multiple period volatility forecasts under GARCH models. To examine te rule and to study te tail properties of te aggregate returns, we derive te exact conditional variance, skewness and kurtosis of R t, given Ω t for some GARCH processes. Troug tis variance, we provide teoretical justification for te adaptation of te square root of time rule in some cases suc as te RiskMetrics model of J. P. Morgan. More importantly, te variance, skewness and kurtosis enable us to construct two new metods for estimating multiple period Value at Risk (VaR). VaR is a common measure of risk. It is te loss of a portfolio tat will be exceeded wit a predetermined probability over a time period. In general, if C is te current market value of a portfolio and is te olding period, te -period VaR of tat portfolio is given by VaR = C V, (1) were V is te cutoff value wic is exceeded by -period returns wit probability 1 p. Terefore, estimating te VaR amounts to computing a percentile of te -period portfolio return distribution. Several approaces ave been developed, including te istorical simulation metod, variance-covariance metod and Monte Carlo simulation metod. Danielsson and de Vires (1997) discussed a newly-developed metod wic is based on extreme value teory. Ho, Burridge, Cadle and Teobald (2) applied extreme value teory to some Asian market indices. Lucas (2) considered te misspecification of tail properties in te return distribution and its effect on VaR estimation. For compreensive reviews of te VaR, one can refer to Duffie and Pan (1997), Jorion (1997), Dowd (1998) and Tsay (22).

3 ON CONDITIONAL MOMENTS OF GARCH MODELS 117 Most of te existing researces focus on te one-period VaR estimation, tat is, te time orizon is one unit. For te calculation of te VaR in long orizons, we need to know te distribution of R t, given Ω t, wic is generally not feasible. A traditional metod wic applies te rule treats R t, as a normal variable wit mean zero and variance calculated by scaling σt+1 2 wit. RiskMetrics adopted tis metod in te multiple period VaR estimation. Beltratti and Morana (1999) applied tis metod wit GARCH models to daily and alf-ourly data. Rater tan following te square root of time rule, we make use of te tail beavior of te aggregate return distribution. We developed two new VaR estimation metods based on te exact conditional variance, skewness and kurtosis and a Monte Carlo metod. Simulation and empirical results demonstrate tat our proposed metods outperform te metod in many cases. Te rest of te article is organized as follows. Section 2 gives te derivation of te exact conditional variance. Section 3 derives te exact conditional tird and fourt moments of te aggregate returns in some GARCH processes. Section 4 discusses problems of te multiple period VaR estimation. Tree new metods for estimating long orizon VaR are also introduced in tis section. One approac uses te exact conditional variance derived in Section 2 wile regarding R t, as normal variables. Anoter approac uses te exact conditional variance but assumes R t, as a skewed t-distribution wit te skewness and kurtosis matcing tat of R t,. Te last approac uses some Monte Carlo simulation metods. Section 5 studies te distribution of te aggregate returns R t, in various scenarios. Section 6 presents results for comparing our proposed multiple period VaR estimation metods wit te commonly used metod. Section 7 contains empirical applications using daily returns of seven market indices. 2. Exact Conditional Variance of Aggregates In te general eteroskedastic models considered in Engle (1982) and Bollerslev (1986), te conditional variance of r t is independent of te sign of r t.however, as it is commonly observed in te literature tat te variance of returns responds asymmetrically to te rise and drop in te stock markets, we adopt te Quadratic GARCH model in tis paper (Engle (199), Sentana (1991) and Campbell and Hentscel (1992)). Specifically, te return generating process follows te QGARCH(p,q) model is r t = µ + r t, r t = σ t ɛ t, ɛ t D(, 1), (2) p σt 2 = α + α i ( r t i b i ) 2 + β j σt j 2, (3)

4 118 CHI-MING WONG AND MIKE K. P. SO were D(,1) denotes a distribution wit mean and variance 1. As usual, te random errors ɛ t are uncorrelated. In te model µ is te unconditional mean, r t = r t µ is te centered return and te b i s are te asymmetric variance parameters wose values equal to zero gives te traditional GARCH(p, q) model. An interesting particular case is te IGARCH(1,1) model as adopted in RiskMetrics, were µ = α = b 1 =,α 1 =1 λ, β 1 = λ and D(,1) is te standard normal distribution. Writing (3) as σt 2 = α + q α i r t i 2 + p β jσt j 2 2 q α sb s r t s, α = α + q α sb 2 s, we ave te following results. Proposition 1. Var (R t, Ω t )= k=1 E r t+k 2 Ω t. Proposition 2. Let γ t,s be te conditional expectation E r 2 t Ω s. Define m = max{p, q} and φ i = α i + β i, i =1,...,m were α i =for i>qand β i =for i>p.ten,fork m +1, γ t+k,t = α + m φ i γ t+k i,t. Proofs are given in Appendix A.1 and A.2. Similar forecasting results under GARCH are also discussed in Sections 3.4 and 7.3 of Tsay (22). Using Propositions 1 and 2, we can get te aggregate conditional variance Var (R t, Ω t ) recursively. In particular, if p = q =1, α 1 φ φ 1 σt+1 2 if φ 1 < 1 Var (R t, Ω t )= 1 φ 1 1 φ 1 1 φ 1. (4) ( 1) α 2 + σ2 t+1 if φ 1 =1 For te RiskMetrics model tat as φ 1 =1andα =, te volatility of R t, is given by te volatility at time t +1, tatis σ t+1 multiplied by. Terefore, te square root of time rule adopted by many practitioners is obeyed in te RiskMetrics set-up. Tis is also mentioned in Section 7.2 of Tsay (22). In te stationary case φ 1 < 1, φ 1 as and so Var (R t, Ω t ) α α 1 φ 1 (1 φ 1 ) 2 + σ2 t+1, (5) 1 φ 1 wen is large. Te long orizon forecast variance is rougly times α /(1 φ 1), independent of t, contradicting te rule tat Var (R t, Ω t )is multiples of σ 2 t+1. Needless to say, in te unconditional context, te rule olds in te stationary case because Var (R t, )is Var (r t+1 ). Te above conclusions are in accord wit te argument in Diebold, Hickman, Inoue and Scuermann (1998) tat in stationary GARCH(1,1) models, applying te rule produces wrong fluctuation in te long orizon volatility forecasts. Using te results in Drost and Nijman (1993), Diebold, Hickman, Inoue and

