Asymmetry and Leverage in Stochastic Volatility Models: An Exposition

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1 Asymmery and Leverage in Sochasic Volailiy Models: An xposiion Asai, M. a and M. McAleer b a Faculy of conomics, Soka Universiy, Japan b School of conomics and Commerce, Universiy of Wesern Ausralia Keywords: Asymmery; leverage; sochasic volailiy; condiional volailiy; unsandardized residuals; sandardized residuals. XTDD ABSTRACT The accurae specificaion and modelling of risk are inegral o opimal porfolio and risk managemen. In his conex, a wide variey of condiional and sochasic volailiy models has been used o esimae laen volailiy (or risk. In boh he condiional and sochasic volailiy lieraure, here has been some confusion beween he definiions of asymmery and leverage. This paper examines alernaive univariae SV models ha have recenly been developed and esimaed in o order o undersand he differences and similariies beween he definiions of asymmery and leverage. Five univariae SV models, namely he basic SV model, SV model wih leverage, and hree differen ypes of asymmeric SV models, are analysed in order o clarify he disincion beween asymmery and leverage. Alernaive specificaions of SV models are defined according o he use of sandardized or unsandardized reurns, wih or wihou leverage, in order o evaluae he differenial impacs of posiive and negaive reurns on fuure volailiy, namely symmery, asymmery, ype I asymmery (or leverage, ype II asymmery, and ype III asymmery. 83

2 . ITRODUCTIO The accurae specificaion and modelling of risk are inegral o opimal porfolio and risk managemen, and for calculaing Value-a-Risk (VaR forecass and opimal capial charges under he Basel Accord. In his conex, a wide variey of condiional and sochasic volailiy models has been used o esimae laen volailiy (or risk. McAleer (005 provides a comprehensive discussion of boh univariae and mulivariae Sochasic Volailiy (SV models in he lieraure. In boh he condiional and sochasic volailiy lieraure, here has been some confusion beween he definiions of asymmery and leverage. This paper examines alernaive univariae SV models ha have recenly been developed and esimaed in o order o undersand he differences and similariies beween he definiions of asymmery and leverage. The plan of he paper is as follows. Secion presens five univariae SV models, namely he basic SV model, SV model wih leverage, and hree differen ypes of asymmeric SV models, in order o clarify he disincion beween asymmery and leverage. Alernaive specificaions of SV models are defined in Secion 3 according o he use of sandardized or unsandardized reurns, wih or wihou leverage, in order o evaluae he differenial impacs of posiive and negaive reurns on fuure volailiy, namely symmery, asymmery, ype I asymmery (or leverage, ype II asymmery, and ype III asymmery. Some concluding remarks are given in Secion 4. where μ ( y = I denoes expeced reurns on he financial asse, and I is he pas informaion available a ime. In order o undersand he differences and similariies among alernaive univariae SV models ha have been developed recenly, consider he following SV models for h and α = log h : Model : Basic SV Model Wihou Leverage α = μ + φα + +, : ( ε ( The basic SV model is symmeric as posiive and negaive reurns have idenical effecs on fuure volailiy. Model : SV Model Wih Leverage α = μ + φα + +, : ( ε = σ. (3. MODL SPCIFICATIO Le he reurns on a financial asse, by y = μ + e, e = h ε, ε : (0,, / y, be given ( This asymmeric model was suggesed by Harvey and Shephard (996 as a discree ime SV model. Given curren reurns and volailiy in equaion (3, he leverage SV model is given as ( ε ( ( ( α = μ+ φα + σ e exp 0.5 α +, + : 0, σ, (4 84

3 where e is defined in equaion (. Model 3: Asymmeric SV Model - Unsandardized Reurns Wihou Leverage α = μ + φα + e + + e +, : (5 unsandardized reurns in forecasing fuure volailiy.. Model 5: Asymmeric SV Model - Sandardized Reurns Wihou Leverage An alernaive asymmeric SV model ha is differen from he wo previous asymmeric SV models can be proposed as follows: α = μ+ φα + σε+ σ ε +, + ( ε ( α ε = e exp 0.5, (8 This asymmeric SV model was proposed by Danielsson (994, and was esimaed in Asai and McAleer (005. quaion (5 uses he unsandardized reurns in forecasing fuure volailiy. Model 4: Asymmeric SV Model - Unsandardized Reurns Wih Leverage ( ( : 0, σ, ε This new model is an adapaion of he exponenial GARCH (GARCH model of elson (99 o he SV lieraure. In conras o Model 3, his model uses he sandardized reurns, ε, in forecasing fuure volailiy, and can capure various ypes of asymmery and leverage. α = μ + φα + e + +, : (6 3. SV MODL DFIITIOS AD COMPARISO ( ε = σ. This asymmeric SV model was suggesed by Asai and McAleer (005 o capure boh leverage and asymmeric effecs. Similar algebraic manipulaion as for Model yields ( ε ( ( ( α = μ+ φα + σ e exp 0.5 α + e +, + : 0, σ, (7 Model 4 is an alernaive o Model 3 for capuring he asymmeric effecs of posiive and negaive reurns. quaion (6 also uses he Given he developmens presened above, consider he following caegories of symmery and asymmery SV models, condiional on a negaive shock increasing volailiy: Symmery: Posiive and negaive reurns have idenical effecs on fuure volailiy; Asymmery: Posiive and negaive reurns have differen effecs on fuure volailiy; Type I Asymmery (Leverage: A negaive correlaion exiss beween curren reurns and fuure volailiy; Type II Asymmery: Posiive and negaive shocks increase fuure volailiy, bu a negaive shock has a larger effec han does a posiive shock. 85

