Modelling volatility and correlation

Transcription

1 Modelling volailiy and correlaion

2 An Excursion ino Non-lineariy Land Moivaion: i he linear srucural l( (and ime series) models canno explain a number of imporan feaures common o much financial daa - lepokurosis (fa ails and excess peak a he mean) - volailiy clusering or volailiy pooling (large re. followed by large re. and small re folllowed by small re.) - leverage effecs (volailiy rises more following a price fall han following a price increase of he same magniude) Our radiional srucural model could be somehing like: y = + x k x k + u, or more compacly y = X + u. We also assumed u N(0,σ ).

3 A Sample Financial Asse Reurns Time Series Daily S&P 500 Reurns for January 990 December 999 Reurn /0/90 /0/93 Dae 9/0/97 3

4 Non-linear Models: A Definiion Campbell, Lo and MacKinlay (997) define a non-linear daa generaing process as one ha can be wrien y = f(u, u -, u -, ) where u is an iid error erm and f is a non-linear funcion. They also give a slighly more specific definiion as y = g(u -, u -, )+ u σ (u -, u -, ) where g is a funcion of pas error erms only and σ is a variance erm. Models wih nonlinear g( ) are non-linear in mean, while hose wih nonlinear σ ( ) are non-linear in variance. 4

5 Types of non-linear models The linear paradigm is a useful one. Many apparenly non-linear relaionships can be made linear by a suiable ransformaion. On he oher hand, i is likely l ha many relaionships in finance are inrinsically non-linear. There are many ypes of non-linear models, e.g. - ARCH / GARCH -swiching models dl - bilinear models 5

6 Tesing for Non-lineariy The radiional ools of ime series analysis (acf s, specral analysis) may find no evidence ha we could use a linear model, bu he daa may sill no be independen. Pormaneau ess for non-linear dependence have been developed. The simples is Ramsey s RESET es, which ook he form: 3 u y y... y p = v 0 p Many oher non-lineariy ess are available, e.g. he BDS es and he bispecrum es. Always ess he whie noise propery of he residuals One paricular non-linear model ha has proved very useful in finance is he ARCH model dueo Engle (98). 6

7 H Heeroscedasiciy ii Revisied iid An example of a srucural model is y = + x + 3 x x 4 + u wih u N(0, σ u ). The assumpion ha he variance of he errors is consan is known as homoscedasiciy, i.e. Var (u )= σ u. Wha if he variance of he errors is no consan? - heeroscedasiciy -would imply ha sandard derror esimaes could be wrong. Is he variance of he errors likely o be consan over ime? No for financial daa. 7

8 Auoregressive Condiionally Heeroscedasic (ARCH) Models Souseamodel dlwhich h does no assume ha hevariance is consan. Recall he definiion of he variance of u : σ =Var(u u -,u -,...) =E[(u -E(u )) u -,u -,...] Weusuallyassumeha E(u ) = 0 so σ = Var(u u -,u -,...) =E[u u -,u -,...]. Wha could he curren value of he variance of he errors plausibly depend upon? Previous squared error erms. This leads o he auoregressive condiionally h heeroscedasic i model dl for he variance of he errors: σ = α 0 + α This is knownn as an ARCH() model. u 8

9 Auoregressive Condiionally Heeroscedasic (ARCH) Models (con d) The full model dlwould be y = + x k x k +u, u N(0, σ ) where σ = α 0 + α u We can easily exend his ohe general case where he error variance depends on q lags of squared errors: σ = α 0 + α u +α u +...+α q u q This isanarch(q) model. dl σ Insead of calling he variance, in he lieraure i is usually called h, so he model is y = + x k x k +u, u N(0,h ) where h = α 0 + α +α +...+α q u q u u 9

10 Anoher Way of Wriing ARCH Models For illusraion, consider an ARCH(). Insead of he above, we can wrie y = + x k x k +u, u = v σ σ α α, v N(0,) = 0 + u The wo are differen ways of expressing exacly he same model. The firs form is easier o undersand while he second form is required for simulaing from an ARCH model, for example. 0

11 Tesing for ARCH Effecs. Firs, run any posulaed linear regression of he form given in he equaion above, e.g. y = + x k x k +u saving he residuals,. û. Then square he residuals, and regress hem on q own lags o es for ARCH of order q, i.e. run he regression u ˆ ˆ ˆ ˆ = γ 0 + γ u + γ u γ qu q + v where v is iid. Obain R from his regression 3. The es saisic is defined as TR (he number of observaions muliplied by he coefficien of muliple correlaion) from he las regression, and is disribued as a χ (q).

