Discussion Papers. Sven Husmann Andreas Stephan. On Estimating an Asset s Implicit Beta

Transcription

1 Discussion Ppers Sven Husmnn Andres Stephn On Estimting n Asset s Implicit Bet Berlin, November 2006

2 This is preprint of n rticle ccepted for publiction in the Journl of Futures Mrkets, Copyright 2007 Wiley Periodicls, Inc, A Wiley Compny Opinions expressed in this pper re those of the uthor nd do not necessrily reflect views of the institute IMPRESSUM DIW Berlin, 2006 DIW Berlin Germn Institute for Economic Reserch Königin-Luise-Str Berlin Tel +49 (30) Fx +49 (30) ISSN print edition ISSN electronic edition Avilble for free downloding from the DIW Berlin website

3 On Estimting n Asset s Implicit Bet Sven Husmnn (Europen University Vidrin) Andres Stephn (Europen University Vidrin nd DIW Berlin) Abstrct Siegel (1995) 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 We provide empiricl evidence on implicit bets using prices of exchnge options from the EUREX over yers 2000 to 2004 JEL-Clssifiction: G12 Keywords: Cpitl Asset Pricing Model; Bet; Option Pricing Corresponding uthor: Prof Dr Sven Husmnn, Assistnt Professor for Interntionl Accounting, Deprtment of Business Administrtion nd Economics, Europ-Universität Vidrin Frnkfurt (Oder), Große Schrrnstrße 59, Frnkfurt (Oder), Germny E-mil: husmnn@euvfrnkfurt-ode

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5 1 Introduction The Cpitl Asset Pricing Model (CAPM) of Shrpe (1964) nd Lintner (1965) continues to be of centrl importnce to the vlution of risk-bering securities, in theory s well s in prctice The CAPM is still widely used in estimting the cost of cpitl of firms nd evluting their performnce Unfortuntely, empiricl findings of the CAPM re poor Empiricl problems my be cused by theoreticl problems, such s simplifying ssumptions, or by difficulties in implementing tests of the CAPM 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 ttention (1992) in prticulr generted much 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 (1995) showed nonetheless tht these effects cn lso be trced bck to flwed mesurement of bet 1 Sttisticl errors cn be cused in prticulr by the fct tht bet chnges through time 2 In order to void this problem, Siegel (1995) 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 (1995) method is bsed on estimtion of implicit voltility ccording to Ltné nd Rendlemn (1976), whose technique is considered the stndrd in option pricing tody 3 Siegel (1995) ties this technique together with the vlution of exchnge options ccording to Mrgrbe (1978) in order to estimte implicit bet Siegel (1997), Cmp nd Chng (1998), 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 1 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 (2001), for exmple, observed tht correltions of stocks returns increse in highly voltile or ber mrkets 3 See Blir et l (2001) for recent studies on the predictive bility of implied voltility 1

6 correltion clled implied correltion index tht is lso bsed on current option prices In this rticle the Siegel (1995) 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 (1976) nd Breeden nd Litzenberger (1978) nd hs been used in numerous works to this dy: Rubinstein (1994), Drmn nd Kni (1994), Jckwerth nd Rubinstein (1996), nd Brown nd Toft (1999) estimte implicit risk-neutrl density functions with the help of modified binomil model (Implied Binomil Trees) Shimko (1993), Jrrow nd Rudd (1982), nd Longstff (1995) estimte the price functions of options directly from their observed mrket prices, in dependence on 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 In section 5 we pply this pproch to estimting bets from cll options trded t the EUREX Section 6 summrizes the results 2 Assumptions nd Nottion The vlution of options in section 3 is bsed on the ssumptions of the one-period CAPM: 1 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 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 5 For definition of bivrite norml distribution, see Appendix A 2

7 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 n m Number of shres of sset S to be exchnged under the exchnge option 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 r f r s r m Bet of n underlying sset S with respect to the mrket index Instntneous risk-free rte of interest Instntneous rte of return on sset S Instntneous rte of return on the mrket index µ s Expected instntneous rte of return on sset S µ m Expected instntneous rte of return on the mrket index σ S σ m ρ Instntneous vrince of the rte of return on sset S Instntneous vrince of the rte of return on the mrket index 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 µ s+ 1 2 σ2 s, (1) E [ R m ] = e µ m+ 1 2 σ2 m, (2) ) Vr [ R m ] = e 2µm+σ2 m (e σ2 m 1, (3) Cov [ X s, R m ] = p( X s ) e µ m+ 1 2 σ2 m+µ s σ2 s (e ρ σ m σ s 1) (4) 6 The moments of lognorml distribution cn be clculted with the help of the integrls (38), (39) nd (40) indicted in Appendix A 3