5 ON CONDITIONAL MOMENTS OF GARCH MODELS 119 Scuermann (1998) demonstrated tat aggregation diminises te volatility fluctuation as increases. For instance, if {r t } is GARCH(1,1) wit µ = b 1 =, te aggregates R t, = r t r t+ will follow an implied GARCH(1,1) process wit te conditional variance σ ()2 t σ ()2 t =Var(R t, Ω () t )givenby = α () + α () 1 R2 (t 1), + β() 1 σ()2 t 1, (6) were Ω () t is te set of aggregate returns R,,...,R (t 1),. As tends to, α () tends to α /(1 φ 1 )andbotα () 1 and β () 1 tend to zero (see Drost and Nijman (1993)). Terefore, te volatility fluctuation disappears and te conditional variance σ ()2 t converges to α /(1 φ 1 ). Tis finding is consistent wit (5) in tat Var (R t, Ω t )convergesto times te unconditional variance of r t as tends to. Altoug we ave te coerent limit result for te variance of R t, from te implied -period volatility model in (6) and te associated 1- period model, te variance forecast of R t, given in (4), tat is Var (R t, Ω t ), is different from Var (R t, Ω () t ). Te former incorporates information of all 1-period returns up to time t wereas te latter uses te -period returns R,,...,R (t 1),. Te advice put forward in Diebold, Hickman, Inoue and Scuermann (1998) is tat a -period volatility model sould be used if we are interested in te -period volatilities. For example, if we ave 25 daily observations and we want montly volatility forecasts or a olding period of =2days,wecanonly use 125 montly observations instead of te 25 daily observations to construct a montly return model. As far as te parameter accuracy is concerned, tis reduction in te number of observations is certainly not desirable. As Ω t contains more information tan Ω () t,ifvar(r t, Ω t ) can be worked out numerically or analytically, wic is feasible for te QGARCH processes in (2) and (3), it is more natural and appropriate to use Var (R t, Ω t ) rater tan Var (R t, Ω () t ) to forecast te variance of R t,. Hence, we suggest fitting models of 1-period returns rater tan models of aggregate returns for multiple period volatility forecasting. 3. Exact Conditional Tird and Fourt Moments of Aggregates Common conditional eteroskedastic models, suc as GARCH models, are defined by te predictive distribution of r t+1 conditional on Ω t. Altoug te conditional distribution f(r t+1 Ω t ) is fixed in te model formulation, f(r t+ Ω t )orevenf(r t, Ω t ) are usually very complicated and unknown if >1. As te construction of te -period VaR is based on te percentiles of f(r t, Ω t ),

6 12 CHI-MING WONG AND MIKE K. P. SO some properties of f(r t, Ω t ) are likely to be elpful in improving te VaR estimation. In tis section, we focus on te conditional tird and fourt moments of R t, under te QGARCH(p, q) modelin(2)and(3)witsymmetricɛ t.take te kurtosis of ɛ t to be K = E ɛ 4 t >E ɛ 2 2 t = 1. Define te aggregate centered return as R t, = r t r t+, wic relates to te aggregate return troug R t, = µ + R t,. Under te symmetry of ɛ t,weave E R3 t, Ω t =3 L t,i, 2, (7) i=2 and te conditional fourt moment A t, = E R4 t, Ω t given by A t, = Kσt E t,j + P t+j,t+j, 2, (8) j=2 j=2 were L t, = E Rt, 1 r t+ t 2 Ω, E t, = E R2 t, 1 r t+ t 2 Ω and P t+l,t+k = E r t+l r2 2 t+k Ω t can be computed via some recursions. A proof of (7) and (8) and detailed procedures in calculating A t, are given in Appendix A.3 and A.4. If tere is no variance asymmetry, tat is, b i =,tetirdmomente R3 t, Ω t will vanis and so te skewness of R t, is zero. Terefore te conditional skewness of te aggregate returns is induced by te presence of a variance asymmetry effect. Furtermore, E Rt, 3 Ω t = 3 µ 3 +3µ E R2 t, Ω t + E R3 t, Ω t and E Rt, 4 Ω t = 4 µ µ 2 E R2 t, Ω t +4µ E R3 t, Ω t + E R4 t, Ω t, for 1. Te availability of E Rt, 3 Ω t and E Rt, 4 Ω t elps us understand te tail beavior of f(r t, Ω t ), important for working out te percentiles of R t, accurately given te information up to time t. Given te exact conditional variance Var (R t, Ω t ) we introduce, in te next section, a new multiple-period VaR estimation metod tat is likely to outperform oter metods tat do not make use of te tail properties of f(r t, Ω t ). An important special case of (3) is te RiskMetrics model: r t = σ t ɛ t, σt 2 =(1 λ) rt λσt 1. 2 (9) In tis case, p = q =1,µ = α = b 1 =,α 1 =1 λand β 1 = λ. Following (8), te conditional kurtosis of te 1-period return r t+ and te aggregate return R t, given Ω t, denoted by K rt+ Ω t and K Rt, Ω t respectively, can be written down in closed form as K rt+ Ω t = KG 1, (1) K Rt, Ω t = K ( ) ( G ) 1 6H 1+ (G 1) 1 G 1 +1, (11)

