(Almost) Model-Free Recovery

Transcription

1 (Almos) Modl-Fr covry Paul Schnidr and Fabio Trojani January 9, 6 Absrac Basd on mild conomic assumpions, w rcovr h im sris of condiional physical momns of mark indx rurns from a modlfr projcion of h pricing krnl on h rurn spac. Ths momns idnify h minimum varianc pricing krnl projcion and ar suppord by a corrsponding s of physical disribuions. Th rcovrd momns prdic S&P 5 rurns, spcially for longr horizons, giv ris o rfind condiional vrsions of Hansn-Jagannahan bounds, and can b radd using dla-hdgd opion porfolios. Thy also imply condiional pricing krnl projcions ha ar ofn far from bing uniformly monoonic and convx. Inroducion A sminal finding in Brdn and Liznbrgr (978) shows ha in an arbirag-fr mark h dnsiy of h condiional forward-nural disri- W ar hankful for hlpful discussions wih Jaroslav Borovička, Pr Carr, Gianluca Casss, Jrom Dmpl, Damir Filipovic, Parick Gagliardini, Pr Grubr, Lonid Kogan, Eric naul, Sv oss, Olivir Scaill, Chrisian Wagnr, Liurn Wu and workshop paricipans a Boson Univrsiy, Brown Univrsiy, EPFL, MIT, Morgan Sanly, and USI. Financial suppor from h Swiss Financ Insiu (Projc Trm srucurs and cross-scions of ass risk prmia ), and h SNF (Projc Trading Ass Pricing Modls ) is grafully acknowldgd. Boson Univrsiy, Univrsiy of Lugano, and Swiss Financ Insiu. . paul.schnidr@usi.ch Univrsiy of Gnva, Univrsiy of Lugano, and Swiss Financ Insiu. . fabio.rojani@usi.ch

2 buion (Q T ) of an ass rurn coincids wih h scond drivaiv of h pric funcion of Europan call opions wih rspc o h opion s srik. Thr is no such gnral and modl-fr rsul for h condiional disribuion of ass rurns undr h physical probabiliy masur (P). To larn mor abou physical disribuions or h chang of masur bwn Q T and P, rsarchrs ofn hav mad us of paramric modlling approachs (.g., Jons, 3; Erakr, 4), or combinaions of non-paramric and paramric approachs (for xampl Gagliardini al., ). In hs modls, probabiliis P and Q T ar usually linkd by a paramric (forward) sochasic discoun facor dq T /dp, which can b simad from hisorical rurn informaion and allows o uniquly rcovr P probabiliis from Q T probabiliis using his addiional informaion. Mor rcnly, som auhors hav drivd rcovry horms basd on a diffrn s of nonparamric assumpions. In a compl mark sing wih Markovian and saionary sa dynamics on a (boundd) fini sa spac, oss (5) uniquly rcovrs a pah-indpndn sochasic discoun facor dq T /dp using Prron Frobnius hory. Borovička al. (4) discuss in mor dail h implicaions of pah-indpndnc and mphasiz ha in gnral only a misspcifid probabiliy masur can b rcovrd, diffrn from h physical probabiliy, which incorporas long-run risk adjusmns. Hnc, h physical probabiliy rmains unidnifid wihou inroducing addiional rsricions or using addiional daa. W propos o idnify h main characrisics of h physical probabiliy P wih a concpually diffrn approach from h on adopd in xising rcovry horms. Wihou making sringn assumpions abou h undrlying conomy or pric procsss, w sar from a s of plausibl conomic assumpions on h sign of h risk prmia for rading paricular nonlinar risks in opion marks. Whil horically h forward quiy prmium, h firs condiional P momn of forward mark rurns, nds o b posiiv in quilibrium, addiional naural assumpions can b moivad for h risk prmia on highr momns. For insanc, i is widly rcognizd in h horical and mpirical liraur ha h pric of mark varianc risk, and mor gnrally vn mark divrgnc risk, is ngaiv; s Carr and

3 Wu (9), Marin (3) and Schnidr and Trojani (4), among ohrs. Similarly, h risk prmium for xposur o odd mark divrgnc risk, such as skwnss risk, is naurally posiiv, bcaus i is gnrad by risks ha ar monoonic ransformaions of mark rurns; s Kozhan al. (3) and Schnidr and Trojani (4), among ohrs. A sign rsricion on an ass risk prmium is a consrain on h covarianc bwn h pricing krnl and h rurn of ha paricular ass. Givn a s of obsrvd prics of suiabl opion porfolios and a modl-fr no-arbirag condiion, w show ha his rsricion implis usful modlfr consrains on h physical condiional momns of mark rurns. In his way, w obain a family of modl-fr uppr and lowr bounds on diffrn condiional momns of mark rurns, which xnd h lowr bound in Marin (5) for h mark quiy prmium. W show ha hs bounds ar highly im-varying, rflcing a rich condiional disribuion of mark rurns, and ha hy imply rlaivly igh inrvals for h unknown physical momns of mark rurns. Our modl-fr bounds on h physical condiional momns ffcivly consrain h s of physical probabiliis ha in abirag-fr marks can suppor (i) our conomic risk prmium consrains and (ii) h obsrvd prics of suiabl opion porfolios. W characriz h admissibl momns suppord by a probabiliy masur saisfying condiions (i), (ii), using known rsuls on h soluion of (runcad) momn problms. In his way, w obain a parsimomious dscripion of h family of physical momns for which a modl-fr rcovry rsul can b moivad. W obain rcovry basd on a modl-fr L projcion of h pricing krnl on mark rurns. This projcion is paramrizd by forwardnural and physical momns alon. Thrfor, any paramrizaion consisn wih our physical momn consrains and wih h prics of suiabl opion porfolios dfins an admissibl physical masur P in our incompl mark sing. W focus on modl-fr rcovry of h minimal varianc Momn problms (runcad momn problms) dal wih h qusion of whhr for a givn counabl (fini) squnc of numbrs hr xiss a probabiliy masur having hs numbrs as is momns. As h numbr of consraind physical and forward-nural momns gos o infiniy, 3

