Recovering Risk Aversion from Options

Transcription

1 Recoverng Rsk Averson from Optons by Robert R. Blss Research Department Federal Reserve Bank of Chcago 230 South La Salle Street Chcago, IL U.S.A. (312) (312) Fax and Nkolaos Pangrtzoglou Monetary Instruments and Markets Dvson Bank of England Threadneedle Street London EC2R 8AH U.K Fax November 29, 2001 Frst Draft: November 2, 2001 JEL Classfcatons: G13, C12 The authors thank Lars Hansen and semnar partcpants at the Unversty of Georga. Any remanng errors are our own. The vews expressed heren are those of the authors and do not necessarly reflect those of the Federal Reserve Bank of Chcago or the Bank of England. 1

2 Recoverng Rsk Averson from Optons Abstract Cross-sectons of opton prces embed the rsk-neutral probablty denstes functons (PDFs) for the future values of the underlyng asset. Theory suggests that rsk-neutral PDFs dffer from market expectatons due to rsk prema. Usng a utlty functon to adjust the rsk-neutral PDF to produce subjectve PDFs, we can obtan measures of the rsk averson mpled n opton prces. Usng FTSE 100 and S&P 500 optons, and both power and exponental utlty functons, we show that subjectve PDFs accurately forecast the dstrbuton of realzatons, whle rsk-neutral PDFs do not. The estmated coeffcents of relatve rsk averson are all reasonable. The relatve rsk averson estmates are remarkably consstent across utlty functons and across markets for gven horzons. The degree of relatve rsk averson declnes wth the forecast horzon and s lower durng perods of hgh market volatlty. JEL Classfcatons: G13, C12 2

3 Introducton Cross sectons of opton prces have long been used to estmate mpled densty functons (PDFs). These PDFs represent a forward-lookng forecast of the dstrbuton of the underlyng asset. Opton-derved dstrbutons have the dstnct advantage of (usually) beng based on data from a sngle pont n tme, rather than data taken from an hstorcal tme-seres. As a result, these PDFs are theoretcally much more responsve to changng market expectatons than are densty forecasts estmated from hstorcal data usng statstcal densty estmaton methods or dervng densty forecasts from the parameterzed tme seres models. Unfortunately, theory also tells us that the PDFs estmated from optons prces are rsk-neutral PDFs. If the representatve nvestor determnng optons prces s not rsk neutral, these PDFs need not correspond to the representatve nvestor s (.e. the market s) actual forecast of the future dstrbuton of underlyng asset values. If one assumes that nvestors are ratonal the subjectve densty forecasts should correspond, on average, to the dstrbuton of realzatons. Thus, one way to test whether rsk-neutral denstes reflect market expectatons s to test whether they provde accurate densty forecasts. If rsk-neutral PDFs do not forecast accurately we may nfer that the dfference between the rsk-neutral and an accurate or subjectve forecast arses from the rsk averson of the representatve agent. We can then use ths dfference to nfer the degree of rsk averson. A number of papers have examned the densty forecast accuracy for dfferent opton-derved rsk-neutral PDFs. 1 Most of these studes have rejected the hypothess that opton-derved rsk-neutral PDFs are accurate forecasts of the dstrbuton of future values of the underlyng asset. Thus, evdence suggests that mpled PDFs can not relably be used to nfer market expectatons concernng the future dstrbuton of the underlyng asset. Ths s not entrely surprsng as there s a large lterature establshng the exstence of rsk prema n market prces, partcularly equty markets. Nonetheless, 1 Anagnou, Bedendo, Hodges, and Tompkns (2001) provde an excellent revew of prevous papers before addng ther own results. 3

4 numerous other papers have proceeded to nterpret rsk-neutral PDFs as market expectatons. 2 Whle estmatng the representatve agent s or market s degree of rsk averson from securtes prces has a long hstory, t s only recently that scholars have begun usng optons data to do so. The methodology used n prevous studes has been to separately estmate the rsk-neutral and subjectve (or statstcal) densty functons, use these two separately-derved functons to nfer the rsk averson functon, and then draw conclusons from the mpled rsk averson functon. Some of these papers permtted the rsk-neutral densty functon to vary from observaton to observaton, however all mposed an assumpton of statonarty on the statstcal densty functon or underlyng stochastc process to facltate estmatng the subjectve densty functon from hstorcal data. Jackwerth (2000) assumes subjectve PDF that constant wthn a 4-year movng wndow. The tme seres of subjectve PDFs are then compared to a tme seres of rskneutral PDFs derved from S&P 500 ndex optons. At-Sahala and Lo (2000), compare an average rsk-neutral PDF over a year-long sample perod wth an average subjectve PDF for S&P 500 ndex, to derve an average mpled rsk averson functon. They use the kernel densty method to construct an average subjectve PDF. To construct an average rsk-neutral PDF, they used a two-dmensonal kernel smoothng method to fnd the mpled volatlty smle functon wth respect to exercse prce and maturty, and then they derve an average rsk-neutral PDF. The dsadvantage of ths approach s that an average rsk-neutral PDF gnores the actual daly movements n the rsk-neutral densty. At Sahala, Wang and Yared (2001) test the jont hypothess of the effcent prcng of optons on the S&P500 ndex and that the ndex tself follows a one-factor dffuson. They compare an average rsk-neutral PDF over the sample perod wth the rsk-adjusted stochastc process of the ndex. The later s derved by adjustng the drft of the true S&P500 one-factor dffuson process by applyng Grsanov s change-of-measure theorem. The dsadvantages of ths approach s that the stochastc process of the ndex s restrcted to be a one-factor dffuson, and that the rsk-neutral densty s nvarant over the sample perod. 2 See Blss and Pangrtzoglou (2001) for a partal lst. 4

