Empirical pricing kernels

Transcription

1 Empirical pricing kernels Joshua V. Rosenberg a,* and Rober F. Engle b a Federal Reserve Bank of New York, New York, NY 10045, USA b Sern School of Business, New York Universiy, New York, NY 10012, USA June 2001 Absrac This paper invesigaes he empirical characerisics of invesor risk aversion over equiy reurn saes by esimaing a ime-varying pricing kernel, which we call he empirical pricing kernel (EPK). We esimae he EPK on a monhly basis from 1991 o 1995, using S&P 500 index opion daa and a sochasic volailiy model for he S&P 500 reurn process. We find ha he EPK exhibis counercyclical risk aversion over S&P 500 reurn saes. We also find ha hedging performance is significanly improved when we use hedge raios based he EPK raher han a ime-invarian pricing kernel. JEL classificaion: G12, G13, C50 Keywords: Pricing kernels; Risk aversion; Derivaives; Hedging We are graeful for commens from David Baes, Mark Broadie, Peer Bossaers, Peer Carr, Ravi Bansal, Seve Figlewski, Joel Hasbrouck, Gur Huberman, Jens Jackwerh, Bruce Lehmann, Jose Lopez, Jayendu Pael, Mark Rubinsein, William Schwer, Allan Timmermann, wo anonymous referees a he Journal of Financial Economics, and seminar paricipans a New York Universiy, Columbia Universiy, Indiana Universiy, Georgeown Universiy, Norhwesern Universiy, Boson Universiy, Case Wesern Universiy, he Universiy of Iowa, he Federal Reserve Board of Governors, and he Federal Reserve Bank of New York. The auhors also appreciae he commens of paricipans a he Conference on Risk Neural and Objecive Probabiliy Disribuions, he 1998 Wesern Finance Associaion meeings, he 1998 Compuaional and Quaniaive Finance conference, and he 1997 Time Series Analysis of High Frequency Daa Conference. The auhors wish o hank David Hai for research assisance. The views expressed here are hose of he auhors and no necessarily hose of he Federal Reserve Bank of New York or he Federal Reserve Sysem. * Corresponding auhor. Tel: ; fax: address: joshua.rosenberg@ny.frb.org (J.V. Rosenberg).

2 1. Inroducion The asse pricing kernel summarizes invesor preferences for payoffs over differen saes of he world. In he absence of arbirage, all asse prices can be expressed as he expeced value of he produc of he pricing kernel and he asse payoff. Thus, he pricing kernel, when i is used wih a probabiliy model for he saes, gives a complee descripion of asse prices, expeced reurns, and risk premia. In his paper, we esimae he pricing kernel using curren asse prices and a prediced asse payoff densiy. We define he empirical pricing kernel (EPK) as he preference funcion ha provides he bes fi o asse prices, given he forecas payoff densiy. By esimaing he EPK a a sequence of poins in ime, we can observe and model he dynamic srucure of he pricing kernel iself. From his analysis, we obain improved opion pricing relaions, hedging parameers, and a beer undersanding of he paern of risk premia. We esimae he EPK each monh from 1991 o 1995, using S&P 500 index opion daa and a sochasic volailiy model for he S&P 500 reurn process. We find subsanial evidence ha he pricing kernel exhibis couner-cyclical risk aversion over S&P 500 reurn saes. Empirical risk aversion is posiively correlaed wih indicaors of recession (widening of credi spreads) and negaively correlaed wih indicaors of expansion (seepening of erm srucure slope). We develop an opion hedging mehodology o compare he accuracy of several pricing kernel specificaions. Our ess measure relaive performance in hedging ou-of-he-money S&P 500 pu opions using a-he-money S&P 500 pu opions and he S&P 500 index porfolio. We find ha hedge raios formed using a ime-varying pricing kernel reduce hedge porfolio volailiy more han hedge raios based on a ime-invarian pricing kernel. 2

3 Alhough here is a large lieraure on pricing kernel esimaion using aggregae consumpion daa, problems wih imprecise measuremen of aggregae consumpion can weaken he empirical resuls of hese papers. Hansen and Singleon (1982, 1983) posulae ha he pricing kernel is a power funcion of aggregae U.S. consumpion. They use maximum-likelihood esimaion and he generalized mehod of momens o esimae he pricing kernel. Chapman (1997) uses funcions of consumpion and is lags as pricing kernel sae variables, and he specifies he pricing kernel funcion as an orhogonal polynomial expansion. Hansen and Jagannahan (1991) derive bounds for he mean and sandard deviaion of he consumpion-based pricing kernel in erms of he mean and sandard deviaion of he marke porfolio excess reurns. Recenly, Ai-Sahalia and Lo (2000) have used opion daa and hisorical reurns daa o nonparamerically esimae he pricing kernel projeced ono equiy reurn saes. This echnique avoids he use of aggregae consumpion daa or a parameric pricing kernel specificaion. Along similar lines, Jackwerh (2000) nonparamerically esimaes he risk aversion funcion using opion daa and hisorical reurns daa. Ai-Sahalia and Lo (2000) and Jackwerh (2000) esimae invesor expecaions abou fuure reurn probabiliies by smoohing a hisogram of realized reurns over he pas four years. Implicily, hese papers assume ha invesors form probabiliy beliefs by equally weighing evens over he prior four years and disregarding previous evens. 1 These assumpions are inconsisen wih evidence from he sochasic volailiy modeling lieraure e.g. Bollerslev, Chou, and Kroner (1992) indicaing ha fuure sae probabiliies depend more on he recen evens han long-ago evens, bu ha longago evens sill have some predicive power. Misspecificaion of sae probabiliies induces error in 1 For example, using a four-year window, he Ocober 1987 sock marke crash influences probabiliy beliefs unil Ocober In November 1991, he crash no longer has an effec on beliefs. 3