12 Tesing for ARCH Effecs (con d) 4. The null and alernaive hypoheses are H 0 : γ = 0andγ = 0and γ 3 = 0 and... and γ q = 0 H : γ 0 or γ 0 or γ 3 0 or... or γ q 0. If he value of he es saisic is greaer han he criical value from he χ disribuion, ib i hen rejec henull hypohesis. Noe ha he ARCH es is also someimes applied direcly o reurns insead of he residuals from Sage above.

13 Problems wih ARCH(q) )Models How do we decide on q? The required value of q migh be very large Non-negaiviy consrains migh be violaed. When we esimae an ARCH model, we require α i >0 i=,,...,q (since variance canno be negaive) A naural exension of an ARCH(q) model which ges around some of hese problems is a GARCH model. 3

14 Generalised ARCH (GARCH) Models Due o Bollerslev (986). Allow he condiional variance o be dependen upon previous own lags The variance equaion is now σ = α 0 + α u +σ - () This is a GARCH(,) model, which is like an ARMA(,) model for he variance equaion. We could also wrie σ - = α 0 + α u +σ - σ - = α 0 + α u +σ Subsiuing ino () for σ - : σ = α 0 + α u +(α 0 + α u +σ - ) = α 0 + α u +α 0 + α u +σ - 4

15 Generalised ARCH (GARCH) Models (con d) Now subsiuing ino () for σ - σ =α 0 + α u +α 0 + α u + (α 0 + α u 3 +σ -3 ) σ = α 0 + α u +α 0 + α u +α 0 + α u σ -3 σ = α 0 (++ ) + α u (+L+ L ) + 3 σ -3 An infinie number of successive subsiuions would yield σ = α 0 ( ) + α u (+L+ L +...) + σ 0 So he GARCH(,) model can be wrien as an infinie order ARCH model. We can again exend he GARCH(,) model o a GARCH(p,q): σ = α 0 +α u +α u +...+α q u + σ - + σ p σ -p q σ = α 0 + α iu i + jσ p i= j= j q 5

16 Generalised ARCH (GARCH) Models (con d) Bu in general a GARCH(,) model will be sufficien o capure he volailiy clusering in he daa. Why is GARCH Beer han ARCH? - more parsimonious - avoids overfiing - less likelyl o breech non-negaiviy i consrains 6

17 The Uncondiional Variance under he GARCH Specificaion The uncondiional i variance of u is given by when Var(u ) = α + < α0 ( α + ) α + is ermed non-saionariy in variance α + = is ermed inergraed GARCH For non-saionariy in variance, he condiional variance forecass will no converge on heir uncondiional value as he horizon increases. 7

18 Esimaion of ARCH / GARCH Models Since he model is no longer of he usual linear form, we canno use OLS. We use anoher echnique known as maximum likelihood. The mehod works by finding he mos likelyl values of he parameers given he acual daa. More specifically,we form a log-likelihood funcion and maximise i. 8

19 Esimaion of ARCH / GARCH Models (con d) The seps involved in acually esimaing i an ARCH or GARCH model dl are as follows. Specify he appropriae equaions for he mean and he variance - e.g. an AR()- GARCH(,) model: y = μ + φy - + u, u N(0,σ ) σ = α 0 + α u +σ -. Specify he log-likelihood funcion o maximise: L = T T log(π ) log( σ ) = 3. The compuer will maximise he funcion and give parameer values and heir sandard errors T = ( y μ φy ) / σ 9

20 Parameer Esimaion using Maximum Likelihood Consider he bivariae regression case wih homoscedasic errors for simpliciy: y + x + u = Assuming ha u N(0,σ ), hen y N(, σ + x )sohahe probabiliy densiy funcion for a normally disribued random variable wih his mean and variance is given by ( y x ) () f ( y + x, σ ) = exp σ π σ Successive values of y would race ou he familiar bell-shaped curve. Assuming ha u are iid, hen y will also be iid. 0

21 Parameer Esimaion using Maximum Likelihood Parameer Esimaion using Maximum Likelihood (con d) Then he join pdf for all he y s can be expressed as a produc of he individual densiy funcions () = + T T T X y f X y f X y f X y y y f 4 ), ( )..., ( ), ( ),,...,, ( σ σ σ σ Sbii i i () f f i () = + = y X f ), ( σ Subsiuing ino equaion () for every y from equaion (), (3) + T y x f ) ( ) ( = + = T T T y x y y y f ) ( exp ) ( ),,...,, ( σ π σ σ