5 Coutant (2001) compares a seres of rsk-neutral denstes for CAC40 ndex wth a seres of subjectve denstes derved from usng a parameterzed model for the underlyng stochastc process. As n At Sahala, Wang, and Yared (2001), the stochastc process of the ndex s restrcted to follow one-factor dffuson. The volatlty structure of the dffuson s the same as the rsk-neutral denstes volatlty structure, n lne wth Grsanov s theorem. The mean, whch s allowed to vary over tme, s dfferent and s estmated by maxmum lkelhood usng hstorcal data of the ndex. Comparng the seres of rsk-neutral denstes and subjectve denstes derves a tme-seres of rsk averson coeffcents. The aforementoned papers produced mpled rsk-averson functons under statstcal assumptons (statonarty; partcular stochastc processes), but mposed no theoretcal restrctons on the mpled rsk-averson functons. Unfortunately, the derved rsk-averson functons are somewhat nconsstent wth theory. Both At-Sahala and Lo (2000) and Jackwerth (2000) estmated U-shaped rsk averson functons. These suggest that nvestors are more rsk averse at both hgher and lower levels of wealth, whle theory suggests that they should be more rsk averse at lower levels of wealth and less rsk averse as wealth ncreases. Jackwerth also estmated rsk averson functons that had the coeffcent of absolute rsk averson changng sgns across the dstrbuton, wth large negatve values n the mddle of the dstrbuton and large postve values at the tals. Ths s also nconsstent wth theoretcal predctons. The statonarty assumptons and/or stochastc process assumptons made n these same papers are also doubtful. Estmated rsk-neutral PDFs are rarely consstent wth the smple functonal forms mpled by smple one-factor dffuson models, nor are changes n PDFs over tme consstent wth smple shfts n the mean of the stochastc process. When faced wth changng rsk-neutral PDFs, the statonarty of the subjectve PDF becomes questonable. The assumpton made n the prevously dscussed papers that the true statstcal dstrbuton s constant begs the queston of why then the rsk-neutral dstrbuton s not. A statonary statstcal dstrbuton requres ether that the rsk averson functon s tme-varyng or that nvestors are rratonal, that s that they do not account for the supposedly statonary dstrbuton of prces, n order to explan the clearly tmevaryng rsk-neutral dstrbutons that we observe. Drectly testng the statonarty of the 5

6 statstcal dstrbuton generally requres more data than s usually avalable, though volatlty clusterng, spkes and the frequent applcaton of tme-varyng volatlty models and regme-swtchng models to descrbe fnancal tme-seres ponts to the strong possblty that true underlyng statstcal dstrbutons are tme-varyng. The rsk-averson functons prevously estmated by At-Sahala and Lo (2000) and Jackwerth (2000) under the assumpton of statstcal dstrbuton statonarty are smply too mplausble to support such an assumpton. An alternatve to assumng statstcal-dstrbuton statonarty s to assume rskaverson functon statonarty. We can do ths by assumng some well-behaved functonal form for the underlyng utlty functon, consstent wth most papers that have examned the queston of market rsk averson. Bartunek and Chowdhury (1997) follow ths approach and assume a power utlty functon wth constant relatve rsk averson, and a one-factor return generatng process wth constant mean return and varance for the equty ndex. The mean return s estmated hstorcally usng S&P 500 ndex hstorcal values. The remanng parameters of the combned model are estmated usng the method of smulated moments, and the ftted opton prces compared to the actual prces. Ths method produces theoretcally reasonable rsk averson functons by constructon, but contnues to mpose restrctve statonarty assumptons on the statstcal dstrbuton. Rather than mposng statonarty restrctons on the underlyng statstcal processes to permt estmatng the subjectve densty from a tme seres of hstorcal prces, we mpose an alternatve restrcton on the rsk averson functon and permt the subjectve densty to tme-vary. We assume a parametrc form for the rsk averson functon, estmate the approprate rsk averson under the assumpton that ths value s statonary over the sample perod and then usng tme-varyng rsk-neutral densty functons estmated from optons prces to derve the tme-varyng mpled subjectve densty functons. Our goal s to fnd mpled subjectve densty functons that are consstent wth both utlty theory and ratonal expectatons. We nvestgate these questons usng FTSE 100 and S&P 500 optons and two dfferent methods of estmatng the subjectve PDFs usng dfferent utlty functons to adjust rsk-neutral PDFs. We fnd, as others have, that rsk-neutral PDFs are poor 6

7 forecasters of the dstrbuton of future values of the underlyng ndces. We then fnd the optmal value for the parameters of the utlty functons used to construct the subjectve PDFs and show that these subjectve PDFs are good forecasters of the dstrbuton of future values of the underlyng ndces. The measures of rsk averson mplct n these adjustments are well behaved, of reasonable magntude, remarkably consstent across the two markets and the two utlty functons consdered. The Methodology secton of ths paper outlnes the theory underlyng the comparson of rsk-neutral and subjectve denstes, detals how we estmated rsk-neutral PDFs, adjust them to get the subjectve PDFs, and then test ther densty forecasts. The secton concludes wth a descrpton of the data. The emprcal results are presented and analyzed n the Results secton, and the concluson follows. The Appendx dscusses alternate methods of testng densty forecasts, together wth the Monte Carlo tests we used to select the method used n ths paper. Methodology Our approach to studyng the rsk premum mplct n optons prces nvolves lookng at the ablty of rsk-neutral and rsk-adjusted or subjectve PDFs to forecast future realzatons of the underlyng asset. Our assumpton s that nvestors are ratonal and perhaps rsk averse. For nstance, f we were nterested only n pont forecasts ths would mean that the degree of bas n the forecast could be nterpreted as an ndcaton of the degree of market rsk averson, provded the bas s of the correct sgn, rather than an ndcaton that nvestors are rratonal. In ths study we are nterested n forecasts of dstrbutons rather than of a sngle pont estmate. We wll therefore examne whether the realzatons over tme are consstent wth the PDFs mplct n optons prces at some horzon pror to the respectve realzatons. Opton prces embed rsk-neutral PDFs. If these rsk-neutral PDFs provde good forecasts of the dstrbuton of future realzatons then we must conclude that there s no evdence of rsk prema n the prcng of optons. On the other hand f rsk-neutral PDFs are not good forecasters, we can test whether rsk-adjusted PDFs provde better forecasts. If ths s the case, the relatve rsk averson of the utlty functon used to adjust the rsk-neutral PDF provdes a measure of the degree of rsk averson. 7