4 physical masur, which implis h ighs uppr bound on h Sharp raio of any porfolio of ass rurns ha ar xacly pricd by h projcion. Using our conomically moivad modl-fr rcovry, w avoid a numbr of srong chnical assumpions on h undrlying conomy, which migh b difficul o moiva or o s in pracic. For insanc, w do no nd srong assumpions on h conomy sa spac or h undrlying sochasic procsss, such as h Markovianiy or h saionariy of ass rurns, nor do w nd o assum pah-indpndn pricing krnls. W can assum a wak modl-fr dfiniion of arbirag opporuniis o invok modl-fr vrsions of h fundamnal horm of ass pricing (Acciaio al., 3) and nsur xisnc of a forward-nural masur in our sing. For h xisnc of our L pricing krnl projcion, w nd h xisnc of all polynomial momns of mark rurns, which is nsurd, for xampl, if h sa spac of mark rurns is condiionally boundd. Bounddnss of h sa spac is assumd in virually all rcovry horms in h liraur. Morovr, from our ramn basd on runcad momn problms, h rcovrd physical probabiliy in our incompl mark sing can indd b nsurd o hav boundd suppor. Clarly, h cos of h gnraliy of our approach in rms of wak chnical condiions ariss from h conomic assumpions abou h risk prmia of paricular ass rurns, which migh howvr by asir o moiva and s in som cass. W find ha h condiional momns implid by our rcovrd pricing krnl projcions ar highly im-varying and informaiv abou fuur mark ralizaions, spcially for longr horizons. Our chnology also naurally rcovrs scond condiional momns of nonlinar minimum varianc pricing krnl projcions, which xnd h linar projcion approach in Hansn and Jagannahan (997). W documn larg Sharp raios from opion sragis rading h diffrn momns of h pricing krnl projcion. Th condiional projcions hmslvs suggs frqun dparurs from monooniciy and convxiy. Prcisly, highr-ordr projcions frqunly xhibi a u-shap a shor mauriis of monh, in lin wih Bar and Schmid our L projcion paramrizaion convrgs o h physical xpcaion of h pricing krnl condiional on forward mark rurns. 4

5 (4) and Bakshi al. (), implying an avrag uncondiional projcion ha is concav (convx) in rgions of low (larg) rurns. For longr mauriis, h avrag uncondiional projcion is concav vrywhr. Our papr borrows from svral srands in h liraur. Chapman (997) invsigas consumpion-basd ass pricing modls uncondiionally using chnology similar o ours. Aï-Sahalia and Lo (998) sima h sa pric dnsiy (h produc of h pricing krnl projcion and h physical probabiliy masur) using krnl rgrssion. Thir approach rquirs a choic of rgrssors and uss informaion from h nir sampl hisory of opion prics. Song and Xiu (4) xploi also h informaion conaind in VIX opions. Jackwrh () invsigas h marginal ra of subsiuion in a compl mark. Aï-Sahalia and Duar (3) and Birk and Pilz (9) dvlop a polynomial krnl projcion which mainains convxiy of opion prics in srik. Aï-Sahalia and Lo () larn abou h marginal ra of subsiuion from im sris daa and opion prics. Thr is also a rlad liraur on inqualiis for momns of ass prics and h pricing krnl. Alvarz and Jrmann (5) dvlop bounds on h pricing krnl along wih a dcomposiion undr h mainaind assumpion ha i is a saionary sochasic procss. Hansn and Schinkman (9) obain a similar dcomposiion undr an addiional Markov assumpion. Marin (5) drivs a lowr bound of h quiy prmium from a ngaiv covarianc condiion, a join rsricion on h pricing krnl and h mark rurn. Julliard and Ghosh () dvlop an mpirical liklihood simaor of a consumpion-basd pricing krnl. This approach is xndd in Gosh al. (3) along wih nropy bounds for h pricing krnl. Wihin h sam mhodological framwork, Almida and Garcia (5) compu a family of discrpancy bounds for pricing krnls. Carr and Yu () rad in h discr sa assumpion in oss (5) mniond abov for h family of boundd diffusion procsss. Borovička al. (4) show ha oss (5) rcovry rvals h physical condiional dnsiy only undr addiional chnical condiions. Chrisnsn (4) provids a gnral conomric Markovian framwork o larn abou h ransiory and prmann componns of h pricing krnl. Linzky and Qin (5a) xnd 5

6 h Markovian framwork in oss and Carr and Yu () o gnral unboundd sa spacs. Linzky and Qin (5b) inroduc xnsions of h Alvarz and Jrmann (5) and Hansn and Schinkman (9) facorizaions in a gnral smi-maringal sing, wihou imposing Markovianiy or im-discrnss on h sa variabls. This papr firs dvlops ass pricing bounds on momns of h S&P 5 in Scion. Subsqunly i pus hs bounds o us o paramriz pricing krnl projcions in Scion 3. An mpirical sudy uss hs projcions in an mpirical sudy in Scion 4. Scion 5 concluds and h Appndix conains addiional compuaions B, figurs and abls in Scion C. Bounds on Condiional Polynomial Momns of h Mark In his Scion w dvlop uppr and lowr bounds on polynomial momns undr h ru, unobsrvd physical P masur of gross forward rurns := F T,T F,T D +, whr F,T is h forward pric a im of h spo S&P 5 for dlivry a im T. W rfr o as h gross mark rurn. Undr no-arbirag h ru and unobsrvd forward pricing krnl and is xpcaion condiional on a im-t masurabl random variabl ar dnod by M P = dq T dp, and M P() := E P M P, () whr Q T dnos h forward maringal masur associad wih h zro coupon bond numrair. W inroduc his noaion anicipaing our focus on P, and ha a rprsnaiv Q T of forward-nural masurs can b infrrd from opion prics. This papr is basd on modl-indpndn argumns in h sns ha w do no assum an undrlying sochasic procss for. In our conx i is hrfor insruciv o ak D o b a compac subs of +, as his nsurs, oghr wih sufficinly many opions wrin on, a modl-fr fundamnal horm of ass pricing (Acciaio al., 3, 6

7 mark.4). W nx inroduc noaion ha will hlp us in h conx of financial marks quippd wih Europan opions. Th im-dpndn opraor J aks a funcion f, wic diffrniabl almos vrywhr, and approximas i in a pic-wis linar fashion insid a crain corridor, and linarizs h funcion ousid of h corridor 3 b J f() := f (K)(K ) + dk + f (K)( K) + dk a f(a ) + f (a )( a ) < a = f() a b f(b ) + f (b )( b ) > b. No ha J =. W inroduc his linarizaion opraor o us h spanning rsuls from Carr and Madan () in opion marks wih limid monynss. Thr is a sric ordring bwn J f() and f() which hlps rlaing unobsrvd forward-nural xpcaions of f() o obsrvd forward-nural xpcaions of J f(). Lmma. (Obsrvd and Thorical Momns). Th diffrnc f() J f() is posiiv (ngaiv) for f sricly convx (concav). Imporanly, opraor J is curvaur-prsrving in h sns ha if f is convx (concav) also J f() is convx (concav). W will in h mpirical scion work wih a sa spac D = a ɛ l,, b + ɛ u, for som ɛ l,, ɛ u, >, no oo big, in which cas J f() f(). I is mping o assum ha h sa spac of agrs xacly wih h obsrvd opion monynss, bu 3 In pracic w will compu Eq. from a fini ou-of-h-mony opion porfolio wih sriks a = K < K < < K n = F,T < K n+ < < K N = b as wih n N f (K i )(K i F T,T ) + K i + f (K i )(F T,T K i ) + K i i= K := K K, K i := /(K i+ K i ) for < i < N, and K N := K N K N. i=n 7

