Variance Risk Premium and VIX Pricing: A Simple GARCH Approach

Transcription

1 Variance Risk Premium and VIX Pricing: A Simple GARCH Approach Qiang iu a Professor, School of Finance Souhwesern Universiy of Finance and Economics Chengdu, Sichuan, P. R. China. Gaoxiu Qiao Graduae suden, School of Finance Souhwesern Universiy of Finance and Economics Chengdu, Sichuan, P. R. China. a) Qiang iu, Corresponding Auhor. School of Finance Souhwesern Universiy of Finance and Economics A5 Tongbo Hall, 555 iuai Boulevard, Wenjiang, Chengdu, Sichuan , P. R. China. qiangliu@swufe.edu.cn. Phone: Fax: Acknowledgemen The work is suppored by a key research gran from Projec 11 (Phase III) of Souhwesern Universiy of Finance and Economics, by Huaxi Fuures Co., d., and by Tri-Spring Seel Trades Co., d.

2 Variance Risk Premium and VIX Pricing: A Simple GARCH Approach Absrac: This sudy proposes a simple GARCH mehod for pricing VIX. Closed-form formulas for VIX are derived for GARCH(1,1), asymmeric GJR GARCH(1,1), and Heson-Nandi GARCH(1,1) models. Wih he empirical GARCH parameers esimaed from 3500 daily reurns for he S&P 500 Index, hese formulas under-price VIX by 1.0~33.9% on average for he wo periods of January 1996-Sepember 003 and Sepember 003-January 01. This underesimaion could be inerpreed as variance risk premium. On he oher hand, he parameers in hese formulas can be calibraed by he marke VIX of he previous rading day o obain he risk-neural measure. For he same wo periods, he risk-neural parameers price VIX accuraely wih a mean pricing error of no more han 0.%. Keywords: Variance Risk Premium; VIX Pricing; VIX formulas; GARCH(1,1); GJR GARCH; Heson-Nandi GARCH JE Classificaion: G13, G1 Firs version: April 3, 01. This version: April 4, 01.

3 1. INTRODUCTION The CBOE Volailiy Index (VIX), proposed by Whaley (1993) and inroduced in 1993, has become he sandard gauge of he invesor fear and marke senimen. I is befiingly referred o as he fear index by popular news media, such as he New York Times and he Wall Sree Journal. Nowadays, CBOE rades VIX fuures, VIX opions, VIX Binary Opions, and Mini-VIX fuures. As a forecas for he 30-day volailiy of he S&P 500 Index, VIX and is pricing (or is own forecasing) have no overlooked by academia. Using he Heson sochasic variance model, Zhang and Zhu (006) iniiaed he sudy of VIX fuures pricing, bu did no ry o forecas VIX. Closed-form formulas for he fair value of VIX fuures were derived for several sochasic variance models wih jumps in boh he asse and variance processes (in, 007). Duan and Yeh (010) uilized he maximum likelihood mehod o esimae he parameers of sochasic volailiy models wih or wihou jumps. The Heson sochasic variance model was employed by Zhang and Huang (010) in a sudy of he CBOE S&P500 hree-monh variance fuures. VIX fuures were invesigaed by assuming a square roo mean-revering process for he variance (Zhang, Shu & Brenner, 010). Zhu and ian (011) obained a closed-form exac soluion for VIX fuures in a sochasic volailiy model wih simulaneous jumps in boh he asse and volailiy processes. All hese researches menioned above deal wih coninuous-ime variance or volailiy models, he parameers of which are calibraed by he marke VIX level of a previous rading day. 1