8 To execute our study we need to: 1. Compute rsk-neutral PDFs from opton prces, 2. Test the forecast ablty of PDFs, both rsk-neutral and subjectve. 3. Adjust rsk-neutral PDFs to derve subjectve PDFs, Estmatng the Rsk-Neutral Probablty Densty Functon Breeden and Ltzenberger (1978) showed that the PDF for the value of the underlyng asset at opton expry, f( S T ), s related to the European call prce functon by f( S ) = e T CS (, KTt,, ) 2 K 2 rt ( t) t where S t s the current value of the underlyng asset, K s the opton strke prce and T-t s the tme to expry. Unfortunately, avalable opton quotes do not provde a contnuous call prce functon. To construct such a functon we must ft a smoothng functon to the avalable data. In ths paper we employ a refnement of the smoothed mpled volatlty smle method developed by Pangrtzoglou and presented n Blss and Pangrtzoglou (2001). 3 The essence of the Pangrtzoglou and related methods s to smooth mpled volatltes rather than opton prces and then convert the smoothed mpled volatlty functon nto a smoothed prce functon, whch can be numercally dfferentated to produce the estmated PDF. The Black-Scholes formula s used to extract mpled volatltes for European optons (FTSE 100) and the Barone-Ades-Whaley formula s used for Amercan optons (S&P 500). At the same tme strke prces are converted nto deltas usng the Black- Scholes delta and the approprate at-the-money mpled volatlty, thus producng a seres of transformed raw data ponts n mpled volatlty/delta space. It s mportant to note that the use of the Black-Scholes and Barone-Ades-Whaley formulae s solely to convert data from one space (prce/strke) to another (mpled volatlty/delta) where smoothng K= ST 3 Numerous methods have been developed for extractng PDFs from opton prces. Blss and Pangrtzoglou (2001) provde a revew of many of these. The Pangrtzoglou method tself derves from prevous work as dscussed n Blss and Pangrtzoglou. The Pangrtzoglou method was selected for ths paper because Blss and Pangrtzoglou found t to be relatvely robust and the method permts calbratng the desred smoothness of the extracted PDF. 8

9 can be done more effcacously. Dong so does not presume that ether formula correctly prces optons. A weghted natural splne s used to ft a smoothng functon to the transformed raw data. The natural splne mnmzes the followng functon: N λ = 1 ( ) mn w IV IV(, g x dx where IV s the mpled volatlty of the th opton n the cross-secton; IV (, s the ftted mpled volatlty whch s a functon of the th opton s delta,, and the parameters, that defne the smoothng splne, gx ( ; and w s the weght appled the th opton s squared ftted mpled volatlty error. In ths paper we use the opton vegas, & to weght the observatons. The parameter λ s a smoothng parameter that controls the tradeoff between goodness-of-ft of the ftted splne and ts smoothness measured by the ntegrated squared second dervatve of the mpled volatlty functon. In our prelmnary tests we used values of λ rangng from 0.99 to to check the senstvty of our results to the degree of smoothness we mpose on the estmated PDF. These tests ndcated that forecast results were nsenstve to the choce of λ. We therefore report results based on λ = When fttng a PDF t s necessary to extrapolate the splne beyond the range of avalable data. 4 The natural splne s lnear outsde the range of avalable data ponts and can thus result n negatve or mplausbly large postve ftted mpled volatltes. To prevent ths happenng we force the splne to extrapolate smoothly n a horzontal manner. We do ths by ntroducng two pseudo-data ponts spaced three strke ntervals 5 above and below the range of strkes n the cross sectons and havng mpled volatltes equal to the mpled volatltes of the respectve extreme-strke optons. These pseudodata ponts are added to the cross-sectons before the above transformatons and splnefttng take place. 4 Anagnou, Bedendo, Hodges, and Tompkns (2001) use PDFs truncated to the range of avalable strkes and then rescaled. Ths unusual procedure avods extrapolatng the tals of the PDF, but cannot handle realzatons fallng outsde the range of strkes avalable when the PDF was constructed. 5 Strke ntervals refers to the nterval between adjacent quoted opton strkes. 9

10 Once the splne, gx ( ; s ftted, 5,000 ponts along the functon are converted back to prce/strke space usng the Black-Scholes formula. The delta-to-strke converson uses the same at-the-money mpled volatlty used for the earler strke-todelta converson, thus preservng the consstency n the ntal data transformaton and ts nverse. The mpled volatlty-to-call prce converson uses the mpled volatlty provded by the ftted mpled volatlty functon to produce a ftted European call prce functon. The 5,000 ponts are selected to produce equally spaced strkes over the range where the PDF s sgnfcantly dfferent from zero. Ths range vares wth each crosssecton, prmarly as the prce level of the underlyng changes. Fnally, we use the 5,000 call prce/strke data ponts to numercally dfferentate the call prce functon to obtan the estmated PDF for the cross-secton. Testng PDF Forecast Ablty Each opton cross-secton produces an estmated PDF, f ˆ (), for a sngle opton expry date. Our goal s to test the hypothess that the estmated PDFs, f ˆ (), are equal to the true PDFs, f ( ). The tme-seres of PDFs generated for a gven forecast horzon are all dfferent. Only one realzaton, X, s observed for each opton observaton/expry date par. Under the null hypothess that the X are ndependent and that estmate PDFs are the true PDFs,.e. f ˆ () = f (), the nverse probablty transformatons of the realzatons, y X = fˆ ( u ) du, wll be ndependently and unformly dstrbuted: y ~..d. U(0,1). 6 The range of the transformed data s guaranteed by the nverse probablty transformaton tself, but the 6 Kendal and Stuart (1979), secton 30.36, dscusses the case where the X are..d. and the estmated denstes do not depend on the X. Where the estmated denstes do depend on the X, problems may ensue and the nverse probablty transform need not be ndependent or unform. Debold, Gunther, and Tay (1998) show that for a specal case (arsng n GARCH processes) where the true denstes depend only on past values of X (and no other condtonng nformaton) the..d. unform result holds. However, n the problem addressed n ths paper the PDFs are estmated from opton prces and values of the underlyng, whch do not nclude the X. We therefore rely on Kendal and Stuart. 10

