BANCO DE PORTUGAL Economics Research Department

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1 BANCO DE PORUGAL Economcs Research Department he Estmaton of Rsk Premum Implct n Ol Prces Jorge Barros Luís WP -00 February 000 he analyses, opnons and fndngs of ths paper represent the vews of the author, they are not necessarly those of the Banco de Portugal. Please address correspondence to Jorge Barros Luís, Economcs Research Department, Banco de Portugal, Av. Almrante Res, nº 7, 50-0 Lsboa, Portugal el.# ; Fax# ;e-mal:jhclus@bportugal.pt.

2 he estmaton of rsk premum mplct n ol prces Jorge Barros Luís # Abstract: Futures prces can be seen as a sum of the expected value of the underlyng asset prce wth a rsk premum. In order to dsentangle those two components of the futures prces, one can try to model the relatonshp between spot and futures prces, n order to obtan a closed expresson for the rsk premum, or to use nformaton from spot and opton prces to estmate rsk averson functons. Gven the hgh volatlty of the ratos between futures and spot prces, we opted for the latter, estmatng rskneutral and subjectve probablty densty functons, respectvely from opton and spot prces observed. Lookng at the prces of Brent and West exas Intermedate Lght/Sweet Crude Ol optons, evdence obtaned suggests that the rsk premum s typcally very low for levels near the futures prces. JEL Classfcaton codes: G4. Key words: Rsk averson functon, Rsk neutral densty, Subjectve densty, Kernel estmaton, Ol opton prcng. # Banco de Portugal and Unversty of York. Specal acknowledgements are due to Bernardno Adão, Menelaos Karanasos and Mke Wckens, for ther extensve comments, and to Pedro Neves, for hs nctement to wrte ths paper and comments and suggestons made. he author also thanks to Maxmano Pnhero and Nuno Rbero for ther comments. he support of the Lsbon Stock Exchange s acknowledged.

3 . Introducton Ol prces are mportant determnants of nflaton. hus, the assessment of expectatons on future values of ol prces s a relevant ssue for macroeconomc modellng. ypcally, some scenaros about ol prces are consdered wthn ths framework, or ol futures prces are used as the best guess of future prce values. However, t s known that futures prces, regardless the underlyng asset, may also ncorporate a rsk premum component. In ths paper, we propose to estmate rsk-averson functons for Brent and West exas Intermedate ol usng opton and spot prces. As t s well establshed (see, e.g. Sö derlnd and Svensson (997) or Bahra (996)), opton prces are related to the rsk-neutral probablty densty functon (RND) of the underlyng asset prce n a future moment correspondng to the settlement date. Another mportant theoretcal result concerns the relatonshp between the subjectve or true probablty densty functon (SD) and the RND, correspondng the former to the product of the latter by the rsk-averson functon. herefore, f we are able to estmate SD, we get an estmate for the rskaverson functon. Naturally, the proxy for the SD gves drectly an estmate for the expected value. However, t doesn t provde any nformaton on the nvestors behavour regardng rsk-averson. hus, the estmaton of rsk-averson functons s a relevant ssue, n addton to the computaton of the expected value under the SD measure.

4 Generally, a proxy for SD s used, based on the observed prces, though the estmaton technques may dffer. For nstance, n At-Sahala and Lo (000) and Jackwerth (997) a kernel estmator of the past returns s used, whle Coutant (999) uses Hermte polynomals expansons and Rosenberg and Engle (997) use a GARCH model. he methods for the estmaton of the RND functons are also numerous, but generally are based on the relatonshp between RND functons and the opton prces, whle the SD estmator has not that theoretcal background. In ths paper, a mxture of two log-normal denstes s used to estmate the RND, whle a kernel estmator s the tool adopted for extractng the SD from observed prces. he paper s organsed as followng: n the next secton, the concept of rsk averson s approached; the thrd secton s devoted to the relatonshp between rsk-averson and densty functons; the representatve agent s preferences are analysed n the fourth secton; n the ffth secton the estmaton methodology s detaled; data and results are presented n the sxth secton and the man conclusons are stated at the end.. he concept of rsk-averson he averson to rsk s one of the most mportant concepts n fnancal economcs. As stated n Ingersoll (987), t s generally sad that a decson maker wth a von Neumann-Morgenstern utlty functon s rsk averse at a gven wealth level f he s unwllng to accept every actuarally far and mmedately resolved gamble A von Neumann-Morgenstern utlty functon (u(z)) s a functon on sure thngs: u(z) = H(P z), z Z, beng z a consumpton plan defned on a collecton of consumpton plans, P Z the sure consumpton plan and H an utlty functon (for more detals see, e.g., Huang and Ltzenberger (988), chapter ).

