Economic Computation and Economic Cybernetics Studies and Research, Issue 4/2017; Vol. 51

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1 Economc Compuaon and Economc Cybernecs Sudes and Researc, Issue 4/07; Vol. 5 Assocae Professor Xnyu WU, PD E-mal: xywu@omal.com Scool of Fnance, Anu Unversy of Fnance and Economcs, Cna Professor Sencun REN, PD E-mal: rsc00@63.com Scool of Fnance, Anu Unversy of Fnance and Economcs, Cna Professor Haln ZHOU, PD E-mal: aln_zou@6.com Scool of Fnance, Anu Unversy of Fnance and Economcs, Cna EMPIRICAL PRICING KERNELS: EVIDENCE FROM HE HONG KONG SOCK MARKE Absrac. In s paper, we nvesgae e emprcal prcng kernels for e Hong Kong sock marke. We deal w semparamerc esmaon of e emprcal prcng kernel as e rao of e objecve and rsk-neural denses, under a conssen paramerc framework of e non-affne GARCH dffuson model. An effcen mporance samplng (EIS-based jon maxmum lkelood esmaon meod s developed for e objecve and rsk-neural denses, usng e Hang Seng Index (HSI and ndex warrans daa. Emprcal resuls sow a ere exss a reference pon and around s reference pon e emprcal prcng kernel exbs a ump. e marke uly funcon does no correspond o sandard specfcaon of uly funcon n e classcal expeced uly eory, bu exbs a convex form below e reference pon and a concave form above, and e nvesors ac rsk seekng around e reference pon. Keywords: prcng kernel; uly funcon; rsk averson; GARCH dffuson model; maxmum lkelood esmaon. JEL Classfcaon: C3, C3, C58, G3. Inroducon e beavour of marke nvesors as always been n focus n e leraure on fnancal economcs. Naurally, nvolves e emprcal prcng kernel (Rosenberg and Engle, 00. e asse prcng kernel conans a weal of nformaon, wc summarzes e paern of e marke uly funcon and nvesor rsk preference. In sandard economc eory, e prcng kernel s a monooncally decreasng funcon of e marke reurn, corresponds o a concave uly funcon and nvesor rsk averson. However, ere as been a lo of dscusson abou e relably of s eory. In parcular, several recen emprcal sudes sowed a ere s a reference pon near e zero reurn and around s reference pon e 63

2 Xnyu Wu, Sencun Ren, Haln Zou emprcal prcng kernel exbs a ump (see e.g., Jacwer, 000; Delefsen e al., 007. Hence, e nvesors ac rsk seekng around e reference pon. e non-monooncy of e emprcal prcng kernel as become known as e "prcng kernel puzzle" or "rsk averson puzzle". Numerous aemps ave been underaken o explan e reason for e prcng kernel puzzle from dfferen perspecves (see e.g., Delefsen e al., 007; Zegler, 007; Cab-Yo e al., 008; Baks e al., 00; Goller, 0; Cab-Yo, 0; Crsoffersen e al., 03; Hens and Recln, 03; Barone-Ades e al., 05, and among many oers. On e oer and, Beare and Scmd (04 and Golubev e al. (04 fnd e evdence of non-monooncally decreasng prcng kernel by conducng formal sascal es abou e sape of e prcng kernel. er resuls provde emprcal suppor for e fnancal economcs leraure on e prcng kernel puzzle. In e las decades, ere s a large leraure on e esmaon of e prcng kernel. A number of earler papers esmae e prcng kernel usng aggregae consumpon daa (see e.g., Hansen and Jagannaan, 99; Capman, 997, problems w mprecse measuremen of aggregae consumpon can weaken e emprcal resuls of ese papers. Recenly, many auors ave used e sorcal reurns and opon prces daa o esmae e prcng kernel. s approac avods e use of aggregae consumpon daa. Based on e reurns and opon prces daa, ree ypes of esmaon approaces for esmang e prcng kernel ave been developed: paramerc approaces (e.g., Rosenberg and Engle, 00, nonparamerc approaces (e.g., Aï-Saala and Lo, 000; Jackwer, 000; Song and Xu, 06 and semparamerc approaces (e.g., Cernov, 003; Delefsen e al., 007. However, e paramerc approaces wc mpose a src srucure on e kernel are oo resrcve o accoun for e dynamcs of e rsk preference, wle e nonparamerc approaces depend a lo on e bandwd selecon wc nfluences e sape of e prcng kernel. e semparamerc approaces avod e use of paramerc prcng kernel specfcaon and bandwd selecon, wc s flexble and smple o mplemen. erefore, we derve e emprcal prcng kernel n s paper by employng a semparamerc approac based on e objecve and rsk-neural denses. Prevous economercs sudes are concerned w dervng e emprcal prcng kernel by esmang e objecve and rsk-neural denses separaely, and relyng on e dscree-me GARCH model or/and Heson (993 model. Our esmaon procedure s based on e objecve and rsk-neural denses and ese dsrbuons are derved jonly w a conssen paramerc socasc volaly framework of non-affne GARCH dffuson model. From ese denses we consruc e correspondng prcng kernel. e GARCH dffuson model s a non-affne socasc volaly model, wc as been found o capure e dynamcs of e fnancal me seres muc beer an e popular affne socasc volaly model of Heson (993. Moreover, a number of recen papers ave provde srong evdence for e GARCH dffuson model no only for reurns 64

