On Estimating an Asset's Implicit Beta

Transcription

1 On Estimting n Asset's Implicit Bet Sven Husmnn Europen University Vidrin Frnkfurt Oder Deprtment of Business Administrtion nd Economics Discussion Pper No ISSN

2 Europen University Vidrin Frnkfurt Oder, Germny Deprtment of Business Administrtion nd Economics Discussion Pper No. 238 June 2005 On Estimting n Asset s Implicit Bet Sven Husmnn Abstrct Siegel 995 hs developed technique with which the systemtic risk of security bet cn be estimted without recourse to historicl cpitl mrket dt. Insted, bet is estimted implicitly from the current mrket prices of exchnge options tht enble the exchnge of security ginst shres on the mrket index. Becuse this type of exchnge options is not currently trded on the cpitl mrkets, Siegel s technique cnnot yet be used in prctice. This rticle will show tht bet cn lso be estimted implicitly from the current mrket prices of plin vnill options, bsed on the Cpitl Asset Pricing Model. Prof. Dr. Sven Husmnn, Assistnt Professor for Interntionl Accounting, Deprtment of Business Administrtion nd Economics, Europ-Universität Vidrin Frnkfurt Oder, Große Schrrnstrße 59, 5230 Frnkfurt Oder, Germny. E-mil: husmnn@euv-frnkfurt-o.de.

3 Introduction The Cpitl Asset Pricing Model CAPM of Shrpe 964 nd Lintner 965 continues to be of centrl importnce to the vlution of risk-bering securities, in theory s well s in prctice. According to the CAPM, the expected rte of return on security depends primrily on its systemtic risk bet, which is normlly estimted by mens of regressive nlysis of historicl cpitl mrket dt. Of ll of the numerous empiricl tests of the CAPM, the study by Fm nd French 992 in prticulr generted much ttention. According to this study, bet hs hrdly ny explntory power for the expected rte of return on security. In fct, the expected rte of return depends much more on the size of compny nd the book-to-mrket rtio. Berk 995 showed nonetheless tht these effects cn lso be trced bck to flwed mesurement of bet. Sttisticl errors cn be cused in prticulr by the fct tht bet chnges through time. 2 In order to void this problem, Siegel 995 proposes method with which bet cn be estimted from current options prices, without recourse to historicl cpitl mrket dt. However, prcticl ppliction of this method requires tht exchnge options be trded tht entitle the exchnge of securities for shres on the mrket index. Presently, such options re not trded on the cpitl mrket. The purpose of this pper is to universlize Siegel s method so tht bet cn lso be estimted from plin vnill options. The Siegel 995 method is bsed on estimtion of implicit voltility ccording to Ltné nd Rendlemn 976, whose technique is considered the stndrd in option pricing tody. 3 Siegel 995 ties this technique together with the vlution of exchnge options ccording to Mrgrbe 978 in order to estimte implicit bet. Siegel 997, Cmp nd Chng 998, nd Wlter nd Lopez 2000 use similr pproches to obtin implied correltion of currencies from currency options. Recently, Skintzi nd Refenes 2005 propose method in forecsting future index correltion clled implied correltion index tht is lso bsed on current option prices. In this rticle the Siegel 995 method will be universlized in tht the implicit density function of n underlying sset is estimted implicitly from the theoreticl CAPM prices of plin vnill options. 4 The bet of n underlying sset results from the moments of this density function. The theoreticl bsis for clcultion of implicit, risk-neutrl density functions origintes from Ross 976 nd Breeden nd Litzenberger 978 nd hs been used in numerous works to this dy: Rubinstein 994, Drmn nd Kni 994, Jckwerth nd Rubinstein 996, nd Brown nd Toft 999 estimte implicit riskneutrl density functions with the help of modified binomil model Implied Binomil Trees. Shimko 993, Jrrow nd Rudd 982, nd Longstff 995 estimte the price functions of options directly from their observed mrket prices, in dependence on Fm nd French 2004 discuss the empiricl problems tht my be cused by difficulties in implementing vlid tests of the CAPM. 2 Skintzi nd Refenes 2005 nd Longin nd Solnik 200, for exmple, observed tht correltions of stocks returns increse in highly voltile or ber mrkets. 3 See Blir et l. 200 for recent studies on the predictive bility of implied voltility. 4 Dennis nd Myhew 2002 investigted the reltive importnce of bet in explining the prices of stock options trded on the Chicgo Bord Options Exchnge.