5 wth only wealth consequences, that s, those that leave consumpton good prces unchanged. It can be shown that a necessary and suffcent condton for a decson maker beng strctly rsk averse corresponds to hs utlty functon beng strctly concave at all relevant wealth levels: () EUW+ ε < U EW+ ε = UW beng W the wealth level, ε the outcome of a lottery and U a utlty functon. As stated n Ingersoll (987), a rsk averse agent accepts to pay an nsurance rsk premum (Π) n order to avod a lottery: () EUW+ ε = UWΠ Usng a aylor expanson of both sdes of (), we wll be able to determne the rsk premum, as followng: 5 (3) " $# = 3 EUW + εu W + ε U W + ε U W+ αε! 6 = UW 0 5 ΠU 0W5+ ΠU W Π6 Global rsk averson arses when the decson-maker s rsk averse at all relevant wealth levels. 3 See appendx A. If the decson-maker s rsk averse, though not strctly, the nequalty sgnal becomes. 4 Conversely, a rsk averse agent would demand a compensatory premum (Π c) n order to partcpate n a lottery: EUW + Π c + ε6 = UW 6. hs defnton s more usual n fnance lterature, whle the former s more representatve n economc analyss. If the rsk s small and the utlty functon s suffcently smooth, the two rsk premums are roughly equal. 5 he coeffcents α and are such that the aylor expansons of both sdes of equaton () may be represented by the thrd- and the second-order expansons presented n equaton (3). 3

6 beng α a coeffcent of the expanson s remander ( 0 α ). Gven that the utlty functon s assumed to be suffcently smooth, the last term n the left-hand sde (LHS) wll approxmately be nl. Besdes, assumng that the rsk premum s small, the last term of the rght hand sde (RHS) s approxmately nl too. hus, under the assumpton of E ε = 0 we get the followng expresson from (3): (4) ε U W ΠU W.e "! 0 5$ # U W (5) Π = var ε U W he term n brackets n equaton (5) corresponds to the usually known Arrow- Pratt absolute rsk-averson functon and t s a measure of local rsk-averson, ndependent from the scalng factor var ε If the representatve agent s rskaverse, ths functon must always by postve,.e., the utlty functon must be concave. As the Arrow-Pratt rsk-averson functon s a measure of the relatve change n the slope of the utlty functon at a gven wealth level, t must be decreasng and convex n the wealth level under the rsk-averson assumpton. 6 he varance n the scalng factor may be approached by the varance of the returns of a fnancal asset. 4

7 3. Rsk-averson functons he rsk-averson functons can be derved n a context of a sngle-perod or of an ntertemporal model. Wthn the former framework, let us assume a complete market economy, 7 where all nvestors take decsons on the quantty of state-contngent clams to purchase. he target s the maxmsaton at the current date t of ther expected utlty n, subject to the constrant of the dscounted expected value of wealth at beng equal to ther ntal wealth: 8 (6) Max p S U ζ, S ds ζ S I S I + rτ τ subject to e ζ qs 6dS = W S t beng p = subjectve or true densty (n order to wealth states, henceforth denoted by SD), representng the probablty measure P; q = RND (n order to wealth states), representng the probablty measure Q, or the prce vector of the state-contngent clams; ζ S = quantty of state-contngent clams purchased; U() = utlty functon, dependng on the quantty of state-contngent clams purchased and the future state of nature; S = state of nature or underlyng asset prce n ; W t = endowed or ntal wealth; r = τ-maturty rsk-free nterest rate, wth τ = t; 7 A market n whch each state s nsurable,.e., for each state one can fnd a portfolo of assets wth a non-zero return only n that state (lke Arrow-Debreu securtes). As stated n Ingersoll (987), a complete market guarantees that a representatve nvestor exsts, though not necessarly the utlty functon of the representatve nvestor. 8 he constrant may also be read as the dentty between the ntal wealth and the value of the state-contngent clams purchased. 5

8 W S 6= ζ,.e., the wealth n as a functon of each state corresponds to the S number of state-contngent clams purchased, as prevously stated. 9 As usually, the equlbrum wll be gven by the frst order condtons, dfferentatng the Lagrangean of the optmsaton problem (denoted by L) n order to the consumpton: (7a) (7b) L ζ S 6 ζ 3 S 8 6 = rτ ps U ζ, S λe qs = 0 I + L = r W e q S ds = t ζ S λ τ 6 0 From (7a), one can easly conclude that n equlbrum the followng condton must hold: ζ 3 S 8 (8) U ζ, S = λe rτ 6 qs ps In order to estmate the Arrow-Pratt absolute rsk averson measure, the second dervatve of the utlty functon wll be requred. From (8) ths corresponds to: ζ 3 S 8 (9) U ζ, S = λe q S p S q S p S rτ ps herefore, the Arrow-Pratt absolute rsk averson wll be: 9 Notce that n the objectve functon t s consdered the SD n order to weght the utlty values, whle n the budget constrant t s used the rsk-neutral densty, gven that t corresponds to the prce of the state-contngent clams, as prevously stated. 6