3 Emprcal Prcng Kernels: Evdence from e Hong Kong Sock Marke daa bu also for opons daa (e.g., Crsoffersen e al., 00; Wu e al., 0; Kaeck and Alexander, 03. us, e model s well sued for our esmaon of e prcng kernel. e objecve and rsk-neural denses are derved by esmang jonly e objecve and rsk-neural parameers of e GARCH dffuson model. In s paper, we develop an jon esmaon procedure for esmang e model usng e Hong Kong Hang Seng Index (HSI and ndex warran prces daa. e fundamenal advanage of s approac s a all e parameers of e model can be relably denfed n a way a manans e nernal conssency of e objecve and rsk-neural measures. e jon esmaon procedure we adop n s paper s based on e maxmum lkelood meod were e lkelood funcon s evaluaed usng e effcen mporance samplng (EIS ecnque of Rcard and Zang (007. e EIS-based jon maxmum lkelood meod s easy o mplemen and enables us o esmae e parameers of e GARCH dffuson model effcenly. e res of e paper s organzed as follows. In Secon, we descrbe e eorecal relaonsp beween e prcng kernel, marke uly funcon and absolue rsk averson and e objecve and rsk-neural denses. In Secon 3, we presen under bo e objecve and rsk-neural measures e GARCH dffuson model, wc serves as e bass for e esmaon of e objecve and rsk-neural denses, and dscuss ow o esmae jonly e objecve and rsk-neural parameers of e GARCH dffuson model usng daa on e HSI reurns and ndex warran prces. In Secon 4, we dscuss e emprcal prcng kernels obaned from e HSI daa, and we conclude n Secon 5. ecncal deals are provded n appendces o e paper.. Prcng kernel, marke uly funcon and absolue rsk averson In e absence of arbrage, ere exss one posve random varable M, suc a e curren prce P of an asse w payoff a me s P E P [ M ( X ( X F ] (, were X s e sae varable of e economy (e.g., log aggregae consumpon, E P s e expecaon w respec o e objecve measure P, M, s called e prcng kernel, and F s e nformaon up o and ncludng me. Accordng o e rsk-neural valuaon prncpal, e prce P of e asse can be equvalenly represened as r P E [ e ( X F ] ( were E s e expecaon w respec o e rsk-neural measure, r s 65

4 Xnyu Wu, Sencun Ren, Haln Zou e rsk free neres rae,. Assumng a p, ( X and q, ( X are e objecve densy and rsk-neural densy of X, respecvely. From Eq. (, we ave r P e r q, ( x ( x q, ( x dx e ( x p, ( x dx p ( x r q, ( X P E e ( X F (3 p, ( X Compare Eqs. ( and (3, we ge r q, ( X M, ( X e (4 p ( X, In a dynamc equlbrum model, e prcng kernel s equal o e neremporal margnal rae of subsuon,.e., U( X M, ( X (5 U( X Here e sae varable, X, s log aggregae consumpon, wc can be subsued w log equy ndex or equy ndex reurn (e.g., Rosenberg and Engle, 00. us, from Eqs. (4 and (5, we ave r q, ( X U( X e (6 p ( X U( X, en we can derve e marke uly funcon as X, ( X r q x U( X U( X e U( X dx U ( X U( X M, ( x dx X p ( x X, (7 Besdes e prcng kernel and marke uly funcon, we are also neresed n e nvesor rsk preference n e marke. Suc rsk preference s ofen descrbed n erms of Arrow-Pra measure of absolue rsk averson a s defne by U( X ARA( X (8 U ( X From Eq. (6, we ge r q, ( X U( X e U( X (9 p ( X and,, 66