4 the exercise price, nd from there derive risk-neutrl density functions. Aït-Shli nd Lo 2000 nd Jckwerth 2000 determine cler difference between risk-neutrl nd subjective expecttions nd ttempt to drw conclusions from this regrding the risk version of mrket prticipnts. Jckwerth 2000 rrives furthermore t the result tht the historicl cpitl mrket rtes of return re pproximtely lognormlly distributed. The nottion nd model ssumptions re explined in section 2. In section 3, model is presented with which clls cn be evluted bsed on the CAPM when rtes of return re distributed lognormlly. On this bsis, it is possible to estimte bet implicitly from the prices of ordinry clls in section 4. Section 5 summrizes the results of the rticle. 2 Assumptions nd Nottion The vlution of options in section 3 is bsed on the ssumptions of the one-period CAPM:. Risk-verse investors mximize the µ-σ-utility of their end-of-period welth. 2. Investors hve homogeneous expecttions bout ssets returns; the instntneous rte of return on ny sset nd the mrket portfolio hve joint norml distribution. 5 Investors my borrow or lend unlimited mounts t the risk-free rte. 3. Mrkets re frictionless. Informtion is costless nd simultneously vilble to ll investors. There re no mrket imperfections such s trnsction costs, txes, or restrictions on short selling. The following nottion is used throughout the pper: K Exercise price on n option p X c Price of cll on n sset S with cshflow X c p X cm Price of cll on the mrket index X m with cshflow X cm p X ce Price of n exchnge option cll with cshflow X ce p X s Price of n underlying sset S with cshflow X s p X m Current Mrket index n s Number of shres of sset S to be exchnged under the exchnge option n m Number of shres of the mrket index under the exchnge option R s Stndrdized cshflow of n underlying sset, R s = X s / p X s R m Stndrdized cshflow of the mrket portfolio, R m = X m / p X m β s Bet of n underlying sset S with respect to the mrket index r f Instntneous risk-free rte of interest r s Instntneous rte of return on sset S r m Instntneous rte of return on the mrket index µ s Expected instntneous rte of return on sset S 5 For definition of bivrite norml distribution, see Appendix A. 2

5 µ m Expected instntneous rte of return on the mrket index σ S Instntneous vrince of the rte of return on sset S σ m Instntneous vrince of the rte of return on the mrket index ρ Instntneous correltion between the rtes of return on sset S nd on the mrket index In the cse of the given prmeters for bivrite norml distribution of rtes of return, the following pplies for the expected vlues, vrinces nd covrinces of the securities csh flow nd mrket portfolio s stndrdized csh flow 6 E [ X s ] = p X s e µs+ 2 σ2 s, E [ R m ] = e µm+ 2 σ2 m, 2 Vr [ R m ] = e 2µm+σ2 m e σ2 m, 3 Cov [ X s, R m ] = p X s e µ m+ 2 σ2 m+µ s + 2 σ2 s e ρ σ m σ s. 4 For the stndrd definition of bet, the following results in the cse of bivrite norml distribution 7 β s = Cov [ R s, R m ] Vr [ R m ] = eµ s+ 2 σ2 s e ρ σ s σ m e µ m+ 2 σ2 m e σ 2 m. 5 3 The Model 3. Option Pricing in n Incomplete Lognorml Mrket In n incomplete lognorml mrket the CAPM my be used for option pricing. 8 well-known certinty equivlent vlution formul of the single-period CAPM is 9 The p X c = E [ X c ] λ Cov [ X c, R m ] + r f where λ = E [ R m ] + r f Vr [ R m ]. 6 In order to be ble to pply this eqution to the vlution of cll, the expected csh flow of the cll nd the covrince between the csh flow of the cll nd the rtes of return on the mrket portfolio must first be determined. Under the ssumption of lognormlly 6 The moments of lognorml distribution cn be clculted with the help of the integrls 38, 39 nd 40 indicted in Appendix A. 7 For generl definition of bet, see Copelnd nd Weston 988, pg Options re redundnt securities in complete mrket. However, the empiricl results of Vnden 2004 indicte tht options re nonredundnt for explining the returns on risky ssets. 9 See Copelnd nd Weston 988, pg