4 The calibraed (risk-neural) parameers are hen used o forecas he price of fuures. As a discree ime model for volailiy, GARCH models seem o be a naural choice for sudying VIX, bu nobody appears o have paid serious aenion o i so far. Barone-Adesi e al. (008) alluded o he pricing of VIX briefly, when hey proposed he filered hisorical simulaion GJR GARCH mehod for pricing European opions. They used he marke prices of S&P 500 Index opions from a previous rading day o calibrae GARCH parameers in order o obain he risk-neural GARCH model. Their idea of pah simulaions was laer applied o VIX pricing wih some more deails (Byun & Min, 01). VIX formulas under he empirical measure for five GARCH models were presened by Hao and Zhang (010), who did no ry o obain he risk-neural parameers for forecasing VIX. This paper proposes a simple GARCH based VIX pricing mehod. For GARCH(1,1), GJR, and Heson-Nandi variance models, simple closed-from formulas are derived firs. Then insead of using opions marke prices for he calibraion of parameers, he new approach uilizes direcly he marke VIX of he previous rading day o obain risk-neural GARCH parameers. Wih he risk-neural parameers, he VIX formulas can be applied o VIX pricing. aer, empirical invesigaions are carried ou for he periods of January Sepember 003 and of Sepember January 01. The empirical resuls reveal he variance risk premium under he empirical measure and provide adequae proof of concep for he new pricing mehod. Finally, he paper concludes wih commens.

5 . VIX FORMUAS UNDER GARCH(1,1) MODES Several GARCH(1,1) models can be wrien as follows under eiher hisorical or risk-neural measures: ln( S 1 S ) = µ + ε v ω βv ⑴. = u 1 where v = σ is he variance, and ε = σ z is he innovaion for dae. Here z is eiher a sandard normal variable or he empirical random variable wih a mean of zero and a variance of one. For GARCH(1,1) or G11 hereafer, u = αε (Hull, 009); for asymmeric GJR GARCH(1,1) or GJR hereafer, u ( α + γi{ z < 0}) ε =, where I{ z < 0} is he indicaor funcion (Barone-Adesi, Engle & Mancini, 008); for HN GARCH(1,1) or HN hereafer, u = α( z γσ ) (Heson & Nandi, 000). By definiion, z and σ are independen, and he condiional expecaions are E [ 1 ] = z + 0, E [ 1 ] = 1, E [ ] z + 1 σ + 1 = 0, and E [ ] v + 1 = v + 1. Furher, E [ z I{ z < 0}] = z Therefore using GJR as an example, one has: E v + k = ω + E [ βv + k 1 = ω + E [( β + E = ω + E v + k 1 + ( α + γi{ z + k + k 1 [( α + γi{ z < 0}) ε + k 1 + k 1 < 0}) z ] + k 1 ]) v + k 1 ] where = α + β γ. eω = ( 1 )V, where V represens he long-erm average variance. Then from he above expression i is easy o show (following Hull (009)) ha: E [ v + k V ] = E [ v = k 1 + k 1 [ v + 1 V V ] ] Now he CBOE 30-day volailiy index VIX can be compued according o 3

6 Equaion (1) 1 of Barone-Adesi e al. (008) as: VIX = ( = 84000[30V = 84000[30V + ( v + ( v E [ v k = 1 + k V V ) 30 k = ) ] 1 ]) k 1 ] ⑵. = α + β + 0.5γ, V = ω (1 ) (for GJR) ⑶. Noe ha Equaion () has exacly he same linear form as hose given by he coninuous-ime sochasic volailiy models (Zhang & Zhu, 006; in, 007; Zhu & ian, 011). Similar bu a lile bi more complicaed resuls for GARCH(1,1), EGARCH, TGARCH, AGARCH and CGARCH models were derived by Hao and Zhang (010). One nice feaure of Equaion () is is independence of he parameers of he sock process, while boh he resuls of coninuous-ime models and Hao and Zhang (010) are no. Ineresingly Equaion () is also correc for G11 and HN given he following parameers: = α + β, V = ω (1 ) (for G11) ⑷. = αγ + β, V = ( ω + α) (1 ) (for HN) ⑸. Wih only GARCH parameers ω, α, β and γ (for GJR and HN only) under eiher he empirical or risk-neural measure, VIX can hen be compued direcly using Equaion 1 GARCH variance is measured in rading days, while VIX is compued in calendar days. Therefore, he righ hand-side of Equaion (1) of Barone-Adesi e al. (008) shall be muliplied by a facor of 5/365. 4