5 Emprcal Prcng Kernels: Evdence from e Hong Kong Sock Marke r q ( X p ( X q ( X p ( X U ( X e U( X (0,,,, p, ( X Pluggng Eqs. (9 and (0 no Eq. (8, we ge e absolue rsk averson n erms of e objecve and rsk-neural denses: r e U( X ( q, ( X p, ( X q, ( X p, ( X / p, ( X ARA( X r e U( X q, ( X / p, ( X p, ( X q, ( X ( p ( X q ( X,, 3. Esmaon meodology We adop e non-affne GARCH dffuson model o caracerze e dynamcs of e HSI ndex, and form e bass for e esmaon of e objecve and rsk-neural denses. We frs descrbe e model under e objecve and rsk-neural measures n Secon 3., and en dscuss ow o esmae jonly e objecve and rsk-neural parameers of e GARCH dffuson model usng daa on e HSI reurns and ndex warran prces n Secon 3.. Addonal nformaons abou lkelood approxmaon and unobservable sae varables esmaon are gven n Appendces A and B. 3. e model In e GARCH dffuson model, e dynamcs under e objecve measure of e HSI ndex prce S and e assocaed volaly V are assumed o be gven by ( ds S d V S dw dv ( V d V dw (3 were s e mean of e HSI reurns, / s e long-run mean of volaly, s e mean reverson rae of volaly, s e volaly of volaly, and W and W are wo sandard Brownan moons w Corr ( dw, dw. Smlar o Cernov and Gysels (000, we assume a e GARCH dffuson model ave e same form under e rsk-neural measure as under e objecve measure, and e dynamcs of ( S, V under e rsk-neural measure are of e form * (4 ds rs d V S dw dv ( V d V dw * * * (5 67

6 Xnyu Wu, Sencun Ren, Haln Zou were r s e rsk-free neres rae, * W and * W are wo sandard Brownan * * moons under e rsk-neural measure w Corr ( dw, dw. Followng Wu e al. (0, e caracersc funcon for e log HSI ndex X ln S can be derved. en e objecve/rsk-neural densy for X can be obaned by nverng e correspondng caracersc funcon. a s, X p, ( X e f, ( d (6 X * q, ( X e f, ( d (7 * were f, and f, are e caracersc funcons for X under e objecve and rsk-neural measures, respecvely, and e negrals n Eqs. (6 and (7 can be easly compued by usng some numercal meods. 3. Jon maxmum lkelood esmaon In s subsecon, we develop a maxmum lkelood meod o esmae jonly e objecve and rsk-neural parameers of e GARCH dffuson model usng daa on e HSI reurns and ndex warran prces. akng e sablzng ransformaon X ln S, lnv. By Iô's lemma, we ave / dx ( e d e dw (8 d ( e d dw (9 In e emprcal leraure, e above connuous-me model mus be dscrezed o faclae e parameer esmaon. A smple Euler sceme leads o e followng dscree-me socasc processes / y ( e e (0 ( e ( were y X X s e HSI reurn, s e me nerval, and are ndependen and dencally dsrbued (..d. sandard normal random varables w Corr (,. o perform jon esmaon of e objecve and rsk-neural parameers, we consder e addonal nformaon provded by e HSI warran prces. We assume a e observed warran prce s equal o e eorecal value plus a prcng error: 68