7 ON CONDITIONAL MOMENTS OF GARCH MODELS 121 were G =(K 1)(1 λ) 2 +1and H =1 λ + λ K. Te derivations of (1) and (11) are presented in Appendix A.5. It is interesting to see tat bot K rt+ Ω t and K Rt, Ω t are independent of t. Since G is greater tan one (as K>1), te conditional kurtosis of r t+ increases exponentially wit, wile te conditional kurtosis of R t, tends to infinity as tends to infinity. Tis long-orizon beavior of K Rt, Ω t indicates tat te distribution of R t, becomes more eavy tailed as te forecast orizon or te olding period get longer. Terefore, we cannot be surprised if a small percentile of R t, is poorly estimated under te normality assumption of f(r t, Ω t ), especially wen is large. 4. Multiple Period VaR Estimation Value at Risk is a measure of te maximum loss of a portfolio over a predetermined orizon. More precisely, it is te loss tat will be exceeded wit probability p over a time orizon of periods. According to tis definition, te Value at Risk can be formulated as in (1), were C is te current market value of te portfolio and V is te -period return pt percentile. Obviously, an VaR estimate depends very muc on te parameters p and. Te coices of p and can be subjective. For example, Jorion (1997, p.2) stated tat p can range from 1% to 5% according to te individual preference of different commercial banks. Moreover, te time orizon or te olding period can vary quite a lot in different applications (see Cristoffersen, Diebold and Scuermann (1998) and Jorion (1997)). In 1996, te Bank for International Settlements (BIS) put forward an amendment to te Capital Accord to Incorporate Market Risks. According to te guidelines of te amendment, te VaR associated wit p =1%and =1 days sould be calculated for te determination of te market risk capital. In practice, te selection of 1 leads to muc complication in te estimation of VaR. In te actual calculation of VaR, we usually assume a time series model for 1-period returns, suc as te one given in (2) and (3). Te time unit for a single period depends on te frequency of te available related financial data. For example, for equity indices data, daily or even ourly returns can be collected and so te time unit can be set at 1 day. To implement te BIS regulation based on a model of daily return data, we set = 1. Suppose tat a model for 1-period returns is formulated as in (2) and (3). Using te notations set out in Section 2, given te information up to time t, we can determine V as V = Ft, 1 (p), were F t,( ) is te probability distribution of te -period return R t, given Ω t, i.e., F t, (x) =Pr(R t, x Ω t ). If we want to obtain VaR as in (1), we need te inverse of F t, ( ) evaluated at p. Inparticular, if =1,V is σ t+1 D 1 (p), were D 1 ( ) is te inverse of te error distribution

8 122 CHI-MING WONG AND MIKE K. P. SO D(, 1). However, F t, ( ) is generally analytically intractable, especially wen is large. Even toug for >1, te conditional distribution of R t, given Ω t can be written as F t, (x) = R t, x 1 f(r t, Ω t+ 1 ) f(r t+i Ω t+i 1 ) d(r t+1,,r t+ ), te evaluation of V as to involve ig-dimension integration. Terefore, te exact value of V is usually unavailable wen is greater tan one. A commonly 1 used estimator for V is = µ + σ t+1 Φ 1 (p), were Φ( ) is te standard 1 normal distribution function. RiskMetrics adopts wit µ =forteperiod VaR estimation. Te rationale is based on a normality assumption and te square root of time rule. In te model assumed by RiskMetrics, D(, 1) 1 is te standard normal and so 1 = σ t+1 Φ 1 (p) gives te exact value of V 1. Borrowing te idea of te 1 1 rule, is constructed by scaling 1 wit. Tis scaling metod as been widely accepted by practitioners, for example, te BIS suggested using te rule to convert a 1-day VaR estimate to a 1- day VaR estimate for calculating capital requirement. If te distribution F t, ( ) is normal and Var (R t, Ω t )=σt+1 2, 1 is equivalent to V. According to (4), Var (R t, Ω t )=σt+1 2 olds in te QGARCH(1,1) model wit µ = b 1 = 1 α =andα 1 + β 1 = 1. Terefore, using under te RiskMetrics model setting can provide a good estimate of V if F t, ( ) is reasonably close to normal. 1 To investigate weter is an appropriate estimator for V,weexaminete discrepancy between F t, ( ) and a normal distribution aving te same variance in te next section. Since in most cases, suc as modeling te return r t wit a stationary QGARCH(1,1) model, neiter Var (R t, Ω t )=σt+1 2 nor F t,( ) is normal, using 1 to provide a good estimate of V is questionable. In tis paper, we propose a 1 2 natural alternative to as = µ+ Var (R t, Ω t )Φ 1 (p). Tis estimator is constructed by treating F t, ( ) as normal wit variance Var (R t, Ω t ). An advantage of over is tat using te exact variance of F t, ( ) in 1 bypasses te potential bias of due to mis-scaling. For example, under a 1 stationary QGARCH(1,1) model, using for te long orizon VaR estimation can be problematic because, wen is large, Var (R t, Ω t ) α /(1 φ 1). Tis can be very different from σt Te new estimator is expected to be 1 superior to in many cases. Obviously, wen Var (R t, Ω t )=σt+1 2 2, = 1.

9 ON CONDITIONAL MOMENTS OF GARCH MODELS Altoug can overcome te mis-scaling problem in using, te error in te VaR estimation due to te departure of F t, ( ) from normal can be very 2 significant. We propose anoter estimator for V wic is more general tan by incorporating also te skewness and tail properties of F t, ( ). A new estimator is constructed using te skewed t-distribution introduced in Teodossiou (1998). Its probability density function is C f(x) = C 1+ 2 ν ν 2 ( x + a θ(1 τ) ( x + a θ(1 + τ) ) 2 (ν+1) 2 ) 2 (ν+1) 2 were τ and ν are parameters of te distribution, C = B( 3 2, ν 2 2 ) 1 2 S(τ) 2 B( 1 2, ν 2 ), θ = 3 2 S(τ), 2τB(1, ν 1 2 a = ) S(τ)B( 1 2, ν 2 ) 1 2 B( 3 2, ν 2 2 ), S(τ) = 1 2 if x< a, if x a, 1 1+3τ 2 4τ 2 B(1, ν 1 2 )2 B( 1 2, ν 2 )B( 3 2, ν 2 2 ) and B( ) is te beta function. Te above distribution as mean, variance 1, Ex 3 = 4τ(1 + τ 3 )B(2, ν 3 2 )B( 1 2, ν 2 ) 1 2 B( 3 2, ν 2 2 ) 3a a 3 and 3 2 S(τ) 3 Ex 4 = 3(ν 2)(1 + 1τ 2 +5τ 4 ) (ν 4)S(τ) 4 4aEx 3 6a 2 a 4. If τ =, te skewed t is te usual symmetric t-distribution. By encompassing te tird and fourt moments structure of te aggregate returns, te new estimator is 3 = µ + (12) 1 2 Var (R t, Ω t ) f 1 (p), (13) were f 1 (p) istept percentile of (12). We coose τ and ν to matc te skewness and kurtosis of te skewed t-distribution and tat of te aggregate returns. In oter words, te two parameters are found by solving te two equations: E R3 t, Ω t = Ex 3, (14) Var (R t, Ω t ) 3 2 E R4 t, Ω t Var (R t, Ω t ) 2 = Ex4. (15)

