Asymmetry and Leverage in Conditional Volatility Models*

Transcription

1 Asymmery and Leverage in Condiional Volailiy Models* Micael McAleer Deparmen of Quaniaive Finance Naional Tsing Hua Universiy Taiwan and Economeric Insiue Erasmus Scool of Economics Erasmus Universiy Roerdam and Tinbergen Insiue Te Neerlands and Deparmen of Quaniaive Economics Compluense Universiy of Madrid Spain EI Sepember 014 * For financial suppor, e auor wises o acknowledge e Ausralian Researc Council and e Naional Science Council, Taiwan. 1

2 Absrac Te ree mos popular univariae condiional volailiy models are e generalized auoregressive condiional eeroskedasiciy (GARCH) model of Engle (198) and Bollerslev (1986), e GJR (or resold GARCH) model of Glosen, Jagannaan and Runkle (199), and e exponenial GARCH (or EGARCH) model of Nelson (1990, 1991). Te underlying socasic specificaion o obain GARCH was demonsraed by Tsay (1987), and a of EGARCH was sown recenly in McAleer and Hafner (014). Tese models are imporan in esimaing and forecasing volailiy, as well as capuring asymmery, wic is e differen effecs on condiional volailiy of posiive and negaive effecs of equal magniude, and leverage, wic is e negaive correlaion beween reurns socks and subsequen socks o volailiy. As ere seems o be some confusion in e lieraure beween asymmery and leverage, as well as wic asymmeric models are purpored o be able o capure leverage, e purpose of e paper is wo-fold, namely: (1) o derive e GJR model from a random coefficien auoregressive process, wi appropriae regulariy condiions; and () o sow a leverage is no possible in ese univariae condiional volailiy models. Keywords: Condiional volailiy models, random coefficien auoregressive processes, random coefficien complex nonlinear moving average process, asymmery, leverage. JEL classificaions: C, C5, C58, G3.

3 1. Inroducion Te ree mos popular univariae condiional volailiy models are e generalized auoregressive condiional eeroskedasiciy (GARCH) model of Engle (198) and Bollerslev (1986), e GJR (or resold GARCH) model of Glosen, Jagannaan and Runkle (199), and e exponenial GARCH (or EGARCH) model of Nelson (1990, 1991). Te underlying socasic specificaion o obain GARCH was demonsraed by Tsay (1987), and a of EGARCH was sown recenly in McAleer and Hafner (014). Tese models are imporan in esimaing and forecasing volailiy, in capuring asymmery, wic is e differen effecs on condiional volailiy of posiive and negaive effecs of equal magniude, and (possibly) in capuring leverage, wic is e negaive correlaion beween reurns socks and subsequen socks o volailiy. Te purpose of e paper is wo-fold, namely: (1) o derive e GJR model from a random coefficien auoregressive process, wi appropriae regulariy condiions; and () o sow a leverage is no possible in ese univariae condiional volailiy models. Te derivaion of ree well known condiional volailiy models, namely GARCH, GJR and EGARCH, from eir respecive underlying socasic processes raises wo imporan issues: (1) e regulariy condiions for eac condiional volailiy model can be derived in a sraigforward manner; and () e GJR and EGARCH models can be sown o capure asymmery, bu ey can also be sown o be unable o capure leverage. Te paper organized is as follows. In Secion, e GARCH, GJR and EGARCH models are derived from differen socasic processes, e firs wo from random coefficien auoregressive processes and e ird from a random coefficien complex nonlinear moving average process. I is sown a asymmery is possible for GJR and EGARCH, bu a leverage is no possible. Some concluding commens are given in Secion 3.. Socasic Processes for Condiional Volailiy Models 3

4 .1 Random Coefficien Auoregressive Process and GARCH Consider e condiional mean of financial reurns as in e following: y = E ( y I ) 1 + ε (1) were e reurns, y = log P, represen e log-difference in sock prices ( P ), I 1 is e informaion se a ime -1, and ε is condiionally eeroskedasic. In order o derive condiional volailiy specificaions, i is necessary o specify e socasic processes underlying e reurns socks, ε. Consider e following random coefficien auoregressive process of order one: ε = φ ε 1 + η () were φ ~ iid ( 0, α ), η ~ iid ( 0, ω ). Tsay (1987) sowed a e ARCH(1) model of Engle (198) could be derived from equaion () as: = E( I 1) = ω + αε 1 ε. (3) were is condiional volailiy, and I 1 is e informaion se a ime -1. Te use of an infinie lag leng for e random coefficien auoregressive process in equaion (), wi appropriae resricions on e random coefficiens, can be sown o lead o e GARCH model of Bollerslev (1986). 4

5 As e ARCH and GARCH models are symmeric, in a posiive and negaive socks of equal magniude ave idenical effecs on condiional volailiy, ere is no asymmery, and ence also no leverage, wereby negaive socks increase condiional volailiy and posiive socks decrease condiional volailiy (see Black (1976)). I is wor noing a a leas one of ω or α mus be posiive for condiional volailiy o be posiive. From e specificaion of equaion (), i is clear a bo ω and α sould be posiive as ey are e variances of wo differen socasic processes.. Random Coefficien Auoregressive Process and GJR Te GJR model of Glosen, Jagannaan and Runkle (199) can be derived as a simple exension of e random coefficien auoregressive process in equaion (), wi an indicaor variable I ε ) ( 1 a disinguises beween e differen effecs of posiive and negaive reurns socks on condiional volailiy, namely: ε ( + (4) = φ ε 1 + ψ I ε 1 ) ε 1 η were φ ~ iid ( 0, α ), ψ ~ iid ( 0, γ ), η ~ iid ( 0, ω ), I( ε ) 1 = 1 wen ε 1 < 0, I( ε ) 1 = 0 wen ε