22 Parameer Esimaion using Maximum Likelihood (con d) The ypical siuaion we have is ha he x and y are given and we wan o esimae,, σ. If his is he case, hen f( ) is known as he likelihood funcion, denoed LF(,, σ ), so we wrie LF(,, σ T ( y ) = exp T T σ ( π ) = x Maximum likelihood esimaion involves choosing parameer values (,,σ ) ha maximise his funcion. We wan o differeniae (4) w.r..,,σ, bu (4) is a produc conaining T erms. σ ) (4)

23 Parameer Esimaion using Maximum Likelihood (con d) Since max f ( x ) = maxlog( f ( x )), we can ake logs of (4). x x Then, using he various laws for ransforming funcions conaining logarihms, we obain he log-likelihood funcion, LLF: T T ( y = x LLF T ) logσ log(π ) = σ which is equivalen o T T T ( y = x LLF logσ log(π ) Differeniaing (5) w.r..,σ = σ,,, we obain ) LLF ( y ).. = x σ (5) (6) 3

24 Parameer Esimaion using Maximum Likelihood (con d) LLF ( y x ).. x = (7) σ LLF T ( y x = + 4 σ σ σ (8) Seing (6)-(8) o zero o minimise he funcions, and puing has above he parameers o denoe he maximum likelihood esimaors, ) From(6), ( y ˆ ˆ x ) = 0 ˆ ˆ y x = y ˆ ˆ T x = T T y ˆ ˆ x = ˆ = y ˆ x (9) 4

25 Parameer Esimaion using Maximum Likelihood (con d) From (7), ˆ ˆ ( y x ) x = 0 From(8), ( y y ˆ x x ˆ ˆ x x = ˆ ˆ x x = y x ( y ˆ 0 = 0 x ) ˆ T ˆ σ x ( x y x = ( x = x x ˆ = y x Txy T x ˆ ˆ = 4 ˆ σ Tx ) = Tx Txy ) y x ( y ˆ ˆ x ) Txy (0) 5

26 Parameer Esimaion using Maximum Likelihood (con d) Rearranging, ˆ ˆ ˆ σ = ( y x ) T σ = u T () How do hese formulae compare wih he OLS esimaors? (9) & (0) are idenical o OLS () is differen. The OLS esimaor was σ = T k u Therefore he ML esimaor of he variance of he disurbances is biased, alhough i is consisen. Bu how does his help us in esimaing heeroscedasic models? 6

27 Esimaion of GARCH Models Using Maximum Likelihood Now we have y = μ + φy - + u, u N(0, ) σ = α 0 + α u +σ - L = σ T T T log(π ) log( σ ) ( y μ φy ) / σ = = Unforunaely, he LLF for a model wih ime-varying variances canno be maximised analyically, excep in he simples of cases. So a numerical procedure is used o maximise he log-likelihood likelihood funcion. A poenial problem: local opima or mulimodaliies in he likelihood surface. The way we do he opimisaion is:. Se up LLF.. Use regression o ge iniial i i guesses for he mean parameers. 3. Choose some iniial guesses for he condiional variance parameers. 4. Specify a convergence crierion - eiher by crierion or by value. 7

28 Non-Normaliy Normaliy and Maximum Likelihood Recall ha he condiional normaliy assumpion for u is essenial. We can es for normaliy using he following represenaion u = v σ v N(0,) v 0 σ = α + α u + α σ u = σ The sample counerpar is uˆ vˆ = σˆ σ Are he normal? Typically are sill lepokuric, alhough lesssohan he. Isvˆ his a problem? No really, as we can use he ML wih a robus vˆ variance/covariance esimaor. ML wih robus sandard errors is called Quasi- Maximum û Likelihood or QML. 8

29 Exensions o he Basic GARCH Model Since he GARCH model was developed, a huge number of exensions and varians have been proposed. Three of he mos imporan examples are EGARCH, GJR, and GARCH-M models. Problems wih GARCH(p,q) Models: - Non-negaiviy consrains may sill be violaed - GARCH models canno accoun for leverage effecs Possible soluions: he exponenial GARCH (EGARCH) model or he GJR model, which are asymmeric GARCH models. 9