10 124 CHI-MING WONG AND MIKE K. P. SO In particular, wen te volatility responds symmetrically to good and bad news, tat is b i =,weavee R3 t, Ω t = and tus solving (14) and (15) gives τ =and ν = 6 4 K R t, Ω t 6 or K Rt, Ω t K Rt, Ω t 3, (16) were K Rt, Ω t = Ex 4. Ten, te estimator in (13) is simplified to = µ + Var (R t, Ω t ) t 1 ν (p), were t ν( ) is te standardized t-distribution wit variance 1 and degrees of freedom ν. Under te RiskMetrics model specification in (9), ν in (16) depends on λ, K and only because, according to (11), K Rt, Ω t is time-independent under te RiskMetrics model. If ɛ t is standard normally distributed (K = 3), we ave te following values of K Rt, Ω t, ν and t 1 ν (p) forp =1%and5%, = 5, 1 and 5 and λ =.94 and Table 1. K Rt, Ω t ν t 1 ν (.1) d ν(.1) t 1 ν (.5) d ν(.5) λ = λ = Knowing te standard normal percentiles, we also present d ν (p) =t 1 ν (p) Φ 1 (p) in various scenarios. We observe from te table tat t 1 ν (.1) decreases wit wereas t 1 ν (.5) increases wit. Te magnitude of d ν(p) grows wit for bot p = 1% and 5%, implying tat tere is greater discrepancy between te standard normal and te t-distribution used to matc K Rt, Ω t. Altoug all 2 3 d ν (.5) reported are positive, teir magnitude is so small tat and are likely to be close in data implementation. Terefore, it is not surprising to see satisfactory performance in using te RiskMetrics metod for p = 5% even toug te fat-tailed caracteristics of f(r t, Ω t ) ave not been accounted for. On te 2 oter and, all d ν (.1) are large and negative, implying tat is substantially greater tan for p = 1%. Hence, replacing wit t 1 ν (.1), our tird 3

11 ON CONDITIONAL MOMENTS OF GARCH MODELS 125 estimator offers a simple way to reduce te usual upward bias in te RiskMetrics metod for estimating te 1% V. Te last estimator we propose is based on some Monte Carlo samples of R t, from F t, ( ). Tis metod avoids making any assumptions on te distribution of F t, ( ). If te number of Monte Carlo samples obtained is large enoug, tis metod is likely to produce a good estimate of V. Because of te decomposition f(r t+1,,r t+ Ω t )= f(r t+i Ω t+i 1 ), i.i.d. samples from te joint density can be simulated by te metod of composition (Tanner (1993, pp.3-33)). Given Ω t, σt+1 2 is known. For i =1,...,N were N is te number of replications, we 1. simulate r (i) 2. calculate σ (i) t+j t+1 µ + σ t+1d(, 1) and set j =2, from (3) using r(i) t+j 1,...,r(i) t+1 and Ω t, 3. simulate r (i) t+j µ + σ(i) t+jd(, 1), 4. repeat steps 2 and 3 for j =3,...,. Ten (r (i) t+1,...,r(i) t+ ) is a draw from te joint density f(r t+1,,r t+ Ω t )and R (i) t, =r(i) t+1 + +r(i) t+,,...,n, forms an independent sample from f(r t, Ω t ). Finally, we propose a Monte Carlo estimator for V given by =samplep 4 percentile of R t,. It was sown in Serfling (198, pp.74-75) tat,constructed by te i.i.d. sample R (i) t,, i =1,...,N,convergestoV wit probability one. So tis fourt estimator converges to V as N increases. Wen N is sufficiently large, te empirical distribution of te Monte Carlo sample can well approximate te target distribution F t, ( ) and te sample percentile give us a good estimate of te desired VaR. 4 4 can 5. Distribution of te Aggregates R t, In Sections 2 and 3, we ave sown ow to calculate te exact conditional variance, skewness and kurtosis of te aggregate return R t, given Ω t for QGARCH(p, q) models. In tis section, we study in detail te fourt moment properties of te aggregates distribution. We also examine by simulations ow close te distribution of te aggregates is to te normal and t-distributions used 2 3 to construct and respectively, for different orizons. Te following two sub-sections describe te design of te simulation study and report te results Simulation design We ave considered te QGARCH(1,1) model defined in (2) and (3) wit µ = andb 1 =. Te focus is on te symmetric GARCH model as it is commonly adopted in financial researc. Te parameters α =1,α 1 =.1 and tree values

12 126 CHI-MING WONG AND MIKE K. P. SO of β 1 (β 1 =.8,.85 and.895) were cosen in te simulations. Te parameters µ and α of te GARCH(1,1) model are only location and scaling factors wic would not affect te sape of te aggregates distribution. Te parameters α 1 and β 1 were set to comport wit common results from data analyses tat β 1 is large and α 1 + β 1 is close to one. Te coice of α 1 + β 1 close to one is to capture te stylized fact of ig persistent volatility. We focus on te forecast orizons =1,...,15 and two distributions of errors ɛ t in (2), namely te standard normal distribution and te t-distribution wit 5 degrees of freedom. For eac model considered, a series of sample size t = period returns, togeter wit teir conditional variances up to σt+1 2, was generated. Starting from time t +1 wit σt+1 2 being fixed, N = 2, replications were formed. In te it replication, a sample pat consisting of r (i) t+1,r(i) t+2,...,r(i) t+ was generated, and aggregates R (i) t,1,r(i) t,2,...,r(i) t, were calculated. We computed te Kolmogorov-Smirnov (K-S) one-sample goodness-of-fit test statistic sup x S N (x) F (x), weres N (x) is te empirical distribution of R (1) t,,...,r (N) t, and F (x) is te null distribution. Here, F (x) stands for eiter te normal distribution wit mean and variance Var (R t, Ω t ) for constructing 2, or te t-distribution wit mean, variance Var (R t, Ω t )andkurtosisk Rt, Ω t 3 for defining. Te K-S test statistic was used to measure te maximum distance between te empirical distribution of te aggregate return R t, and te null distribution F (x) Simulation results Figures 1 and 2 summarize te simulation results of te distributions of te 1-period return r t+ and te aggregate return R t, for orizons =1,...,15. In Figure 1, te excess kurtosis (kurtosis 3) was plotted against te orizon. Excess kurtosis measures te tail tickness of te distribution and a positive excess kurtosis indicates a leptokurtic distribution. Te orizontal line in eac plot locates te zero excess kurtosis wic corresponds to normality. We can see from Figure 1 tat all excess kurtoses are positive. In parts (a) to (d), te excess kurtosis of te 1-period return r t+ (dotted line) converges to some value as te orizon increases. Te larger te value of β 1, te furter tat value is above zero and te longer it takes to converge. Te excess kurtosis of aggregate return R t, (solid line) tends to decrease over time orizons were te 1-period return r t+ as similar kurtosis. In parts (e) and (f), corresponding to te near nonstationary case of β 1 =.895, bot te excess kurtoses of r t+ and R t, seem to increase exponentially wit. Tis particular finding agrees wit