7 () and (3), (4), or (5). 3. VARIANCE RISK PREMIUM AND VIX-CAIBRATED GARCH PARAMETERS Obviously given GARCH parameers esimaed from he hisorical prices of he underlying asse, VIX can be compued via Equaion () and (3), (4), or (5). Unforunaely, GARCH under empirical measure does no price eiher opions (Barone-Adesi, Engle & Mancini, 008) or VIX accuraely. As a maer of fac, empirical GARCH under-esimaes VIX consisenly (Hao & Zhang, 010). The difference beween he observed marke VIX and GARCH-esimaed VIX is ermed he variance risk premium by Hao and Zhang (010). The underesimaion is also confirmed by he empirical invesigaion of his paper. Risk-neural GJR GARCH parameers calibraed by he marke prices of raded opions do a much beer job in pricing opions (Barone-Adesi, Engle & Mancini, 008; iu & Xiang, 01). Furher, Barone-Adesi e al. (008) menioned briefly ha he Risk-neural GJR GARCH parameers can be used o simulae daily variances in order o price VIX. Their idea was adoped by a laer paper (Byun & Min, 01). Unforunaely, i is quie expensive compuaionally o obain he risk-neural GARCH parameers by simulaing price pahs, pricing he opions, and minimizing pricing errors (Barone-Adesi, Engle & Mancini, 008; Byun & Min, 01). Since VIX is by now well-esablished, his paper proposes o calibrae he GARCH parameers by VIX direcly via he closed-form Equaion of (). The saving of doing 5

8 his is wofold. Firs, prices of raded opions are no needed. Second, price pahs and daily variances do no have o be simulaed. Furhermore, because marke VIX is compued from prices of raded opions, he new scheme obains he risk-neural measure direcly hrough Equaion (), wihou having o use he risk-free ineres rae. This makes he new approach simpler addiionally. Wih he VIX-calibraed risk-neural GARCH parameers, VIX can be priced direcly by Equaion (). Once again, daily variances do no have o be simulaed. 4. EMPIRICA STUDY 4.1. Daa Descripion Since CBOE does no provide hisorical daa for he S&P 500 Index, boh daily VIX and he S&P 500 Index from Yahoo!Finance are used in his sudy. A comparison of VIX from CBOE and Yahoo shows only very minor differences for a few days. I seems ha he qualiy of Yahoo!Finance daa should no be a problem for our purpose. On Sepember 003, CBOE modified he mehod for compuing VIX. Accordingly, his paper divides he ime series of VIX ino wo sub-periods. Phase I is beween January 1996 and 19 Sepember 003; Phase II is from Sepember 003 o 31 January 01. Table I provides a summary descripion of he VIX daa. Apparenly, he long-erm average of VIX is somewha sable, bu VIX shows more pronounced variaion in Phase II. Table I here This choice makes he wo pars have roughly he same rading days. 6

9 4.. Compuaional Deails and Resuls Analyses Assume is he valuaion dae, and -1 is he calibraion dae. Following Barone-Adesi e al. (008), he paper uilizes 3500 daily reurns of he S&P 500 Index beween and = (ln v i ε i v i ), he negaive of he maximum-likelihood i 1 funcion (Hull, 009), is minimized via he Nelder-Mead algorihm (Press, Teukolsky, Veerling & Flannery, 00) o obain he opimal se of empirical GARCH parameers. The repored average parameers for he wo phases are quie close (Table II). Table II here m Denoe he marke VIX level for he calibraion dae byvix 1. The objecive funcion{ 84000[30V + ( v V )(1 30 m ) (1 )] VIX 1}, where v (and ε as well) is compued via Equaion (1) using he empirical GARCH parameers, is again minimized via he Nelder-Mead algorihm 3 o obain he risk-neural GARCH parameers (Table III). The risk-neural parameers are markedly differen from heir corresponding empirical ones. Furher, he parameer for Phase I for all hree GARCH models is larger han one, which means ha he risk-neural models are mean-fleeing (Hull, 009). Table III here 3 Nelder-Mead is sensiive o he iniial choice of he variables characerisic lenghs, which are obained hrough exensive esing of he hree GARCH models and differen for esimaing he empirical GARCH and calibraing. 7

