Dynamic Hedging using a Bivariate Markov Switching FIGARCH model

Transcription

1 Dynamic Hedging using a Bivariae Markov Swiching FIGARCH model Jonahan Dark Deparmen of Finance, The Universiy of Melbourne Absrac This paper develops a bivariae Markov Swiching FIGARCH (MS-FIGARCH) process wih consan and ime varying ransiion probabiliies as a way of modeling spo fuures dynamics. An applicaion of he model illusraes ha he S&P500 and is fuures exhibi long memory in volailiy and srucural breaks ha are driven by changes in he cos of carry. The model wih consan ransiion probabiliies provides a sligh improvemen in hedging performance relaive o an approach ha does no allow for regime swiching. On allowing he ransiion probabiliies o be a funcion of he risk free rae, regime shifs are more effecively idenified and improvemens in hedging oucomes around he ime of a break may be achieved. jdark@unimelb.edu.au. I hank Bruce Grundy and Paul Kofman for heir commens and helpful suggesions.

2 . Inroducion The las few decades have seen a proliferaion of lieraure seeking o esimae fuures hedge raios (for comprehensive reviews see Chen e al, 2003 and Lien and Tse, 2002). The convenional approach assumes a given spo posiion and an expeced mean variance uiliy funcion. In his conex, he opimal hedge raio maximises he risk adjused expeced profi from he hedge. The minimum variance hedge raio (MVHR) is a special case of he opimal hedge raio. The wo are equal if he fuures price follows a maringale process or if he hedger is exremely risk averse. Much of he recen lieraure has quesioned he use of he mean variance framework or sough o improve he economeric mehods of esimaion. Alernaive risk measures like he lower parial momen and he mean-gini approach have been used when hedging (Lien and Tse, 998; Kolb and Okunev, 992). The improved economeric mehods have allowed for feaures such as coinegraion and ime varying covariances (Kroner and Sulan, 993), fracional coinegraion (Lien and Tse, 999) and long memory in volailiy and basis convergence (Dark, 2007). Oher economeric mehods have allowed regime swiches (RS) o affec he spo fuures dynamics. I is argued ha if regime swiches occur, he MVHR is also likely o be sae dependen. Hedge raio esimaion ha ignores he shifs in dynamics beween saes is herefore likely o resul in sub-opimal hedging oucomes. Despie he concepual superioriy of uilizing a model ha allows for regime swiches, here is mixed suppor for he use of regime swiching models when esimaing MVHRs. Alizadeh and Nomikos (2004) find ha consan volailiy markov swiching (MS) models for he S&P500 and FTSE00 ouperform convenional measures of MVHR esimaion in sample. Ou of sample, he MS model performs he bes for he FTSE00, however a mulivariae GARCH model provided he greaes variance reducion for he S&P500. Lee and Yoder (2007) used a bivariae MS-

3 GARCH process o esimae dynamic MVHRs for corn and nickel. They show ha allowing he covariance marix o be sae dependen improves ou of sample hedging effeciveness, however furher analysis finds he improvemens o be saisically insignifican. Kofman and McGlenchy (2005) idenify srucural breaks via a reverse order CUSUM-squared (ROC) es. Forecass of MVHRs are condiioned on observaions ha occurred subsequen o he idenified break. An applicaion o hree Hong Kong sock index porfolios found limied improvemen for in and ou of sample risk reducion. The mixed suppor for RS models from previous research may be due o model misspecificaion, failure o forecas he regime or esimaion error. Dacco and Sachell (999) show ha even if he underlying process exhibis regime swiching, a sligh misclassificaion (eiher from failure o forecas he regime or esimaion error) may mean ha he forecass from he appropriae regime swiching model have a higher mean square error han forecass from a random walk. This resul suggess ha he search for RS models ha more adequaely capure he underlying dynamics may sill prove o be very useful when hedging. This paper herefore proposes a new model, a bivariae MS-FIGARCH process wih mauriy effecs o model he spo fuures dynamics of he S&P500. The proposed modeling approach differs from he previous approaches in hree imporan respecs. Firs, he regime swiches in he model occur beween high cos of carry (COC) and low cos of carry saes. This is in conras o previous research where he swiches in spo fuures dynamics occur beween high and low volailiy saes (Alizadeh and Nomikos, 2004; Swiches in he dynamics may also occur according o wheher here are violaions in he no arbirage bounds implied by he cos of carry. Wihin he no arbirage bounds, no adjusmen akes place and deviaions from he cos of carry exhibi uni roo or near uni roo behaviour. Ouside he no arbirage bounds, he basis swiches o a saionary auoregressive process. Early work esimaed hreshold error correcion models which allowed he dynamics o differ according o wheher he basis was inside or ouside he bounds (Dwyer e al, 996; Marens e al, 998). Monoyios and Sarno (2002) argue ha proporional ransacion coss, ime aggregaion and nonsynchronous adjusmen by heerogeneous agens produce a nonlinear adjusmen of he basis ha occurs in a smooh raher han a discree manner. Monoyios and Sarno (2002) herefore model he daily S&P500 basis dynamics using an exponenial smooh ransiion auoregressive (ESTAR) specificaion. The imporance of hese swiches has no been considered when esimaing MVHRs. This is an area for furher research.