8 hn h pric of h farhs ou-of-h-mony pu and call opions would nd o b zro. Our mhodology is basd on momns. To nsur ha wha w do is wlldfind w nd momns of all ordrs. Dno by P h im--condiional physical probabiliy masur (and likwis Q,T h T -forward masur condiional on im informaion) gnraing h condiional xpcaions E P, rspcivly E Q T. Lmma. (Exisnc of Momns). Th momn-gnraing funcions E P u, and E Q T u xis for u +. Th compacnss of h sa spac D guarans ingrabiliy vn for fa-aild disribuions. 4 A compuaion shows ha Lmma. also guarans xisnc of momns of discr rurns :=, and corridor momns J n. Dno h condiional monomial momns by µ P,n := E P n and µ Q T,n := E Q T n, rspcivly. Th nx assumpion is on xpcd profis of rading sragis wih xposur o nonlinar funcions of. Assumpion (Ngaiv Divrgnc Prmium (NDP)). Wih powr divrgnc funcion D p () := p p + p, p p D () := log() +, and D () := log(), (4) w dfin h ngaiv n-powr divrgnc prmium NDP(p,n) assumpion of ordrs p and n as h inqualiy Cov P M, J D p ( n ) = Cov P M(), J D p ( n ). (5) 4 Sandard modls on unboundd sa spacs such as Black-Schols or Hson (993) do no saisfy h rquirmn of a compac sa spac ncssary for Lmma., bu hy ar rlaivly asy o compacify. W us hm in Figur 4 o illusra modl liklihood raios. 8

9 Powr divrgnc swaps inroducd by Schnidr and Trojani (5) paying h diffrnc bwn ralizd and implid divrgnc 5 can b rplicad from opion daa. Toghr wih h idniy D p ( n ) = n p (np )D pn() (n )D n (), wih D ( n n ) := lim p p (np )D pn() (n )D n () = n n log() n +, (6) Assumpion is hrfor mpirically sabl uncondiionally, sinc boh D pn () and D n () ar radabl quaniis. Jnsn s inqualiy w dircly g From Assumpion and Proposiion.3 (Uppr Bounds on Condiional P Momns). N DP (p, n) holds if and only if S(J D p ( n )) := E Q T J D p ( n ) E P J D p ( n ) J D p (E P n ). (7) From Proposiion.3 abov w dfin implicily h uppr bound on h n-h momn of, µ Puppr,n, as h soluion o S(J D p ( n )) = J D p (µ Puppr,n ). (8) Th NDP from Assumpion sablishs rlaions bwn Q T and P momns of diffrn ordrs by varying p and n and hrfor nails conomic informaion byond Proposiion.3. Th nx assumpion is hardr o s mpirically. Assumpion (Ngaiv Covarianc Condiion (NCC)). For p, q w dfin h ngaiv covarianc condiion N CC(p, q) as h inqualiy Cov P M q, p = Cov P M() q, p = E Q T p+q E Q T q E P p. (9) 5 In hir consrucion ralizd divrgnc dpnds on h pah of h forward pric from im o im T, bu all payoffs aris a im T, such ha hr is no diffrnc bwn pricing wih M P, or M P (). 9

10 To adap Assumpion o fini opion marks w mploy Lmma. and dfin for q (, and p + q > L(p, q) := EQ T J p+q E Q T J q EQ T p+q E Q T q E P p. () This inqualiy is no binding, whnvr h quiy prmium is assumd posiiv, for q. In conras, i provids informaiv bounds for q >. In h mpirical invsigaion w mploy L(, q), for which hr is ampl vidnc for is validiy from a bary of conomic modls and mpirical ss. 6 A wakr lowr bound, assuming only h NCC for p =, is givn by Proposiion.4 (Lowr Bounds on Condiional P Momns). Suppos NCC(, q) holds for q (,, hn L(, q) p E P p. () Analogously o h dfiniion of h uppr bound, w dfin h lowr bound µ Plowr,n corrsponding o Proposiion.4 basd on L(, ). Whil uppr and lowr bounds on gross rurns of h mark ar inrinsically inrsing, w hav on paricular applicaion in mind in h conx of pricing krnl projcions ha w will labora in h Scion blow. 3 Nonlinar Pricing Krnl Projcions This Scion xplains how h pricing krnl can b mad visibl hrough h momn bounds and opion prics and how his projcion rlas o h xan liraur. Hr w adop h framwork of Filipović al. (3) o xpand h unobsrvabl pricing krnl in rms of h momns of h forwardnural dnsiy Q T and a candida physical masur M. This mhod is 6 Marin (5) mnions for h NCC(, ): ) a joinly log-normal pricing krnl and mark rurn and a Sharp raio on h mark ha is grar han is volailiy, ) a rprsnaiv agn who maximizs xpcd uiliy and whos risk avrsion is a las 3) an Epsin-Zin rprsnaiv agn wih risk avrsion grar han, and lasiciy of inr-mporal subsiuion grar han. From Schmid (3) hs rsuls carry ovr for -masurabl pricing krnls also o NCC(p, ), p >.

11 prfrabl ovr Taylor xpansions in h conx of approximaing liklihood raios for a numbr of rasons. Firsly, conomic rsricions such as pricing consrains ar auomaically buil in. Scondly, h projcion is by consrucion a M maringal. Thirdly, i accommodas asily conomis wih pricing krnls ha ar no masurabl wih rspc o h ass h pricing krnl is projcd on. W work hr wih polynomial projcions, alhough ohr bass ar hinkabl and mayb vn prfrabl for som applicaions. For h purpos of using h projcion horm w inroduc h noion of squar ingrabiliy. Dfiniion 3.. Dfin h wighd Hilbr spac L M as h s of (quivalnc classs of) masurabl ral-valud funcions f on wih fini L M -norm f L = M Accordingly, h scalar produc on L M is dnod by f, h L M = f(ξ) dm(ξ) <. f(ξ) h(ξ) dm(ξ). (3) Assumpion 3 (Fini Pricing Krnl Varianc). For givn probabiliy masur M M M L M. (4) Assumpion 3 sas ha h varianc of h pricing krnl is fini, a common assumpion ncssary for xampl for h xisnc of Hansn- Jagannahan and good dal bounds (Cochran and Saa-qujo, ; Črný, 3). Wih Assumpions. and 3 in plac w can rprsn h condiional xpcaion of h pricing krnl as an infini sris M M () = M ( ) M (), whr h quals sign is o b inrprd in an L sns. 7 wih M (J) J M () := + c i H i (). (5) 7 In paricular on compac sa spacs h convrgnc may vn b poinwis and uniform. i=