4 he esimaion of he pricing kernel, since he denominaor of he sae-price-per-uni probabiliy is incorrecly measured. In Ai-Sahalia and Lo (2000) and Jackwerh (2000), sae prices and probabiliies are averaged over ime, so heir esimaes are perhaps bes inerpreed as a measure of he average pricing kernel over he sample period. Since he sample periods used are a leas one year in lengh, neiher paper deecs ime-variaion a less han an annual frequency. Average pricing kernels are also limied in heir abiliy o price and hedge asses on an ongoing basis, since asses are correcly priced only when risk aversion and sae probabiliies are a heir average level. 2 The remainder of he paper is organized as follows. Secion 2 describes he heory and previous research relaed o he pricing kernel. Secion 3 presens he empirical pricing kernel esimaion echnique, EPK specificaion, and hedge raio specificaion. Secion 4 describes he daa used for esimaion, and Secion 5 presens he esimaion resuls. Secion 6 conains he hedging es resuls, and Secion 7 concludes he paper. 2. Theory and previous research Our iniial discussion of asse pricing kernel heory and previous research inroduces several pricing kernel specificaions and discusses some poenial esimaion problems. We hen consider he characerisics of pricing kernel projecions. 2 Ai-Sahalia and Lo (2000) discuss he issue of ime-variaion of he sae price densiy (SPD) wihin he averaging period: In conras, he kernel SPD esimaor is consisen across ime [emphasis added] bu here may be some daes for which he SPD esimaor fis he cross secion of opion prices poorly and oher daes for which he SPD esimaor performs very well. 4

5 2.1. The asse pricing kernel The asse pricing kernel is also known as he sochasic discoun facor, since i is a saedependen funcion ha discouns payoffs using ime and risk preferences. 3 In he absence of arbirage, he curren price of an asse equals he expeced pricing-kernel-weighed payoff: P E M X 1 (1) where P is he curren asse price, M is he asse pricing kernel, and X +1 is he asse payoff in one period. In Lucas s (1978) consumpion-based asse pricing model, he pricing kernel is equal o he ineremporal marginal rae of subsiuion, so M = U (C +1 )/U (C ). Under he assumpion of power uiliy, he pricing kernel is M = e - (C +1/C ) -, wih a rae of ime preference of and a level of relaive risk aversion of. One of he basic characerisics of he pricing kernel is is slope, and sandard risk-aversion measures are usually funcions he pricing kernel slope. For example, he Arrow-Pra measure of absolue risk aversion is he negaive of he raio of he derivaive of he pricing kernel o he pricing kernel. The Arrow-Pra measure of relaive risk aversion is absolue risk aversion muliplied by curren consumpion. ' C M C ) / M ( C ) 1 ( 1 1 (2) 5

6 Generally, he pricing kernel will depend no only on curren and fuure consumpion, bu also on all variables ha affec marginal uiliy. In he habi persisence models of Abel (1990), Consaninides (1990), or Campbell and Cochrane (1999), he pricing kernel depends on boh pas and curren consumpion. Eichenbaum, Hansen, and Singleon (1988) le he pricing kernel depend on leisure, while Sarz (1989) uses durable goods purchases. Bansal and Viswanahan (1993) specify he pricing kernel as a funcion of he equiy marke reurn, he Treasury bill yield, and he erm spread. When he pricing kernel is a funcion of muliple sae variables, he level of risk aversion can also flucuae as hese variables change. Campbell (1996) shows ha a habi persisence uiliy funcion exhibis ime-varying relaive risk aversion, where relaive risk aversion is decreasing in he amoun ha consumpion exceeds he habi (he surplus consumpion raio). In Campbell s (1996) model, we observe decreases in relaive risk aversion during economic expansions when consumpion is high relaive o he habi. Furhermore, we observe increases in relaive risk aversion during economic conracions when consumpion falls closer o he habi. In conras, he power uiliy funcion exhibis relaive risk aversion ha is ime-invarian. To invesigae he characerisics of invesor preferences, many researchers have used Eq. (1) as an idenifying equaion for he pricing kernel. For example, Hansen and Singleon (1982) idenify he pricing kernel wih an uncondiional version of his equaion: 0 E ( P 1 / P ) M 1 (3) 3 Campbell, Lo, and MacKinlay (1997) and Cochrane (2001) provide comprehensive reamens of he role of he pricing kernel in asse pricing. Oher relaed papers include Ross (1978), Harrison and Kreps (1979), Hansen and Richard (1987), and Hansen and Jagannahan (1991). 6

7 Hansen and Singleon (1982), using an approach followed in many subsequen papers, specify he aggregae consumpion growh rae as a pricing kernel sae variable. They measure consumpion using daa from he Naional Income and Producs Accouns (NIPA). However, measuremen error in he NIPA consumpion daa can pose a significan problem. Ermini (1989), Wilcox (1992), and Slesnick (1998) discuss issues such as coding errors, definiional problems, impuaion procedures, and sampling error. Ferson and Harvey (1992) consider problems inroduced by he Commerce Deparmen s seasonal adjusmen echnique. Breeden, Gibbons, and Lizenberger (1989) address problems induced by use of ime-aggregaed raher han insananeous consumpion Projecions of he pricing kernel Because here is considerable debae among researchers over he sae variables ha ener ino he pricing kernel, we examine a pricing kernel projecion ha can be esimaed wihou specifying hese variables. We are paricularly ineresed in projecing he pricing kernel ono he payoffs of a raded asse (X +1 ). As discussed in Cochrane (2001), his projeced pricing kernel has exacly he same pricing implicaions as he original pricing kernel for asses wih payoffs ha depend on X +1. To allow complee generaliy, we wrie he original pricing kernel as M = M (Z, Z +1 ), where Z is a vecor of pricing kernel sae variables. We hen rewrie Eq. (1) by facoring he join densiy f (X +1,Z +1 ) ino he produc of he condiional densiy f (Z +1 X +1 ) and he marginal densiy f (X +1 ). We evaluae he expecaion in wo seps. Firs, he pricing kernel is inegraed using he condiional 7