6 distributed rtes of return, we derive 0 E [ X c ] = p X s e µ s+ 2 σ2 s Φ d K Φ d 2, 7 Cov [ X c, R m ] = p X s e µ s+ 2 σ2 s+µ m + 2 σ2 m e ρ σ s σm Φ d 3 Φ d K e µ m+ 2 σ2 m Φ d4 Φ d 2, 8 d = lnp X s /K + µ s /σ s + σ s, 9 d 2 = lnp X s /K + µ s /σ s, 0 d 3 = lnp X s /K + µ s /σ s + σ s + ρ σ m, d 4 = lnp X s /K + µ s /σ s + ρ σ m. 2 If we insert 7 nd 8 in 6, fter further conversion we get representtion tht llows comprison with the vlution eqution ccording to Blck nd Scholes 973 p X c = p X s θ K e r f θ 2 3 where θ = e µ s+ 2 σ2 s r f Φ d λ e µ m+ 2 σ2 m e ρσ s σ m Φ d 3 Φ d 4 θ 2 = Φ d 2 λ e µm+ 2 σ2 m Φ d4 Φ d 2. 5 This model cn be pplied to the specil cse of complete mrkets. On complete mrkets, risk-neutrl vlution lwys leds to the correct vlution result. 2 In risk-neutrl world, the rte of return of the expected csh flow of given risk-bering finncil title nd tht of the mrket portfolio equl the risk-free interest rte 3 µ s + σ 2 s/2 = r f, 6 µ m + σ 2 m/2 = r f. 7 This correltion cn lso be intuitively justified. Mrket prticipnts my only expect risk premium for their risk-bering finncil title if they cnnot nullify the risk through diversifiction of their portfolio. Becuse systemtic risk cn be nullified through diversifiction in complete mrkets, the mrket price of the risk is zero. From 6 nd 7 follows λ = 0, θ = Φ d nd θ 2 = Φ d 2. The vlution eqution 3 is reduced ccordingly with risk-neutrl vlution to p X c λ = 0 = p X s Φ lnp Xs/K+r f +σ 2 s/2 σ s Ke r f Φ which equls the vlution eqution of Blck nd Scholes See Appendix B. Put prices follow from put-cll prity. lnp Xs/K+r f σs/2 2 σ s, 8 Ritchken 985 developed similr vlution eqution for options bsed on the CAPM. This model is not consistent with the Blck nd Scholes 973 model in the cse of risk-neutrl vlution, however. 2 See Cox nd Ross If the expected instntneous rte of return on security equls µ s nd the rte of return is lognormlly distributed, then the rte of return on the expected csh flow equls µ s + σ 2 s. 4