13 ON CONDITIONAL MOMENTS OF GARCH MODELS 127 te caracteristics of te RiskMetrics model documented in (1) and (11). Te simulation results in Figure 1 indicate tat te predictive density f(r t, Ω t ) deviates substantially from normality, especially wen α 1 + β 1 1. Terefore, assuming f(r t, Ω t ) to be normal in constructing V 1 and V 2 is arguable. (a) (b) TrueKurt-3 TrueKurt (c) 5 1-period Returns Aggregate Returns 1 15 TrueKurt-3 TrueKurt (d) (e) (f) TrueKurt TrueKurt Figure 1. Plots of te true excess kurtosis (kurtosis 3) as a function of orizon for bot 1-period return r t+ (dotted line) and aggregate return R t, (solid line) generated from a GARCH(1,1) process. Parts (a) and (b) are for β 1 =.8; parts (c) and (d) are for β 1 =.85, and parts (e) and (f) are for β 1 =.895. Parts (a), (c) and (e) are for normal distributed ɛ t ; parts (b), (d) and (f) are for t-distributed ɛ t wit 5 degrees of freedom.

14 128 CHI-MING WONG AND MIKE K. P. SO (a) (b) T-stat T-stat (c) 5 1 Normal t-dist 15 (d) T-stat T-stat (e) (f) T-stat.2 T-stat Figure 2. Plots of te K-S test statistic (T-stat) as a function of orizon for te aggregate return R t, generated from a GARCH(1,1) process. Te orizontal line is te critical value of te K-S test at 1% significance level. Te dotted line represents T-stat of te null normal distribution wit variance Var (R t, Ω t ) and te solid line represents T-stat of te null t-distribution wit variance Var (R t, Ω t )andkurtosisk Rt, Ω t. Parts (a) and (b) are for β 1 =.8; parts (c) and (d) are for β 1 =.85; parts (e) and (f) are for β 1 =.895. Parts (a), (c) and (e) are for normal distributed ɛ t ; parts (b), (d) and (f) are for t-distributed ɛ t wit 5 degrees of freedom. In Figure 2, we want to see ow close te conditional distribution of te aggregate return f(r t, Ω t ) is to te normal distribution wit te same variance,

15 ON CONDITIONAL MOMENTS OF GARCH MODELS 129 and to te t-distribution wit te same variance and kurtosis. In oter words, tese normal and t-distributions are te null distributions for computing te K-S test statistic. Te orizontal line in eac plot marks te 1% critical value of te K- S test for reference. Parts (a), (c) and (e) of Figure 2 correspond to GARCH(1,1) models wit standard normally distributed ɛ t. Te K-S test statistic associated wit te null t-distribution lies very close to te 1% critical value wile te K- S test statistic associated wit te null normal distribution is well above te 1% critical value. Tis indicates tat te t-distribution tat matces te true conditional variance and kurtosis of f(r t, Ω t ) is a good approximation to te desired conditional distribution. For GARCH(1,1) models wit ɛ t distributed as standardized t wit 5 degrees of freedom, parts (b), (d) and (f) of Figure 2 sow tat te null t-distribution is still closer to f(r t, Ω t ) tan te normal. Since te case wit β 1 =.895 resembles te RiskMetrics model, it is anticipated tat te pattern of te K-S test statistics for te RiskMetrics model is very similar to, based on te null in VaR estimation under te GARCH Figures 2(e) and (f). Hence, we sould not be surprised if V 3 t-distribution, outperforms V 1 and V 2 and RiskMetrics framework. 6. Comparing te Four VaR Estimation Metods 4 Since te Monte Carlo estimator approaces V as te number of replications N tends to infinity, it can be regarded as te bencmark among te four estimators discussed in Section 4 if N is large enoug. In tis section, we set N = 2, and use te same simulation setup in Section 5 to compare te 2 four VaR estimation metods. To facilitate te comparison of, and wit te cosen bencmark, we compute te percentage difference between eac i 4 of te first tree metods and te Monte Carlo metod, ( / 1) 1% for i =1, 2, 3. We expect tat good estimation metods are able to produce VaR estimates tat are close to tat generated by te Monte Carlo metod, so small absolute percentage differences are desirable. In Figure 3, te percentage differences of te tree estimation metods are plotted against te orizon for te GARCH(1,1) model were ɛ t is t-distributed wit 5 degrees of freedom, p = 1% and 5%, and β 1 =.8,.85 and.895. From 1 parts (a) to (f) of Figure 3, (dased line) as te largest magnitude in 1 percentage difference among te tree metods. Te large deviation of from 4 is due to te mis-scaling problem of using te rule. By incorporating te 2 1 exact variance, (dotted line) sows great improvement over. However, 2 systematic bias is recorded in by aving negative and positive percentage differences wen p = 1% and 5% respectively. Tis is due to te fact tat te distribution of R t, is leptokurtic (see Figure 1) and R t, is assumed to be normal 1 3

16 13 CHI-MING WONG AND MIKE K. P. SO (a) p = 1* (b) p = 5* * difference * difference * difference (c) (e) V1 V2 V * difference * difference * difference (d) (f) Figure 3. Plots of te percentage difference between and (dased line), and (dotted line), and and (solid line) as a function of te orizon for GARCH(1,1) model, ɛ t is t-distributed wit 5 degrees of freedom. Parts (a) and (b) are for β 1 =.8; parts (c) and (d) are for β 1 =.85; parts (e) and (f) are for β 1 =.895. Parts (a), (c) and (e) are for p = 1%; parts (b), (d) and (f) are for p =5% wen deriving. In terms of te magnitude of te percentage differences, 3 2 (solid line) generally performs better tan. In te simulations using 3 normal distributed ɛ t, we observe similar results as above, tat is able to 4 produce estimates tat are closest to among te tree estimators in most 2 1 orizons. For te RiskMetrics model, te estimator is identical to as