8 densiy, which gives he projeced pricing kernel, M * (X +1 ). 4 Second, he produc of he projeced pricing kernel and he payoff variable is inegraed using he marginal densiy, which gives he asse price, P. P E * * M X ) X M ( X ) E M ( Z, Z ) X ( (4) The original pricing kernel depends on he realizaion of he sae vecor (Z +1 ), while he projeced pricing kernel depends on he realizaion of he asse payoff (X +1 ). Thus, for he valuaion of an asse wih payoffs ha depend only on X +1, he pricing kernel is summarized as a funcion of he asse payoff. This univariae funcion can vary over ime, reflecing ime-variaion in he pricing kernel sae variables. Eq. (4) can also be used o idenify he projeced pricing kernel. For example, Ai-Sahalia and Lo (2000) and Jackwerh (2000) esimae pricing kernels projeced ono equiy reurn saes using equiy index opion prices. These papers assume ha invesors have a finie horizon and ha he equiy index level is equal o he aggregae wealh. Under hese assumpions, a pricing kernel ha is projeced ono he equiy index level is equal o he original pricing kernel. 5 In general, we can inerpre he projeced pricing kernel he same way we inerpre he original pricing kernel, even hough he wo are no necessarily idenical. When M * (X +1 ) is consan, 4 By aking his condiional expecaion, we do no assume ha X +1 is known a dae. Raher, we are making a saemen abou he value of nex period s pricing kernel, for each possible realizaion of nex period s payoff variable. Since here is no necessarily a deerminisic relaion beween nex period s pricing kernel and he payoff variable, we measure his relaion by aking he expecaion based on informaion known a dae. The projeced pricing kernel depends on he sae of he world nex period hrough X +1 in he same way ha he original pricing kernel depends on he sae of he world nex period hrough Z Earlier papers, such as Rubinsein (1976) and Brown and Gibbons (1985), derive condiions such ha a pricing kernel ha has he consumpion growh rae as a sae variable is equivalen o a pricing kernel ha has he equiy index reurn as a pricing kernel sae variable. 8

9 invesors are indifferen o a uni payoff across payoff saes. When M * (X +1 ) is decreasing (increasing), invesors show a decreasing (increasing) desire for a uni payoff across payoff saes. We define a measure of risk aversion ( * ) for he projeced pricing kernel ha is relaed o he Arrow-Pra measure of relaive risk aversion. We se he projeced pricing kernel risk aversion equal o he opposie of he normalized slope of he projeced pricing kernel. 6 * *' * X M X ) / M ( X ) (5) 1 ( 1 1 The level of projeced risk aversion deermines he relaive preference for a uni payoff across payoff saes. High levels of * correspond o a seep, negaively sloped pricing kernel projecion, i.e., a srong demand for hedging securiies ha pay off when he asse is low. 3. Empirical pricing kernel esimaion sraegy In his secion, we describe our mehodology for esimaion of a ime-varying pricing kernel projeced ono asse reurn saes. We propose an opimizaion echnique ha selecs he pricing kernel ha bes fis raded asse prices, and we sugges wo pricing kernel specificaions. We hen presen he sochasic volailiy model used o esimae payoff probabiliies, and we derive opion hedge raios in a seing wih ime-varying probabiliies and a ime-varying pricing kernel Esimaion echnique 6 Ai-Sahalia and Lo (2000) and Jackwerh (2000) use similar formulas in heir definiions of relaive and absolue risk aversion funcions. 9

10 The accuracy of a candidae pricing kernel can be judged by how well i reproduces prices of raded asses. This crierion moivaes our esimaion procedure. We selec he empirical pricing kernel as he funcion ha provides he bes fi o curren derivaive prices, given curren expecaions abou fuure payoffs. Therefore, he EPK represens an esimae of he pricing kernel projecion on a paricular dae, raher han an esimae of an average pricing kernel over a period of a year or more. We begin by wriing Eq. (4) for a derivaive wih a payoff ha depends on he reurn o he underlying asse (r +1 ): P * * M ( r 1) gi ( r 1) M ( r 1) gi ( r 1) f ( r 1 dr 1 i, E ) (6) In Eq. (6), P i, is he price of he i h asse wih a payoff funcion of g i (r +1 ), and f (r +1 ) is he probabiliy densiy of one-period underlying asse reurns. Eq. (6) also shows ha he pricing kernel projecion, M * (r +1 ), is an implici funcion of prices, payoffs, and probabiliies. Nex, we rewrie Eq. (6) o find he formula for he fied asse price ( * pricing kernel projecion ( M ) ) and esimaed payoff densiy ( f ˆ ( ) ). ( r 1 r 1 ˆ ) using an esimaed P i, Pˆ * * M ˆ ˆ ( r 1) gi ( r 1) M ( r 1) gi ( r 1) f ( r 1 dr 1 i, E ) (7) We hen esimae he pricing kernel projecion as he funcion ha makes fied prices closes o observed prices, using he esimaed payoff densiy. To simplify he esimaion problem, we le he * pricing kernel projecion be a parameric funcion, M ; ), where is an Nx1 parameer ( r 1 1 vecor. We use he sum of squared errors as a disance measure. 10

11 EPK: We refer o he projeced pricing kernel ha solves he following opimizaion problem as he L Min i1 2 P ˆ i P (, i, ) (8) where L represens he number of asse prices, and P ( ) is he fied price as a funcion of he pricing kernel parameer vecor. To idenify he pricing kernel parameer vecor, we mus observe a leas as many derivaive prices as here are parameers. If we define he payoff densiy using a se of J realized (or simulaed) reurns, hen we can esimae he fied asse price using he following approximaion o Eq. (7), where averaging replaces inegraion: ˆ i, Pˆ J * M ( r 1, j ; ) gi ( r 1, j 1 i, ( ) J ) j1 (9) 3.2. Pricing kernel specificaions We consider wo specificaions for he projeced pricing kernel. In he firs specificaion, he pricing kernel is a power funcion of he underlying asse s gross reurn: M * (r +1 ; ) = 0, (r +1 ) - 1, (10) 11

12 In Eq. (10), he firs parameer ( 0, ) is a scaling facor and he second parameer ( 1, ) deermines he slope of he pricing kernel a dae. When 1, is posiive, he pricing kernel is negaively sloped, which implies ha he value of a uni payoff increases as he underlying asse reurn decreases. The level of projeced risk aversion is * = 1,.. If 1, changes over ime, hen his specificaion exhibis ime-varying risk aversion. Our second specificaion permis more flexibiliy in he shape of he pricing kernel, and also allows ime-variaion in risk aversion. 7 We consider a pricing kernel wih N+1 parameers ( 0, N, ) and N+1 polynomial erms (T 0 (r +1 ) T N (r +1 )): M * (r +1 ; ) = 0, T 0 (r +1 ) + 1, T 1 (r +1 ) + 2, T 2 (r +1 ) + + N, T N (r +1 ) (11) If here are an infinie number of erms in his polynomial expansion, hen he specificaion will accuraely approximae any coninuous funcion. However, in he conex of our esimaion problem, he number of observed asse prices places an upper bound on he order of he approximaing polynomial. Thus, we use a class of polynomials ha provides he mos accurae approximaions wih he smalles possible number of erms. Orhogonal polynomials are designed o provide more precise approximaions using lower order expansions han alernaive classes of polynomials. In an orhogonal polynomial expansion, each erm is muually orhogonal o all oher erms. The number of required erms is minimized, since each erm provides unique informaion ha is no conained in previous erms. Alhough here are several families of orhogonal polynomials (e.g. Legendre, Chebyshev, Laguerre, Hermie), he Chebyshev family provides an approximaion ha comes close o minimizing 12