11 unformty need obtan only f the estmated PDF equals the true PDF. Independence must also be establshed as most dstrbutonal tests assume ndependence and would generate ncorrect nferences f ths were not the case, though ndependence s not always verfed n practce. Several non-parametrc methods have been proposed for testng the unformty of the nverse probablty transformed data, ncludng the Kolmogorov-Smrnov, Chsquare, and Kuper tests. None of these methods provdes a jont test of the assumpton that the y are..d. Berkowtz (2001) has proposed a parametrc methodology for jontly testng unformty and ndependence. He frst defnes a further transformaton, z t, of the nverse probablty transformaton, y t, usng the nverse of the standard normal cumulatve densty functon, Φ (): X 1 1 z ( ) ˆ =Φ y =Φ f( u) du. Under the null hypothess, f ˆ () = f (), z ~..d. N(0,1). Berkowtz tests the ndependence and standard normalty of the z t by estmatng the followng model z t = zt 1 + t usng maxmum lkelhood and then testng restrctons on the estmated parameters usng a lkelhood rato test. 7 Under the null, the parameters of ths model should be: = 9DU t = Denotng the log-lkelhood functon as = 2 L ( the lkelhood 2 rato statstc, LR = 2 L(0,1, 0) L( ˆˆ s dstrbuted 2 under the null hypothess. In practce, t s sometmes necessary to test overlappng forecasts, for example 60-day-ahead forecasts of monthly realzatons. In ths case, f the above test rejects t s possble that the rejecton arses from the overlappng nature of the data, whch may 3 ˆ 7 The log-lkelhood functon for ths model s gven n Hamlton (1994), equaton (5.2.9). Ths test does not test the normalty of the transformed data per se, but tests that the data s standard normal under the assumpton that t s normally dstrbuted. Rejectng ths test s suffcent to establsh that the null hypothess does not hold, however t s not necessary. 11

12 nduce autocorrelaton, rather than from problems wth the estmated PDFs. Ths s also true for non-overlappng, but serally correlated, data. Berkowtz therefore tests the ndependence assumpton separately by examnng 2 2 ˆˆ ˆˆ ˆ 1 = L L LR 2 ( whch has a 2 dstrbuton under the null. If LR 3 rejects the hypothess that the z ~..d. N (0,1), falure to reject LR 1 provdes evdence that the estmated PDFs are not provdng accurate forecasts of the true tme-varyng denstes. On the other hand f both LR 3 and LR 1 reject, we cannot determne whether the problem arses from a lack of forecast ablty or seral correlaton n the data. Falure to reject both LR 3 and LR 1 s consstent wth forecast power, though as n all statstcal tests falure to reject the null hypothess does not necessarly mean that the null hypothess s true. The smple AR(1) model used n the above Berkowtz test captures only a specfc sort of seral dependence n the data, though ths s the dependence most lkely to occur n ths case. Berkowtz (2001) shows how to expand the model and assocated tests to hgher order AR(p) processes. However, ths results ncreasng numbers of model parameters and reduced power. Other tests for ndependence, for nstance runs tests, could be appled to the y pror to conductng the LR test, f more complex temporal dependences are suspected. The LR test s unformly most powerful only n a sngle-sded hypothess test. However, as we show n Appendx A n Monte Carlo smulatons the Berkowtz test s more relable than the Ch-squared and Kuper tests n large and small samples under the null hypothess, and s addtonally superor to the Kolmogorov-Smrnov test n small samples when the data are autocorrelated. We therefore use the Berkowtz test n ths paper. Estmatng the Subjectve Densty Functon To compute and test the forecast ablty of a subjectve densty functon t s frst necessary to hypothesze a utlty functon for the representatve agent and then, followng At-Sahala and Lo (2000), use ths to convert the estmated rsk-neutral densty functon nto a subjectve densty functon. The forecast ablty of the resultng 12

13 subjectve densty functon s then tested n the same manner as the rsk-neutral densty functon. At-Sahala and Lo (2000) show that, subject to certan condtons such as complete and frctonless markets and a sngle asset, the rsk-neutral densty functon, ps ( T ), s related to the subjectve densty functon, qs ( T ), by a thrd functon, ST whch s n turn related to the representatve nvestor s utlty functon, U( S T ), as follows: ps ( T) U( ST) ST = qs ( ) =λ U ( S) where λ s a constant. The resultng subjectve densty functon must be normalzed to ntegrate to one. Thus, ps ( T) U ( St) ps ( T ) ps ( T ) ζ ( ST; St) λu ( ST) U ( ST) qs ( T ) = = =. px ( ) U ( St ) px ( ) dx pxdx ( ) dx ζ( xs ; ) λu ( x) U ( x) t T In ths paper we test subjectve densty functons derved usng two representatve agent utlty functons: the power utlty functon, and the exponental utlty functon. In both cases the utlty functons, and thus the resultng subjectve densty functons, are condtoned on the value of the sngle parameter, In testng the subjectve densty t functons we frst selected the value of to maxmze the forecast ablty of the resultng subjectve PDFs as measured by the Berkowtz LR 3 statstc p-value. Table 1 provdes the functonal forms of the power and exponental utlty functons and the margnal utlty functon used to transform the rsk-neutral densty nto the correspondng subjectve densty, together wth the measure of relatve rsk averson (RRA) for each utlty functon. The power utlty functon has constant relatve rsk averson, and the measure of RRA s smply equal to the parameter However, the exponental utlty functon exhbts constant absolute rsk averson, the parameter rather than constant relatve rsk averson. For exponental utlty, the RRA s dependent on both and the realzaton S T, whch s tme varyng. Therefore, for exponental utlty RRA we report the dstrbuton of the RRA across the sample observatons. 13