17 ON CONDITIONAL MOMENTS OF GARCH MODELS 131 Var (R t, Ω t )=σt+1 2, so we present only two curves in eac plot of Figure Again, we can observe tat (solid line) is muc better tan (dotted line) 3 if p = 1% or 5%, in te sense tat is closer to te bencmark in most cases We can also see from te percentage difference of tat < wen p = 2 1%, and te opposite is true wen p = 5%. Te large discrepancy between 4 and for p = 1% explains te usual upward bias observed wen applying te RiskMetrics VaR estimator to real data. 3 To conclude, te VaR estimation metod, wic uses t-distribution to matc te conditional variance and kurtosis, is te best among te tree estimation metods. It as performance similar to tat of te Monte Carlo metod 4 3, but can be calculated instantly. In practice, we can use as a substitute 4 of, to avoid long execution time for large N. (a) p = 1* (b) p = 5* * difference -4 * difference (c) 5 1 V2 V3 15 * difference * difference (d) Figure 4. Plots of te percentage difference between 2 and 4 (dotted 3 4 line), and and (solid line) as a function of te orizon for te RiskMetrics model, ɛ t is normal. Parts (a) and (b) are for λ =.94; parts (c) and (d) are for λ =.97; parts (a) and (c) are for p = 1%; parts (b) and (d) are for p =5%. 7. Empirical Applications In tis section, we apply te four VaR estimation metods wit two

18 132 CHI-MING WONG AND MIKE K. P. SO QGARCH(1,1) models and te RiskMetrics model to te daily returns of seven market indices. Te indices we ave used are te AOI (Australia) from 199 to 1998; te CAC 4 (France) from 1991 to 1998; te DAX (Germany) from 1991 to 1998; te FTSE 1 (UK) from 199 to 1998; te HSI (Hong Kong) from 199 to 1998; te Nikkei 225 (Japan) from 199 to 1998; te S & P 5 (USA) from 199 to For eac market index, we ave its daily returns for te period of 199 to 1998 (1991 to 1998 for te France CAC 4 and Germany DAX). Te models we considered ere are: (a) QGARCH, QGARCH(1,1) model wit t-distributed ɛ t ;(b)garch, QGARCH(1,1) model wit µ = b 1 = and t-distributed ɛ t ;(c) RiskMetrics model wit normally distributed ɛ t.forteqgarch and GARCH models, te parameters were obtained by maximum likeliood estimation using te initial five years daily data r j were j =1,...,t,andt 1, 25 (initial four years for CAC 4 and DAX: t 1, ). Te number of trading days in eac year is sligtly different from market to market but is rougly equal to 252 days. For te RiskMetrics model, te decay factor was set to λ =.94, as suggested by J. P. Morgan (1996) for daily data. i Te four types of VaR estimates, i =1,...,4, were computed based on te models (a) to (c) for =5, 1 and 5 and probabilities p = 1%, 2.5% and 5% at te time point t. Te actual -period returns R t, for =5, 1 and 5 were also computed from te daily returns of te market indices. Ten te estimation window was sifted forward by one day and te QGARCH and GARCH parameters were re-estimated using te daily returns r j, j =2,...,t+1. Te computation of VaR estimates and actual multiple period returns were performed again at te time point t + 1. Tis rolling sample analysis was repeated until te wole validation period ( ) was covered. At te end, te VaR estimates i, i =1,...,4, togeter wit te actual multiple period returns R t, for = 5, 1 and 5 were obtained at te time points t,...,t+ n were n 1, 8 (four year validation period: 1995 to 1998). For eac combination of values of p, type of VaR estimates i and orizon, te proportion of R t, tat falls below its i VaR estimates denoted by ˆp was calculated. If te assumed model for te 1-period returns is correct, we expect tat a good VaR estimation metod will ave ˆp close to p or te ratio ˆp/p close to 1. Table 2 lists te ratio ˆp/p of te seven market indices for = 1 in te four year validation period (1995 to 1998). For eac market index and given p, te ratios closest to 1 were put in boxes. For p = 2.5% and 5%, te ratios ˆp/p do not vary muc and are similar. Te major factor tat determines te difference in te ratios seems to be te underlying dynamic model we assumed for te 1-period returns. For tese two moderately small p, itisevidenttatgarch produces more reliable VaR estimates tan QGARCH and RiskMetrics. Te differences among te four VaR estimation metods are small witin eac model, except

19 ON CONDITIONAL MOMENTS OF GARCH MODELS 133 for some cases of QGARCH. It is also interesting to note te extraordinary large variation in te ratios of te Nikkei 225. Table 2. Ratio of te proportion ˆp of 1-day returns less tan te estimated V to te actual probability p ( = 1). 1 QGARCH GARCH RiskMetrics p =1% HSI Nikkei SP AOI FTSE CAC DAX p =2.5% HSI Nikkei SP AOI FTSE CAC DAX p =5% HSI Nikkei SP AOI FTSE CAC DAX Figures in te boxes are te ratios ˆp/p closest to For p = 1%, te differences in ˆp/p among te four estimation metods can be substantial witin eac model. For example, te ratios of QGARCH vary from 1.84 to 2.86 for HSI, and from.7 to 1.89 for AOI. In te estimation of tis extreme percentile, te boxes cluster in GARCH and locate mostly in and Tis indicates tat tey are superior to and. Incorporating also te skewness and kurtosis of R t, in significantly improves te VaR estimation 3 3

20 134 CHI-MING WONG AND MIKE K. P. SO results. Wile and work equally well in tis competition, costs muc less in computational time and so is recommended for applications. In Table 3, te olding period is sortened to 5 days ( = 5). Te estimation 3 metod associated wit GARCH is consistently te best for p =1%and 2.5% (except for Nikkei and AOI wit p = 1%). In Table 4, te olding period is increased to 5 days ( = 5). In tis case, all VaR estimation metods perform equally poorly wen p = 1%. Allowing te mean and asymmetric parameters in QGARCH seems to ave some advantages in estimating te fift percentile, but it does not lead to any noticeable improvement for p = 1% and 2.5%. Table 3. Ratio of te proportion ˆp of 5-day returns less tan te estimated V to te actual probability p ( =5). 1 QGARCH GARCH RiskMetrics p =1% HSI Nikkei SP AOI FTSE CAC DAX p =2.5% HSI Nikkei SP AOI FTSE CAC DAX p =5% HSI Nikkei SP AOI FTSE CAC DAX Figures in te boxes are te ratios ˆp/p closest to