13 he maximum approximaion error. 8 As shown by Judd (1998), Chebyshev polynomials are nearly opimal polynomial approximaions under he L norm. The Chebyshev polynomial is defined over he domain [-1,1] wih erms given by T n (x) = cos( n * cos -1 (x) ). The firs and second Chebyshev erms are T 0 = 1 and T 1 = x. The higher order erms are periodic funcions. To obain an accurae approximaion over a closed domain [a,b], we use he generalized Chebyshev polynomial. In his polynomial, x = ((2r +1 a b)/(b-a)), where a and b are he endpoins of he approximaion inerval. We adop he generalized Chebyshev polynomial approximaion for our second pricing specificaion. To ensure ha he pricing kernel is sricly posiive, we ake he exponenial of he polynomial expansion. M * (r +1 ; ) = [ 0, T 0 (r +1 )]exp[ 1, T 1 (r +1 ) + 2, T 2 (r +1 ) + + N,T N (r +1 )] (12) 3.3. Sae probabiliy densiy specificaion In he subsequen empirical porions of he paper (Sec. 4, 5, 6), we use he equiy index reurn probabiliy densiy for pricing kernel esimaion. So, in his secion of he paper, we develop a sochasic volailiy model ha incorporaes he mos imporan feaures of equiy index reurn process. Previous sudies, such as Ghysels, Harvey, and Renaul (1996), have documened ha equiy index reurn volailiy is sochasic and mean-revering, reurn volailiy responds asymmerically o posiive and negaive reurns, and reurn innovaions are non-normal. 7 The power specificaion is only nesed in an orhogonal polynomial specificaion wih an infinie number of erms. Wih a finie number of orhogonal polynomial erms, he power specificaion migh be a more accurae represenaion of he pricing kernel. 13

14 Researchers ofen capure sochasic volailiy in a discree-ime seing, using exensions of he auoregressive condiional heeroskedasiciy (ARCH) model proposed by Engle (1982). Comprehensive surveys of ARCH and relaed models are given by Bollerslev, Chou, and Kroner (1992) as well as Bollerslev, Engle, and Nelson (1994). In a coninuous-ime seing, researchers commonly use sochasic volailiy diffusions. Surveys of his lieraure include Ghysels, Harvey, and Renaul (1996) and Shephard (1996). Our model of he equiy index reurn process uses an asymmeric GARCH specificaion wih an empirical innovaion densiy. The GARCH specificaion of Bollerslev (1986) incorporaes sochasic, mean-revering volailiy dynamics. The asymmery erm in our model is based on Glosen, Jagannahan, and Runkle (1993). Our empirical innovaion densiy capures poenial non-normaliies in he rue innovaion densiy. The asymmeric GARCH model is specified as follows: ln( S / S ) rf 2, ~ f (0, ) (13) I Max[ 0, 1 ] (14) In Eq. (13), he log-reurn ne of he riskless rae of ineres, ln(s /S -1 ) rf, has a consan mean (). Alhough a consan expeced reurn is no usually compaible wih ime-varying risk aversion, he effec over a shor period (e.g. up o one monh) has a negligible effec on probabiliy esimaes. 8 Chapman (1997) esimaes he asse pricing kernel as a funcion of aggregae consumpion using a five erm Legendre polynomial expansion. Bansal, Hsieh, and Viswanahan (1993) esimae an inernaional asse pricing kernel as a funcion of powers of he Eurodollar ineres rae and a world equiy index reurn. 14

15 Therefore, Eq. (13) works as an approximaion. We draw reurn innovaions ( ) from an empirical densiy funcion (f) wih sochasic variance ( 2-1). Eq. (14) defines condiional reurn variance ( 2-1) as a funcion of wo consans ( and ), he lagged squared innovaion ( -1), and a non-linear funcion of he lagged reurn (Max[0,- -1 ] 2 ). The second consan ( ) permis a shif in long-run volailiy using an indicaor variable (I = 0 or 1) o mark he differen ime periods. We esimae he model parameers using maximum likelihood wih a normal innovaion densiy. Bollerslev and Wooldridge (1992) show condiions ha allow his echnique o provide consisen parameer esimaes even when he rue innovaion densiy is non-normal. We model he empirical innovaion densiy (f) by facoring he innovaion densiy ino imevarying and ime-invarian componens. To separae hese componens, we define a sandardized innovaion as he raio of a reurn innovaion ( ) and is condiional sandard deviaion ( -1 ). The sandardized innovaion densiy i.e. he se of sandardized innovaions is he ime-invarian componen of empirical innovaion densiy. The condiional sandard deviaion ( -1 ) is he imevarying componen of he empirical innovaion densiy. On a paricular dae, we consruc he empirical innovaion densiy by muliplying each sandardized innovaion by he condiional sandard deviaion. To esimae he sandardized innovaion densiy, we ake he raio of each reurn innovaion and is condiional sandard deviaion using he esimaed sochasic volailiy model. This collecion of esimaed sandardized innovaions forms a densiy funcion ha incorporaes excess skewness, kurosis, and oher exreme reurn behavior ha is no capured in a normal densiy. Afer we esimae he sochasic volailiy model, we use Mone-Carlo simulaion o deermine he fuure reurn densiy over any desired ime horizon. For example, we can creae he one-period reurn densiy by simulaing many one-period reurn realizaions. We obain a simulaed one-period log- 15

16 reurn ( + rf + +1 ) and a simulaed one-period simple reurn [exp( + rf + +1 )-1] by randomly selecing an innovaion ( +1 ) from he empirical innovaion densiy. We can creae a muli-period reurn densiy by simulaing many muli-period reurn pahs. We obain a 20-period reurn by drawing he firs reurn innovaion ( +1 ), updaing he condiional variance ( ), drawing he second reurn innovaion ( +1 ), updaing he condiional variance ( ), and coninuing hrough he wenieh innovaion. The one-period simulaed log-reurn is equal o i= rf + +i ), and he one-period simulaed simple reurn is equal o exp[ i= rf + +i )]. 16