9 (0) RA S U ζ S, S p S q S = = U ζ, S ps qs 3 S As referred at the begnnng of the secton, the rsk-averson functon can also be derved from an ntertemporal equlbrum model. In fact, let us consder a standard dynamc exchange economy wth dynamcally complete securtes markets, a sngle consumpton good, no exogenous ncome, where a representatve agent maxmses hs expected utlty at each date t n order to hs consumpton and savng decsons, subject to the usual budget constrants. 0 hs problem may be formalsed as follows: j () Max E U C t! 3 8# " $ δ t+ j j= 0 where δ s the tme constant dscount factor, C t+ j s the nvestor s consumpton n the perod t+j andu C t 3 + j 8 s the utlty of consumpton n that perod. One of the Euler condtons of the problem stated n () conssts n the soluton of the same optmsaton problem n a two-perod settng: () Max ζ UC t6+ δeuc t t+ 6 R s.t. C = e P ζ t t t C = e + P ζ R t+ t+ t+ beng e the ncome, P R the prce of a fnancal asset provdng real cash-flows and ζ s agan the number of asset unts bought. Insertng the constrants n the objectve functon, the followng optmsaton problem arses: 0 As stated n Ingersoll (987), a complete market guarantees that a representatve nvestor exsts, though not necessarly the utlty functon of the representatve nvestor. 7

10 (3) Max ζ t, t+ 6 t t 7 t t+ t+ 7 UC C = Ue R Pζ + R δeue + P ζ One of the Euler condtons wll then be: (4) U ζ R R = PU t Ct = δet U Ct+ Pt+ 0 herefore, n equlbrum the margnal utlty of consumng one real monetary unt less at tme t must be equal to the dscounted expected value of the margnal utlty of consumng at tme t+ the proceeds of an nvestment of P R unts at tme t n the fnancal asset. monetary Followng equaton (4), the basc prcng or the consumpton CAPM (CCAPM) equaton s easly obtaned: R (5) P E P R D t = t t+ t + where D t+ 6 U Ct = δ U C + t6 s the ntertemporal margnal rate of substtuton, the stochastc dscount factor (sdf) or the prcng kernel. Gven that real assets are usually scarce, equaton (.5) f often adapted to nomnal assets. Followng Campbell et al. (997), f the nomnal prce ndex at he most well known are nflaton-ndexed Government bonds and they exst only n a few countres. he UK nflaton-ndexed Government bond market s the most promnent and ts nformaton content has been studed n several papers (see, for nstance, Deacon and Derry (994) and Remolona et al. (998)). 8

11 tme t s denoted by Qt, wth Π t we get: + Qt + = beng the rate of nflaton from t to t+, Q t (6) P Q E P t t + Q D t t t " =! + $ # t + Multplyng (6) by Qt, an equaton smlar to (5) s obtaned for nomnal bond prces, wth the stochastc dscount factor M t + Dt + = Π. t + For call-opton prces, equaton (6) corresponds to: (7) C = E max S 0 M t Pt,, where C()t s the prce n t of an European call-opton wth expry date, S s the underlyng asset prce at the expry date and the subscrpt P denotes that the expected value s computed usng the true or orgnal probablty measure P, represented by a densty functon p. As stated above, wthn the consumpton-based CAPM, the stochastc dscount factor s the nomnal ntertemporal margnal rate of substtuton, denoted by MRS t,. herefore, from equatons (6) and (7) t s obtaned: (8)! U C C 0 5 E S t t 6 6 max, 0 δ U C Q t6 " =! Q $ # t 6 t, t, = E max S 0 MRS " $ # 9

12 where MRS t, U C 6 = δ U C t6 Q Q t. In order to compute the expected value n (8), one has to use the densty p t related to the probablty measure P: I C = max S MRS p S ds t, 0 t, t( ) 0 I MRSt, pt( S) = maxs, 06 e 0 (9) MRS p ( S ) I τ 0 rt, τ = e max S, 0 q ( S ) ds rt, ττ = e E max S, 0 Qt, I0 6 t, t t 6 rτ τ ds where r t,τ s the rsk-free rate n t for maturty τ (τ = Ττ) and q t MRStpt S = I, ( ) s alternatvely known as the rsk-neutral MRSt, pt( S) 0 probablty densty assocated to the probablty measure Q (see Cox and Ross (976)), the equvalent martngale measure (see Harrson and Kreps (979)) or the state-prce densty (SPD), beng, as referred n At-Sahala and Lo (000), the contnuous-state counterpart to the prces of Arrow-Debreu state-contngent clams. It can be easly concluded that qt(s ) s a probablty densty functon, as t assumes values only n the nterval between 0 and and ts ntegral s equal to. Moreover, as accordng to (9) the call-opton prce s the dscounted expected value of ts future pay-off, S t s a martngale n the probablty measure Q. Dfferentatng (9) n order to the strke prce, we obtan: hese assets were ntroduced n economcs by Arrow (964) and Debreu (959). hey are charactersed by payng one monetary unt n a gven state and nothng n all other states. Probablty densty functons could be drectly obtaned from the prces of Arrow-Debreu securtes f these were traded for every state. 0

13 (0a).e., C = e = e rτ τ rτ τ q( S ( ) ds q( S = ) ds ) rτ τ (0b) + e = P [ S ] C Q hus, we have the cumulatve probablty dstrbuton functon. Obvously, the densty functon wll be obtaned by the dfferentaton of the LHS of (0b): 3 r C( ) ττ () q( ) = e In order to get nformaton on the rsk-averson of the representatve agent, one has to compare the SD and the RND, for nstance, computng the rato between them: 6 6 qt Y () ζ t Y = p Y t 6 From (9) we see that ths rato s proportonal to MRS,.e.: 6 = = U ( Y) (3) ζ t Y θmrst, θ U ( Y ) t t 3 Notce that for put opton prces, the result s the same, as the only dfference s n the sgn of the frst element of the LHS of (0b).