8 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+ 1 2 σ2 s (e ρ σ s σ m 1) e µ m+ 1 2 σ2 m (e σ 2 m 1) (5) To simplify mtters the time-to-mturity of n option is set equl to one throughout the pper 8 3 The Model 31 Option Pricing in n Incomplete Lognorml Mrket In n incomplete lognorml mrket the CAPM my be used for option pricing 9 The well-known certinty equivlent vlution formul of the single-period CAPM is 10 p( X c ) = E [ X c ] λ Cov [ X c, R m ] 1 + r f where λ = E [ R m ] (1 + 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 distributed rtes of return, we derive 11 E [ X c ] = p( X s ) e µs+ 1 2 σ2 s Φ (d1 ) K Φ (d 2 ), (7) Cov [ X c, R m ] = p( X s ) e µ s+ 1 2 σ2 s+µ m σ2 m (e ρ σ s σm Φ (d 3 ) Φ (d 1 )) K e µ m+ 1 2 σ2 m (Φ (d4 ) Φ (d 2 )), (8) d 1 = (ln(p( X s )/K) + µ s )/(σ s ) + σ s, (9) d 2 = (ln(p( X s )/K) + µ s )/(σ s ), (10) d 3 = (ln(p( X s )/K) + µ s )/(σ s ) + σ s + ρ σ m, (11) d 4 = (ln(p( X s )/K) + µ s )/(σ s ) + ρ σ m (12) 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 7 For generl definition of bet, see Copelnd et l (2005), p However, one cn esily djust the model to ny time-to-mturity t different from one yer using the following trnsformtions: µ = t µ p, σ 2 = t σ 2 p nd r f = t r f,p 9 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 10 See Copelnd et l (2005), p See Appendix B Put prices follow from put-cll prity 4

9 (1973) 12 p( X c ) = p( X s ) θ 1 K e r f θ 2 (13) ) where θ 1 = e µs+ 1 2 σ2 s r f (Φ (d 1 ) λ e µm+ 1 2 σ2 m (e ρσ sσ m Φ (d 3 ) Φ (d 1 )) (14) θ 2 = Φ (d 2 ) λ e µ m+ 1 2 σ2 m (Φ (d4 ) Φ (d 2 )) (15) 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 13 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 14 This correltion cn lso be intuitively justified µ s + σ 2 s/2 = r f, (16) µ m + σ 2 m/2 = r f (17) 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 (16) nd (17) follows λ = 0, θ 1 = Φ (d 1 ) nd θ 2 = Φ (d 2 ) The vlution eqution (13) is reduced ccordingly with risk-neutrl vlution to p( X c λ = 0) = p( X ( ) ( ) ln(p( Xs)/K)+r s ) Φ f +σs/2 2 σ s Ke r f ln(p( Xs)/K)+r Φ f σs/2 2 σ s which equls the vlution eqution of Blck nd Scholes (1973), (18) 32 Pricing Options on the Mrket Index Jrrow nd Mdn (1997) 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 12 Ritchken (1985) developed similr vlution eqution for options bsed on the CAPM This model is not consistent with the Blck nd Scholes (1973) model in the cse of risk-neutrl vlution, however 13 See Cox nd Ross (1976) 14 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/2 5

10 mrket index equls 15 p( X cm ) = p( X m ) θ m1 K e r f θ m2 (20) where θ m1 = e µm+ 1 2 σ2 m r f (Φ (dm 1 ) λ e µm+ 1 2 σ2 m ( )) e σ2 m Φ (dm3 ) Φ (dm 1 ), (21) θ m2 = Φ (dm 2 ) λ e µ m+ 1 2 σ2 m (Φ (dm1 ) Φ (dm 2 )), (22) dm 1 = (ln(p( X m )/K) + µ m + σ 2 m) / (σ m ), (23) dm 2 = (ln(p( X m )/K) + µ m ) / (σ m ), (24) dm 3 = (ln(p( X m )/K) + µ m + 2σ 2 m) / (σ m ) (25) This vlution eqution is solely specil cse of (13); for clls on the mrket index, the following pply: ρ = 1, µ s = µ m nd σ s = σ m 4 Implicit Bet 41 Estimting Bet Using Exchnge Options Siegel (1995) ssumes tht continuous security trding on perfect cpitl mrkets is possible 16 This stndrd ssumption of options price theory enbles riskneutrl vlution of options nd is equivlent to the ssumption of complete cpitl mrkets 17 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 15 Jrrow nd Mdn (1997) 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 (1997), p( X cm ) = ( + b K) p( X m ) e µ m+ 1 2 σ2 m Φ (dm1 ) KΦ (dm 2 ) b p( X m ) 2 e 2(µ m+σ 2 m ) (dm 3 ) (19) where = (e σ2 m r f e (µm+ 1 2 σ2 m ) ) / (e σ2 m 1) b = (e µ m+ 1 2 σ2 m r f 1) / (p( X m ) e 2(µ m+σ 2 m ) (e σ2 m 1)) 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 (19) However, for the specil cse of clls on the mrket index, the Ritchken (1985) model is not identicl with the Jrrow nd Mdn (1997) model 16 See Assumption 1 in Siegel (1995) 17 See Cox et l (1979) 6