12 and c i and H i dpnd on h polynomial momns µ M,,..., µ M,i, and µ Q T,,..., µ Q T,i. Appndix B dscribs how c i and H i can b compud for a gnric probabiliy masur M, givn Q T. Appndix B. works ou h funcional form for J =,. Thr ar imporan addiional facs abou h polynomial rprsnaion worh mnioning. Firsly, for ach J, h krnl projcion is h bs approximaion o M in rms of polynomials in a las-squars sns and can b undrsood inuiivly as a linar rgrssion on powrs M (J) M of using h condiional probabiliy masur. Scondly, as mniond alrady abov, for vry J, M (J) M () is a M maringal. If in addiion () > M-almos surly, i is a valid pricing krnl dspi is polynomial form. This can b asily chckd givn h cofficins of h xpansion. Thirdly, by consrucion M (J) M () prics powrs of prfcly up o ordr J: E M M (J) M () n = E Q T n, n =,..., J, such ha pricing rsricions ar auomaically incorporad, whr condiional momns such as E Q T n can b compud from opion prics, so ha w can ak hm as givn. Finally, h funcional form (5) rvals ha for ach ordr J, h sris approximas h ru unobsrvd funcional form of h condiional xpcaion conforming wih is vry dfiniion in paricular for changs of masur which ar no masurabl wih rspc o. W show in Figur 4a for h Black-Schols pricing krnl how an ordr J = liklihood xpansion dvias from a scond-ordr Taylor sris xpansion. Th liklihood xpansion nsurs for ach J ha h projcion ingras o on, and ha h firs J momns of ar prfcly pricd. As such i likly convrgs slowr han a Taylor sris xpansion. A h sam im h conomic faurs of h L M xpansion mor han compnsa for h slowr convrgnc. discuss hs issus in mor dail in Scion 4.4 blow. In Figur 4b i can b sn ha h xpansion works xrmly wll alrady for low ordrs also in h prsnc of sochasic volailiy. Th abiliy o dvlop an orhonormal basis of ordr J wih rspc o M wihou knowing xplicily M and Q,T, jus in rms of h momns µ M,,..., µ M,J and µq T,,..., µ Q T,J, pus us in h posiion o us our ass pricing bounds in dvloping h projcion. Wih µ Q T,,..., µ Q T,J fixd from opion W

13 prics, any projcion wih probabiliy masur M such ha h momns saisfy µ Plowr,i µ M i µ Puppr,i, (6) is a viabl candida for P. To nsur ha hr xiss such a masur w can us 8 Proposiion 3. (Truncad Hausdorff Problm). (Curo and Fialko, 99, Thorm 4.3) Givn a squnc µ i, i =,..., n, µ i > h following wo ar quivaln. hr xiss a probabiliy masur M suppord on a, b such ha µ i = µ M i = E M i. h Hankl marix G n (µ) := µ µ µ n µ µ µ n+.. (7) µ n µ n µ n is posiiv, and hr xiss µ n+ such ha µ µ µ n+ µ H n (µ) := µ 3 µ n+.. (8) µ n µ n+ µ n+ is posiiv and hy joinly saisfy ag n (µ) H n (µ) bg n (µ). 8 In h original papr (Curo and Fialko, 99) hr is a vrsion of h horm which opras on posiiv smi-dfininss of G n (µ) and H n (µ), which nails also addiional uniqunss rsuls for h masurs. In our sing h rquird xac valus of a and b ar unknown. In ligh of his, numrical imprcisions, and h propry ha drminans of G n (µ) appar in h dnominaor of h xpansion (5), w sa a slimmd down vrsion of h rsul for h non-singular cas. Furhrmor, sinc w ar opraing in an incompl mark, h forward-nural masur is no uniqu and hrfor h pricing krnl would b non-uniqu vn in h singular cas. 3

14 Th abov proposiion is imporan in idnifying whhr a s of givn momns is admissibl in h sns ha a probabiliy masur could hav gnrad i. For a givn im dfin h s of fasibl probabiliy masurs { M (J) := M µ Plowr,i µ M i µ Puppr,i ; i =,..., J, } < (a ɛ l, )G J (µ M ) H J (µ M ) (b + ɛ h, )G J (µ M ). (9) Th dpndnc of an ordr J xpansion on up o h J-h momn ariss hrough h normalizaion, ncssary o hav h pricing krnl projcion ingra o on (Scion B). To idnify on s of momns w finally solv min E M µ M,...,µM J M (J) M (), () subjc o M M (J), and Cov M M (J) M (), J D / ( i ), for i =,..., J. Th inqualiy consrains corrspond o Assumpions and, rspcivly, and ncod h dsir o impos h sam conomic mchanism ha has gnrad h bounds also on h projcion. W will dno h momns ha solv his opimizaion problm by µ P(J),..., µ P(J) J, whr P (J) M (J) dscribs h s of fasibl condiional probabiliy masurs a im and w hav by h dfiniion of h condiional xpcaion ha P ( ) = P. I is imporan o no ha in absnc of h ass pricing bounds in Scion h soluion o h minimizaion program () is rivially P (J) = Q T, and ha a non-rivial soluion ariss solly hrough a combinaion of, boh, uppr, and lowr bounds. Th variancs of h pricing krnl projcion ar incrasing in J as h following rsul shows. Lmma 3.3 (Varianc of Projcion). For any J, J N J > J V M By consrucion w hav E P M (J ) M () V M M (J) M (). ( ) M (J) M () M M() = for vry J. 4

15 4 Empirical covry W bas our mpirical sudis on a panl of S&P 5 Europan opions in h sampl priod from January 99 o January 4. Opion daa ar from MarkDaaExprss, a vndor conncd o h CBOE. Th daa s includs closing bid and ask quos for ach opion conrac from which w compu mid prics, along wih h corrsponding srik pric. From h daa w filr ou all nris wih non-sandard slmns and hos which viola basic no-arbirag condiions. Opions maur on h hird Friday ach monh, and w us his mauriy on a monhly im grid o compu forward prics of divrgncs. For h sam im grid w also us opion panls for on quarr of a yar, on half, and a full yar. Ths longr-mauriy panls us inrpolaion. W rplica im sris of forward prics on h S&P 5 spo indx using pu call pariy, consisnly wih h procdur in h CBOE (9) whi papr for h compuaion of h VIX implid volailiy indx. In h following Scion w dscrib h bhaviour and propris of h ass pricing bounds implid by h daa and Assumpions and. 4. alizd Condiional Ass Pricing Bounds In his Scion w discuss h mpirical implmnaion and propris of h bounds dvlopd in Scion abov. W sar by chcking mpirically h validiy of Assumpion uncondiionally. 9 Unforunaly hr is no condiional modl-fr s, bu w can rly on h fac ha for a random variabl ha is posiiv almos surly (h condiional xpcaion in qusion), also is uncondiional xpcaion will b posiiv. A sampl avrag ha suppors h assumpions wih high probabiliy uncondiionally is hrfor vidnc ha h assumpions may also hold condiionally. For his purpos w compu avrags of xcss rurns on ralizd divrgnc using dcomposiion (6). Figur shows summary saisics of his dcomposiion. In h firs Panl a w s ha uncondiionally i is profiabl o sll divrgnc swaps uniformly across diffrn p. This rsul is known alrady from Schni- 9 For Assumpion w rly on h ss prformd in Marin (5). 5