4 Lee and Yoder, 2007). Two models are esimaed, he firs imposes consan ransiion probabiliies and he second models ime varying ransiion probabiliies as a funcion of he risk free rae. The resuls illusrae ha on allowing he ransiion probabiliies o be a funcion of he risk free rae, he regime shifs are more effecively idenified and significan improvemens in hedging oucomes may be achieved around he ime of he break. Second, he model allows for regime swiches and long memory in volailiy (via he FIGARCH process). To dae, he hedging lieraure has uilized eiher swiching models (Alizadeh and Nomikos, 2004, Lee and Yoder, 2007) or long memory models (Dark, 2007) bu nohing ha combines hese wo feaures. This is consisen wih he lieraure on long memory and regime swiching which unil he lae 990s evolved independenly. Long memory has been widely documened in equiy, currency and commodiy marke volailiies (see for example Baillie, 996; Andersen and Bollerslev, 997b; Elder and Jin, 2007). 2 The alernaive view however rejecs long memory and shows ha if occasional breaks occur, he process may easily be confused wih a long memory process (Diebold and Inoue, 200; Granger and Hyung, 2004). 3 This is suppored by Mikosch and Sarica (998) who sugges 2 There are a number of explanaions for he exisence of long memory in volailiy. Mos focus on he role of aggregaion, exending he mixure of disribuions hypohesis (MDH) and he work of Granger (980). The MDH was proposed by Clarke (973) and subsequenly developed by Epps and Epps (976) and Tauchen and Pis (983). Granger (980) demonsraed ha he cross-secional aggregaion of a small number of N dependen auoregressive processes will resul in he aggregae series following an ARMA(N,N-) process. If he auoregressive coefficiens come from he bea disribuion, hen in he limi, he aggregae process is I(d). Ding and Granger (996) exend his o he second momen, demonsraing how long memory in volailiy may arise from he cross-secional aggregaion of muliple volailiy componens. Subsequen research has shown ha long memory in volailiy may arise from he aggregaion of muliple volailiy componens caused by heerogeneous informaion flows (Andersen and Bollerslev, 997a) or heerogeneous raders (Muller e al, 997). Long memory in sock indices may also be due o he aggregaion of individual socks which are only weakly dependen (Lobao and Savin, 998). Granger s (980) work has also been generalized by Chambers (998) who considers emporal aggregaion and Lippi and Zaffaroni (999) who weaken he assumpion of he bea disribuion. The aggregaion argumen however is generally criicized given is resricive assumpions and he lack of evidence ha long memory is more eviden in aggregaed versus disaggregaed daa. Forunaely, for he long memory proponens a leas, aggregaion is no he only pah o long memory. Chen e al (999b) show how long memory may arise from Markov processes. Liu (2000) shows how a heavy ailed regime swiching sochasic volailiy model generaes a process consisen wih long memory. 3 The occasional breaks inerpreaion can herefore be seen as an exension of he research which reveals ha if a srucural break resuls in a shif in he uncondiional variance and his is ignored, his may resul in he spurious deecion of an IGARCH process (Diebold, 986, Lasrapes, 989; Lamourex and Lasrapes, 990). See Sock (994) for a summary of he lieraure on uni roos and srucural breaks.

5 ha he findings of long memory in S&P500 volailiy are spurious and a resul of ignoring srucural breaks. 4 More recen lieraure however acknowledges he presence of occasional breaks and long memory in S&P500 volailiy. Lobao and Savin (998) avoid spurious deecion of long memory from srucural breaks by spliing daily S&P500 reurns ino sub-periods ha are saionary and show ha he sub-series exhibi long memory in volailiy. 5 Andersen and Bollerslev (997b) find long memory in he absolue five minue reurns of he S&P500 over shor ime periods ha are also no subjec o breaks. Belrai and Morana (2006) show ha once srucural breaks are removed from he S&P500, he break free series is consisen wih long memory in volailiy. 6 Baillie and Morana (2007) develop he Adapive FIGARCH process as a way o joinly model long memory and srucural breaks. Applicaion of he model o he S&P500 suppors long memory in volailiy and srucural breaks. Marens e al (2004) also suppor he presence of srucural breaks and long memory in S&P500 realised volailiy. The forecasing performance of models ha allow for long memory and occasional breaks is mixed. Marens e al (2004) find ha allowing for srucural breaks does no improve ou of sample forecasing performance. Morana and Belrai (2004) show ha neglecing he breaks and employing a long memory model does no maer for shor erm forecasing, however over longer horizons, superior forecass are obained when using a model ha capures breaks and long memory. I herefore seems likely ha he proposed MS- 4 Unforunaely he esing and esimaion procedures are unable o differeniae beween long memory and occasional break models. If he DGP is an occasional break model, long memory esing and esimaion procedures will spuriously deec long memory (Granger and Hyung, 2004; Breid and Hsu; 2002). If he DGP is a long memory process, srucural break ess will spuriously idenify breaks (Granger and Hyung, 2004). 5 Furher, he aggregaion argumen is addressed by esablishing he presence of long memory for he sub-series of individual socks on he Dow Jones Indusrial Average. 6 They also show ha in order o uncover he relaionships beween macro-economic volailiy and S&P500 volailiy, an approach ha accouns for srucural breaks and long memory is required.

6 FIGARCH process will adequaely capure he dynamics of he S&P500. I is unclear hough, wheher he proposed model will produce significan improvemens in hedging performance. The hird difference in he modeling approach is ha i models he dynamics via he spo and he basis, raher han he spo and he fuures. By direcly modeling he basis, wo imporan feaures can be incorporaed ino he approach. Firsly, he model and he MVHR are able o explicily accoun for basis convergence. This has imporan implicaions when hedging given ha failure o allow for basis convergence will on average produce a saisically significan increase in porfolio variance (Dark, 2007). Secondly, spo-fuures dynamics are srongly relaed o he sign and size of he basis. Lee (994), Ng and Pirrong (994) and Lien and Yang (2007) show ha as he absolue value of he basis increases, he spo and fuures become more volaile and less correlaed. This resul suggess ha a high cos of carry sae is likely o experience lower and more volaile MVHRs. By modeling he basis direcly and allowing for regime swiches, his imporan relaionship can be explicily accouned for. The srucure of he paper is as follows. Secion 2 derives an expression for he bias in he MVHR ha arises from failing o recognize srucural breaks. Secion 3 presens he daa for he S&P500 and illusraes he need o allow for regime swiches when modeling he spo fuures dynamics. Secion 4 presens he proposed bivariae MS-FIGARCH model wih mauriy effecs wih consan and ime varying ransiion probabiliies. The resuls illusrae ha in high cos of carry saes, he higher basis volailiy resuls in a lower spo fuures correlaion and herefore lower bu more volaile hedge raios. The MS FIGARCH model wih consan ransiion probabiliies provides a sligh improvemen in hedging performance relaive o a MVHR ha does no allow for regime swiching. On allowing he ransiion probabiliies o be a funcion of he risk free rae, regime shifs are more effecively idenified