7 Emprcal Prcng Kernels: Evdence from e Hong Kong Sock Marke C C(,, K, S, V ( were e nonlnear funcon C(,, K, S, V s e prcng formula for European warrans n e GARCH dffuson model (see Wu e al., 0, and are..d. sandard normal random varables and ndependen of and. I s obvous a Eqs. (0-( consue a nonlnear and non-gaussan sae-space model w volaly s e unobservable sae varable. o esmae s model usng maxmum lkelood meod, we need o negrae ou e unobservable sae varables from e jon densy of e observaons and unobservable sae varables and derve an explc expresson for e margnal lkelood of observaons. Le C ( C,, C be a vecor of e N observed HSI ndex warran prces, N Y ( y,, y be a vecor of e N observed HSI reurns and N N H (,, be a vecor of e unobservable sae varables (log volales. e lkelood funcon of e model can be expressed as L ( C, Y;, p( C, Y, H;, dh (3 were 0 0 * * (,,,,,,, s e parameer vecor, wc consss of * * e objecve and rsk-neural parameers (,,,,,, of e GARCH dffuson model and e parameer n measuremen equaon (, and p( C, Y, H;, s e jon densy of C, Y and H, wc can be wren as 0 p( C, Y, H;, p( C Y, H, p( Y, H;, N 0 0 p( C y,, (, (,, p y p y (4 were p( C y,, s e normal densy of C(,, K, S, V and e condonal varance normal densy of condonal varance e and w e condonal mean by C w e condonal mean, p ( y, s e y w e condonal mean ( e and e p( y,, s e normal densy of and e condonal varance wc are gven 69

8 Xnyu Wu, Sencun Ren, Haln Zou ( y e ( e / e (5 ( (6 Gven e lkelood funcon n Eq. (3, e ML esmaes of parameers of e sae-space model n Eqs. (0-( are en gven by ( ˆ, ˆ arg max ln L ( C, Y;, (, As a ypcal fnancal me seres as a leas several undreds of observaons, e g-dmensonal negral n e rg and of Eq. (3 rarely as analycal expresson. Meanwle, usng e radonal numercal negraon meods o approxmae e negral s also nfeasble. In order o overcome s problem, we adop e EIS ecnque o compue e lkelood funcon. e EIS algorm for lkelood approxmaon s presened n Appendx A. o exrac e laen spo volaly, we use a parcle fler algorm wc s gven n Appendx B. 4. Emprcal analyss In conras o many prevous sudes a ave focused manly on e S&P 500 daa, we nvesgae n s paper e emprcal prcng kernels by focusng on e HSI daa (HSI ndex and s warrans. e HSI ndex serves as an approxmaon o e Hong Kong economy, and can be used as a proxy for marke porfolo. e HSI ndex warrans were cosen over e HSI ndex opons because e HSI warrans marke s a more lqud/acve marke an e HSI opons marke n Hong Kong. 4. e daa In e emprcal analyss we use daly daa on e HSI reurns and ndex warran prces from July, 0 o May 3, 03. e HSI reurns compued are logarmc,.e., x log p log p, were p s e closng prce. e HSI ndex warran s cosen as e HS-HSI@EC309 wc s one of e mos acvely raded HSI ndex warrans. e seleced warran s European-syle call warran wc s smlar o a European-syle call opon. Is maury dae s Sepember 7, 03, e exercse prce s 5,000 and e exercse rao s,000. e sample sze s 98 for e jon daa. e me-seres of HSI reurns and HS-HSI@EC309 prces are ploed n Fgure. Fnally, we use e -year Hong Kong Inerbank Offer Rae (HIBOR as a proxy for e rsk-free neres rae. All of e daa are obaned from e Wnd Daabase of Cna. Summary sascs for e HSI reurns are sown n able. As can be seen from e able, e HSI reurns are skewed and lepokurc. Jarque-Bera sascs suggess a e assumpon of normaly s rejeced for e HSI reurn seres. Furermore, from Fgure we can observe a e HSI reurns exb 70

9 Emprcal Prcng Kernels: Evdence from e Hong Kong Sock Marke me-varyng volaly and volaly cluserng durng e sample perod. Fgure : me seres of HSI reurns and HS-HSI@EC309 prces for e sample perod from July, 0 o May 3, 03 able. Summary sascs of HSI reurns Mean Max Mn Sd. Skew Kur Noe: e number n pareness s e p-values of Jarque-Bera ess. Jarque- Bera ( Esmaon resuls Based upon daa on e HSI reurns and HS-HSI@EC309 prces, e objecve and rsk-neural parameers of e GARCH dffuson model are esmaed jonly by applyng e maxmum lkelood meod descrbed n Secon 3. able repors e esmaon resuls. e esmaed parameers allow us o esmae e volaly, V, va e parcle fler algorm. e number of parcles used n e emprcal sudes s 000. Fgure plos e esmaed volales. 7