14 Instead of lookng at ζ, we can extract nformaton on rsk-averson usng the Arrow-Pratt absolute rsk averson n (0), whch can be easly computed from ζ and ts frst dervatve, whch corresponds to: t t t t t 6 U ( Y) q Y p Y q Y p Y (4) ζ t Y = θ = Ut ( Yt) p Y Computng the symmetrc of the rato between ζ and ζ, we get: 6 ζ t Y U ( Y) U ( Y) U ( Y) (5) = θ θ = = RA Y ζ ty6 6 Ut ( Yt) Ut ( Y ) t U ( Y ) Consequently, usng equatons () and (4), we can estmate the rsk-averson functon from the probablty densty functons p and q: (6a).e. (6b) ζ t Y q Y p Y q Y p Y RAY 6 6 = = ζ t Y p Y 6 pt Y qt Y RAY 6 6 = p Y q Y t 6 6 t t t t t t 6 t t 6 q Y p Y 4. he representatve agent s preferences Rubnsten (994) shows that, n a general equlbrum model, any two of the followng mply the thrd: () the preferences of the representatve agent; () the asset s stochastc process; and ) the RND. Consequently, one can dentfy the asset s stochastc process n accordance wth a gven utlty functon (see, e.g., Bck (990), Wang (993) and He and Leland (993)). Instead, the stochastc

15 process may be obtaned f specfc preferences are assumed (see, e.g., Derman and Kan (994) and Dupre (994)). herefore, a relevant queston arsng s the dentfcaton of the utlty functon compatble wth the rsk-averson functon to be estmated. For nstance, f the RND shape s smlar to a log-normal densty, then there s evdence that Black- Scholes model holds. As shown n the lterature (see, e.g., Bck (987)), ths mples that the representatve agent has Constant Relatve Rsk Averson (CRRA) preferences, resultng from the followng utlty functon: 05 (7) UY = % & K ' K λ Y, f λ λ ln05 Y, f λ = beng λ a nonnegatve parameter representng the level of relatve rsk averson, as can be easly seen from the resultng rsk-averson functon: (8) RA Y 0 5 = λ Y Other useful utlty functons to consder are those ncluded n the lnear rsk tolerance (LR) or hyperbolc absolute rsk averson (HARA) class. 4 hese are defned as: (9) UY 0 5 = λ λ ay + b λ λ, wth b >0, and the rsk-averson functons have the followng specfcaton: 3

16 (30) RA 05 a 0 γ 5 Y = ay + b γ 0 5. Another nterestng class of utlty functons s the negatve exponental class: (3) UY 05 =e λy whch gves a constant absolute rsk-averson λ. 5. he estmaton methodology In order to estmate the rsk-averson functons, the estmaton of both probablty denstes above mentoned s requred. Regardng the SD, a kernel estmaton of the returns densty was performed. 5 As referred n Rosenberg and Engle (997), assumng a kernel smoothng of the past returns as a good proxy for future returns may be seen as a martngale assumpton for prcng kernel. hs technque s very useful as t permts to estmate the SD wthout avodng to mpose a parametrc structure on the representatve agent s utlty functon. As we are nterested n the densty of prces, nstead of returns, some transformatons were requred. Usng the densty functon of the returns, denoted by p v(v), t s possble to compute the dstrbuton functon of Y : 6 4 he rsk-tolerance functon s just the nverse of the rsk-averson functon. 5 hs method was also used n Jackwerth (997) and At-Sahala and Lo (000). 6 For detals on the kernel estmaton of the returns densty, see appendx B. 4

17 logyyt 6 (3) Prob Y < Y ut = Prob Y e Y = Prob u log Y Y = p v dv 6 t 7 t t6 v j I j Usng the defnton of probablty densty functon and the Lebnz rule for dfferentatng an ntegral, we get: (33) ProbY Y py ( ) = 6 = Y 3 8 log YYt = pv vj Y 6 YYt 6 p v dv log I v3 j8 Y v3 j8 = p v YY = Y t j t p v 3vj8 Y Regardng the estmaton of the RND, several methods may be found n the lterature. Some estmate the RND usng non-parametrc technques,.e., avodng to mpose any specfcaton on the stochastc process of the fnancal asset, the opton premum functon n order to the strke prce, the mpled volatlty or the RND. he most straghtforward way to get the RND from opton prces s usng the prces of state-contngent clams or Arrow-Debreu securtes. hough these securtes are not usually avalable n fnancal markets, one can construct them from the opton prces. 7 Another non-parametrc technque s presented n Jackwerth and Rubnsten (996), where a dscrete approxmaton to the fourth dervatve of the opton prce functon n order to the strke prce s mnmsed. hough these methods provde more mmedate results, the densty functons obtaned are frequently too rregular. 8 7 See appendx C. 8 For more detals see, e.g., Adão et al. (997), Bahra (996), Melck and homas (997) or S derlnd and Svensson (997). 5