14 Data Descrpton Two sets of equty optons contracts are used n ths study S&P 500 optons traded on the Chcago Mercantle Exchange (CME) and FSTE 100 optons traded on the London Internatonal Fnancal Futures Exchange (LIFFE) together wth data on the underlyng asset and the rsk free nterest rated needed to prce optons. 8 Data ncluded optons expres from February 18, 1983 through June 15, 2001 for the S&P 500 optons and June 19, 1992 through March 16, 2001 for the FTSE 100 ndex optons. The CME S&P 500 optons contract s an Amercan opton on the CME S&P 500 futures contract. S&P 500 optons trade wth expres on the same expry dates as the futures contracts, whch trade out to one year wth expres n March, June, September, and December. In addton, there are monthly seral optons contracts out to one quarter. Thus, at the begnnng of January optons are tradng wth expres n January, February, March, June, September, and December; at the begnnng of February optons trade wth expres n February, March, Aprl, June, September, and December. Optons expre on the 3 rd Frday of the expry month, as do the futures contracts n ther expry months. Pror to March 1987 the S&P 500 futures settled to the value of the S&P 500 ndex at the close on Frday. Begnnng n March 1987 the futures settled to an exchange-determned Specal Openng Prce on the expry Frday. For seral months there s no futures expry and the optons settle to the closng prce on the opton expry date of the next maturng S&P 500 futures contract. The S&P 500 optons realzatons used n ths study are the Specal Openng Quote for quarterly contracts begnnng n March 1987 and the S&P 500 futures closng prce for seral contracts and all contracts pror to Opton quotatons used to compute PDFs are the closng prces, the assocated value of the underlyng s the closng prce of the S&P 500 futures contract maturng on or just after the opton expry date. The LIFFE FTSE 100 opton contract used n ths study s a European opton on the FTSE 100 equty ndex. Optons are traded wth expres n March, June, September and December. Addtonal seral contracts are ntroduced so that optons trade wth expres n each of the nearest three months. FTSE 100 optons expre on the thrd Frday 8 Short Sterlng optons were also examned but faled to produce enough usable cross-sectons for meanngful analyss. 14

15 of the expry month. FTSE 100 optons postons are marked-to-market daly based on the daly settlement prce, whch s determned by LIFFE and confrmed by the Clearng House. The FTSE 100 optons prces used n ths study are the LIFFE-reported settlement prces. The quarterly FTSE 100 futures contract expre on the same date as the optons and therefore wll have the same value as the ndex when the opton expres. The European-style FTSE 100 contract may thus be vewed as an opton on the futures, f one assumes that mark-to-market effects are nsgnfcant. LIFFE reports the futures prces as the value of the underlyng n ther optons data. For seral months, LIFFE constructs a theoretcal futures prce based on a far value spread over the current futures front quarterly delvery month. In computng FTSE 100 mpled volatltes, the value of the underlyng asset correspondng to each cross-secton of opton quotes used n ths study s the actual or theoretcal futures prce reported by LIFFE for that contract. At expry the optons settle to the Exchange Delvery Settlement Prce determned by LIFFE by takng the average level of the FTSE 100 ndex sampled every 15 seconds between 10:10 and 10:30 on the last tradng day, after frst dscardng the hghest and lowest 12 observatons. Ths seres was used to compute opton realzatons for ths study. The rsk free rates used n ths study are the Brtsh Bankers Assocaton s 11 a.m. fxngs of the 3-month EuroDollar and EuroSterlng LIBOR rates reported by Bloomberg. 9 A target observaton date was determned for horzons of 1, 2, 3, 4, 5, 6 weeks, 1, 2, 3, 4, 5, 6, 9 months and one year, by subtractng the approprate number of days (weekly horzons) or months (monthly and 1-year horzons) from the opton expry date. If no optons traded on the target observaton date, the nearest optons tradng date was determned. If ths nearest tradng date dffered from the target observaton date by no more than 3 days for weekly horzons or 4 days for monthly and 1-year horzons, that date was substtuted for the orgnal target date. If no suffcently-close optons tradng date exsted, that expry was excluded from the sample for that horzon. 9 Duffee (1996) provdes evdence that short maturty U.S. Treasury securtes exhbt dosyncratc varatons that makes them unsutable proxes for the U.S. rsk free rate. The U.K. does not have a lqud Treasury Bll market. The LIBOR market has the dual advantages of lqudty and approxmatng the actual market borrowng and lendng rates faced by optons market partcpants. 15

16 Optons quotes for the target dates are then fltered. Because tradng n optons markets s asymmetrcally concentrated n at- and out-of-the-money strkes, and because the splne algorthm wll not accommodate duplcate strkes n the data, we dscard nthe-money optons. Optons for whch t was mpossble to compute an mpled volatlty (usually far-away-from-the-money optons quoted at ther ntrnsc value) or optons wth mpled volatltes of greater than 100 percent were also dscarded. If there were fewer than fve remanng usable strkes n a gven cross-secton the entre cross-secton was dscarded. Table 2 presents the resultng cross-secton counts and the range and mean of the strkes per cross-secton of the remanng data. In practce, too few cross-sectons leads to nsuffcent power to conduct meanngful tests. FTSE 100 horzons greater than 6 weeks, and S&P 500 horzons greater than 2 months, were found to have too few usable cross-sectons for our study. Furthermore, overlappng data produced serous autocorrelaton problems for longer maturtes. Our fnal sample therefore consstent of fltered cross-sectons for horzons of between 1 and 6 weeks. Emprcal Results The analyss of the emprcal results conssts of three sequental steps. We frst examne the rsk-neutral PDFs to determne whether there s evdence that they adequately capture the dstrbuton of ex post realzatons. We next rsk adjust the rskneutral PDFs and then test these subjectve PDFs n the same manner. Condtonal on the subjectve PDFs provdng a better forecast of the dstrbutons of future realzatons, we examne the measures of RRA mplct n these rsk-adjusted PDFs. Table 3 provdes the evdence on our frst two questons. We cannot reject the hypothess that the rsk-neutral PDFs provde accurate forecasts of the dstrbutons of future realzatons for ether FTSE 100 or S&P 500 contracts at the 1-week horzon. Nether can we reject the hypothess for the S&P 500 contracts at the 2-week horzon. However, the p-values for the 1- and 2-week S&P 500 optons tests are only slghtly hgher than 10 percent, whch s a reasonable threshold gven the lack of power n tests of goodness-of-ft and the small number of observatons. The 1-week FTSE 100 results gve no comfort however to skeptcs of usng rsk-neutral PDFs as forecasts. 16