10 Xnyu Wu, Sencun Ren, Haln Zou able. Esmaon resuls ( ( ( ( (0.055 * * Log-lk ( (0.03 ( Noe: e EIS-ML meod s mplemened by usng S=3 Mone Carlo draws and 5 EIS eraons. Log-lk s e log-lkelood value. e number n pareness s e sandard error. Fgure : Esmaed volales Based on e esmaes of e objecve and rsk-neural parameers and volales, e objecve and rsk-neural denses of e HSI reurns can be obaned by usng Eqs. (6 and (7. e esmaon resuls of e objecve and rsk-neural denses are presened n Fgure 3 for e day May 3, 03 and for wo me o maures: 0.5 and years. I can be seen a ere are large dscrepances n e esmaon resuls of e objecve and rsk-neural denses. By usng e Eqs. (4, (7 and (, we derve e esmaed emprcal prcng kernels, marke uly funcons and absolue rsk averson funcons of HSI reurns 7

11 Emprcal Prcng Kernels: Evdence from e Hong Kong Sock Marke on May 3, 03 for wo me o maures: 0.5 and years, wc are presened n Fgures 4-6. As can be seen from Fgure 4, our esmaed emprcal prcng kernels are no monooncally decreasng, and ese are no n accordance w e classcal economc eory. e esmaed emprcal prcng kernels ave umps locaed a small losses (correspondng o a HSI reurn of abou -0% for me o maury 0.5 and a HSI reurn of abou -% for me o maury, ereafer referred o as reference pons. Our resuls provde emprcal suppor for e leraure on e prcng kernel puzzle. Fgure 3: Esmaed objecve and rsk-neural denses on May 3, 03 for me o maures 0.5 and years Fgure 4: Emprcal prcng kernels on May 3, 03 for me o maures 0.5 and years 73

12 Xnyu Wu, Sencun Ren, Haln Zou Fgure 5: Marke uly funcons on May 3, 03 for me o maures 0.5 and years Fgure 6: Absolue rsk averson funcons on May 3, 03 for me o maures 0.5 and years e prcng kernels are e lnk beween e absolue rsk aversons and e marke uly funcons a are presened n Fgure 5. As can be seen from e fgure, e esmaed marke uly funcons are ncreasng bu do no correspond o sandard specfcaon of uly funcon n e classcal expeced uly eory. Specfcally, e esmaed marke uly funcon exbs a convex form below e reference pon and a concave form above, wc s n accordance w e uly funcon form proposed by Kaneman and versky (979. Fnally, we consder e absolue rsk aversons n e Hong Kong sock marke. e esmaed absolue rsk averson funcons are presened n Fgure 6. I can be seen from e fgure a e absolue rsk averson s negave around e 74