7 3.2 Pricing Options on the Mrket Index Jrrow nd Mdn 997 developed vlution model for clls on the mrket index tht is lso bsed on the ssumptions of the CAPM nd lognormlly distributed rtes of return. If we use the nottion estblished bove, then the vlue of cll on the mrket index equls 4 p X cm = p X m θ m K e r f θ m2 20 where θ m = e µ m+ 2 σ2 m r f Φ dm λ e µ m+ 2 σ2 m e σ2 m Φ dm3 Φ dm, 2 θ m2 = Φ dm 2 λ e µm+ 2 σ2 m Φ dm Φ dm 2, 22 dm = lnp X m /K + µ m + σ 2 m / σ m, 23 dm 2 = lnp X m /K + µ m / σ m, 24 dm 3 = lnp X m /K + µ m + 2σ 2 m / σ m. 25 This vlution eqution is solely specil cse of 3; for clls on the mrket index, the following pply: ρ =, µ s = µ m nd σ s = σ m. 4 Implicit Bet 4. Estimting Bet Using Exchnge Options Siegel 995 ssumes tht continuous security trding on perfect cpitl mrkets is possible. 5 This stndrd ssumption of options price theory enbles risk-neutrl vlution of options nd is equivlent to the ssumption of complete cpitl mrkets. 6 Becuse the theoreticl option prices in the cse of risk-neutrl vlution re independent of the correltion of csh flow of the underlying sset with tht of the mrket portfolio, bet cnnot be implicitly estimted from simple options. Siegel 995 therefore recourses to 4 Jrrow nd Mdn 997 define the prmeter µ m s the rte of return of the expected vlue, while we use it to identify the expected rte of return. In order to estblish comprbility with our results, the prmeter µ must be replced with µ + σ 2 /2 in the work by Jrrow nd Mdn 997, p X cm = + b K p X m e µ m+ 2 σ2 m Φ dm KΦ dm 2 b p X m 2 e 2µ m+σ 2 m dm 3 9 where = e σ2 m r f e µ m+ 2 σ2 m / e σ2 m b = e µm+ 2 σ2 m r f / p X m e 2µm+σ2 m e σ2 m. If we furthermore ssume tht the investor s plnning horizon nd the time to mturity of the option re identicl, following elementry conversions, the vlution eqution 20 results from the vlution eqution 9. However, for the specil cse of clls on the mrket index, the Ritchken 985 model is not identicl with the Jrrow nd Mdn 997 model. 5 See Assumption in Siegel See Cox et l

8 exchnge options, which securitize the right for exchnge of finncil title for shres on the mrket portfolio. The theoreticl price of n exchnge option in terms of risk-neutrl vlution depends on the correltion of the csh flow of finncil title with the rtes of return of the mrket portfolio nd is therefore generlly suitble for determining implicit bet fctors. The risk-neutrl vlution of exchnge options is bsed on Mrgrbe 978, p X ce = n s p X n ln s p Xs + σ nm p Xm e/2 2 s Φ σ e n m p X n ln s p Xs σ nm p Xm e/2 2 m Φ σ e, 26 whereby the voltility σ e depends on the voltilities of the underlying ssets nd the correltion of their rtes of return, σ 2 e = Vr [ r s r m ] = σ 2 s + σ 2 m 2 ρ sm σ s σ m. 27 Siegel 995 ssumes tht three types of options re trded on the cpitl mrket: options on common sset, options on the mrket index, nd options tht entitle the exchnge of securities for shres on the mrket index. His ide for determintion of implicit bet fctors consists of first estimting the voltilities of the two underlying ssets nd the voltility σ e of the exchnge option implicitly from trded options. The correltion coefficient is then derived from correltion 27, ρ sm = σ 2 s + σ 2 m σ 2 e / 2 σ s σ m. 28 According to Siegel 995, this results in the bet fctor of the sset, β Siegel s := ρ sm σ s / σ m = σ 2 s + σ 2 m σ 2 e / 2 σ 2 m. 29 Lelnd 999 describes definition 29 s modified bet. Even in risk-neutrl vlution, this definition does not equl the stndrd definition of bet 7 β s = Cov [ R s, R m ] Vr [ R m ] = eρ σsσm e σ2 m. 30 Regrdless of this, from prcticl view there is the problem - s Siegel 995 himself notes - tht exchnge options re not currently trded on the cpitl mrkets. 4.2 Estimting Bet Using Plin Vnill Options On incomplete mrkets, bet cn be estimted implicitly with the vlution equtions 3 nd 20. As result of the stte of dt typiclly given on the cpitl mrket, two-stge process for estimting implicit bet is dvisble. In first step, expecttions of the mrket prticipnt with regrd to the mrket index re estimted. Bsed on the vlution eqution 20 for options on the mrket index, the sum of the squred reltive differences between the empiricl options prices p Xcm nd theoreticl options prices 20 is 7 Inserting 6 nd 7 in 5 results in 30. 6