17 However, we are usually nterested n forecast horzons beyond one or two weeks. At the 6-week horzon, rsk-neutral PDFs for both FTSE 100 and S&P 500 clearly reject the hypothess that the PDFs accurately forecast future dstrbutons of the underlyng ndces; however, the supplementary LR 1 tests also reject the hypothess that the nverse probablty transforms are ndependent. We therefore can draw no conclusons from the 6-week results. Thus, the falure to reject for the 1- and 2-week horzons and the rejecton at the 6-week horzon are ambguous results that may or may not mean poor densty forecast ablty, recallng that falure to reject should not be nterpreted as acceptng. The ntermedate 2 5 week results for FTSE 100 and 3 5 weeks for S&P 500 provde evdence that rsk-neutral PDFs do not provde accurate forecasts of the future dstrbutons of realzatons of the underlyng ndces. The Berkowtz LR 3 statstc rejects at conventonal sgnfcance levels, whle the supplementary LR 1 statstc fals to reject the hypothess of ndependence. Ths s consstent wth the estmated rsk-neutral PDFs falng to provde accurate forecasts, rather than a falure of the ndependence assumpton. Unlke the 1-, 2-, and 6-week results, we here have evdence that s consstently nconsstent wth the hypothess that rsk-neutral PDFs forecast the dstrbuton of the underlyng asset. These results also demonstrate that the Berkowtz test has suffcent power to reject the good-forecast null, exceptng perhaps n the extreme short horzons. Ths observaton becomes mportant when we examne the forecast ablty of the rsk-adjusted PDFs and fnd very dfferent results. Havng prevously establshed that our tests are able to reject n the rsk-neutral case, we are more secure n nterpretng the falure of the same test to reject n the subjectve cases as arsng from superor performance of the rskadjusted PDFs rather than lack of power n our test methodology. For the 3 5 week horzons the Berkowtz test faled to reject the hypothess that the rsk-adjusted PDFs provded good forecast of the dstrbuton of the underlyng asset values at those horzons. Ths was equally true for FTSE 100 and S&P 500, and for power-utlty-adjusted and exponental-utlty-adjusted PDFs. P-values were well above conventonal thresholds. The exponental-utlty-adjusted PDFs p-values were consstently hgher than those of the power-utlty-adjusted PDFs. Ths result s statstcally sgnfcant at the 10% level when taken n aggregate, though perhaps not for 17

18 ndvdual cases. 10 However, we offer ths as merely suggestve rather than conclusve, and contnue to examne both methods of rsk adjustment. The notable excepton to ths pcture of rsk-adjusted PDFs out-performng rskneutral PDFs s the 2-weeks FTSE 100 case. For ths sample the rsk-neutral and rskadjusted PDFs were rejected as accurate forecasts of the dstrbutons of the forecasts of future values of the value of the FTSE 100 ndex. The supplementary LR1 test p-values fal to provde evdence that ths s assocated wth rejecton of the jont ndependence assumpton. We must therefore conclude that for the 2-week FTSE 100 contracts nether the power utlty nor the exponental-utlty-adjusted PDFs captured the market rsk prema. Havng determned that the 3 5 week rsk-adjusted PDFs appear to do a reasonable job of forecastng the dstrbuton of future values of the underlyng asset values, whle the rsk-neutral PDFs do not, we next examne the degree of relatve rsk averson reflected n the rsk-adjusted PDFs. The top panel of Table 4 presents the all observatons RRAs correspondng to the results just dscussed. There s close agreement between the power utlty RRAs and the mean exponental utlty RRAs. Furthermore, the RRAs for FTSE 100 and S&P 500 are nearly dentcal for matched horzons. 11 Ths s not an artfact of the methodology as the samples are entrely dstnct and we see varaton between RRAs for dfferent horzons. The medan exponental utlty RRAs are slghtly lower than the mean, reflectng the postve skew n the dstrbuton of ndex values. Thus, whle the exponental-utlty-adjustment appears to produce somewhat superor forecasts of densty, the (mean) measured RRAs are broadly consstent between the two rsk adjustment functons. However, the range of 10 Monte Carlo was used to determne the probablty of observng 8 nstances of exponental utlty PDFs achevng hgher Berkowtz statstcs out of 8 cases (3 5 week horzons) or 11 of 14 cases (all 7 horzons), under the assumpton that the data were drawn from dentcally dstrbuted, but cross-horzon correlated data. Pared sets (A and B) of unformly dstrbuted data were generated havng the same correlaton structure as the actual nverse probablty transforms and wth seres lengths also matchng the data. However, the pared data sets were constructed to have otherwse dentcal dstrbutons. Pars of Berkowtz statstcs were then computed for each par of constructed seres. The process was repeated 10,000 tmes and the frequency of Berkowtz(A) > Berkowtz(B) n 8 of 8 cases (6.5% of smulatons) and 11 of 14 cases (6.3% of smulatons) was noted. 11 Ths s true for the power utlty and mean and medan values for the exponental utlty. The S&P exponental utlty RRA ranges are greater than the correspondng FTSE 100 ranges because of the greater range found n the values of the S&P ndex, whch n turn arses from the longer tme-seres avalable for S&P data. 18