7 and significan improvemens in hedging oucomes around he ime of he break may be achieved. Secion 5 concludes. 2. The Bias in he MVHR from failing o allow for srucural breaks To illusrae he imporance of allowing for regime shifs when hedging, his secion derives an analyical expression for he bias in he MVHR from failing o allow for srucural breaks. For illusraive purposes, a simple MS model wih wo saes, consan volailiy and consan ransiion probabiliies is employed. To define he model le spo and fuures prices a ime, and le he basis be B F C C and F represen he =. Le {, 2} s = denoe he unobserved sae variable a ime, which follows a firs order wo sae Markov process. Le Ω denoe he informaion se a ime - and le m denoe he number of days o mauriy. The following daa generaing process is assumed ΔC = μ + ε () cs, j cs,, j C- Δ B = + j =, 2 (2) ( ) μbs, ε j bs,, m λ j C- where 2 μcs, σ cs, σ cbs, Sae N μs, H, s = 2 μ bs, σ cb, s σ b, s (3) 2 μcs, σ 2 cs, σ 2 cbs, 2 Sae 2 N μs, H, 2 s2 = 2 μ bs, σ 2 cb, s σ 2 b, s 2 (4) and λ capures he effec of basis volailiy convergence over he life of he fuures conrac (assumed o be he same across boh saes). This specificaion means ha ( ) σ Var B C m λ Δ = and (, ) s, j bs, j Cov ΔC C Δ B C = σ m λ. Expressing he volailiy of s, j cbs, j he normalized change in he basis and he covariance as a power funcion of mauriy is consisen wih he specificaion developed in Chen e al (999a). Assume ha he hedger

8 seeks o esimae a consan MVHR a ime by using uncondiional momens. If he hedger allows for regime swiches and correcly idenifies he regime, he MVHR will be equal o 7 θ s j Vars [ Δ C] + Cov, [, ] j s ΔC ΔB j [ Δ ] + [ Δ ] + 2 [ Δ, Δ ] = Var C Var B Cov C B sj, sj, sj ΔC ΔC ΔB Vars,, j Cov sj C + C C = ΔC ΔB ΔC ΔB Vars Var, 2,, j + s + Cov j sj C C C C σ + σ m = σ σ σ 2 cs, j cbs, j λ 2 2 2λ λ cs, +, 2 j bsm j + cbs, m j (5) To illusrae he imporance of allowing for regime swiches, consider a hedger ha ignores regime swiching. This hedger calculaes he MVHR based on an esimae of he variance covariance marix for he daa se available a ime. The esimae of he variance covariance marix will herefore be equivalen o ha obained via a mixure of wo mulivariae normal disribuions. The wo componen mulivariae normal mixure (MNM) disribuion is given by ( ) πsφ(, μs, s ) πsφ(, μs, s ) f x = x H + x H (6) where φ denoes he mulivariae normal densiy (,, ) exp ' x s H ( ) ( ) j s = x j s H j s x j sj φ μ μ μ 2π H 2 s j (7) and π s and π s 2 represen he mixing weighs so ha π + π =. I is well known ha he s s2 MNM densiy has an expeced value and covariance marix equal o ( ) s s s2 s2 E X = π μ + π μ (8) ' ' ( ) = π s ( ) ( ) s + μ s μ s + π s2 s + μ 2 s μ 2 s2 ' ( πs μs + πs μs )( πs μs + πs μs ) Cov X H H (9) 7 The MVHR is derived by aking he firs order condiions of he hedged profi (expressed as a funcion of he spo and he basis) wih respec o he MVHR. See Chen e al (999a) or Dark (2007) for deails.

9 see for example McLachlan and Peel (2000). The hedger who ignores regime shifs would herefore esimae a MVHR (θ % ), which is relaed o he underlying DGP as follows % % σ + % σ m θ = % σ % σ % σ 2 λ c cb 2 2 2λ λ c + bm + 2 cbm (0) where ( ) ( ),, 2, 2, ( πs μcs, 2πsπs μcs, μcs, πs μcs, ) % σ = π σ + μ + π σ + μ c s c s c s s c s c s ( ) ( ),,, 2, 2, 2, ( πs μb, s μc, s πsπs μc, s μb, s πsπs μb, s μc, s πs μc, s μb, s ) % σ = π σ + μ μ + π σ + μ μ cb s cbs bs cs s cbs bs cs ( ) ( ),, 2, 2, ( πs μbs, 2πsπs μbs, μbs, πbμbs, ) % σ = π σ + μ + π σ + μ b s bs bs s bs bs () (2) (3) Given ha spo and fuures markes are generally found o be coinegraed (see for example Koumos and Tucker, 996; Tse 999), he following simplifying assumpion is made ( F S ) ( F S ) ( F F ) ( S S ) Δ B = = = μbs, 0 j. (4) C C C Equaions 2 and 3 herefore simplify o % σ = π σ + π σ (5) cb s cb, s s2 cb, s2 % σ = π σ + π σ (6) b s b, s s2 b, s2 The bias in he MVHR from failing o allow for regime swiches ( % θ θ s j ), is herefore a complex funcion of π, μ and s j s j H for j =, 2. To gain some insighs ino he behaviour of s j he bias, wo alernaive MNM disribuions are considered. The relevan parameer values are based on acual values for he S&P500 deailed in Table 4 below. The firs DGP has he following parameers

10 Sae N,, Sae 2 N, (7) The second DGP has he following parameers Sae N, , Sae 2 N, (8) Boh DGPs assume λ = 0.5, m = 0.5 and π = Applicaion of Equaion 5 yields s θ = ( ) and ( ) s θ = for he firs (second) DGP. Using Equaions s2 0,, 5 and 6, he hedger who ignores regime shifs will esimae a MVHR equal o % θ = ( ). If he acual process is in sae over he period of he hedge, he hedger who ignores he regime shifs will have an upwardly biased MVHR of magniude ( ) % θ θ s = = = If however he acual process is in sae 2 over he period of he hedge, he hedger who ignores regime shifs will have a % θ θ s. downwardly biased MVHR of magniude = ( ) (Inser Figure ) 2 Figure examines he effecs of changing π, m or λ on boh biases whils holding s everyhing else consan. The firs column represens he bias in he MVHR for he firs DGP, he second column examines he bias for he second DGP. The firs row demonsraes he effec of increases in π s on he bias in he MVHR ha ignores regime swiches. To illusrae, consider he graph in he firs row and column. When π = 0, he MVHR from he mixure is equivalen o he MVHR for sae 2 % θ = θ s = If he process is in sae 2 over he life 2 of he hedge, he bias is zero % θ θ s = 0. If however he process is in sae, he bias is a is 2 s maximum of % = = As s θ θ s π increases, θ % decreases. If for example π = 0.5, % θ = If he process is in sae over he life of he hedge, he bias s