16 dr and Trojani (5) along wih h fac ha condiionally, xcss rurns from diffrn divrgnc swaps co-mov (Panl b). Panls c and d suggs ha Assumpion appars rasonabl. Indpndnly of h convxiy paramr p, divrgnc swaps on powrs of h S&P 5 los mony. Wih high probabiliy h uncondiional prmium is ngaiv, making i concivabl ha his is valid also for h condiional on. Marin (5) provids mpirical and horical vidnc for h validiy of h N CC(, ). Upon accping h mpirical validaion of h horical assumpions w can now mov on o hir implicaions. Figur a indicas how h choic of paramr p drmins h ighnss of h uppr bound considrably. In paricular a crisis das, Panls b, c, and d show diffrncs of up o 5% bwn bounds of diffrn srngh. Ovr im h bounds of various srngh ar im-varying and hy nvr ovrlap (Figur 3a). A similar bhavior can b sn also for h lowr bounds in panl 3b. Thr is a im-consisn ranking bwn h bounds drmind by h paramr q. This consisncy is also bwn uppr and lowr bounds. Puing lowr and uppr bounds oghr Figur 3c shows a band in which h quiy prmium can b lockd in. Thr is considrabl im variaion in h diffrnc bwn uppr and lowr bounds (Figur 3d). Ou of all availabl paramrizaions w coninu our analysis using p = / for h uppr bounds xclusivly, and q = for h lowr bounds in conncion wih Proposiion.4, rspcivly, for simpliciy, o conform wih h prvailing liraur, and in ligh of h applicaions o com. In h nx Scion w mak us of h ass pricing bounds dscribd abov o guid our sarch for a family of condiional probabiliy masurs. 4. An (Almos) Modl-Fr Projcion For vry hird Friday ach monh from January 99 o January 4 w solv h opimizaion program () for J =,, 3 for h mauriis, 3, 6, and monhs. This ask may numrically bcom challnging in paricular If h assumpions do no sm rasonabl on a givn day, h rsarchr has h frdom o rfrain from using h bounds, or mploy hm wih vn mor consrvaiv paramrs. 6

17 for highr-ordr xpansions in ha h Hankl marix (7) may only b barly posiiv, wih h compuaion bing subjc o numrical imprcisions hrough adding vry larg o vry small numbrs. Th objciv funcion is highly nonlinar in h momns, and wih h addiional burdn of highly nonlinar consrains w minimiz using h Baysian MCMC mhod from Chrnozhukov and Hong (3). Thr ar crain das for which hr is no fasibl paramr consllaion, for insanc whn h Hankl marix of h Q T momns is no posiiv, or h consrains lav only an mpy s, in which cas w skip h projcion for ha day and do no rcord implid P (J) momns. W prform h minimizaion in rms of gross momns. Afr obaining µ P(J) w convr hm ino simply compoundd momns µ P(J) by adding and subracing h gross momns according o h binomial formula. W obain gross momns sparaly for ach ordr J. Gross momns ar numrically vry clos for ach J, suggsing ha µ P() could b rusd for h compuaion of µ P and so forh, bu h consrains ar oo sringn oo allow for his rcursiv program. Figurs 5, 6, 7, and 8 show h gross momns ha solv program () ovr im for mauriis, 3, 6, and monhs, rspcivly. Thy show ha h implid µ P(J), momns ar vry clos o h lowr bounds, suggsing ha h lowr bound for h quiy prmium is xrmly informaiv abou h conomy. Highr momns ar locad wll wihin h band for all mauriis. A naural qusion ha ariss wih a s of condiional momns undr h physical masur a hand is whhr i is possibl o prdic ralizaions. Wih no modl o b simad, any prdiciv rgrssion will b ou-ofsampl. Tabl shows corrsponding compud according o Campbll and Thompson (8) agains h sampl man. I shows ha hr is lil vidnc for prdicabiliy across mauriis wih h xcpion of h firs momn for all ordrs (J =,, 3). On h longr mauriis, howvr, hr ar som rmarkabl improvmns ovr h sampl avrag. No only do h implid P (J) momns prdic firs momns (bwn % and 3%). For h firs hr momns a h monh horizon, hr is a vry pronouncd parn ha h P (J) momns ouprform h sampl avrag by a larg In h numrical implmnaion w us h summaion algorihm from Nal (5). 7

18 margin. Th nx naural applicaion ar risk prmia in hir ru sns of xpcd profis of rading sragis. Figurs 9 hrough show im sris of risk prmia on highr-ordr momns. Thr is significan im variaion for all mauriis and all momns. Prmia on scond momns ar prdominanly ngaiv wih fw xcpions. Prmia on hird momns ak boh signs, wih a posiiv avrag. From momn 4 on, his parn swichs such ha prmia on vn momns ar posiiv, and ngaiv for odd momns. Alhough h prmia implid by J =,, 3 clarly co-mov, hr ar significan diffrncs. Th condiional scond momn implid by J =, for insanc, is almos as big as h implid on, bu his is no surprising, sinc h linar projcion dos no pric scond momns prfcly lik projcions wih J. In h nx Scion w discuss in grar dail h conomics of h projcion of h pricing krnl. 4.3 covring and Trading h Shap of isk Th P (J) condiional momns ar soluion of opimizaion program (), wih h scond condiional momn of h pricing krnl projcion as objciv. Solving his program vry day hrfor rvals a im sris of minimum varianc pricing krnl projcions. 3 Kping in mind ha h limi of h projcion M (J) () as J aains h ru condiional xpcaion, w can xpc ha h projcion wih ordr J = 3 will b closr o M P () han h linar on. Collcing cofficins in of h runcad xpansion (5) compud a h opimal P (J) momns givs M (J) P () = J i= b (J),i i. () Th cofficins a carry a suprscrip J, bcaus h polynomials H i in sris (5) ar of ordr i, which mans ha hy also dpnd on monomials of ordr Prmia on scond momns ar diffrn from divrgnc prmia, which conain an addiional dla hdging rm. 3 Th xpcd valu of h projcion is by consrucion uniy. 8

19 j i. This in urn mans ha b (J),i will in gnral b diffrn from b (K),i if and b (J+),i indicas convrgnc, K J. In fac, a small chang bwn b (J),i sinc conribuions from highr-ordr momns do no chang h projcion any mor. Whil i is also o b xpcd ha Cov P ( n, m ) > for any n, m, h cofficins b (J),i of h xpansion can ak boh ngaiv and posiiv signs. W invsiga his in grar dail in Scion 4.4 blow. From Proposiion 3.3 w know howvr a-priori ha h condiional scond momn of h projcion will b incrasing in J. 4 W can also asily convr from momns of gross rurns o momns of discr rurns := by using h binomial formula, rcompuing h basis and r-collcing h cofficins M (J) P ( ) = J i= a (J),i i. (3) Figur 3 rvals h corrsponding sampl pahs, and suggss ha mpirically h varianc is indd grar for highr-ordr xpansions. Whil for monh mauriy h scond momns ar almos idnical, h diffrncs widn subsanially wih mauriy, howvr. Th rajcoris of h linar projcion in Panl 3a, 3b, 3c, and 3d corrspond o im-varying Hansn-Jagannahan bounds for mulipl mauriis, whil h ons for h highr-ordr projcions corrspond o nonlinar xnsions which ar likly closr o h ru, unobsrvd projcion. From Schnidr (5) w know ha hr is also a radabl aspc of his. Sinc w can rplica hrough opions conracs on i, h sam is possibl also for wighd sums hrof. Th xpcd profi of h projcion is minus h varianc of h pricing krnl. Tabl givs h corrsponding sampl avrags, which corrspond roughly o h minus h krnl projcion scond momns in Figur 3 plus. 4 This rsul holds if h firs (J ) momns of h ordr J projcion ar qual o h ons of h ordr J projcion. This is no h cas in gnral, bu h momns of h diffrn-ordr projcions ar sufficinly clos oghr o jusify a rough comparison. 9