18 6 Alternatvely, one can estmate the RND mposng a parametrc specfcaton to some functon related to the RND, such as the prce or the volatlty curve (see, e.g., Shmko (993)). A very popular parametrc technque, gven ts flexblty and promptness, has been the fttng of a mxture of two log-normal dstrbutons, solvng the followng optmsaton problem: (34) ( ) ( ) 0 ^ 0 ^,,,, ) (,, = = S e e e P P C C Mn rt N N α α θ α α θ θ τ τ s.t., >0 e 0 θ. beng (35) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = + = + + ln ln ln ln ) ))( ;, ( ) ( ) ;, ( (, ˆ α α θ α α θ α θ α θ τ α τ α τ τ r r r N N e e N N e e ds S S L S L e C (36) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) + + = = + = + + ln ln ln ln ) ))( ;, ( ) ( ) ;, ( (, ˆ α α θ α α θ α θ α θ τ α τ α τ τ r r r N N e e N N e e ds S S L S L e P and where ( ) L S α, ; s the log-normal densty functon ( =, ), the parameters α and α are the means of the respectve normal dstrbutons, and

19 are the standard-devatons of the latter and θ the weght attached to each dstrbuton. Melck and homas (997) developed a smlar method for the estmaton of the RND usng Amercan opton prces, based on a mxture of three log-normal denstes and on bounds constructed upon European opton values. he RND estmates presented n the followng secton were obtaned wth a lnear combnaton of two log-normal dstrbutons, both for European and Amercan optons, usng call and put opton prces for each market and maturty dates Data and results he SD functons were estmated usng monthly average Brent and WI ol prces collected by Datastream and Bank of Portugal, from January 977 and February 98, respectvely, to each maturty date consdered. hough there are some more complcated methods to determne the optmal bandwdth (h*), we adopted a smlar rule to Jackwerth (997),.e., h* = σn 5, where σ s the standard devaton of the seres to be smoothed and n the number of observatons. 0 he RND functons for the Brent prce were estmated usng call and put opton prces of the Brent Crude Optons traded at Internatonal Petroleum Exchange, for the followng dates of 999: Aprl, May, July and August. For 9 Some of the observed prces were elmnated from the database, whenever a concavty n the opton prce functon appeared. 0 As t s referred n Jackwerth (997), solvng an optmsaton problem, n order to the bandwdth, to obtan the smoothest dstrbuton consstent wth the observed returns gves smlar results but s sgnfcantly slower. hough these optons are Amercan, they may be treated as European, gven that the premum s pad only at expry date and margns are adjusted daly on a market-to-market bass. herefore, 7

20 WI prce, call and put opton prces observed on September 999 for several maturtes were used. Gven the restrctons on data avalablty, Brent ol data was consdered for the analyss of tme and maturty evoluton of RND and rsk-averson functons, whle WI ol data s rcher for gettng nformaton on longer maturtes. herefore, our goal was not to compare results obtaned for both ol prces, but nstead to use them as complementary. Frst, we dentfed the evoluton of rsk averson functons along tme, unquely usng optons wth two months to maturty for the Brent,.e., respectvely the July, August, October and November contracts for each date referred. Consequently, two-month returns were used. he Brent kernel denstes estmated exhbt a very smlar shape (see chart ). Regardng the prce denstes, as we orgnally estmated the densty of the returns and prces ncreased along the perod consdered, they evdence sgnfcant dfferences (chart ), as well as notceable departures from the lognormal shape. 3 Chart 3 shows the kernel smoothng of Brent two-month returns up to August. he Brent RND functons estmated are presented n chart 4. hey show a rghtward movement, wth some varance reducton. As we can see from charts 5 to 8, the Brent denstes on August evdence a strkng dfference regardng the frst dervatve n the left tal of the dstrbuton, whch, accordng to equatons (0) or (7b), can be taken as an ndcator of a rsk premum ncrease. they may be consdered as ncludng a futures contract and a plan-vanlla European opton for the maturty date. Notce that ol opton contracts expry n the prevous month to the settlement. 3 Accordng to Sundaresan (984), equlbrum spot prces are lognormally dstrbuted along the optmal extracton path. 8

21 In Chart 9, Brent rsk-averson functons for several dates and two-month maturty are presented. 4 It can be seen that these functons have a very rregular shape and some negatve values, whch s n contrast wth the result expected for a rsk-averse representatve agent. However, these values are consstently around zero. hese rregulartes may arse from some of the assumptons mposed, concernng namely the RND estmaton technque, the kernel bandwdth or the proxy chosen for the SD, and/or from optons msprcng. 5 he most strkng result s the rsk-averson ncrease for lower wealth levels n July 999. In ths date, our estmatons suggest that for an expected Brent prce of USD 8, the rsk averson would be crca USD 3,5, whle t was around zero n Aprl 999. In order to transform the rsk-averson n a value for the rsk premum, a fgure for the scalng factor n equaton (5) must be used. Approachng var05 ε by the product between the varance of returns and the squared strke prces, the rsk premum would be near.4 USD for the same expected Brent prce n July 999. For hgher prces, closer to the futures levels, the rsk premum s vrtually nl, though t decreases at a slower pace on August 999. When comparng Brent RNDs for several maturtes n the last day reported, one can conclude that rsk-neutral expected values pont to a Brent prce decrease (chart 0). Another relevant dfference regards the hgher postve skewness n the longest maturtes consdered. Smlar conclusons are obtaned n what concerns to expectatons of WI prces (chart ). 4 he densty dervatves were computed by arthmetc approxmaton. Gven that a very thn grd was used, t can be taken as a reasonable proxy for the analytc dervatve. 5 See Jackwerth (997) for possble explanatons of hs estmaton results. he shape of rskaverson functons estmated s smlar to those obtaned by At-Sahala and Lo (000). 9