30 The EGARCH Model Suggesed by Nelson (99). The variance equaion is given by u u = π σ α σ γ σ ω σ ) log( ) log( u u Advanages of he model - Since we model he log(σ ), hen even if he parameers are negaive, σ will be posiive. - We can accoun for he leverage effec: if he relaionship beween volailiy and reurns is negaive γ will be negaive volailiy and reurns is negaive, γ, will be negaive. 30

31 The GJR Model Due o Glosen, Jaganahan and Runkle σ = α 0 + α u +σ - +γu - I - where I - =ifu - <0 = 0 oherwise For a leverage effec, we would see γ >0. We require α + γ 0 and α 0 for non-negaiviy. 3

32 An Example of fhe use of a GJRM Model Using monhly S&P 500 reurns, December 979- June 998 Esimaing a GJR model, we obain he following resuls. y = 0.7 (3.98) σ = u σ u I (6.37) (0.437) (4.999) (5.77) 3

33 News Impac Curves The news impac curve plos he nex period volailiy (h ) ha would arise from various posiive and negaive values of u -, given an esimaed model. News Impac Curves for S&P 500 Reurns using Coefficiens from GARCH and GJR 0.4 Model Esimaes: 0. GARCH GJR Value of Condiional Varia ance Value of Lagged Shock 33

34 GARCH-in Mean We expec a risk o be compensaed by a higher reurn. So why no le he reurn of a securiy be parly deermined by is risk? Engle, Lilien and Robins (987) suggesed he ARCH-M specificaion. A GARCH-M model would be y = μ + δσ - + u, u N(0,σ ) σ = α 0 + α u +σ - δ can be inerpreed as a sor of risk premium. I is possible o combine all or some of hese models ogeher o ge more complex hybrid models - e.g. an ARMA-EGARCH(,)-M model. dl 34

35 Wha Use Are GARCH-ype Models? GARCH can model he volailiy clusering effec since he condiional variance is auoregressive. Such models can be used o forecas volailiy. Wecould show ha Var (y y -,y -,...) = Var (u u -,u -,...) So modelling σ will give us models and forecass for y as well. Variance forecass are addiive over ime. 35

36 Forecasing Variances using GARCH Models Producing condiional variance forecass from GARCH models uses a very similar approach o producing forecass from ARMA models. I is again an exercise in ieraing wih he condiional expecaions operaor. Consider he following GARCH(,) model: y, u N(0,σ ), = μ + u σ = α 0 + αu + σ Wha is needed is o generae are forecass of σ T+ Ω T, σ T+ Ω T,..., σ T+s Ω T where Ω T denoes all informaion available up o and including observaion T. Adding one o each of he ime subscrips of he above condiional variance equaion, and hen wo, and hen hree would yield he following equaions σ T+ = α 0 + α +σ T, σ T+ = α 0 + α +σ T+, σ T+3 = α 0 + α +σ T+ 36

37 Forecasing Variances using GARCH Models (Con d) f Le σ be he one sep ahead forecas for σ, T made a ime T. This is easy o calculae since, a ime T, he values of all he erms on he RHS are known. f σ would be obained by aking he condiional expecaion of he, T firs equaion a he boom of slide 36: f σ, T = α 0 + α u T +σ T f f Given, σ how is, he -sep ahead forecas for σ, T σ made a ime T,, T calculaed? Taking he condiional expecaion of he second equaion a he boom o of slide 36: f f σ = α 0 + α E( u Ω T ) +, T T + σ, T where E( u Ω T ) is he expecaion, made a ime T, of u T + T +, which is he squared disurbance erm. 37

38 Forecasing Variances using GARCH Models (Con d) We can wrie E(u T+ Ω ) = σ T+ Bu σ T+ is no known a ime T, so i is replaced wih he forecas for f i, σ, T, so ha he -sep ahead forecas is given by f f = α 0 + α + f σ, T σ, T σ, T f f σ = α 0 + (α +) σ, T, T By similar argumens, he 3-sep ahead forecas will be given by f σ 3, T = E T (α 0 + α + σ T+ ) f = α 0 + (α +) σ, T f = α 0 + (α +)[ α 0 + (α +) σ ], T = α 0 + α 0 (α +) + (α +) f σ, T Any s-sep ahead forecas (s ) would be produced by s f i s f h s, T = α 0 ( α + ) + ( α + ) h, T i= 38