6 Te condiional expecaion of e squared reurns socks in (3), wic is ypically referred o as e GJR (or resold GARCH), can be sown o be an exension of equaion (3), as follows: = E( I 1) = ω + α ε 1 + γ I( ε 1 ) ε 1 ε. (5) Te use of an infinie lag leng for e random coefficien auoregressive process in equaion (4), wi appropriae resricions on e random coefficiens, can be sown o lead o e sandard GJR model wi lagged condiional volailiy. I is wor noing a a leas one of ( ω, α, γ ) mus be posiive for condiional volailiy o be posiive. From e specificaion of equaion (4), i is clear a all ree parameers sould be posiive as ey are e variances of ree differen socasic processes. Te GJR model is asymmeric, in a posiive and negaive socks of equal magniude ave differen effecs on condiional volailiy. Terefore, asymmery exiss for GJR if: Asymmery for GJR: γ > 0. A special case of asymmery is leverage, wic is e negaive correlaion beween reurns socks and subsequen socks o volailiy. Te condiions for leverage in e GJR model in equaion (5) are: Leverage for GJR: α < 0 and α + γ > 0. I is clear a leverage is no possible for GJR as bo α and γ, wic are e variances of wo socasic processes, mus be posiive..3 Random Coefficien Complex Nonlinear Moving Average Process and EGARCH 6

7 Anoer condiional volailiy model a can accommodae asymmery is e EGARCH model of Nelson (1990, 1991). McAleer and Hafner (014) sowed a EGARCH could be derived from a random coefficien complex nonlinear moving average (RCCNMA) process, as follows: ε + η (6) = φ η 1 + ψ η 1 were φ ~ iid ( 0, α ), ψ ~ iid ( 0, γ ), η ~ iid ( 0, ω ), η 1 is a complex-valued funcion of 1 η. Te condiional variance of e squared reurns socks in equaion (6) is given as: = E( I 1) = ω + α η 1 + γ η 1 ε. (7) I is wor noing a e ransformaion of approximaion given by: in equaion (7) is no logarimic, bu e log = log(1 + ( 1)) 1 can be used o replace in equaion (7) wi 1 + log. Te use of an infinie lag for e RCCNMA process in equaion (6) would yield e sandard EGARCH model wi lagged condiional volailiy. EGARCH differs from GARCH and GJR in a, given e logarimic ransformaion, no sign resricions on ( ω, α, γ ) are necessary for condiional volailiy o be posiive. However, i is clear 7

8 from e RCCNMA process in equaion (6) a all ree parameers sould be posiive as ey are e variances of ree differen socasic processes. Terefore, asymmery exiss for EGARCH if: Asymmery for EGARCH: γ > 0. Te condiions for leverage in e EGARCH model in equaion (7) are: Leverage for EGARCH: γ < 0 and γ < α < γ. As acknowledged in McAleer and Hafner (014), leverage is no possible as bo α and γ, wic are e variances of wo socasic processes, mus be posiive. 3. Concluding Remarks Te paper was concerned wi e ree mos widely-used univariae condiional volailiy models, namely e GARCH, GJR (or resold GARCH) and EGARCH models. Tese models are imporan in esimaing and forecasing volailiy, as well as in capuring asymmery, wic is e differen effecs on condiional volailiy of posiive and negaive effecs of equal magniude, and in capuring leverage, wic is e negaive correlaion beween reurns socks and subsequen socks o volailiy. As ere seems o be some confusion in e lieraure beween asymmery and leverage, as well as wic asymmeric models are purpored o be able o capure leverage, e purpose of e paper was wo-fold, namely: (1) o derive e GJR model from a random coefficien auoregressive process, wi appropriae regulariy condiions; and () o sow e GJR and EGARCH models are able o capure asymmery, bu are unable o capure leverage. 8

9 References Black, F. (1976), Sudies of sock marke volailiy canges, 1976 Proceedings of e American Saisical Associaion, Business and Economic Saisics Secion, pp Bollerslev, T. (1986), Generalised auoregressive condiional eeroscedasiciy, Journal of Economerics, 31, Engle, R.F. (198), Auoregressive condiional eeroscedasiciy wi esimaes of e variance of Unied Kingdom inflaion, Economerica, 50, Glosen, L., R. Jagannaan and D. Runkle (199), On e relaion beween e expeced value and volailiy of nominal excess reurn on socks, Journal of Finance, 46, McAleer, M. and C. Hafner (014), A one line derivaion of EGARCH, Economerics, (), Nelson, D.B. (1990), ARCH models as diffusion approximaions, Journal of Economerics, 45, Nelson, D.B. (1991), Condiional eeroskedasiciy in asse reurns: A new approac, Economerica, 59, Tsay, R.S. (1987), Condiional eeroscedasic ime series models, Journal of e American Saisical Associaion, 8,