22 Comparng the Brent rsk-averson functons, our estmates show sgnfcant rskaverson only for the one-month maturty and for the lowest strke prces (chart ). In fact, for an expected value around USD 4, the rsk-averson s around USD 4. Usng the prevously mentoned approach for the rsk premum, ts value for the same expected prce s close to USD 3. Addtonally, t can be concluded once more that the rsk averson s around zero for expected values above USD 5. Decreasng rsk-averson functons n the terms to maturty are n accordance wth the stylsed fact that n most market days the ol futures curve exhbts a negatve slope. 6 It can be also nterpreted as evdencng expectatons that the current prce ncrease s perceved as a short-term phenomenon. he fgures around zero for the rsk-averson may be consdered n lne wth what could be antcpated, as they respect to short maturtes. 7 However, low rsk-averson levels are also obtaned when WI optons wth longer maturtes are consdered (chart 3), wth slghtly hgher values for shorter maturtes. Our results of roughly nl rsk-averson levels seem to be corroborated by the dfferences between the Brent futures prces and the ex-post realsed spot prces. In fact, accordng to chart 4, the average dfferences are around zero for the whole term spectrum avalable and there s an upward sloped term structure of standard-devaton (chart 5). hus, assumng estmaton errors wth zero mean and a gven standard devaton, those average dfferences may be taken as a proxy for the rsk premum. 6 Accordng to Ltzenberger and Rabnowtz (995), between February 984 and Aprl 99, the nne months West exas Intermedate futures prce was backwardated (.e. had a negatvely sloped futures prce curve) most of the tme (more than 75% of the tme). An dentcal concluson s stated n Consdne and Larson (996) for

23 hough almost all rsk-averson functons vary around zero, the curvature of the Brent rsk-averson functon estmated for the one-month maturty n the th August seems to be too pronounced to allow an adequate fttng by a rskaverson functon mpled by a CRRA utlty functon. Conversely, ts shape s not too far from that mpled by a HARA utlty functon (chart 6) Conclusons Informaton on nvestor s preferences may be extracted from spot and opton prces. From Brent ol data we concluded that rsk averson s typcally small for levels near the futures prces, for terms to maturty up to four months. Usng data on WI ol prces, the results obtaned suggest that the rsk premum s stll close to zero for terms up to around 4 months. Consequently, one can argue for usng futures prces as a proxy for the expected value of ol prce. he results seem to be confrmed by the average dfferences between the Brent futures prces and the ex-post realsed spot prces, whch vary around zero. he rsk-averson functons estmated seem to be compatble wth HARA utlty functons, though they exhbt hgh volatlty n the tals. More robust conclusons may be obtaned by estmatng non-parametrcally the dstrbuton of the rsk-averson functons, e.g. usng bootstrappng methods. 7 Brent optons for hgher maturtes are not suffcently lqud to be consdered. 8 he mpled HARA utlty functon results from the parameters offerng the lowest sum of squared dfferences to the estmated rsk-averson functon.

24 References Adão, Bernardno, Nuno Cassola and Jorge Barros Luís (997), Extractng nformaton from optons prema: the case of the return of the Italan Lra to the ERM of the EMS, Banco de Portugal, Quarterly Economc Bulletn, December. At-Sahala, Yacne and Andrew W. Lo (000), Nonparametrc Rsk Management and Impled Rsk Averson, Journal of Econometrcs, No. 94, pp Arrow, Kenneth (964), he Role of Securtes n the Optmal Allocaton of Rsk Bearng, Revew of Economc Studes, 3, pp Bahra, Bhupnder (996), Impled Rsk-Neutral Probablty Densty Functons from Opton Prces: heory and Applcaton, Bank of England Economc Bulletn, August 996. Bck, Av (990), On Vable Dffuson Prce Processes of the Market Portfolo, he Journal of Fnance, Vol. LV, No., June. Bck, Av (987), On the Consstency of the Black-Scholes Model wth a General Equlbrum Framework, Journal of Fnancal and Quanttatve Analyss, No., pp Breeden, Douglas e Robert Ltzenberger (978), Prces of State-Contngent Clams Implct n Opton Prces, Journal of Busness, October 978, 5(4), pp. 6-5.