19 RRAs permtted by the exponental-utlty-adjustment, whch s qute substantal, coupled wth the suggeston of better ft, suggests that the constant relatve rsk averson nherent n the power-utlty-adjustment may be unduly restrctve, and that constant absolute rsk averson seems to be more consstent wth the data. In all cases the RRAs are consstent wth the moderate values found n most other studes shown n Table 5. There s no evdence n these results of the extreme, puzzle values found n Mehra and Prescott (1985) and Cochrane and Hansen (1992). The RRAs are generally declnng wth the forecast horzon. If we focus on the 3 5 week results, whch show the clearest contrast between rsk-neutral and rsk-adjusted forecast performance, the RRAs declne monotoncally by a factor of slghtly less than 2 over that range of forecast horzons. Ths strong horzon-dependence n estmated RRAs suggests that nvestors are more rsk averse when nvestng at shorter horzons. Our methodology necessarly mposes the assumpton that the utlty functon parameter s constant across the sample. Wth only one realzaton per observaton t s dffcult, f not mpossble, to estmate tme-varyng values for the utlty functon parameters. However, we can examne the robustness of ths constant parameter assumpton by dvdng the sample nto sub-samples and re-estmatng the parameters for each sub-sample. Recallng that the data dd not reject the full sample rsk-adjusted PDFs, allowng sub-sample varaton s unlkely to be statstcally sgnfcantly better on a caseby-case bass usng a Wald or smlar test. However, the patterns that emerge are consstent and nstructve. Rather than dvde the sample by tme-perod we elected to dvde t nto two equal-szed sub-samples correspondng to perods of hgh and low volatlty as measured by the mpled volatlty of at-the-money optons. The ratonale was the rsk averson s more lkely to vary wth the degree of rsk than n a smple lnear tme-trend. The mddle and lower panels n Table 4 present the RRAs measured over these two sub-samples. The results are marked and consstent. For every horzon, and for both FTSE 100 and S&P 500, the low-volatlty RRAs exceed the hgh-volatlty RRAs by a factor of between 3 and 5. Ths s consstent wth what s observed n equty markets. When market volatlty, usually measured by at-the-money opton mpled volatltes, spkes durng 19

20 crses we do not observe falls n equty prces whch would be consstent wth the equty rsk premum rsng n lne wth volatltes. 12 Ths s evdence that the rsk averson mpled n equty rsk prema s nversely related to the level of rsk. Our results confrm that ths phenomenon s measured n the prcng of optons as well. A possble explanaton for ths nverse relaton between equty rsk and measures of rsk averson les n our usng equty rsk as a proxy for consumpton rsk. Equty prces and volatlty are much more volatle than consumpton and consumpton uncertanty. Thus, use of equty returns as a proxy for consumpton ntroduces excess volatlty, whch s reflected n volatlty dependence n the derved measures of rsk averson. Our full-sample results show that, at least for forecast horzons of 3 to 5 weeks, the rsk-neutral dstrbutons provde poor forecasts of future denstes whle the subjectvely-adjusted denstes provde reasonably good (.e. not statstcally rejectable) forecasts. An obvous queston s how much the rsk-neutral and subjectve denstes dffer. One measure of ths s to look at the tal percentle ponts under the rsk-neutral and subjectve dstrbutons. The estmaton of tal-percentle ponts s of partcular mportance n rsk management where value-at-rsk s wdely used. Suppose we were to compute the 1-percentle value under the (rejectable) rsk-neutral densty forecast each perod. These values of the underlyng wll have dfferent percentle values under the (not-rejectable) subjectve denstes, and the correspondng subjectve percentles of the rsk neutral 1-percentle values may vary from observaton to observaton. Table 6 presents the results of these computatons. Thus, at the 2 week horzon the values of the FTSE 100 correspondng to the 1-percentle of the rsk-neutral densty measured each observaton perod have subjectve cumulatve probabltes (percentles) rangng between 0.2 and 0.8 percent for the power-utlty-adjusted denstes and between 0.2 and 0.9 percent for the exponental-utlty-adjusted denstes, wth means of 0.6 and 0.7 percent respectvely. For all forecast horzons, percentle ponts, both utlty functons and both contract types, the rsk-neutral percentle ponts have lower probabltes under the subjectvely-adjusted denstes than under the rsk-neutral denstes. Thus, relance on 12 In a standard consumpton CAPM, the equty rsk premum s proportonal to the covarance of equty returns wth the margnal utlty of consumpton. Ths covarance s, n turn, proportonal to the standard devaton (volatlty) of equty returns. 20

21 rsk-neutral denstes to estmate and hold captal aganst a 1 percent value-at-rsk would be unduly conservatve (and expensve) for long equty postons, and would understate the rsk and requred captal for short postons. Whether these dfferences are materal depends on the partcular applcaton. For nstance, dfferences may be economcally unmportant for an unlevered equty portfolo, whle for a hghly levered or equty dervatve portfolos these dfferences could be crtcal to the sound management of rsk. 13 These are, of course, average results. The hgh-low mpled volatlty results presented n Table 4 show that the relance on rsk-neutral denstes would be less problematcal durng perods of hgh volatlty and more problematcal durng perod of low volatlty. The dfference between the mean of the rsk-neutral and the subjectve PDFs, normalzed by one of the means (we use the rsk-neutral PDF mean), s an approxmate measure of the equty rsk premum. Fgure 1 plots the tme seres of the 1-month forecast horzon rsk prema for the S&P 500 contract. The same data s presented n both panels wth dfferng scales for clarty. Untl 1997 the exponental-utlty-estmated rsk premum was less than that estmated usng a power-utlty adjustment. Snce 1997 ths relaton has been reversed. Changes n the rsk prema appear to be correlated across rskadjustment methods, as one would expect. Dfferences n estmated rsk prema can be large. For nstance durng the 1987 stock market crash the power-utlty-adjusted PDF suggested a rsk premum nearly three tmes as large as that estmated usng an exponental utlty functon to adjust PDFs. Ths spke results from the subjectve PDFs havng markedly hgher varances durng the 1987 crash (power: 0.33; exponental: 0.31) than the correspondng rsk-neutral PDF (0.27). 14 Fgure 2 compares the standard devatons and skewness coeffcents mpled by the subjectve PDFs aganst those from the rsk-neutral PDFs for one contract/horzon 13 The dfferences between the 2-week horzon mean rsk-neutral 1 st percentle pont (3975) and the correspondng power and exponental-utlty-adjusted 1-percentle ponts (4010 and 4015) s a small percentage of the mean level of the ndex. However, when compared to the mean absolute change n the ndex level over the 2-week horzon (85) the 1-percentle pont dfferences (35/45) are large. Comparsons for other horzons /contracts are smlar. 14 Campbell, Lo and MacKnlay (1997) pont out that the market rsk premum s proportonal to the market volatlty. 21