11 in he MVHR is % θ θ s = = If he process is in sae 2 over he life of he hedge, he bias in he MVHR is % θ θ s = = Clearly as π, 2 s % θ θ s and he bias if in sae ( % θ θ s ) decreases whils he bias if in sae 2 % θ θ s2 increases. 8 The second row reses π = 0.67 and examines he effecs on he bias in each sae s from changing he number of days o mauriy. The hird row examines he effecs on he bias in each sae as λ increases. Boh resuls show ha he absolue value of he bias decreases slighly as he conrac approaches mauriy/as λ increases. The changes in he bias from changing hese parameer values however are relaively small. In summary, failing o allow for regime swiches can resul in a significan upward or downward bias in he MVHR. If he hedger ignores regime swiches, he esimaed MVHR will represen a combinaion of he MVHRs in each sae. The magniude of he bias is a funcion of he differences in he dynamics beween regimes, he weighs applied o he mixure and he sae of he process over he life of he hedge. Ceeris paribus, if in sample, he process spends a high proporion of he ime in sae ( π > π ), and over he life of he s s2 hedge he process is in sae, he bias is likely o be small. If however, he process is in sae 2 over he life of he hedge, he bias is likely o be much larger. If in conras he process spends comparable amouns of ime in each regime (he mixure weighs are comparable), hen he bias in he MVHR is likely o be somewhere beween hese wo exremes. 8 This analysis assumes ha if a MS model is applied, i will correcly idenify he sae wih he model inferring a 00% probabiliy of being in ha sae. This of course is unrealisic. MS models make probabilisic saemens abou he likelihood of each sae. Therefore even a well specified MS model will generae covariance forecass ha are a probabilisically weighed average of he covariances in each sae. The acual bias from failing o allow for regime swiches is herefore likely o be less han ha presened here.

12 3. Daa The daa se consiss of daily spo and fuures prices on he S&P500 from January 5, 988 o November 9, A oal of 400 daily observaions. The fuures series is consruced by linking he nearby conracs, wih rollover five rading days prior o conrac expiraion. 9 Figure a presens he S&P500 spo price. Figure b presens he normalized basis B C (muliplied by 00), where B F C = and F and C represen he fuures and spo prices a ime. Figure c presens he normalized change in he basis Δ B C (also muliplied by 00). The jagged paern in he normalized basis is caused by he jump in he basis on rollover and he subsequen convergence o zero over he life of he fuures conrac. This paern of basis convergence is consisen wih arbirage and he cos of carry. Imporanly, he normalized basis shows signs of regime swiching. In he periods from approximaely January 988 o January 992 and January 995 o January 200 he basis is slighly higher, more volaile and experiences a larger jump on rollover han he oher periods (January 992 o January 995 and January 200 o December 2003). The regime swiching in he basis dynamics may be caused by regime swiches in he cos of carry. For sock index fuures, level and/or volailiy shifs in he basis could herefore be explained by level and/or volailiy shifs in ineres raes and/or dividend yields. (Inser Figure 2) To invesigae he source of regime swiches, Figure 3 presens he normalized cos of carry and is consiuen pars (he annualized risk free rae and dividend yield) over he period. The normalized cos of carry (NCOC) is calculaed in he following way 9 Creaing a coninuous fuures series in his way is a commonly employed approach. To avoid he jumps creaed by conrac rollover, Alizadeh and Nomikos (2004) creae a perpeual fuures conrac by weighing near and disan conracs, hereby creaing a fuures series wih a consan mauriy. This approach however is unrealisic in pracical erms. Hedgers will seek o hedge heir exposure using he nearby conrac and herefore need an approach ha explicily akes ino accoun conrac rollover and he convergence of he basis.

13 where T C exp ( i q) C 365 NCOC = 00 (9) C C is he spo price a ime, i is he annualized risk free rae proxied by he 3 monh US reasury bill rae (wih consan mauriy) and q is he dividend yield. 0 The dividend yield on he S&P500 is consruced by aking he difference in he Value Weighed Index reurn (including all disribuions) and he Value Weighed Index reurn (excluding dividends) from CSRP and annualizing ha difference by muliplying by 365. Given ha dividends are received in clusers, he dividend yield series is hen smoohed by creaing a new series ha represens he average over he las 5 days. (Inser Figure 3) Figure 3 reveals ha he normalized cos of carry exhibis dynamics ha are similar o he normalized basis. (The correlaion coefficien is When he dividend yield is smoohed using a 20 day average, he correlaion increases o 0.77). Figure 3b reveals ha he swiches in dynamics are primarily driven by he changes in he risk free rae. The high cos of carry periods (January 988 o January 992 and January 995 o January 200) coincide wih a relaively high ineres rae. The low cos of carry periods (January 992 o January 995 and January 200 o December 2003) coincide wih a relaively low ineres rae. Dividend yields in comparison appear o exhibi a genle downward sloping rend and herefore seem o play a very minor role in he changing basis dynamics. 0 This calculaion is based on he cos of carry model. If he fuures price a ime wih conrac mauriy T (denoed by F T, ) is an unbiased predicor of he spo, he fuures can be priced according o FT, = E CT Ω where C ( T ) is he spo price a ime T and ime. If he spo price evolves according o dc ( ) C ( )( i q) d σ dw ( ) Ω denoes he informaion se available a = + C C where σ C is he index volailiy and W () represens Brownian moion, hen he cos of carry model can be derived C ( )( ) FT, = Cexp i q T.