9 minimized through selection of the prmeter ˆµ m nd ˆσ m, 8 J 2 p Xcm p Xcm min µ m,σ m p. 3 Xcm j= Bsed on the prmeters ˆµ m nd ˆσ m, estimted in the first step, the prmeters ˆµ s nd ˆσ s cn be determined with the sme method for ny sset S, J 2 p Xc p Xc min µ s,σ s p 32 Xc j= Through the ppliction of reltive insted of bsolute differences, it is voided tht in-the-money options influence estimtions of the prmeters much stronger thn out-ofthe-money options. In the minimiztion, the correltion coefficient ˆρ cnnot be estimted independently of the prmeters ˆµ s nd ˆσ s, s the CAPM equilibrium condition must be considered s n dditionl condition for the underlying sset, p X s = E [ X s ] λ Cov [ X s, R m ] + r f where λ = E [ R m ] + r f Vr [ R m ] Following severl conversions, inserting, 2, 3 nd 4 in 33 results in e σm 2 e µ s+σ 2 /2 = e r f. 33 e σ 2 m + e µ m+ 2 σ2 m r f e ρ σ mσ s. 34 Implicit bet 5 of n sset S cn be clculted with the estimted prmeters. 9 5 Summry nd Conclusions This rticle presents technique with which bet cn be estimted implicitly from the prices of plin vnill options, without recourse to historicl cpitl mrket dt. The fundmentl ide resembles tht of Ltné nd Rendlemn 976 in the estimtion of implicit voltilities from options prices: bet is estimted implicitly from options trded on the cpitl mrket, under the ssumption of normlly distributed rtes of return bsed on the CAPM. Sttisticl errors tht result from conventionl regressive nlysis of historicl dt, tht bet vlues chnge through time, cn hereby be voided. Likewise, s in the estimtion of implicit voltilities, the qulity of the implicit bet depends on how well the options price model tht is pplied cn explin the prices of the trded options. 8 This technique is lso pplied by Rubinstein 994 for the estimtion of implicit risk-neutrl density functions. 9 In order to be ble to clculte bet, 34 must be resolved ccordingly, e 2µ m+σ 2 m e σ2 m e ρ σ m σ s = ln µ s+ 2 σ2 s e r f + e µ m+ 2 σ2 m e r f nd inserted in 5. e µ m+ 2 σ2 m +µ s+ 2 σ2 s, 35 7

10 Appendix A: The Lognorml Distribution The definition of density of norml distribution is fx = x µ2 e 2σ πσ 2 Φ is the stndrd norml distribution µ = 0 nd σ =. A vrite is lognormlly distributed if its nturl logrithm is normlly distributed. The definition of bivrite norml distribution is fx, y = e 2 ρ 2 2π σxσ 2 y 2 ρ 2 x µx 2 σ 2 x «2ρ x µ xy µy + y µ y 2 σxσy σy Two vrites re bivrite lognormlly distributed if their nturl logrithms re bivrite normlly distributed. In order to be ble to clculte the moments of lognormlly distributed vrites, 2, 3 nd 4, the simplifictions of the following specil integrls re required: e cx fx dx = e cµ+ 2 cσ2 Φ e cx 2 fx, x 2 dx dx 2 = e cµ 2+ 2 cσ 2 2 Φ +µ+cσ 2 σ +µ +cρ σ σ 2 σ e x e cx 2 fx, x 2 dx dx 2 = e µ + 2 σ2 +cµ 2+ 2 cσ 2 2 +cρ σ σ2 +µ +σ Φ 2+cρ σ σ 2 σ In order to keep the proofs of 38, 39 nd 40 concise in the following, it is convenient to use the conditionl density. The definition of the conditionl density is fx 2 x = fx, x 2. fx If we pply this definition to the bivrite norml distribution, we get fx 2 x = σ 2 2 σ x µ e x2 µ2+ρ 2πσ 2 2 ρ 2 2σ 2 2 ρ2. 4 Note tht the conditionl density of the bivrite norml distribution equls the density of the norml distribution with the prmeters µ x2 x = µ 2 + ρ σ 2 x µ und σ 42 σx 2 2 x = σ2 2 ρ