21 ON CONDITIONAL MOMENTS OF GARCH MODELS 135 Table 4. Ratio of te proportion ˆp of 5-day returns less tan te estimated V to te actual probability p ( = 5). 1 QGARCH GARCH RiskMetrics p =1% HSI Nikkei SP AOI FTSE CAC DAX p =2.5% HSI Nikkei SP AOI FTSE CAC DAX p =5% HSI Nikkei SP AOI FTSE CAC DAX Figures in te boxes are te ratios ˆp/p closest to 1. Te overall picture we get from te tables is as follows. First, te data generating model of te return is important in te estimation of VaR. Broadly speaking, suitable coices are eiter te RiskMetrics model or te symmetric GARCH model. For te orizons = 5 and 1, te GARCH model is likely to be a promising alternative to te RiskMetrics model. Wile te QGARCH model is able to capture te volatility asymmetry in financial markets, it seems to be too complicated for predicting te return percentiles and yields poorer performance tan te GARCH model. Te additional conditional skewness of R t, induced by te parameters b i does not evidently elp forecast te VaR. Second, wen p

22 136 CHI-MING WONG AND MIKE K. P. SO is small, te VaR estimation metod becomes important and is usually te best or at par wit oter metods. So even if we follow te RiskMetrics model, 1 our proposed tird estimator is likely to outperform te classical based on te rule. Acknowledgements Te autors would like to tank Professor Ruey S. Tsay and two anonymous referees for teir valuable suggestions and comments. Financial support by te Hong Kong RGC Direct Allocation Grants /1.BM25 and /1.BM28 is gratefully acknowledged. Appendix A.1 Proof of Proposition 1 For i>j>, E r t+i r t+j Ω t =E E r t+i r t+j Ω t+i 1 Ω t =E r t+j E r t+i Ω t+i 1 Ω t =, as E r t+i Ω t+i 1 =. Obviously, te above result implies tat E r t+i r t+j Ω t =,fori, j > andi j. Te proposition follows as E r t+i Ω t =,fori>. A.2. Proof of Proposition 2 Recall tat we ave te general model r t = σ t ɛ t, ɛ t D(, 1), σt 2 = α + q α i r t i 2 + p β j σt j 2 2 q α sb s r t s. Using te notation γ t,s = E r t 2 Ω s,weavefork m +1, γ t+k,t = E r t+k 2 Ω t = E E r t+k 2 Ω t+k 1 Ω t 3 p = E α + α i r t+k i 2 + β j σt+k j 2 2 α s b s r t+k s Ω t p = α + α i γ t+k i,t + β j γ t+k j,t = α + m φ i γ t+k i,t. Te second last equality is valid because Eσ 2 t+k j Ω t=ee r 2 t+k j Ω t+k j 1 Ω t =E r 2 t+k j Ω tande r t+k j Ω t =wenk j 1. A.3. Derivation of te exact conditional tird moment of aggregates Define T t+k,t+ = E r t+k r t+ 2 Ω t and L t, = E Rt, 1 r t+ t 2 Ω.For 2, E R3 t, Ω t = E ( R t, 1 + r t+ ) 3 Ω t = E R3 t, 1 +3 R t, 1 r 2 t+ +3 R t, 1 r t+ 2 + r t+ 3 Ω t=e R3 t, 1 Ω t +3L t,. From tis recursion, te conditional tird

23 ON CONDITIONAL MOMENTS OF GARCH MODELS 137 moment of aggregates is E R3 t, Ω t E r t+1 3 Ω t =, were =3 i=2 L t,i, 2, as E R3 t,1 Ω t = 1 L t, = E Rt, 1 r t+ 2 Ω 1 t = E r t+i r t+ 2 Ω t = T t+i,t+. Terefore, to find te conditional tird moments, it suffices to compute T t+k,t+. Wen =k, T t+,t+ =E r t+ r t+ 2 Ω t =. If<k, T t+k,t+ =E r t+k r t+ 2 Ω t = E r t+ 2 E r t+k Ω t+k 1 Ω t =.For>kand m +1, T t+k,t+ = E r t+k r t+ 2 Ω t = E E r t+k r t+ 2 Ω t+ 1 Ω t = E r t+k σt+ 2 Ω t (as >k) p ) = E r t+k (α + α i r t+ i 2 + β j σt+ j 2 2 α s b s r t+ s Ω t p = α E r t+k Ω t + α i E r t+k r t+ i 2 Ω t + β j E r t+k σt+ j 2 Ω t 2 α s b s E r t+k r t+ s Ω t (17) p = α i T t+k,t+ i + β j T t+k,t+ j 2α k b k γ t+k,t I(1 k q). Using (17), T t+k,t+ can be computed recursively. In te particular case of no variance asymmetry, i.e., b i =,T t+k,t+ = L t, = and so te conditional tird moment E R3 t, Ω t vanises. A.4. Derivation of te Exact Conditional Fourt Moment of Aggregates Recall tat γ t+,t = E r t+ 2 Ω t, K = E ɛ 4 t, m = max{p, q}, At, = E R4 t, Ω t, E t, = E R2 t, 1 r t+ t 2 Ω and P t+l,t+k = E r t+l r2 2 t+k Ω t. In addition, we define Q t+l,t+k = E r t+l 2 σ2 t+k Ω t.for 2, A t, = E R4 t, Ω t ( ) 4 = E Rt, 1 + r t+ Ωt