39 Wha Use Are Volailiy Forecass?. Opion pricing ii. Condiional beas C = f(s, X, σ,t,r f ) i, σ im, = σm, 3. Dynamic hedge raios The Hedge Raio - he size of he fuures posiion o he size of he underlying exposure, i.e. he number of fuures conracs o buy or sell per uni of he spo good. 39

40 Wha Use Are Volailiy Forecass? (Con d) Wha is he opimal value of he hedge raio? Assuming ha he objecive of hedging is o minimise he variance of he hedged porfolio, he opimal hedge raio will be given by σ s h = p σs σf where h = hedge raio p = correlaion coefficien beween change in spo price (S) and change in fuures price (F) σ S = sandard deviaion of S σ F = sandard deviaion of F Wha if he sandard deviaions and correlaion are changing over ime? Use h = p σ σ s, F, 40

41 Tesing Non-linear Resricions or Tesing Hypoheses abou Non-linear Models Usual - and F-ess are sill valid in non-linear models, bu hey are no flexible enough. There are hree hypohesis esing procedures based on maximum likelihood principles: Wald, Likelihood Raio, Lagrange Muliplier. Consider a single parameer, θ o be esimaed, Denoe he MLE as ~ and a resriced esimae as θ. θˆ 4

42 Likelihood Raio Tess Ei Esimaeunder henull hypohesisand under healernaive. i Then compare he maximised values of he LLF. So we esimae he unconsrained model and achieve a given maximised value of he LLF, denoed L u Then esimae he model imposing he consrain(s) and ge a new value of he LLF denoed L r. Which will be bigger? L r L u comparable o RRSS URSS The LR es saisic is given by LR =-(L r - L u ) χ (m) where m = number of resricions 4

43 Likelihood Raio Tess (con d) Example: We esimae a GARCH model and obain a maximised LLF of We are ineresed in esing wheher = 0 in he following equaion. y = μ + φy - + u, u N(0, σ ) = α 0 + α + σ σ u We esimae he model imposing he resricion and observe he maximised LLF falls o Can we accep he resricion? LR = -( ) = 4.6. The es follows a χ () = 3.84 a 5%, so rejec he null. Denoing hemaximised i value of he LLF by unconsrained dml as L( ) and he consrained opimum as. Then we can illusrae he 3 esing θˆ procedures in he following diagram: L( ~ θ ) 43

44 Comparison of Tesing Procedures under Maximum Likelihood: Diagramaic Represenaion L ( θ ) L ( θˆ ) A L ( ~ θ ) B θ ~ θˆ θ 44

45 Hypohesis Tesing under Maximum Likelihood The verical ldisance forms he basisof he LR es. The Wald es is based on a comparison of he horizonal disance. The LM es compares he slopes of he curve a A and B. We know a he unresriced MLE, L( θˆ ), he slope of he curve is zero. Bi Bu is i i significanlyifi seep a L( ~ θ )? This formulaion of he es is usually easies o esimae. 45

46 An Example of he Applicaion of GARCH Models - Day & Lewis (99) Purpose To consider he ou of sample forecasing performance of GARCH and EGARCH Models for predicing sock index volailiy. Implied volailiy is he markes expecaion of he average level of volailiy of an opion: Whichh is beer, GARCH or implied volailiy? Daa Weekly closing prices (Wednesday o Wednesday, and Friday o Friday) for he S&P00 Index opion and he underlying March 83-3 Dec. 89 Implied volailiy is calculaed using a non-linear ieraive procedure. 46

47 The Models The Base Models For he condiional mean R λ 0 λ h + u M RF = + And for he variance () h = + + α 0 αu h or / (3) u u ln( h ) = α 0 + ln( h ) + α( θ + γ ) where h π h R M denoes he reurn on he marke porfolio R F denoes he risk-free rae h denoes he condiional variance from he GARCH-ype models while σ denoes he implied variance from opion prices. () 47

48 The Models (con d) Add in a lagged value of he implied volailiy parameer o equaions () and (3). () becomes (4) h = α + α u + h + δσ and (3) becomes = 0 + / u u ln( h ) = α + ln( h ) + α( θ + γ ) + δ ln( σ h h π We are ineresed in esing H 0 : δ =0in(4) or (5). Also, we wan o es H 0 : α =0and = 0 in (4), andh 0 : α =0and =0andθ =0andγ = 0 in (5). 0 ) (5) 48