17 3.4. Hedge raio specificaion We develop a hedge raio esimaion echnique ha does no depend on a paricular specificaion of he sae probabiliy densiy or he pricing kernel. Our hedge raios neuralize an opion porfolio o he firs- and second-order effecs of changes in he underlying price. In a coninuous-ime diffusion seing, a firs-order hedge will eliminae all randomness in he hedge porfolio and provide a minimum-variance hedge. In a discree-ime seing wih sochasic volailiy, firs- and second-order hedges will reduce, bu no eliminae hedge porfolio variabiliy. We derive hese hedge raios using a Taylor series expansion of he opion pricing formula. The pu opion price change afer one day (Pu +1 Pu ) is approximaely equal o he following funcion of he underlying price change (S +1 S ): Pu Pu 1 Pu Pu ( S 1 S ) ( S 2 1 S ) S 1 2 S 1 (15) The firs and second parial derivaives (Pu +1 /S +1, Pu +1 /S 2 +1 ) in Eq. (15) measure he sensiiviy of he pu price o firs- and second-order changes in he underlying price. These price sensiiviies are commonly called he opion dela and he opion gamma. To form an opion porfolio ha is hedged agains a firs-order underlying price change, anoher securiy mus be purchased ha moves in he opposie manner from he opion. This hedging securiy has sensiiviy o a firs-order underlying price change equal o -Pu +1 /S +1. To furher hedge agains second-order effecs of he underlying price change, a second securiy wih second-order price sensiiviy equal o -Pu +1 /S 2 +1 is required. 17

18 We refer o he number of unis of he hedging securiy ha are purchased o hedge a single opion as he hedge raio. To esimae hedge raios in a seing wih an arbirary pricing kernel, we generalize he Engle and Rosenberg (1995) mehodology for esimaion of price sensiiviies. Therefore, we consider hree possible one-day underlying price changes. The sock price could rise by one-sandard deviaion o S +, remain consan a S, or fall by one sandard deviaion o S -. Each underlying price change resuls in a differen dae +1 pu opion price: Pu Pu, or Pu S 1 S, 1 S 1. We calculae approximaions o he firs and second parial derivaives of he opion pricing formula using cenered finie difference approximaions: Pu S 2 Pu S Pu S Pu Pu S 2 1 S 1 Pu 2 1 S 2 Pu 1 S (16) To evaluae hese approximae derivaives, we find he value of he pu opion nex period a differen underlying price levels. We measure he curren pu price using he pricing kernel projeced ono he T-period underlying asse reurn (r,+t ) and he pu s payoff funcion: Pu * E M r ; ) Max[0, K S ] (17) (, T T Nex period, he underlying price is equal o S +1. The new underlying price affecs nex period s pu price hrough he payoff probabiliy densiy and he pricing kernel. We wrie he condiional 18

19 expecaion using he updaed payoff densiy as E +1 S+1 [ ]. 9 The parameers of he pricing kernel can also change when he underlying price changes, so we wrie nex period s parameer vecor as +1 S+1. Pu * E ( ; ) [0, ] 1 1 S M r 1 1, T 1 S Max K S 1 (18) 1 S T We hen form an approximae pricing equaion by replacing nex period s parameer vecor wih is condiional expecaion in he curren period. Pu * E 1 ( 1, ; [ 1]) [0, ] 1 S M r T E Max K S T (19) 1 S 1 We represen he pricing kernel parameer vecor as a linear funcion of a consan vecor (), lagged parameer vecors ( -1 -K ), and an error vecor (e +1 ):... e 1 0 K K... 1 (20) Thus, we can evaluae he condiional expecaion E [ +1 ]. E[ 1 ] 0... K K (21) We use Mone-Carlo simulaion o esimae omorrow s opion price, condiional on omorrow s underlying asse price, as: 9 If he underlying price follows a geomeric Brownian moion, hen he underlying price a dae +1 (S +1 ) provides addiional informaion abou he expeced price a dae +T (S +T ). If he underlying price follows an asymmeric GARCH 19

20 Pu J * M ( r 1, T, j ;[ 0... K K ]) Max[0, K ST, j 1 1 S J ] j1 (22) 4. Daa We develop our opions daase from a subse of Berkeley Opions Daabase covering he period One of he advanages of his daase is ha opion quoes are ime-samped and recorded along wih he simulaneously measured underlying price making i easier o consruc a daabase of ime-synchronized daily opion closing prices. To creae our daabase of opion closing prices, we firs collec end-of-day opion prices for all conracs. We do so by averaging he las recorded bid-ask quoe of he day beween 2:00 and 3:00 PM Cenral Time. The cross-secion of midquoes from he las hour of rading is no enirely synchronized, since he S&P 500 index level can change over he las hour of rading. To correc for his effec, we calculae a Black-Scholes (1973) implied volailiy each day for each opion conrac. We hen find he closing price for each conrac by evaluaing he Black-Scholes formula. We use he same inpus and he implied volailiy, excep ha he closing S&P 500 index level replaces he synchronized S&P 500 level. Finally, we average each call (pu) price wih he synheic pu (call) price ha we deermine using pu-call pariy adjused for dividends. This average price is used in he esimaion procedure. This echnique does no require he Black-Scholes model o be correc. I simply uses he Black- Scholes formula as an exrapolaion device o calculae an opion price adjusmen when he S&P 500 level changes from he ime of he las opion quoe o he close of opion rading. process, hen he underlying price a dae +1 provides addiional informaion abou he expeced price as well as higher 20