24 138 CHI-MING WONG AND MIKE K. P. SO = E R4 t, 1 +4 R t, 1 r 3 t+ +6 R t, 1 r2 2 t+ +4 R t, 1 r t+ 3 + r4 t+ t Ω = A t, 1 +6E t, + P t+,t+. (18) Te last equality in (18) follows because E R3 t, 1 r t+ Ω t Ω t+ 1 Ω t = E R3 t, 1 E r t+ Ω t+ 1 Ω t = E E R3 t, 1 r t+ =ase r t+ Ω t+ 1 =, and E Rt, 1 r t+ t 3 Ω = E E Rt, 1 r t+ t+ 1 3 Ω Ω t = E Rt, 1 E r t+ 3 Ω t+ 1 Ω t =,asɛ t+ is symmetric about. From (18), it is not difficult to see tat A t, = A t,1 +6 E t,j + P t+j,t+j, 2, (19) j=2 were A t,1 = Kσt+1 4. Terefore, it suffices to calculate E t,j and P t+j,t+j, j = 2,...,, for evaluating te conditional fourt moment E R t+ 4 Ω t. Since P t+l,t+k = P t+k,t+l, we only need to consider te two cases (i) k = l, and (ii) k<l for P t+l,t+k. Assume tat k m +1. Case 1. k = l P t+l,t+k = E r t+l r2 2 t+k Ω t = E r t+k 4 Ω t = E E r t+k 4 Ω t+k 1 Ω t = E Kσt+k 4 Ω t j=2 p ) 2 = KE (α + α i r t+k i 2 + β j σt+k j 2 2 α s b s r t+k s Ωt ( q = KE α 2 + ) 2 ( α i r t+k i 2 p ) 2 + β j σt+k j 2 +2α α i r t+k i 2 p ( +2α β j σt+k j 2 +2 q )( α i r 2 p ) ( t+k i β j σ 2 q ) 2 t+k j +4 α s b s r t+k s ( 4α q ) ( q )( q ) α s b s r t+k s 4 α s b s r t+k s α i r t+k i 2 ( q )( p ) 4 α s b s r t+k s β j σt+k j 2 Ω t = KE α 2 + α 2 i r t+k i 4 +2 p α i α i r t+k i r 2 t+k i 2 + βj 2 σt+k j 4 i>i +2 p β j β j σt+k jσ 2 t+k j 2 +2α α i r t+k i 2 +2α β j σt+k j 2 j>j

25 ON CONDITIONAL MOMENTS OF GARCH MODELS 139 p +2 α i β j r t+k i 2 σ2 t+k j +4 α 2 s b2 s r2 t+k s +8 α s α s b s b s r t+k s r t+k s 4α α s b s r t+k s s>s p 4 α s b s r t+k s α i r t+k i 2 4 α s b s r t+k s β j σt+k j 2 Ω t = K α 2 + α 2 i P t+k i,t+k i +2 α i α i P t+k i,t+k i i>i + 1 p βj 2 K P t+k j,t+k j +2 β j β j Q t+k j,t+k j j>j p p +2α α i γ t+k i,t +2α β j γ t+k j,t +2 α i β j Q t+k i,t+k j +4 α 2 sb 2 sγ t+k s,t 4 α s b s α i T t+k s,t+k i p 4 α s b s β j T t+k s,t+k j. (2) s>j Te last equality in (2) is valid because of (a) (e) below. (a) For k m +1, E σt+k j 4 Ω t = E (1/K)E r t+k j 4 Ω t+k j 1 Ω t = (1/K)E r t+k j 4 Ω t =(1/K)P t+k j,t+k j,ast + k j 1 t + m j t. (b) For k m+1, j>j, E σt+k j 2 σ2 t+k j Ω t =E σ t+k j 2 E r t+k j 2 Ω t+k j 1 Ω t = E E r t+k j 2 σ2 t+k j Ω t+k j 1 Ω t = E r t+k j 2 σ2 t+k j Ω t = Q t+k j,t+k j, asj>j and t + k j 1 t + m j t. (c) E r t+k i 2 Ω t = γ t+k i,t,ast + k i t + m +1 i t +1. (d) E σt+k j 2 Ω t = γ t+k j,t, as t + k j t + m +1 j t +1. (e) For s>j, E r t+k s σt+k j 2 Ω t = E r t+k s E r t+k j 2 Ω t+k j 1 Ω t = E E r t+k s r t+k j 2 Ω t+k j 1 Ω t = T t+k s,t+k j,ast + k s<t+ k j. For s j, E r t+k s σt+k j 2 Ω t = E E r t+k s σt+k j 2 Ω t+k s 1 Ω t = E σt+k j 2 E r t+k s Ω t+k s 1 Ω t =,ast + k j<t+ k s.

26 14 CHI-MING WONG AND MIKE K. P. SO Case 2. k<l P t+k,t+l = E r t+k r2 2 t+l Ω t = E E r t+k r2 2 t+l Ω t+l 1 Ω t = E r t+k 2 σ2 t+l Ω t ( p ) = E r t+k 2 α + α i r t+l i 2 + β j σt+l j 2 2 α s b s r t+l s Ω t p = α γ t+k,t + α i E r t+k r 2 t+l i 2 Ω t + β j E r t+kσ 2 t+l j 2 Ω t 2 α s b s E r t+k r 2 t+l s Ω t p = α γ t+k,t + α i P t+k,t+l i + β j Q t+k,t+l j 2 α s b s T t+l s,t+k. (21) Since t + l 1 t and l>k. Terefore, recursive formulas for P t+k,t+l are establised in (2) and (21). For k, l m + 1, te following equation is used to evaluate Q t+l,t+k : Q t+l,t+k = E r t+l 2 σ2 t+k Ω t ( p ) = E r t+l 2 α + α i r t+k i 2 + β j σt+k j 2 2 α s b s r t+k s Ω t p = α E r t+l 2 Ω t + α i E r t+l r 2 t+k i 2 Ω t + β j E r t+lσ 2 t+k j 2 Ω t 2 α s b s E r t+l r 2 t+k s Ω t p = α γ t+l,t + α i P t+l,t+k i + β j Q t+l,t+k j 2 α s b s T t+k s,t+l. (22) Given initial values P t+l,t+k and Q t+l,t+k, l, k =1,...,m, we can obtain P t+j,t+j, j = 2,..., in (19) via te recursions in (2), (21) and (22) by calculating P t+1,t+m+1,..., P t+m+1,t+m+1, Q t+m+1,t+1,..., Q t+m+1,t+m+1,q t+1,t+m+1,..., Q t+m,t+m+1, P t+1,t+m+2,..., P t+m+2,t+m+2, Q t+m+2,t+1,..., Q t+m+2,t+m+2, Q t+1,t+m+2,...,q t+m+1,t+m+2,... Te above calculation can be furter simplified by noting tat for k>l 1, Q t+l,t+k = E r t+l 2 σ2 t+k Ω t = E r t+l 2 E r t+k 2 Ω t+k 1 Ω t = E E r t+l r2 2 t+k Ω t+k 1 Ω t = E r t+l r2 2 t+k Ω t = P t+l,t+k, and for l 1, Q t+l,t+l = E r t+l 2 σ2 t+l Ω t = E E r t+l 2 σ2 t+l Ω t+l 1 Ω t = E σt+l 2 E r t+l 2 Ω t+l 1 Ω t = E σt+l 4 Ω t =(1/K)P t+l,t+l. (23)