49 The Models (con d) If his second se of resricions holds, hen (4) & (5) collapse o h α + δσ (4 ) = 0 and (3) becomes ln( h ) = α0 + δ ln( σ ) (5 ) We can es all of hese resricions using a likelihood raio es. 49

50 In-sample Likelihood Raio Tes Resuls: GARCH Versus Implied Volailiy R λ h + u (8.78) h h M RF = 0 + λ = αu h = 0 + αu + h + δσ = 0 + δσ α (8.79) α (8.8) h α (8.8 ) Equaion for λ 0 λ α α δ Log-L χ Variance specificaion (8.79) (0.005) (0.0) (.65) (0.84) (8.7) (8.8) (0.08) (0.0) (.98) (.7) (-0.59) (3.00) (8.8 ) (0.00) 00) ( ) 00) (.50) (.94) Noes: -raios in parenheses, Log-L denoes he maximised value of he log-likelihood funcion in each case. χ denoes he value of he es saisic, which follows a χ () in he case of (8.8) resriced o (8.79), and a χ () in he case of (8.8) resriced o (8.8 ). Source: Day and Lewis (99). Reprined wih he permission of Elsevier Science. 50

51 In-sample Likelihood Raio Tes Resuls: EGARCH Versus Implied Volailiy R λ h + u (8.78) M RF = 0 + λ / u u ln( h ) = α 0 + ln( h ) + α( θ + γ ) h h π (8.80) / u u ln( h ) = α 0 + ln( h ) + α( θ + γ ) + δ ln( σ ) (8.8) h h π ln( h ) = α 0 + δ ln( σ ) (8.8 ) uaion for λ 0 λ α θ γ δ Log-L χ Variance cificaion (c) (-0.03) (0.5) (-.90) (3.6) (-4.3) (3.7) (e) (0.56) (-0.4) (-.8) (.48) (-4.34) (.89) (.8) (e ) (0.7) (-0.43) (-.30) (4.0) Noes: -raios in parenheses, Log-L denoes he maximised value of he log-likelihood funcion in each case. χ denoes he value of he es saisic, which follows a χ () in he case of (8.8) resriced o (8.80), and a χ () in he case of (8.8) resriced o (8.8 ). Source: Day and Lewis (99). Reprined wih he permission of Elsevier Science. 5

52 Conclusions for In-sample Model Comparisons & Ou-of-Sample Procedure IV has exra incremenal power for modelling sock volailiy beyond GARCH. Bu he models do no represen a rue es of he predicive abiliy of IV. So he auhors conduc an ou of sample forecasing es. There are 79 daa poins. They use he firs 40 o esimae he models, and hen make a -sep ahead forecas of he following week s volailiy. Then hey roll he sample forward one observaion a a ime, consrucing a new one sep ahead forecas a each sep. 5

53 Ou-of-Sample Forecas Evaluaion They evaluae he forecass in wo ways: The firs is by regressing he realised volailiy series on he forecass plus a consan: σ = b + b σ + ξ (7) σ f + where is he acual value of volailiy, and σ f is he value forecased for i during period. Perfecly accurae forecass imply b 0 =0andb =. Bu wha is he rue value of volailiy a ime? Day & Lewis use measures. The square of he weekly reurn on he index, which hey call SR.. The variance of he week s daily reurns muliplied by he number of rading days in ha week. 53

54 Ou-of Sample Model Comparisons σ (8.83) + = b0 + bσ f + ξ + (8 83) Forecasing Model Proxy for ex b 0 b R pos volailiy Hisoric SR (5.60) (.8) Hisoric i WV (.90) (7.58) GARCH SR (.0) (.0) GARCH WV (.07) (3.34) 34) EGARCH SR (0.05) (.06) EGARCH WV (-0.48) (.58) Implied Volailiy SR (.) (.8) Implied Volailiy WV (0.389) 0.78 (.95) 0.06 Noes: Hisoric refers o he use of a simple hisorical average of he squared reurns o forecas volailiy; -raios in parenheses; SR and WV refer o he square of he weekly reurn on he S&P 00, and he variance of he week s daily reurns muliplied li by he number of rading days in ha week, respecively. Source: Day and Lewis (99). Reprined wih he permission of Elsevier Science. 54