11 We next prove eqution 38. e cx σ 2π x µ 2 e 2σ 2 dx = = = = σ 2π e σ 2π e σ 2π e σ 2π e = e cµ+ 2 cσ2 x 2 2xµ 2cσ 2 x+µ 2 2σ 2 dx x 2 2xµ+cσ 2 +µ 2 2σ 2 dx x 2 2xµ+cσ 2 +µ+cσ 2 2 µ+cσ 2 2 +µ 2 2σ 2 dx x µ+cσ 2 2 2σ 2 e µ+cσ2 2 +µ 2 2σ 2 dx σ 2π e x µ+cσ 2 2 2σ 2 dx Eqution 38 follows with Φ µ+cσ 2 σ = Φ +µ+cσ 2 σ. The proof for eqution 39 is given under considertion of the conditionl density indicted bove, e cx 2 fx, x 2 dx dx 2 = e cx 2 fx 2 x dx 2 fx dx. The integrl in brckets cn be interpreted in tht the expected vlue nd the vrince ccording to 42 nd 43 re trnsformed nd the eqution 38 is subsequently used, e cx 2 fx 2 x dx 2 fx dx = [e c µ 2 +ρ σ 2 x σ µ + 2 c2 σ2 2 ρ2 ] fx dx = e cµ 2 cρ σ 2 σ µ + 2 c2 σ c2 σ 2 2 ρ2 e cρ σ 2 σ x fx dx. If we define the helping vrible c := cρ σ 2 σ, we rrive t the eqution 39 fter ppliction of 38 nd shortening of the terms in exponents. The proof for eqution 40 cn be shown nlogously, e cx 2 e x fx, x 2 dx dx 2 = e cx 2 fx 2 x dx 2 e x fx dx = e cµ 2 cρ σ 2 σ µ + 2 c2 σ c2 σ 2 2 ρ2 e cρ σ 2 σ x +x fx dx. If we define the helping vrible c := cρ σ 2 σ +, we rrive t the desired result 40 fter repeted ppliction of eqution 38 nd shortening of the terms in exponents. 9

12 Appendix B: Option Pricing Using the CAPM In order to clculte the expected vlue of cll 7, we use eqution 38, E [ X c ] = mx p X s e r s K, 0 fr s dr s = p X s e r s fr s dr s K fr s dr s lnk/p Xs lnk/p Xs = p X s e µs+ 2 σ2 s Φ lnp Xs/K+µ s+σs 2 σ s KΦ lnp Xs/K+µ s σ s. 44 We cn simplify the clcultion of the covrince through ppliction of the decomposition theorem. From equtions 39 nd 40 result E [ X c Rm ] = mxp X s e rs K, 0 e rm fr s, r m dr m dr s = p X s e r s K e r m fr s, r m dr m dr s lnk/p Xs = p X s e r s e r m fr s, r m dr m dr s K e r m fr s, r m dr m dr s lnk/p Xs = p X s e µ s+ 2 σ2 s +µ m+ 2 σ2 m +ρ σ sσm Φ K e µ m+ 2 σ2 lnk/p m Φ X s+µ s+ρ σ sσ m σ s lnk/p Xs lnk/p X s +µ s +σs+ρ 2 σ s σ m σ s. 45 Following the decomposition theorem, we rrive t the covrince 8 with 44 und 45, fter elementry conversions. 0