14 Given ha he basis dynamics appear o be subjec o regime shifs beween high cos of carry and low cos of carry saes, he model parameers and hence he MVHR should ideally be sae dependen. The nex secion proposes a bivariae MS FIGARCH model wih mauriy effecs o capure hese dynamics. 4. Model This secion develops he bivariae MS FIGARCH model wih mauriy effecs. Previous aemps o allow for regime swiches and long memory have aemped o remove he effecs of srucural breaks by eiher breaking he daa ino sub-samples (Lobao and Savin, 998), examining shor spans of high frequency daa (Andersen and Bollerslev, 997a), removing he effecs of breaks and hen esing for long memory on he break free series (Granger and Hyung, 2004; Belrai and Morana, 2006) or by esimaing models ha joinly capure srucural breaks and long memory. Examples of he las approach include Morana and Belrai (2004) who fi a MS-ARFIMA model o realized volailiies, Baillie and Morana (2007) who develop he Adapive FIGARCH process, Marens e al (2004) who augmen he ARFIMA process o include gradual level shifs in volailiy and swiching behaviour in he shor run dynamics, and Haldrup and Nielsen (2006) who develop a RS-seasonal ARFIMA model for hourly elecriciy prices. The approach adoped in his paper also joinly models regime swiching and long memory, esimaing a bivariae MS-FIGARCH process wih mauriy effecs. The model adops a flexible specificaion, allowing all of he condiional mean and variance parameers o be sae dependen. Two models are esimaed. The firs imposes consan ransiion probabiliies (CTP) while he second allows for ime varying ransiion probabiliies (TVTP). The evidence in Secion 3 suggess ha high (low) cos of carry periods coincided wih The srucural change is capured by allowing he inercep in he FIGARCH equaion o follow a smooh ime varying process according o Gallan s (994) flexible funcional form. The model however does no allow he long memory or shor memory dynamics o differ according o he regime.

15 relaively high (low) ineres rae levels. The second model herefore models he ime varying ransiion probabiliies as a funcion of he lagged risk free rae. Gray (996) developed he recombining mehod o overcome he pah-dependency problem associaed wih he esimaion of univariae MS GARCH processes. This approach was exended o a bivariae MS GARCH process by Lee and Yoder (2007). Given he empirical evidence above which suggess he S&P500 exhibis long memory and srucural breaks, exending he bivariae MS GARCH process o a bivariae MS FIGARCH process seems pruden. 2 Furhermore, Dark (2007) argues ha he sandard approaches o MVHR esimaion ha impose shor memory on he volailiy dynamics are mis-specified in he presence of long memory in volailiy. Dark (2007) shows ha failing o allow for long memory over medium o long erm horizons will on average produce saisically significan increases in porfolio variance ha are beween % o 3%. To specify he model, he noaion in secion 2 is employed - spo and fuures prices a ime, he basis is B F C =, {, 2} C and F represen he s = is he unobserved sae variable, Ω is he informaion se and m denoes he number of days o mauriy divided by 00. To capure he jump in he basis on rollover, le I denoe an indicaor variable equal o one on conrac rollover and zero oherwise. The bivariae MS FIGARCH model wih mauriy effecs is specified as follows ΔC C ΔC = a + b + ε (20) cs, cs, cs,, C 2 ΔB B ΔB C C C ε λ ( ) 2, s = a bs, + λ os, I + λ, sm + bbs, + c bs, + ε bs,, m (2) 2 ( H ) Ω ~ N 0, (22) s, s, 2 The pah-dependency problem for he FIGARCH process is simplified because he FIGARCH parameers can be esimaed via he infinie order ARCH represenaion (see equaion 23) wih a runcaion lag of 000 observaions.

16 ( ) dis, ( )( ) ( ) 2 2 σis,, = ω,,,,,,, is β is + φ isl L β isl ε i i = c b (23) σ = ρ σ σ (24) sb,, s sbs, ss,, bs,, where ( ) ε = p ε + p ε i= c, b and (25) i,, is,,, is,, 2 p ( s ) = Pr = Ω, ( ) ( Q) ( p ) ( ) f p f,, 2,, = P + f, p, + f2, p, f, p, + f2, p, (26) where P and Q represen he sae ransiion probabiliies and ( g0 ) ( g ) exp P= Pr ( s = s = ) = (27) + exp 0 ( h0 ) ( h ) exp Q= Pr( s = 2 s = 2) = (28) + exp 0 (, ) f = f r s = j Ω j, ( ) 0.5 ( ) ' π s, ε j s, j s, ε j s, j = 2 H exp 0.5 H j =,2 (29) for r ' ΔC ΔB =,. The proposed model exends he rudimenary model in secion 2 in a C C number of imporan ways. Firs, he condiional mean equaions allow for a much richer se of dynamics. Wih respec o he basis, λ0, s capures he observed increase in he basis on rollover, and λ, s m capures he impac of mauriy effecs on he change in he normalized basis. The erm B C acs like an error correcion erm, hereby ensuring ha he spo and fuures prices implied by he model are coinegraed. Any furher serial correlaion in he

17 condiional mean dynamics is capured by he auoregressive erm. Second, he variance and covariance dynamics are condiionally ime varying and exhibi long memory. 3 To begin he recursion for he regime probabiliy Q Pr( s = Ω 0 ) =. (30) 2 P Q 2 The FIGARCH esimaion ses he pre-sample values of ε i, o an uncondiional variance esimae obained via ( ε ) ( ) ε ( ) Pr s = Ω 0 i, s + Pr s = Ω 0 i, s N i = c, b 2 2 (3) ' where ε = ε, ε,..., ε and N is he sample size. QMLE is employed, he is, j i,, sj i,2, sj ins,, j likelihood funcion is T log,, (, ) 2,. (32) = LL = p f + p f A runcaion lag of 000 observaions is employed. Numerical derivaives are implemened and he non-negaiviy of he FIGARCH esimaes is imposed via he condiions in Bollerslev and Mikkelsen (996). The second model allows for ime varying ransiion probabiliies. This is because previous sudies have indicaed ha consan ransiion probabiliies are generally oo resricive (Filardo, 994; Gray 996; Perez-Quinos and Timmerman, 2000, Marsh 2000). The above model is herefore exended along he lines of Gray (996) and Perez-Quiros and 3 Assume var ( F F ) ~ I( d f ) and var ( S S ) ~ I( ds) where 0 < ds, d f <. Given ( Δ B) = ( F F ) + ( S S ) ( F F S S ) hen he var ( Δ B ) is also I d = I ( d s ) ρ is he correlaion beween he spo and he fuures), var ( Δ B ) ~ I ( 0). While he var var var 2 cov, likely o be I(d). This may no be he case and is ulimaely an empirical maer. If for example ( f ) ρ = (where sf and sf empirical evidence suggess ha similar markes may have common orders of fracional inegraion, he correlaion beween spo and fuures reurns is less han uniy.