20 4.4 isk Avrsion In his Scion w compar our projcion o xan xpcd uiliy modls. Taking rcours o a rprsnaiv agn conomy, w can wri a pricing krnl in rms of h walh of h agn (s, for xampl Garcia al., 9) as M EU = U (W + ) U (W ), (4) whr EU dnos xpcd uiliy and U a Von-Numann Morgnsrn concav uiliy funcion. An xpansion of h abov around W and dnoing by EU := W + W w hav for insanc o ordr M EU = + W U (W ) U (W ) EU + W U (W ) U (W ) ( EU ) ( ) + O EU 3. (5) U From h concaviy of U i follows ha W (W ) U (W ) < and W U (W ) U (W ) >. Th formr sign is associad wih avrsion o uncrainy, whil h lar is associad wih prudnc of h rprsnaiv agn and h signs coninu alrnaing in h sam mannr for highr-ordr rms. Bfor urning o comparing h cofficins of our projcion o h cofficins of h Taylor xpansion abov w gaug signs and magniuds from wll-known noarbirag modls. Indd Black-Schols modl for h sam paramrizaions as in Figur 4a xhibis hs signs, ngaiv for odd ordrs, and posiiv for vn. Hson s modl wih h sam paramrs ha gnrad 4b on h ohr hand faurs a ngaiv cofficin in an ordr four xpansion on h cofficin on. I is hrfor concivabl ha signs and magniuds may b subjc o addiional risk facors. In h cas of Hson s modl i is sochasic volailiy (s also h discussion in Chabi-Yo al., 8; Hansn and naul, ). Figurs 4 o 7 conain h cofficins of h xpansion (3) for mauriis monh o monhs. Th consan a is clos o on for all ordrs and mauriis. Th cofficin a on is pronouncdly ngaiv, consisn wih h noion of firs ordr risk avrsion. Wih h cofficin a (prudnc) on coms a dparur from h xpcd uiliy framwork wih

21 -masurabl pricing krnls. I is clos o zro, bu aks on larg posiiv and ngaiv valus ovr im. Wih an incras in mauriy i bcoms ngaiv on avrag and lss variabl. Wih h cofficin a 3 a similar parn volvs, albi much lss pronouncd, aking ngaiv and posiiv valus on avrag clos o zro for all mauriis. Wih a lo of im variaion in h cofficins, Figurs 8 o b show a rich variy of diffrn shaps of h pricing krnl ovr im. Th quadraic projcion on avrag suggss a concav shap (Figur ). This siuaion changs onc on considrs a cubic projcion, which is concav for ngaiv and convx for posiiv rurns a monh mauriy, whil i appars o b globally concav for longr mauriis. 5 Conclusion This papr dvlops a s of condiional lowr and uppr bounds for xpcd powrs of gross mark rurns from an assumpion abou h sign of xpcd payoffs from crain conracs. Th assumpions can b mpirically sd and ar valid a las on avrag. Th bounds drmin a s of possibl condiional physical probabiliy masurs, h siz of which w furhr rduc by picking a family of momns which minimizs h scond momn of a projcion of h pricing krnl analogously o h noion of minimum varianc maringal masurs in incompl marks. Th rsuling condiional momns rcovr familis of physical probabiliy masurs solly on h basis of h conomic assumpions gnraing h bounds, bu do no mak assumpions abou Markovianiy, saionariy, or h yp of sochasic procss driving h conomy. Thy yild a numbr of novl conomic insighs. Th lowr bounds on h quiy prmium ar xrmly informaiv for all highrordr physical momns and hnc h physical condiional disribuion. In ligh of h consisncy condiions bwn momns of diffrn ordrs his mans ha h lowr bound for h firs momn, h quiy prmium, is of paramoun imporanc. Uppr bounds sablish rlaions bwn physical and forward-nural momns joinly and pu addiional srucur on h pricing krnl. Th rsuling condiional momns prdic ralizaions ou-of-sampl. Th radiional Hansn-Jagannahan bound basd on a lin-

22 ar projcion is sharpnd by accommodaing nonlinariis and rvals a im-varying ass pricing bound along wih a corrsponding opimal Sharp raio rading sragy. Th sam nonlinar projcion on h mark suggss a dparur from convxiy ofn conjcurd by xpcd uiliy hory in paricular for longr mauriis, wih is scond drivaiv bing concav for ngaiv, and convx for posiiv xcss rurns for shorr mauriis. Appndix A Proofs A. Proof of Lmma. Proof. From dfiniion w wri for h firs claim f() J f() = ( < a )(f() f(a ) f (a )( a )) + ( > b )(f() f(b ) f (b )( b )) > if f () > = < if f () <. A. Proof of Lmma. Proof. W wri E P u = D uξ dp(ξ) < sup u dp(ξ) = sup u < D D D (A.) from h compacnss of D. Th sam argumn oghr wih h xisnc of dq T /dp from no-arbirag shows xisnc of E Q T u.

23 A.3 Proof of Proposiion.4 Proof. Th N CC(, q) and momn monooniciy giv us h inqualiis E Q T +q E Q T q E P E P p /p (A.) for p. Employing Lmma. o h numraor and dnominaor of h lfmos xprssion wih q (,, and subsqunly aking h lfmos and righmos xprssions o h powr of p and yilds h dsird rsul. A.4 Proof of Lmma 3.3 Proof. From orhonormaliy and h propry ha E M any J w hav V M M (J) M () = E M = ( + M (J) M () = for J c i H i ()) (A.3) i= J c i H i L = M i= J c i + i= J i=j+ J i= c i = V M c i M (J ) M () (A.4) (A.5) B Orhonormal Polynomial Bass To compu h basis w can mploy h Gram-Schmid procss rviwd blow. B. Orhonormal basis wih rspc o M Th orhonormal polynomials H from (5) can b compud from 3

24 Algorihm B. (Gram-Schmid Procss). H (x) =, i H i (x) = x i H i (x) = j= H i (x) H. i (ξ)dm(ξ) ξ i H j (ξ)dm(ξ) H j (x), (B.6) (B.7) (B.8) Th cofficins in h xpansion (5) ar hn obaind hrough h formula c i = E Q T H i (x), (B.9) which is rplicad from opion daa. Dno by P,T (K) (C,T (K)) a Europan pu (call) pric a im wih mauriy T and srik pric K on h undrlying mark. For wic-diffrniabl f a.., for Eq. polynomial, w hav from Carr and Madan () (B.9) f is E Q T f(f T,T ) = f(f,t ) + + p,t F,T f (K)C,T (K)dK ( F,T f (K)P,T (K)dK ). (B.) B. Polynomial Projcions and Thir Scond Momns For givn Q T and M linar and quadraic projcions sill hav managabl siz. In h linar cas w hav M () = µm,µ QT, µ M, (µ M,) µ M, M () + µm, µ Q T, (µ M,) µ M, (B.) and E M M () M () In h quadraic cas w hav = µm,µ Q T, + µ M, + (µ Q T (µ M,) µ M, 4, ). (B.)