25 Campbell, John. Y., Andrew W. Lo e A. Crag MacKnlay (997), he Econometrcs of Fnancal Markets, Prnceton Unversty Press. Consdne, mothy J. and Donald F. Larson (996), Uncertanty and the Prce for Crude Ol Reserves, mmeo. Coutant, Sophe (999), Impled Rsk Averson n Opton Prces, mmeo. Cox, John and Stephen Ross (976), he Valuaton of Optons for Alternatve Stochastc Processes, Journal of Fnancal Economcs, 3, Debreu, G. (959), heory of Value, John Wley and Sons, New York. Derman, E. and I. Kan (994), Rdng on the Smle, Rsk, 7, February, Dupre, B. (994), Prcng wth a smle, Rsk, 7, January, 8-0. Harrson, M. and D. Kreps (979), Martngales and Arbtrage n Multperod Securtes Markets, Journal of Economc heory, 0, He, Hua and Hayne Leland (993), On Equlbrum Asset Prce Processes, Revew of Fnancal Studes, No. 6, pp Huang, Ch-fu and Robert H. Ltzenberger (988), Foundatons for Fnancal Economcs, North-Holland. Hull, John (997), Optons, Futures, and Other Dervatves (hrd Edton), Prentce- Hall. 3

26 Ingersoll, Jonathan E. (987), heory of Fnancal Decson Makng, Rowman and Lttlefeld. Jackwerth, Jens Carsten (997), Recoverng Rsk Averson from Opton Prces and Realzed Returns, Insttute of Fnance and Accountng Workng Paper 6. Jackwerth, Jens C. e Mark Rubnsten (996), Recoverng Probablty Dstrbutons from Opton Prces, Journal of Fnance, Dec.96, Vol.5, No.5, pp Ltzenberger, Robert H. and Nr Rabnowtz (995), Backwardaton n Ol Futures Markets: heory and Emprcal Evdence, Journal of Fnance, Vol. L, No. 5, December. Melck, Wllam R. and Charles P. homas (997), Recoverng an Asset s Impled PDF from Opton Prces: An Applcaton to Crude Ol durng the Gulf Crss, Journal of Fnancal and Quanttatve Analyss, Vol. 3, No., March. Neuhaus, Hans (995), he nformaton content of dervatves for monetary polcy: mpled volatltes and probabltes, Deutsche Bundesbank Economc Research Group, dscusson paper No. 3/95. Rosenberg, Joshua and Robert F. Engle (997), Opton hedgng usng emprcal prcng kernels, NBER WP No. 6. Rubnsten, Mark (994), Impled Bnomal rees, Journal of Fnance, No. 49, pp Shmko, Davd (993), Bounds on probablty, Rsk, Vol. 6, No. 4, pp

27 S derlnd, Paul e Lars E.O. Svensson (997), New echnques to Extract Market Expectatons from Fnancal Instruments, Journal of Monetary Economcs 40, p.p Sundaresan, Suresh (984), Equlbrum Valuaton of Natural Resources, Journal of Busness, Vol. 57, No. 4. Wang, S. (993), he Integrablty Problem of Asset Prces, Journal of Economc heory, No. 59, pp

28 Appendx A - Condton for rsk-averson If x s a random varable wth densty functon denoted by f(x) and mean represented by x and G s a strctly concave functon of x wth fnte dervatves of n-th order n the regon between x and x (x = x + h), then usng aylor seres we have: (A) n G x = G x + x x G x + x x G x x x = Gx 6+ x xg 6 x6+ x x6 G x* 6 wth x * between x and x. n G 6 n! x hen (A) EGx I + 6 = 66 Gx f xdx I I I = G x f x dx + G x x x f x dx + G x * x x f x dx Gven that the frst ntegral on the RHS s equal to one (as t s the sum of all densty values) and the second one s nl (as t s the dfference between the expected value of x and tself), we have: I (A3) EGx = Gx + G x* x x f xdx 6 6 As G s a strctly concave functon, G < 0 and EGx < Gx herefore, EUW+ ε < U EW+ ε = UW 6

29 Appendx B - Kernel estmaton 9 Our goal s to obtan a suffcently smooth probablty densty functon from the observed values of a seres. herefore, we have to get the hstogram from the seres and smooth the hstogram. 30 he probablty densty functon s the frst dervatve of the dstrbuton functon. hus, for each value of the varable the densty may be computed from the dfference between two close values of the dstrbuton functon: ob V v h ob V v h ob v h V v h (B) pv05 0 < < 5 < < + v = lm Pr Pr = lm Pr h 0 v+ h vh h 0 h beng h usually known as the bandwdth. A common proxy for the probablty of a gven value s the rato between the number of observatons nsde the bn centred on that value (# 05), v wth bandwdth h, and the total number of observatons (n). By defnton, the area of the hstogram s equal to one, as the hstogram s the dscrete representaton of the densty functon: + (B) pv vs h = s= * 6 beng p * v 6 the value of the hstogram n a bn centred on v s. hus, one conclude that v + (B3) pv vs = h s= s * 6 9 hs appendx s based on lecture notes by Yacne At-Sahala. 7

30 As the densty values correspond to the probabltes multpled by a constant (and the sum of all probabltes s one), the hstogram may be estmated as followng: (B4) s6 05 pv v = # h n From (B) or (B4), an estmator for a centred hstogram s: v n v (B5) p * v v = # = # h nh hs densty estmator gves the same postve weght to all observatons n the bn and a zero weght to the remanng observatons. In fact, f we defne a unform densty functon as (B6) % 05= & Ku u < ' / f 0 otherwse the estmator for the centred hstogram n (B5) may be wrtten as (B7) pv *0v5= nh n = K v V h From (B7) t s clear that when the dstance between each V and the v for whch the densty s beng computed s hgher than h,.e., when the observatons are outsde the bn, the weght s nl. Otherwse, the observatons get a unform 30 Kernel smoothng technques can also be mplemented to flter a seres, nstead of ts hstogram 8