18 Timmerman (2000). Given ha swiches in he basis dynamics coincide wih he level of he risk free rae, he ime varying ransiion probabiliies evolve according o ( ) ( 0 ) P = Pr s = s = =Φ g + gi (33) ( ) ( 0 ) Q = Pr s = 2 s = 2 =Φ h + hi (34) where Φ is he cumulaive densiy for he sandard normal and i is he risk free ineres rae. To begin he recursion for he regime probabiliy Q Pr( s = Ψ ) = (35) 2 P Q where T P = P T and = T Q= Q T. One would expec g > 0 given ha he probabiliy of = saying in a high COC sae (sae ) will increase wih rises in he risk free rae. Similarly, he probabiliy of saying in a low COC sae (sae 2) will decrease wih an increase in ineres raes h < Resuls In he firs sage, he proposed model was esimaed wihou regime swiching. Table presens he parameer esimaes. The resuls clearly suppor he presence of long memory in he spo and normalized basis volailiies. The mauriy effec parameers are also consisen wih he convergence in he basis and is condiional variance ( λ, λ 2 > 0). The diagnosics represen Box-Pierce saisics for he sandardized and squared sandardized residuals and heir respecive p values. The sandardized residuals exhibi some remaining serial correlaion paricularly in he basis equaion. (Inser Table ) The second sage allows for regime swiching wih consan ransiion probabiliies (CTP). Table 2 presens he parameer esimaes. The overall model resuls suppor he regime swiching model. There is a subsanial increase in he log likelihood and he plo of

19 ( ) p, = Pr s = Ψ in Figure 4a seems quie reasonable. (Figure 4a also displays he MVHR, his will be discussed below). The probabiliy plo implies ha high COC regimes occurred from January 5, 988 o January, and from November 23, 994 o Sepember 9, 200 a oal of 2734 days. Low COC regimes occurred from January 9, 992 o November 22, 994 and from Sepember 20, 200 o June 6, 2003 a oal of 66 days. 4 Figure 4a reveals ha he probabiliy of being in he high COC regime very closely racks he behaviour of he normalized basis (despie he model being fied o he normalized change in he basis see Figure 2c). The swich from he high COC sae o he low COC sae in Sepember 200 however seems likely o occur prior o ha inferred by he CTP model. (Inser Table 2 and Figure 4) The esimaed parameers for he CTP model in Table 2 appear reasonable. The differences beween he esimaed parameers for he high COC and he low COC regimes sugges ha he basis dynamics are primarily responsible for driving he regime swiches. The mauriy effec parameers λ o, λ, and λ 2 suppor an increase in he basis on rollover wih convergence in he condiional mean and variance. The higher λ o and λ esimaes for he high COC regime are consisen wih he larger increase in he basis on rollover and he subsequen convergence as he conrac approaches mauriy. The λ 2 esimaes sugges a sronger level of volailiy convergence in he low COC regime. The condiional variance esimaes suppor he presence of long memory in boh regimes. For he S&P500 volailiy here is no change in he long memory dynamics (measured via d c ) and only a sligh change in he shor memory dynamics (capured via φ, β ). In conras, he basis volailiy has higher levels of volailiy persisence in he high c c 4 The dae of he swich is assumed o occur when he probabiliy of being in he oher sae exceeds 0.5 and subsequenly remains above 0.5 unil he nex swich of course. The esimaed probabiliies of being in sae (Figure 4a) reveal ha his approach is reasonable given he highly persisen saes.

20 COC regime han in he low COC regime. Furher, he correlaion esimae suggess ha he spo basis correlaion in he high COC regime is larger han he correlaion in he low COC regime. Table 3 presens he esimaed parameers for he TVTP model. The likelihood raio indicaes ha he improvemen in he log likelihood is saisically significan, supporing he ime variaion in he ransiion probabiliies. As noed above, he sign of g and h ( g > 0, h < 0 ) is consisen wih expecaions. Allowing for ime varying ransiion probabiliies has very lile effec on he remaining parameers, bu around he idenified break periods i has a significan effec on he probabiliy of being in sae. The TVTP model idenifies he following regimes: High COC January 5, 988 o January 8, 992, Low COC January 9, 992 o Ocober 3, 994, High COC 2 November, 994 o Augus 22, 200 and Low COC 2 Augus 23, 200 o June 6, Excep for he firs swich from he high COC sae o he low COC sae, hese resuls show ha he TVTP model idenifies he swiches earlier han he CTP model. This resul and is hedging implicaions are invesigaed below. (Inser Table 3) Hedging Oucomes To esimae dynamic MVHRs over he life of he hedge, he MVHR in Equaion 5 is calculaed dynamically via he following θ σ + σ 2 λ c, + cb, Ω = 2 2 2λ λ σc, + + σb, + m+ + 2σcb, + m+ m (36) where one sep ahead forecass of he covariance marix condiional on he informaion se a H ˆ + ime ( Ω ) are obained via ( ˆ ) Hˆ Ω = pˆ Hˆ Ω + p Hˆ Ω (37) +, + +, s, + +, s2 To avoid any significan bias in he forecass from employing he runcaion lag of 000 observaions, he firs 500 observaions are no considered for in sample esimaion of