13 Emprcal Prcng Kernels: Evdence from e Hong Kong Sock Marke reference pon, wc mples a nvesors ac rsk seekng around e reference pon. Our resuls are muc n lne w e prospec eory of Kaneman and versky ( Concluson In s paper, we employ a semparamerc approac o derve e emprcal prcng kernels as e rao of e objecve and rsk-neural denses for e Hong Kong sock marke. e objecve and rsk-neural denses are esmaed jonly by e maxmum lkelood meod based on e EIS ecnque, under a conssen paramerc framework of e non-affne GARCH dffuson model and usng e HSI reurns and ndex warran prces daa. Emprcal resuls sow a ere exss a reference pon (correspondng o a HSI reurn of abou -0%/-% for alf-year/one-year maury and around s reference pon e emprcal prcng kernel exbs a ump. e marke uly funcon does no correspond o sandard specfcaon of uly funcon n e classcal expeced uly eory, bu exbs a convex form below e reference pon and a concave form above, and e nvesors ac rsk seekng around e reference pon. Our resuls are muc n lne w e prospec eory of Kaneman and versky (979 and provde emprcal suppor for e leraure on e prcng kernel puzzle. Acknowledgemens s work was suppored by e Naonal Naural Scence Foundaon of Cna under Gran No , e MOE (Mnsry of Educaon n Cna Projec of Humanes and Socal Scences under Gran No. 4YJC79033, e Cna Posdocoral Scence Foundaon under Gran No. 05M58046, e Naural Scence Foundaon of Anu Provnce of Cna under Gran No QG39, and e Anu Provnce College Excellen Young alens Fund of Cna under Gran No. 03SQRW05ZD. REFERENCES [] Aï-Saala, Y., Lo, A.W. (000, Nonparamerc Rsk Managemen and Impled Rsk Averson; Journal of Economercs 94, 9-5; [] Baks, G., Madan, D., Panayoov, G. (00, Reurns of Clams on e Upsde and e Vably of U-saped Prcng Kernels; Journal of Fnancal Economcs 97, 30-54; [3] Barone-Ades, G., Mancn, L., Sefrn, H. (05, Senmen, Rsk Averson, and me Preference; Workng Paper, Unversy of Lugano; [4] Beare, B.K., Scmd, L.D. (04, An Emprcal es of Prcng Kernel Monooncy; Journal of Appled Economercs 3(, ; 75

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15 Emprcal Prcng Kernels: Evdence from e Hong Kong Sock Marke [] Rcard, J.F., Zang, W. (007, Effcen Hg-dmensonal Imporance Samplng; Journal of Economercs 7(, 385-4; [] Rosenberg, J., Engle, R.F. (00, Emprcal Prcng Kernels; Journal of Fnancal Economcs 64, 34-37; [3] Song, Z.G., Xu, D.C. (06, A ale of wo Opon Markes: Prcng Kernels and Volaly Rsk; Journal of Economercs 90(: 76-96; [4] Wu, X.Y., Ma, C.Q., Wang, S.Y. (0, Warran Prcng under GARCH Dffuson Model; Economc Modellng 9(6, 37-44; [5] Zegler, A. (007, Wy Does Impled Rsk Averson Smle? e Revew of Fnancal Sudes 0, Appendx A. EIS algorm o lkelood approxmaon e EIS algorm for esmang e lkelood funcon s gven as follows: ( s ( s S Sep : Draw nal samples {,, } s from e so-called naural N mporance sampler { p( y,, }. Sep : Calculae a ˆ by esmang e followng regresson model (workng backwards: N ( s ( s ( s ln p( C y,, ln p( y, ln ( y,, aˆ were c a a u s S ( s ( s ( s,, (,,, a a y a a ln (,, ln,, a a, and N N N N N N N a a a, p( y, ( y,, aˆ, a ( a,, a,, are gven n Eqs. (5 and (6. Sep 3: Draw new samples ( s ( s S {,, } s N from e EIS sampler N { m (,, ˆ } y a, were m s e normal densy (called EIS densy of w e condonal mean a and e condonal varance a. Sep 4: Repea Sep and Sep 3, unl a reasonable convergence of e parameers a ˆ s obaned. Sep 5: Calculae e lkelood approxmaon usng, 77

16 Xnyu Wu, Sencun Ren, Haln Zou S ( s ( s ( s ( s µ N p( C y,, (, (,, p y p y L ( C, Y;, 0 ( s ( s S s m (,, ˆ y a Followng Rcard and Zang (007, a same se of Common Random Numbers (CRNs s used o oban e draws from e EIS sampler n order o ensure e lkelood approxmaon be a smoo funcon of e parameer vecor. ypcally, a reasonable convergence can be obaned afer 3-5 eraons. Appendx B. Parcle fler algorm for exracng laen sae varables e parcle fler algorm for exracng e laen sae varables s gven as follows: ( ( M Sep : Gven a se of random samples {,, } from e probably densy funcon p ( F. Sep : Draw samples p (,. (* ( M* {,, } from e probably densy Sep 3: Compue e normalsed weg for eac sample ( j* ( j* ( j p( C y,, (,, p y q j, j, M, M ( l* ( l* ( l p( C y,, p( y,, l us defne a dscree dsrbuon over {,, }. q q M (* ( M* {,, }, w probably mass Sep 4: Resample M mes from e dscree dsrbuon defned above o ( ( M generae samples {,, }. 78