13 References Aït-Shli, Y. nd Lo, A. W Nonprmetric Risk Mngement nd Implied Risk Aversion, Journl of Econometrics, 94, 9 5. Berk, Jonthn B. 995 A Critique of Size Relted Anomlies, Review of Finncil Studies, 8, Blck, Fischer nd Scholes, Myron 973 The Pricing of Options nd Corporte Libilities, Journl of Politicl Economy, 8, Blir, Bevn J.; Poon, Ser-Hung nd Tylor, Stephen J. 200 Forecsting S & P 00 voltility: the incrementl informtion content of implied voltilities nd high-frequency index returns, Journl of Econometrics, 05, Breeden, D. nd Litzenberger, R. 978 Prices of Stte-Contingent Clims Implicit in Option Prices, Journl of Business, 5, Brown, G. nd Toft, B. 999 Constructing Binominl Trees from Multiple Implied Probbility Distributions, Journl of Derivtives, 7, Cmp, Jose Mnuel nd Chng, P.H. Kevin 998 The forecsting bility of correltions implied in foreign exchnge options, Journl of Interntionl Money & Finnce, 7, Copelnd, T.E. nd Weston, J.F. 988 Finncil Theory nd Corporte Policy, 3rd edition, Addison-Wesley, New York. Cox, John nd Ross, Stephen 976 The Vlution of Options for Alterntive Stochstic Processes, Journl of Finncil Economics, 3, Cox, John; Ross, Stephen nd Rubinstein, Mrk 979 Option Pricing: A Simplified Approch, Journl of Finncil Economics, 7, Drmn, E. nd Kni, I. 994 Riding on Smile, RISK, 7, Dennis, Ptrick nd Myhew, Stewrt 2002 Risk-Neutrl Skewness: Evidence from Stock Options, Journl of Finncil nd Quntittive Anlysis, 37, Fm, Eugene F. nd French, Kenneth R. 992 The Cross-Section of Expected Stock Returns, Journl of Finnce, 47, The Cpitl Asset Pricing Model: Theory nd Evidence, Journl of Economic Perspectives, 8, Jckwerth, J. C. nd Rubinstein, M. 996 Recovering Probbility Distribution from Option Prices, Journl of Finnce, 5, Jckwerth, Jens Crsten 2000 Recovering Risk Aversion from Option Prices nd Relized Returns, Review of Finncil Studies, 3,

14 Jrrow, Robert A. nd Mdn, Dilip B. 997 Is Men-Vrince Anlysis Vcuous: Or ws Bet Still Born?, Europen Finnce Review,, Jrrow, Robert. A. nd Rudd, A. 982 Approximte Option Vlution for Arbitrry Stochstic Processes, Journl of Finncil Economics, 0, Ltné, Henry A. nd Rendlemn, Richrd J. 976 Stndrd Devitions of Stock Price Rtios Implied in Option Prices, Journl of Finnce, 3, Lelnd, Hyne E. 999 Beyond Men-Vrince: Performnce Mesurement in Nonsymmetricl World, Finncil Anlysts Journl, 55, Lintner, John 965 The Vlution of Risky Assets nd the Selection of Risky Investments in Stock Portfolios nd Cpitl Budgets, Review of Economics nd Sttistics, 47, Longin, Frnçois nd Solnik, Bruno 200 Extreme Correltion of Interntionl Equity Mrkets, Journl of Finnce, 56, Longstff, F. 995 Option Pricing nd the Mrtingle Restriction, Review of Finncil Studies, 8, Mrgrbe, Willim 978 The Vlue of n Option to Exchnge One Asset for Another, Journl of Finnce, 33, Ritchken, Peter H. 985 Enhncing Men-Vrince Anlysis with Options, Journl of Portfolio Mngement, 40, Ross, S. A. 976 The Arbitrge Pricing Theory of Cpitl Asset Pricing, Journl of Economic Theory, 3, Rubinstein, Mrk 994 Implied Binomil Trees, Journl of Finnce, 49, Shrpe, Willim F. 964 Cpitl Asset Prices: A Theory of Mrket Equilibrium Under Conditions of Risk, Journl of Finnce, 9, Shimko, D. 993 Bounds of Probbility, RISK, 6, Siegel, Andrew F. 995 Mesuring Systemtic Risk Using Implicit Bet, Mngement Science, 4, Interntionl Currency Reltionship Informtion Reveled by Cross-option Prices, Journl of Future Mrkets, 7, Skintzi, Vsiliki D. nd Refenes, Apostolos-Pul N Implied Correltion Index: A New Mesure of Diversifiction, Journl of Futures Mrkets, 25, Vnden, Joel M Options Trding nd the CAPM, Review of Finncil Studies, 7, Wlter, Christin A. nd Lopez, Jose A Is Implied Correltion Worth Clculting? Evidence from Foreign Exchnge Options, Journl of Derivtives, 7,