22 (S&P500; 1-month). Results for other contracts and horzons are smlar. Fgure 2 shows that for most observatons second and thrd moments do not dffer substantally between rsk-neutral and subjectve PDFs. The excepton s the September 16, 1987 observaton whch shows up as an outler on the scatter plots. Nonetheless, the dfferences n the frst 3 moments are suffcent to nduce a statstcally sgnfcant dfference n the forecast ablty of the subjectve and rsk-neutral PDFs and a tme-varyng equty rsk premum of around 10 percent per annum for most of the 1983 to 2001 perod. Conclusons Optons prces embed market expectatons of the dstrbuton of futures values of the underlyng asset. Ths can provde potentally useful nformaton for rsk managers and analysts wshng to extract forecasts from securty market prces. However, the rskneutral densty forecasts that are produced from optons prces cannot be taken at ther face value. We have shown, consstent wth the work of others, that rsk-neutral PDFs estmated from S&P 500 and FTSE 100 optons do not provde good forecasts of the dstrbuton of future values of the underlyng asset, at least at the horzons for whch we can obtan unambguous results. Theory tells us that f nvestors are rsk averse and ratonal the subjectve densty functons they use n formng ther expectatons wll be lnked to the rsk-neutral densty functons used to prce optons by a rsk averson functon. Theory also suggests certan propertes ths rsk averson functon mght be expected to have. We have employed two wdely used, and theoretcally plausble, utlty functons to nfer the unobservable subjectve denstes by adjustng the observed rsk-neutral denstes. Our crteron n makng ths adjustment s to choose the rsk averson parameter that produces subjectve denstes that best ft the dstrbutons of realzed values. That s, we assume that nvestors are ratonal forecasters of the dstrbutons of future outcomes and thus the rsk averson parameter value that best fts the data s most lkely to correspond to that of the representatve agent. In applyng ths methodology we assume that nvestors rsk averson functons are statonary. Ths contrasts wth the assumpton made n prevous papers that the statstcal dstrbuton was statonary. The subjectve densty functons derved under ths 22

23 assumpton cannot be rejected as good forecasters of the dstrbutons of future outcomes (unlke the unadjusted PDFs), and so ths assumpton seems valdated on a practcal level, subject to the caveat that there s some evdence of volatlty dependence n rsk-averson estmates. The coeffcent-of-rsk-averson estmates obtaned by our methodology are comparable to those obtaned n most prevous studes. There s lttle evdence of rsk aversons so hgh as to consttute a puzzle. We have also been able to establsh, we beleve for the frst tme, that the rsk averson estmates are surprsngly robust to dfferences n the specfcaton of the representatve nvestors utlty functon and to the data set used. We also show that the estmated coeffcents of rsk averson declne wth the forecast horzon and are hgher durng perods of low volatlty; both results suggestve that theoretcal models need to evolve to capture these effects. 23

24 Bblography At-Sahala, Yacne and Andrew W. Lo, 2000, Nonparametrc Rsk Management and Impled Rsk Averson Journal of Econometrcs 94(1 2), (January February), At-Sahala, Yacne, Yubo Wang, and Francs Yared, 2001, Do Opton Markets Correctly Prce the Probabltes of Movement of the Underlyng Asset? Journal of Econometrcs 102(1), (May), Anagnou, Ilana, Masca Bedendo, Stewart Hodges, and Robert Tompkns, 2001, The Relaton Between Impled and Realsed Probablty Densty Functons, workng paper, Unversty of Technology, Venna. Arrow, Kenneth J., 1971, Essays n the Theory of Rsk Bearng. North Holland, Amsterdam.. Barone-Ades, Govann and Robert E. Whaley, 1987, Effcent Analytc Approxmaton of Amercan Opton Values, Journal of Fnance 42(2), (June), Bartunek, K S, and M Chowdhury, 1997, Impled Rsk Averson Parameter from Opton Prces, The Fnancal Revew, 32, pages Berkowtz, Jeremy, 2001, Testng Densty Forecasts wth Applcatons to Rsk Management, Journal of Busness and Economc Statstcs 19, Blss, Robert R. and Nkolaos Pangrtzoglou, 2001, Testng the Stablty of Impled Probablty Densty Functons, Journal of Bankng and Fnance forthcomng. Campbell, John Y., Andrew W. Lo, and A. Crag MacKnlay, 1997, The Econometrcs of Fnancal Markets, Prnceton Unversty Press, Prnceton, NJ. Cochrane, John H. and Lars P. Hansen, 1992, Asset Prcng Exploratons for Macroeconomcs, n 1992 NBER Macroeconomcs Annual. NBER, Cambrdge, MA. Coutant, Sophe, 2001, Impled Rsk Averson n Optons Prces, n Informaton Content n Opton Prces: Underlyng Asset Rsk-neutral Densty Estmaton and Applcatons, Ph.D. thess, Unversty of Pars IX Dauphne. Debold, Francs X., Todd A. Gunther, and Anthony S. Tay, 1998, Evaluatng Densty Forecasts wth Applcatons to Fnancal Rsk Management, Internatonal Economc Revew 39(4), (November), Debold, Francs X., Anthony S. Tay, and Kenneth F. Walls, 1998, Evaluatng Densty Forecasts of Inflaton: The Survey of Professonal Forecasters, n R. Engle and H. Whte (eds), Festschrft n Honor of C.W.J. Granger, Oxford Unversty Press, Oxford. Duffee, Gregory R., 1996, Idosyncratc Varaton of Treasury Bll Yelds Journal of Fnance 51,