10 Wih he empirical and risk-neural parameers, he VIX level for he valuaion dae can hen be compued via Equaion (), where v + 1 is once again compued via Equaion (1) using he empirical GARCH parameers. The resuls are summarized in Table IV. Table IV here Table IV repors hree error measures. The mean of pricing errors (MPE) is N c m c defined as = ( VIX j j VIX j 1) N, wherevix 1 j is he compued VIX and m VIX j he marke VIX for dae j. The mean of absolue pricing errors (MAE) is obained via N j = 1 VIX c j VIX m j 1 N. The roo mean of square pricing errors (RMSE) is compued by [ = N c m 0.5 ( VIX ) ] j 1 j VIX j N. Boh MPE and MAE errors under he empirical measure are raher large for all hree GARCH models. MPEs are negaive bu close o MAEs in absolue value, which implies ha on average he empirical GARCH underesimaes VIX. This underesimaion of VIX by empirical GARCH can be regarded as he variance risk premium ha is no presen in he observed prices of he underlying S&P 500 Index. Ineresingly, he mispricing for Phase I is around 5%, bu only abou 14% for Phase II. This seems o sugges ha he variance risk premium becomes smaller in Phase II. Under he calibraed risk-neural measure, all errors are quie small and roughly he same for Phases I and II. MPEs are really small while MAEs are one order of magniude larger bu sill less han 5%. This implies ha on average he errors from under-pricing and overpricing cancel ou. Remarkably, all hree GARCH models price 8

11 VIX accuraely. Among he hree GARCH models, G11 has he smalles errors for pricing VIX under he empirical GARCH measure, bu displays he bigges pricing errors under he risk-neural measure. The laer could be undersandable since G11 uses only hree parameers o fi he daa, bu he former seems hard o explain. Finally for HN, he siuaion of pricing errors under he empirical and risk-neural measure is reversed agains G CONCUSIONS This paper proposes a simple GARCH approach o pricing VIX. Wih closed-form formulas for compuing VIX based on symmeric GARCH(1,1), asymmeric GJR GARCH(1,1), and asymmeric Heson-Nandi GARCH(1,1) models, he GARCH parameers in hese formulas can be calibraed by he marke VIX of he previous rading day. The calibraed parameers are risk-neural, and can be used wih hese formulas o price (or forecas) VIX. Wih empirical GARCH parameers esimaed from 3500 daily reurns for he S&P 500 Index, he formulas under-price VIX by 19.9~33.9% for he period of January Sepember 003 and by 1.0~14.4% for he period of Sepember January 01. This underesimaion could be referred o as variance risk premium. Using risk-neural GARCH parameers calibraed from he marke VIX, hese formulas reduce dramaically he pricing errors o 0.1~0.%. Imporanly, he 9

12 differences among he hree GARCH models and beween he wo sub-periods are almos negligible. In summary, he proposed GARCH mehod can price VIX accuraely. Furher, wih closed-from analyic formulas, he new approach is also compuaionally efficien. Finally, hose resuls could in principle be exended and applied o he pricing of VIX fuures, VIX opions, and even S&P 500 Index opions. 10

13 BIBIOGRAPHY Barone-Adesi, G., Engle, R. F., & Mancini,. (008). A GARCH opion pricing model wih filered hisorical simulaion. Review of Financial Sudies, 1, Byun, S.-J., & Min, B. (01). Condiional volailiy and he GARCH opion pricing model wih non-normal innovaions. Journal of Fuures Markes, forhcoming. Duan, J., & Yeh, C. (010). Jump and volailiy risk premiums implied by VIX. Journal of Economic Dynamics & Conrol, 34, Hao, J., & Zhang, J. (010). GARCH opion pricing models, he CBOE VIX and variance risk premium. Working Paper, Peking Universiy and Universiy of Hong Kong. Heson, S.., & Nandi, S. (000). A closed-form GARCH opion pricing model. Review of Financial Sudies, 13, Hull, J. C. (009). Opions, fuures, and oher derivaives (7h ed.). Upper Saddle River, NJ: Prenice Hall. in, Y. (007). Pricing VIX fuures: evidence from inegraed physical and risk-neural probabiliy measures. Journal of Fuures Markes, 7, iu, Q., & Xiang, Y. (01). FHS-GARCH-SM: A new mehod for pricing American opions. Journal of Fudan Universiy (Naural Science) [in Chinese], forhcoming. Press, W. H., Teukolsky, S. A., Veerling, W. T., & Flannery, B. P. (00). Numerical recipes in C++: The ar of scienific compuing (nd ed.). Cambridge, UK: Cambridge Universiy Press. 11