25 M () = µm,3(µ M,µ QT, + µ M,µ Q T, ) + µ M,µ M,4µ Q T, + (µ M,) µ Q T, µ M,µ M,4 + (µ M,3) (µ M,) µ M,4 µ M,(µ M,µ M,3 + µ M,4) + (µ M,) 3 + (µ M,3) M and E M + µm,(µ M,µ Q T, + µ M,3) + µ M,4(µ M, µ Q T, ) + (µ M,) µ Q T, + µ M,3µ Q T, (µ M,) µ M,4 µ M,(µ M,µ M,3 + µ M,4) + (µ M,) 3 + (µ M,3) + (µm,) µ Q T, µ M,(µ M,µ Q T, + µ Q T, ) µ M,µ M,3 + (µ M,) + µ M,3µ Q T, (µ M,) µ M,4 µ M,(µ M,µ M,3 + µ M,4) + (µ M,) 3 + (µ M,3) M M () = (µ M,) µ M,4 µ M,(µ M,µ M,3 + µ M,4) + (µ M,) 3 + (µ M,3) ( (µ M,) (µ Q T, ) µ M,3(µ Q T, (µ M, µ Q T, ) + µ M,µ Q T, ) (B.3) ( µ M, µ M, µ Q T, µ Q T, + µ M,4 + (µ Q T, ) ) + µ M,4µ Q T, (µ M, µ Q T, ) + (µ M,) ( ) ) (µ Q T, ) + µ Q T, + (µ M,3 ). (B.4) 5

26 C Figurs and Tabls % -.9 man 95% (a) D p p=-3 p=- -.3 p= p=4 -.4 (b) D p % -.8 man 95% (c) NDP, p = / % -.8 man 95% (d) NDP, p = / Figur : Tsing h Assumpions. This collcion of figurs shows avrag profis from rading sragis basd on opion daa on h S&P 5 indx from January 99 o January 4. All confidnc bands ar compud from a block boosrap procdur. Panl (a) shows avrag profis from divrgnc swaps wih conrac funcion (4) as a funcion of h convxiy paramr p. Panl (b) shows im sris of condiional payoffs. Panls (c) and (d) s uncondiionally h validiy of h NDP (p) Assumpion. 6

27 p = p = p = 3 Divrgnc.5. p = Fall 8 Winr 998 Winr Dp().4.3 D () µ P, (a) Bounds of Diffrn Srngh µ P, (b) p =.5. p = Fall 8 Winr 998 Winr.5. p = 3 Fall 8 Winr 998 Winr D().5. D3() µ P, µ P, (c) p = (d) p = 3 Figur : Divrgnc Uppr Bounds on h Equiy Prmium. This figur shows a graphical rprsnaion of h uppr bound of h P-xpcd S&P 5 gross rurn according o Proposiion.3. Th firs panl (a) shows h bound from a hypohical S(D p ()) =. for p =,, 3. Panls (b), (c), and (d) show h ralizd uppr bound for h xrm das Nov 8, 7 Ocobr, and 5 Ocobr 998 for p =,, 3, rspcivly. 7

28 p = p =.5 p = p = 3 p = (a) Uppr Bounds q = 3 q = q = q =.5 q =. (b) Lowr Bounds Equiy Prmium Band (c) Admissibl gion.8 (d) Diffrnc Bwn Bounds Figur 3: Divrgnc Uppr and Lowr Bounds on h Equiy Prmium. Th figur shows uppr (a) bounds from Proposiion.3 for n = and diffrn p, and lowr (b) bounds from Proposiion.4 on h S&P 5 quiy prmium for p = and diffrn choics of q. Panl (c) shows h admissibl rgion implid by h lowr bound wih p = and q =, and h uppr bound implid by p = in annualizd prcnag rms. Panl (d) shows h diffrnc bwn h uppr and h lowr bound from panl (c). Th sampl priod rangs from January 99 o January 4 and h daa ar Europan opions on h S&P 5 indx. 8

29 () (4) ru Taylor (a) Black-Schols Modl () (4) ru (b) Hson Modl Figur 4: Modl Liklihood aios. Th figur shows pricing krnl projcions as a funcion of := F T,T F,T for ordrs J =,..., 4 for Hson s modl (Panl (b)), and Black-Schols modl (Panl (a)). Th forward pric in Black-Schols modl solvs h SDE undr h P and Q T masurs df,t F,T = γσ d + σdw P, df,t F,T = σdw Q T, and h corrsponding pricing krnl γ (γ γ)σ (T ) (C.5) for paramrs T = /, γ =, σ =.5. Th Hson Modl is dscribd hrough h soluion of df,t F,T = γv d + V (ρdw P + ρ dw P ), df,t F,T = V (ρdw Q T + ρ dw Q T ), dv = (b + βv ) + ξ V dw P, dv = (b + (β γρξ)v ) + ξ V dw Q T, wih pricing krnl γ (γ γ) T V sds, (C.6) for paramrs T = /, γ =, b =.5, β =, ξ =., and ρ =.8. Th condiional xpcaion dnod by ru in Panl (b) is gnrad by a simulaion xprimn wih 5,, sampl sampl pahs wih hourly discrizaion. Th probabiliy of h simply compoundd rurn o raliz lowr han -3% or highr han % is xrmly low, wih no a singl ralizaion. 9

30 lowr uppr () (a) E P lowr uppr () (b) E P lowr uppr (c) E P 3 lowr uppr (d) E P lowr uppr () E P 5 lowr uppr (f) E P 6 Figur 5: Implid P Gross Momns ( monh mauriy). This figur shows monhly condiional gross momns of h S&P 5 ha solv h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. Gross momns ar compud for ordr J =,, 3 pricing krnl projcions, dnod by (),, and, rspcivly. 3

31 lowr uppr () lowr uppr () (a) E P (b) E P lowr uppr (c) E P 3 3 lowr uppr (d) E P lowr uppr lowr uppr () E P 5 (f) E P 6 Figur 6: Implid P Gross Momns (3 monhs mauriy). This figur shows monhly condiional gross momns of h S&P 5 ha solv h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. Gross momns ar compud for ordr J =,, 3 pricing krnl projcions, dnod by (),, and, rspcivly. 3

32 lowr uppr () (a) E P lowr uppr () (b) E P.6.4. lowr uppr lowr uppr (c) E P 3 (d) E P lowr uppr 8 7 lowr uppr () E P 5 (f) E P 6 Figur 7: Implid P Gross Momns (6 monhs mauriy). This figur shows monhly condiional gross momns of h S&P 5 ha solv h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. Gross momns ar compud for ordr J =,, 3 pricing krnl projcions, dnod by (),, and, rspcivly. 3