21 Our implied volailiy calculaion uses he end-of-day opion midquoe, he conemporaneous S&P 500 index level, he riskless ineres rae, ime unil expiraion (in rading days), and dividend yield. We measure he riskless ineres rae using Daasream s bid and ask discoun raes for U.S. Treasury Bills wih mauriies of one, hree, and six monhs. The riskless rae for a paricular opion is calculaed by linear inerpolaion of he ineres raes of Treasury Bills ha sraddle he opion expiraion dae. We calculae he dividend yield over he life of each opion conrac by aking he presen value of fuure S&P 500 dividends and dividing by he curren index level. To eliminae daa errors and ensure ha closing opion prices are represenaive of marke condiions a he end of he rading day, we use several screening crieria. We base some of hese on Bakshi, Cao, and Chen (1997). In our sample, we include opions wih moneyness (K/S - 1) 0.10, mid-quoes greaer han $3/8 and less han $50, annualized implied volailiies greaer han 5% or less han 90%, and prices ha saisfy he no-arbirage lower bound (P Max[0, Ke -r(t-) - S + D,T ] or C Max[0, S - Ke -r(t-) - D,T ]). We delee from he sample cross-secions of calls (pus) ha violae he no-arbirage condiion ha opion premia are decreasing (increasing) in he exercise price opions and opions ha violae he maximum verical spread premium condiion (C (K 1,T-) - C (K 2,) K 2 - K 1 ; P (K 2,T-) - P (K 1,) K 2 - K 1 ). Finally, we include in he sample only daes on which a leas eigh opions (boh calls and pus) saisfy he preceding crieria. We use he following procedure o consruc a daase of opions wih one monh (20 rading days) unil expiraion. We firs eliminae all opions wih greaer han 24 or fewer han 16 rading days unil expiraion. For each rading dae, we choose he opion series wih ime unil expiraion closes o 20 days. We are lef wih a single cross-secion of call and pu opions each monh (around he wenieh of he monh) wih a ime-unil-expiraion of approximaely one monh. This sampling mehodology is similar o ha of Chrisensen and Prahbala (1998). momens of he price disribuion. 21

22 In Table 1, we repor he properies of he one-monh opion conracs ha we use for pricing kernel esimaion. In he sample, here are 53 monhs (of 60 oal) for which we have a cross-secion of opions ha saisfies our screening crieria. In 39 monhs, here is an opion series wih exacly 20 days unil expiraion. We use a series wih 21 days unil expiraion eigh imes, and he remaining six daes have opion series wih 18, 19, and 22 days unil expiraion. There is no saisfacory daa in seven of he 60 monhs of he sample. On a given esimaion dae, here are beween 8 and 13 opions available. There are roughly equal numbers of opions wih moneyness (K/S - 1) from 3% o 0%, 0% o 3%, and 3% o -6%. There are somewha fewer opions wih moneyness beween -6% and -10%. The smalles number of opions has moneynesses ranging from 3% o 6%, and here are no opions available wih moneyness greaer han 6%. In our sample, opion conracs wih higher moneyness generally have lower implied volailiies, a paern known as a volailiy skew. Due o pu-call pariy, we gain no addiional informaion if we include a call and a pu wih he same exercise price (or moneyness) in our esimaion procedure. Therefore, we esimae pricing kernels using only ou-of-he-money pu opions (moneyness 0%) and ou-of-he-money call opions (moneyness > 0%). Fig. 1 graphs five represenaive cross-secions of one-monh opion closing prices for June of 1991 hrough June of 1995 agains percen moneyness (K/S -1)*100. The curves exhibi an invered-v shape, since pu premia increase in exercise price and call premia decrease in exercise price. The variaion in he slope and heigh reflecs differences in invesor probabiliy beliefs and risk aversion over ime. Table 2 repors summary saisics for he daily S&P 500 index reurns series ( ) used for esimaion of he sae probabiliy model. Over his period, he average annualized S&P 500 index reurn (capial appreciaion only) is 7.55%, and he annualized S&P 500 reurn sandard deviaion is 22

23 14.79%. S&P 500 reurns exhibi negaive skewness and posiive kurosis, and here is evidence of reurn serial correlaion. 5. Esimaion of he empirical pricing kernel projeced ono S&P 500 reurn saes In his secion, we esimae a monhly pricing kernel using a cross-secion of S&P 500 index opion prices and he S&P 500 reurn densiy funcion. We hen analyze he relaionship beween empirical risk aversion and business condiions Esimaion of S&P 500 reurn sae probabiliy densiies To find he mos accurae model of he S&P 500 reurn densiy, we esimae and es hree nesed GARCH models: ARCH(1), GARCH(1,1), and asymmeric GARCH(1,1). 10 We define he asymmeric GARCH (1,1) model using Equaions (13) and (14). We define he oher wo models using he same equaions, bu se =0 for he GARCH(1,1) model and boh =0 and =0 for he ARCH(1) model. We use a likelihood raio es o measure he saisical significance of he increase in likelihood for each model generalizaion. Table 3 repors he model esimaes. In Table 3, we find ha he GARCH model offers a saisically significan improvemen over he ARCH model wih a likelihood raio es p-value less han Our ess also show ha he asymmeric GARCH model provides a beer fi han he GARCH model. We see ha he robus - saisic for he volailiy asymmery parameer () is 2.41, which confirms he presence of an asymmeric volailiy effec. We se he indicaor variable equal o one during he EPK esimaion 10 We also esimae he GARCH componens model of Engle and Lee (1999) bu do no find ha i is a saisically significan improvemen over he GARCH (1,1) model. To conserve space, we do no repor hese resuls. 23

24 period ( ) and equal o zero for he res of he sample. This variable is no significan in he GARCH or asymmeric GARCH models. We creae he sandardized innovaions ( / -1 ) for he asymmeric GARCH model by aking he raio of each reurn innovaion and is condiional sandard deviaion. If we have correcly specified he sochasic volailiy model, hen he sandardized innovaions will be free of ime-dependence. We perform specificaion ess on he sandardized innovaions o measure auocorrelaion in he sandardized innovaions and in he squared sandardized innovaions. In Table 4, Panel A, we repor resuls of hese specificaion ess. We use he Ljung-Box (1978) Q-saisic o es for auocorrelaion in he sandardized innovaions. The asymmeric GARCH model passes his es wih a p-value of Nex, we use Engle s (1982) ARCH LM es o es for auocorrelaion in he squared sandardized innovaions. The asymmeric GARCH model passes his es wih a p-value of Since he asymmeric GARCH model provides he bes fi o he reurn daa and passes he specificaion ess, we choose his model for sae probabiliy esimaion. We hen esimae sandardized innovaion densiy using he collecion of he sandardized innovaions from asymmeric GARCH model. Table 4, Panel A shows ha his densiy exhibis negaive skewness (-0.36) and posiive excess kurosis (4.26) compared o a normal densiy. In Table 4, Panel B, we compare exreme reurn probabiliies using he sandardized innovaion densiy and a sandard normal densiy. Under he sandard normal assumpion, innovaions of magniude greaer han five or en sandard deviaions almos never occur (less han 1:1,000,000). In pracice, he probabiliy of hese exreme evens is non-negligible. For example, empirical reurn innovaions less han -5 sandard deviaions are observed six imes in 10,000, while empirical reurn innovaions less han -10 sandard deviaions are observed hree imes in 10,