14 Whaley, R. E. (1993). Derivaives on marke volailiy: Hedging ools long overdue. Journal of Derivaives, 1, Zhang, J. E., & Huang, Y. (010). The CBOE S&P 500 hree monh variance fuures. Journal of Fuures Markes, 30, Zhang, J. E., & Zhu, Y. (006). VIX fuures. Journal of Fuures Markes, 6, Zhang, J. E., Shu, J., & Brenner, M. (010). The new marke for volailiy rading. Journal of Fuures Markes, 30, Zhu, S., & ian, G. (011). An analyical formula for VIX fuures and is applicaions. Journal of Fuures Markes, 3,

15 Table I. Descripion of he VIX prices. Minimum Maximum Mean Sd Dev No. Prices Phase I Phase II Phase I: January Sepember 003. Phase II: Sepember January 01.

16 Table II. Empirical GARCH parameers. 6 G11 ω 10 α 10 β Phase I Mean Sd Dev Phase II Mean Sd Dev GJR 6 ω 10 α 10 β γ 10 Phase I Mean Sd Dev Phase II Mean Sd Dev HN 6 ω 10 6 α 10 β γ Phase I Mean Sd Dev Phase II Mean Sd Dev Phase I: January Sepember 003. Phase II: Sepember January 01. The GARCH models are ln( S S ) 1 = µ + ε, v = + + ω βv 1 u 1, where v = σ, σ z ε =, and z is he empirical random variable wih a mean of zero and a variance of one. For G11, u = αε, = α + β ; for GJR, u ( α + γi{ z < 0}) ε =, where I{ < 0} is he indicaor funcion, = α + β γ ; for HN, z u = + = α( z γσ ), αγ β. Parameers are esimaed by minimizing i = (ln v ε v ) wih 3500 daily reurns. i + i i

17 Table III. Risk-neural GARCH parameers. 6 G11 ω 10 α 10 β Phase I Mean Sd Dev Phase II Mean Sd Dev GJR 6 ω 10 α 10 β γ 10 Phase I Mean Sd Dev Phase II Mean Sd Dev HN 6 ω 10 6 α 10 β γ Phase I Mean Sd Dev Phase II Mean Sd Dev Phase I: January Sepember 003. Phase II: Sepember January 01. Parameers are esimaed by minimizing{ 84000[30V 30 + ( v V )(1 ) (1 )] VIX m 1 }, where VIX 1 is he marke VIX m level for he calibraion dae, α + β, V = ω (1 ) for G11, α + β + 0.5γ, V = ω (1 ) for GJR, = = and = αγ + β, V = ( ω + α) (1 ) for HN. Furher, v is compued by using he empirical GARCH parameers via ln( S S ) 1 = µ + ε, v = + + ω βv 1 u 1, where v = σ, σ z ε =, u = αε for G11, u ( α + γi{ z < 0}) ε = (where I{ < 0} is he indicaor funcion) for GJR, and u = α γσ ) for HN. z ( z

18 Table IV. Errors of VIX pricing by GARCH. Empirical Risk-neural MPE (%) MAE (%) RMSE MPE (%) MAE (%) RMSE Phase I G GJR HN Phase II G GJR HN Phase I: January Sepember 003. Phase II: Sepember January 01. For boh he 30 empirical and risk-neural pricing, VIX is compued via 84000[30V ( v V )(1 ) (1 )], where = α + β, V = ω (1 ) for G11, α + β + 0.5γ, V = ω (1 ) for GJR, and = αγ + β, V = ( ω + α) (1 ) for HN. Furher, v + 1 is obained by using he empirical GARCH parameers = via ln( S S ) 1 = µ + ε, v = + + ω βv 1 u 1, where v = σ, σ z ε =, u = αε for G11, u ( α + γi{ z < 0}) ε = (where I{ < 0} is he indicaor funcion) for GJR, and u z = α( z γσ ) for HN. MPE: mean of pricing errors. MAE: mean of absolue pricing errors. RMSE: roo mean of square pricing errors.