31 weght of /. One can also see from (B7) that an estmator for the densty value on each Y corresponds to the average of all terms h K v V h the kernel densty for all V ( =,, n) dvded by the bandwdth.,.e., the values of In order to obtan a smooth densty functon, t s more adequate to use a smooth kernel, beng the Gaussan kernel the most popular: (B8) Ku 05= e π u herefore, n order to get the Gaussan kernel densty value for a gven v, one has just to compute the average of the normal densty values for all observatons, wth v and h, respectvely, as the mean and the varance. (see, for nstance, Campbell et al. (997), chapter. 9

32 Appendx C State-contngent clams and opton prces Followng Breeden and Ltzenberger (978), a portfolo resultng from buyng two call-optons wth strke prce and sellng two put-optons, wth strke prces -ε and +ε, has a pay-off functon usually called butterfly spread. 3, 3 As we can see from Fgure, the pay-off s nl outsde the nterval [ ε + ε], and, when ε approaches zero, t becomes close to the symmetrc of the Drac functon centred on. 33 Fgure Pay-off functon of a butterfly spread pay-off -ε +ε Note: pay-off excludng opton prces. By defnton, the prce of the symmetrc of the butterfly spread presented s: (C) D ( ; ε) = [ C( + ε) C( ) ] [ C( ) C( ε) ] ε 3 In the present case, t s a short butterfly spread. Conversely, the symmetrc butterfly spread charactersed by buyng two call optons wth strke prces -ε and +ε, and sellng two call optons, both wth strke prce s a long butterfly spread (see, for nstance, Hull (997)). hs spread has a non-negatve pay-off, smlar to an nverted butterfly. 3 hese spreads are not traded n structured exchanges, but only n over-the-counter markets. 33 Functon that assumes n and 0 n the remanng strke prces. 30

33 Dvdng (C) by ε, the lmt of (C) when ε tends to zero s an approxmaton to the second dervatve of the call-opton prce functon,.e., accordng to (3), the RND dscounted by the rskless nterest rate: 34 (C) ( ; ε) [ ( + ε) ( )] [ ( ) ( ε) ] D C C C C C ( ) lm = lm = ε 0 ε ε 0 ε In dervatve exchange traded optons, strke prces are spaced by small ntervals, though not necessarly close to zero. hus, Neuhaus (995) proposes the followng dscrete approxmaton: C( ) + (C3) ( ) C( ) C( ) + C( ) C( ) + 34 If the premum s pad only at redempton, the dscount factor n () becomes one. 3

34 Chart Kernel denstes of -month returns of Brent Ol Barrel Probablty densty Returns Chart Kernel denstes of -month forward prces of Brent ol barrel Probablty densty Prces (n USD)

35 Chart 3 Kernel Densty Estmator of -month returns of Brent ol barrel January 977-August Probablty densty Kernel densty Frequency Frequency Returns 0 Chart 4 RND functons of -month forward prce of Brent ol barrel Probablty densty Prces (n USD)

36 Chart 5 RND and SD of -month forward prce of Brent ol barrel n RND RND SD Prces (n USD) SD Chart 6 RND and SD of -month forward prce of Brent ol barrel n RND RND SD SD Prces (n USD) 0.000

37 Chart 7 RND and SD of -month forward prce of Brent ol barrel n RND RND SD SD Prces (n USD) Chart 8 RND and SD of -month forward prce of Brent ol barrel n RND.5 RND SD SD Prces (n USD) 0.000

38 Chart 9 Rsk-averson functons of -month forward prces of Brent ol barrel 5 4 Rsk-averson (n USD) Prces (n USD) Chart 0 RND of forward prces of Brent ol barrel n Probablty densty Prces (n USD) m m 3m 4m 5m 6m

39 Chart Observed WI ol prces and rsk-neutral expectatons $ 5 5 $ Jan-84 Jan-85 Jan-86 Jan-87 Jan-88 Jan-89 Jan-90 Jan-9 Jan-9 Jan-93 Jan-94 Jan-95 Jan-96 Jan-97 Jan-98 Jan-99 Jan-00 Jan-0 Chart Rsk averson and rsk premum mplct n Brent ol futures optons and spot prces n for several terms to maturty Rsk averson (n USD) Rsk premum (n USD) - Prces (n USD) - m m 3m 4m 5m 6m m-rsk premum

40 Chart 3 Rsk-averson functons for WI ol prce n for several maturtes Rsk averson (n USD) Rsk averson (n USD) Dez-99 Mar-00 Jun-00 Dez-00 Dez-0 Chart 4 Average dfferences between futures prces and ex-post realsed spot prces for Brent ol $ erm to maturty (n days)

41 Chart 5 Standard-devatons of the dfferences between futures prces and expost realsed spot prces for Brent ol $ erm to maturty (n days) Chart 6 Rsk-averson and utlty functons n for Brent ol prces (onemonth maturty) 5 Rsk-averson (n USD) 4 estmated rsk-averson HARA Rsk-averson Prces (n USD)