4 Type III Asymmery: Posiive and negaive shocks increase fuure volailiy, bu a posiive shock has a larger effec han does a negaive shock. Type I Asymmery is based on he original framework of Black (976 and Chrisie (98, and is also consisen wih he definiion of leverage in coninuous ime SV models. In he condiional volailiy lieraure, he empirical resuls based on he GJR model of Glosen, Jagannahan and Runkle (99 and he GARCH model of elson (99 ypically fall ino he Type II Asymmery caegory. Leverage effecs are no possible for he GJR model, whereas leverage is possible, hough frequenly no observed, for he GARCH model. 4. COCLUDIG RMARKS The accurae specificaion and modelling of risk are inegral o opimal porfolio and risk managemen. Hence, a wide variey of condiional and sochasic volailiy models has been used o esimae laen volailiy (or risk. In boh he condiional and sochasic volailiy lieraure, here has been some confusion beween he definiions of asymmery and leverage. This paper examined alernaive univariae SV models ha have recenly been developed and esimaed in o order o undersand he differences and similariies beween he definiions of asymmery and leverage. Five univariae SV models, namely he basic SV model, SV model wih leverage, and hree differen ypes of asymmeric SV models, were analysed in order o clarify he disincion beween asymmery and leverage. Alernaive specificaions of SV models were defined according o he use of sandardized or unsandardized reurns, wih or wihou leverage, in order o evaluae he differenial impacs of posiive and negaive reurns on fuure volailiy, namely symmery, asymmery, ype I asymmery (or leverage, ype II asymmery, and ype III asymmery. ACKOWLDGMTS The auhors wish o hank Chrisian Gourieroux, Suhejla Hoi, Marcelo Medeiros, eil Shephard and Jun Yu for helpful discussions. The firs auhor acknowledges he Japan Sociey for he Promoion of Science and he Ausralian Academy of Science. The second auhor is graeful for he financial suppor of he Ausralian Research Council. RFRCS Asai, M. and M. McAleer (005, Dynamic Asymmeric Leverage in Sochasic Volailiy Models, o appear in conomeric Reviews, 4. Black, F. (976, Sudies of Sock Marke Volailiy Changes, 976 Proceedings of he American Saisical Associaion, Business and conomic Saisics Secion, pp Chrisie, A.A. (98, The Sochasic Behavior of Common Sock Variances: Value, Leverage and Ineres Rae ffecs, Journal of Financial conomics, 0, Danielsson, J. (994, Sochasic Volailiy in Asse Prices: simaion wih Simulaed Maximum Likelihood, Journal of conomerics, 64, Glosen, L., R. Jagannahan and D. Runkle (99, On he Relaion Beween he xpeced Value and Volailiy of ominal xcess Reurns on Socks, Journal of Finance, 46, Harvey, A.C. and. Shephard (996, simaion of an Asymmeric Sochasic Volailiy Model for Asse Reurns, Journal of Business and conomic Saisics, 4, McAleer, M. (005, Auomaed Inference and Learning in Modeling Financial Volailiy, conomeric Theory,, 3-6. elson, D.B. (99, Condiional Heeroskedasiciy in Asse Reurns: A ew Approach, conomerica, 59,

5 Table : Parameric Resricions for Various Univariae SV Models Auhors Basic Symmery Type I Asymmery (Leverage Type II Asymmery Type III Asymmery Harvey and Shephard (996 A A A Danielsson (994 < 0 < < 0 + > 0 < < 0 Asai and McAleer (005 σ < σ σ + σ > 0 σ < σ < 0 This paper < + > 0 < oe: σ is defined as he square roo of exp( α. A denoes no applicable. 87