21 he MVHR. The firs in sample forecas of he MVHR is herefore calculaed for a hedge on 27 December 989 (i.e condiional upon he informaion se as a 26 December 989). Ou of sample forecass are calculaed for hedges over he period from 7 June 2003 o 20 November 2003 a oal of observaions. Referring back o Figure 4a (which graphs he dynamic CTP MVHR, he probabiliy of being in sae from he CTP model and he normalized basis), he resuls illusrae ha he differen dynamics in each regime direcly affec he in sample CTP MVHR. Figure 4a suggess ha he CTP MVHR is lower and more volaile in he high COC regime. This is consisen wih he higher levels of basis volailiy and he lower level of spo fuures correlaion illusraed in Table 4. (Inser Table 4) Table 5 Panel A examines descripive saisics for he dynamic MVHRs, i.e he MVHR wihou regime swiching (herein referred o as he MVHR), he CTP MVHR and he TVTP MVHR. The comparison is performed for all in sample observaions, across he idenified regimes and for he ou of sample period. When considering he enire sample, he mean and variance of he RS MVHRs and he MVHR are quie similar. An examinaion of he sub periods however reveals imporan differences. Panel A confirms ha he RS MVHRs are lower and more volaile in he high COC regime. The CTP MVHR (TVTP MVHR) averaged (0.8996) and (0.9053) in he high COC sae, compared o (0.948) and (0.968) in he low COC sae. 5 Furher he variance of he RS MVHRs is approximaely five o en imes higher in he high COC regimes han he low COC regimes. The average MVHR (which does no allow for RS) in conras is very similar across 5 Alizadeh and Nomikos (2004) find ha high variance saes are associaed wih a low hedge raio and viceversa. They also find ha here is a posiive relaionship beween volailiy and he magniude of he basis. This suggess ha high basis amouns are associaed wih low hedge raios. This is consisen wih he findings in his paper.

22 regimes (excep for he las low COC regime). The variance of he MVHR is also higher in he high COC regimes, hough by a smaller facor of beween wo o four. (Inser Table 5) Figure 4b examines he impac of allowing for ime varying ransiion probabiliies on he probabiliy of being in he high COC sae and he MVHR. The probabiliy difference represens he TVTP model s probabiliy of being in he high COC sae a ime minus he corresponding probabiliy from he CTP model. The MVHR difference plos he TVTP MVHR minus he CTP MVHR. Figure 4b reveals ha over mos ime periods he probabiliy of being in he high COC sae from he TVTP and CTP models are very similar. The mos significan differences occur around he firs swich from he low COC sae o he high COC sae around Ocober 994. The second significan difference occurs around he swich from he high COC sae o he low COC sae around Augus 200. In boh cases he TVTP model (and hence he TVTP MVHR) swiches earlier han he CTP model. Panel B of Table 5 compares he porfolio variance achieved using he Unhedged, naïve, OLS, MVHR, CTP MVHR and TVTP MVHR. 6 The panel compares he porfolio variances for each hedging sraegy for he enire in sample period. This is hen broken up ino he performance over each of he idenified regimes (per he TVTP model). 7 Finally he panel repors he ou of sample porfolio variances. The lowes porfolio variance achieved over each period is in bold. The resuls illusrae ha for he enire sample, here is a decrease in porfolio variance when allowing for regime swiching and ha he TVTP MVHR produced he bes variance reducion. The porfolio variance achieved using he MVHR decreased from o when he CTP model was employed. The porfolio variance decreased θ =. The OLS MVHR is he slope coefficien from he following OLS 6 The naïve hedge raio ses regression Δ C C = α + β( Δ F F ) + ε. 7 The porfolio variances were also analysed according o he regimes idenified via he CTP model. The resuls and conclusions are insensiive o he sligh differences in he sub-periods.

23 furher o when he TVTP model was used. When examining sub-periods, he TVTP model performed he bes in hree ou of four sub-periods. Allowing for regime swiching in he high COC saes produced lile or no improvemen in porfolio variance. The mos significan hedging gains from he RS models were achieved in he low COC saes, paricularly he second low COC regime. This is consisen wih he resuls in secion 2. There are approximaely 2.5 imes more observaions in he high COC sae han in he low COC sae. The parameers from he model ha ignore regime swiching are herefore likely o more closely characerize he dynamics from he high COC sae. The mos significan gains from employing RS models are herefore likely o arise in he low COC sae. To furher examine he imporance of ime varying ransiion probabiliies, he performance of he alernaive sraegies was considered around he periods when he CTP and TVTP models produced significan differences in he probabiliy of being in he high COC sae. As discussed above, he firs significan difference occurred around Ocober 994 when here was a swich from a low o a high COC regime. Differences in he probabiliies emerged around 29 Augus 994, widened considerably around 3 Ocober 994 and narrowed around 22 November 994. The performance of he alernaive sraegies over wo sub-periods around his ime are considered in Table 6. The Table also considers he performance of he alernaive sraegies around he oher swich idenified in Figure 4b (around Augus 200). Significan differences in he probabiliy of being in he high COC sae emerged around 26 July 200 and coninued unil 8 Sepember 200. In all hree subperiods, he TVTP model clearly produced he greaes variance reducion. The TVTP model produced porfolio variances ha were lower han he CTP model by 5.4% (Augus 29, 994 o November, 994), 3.5% (Ocober 3, 994 o November 22, 994) and 3.% (July 26, 200 o Sepember 8, 200). These percenage improvemens are considerably higher han hose for each of he six periods examined in Table 5 (which range from -0.6% o 0.%).

24 The resuls herefore sugges ha he earlier swiches in he TVTP model provide significan improvemens in variance reducion paricularly around he ime of a swich. (Inser Table 6) The ou of sample hedging performance shows ha whils he MVHR is ouperformed by he CTP and TVTP models, none of he approaches beas he naïve hedge raio. This resul is broadly consisen wih Dacco and Sachell (999) as discussed above, who show ha even if he underlying process exhibis regime swiching, a sligh misclassificaion may mean ha he forecass from he appropriae regime swiching model have a higher MSE han forecass from a random walk. 8 The above discussion fails o recognize ha he differences in he porfolio variances may no be saisically significan. To assess he saisical significance of he differences in he porfolio variances beween compeing approaches, he moving block boosrapping echnique of Kunsch (989) is employed. 9 To describe he approach le X = ( X X X ),,... n 2 = and represen he observed saionary sequence of porfolio reurns where X j ( X j, b, X j, a) X jb, is he porfolio reurn a ime = j for he benchmark and X ja, is he ime mached porfolio reurn a ime n = j for he alernaive MVHR. Le b, l denoe inegers such ha = bl, where l denoes he block lengh and b denoes he number of blocks. Following Kunsch, n l+ overlapping blocks are produced wih he h k block being (,... ) Β = X X + where k n l N k k k l + = and Xk ( Xk, b, Xk, a) =. The mehod chooses blocks * * Β,..., Β b by resampling randomly wih replacemen from among Β,..., Β N, where 8 For his reason a Muli-period MVHR requiring muli-sep ahead forecass may produce worse hedging oucomes, paricularly if a break occurs during he life of he hedge. Furher, Dark (2007, 2008) employs alernaive muli-period MVHRs and finds ha he single period approach performs as well if no beer han muli-period approaches. I herefore seems pruden o employ a single period MVHR. 9 The serial correlaion in he porfolio reurns requires he use of a block boosrap mehod. The moving block mehod is chosen given ha i provides lower variances han he circular block boosrap of Poliis and Romano (992) and he saionary boosrap of Poliis and Romano (994). See Lahiri (999) for deails.