33 lowr uppr ()..8 lowr uppr () (a) E P (b) E P lowr uppr (c) E P lowr uppr (d) E P lowr uppr 4 lowr uppr () E P 5 (f) E P 6 Figur 8: Implid P Gross Momns ( monhs mauriy). This figur shows monhly condiional gross momns of h S&P 5 ha solv h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. Gross momns ar compud for ordr J =,, 3 pricing krnl projcions, dnod by (),, and, rspcivly. 33

34 () (a) E P E Q T () -.45 (b) E P E Q T (c) E P 3 E Q T 3 -. (d) E P 4 E Q T () E P 5 E Q T (f) E P 6 E Q T Figur 9: isk Prmia ( monh mauriy). This figur shows risk prmia on powrs of simply compoundd S&P 5 rurns. Thy ar compud from condiional gross momns of h S&P 5 ha solv h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. Gross momns ar compud for ordr J =,, 3 pricing krnl projcions, dnod by (),, and, rspcivly. 34 6

35 .6.4 (). () (a) E P E Q T -. (b) E P E Q T (c) E P 3 E Q T 3 -. (d) E P 4 E Q T () E P 5 E Q T (f) E P 6 E Q T Figur : isk Prmia (3 monhs mauriy). This figur shows risk prmia on powrs of simply compoundd S&P 5 rurns. Thy ar compud from condiional gross momns of h S&P 5 ha solv h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. Gross momns ar compud for ordr J =,, 3 pricing krnl projcions, dnod by (),, and, rspcivly. 35 6

36 .6.4 (). () (a) E P E Q T -.6 (c) E P 3 E Q T (b) E P E Q T -. (d) E P 4 E Q T () E P 5 E Q T 5 -. (f) E P 6 E Q T Figur : isk Prmia (6 monhs mauriy). This figur shows risk prmia on powrs of simply compoundd S&P 5 rurns. Thy ar compud from condiional gross momns of h S&P 5 ha solv h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. Gross momns ar compud for ordr J =,, 3 pricing krnl projcions, dnod by (),, and, rspcivly. 36 6

37 .5. (). -. () (a) E P E Q T -. (c) E P 3 E Q T (b) E P E Q T -.6 (d) E P 4 E Q T () E P 5 E Q T 5 (f) E P 6 E Q T Figur : isk Prmia ( monhs mauriy). This figur shows risk prmia on powrs of simply compoundd S&P 5 rurns. Thy ar compud from condiional gross momns of h S&P 5 ha solv h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. Gross momns ar compud for ordr J =,, 3 pricing krnl projcions, dnod by (),, and, rspcivly. 37 6

38 () () (a) monh (b) 3 monhs.5. () () (c) 6 monhs (d) monhs Figur 3: Condiional Scond Momn of Pricing Krnl. Th panls show h imvarying condiional scond momn of h pricing krnl projcion (5) on simply compoundd S&P 5 rurns for ordrs j =,, 3. Th momns ar soluions of h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. 38

39 .8.6 () 4 3 () (a) a -3 (b) a (c) a (d) a 3 Figur 4: Cofficins of Pricing Krnl Projcion ( monh mauriy). Th panls show h cofficins of h projcion (3) on simply compoundd S&P 5 rurns for ordrs j =,, 3. Th momns ar soluions of h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. 39

40 ().95 (a) a () -.8 (b) a (c) a -4 (d) a 3 Figur 5: Cofficins of Pricing Krnl Projcion (3 monhs mauriy). Th panls show h cofficins of h projcion (3) on simply compoundd S&P 5 rurns for ordrs j =,, 3. Th momns ar soluions of h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. 4

41 ().9 (a) a () -.9 (b) a (c) a - (d) a 3 Figur 6: Cofficins of Pricing Krnl Projcion (6 monhs mauriy). Th panls show h cofficins of h projcion (3) on simply compoundd S&P 5 rurns for ordrs j =,, 3. Th momns ar soluions of h opimizaion program () in Scion 3. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. 4

42 ().95 (a) a () -.9 (b) a (c) a (d) a 3 Figur 7: Cofficins of Pricing Krnl Projcion ( monhs mauriy). Th panls show h cofficins of h projcion (3) on simply compoundd S&P 5 rurns for ordrs j =,, 3. Th momns ar soluions of h opimizaion program () in Scion 3. 4

43 .4.3. M(. ) () (a) monh M( ) () (b) monhs Figur 8: Condiional Linar Krnl Projcion ( and monhs). Th figur shows h linar projcion (3) ovr im as a funcion of im and simply compoundd rurns for monh mauriy (Panl (a)) and monhs mauriy (Panl (b)). Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. 43

44 M( ) (a) monh M( ) (b) monhs Figur 9: Condiional Quadraic Krnl Projcion ( and monhs). Th figur shows h quadraic projcion (3) ovr im as a funcion of im and simply compoundd rurns for monh mauriy (Panl (a)) and monhs mauriy (Panl (b)). Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. 44

45 ..8.6 M( ) (a) monh M( ) (b) monhs Figur : Condiional Cubic Krnl Projcion ( and monhs). Th figur shows h cubic projcion (3) ovr im as a funcion of im and simply compoundd rurns for monh mauriy (Panl (a)) and monhs mauriy (Panl (b)). Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. 45

46 .3. 5% man 95%.4.3 5% man 95% M() ()..9 M() () (a) monh (b) 3 monhs M() () (c) 6 monh 5% man 95% M() () (d) monhs 5% man 95% Figur : Uncondiional Linar Krnl Projcion. Th panls show boosrappd sampl mans ovr h condiional linar pricing krnl projcions from Figur 8. Th projcion is compud from simply compoundd rurns on h S&P 5. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. 46

47 .3. 5% man 95%.4.3 5% man 95% M()..9.8 M() (a) monh (b) 3 monhs M() (c) 6 monh 5% man 95% M() (d) monhs 5% man 95% Figur : Uncondiional Quadraic Krnl Projcion. Th panls show boosrappd sampl mans ovr h condiional quadraic pricing krnl projcions from Figur 9. Th projcion is compud from simply compoundd rurns on h S&P 5. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. 47

48 M() (a) monh 5% man 95% M() (b) 3 monhs 5% man 95% M() (c) 6 monh 5% man 95% M() (d) monhs 5% man 95% Figur 3: Uncondiional Cubic Krnl Projcion. Th panls show boosrappd sampl mans ovr h condiional cubic pricing krnl projcions from Figur. Th projcion is compud from simply compoundd rurns on h S&P 5. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. 48

49 M() (a) monh 5% man 95% M() (b) 3 monhs 5% man 95% M() (c) 6 monh 5% man 95% M() (d) monhs 5% man 95% Figur 4: Uncondiional Scond Drivaiv of Cubic Krnl Projcion. Th panls show boosrappd sampl mans ovr scond drivaivs of h condiional cubic pricing krnl projcions from Figur. Th projcion is compud from simply compoundd rurns on h S&P 5. Th daa ar Europan opions on h S&P 5 wrin bwn January 99 and January 4. 49