55 Encompassing Tes Resuls: Do he IV Forecass Encompass hose of he GARCH Models? σ (8.86) + = b0 + bσ I + bσ G + b3σ E + b4σ H + ξ + Forecas comparison b 0 b b b 3 b 4 R Implied vs. GARCH (-0.09) (.03) (0.4) Implied vs. GARCH vs. Hisorical (.5) 0.63 (.0) (-0.8) (7.0) Implied vs. EGARCH (-0.07) (.6) (0.7) Implied vs. EGARCH vs. Hisorical (.37) (.45) (-0.57) (7.74) GARCH vs. EGARCH (0.37) (.78) (-0.00) Noes: -raios in parenheses; he ex pos measure used in his able is he variance of he week s daily reurns muliplied by he number of rading days in ha week. Source: Day and Lewis (99). Reprined wih he permission of Elsevier Science. 55

56 Conclusions of Paper Wihin sample resuls sugges ha IV conains exra informaion no conained in he GARCH / EGARCH specificaions. Ou of sample resuls sugges ha nohing can accuraely predic volailiy! 56

57 Mulivariae i GARCH Models Mulivariae i GARCH models are used o esimae and o forecas covariances and correlaions. The basic formulaion is similar o ha of he GARCH model, bu where he covariances as well as he variances are permied o be ime-varying. There are 3 main classes of mulivariae GARCH formulaion ha are widely used: VECH, diagonal VECH and BEKK. VECH and Diagonal VECH e.g. suppose ha here are wo variables used in he model. The condiional covariance marix is denoed H, and would be. H and VECH(H ) are h h h H VECH ( H = = ) h h h h 57

58 VECH and Diagonal VECH In he case of he VECH, he condiional variances and covariances would each depend upon lagged values of all of he variances and covariances and on lags of he squares of boh error erms and heir cross producs. In marix form, i would be wrien Wriing ou all of he elemens gives he 3 equaions as VECH ( H ) C + A VECH ( Ξ Ξ ) BVECH ( H ) = + Ξ ψ ~ ( 0 H ) N, Such a model would be hard o esimae. The diagonal VECH is much simpler and is specified, in he variable case, as follows: h h h = c + a u + a u + a u u + b + b + b 3 3 = c + au + au + a3u u + bh + bh + b3h = c 3 + a 3u + a 3u + a 33u u + b 3h + b 3h + b 33h The BEKK Model uses a Quadraic form for he parameer marices o ensure a posiive definie variance / covariance marix H. h h h = α 0 + αu + α h = 0 + u + h = γ 0 + γ u u h + γ h h h 58

59 BEKK and Model Esimaion for M-GARCH Neiher he VECH nor he diagonal VECH ensure a posiive definie variancecovariance marix. An alernaive approach is he BEKK model (Engle & Kroner, 995). In marix form, he BEKK model is H = W W + A H A + B Ξ Ξ Model esimaion for all classes of mulivariae GARCH model is again performed using maximum likelihood wih he following LLF: = TN T ' ( θ ) log π ( log H + Ξ H Ξ ) = where N is he number of variables in he sysem (assumed above), θ is a vecor conaining all of he parameers o be esimaed, and T is he number of observaions. B 59

60 An Example: Esimaing a Time-Varying Hedge Raio for FTSE Sock Index Reurns (Brooks, Henry and Persand, 00). Daa comprises 3580 daily observaions on he FTSE 00 sock index and sock index fuures conrac spanning he period January April 999. Several compeing models for deermining he opimal hedge raio are consruced. Define he hedge raio as. No hedge (=0) Naïve hedge (=) Mulivariae GARCH hedges: Symmeric BEKK Asymmeric BEKK In boh cases, esimaing he OHR involves forming a -sep ahead forecas and compuing OHR + = h CF, + h F, + Ω 60

61 OHR Resuls Unhedged = 0 In Sample Naïve Hedge = Symmeric Time Varying Hd Hedge hfc, = h F, Asymmeric Time Varying Hd Hedge hfc, = h Reurn {.373} {-0.035} {0.956} {0.9580} Variance Unhedged = 0 Ou of Sample Naïve Hedge = Symmeric Time Varying Hedge hfc, = h F, F, Asymmeric Time Varying Hedge hfc, = h Reurn {.4958} {0.06} {0.776} {0.9083} Variance F, 6

62 Plo of he OHR from Mulivariae GARCH.00 Time Varying Hedge Raios Conclusions - OHR is ime-varying and less han - M-GARCH OHR provides a beer hedge, boh in-sample and ou-of-sample. - No role in calculaing OHR for asymmeries Symmeric BEKK Asymmeric BEKK 6