25 Fig. 2 uses 200,000 Mone-Carlo simulaion replicaions o graph he esimaed one-monh sae probabiliy densiies each June. The ime-variaion in sae probabiliies is apparen. As we can see, here are higher probabiliies for large negaive reurn saes in June 1991 and June 1992 han in June of subsequen years. 11 Fig. 3 graphs he condiional volailiy forecass using he asymmeric GARCH model. Over his period, he esimaed annualized S&P 500 volailiy ranges from 6.75% o % wih a sandard deviaion of 6.36%. The highes volailiy forecass over his period are around he ime of he Ocober 1987 marke crash Esimaion of he S&P 500 empirical pricing kernel For esimaion of he S&P 500 empirical pricing kernel, we use a power specificaion and a fourparameer orhogonal polynomial specificaion. We se he reurn domain for orhogonal polynomial equal o he range of opion moneyness (-10% o 10%). Ouside of his domain, we se he pricing kernel equal o is esimaed value a -10% or 10%. Once per monh, we idenify he pricing kernel ha bes fis he cross-secion of one-monh S&P 500 opion premia using each specificaion. In he opimizaion procedure, we use he esimaed asymmeric GARCH model o creae a simulaed one-monh probabiliy densiy wih 200,000 replicaions. To ensure ha he esimaed pricing kernel accuraely prices a riskless one-monh bond, we also se he scaling facor ( 0, ) o saisfy he pricing equaion B = E [M * (r +1 ; )]. Table 5 repors he esimaion resuls. The orhogonal polynomial specificaion fis S&P 500 opion prices more closely han he power specificaion. The average pricing error sandard deviaion 11 The annualized reurn sandard deviaion esimaes (in chronological order for June 1991 hrough June 1995) are: 15.55%, 13.03%, 11.74%, 9.98%, and 10.88%. The reurn densiy skewness esimaes are -0.36, -0.43, -0.46, -0.46, and 25

26 for he orhogonal polynomial specificaion is $0.09 wih a minimum of $0.03 and a maximum of $0.24. The average forecas error sandard deviaion for he power specificaion is $0.63 wih a minimum of $0.28 and a maximum of $1.34. In Fig. 4, we graph power pricing kernel esimaes each June of he sample period. We find ha he level of risk aversion varies across ime, as illusraed by he changing slope of he esimaed pricing kernels. The negaively sloped pricing kernel esimaes show ha invesors experience declining marginal uiliy over S&P 500 reurn saes. Fig. 5 graphs orhogonal polynomial pricing kernel esimaes. Compared o he power pricing kernels, he orhogonal polynomial pricing kernels assign greaer value o large negaive S&P 500 reurn saes and lesser value o large posiive S&P 500 reurn saes. The sae-price-per-uni probabiliy for large negaive reurn saes is especially volaile, and could reflec ime-varying demand for insurance agains a significan marke decline. There is also some evidence of a region of increasing marginal uiliy for small posiive S&P 500 reurn saes. We hen esimae an average power pricing kernel and an average orhogonal polynomial kernel by evaluaing each specificaion a he average parameer esimaes. Table 5 repors characerisics of he parameer esimaes (including heir means), and Fig. 6 graphs he average pricing kernels. The average orhogonal polynomial EPK has some similariies o he esimae of Ai-Sahalia and Lo (2000). Boh pricing kernels are seeply upward sloping for large negaive reurns and downward sloping for large posiive reurns, and boh pricing kernels have a region of increasing marginal uiliy. Jackwerh s (2000) absolue risk-aversion funcion is closely relaed o he pricing kernel and can be expressed as he negaive of he raio of he firs derivaive of he pricing kernel and he pricing kernel (-M (r +1 )/M (r +1 )). Jackwerh noes wo key empirical findings for he absolue risk aversion funcions: pos-crash risk aversion funcions are negaive around he cener [reurn saes close o The reurn densiy kurosis esimaes are 5.06, 5.35, 5.66, 5.54, and

27 zero] [and] risk aversion funcions rise for wealh levels greaer han abou 0.99 [reurn saes greaer han 1%]. We do no find eiher of hese characerisics for our esimaes of he power pricing kernel. In his specificaion, he absolue risk aversion funcion equal o he exponen of he power funcion muliplied by he inverse of he gross reurn [ 1, *(r +1 ) -1 ]. Thus, he power pricing kernel exhibis declining (bu posiive) absolue risk aversion as long as he exponen ( 1, ) is posiive. This is he resul ha we find over he period Our esimaes of he orhogonal polynomial pricing kernel exhibi some of he risk-aversion characerisics noed by Jackwerh. We esimae he average absolue risk-aversion funcion using he average orhogonal polynomial pricing kernel graphed in Fig. 6. We find ha here is a region of negaive absolue risk aversion over he range from 4% o 2%, and ha absolue risk aversion increases for reurns greaer han -4%. The shape of our esimaed average absolue risk aversion funcion is similar o Jackwerh s esimae over a similar ime period Linking empirical risk aversion o business condiions We use he risk aversion of he esimaed power pricing kernel ( 1, ) as our measure of empirical risk aversion. We analyze he ime-series of 1, esimaes o gain insigh ino risk aversion dynamics and links beween risk aversion and he business cycle. Table 6 provides summary saisics for empirical risk aversion. Over he sample period, empirical risk aversion averages However, he level flucuaes subsanially, ranging from 2.26 o Empirical risk aversion is posiively auocorrelaed (=0.45) and mean-revering. Fig. 7 graphs he ime-series of empirical risk aversion esimaes. 27