25 draws are from a discree uniform disribuion on {,..., n l } * * * +. If Β (,..., i = X i X il) boosrapped version of X is X * ( X * * ) ( * * * * * *,..., Xn X,.., X l, X2,.., X2l,..., Xb,.., Xbl), he = = where (,,, ) * Xmp = Xmp b X mp a for m b and p l and X, is he mp b h p porfolio reurn for he h m block for he benchmark and mp, a X is he ime mached h p porfolio reurn for he h m block for he alernaive. The es saisic represening he difference in he porfolio variances is ( mp, b ) ( mp, a ) 2 2. (38) f = X + X n Following Hall e al (995), he opimal block size is l n 0.2 =. Each boosrapped densiy is based on replicaions and 95%, 90% and 80% confidence inervals are consruced via he appropriae quaniles. A confidence inerval esimae ha includes zero suggess ha he porfolio variances are equal. A confidence inerval esimae ha is negaive for boh he upper and lower limis indicaes ha he alernaive MVHR provides a lower porfolio variance han he benchmark. For each of he periods idenified in Tables 5 and 6 he following confidence inervals were esimaed: a) Benchmark: Naïve, agains various Alernaives: OLS, MVHR, CTP- MVHR, TVTP-MVHR, b) Benchmark: OLS, agains various Alernaives: MVHR, CTP- MVHR, TVTP-MVHR c) Benchmark: MVHR, agains various Alernaives: CTP-MVHR, TVTP-MVHR and d) Benchmark: CTP-MVHR, agains Alernaive: TVTP-MVHR. Given ha he TVTP-MVHR provides he lowes poin esimaes for he porfolio variances for 7 ou of he 9 periods, he resuls for each of he benchmarks (Naïve, OLS, MVHR, CTP- MVHR) relaive o he TVTP-MVHR are repored in Table 7. The resuls for all confidence inervals are available on reques. The confidence inervals ha are bold include zero for all levels of confidence and herefore sugges ha he difference beween he TVTP-MVHR and

26 he benchmark is no saisically significan. Enries ha include zero a 95% bu no a he lower levels of confidence (90% and or 80%) are no bolded (wih he deails below he able). (Inser Table 7) As idenified above, he poin esimaes indicae ha he TVTP MVHR has he lowes porfolio variance for 7 ou of he 9 periods. The analysis commences by examining he remaining 2 periods where he TVTP MVHR did no provide he lowes poin esimae: he high COC sae (where he MVHR provided he lowes porfolio variance) and he ou of sample period (where he naïve HR provided he lowes porfolio variance). The boosrapped confidence inerval esimaes sugges ha he difference beween hese porfolio variances and he TVTP MVHR however is no saisically significan. The resuls show ha for he remaining 7 periods (where he TVTP MVHR provides he lowes poin esimae) he TVTP MVHR is mached by a leas one of he oher mehods. Imporanly hough, here is no one model ha consisenly maches he TVTP MVHR across all periods. The nex bes performers, OLS and MVHR perform equally well in only 5 ou of he 9 periods considered. Their performance relaive o he TVTP-MVHR in he remaining four periods however is relaively poor. OLS performs as well as he TVTP-MVHR around he break periods bu no as well as TVTP-MVHR over he longer in sample periods (oal, high COC and low COC2). The MVHR in conras performs poorly around he break periods bu generally as well over he longer in sample periods (excep for Low COC2). Boh OLS and MVHR are clearly dominaed by he TVTP-MVHR ou of sample. The resuls herefore consisenly idenify he TVTP MVHR as he bes performing approach. Furher he resuls suppor he use of he model ha allows for TVTPs relaive o one ha imposes CTPs. Sligh in sample improvemens were made over he oal and high COC2 periods. Imporanly, saisically significan improvemens over he CTP MVHR were made

27 around he Ocober 994 break bu no he Augus 200 break. Therefore, here is some evidence ha by more effecively idenifying he locaion of a break, models wih ime varying ransiion probabiliies may provide superior hedging performance around he ime of he break. Finally, when he MVHR is he benchmark, he confidence inerval esimaes (no repored) reveal ha he CTP MVHR ouperforms he MVHR in he low COC2 sae, ou of sample and around he Augus 200 break. The CTP MVHR and he MVHR produced porfolio variances ha were no saisically differen in he remaining periods. This resul suggess ha allowing for regime swiches via he CTP model provides a sligh improvemen over a model ha does no allow for regime swiches. 5. Conclusion This paper developed a bivariae Markov Swiching FIGARCH (MS-FIGARCH) process wih consan and ime varying ransiion probabiliies as a way of modeling spo fuures dynamics. An applicaion of he model o he S&P500 demonsraed he presence of long memory in volailiy and srucural breaks ha are driven by changes in he cos of carry (proxied by he risk free rae). Two saes were idenified, a low cos of carry sae and a high cos of carry sae. Relaive o he low cos of carry regimes, he high cos of carry regimes were associaed wih higher levels of basis volailiy, lower spo fuures correlaions and lower minimum variance hedge raios. The model wih consan ransiion probabiliies provided a sligh improvemen in hedging performance relaive o an approach ha did no allow for regime swiching. On allowing he ransiion probabiliies o be a funcion of he risk free rae, he regime shifs beween high and low cos of carry saes were more effecively idenified. I was shown ha his may resul in significan improvemens in hedging oucomes around he ime of a break.

28 Whils he resuls sugges ha overall he TVTP MVHR was he bes performing approach, i did no consisenly bea all of he alernaive approaches across he sample periods. Imporanly, is ou of sample performance was no saisically differen from he naïve MVHR. The TVTP MVHR provides a number of promising resuls, however he search for he ulimae model coninues..