28 Oher sudies, such as Fama and French (1989), show ha risk premia are correlaed wih he business cycle. Specifically, risk premia are lowes a business cycle peaks and highes a business cycle roughs. We provide evidence of ime-varying risk aversion hrough he business cycle, which suppors he Fama and French (1989) resuls. To measure he relaion beween empirical risk aversion and he business cycle, we consruc several variables ha reflec curren and expeced business condiions. We calculae one-monh percenage changes in business condiion indicaors by using he indicaor measured on he curren and previous pricing kernel esimaion dae. Fama and French (1989) and Lahiri and Wang (1996) use credi spreads (he difference beween he yield on risky and riskless bonds) as an indicaor of business condiions. As he economy weakens (srenghens), credi spreads widen (narrow) o compensae invesors for an increased probabiliy of defaul. We calculae credi spreads using a risky bond yield equal o Moody s long-erm Baa corporae bond yield index. This index measures he average yield-o-mauriy of approximaely 100 seasoned corporae bonds wih mauriies as close as possible o 30 years and a leas 20 years. We se he riskless bond yield equal o he Federal Reserve s hiry-year consan mauriy Treasury yield. We collec boh daa iems from he Federal Reserve s H.15 release. Esrella and Hardouvelis (1991) show ha he slope of he yield curve (erm spread) is procyclical. Seepening of he slope indicaes expansion, while flaening of he slope indicaes conracion. We use he H.15 release informaion o measure he yield curve slope as he hiry-year consan mauriy Treasury yield minus he hree-monh consan mauriy Treasury yield. Esrella and Hardouvelis (1991) also sugges ha he level of shor-erm ineres raes migh reveal he sae of he business cycle. They summarize he view ha high shor-erm raes are associaed wih a igh moneary policy, low curren invesmen opporuniies, and low oupu. They presen empirical 28

29 evidence ha demonsraes his relaion. We use he percenage change in he hree-monh consan mauriy Treasury yield as proxy for his indicaor. In our analysis, we use he one-monh percenage change in he S&P 500 level as repored in he CRSP daabase as a poenial business cycle indicaor. We also include he aggregae U.S. consumpion growh rae, since i is used in many sudies of he sochasic discoun facor such as Hansen and Singleon (1982). We measure consumpion growh using per-capia non-durable goods and services (monhly, real, seasonally adjused) from he Federal Reserve s FRED daabase. Our monhly U.S. residen populaion esimaes are from he Census Bureau. To measure auocorrelaion in risk aversion, we include he one-monh lag of empirical risk aversion. In addiion, we use he difference beween a-he-money implied and objecive volailiy as a proxy for risk aversion. This volailiy spread measures he mark-up of an a-he-money opion price above he price ha a risk-neural invesor would accep. Table 7 repors a muliple regression of empirical risk aversion on all of he above variables. We presen univariae correlaions and heir p-values in he las wo columns. We measure independen variables in percen (excep lagged empirical risk aversion). Therefore, a one basis poin (0.01) increase in he independen variable raises empirical risk aversion by one one-hundredh of he regression coefficien. The Table 7 resuls show ha empirical risk aversion varies couner-cyclically wih business condiions. One business cycle indicaor variable is significan in he muliple regression (credi spread), and wo business cycle variables have significan correlaion wih empirical risk aversion 29

30 (credi spread and erm srucure slope). 12 The signs of he regression parameer esimaes and correlaion coefficien esimaes are all consisen wih couner-cyclical risk aversion. For example, in he muliple regression, he credi spread is saisically significan and has a posiive esimae of Hence, a one-basis poin widening of he credi spread increases empirical risk aversion by The correlaion beween empirical risk aversion and credi spreads is also posiive and saisically significan (=0.50, p-value = ) in he univariae analysis. Since he credi spread is a couner-cyclical indicaor, hese resuls show ha empirical risk aversion is counercyclical. The slope of he erm srucure is no saisically significan in he muliple regression, bu i has a saisically significan negaive correlaion wih risk aversion (=-0.36, p-value = ). This resul is furher evidence of couner-cyclical risk aversion, since risk aversion is negaively correlaed wih a pro-cyclical indicaor. Our empirical findings provide some suppor for habi persisence models of invesor uiliy. Habi models such as Campbell (1996) and Campbell and Cochrane (1999) predic ha Arrow-Pra relaive risk aversion is couner-cyclical. In an economic expansion, he surplus consumpion raio (he proporion ha curren consumpion exceeds he habi) is high and risk aversion is low. In a recession, he reverse is rue. If our measure of empirical risk aversion is posiively correlaed wih Arrow-Pra relaive risk aversion, hen our finding of couner-cyclical empirical risk aversion shows ha Arrow-Pra relaive risk aversion migh be couner-cyclical. In his seing, our finding would provide empirical suppor for a key implicaion of habi models. 12 Two oher variables, lagged risk aversion and volailiy spread, are significan in he regression and correlaion analysis. The posiive coefficien on lagged risk aversion shows ha risk-aversion is posiively auo-correlaed. The posiive 30

31 6. Hedging ess In his secion, we use hedging performance o measure he imporance of ime-variaion in he pricing kernel and o compare he accuracy of he power and orhogonal polynomial specificaions. To implemen our ess, we creae hedge porfolios for a $100 posiion in ou-of-he-money (OTM) S&P 500 index opions. We esimae hedge raios using ime-invarian and ime-varying pricing kernels wih power and orhogonal polynomial specificaions. We use a-he-money (ATM) pu opions and/or he S&P 500 index porfolio as hedging insrumens. We consruc a hedging sample using he same screening crieria as we did for esimaion of he pricing kernels. However, he hedging sample comprises opions wih approximaely one monh (from 16 o 24 days) unil expiraion, insead of exclusively one-monh opions. On each sample dae, we selec a pu wih moneyness (K/S - 1) closes o zero bu no more han 1% in absolue value. This is he a-he-money pu. As he ou-of-he-money pu, we selec he opion wih moneyness closes o 3%, bu no greaer han -3%. When we canno find suiable opions wih closing prices on he sample dae and he nex rading dae, we exclude he sample dae from he analysis. Using hese crieria, we have 243 observaions available for he hedging ess. Table 8 summarizes he sample characerisics. To form one-day ahead opion prices ha are he basis for he hedge raio esimaes, we require esimaes of he expeced one-day ahead pricing kernel. For he ime-invarian pricing kernels, he expeced pricing kernel parameer vecor is equal o he average parameer vecor using he 53 observaions from EPK esimaion. We show he average parameer esimaes in Table 9. coefficien on he volailiy spread shows ha an increase in empirical risk aversion is associaed wih a widening of he spread beween implied and objecive volailiy. 31