11 Siegel (1995) therefore recourses to 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 (1978), ( p( X ce ) = n s p( X ns p( ) ( Xs) ln + σ nm p( Xm) e/2 2 s ) Φ σ e n m p( X ns p( ) Xs) ln 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 (1995) 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 (1995), 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 (1999) describes definition (29) s modified bet Even in risk-neutrl vlution, this definition does not equl the stndrd definition of bet 18 β s = Cov [ R s, R m ] Vr [ R m ] = eρ σsσm 1 e σ2 m 1 (30) Regrdless of this, from prcticl view there is the problem - s Siegel (1995) himself notes - tht exchnge options re not currently trded on the cpitl mrkets 42 Estimting Bet Using Plin Vnill Options On incomplete mrkets, bet cn be estimted implicitly with the vlution equtions (13) nd (20) As result of the stte of dt typiclly given on the cpitl 18 Inserting (16) nd (17) in (5) results in (30) 7

12 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 minimized through selection of the prmeter ˆµ m nd ˆσ m, 19 min µ m,σ m ( ( ) I p Xcm p( ) 2 Xcm ) i i (31) i=1 p ( Xcm ) i 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, min µ s,σ s ( ( ) J p Xc p( ) ) 2 Xc j j p ( (32) Xc )j j=1 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-of-the-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 ] 1 + r f where λ = E [ R m ] (1 + r f ) Vr [ R m ] (33) Following severl conversions, inserting (1), (2), (3) nd (4) in (33) results in e µ s+σ 2 s /2 = ( ) e r f e σm 2 1 ( e σm 2 1 ) + ( e (µ m+ 1 2 σ2 m r f ) 1 )( ) (34) e ρ σ mσ s 1 Implicit bet (5) of n sset S cn be clculted with the estimted prmeters In order to clculte bet, (34) must be resolved ccordingly, ) e 2µm+σ2 m (e σ2 m 1 e ρ σ m σ s = ln 1 µs+ 1 2 σ2 s e r f +, (35) e µm+ 1 2 σ2 m e r f e µm+ 1 2 σ2 m+µ s+ 1 2 σ2 s nd inserted in (5) 19 This technique is lso pplied by Rubinstein (1994) for the estimtion of implicit risk-neutrl density functions 8

13 5 Empiricl Illustrtion We pply the new technique to cll options trded t the Eurex, the Europen electronic exchnge (futures exchnge) bsed in Frnkfurt, Germny It is one of the world s lrgest derivtives exchnges Implicit bets re estimted for those nine cll options on stocks with the highest trding volumes t the Eurex 20 following dt screening procedures were pplied: options with no trnsctions or missing volume were removed Furthermore, options with expirtion dtes smller thn 30 dys or lrger thn two yers were deleted 21 In order to keep the empiricl ppliction s simple s possible, we lso disregrd from dividend pyments Tble 1 displys the nmes of the underlying ssets s well s the frequencies of trde (number of trnsctions), volume of trnsctions in units nd volume of trnsctions in e (turnover) for ech of the nine options over the yers 2000 to is worth noting tht the cll options, eg on Allinz AG, EON AG, Muenchener Rueckversicherung AG or SAP AG show n increse both in terms of trnsctions s well s in volumes over this period, demonstrting the growing importnce of the option trde in generl The highest volumes cn be observed for cll options on the mrket index; the DJ EuroStoXX 50 hd bout 33 billion e turnover in 2004 On the other hnd, there re exmples of cll options, eg AG where trde volume in e decresed during the observtion period The It on Deutsche Telekom For ech trding dy of the yer, the model is estimted using non-liner lest squres 23 In order to reduce the impct of influentil observtions, we ssign weights less thn one to out-of-money clls 24 In the estimtions, we employ the restriction tht µ m + 05σ 2 m > r f to ensure tht the instntneous rte of return of the mrket index is lwys greter thn the instntneous risk-free rte of interest, r f Figure 1 shows the estimtes of µ m nd σ m for ech dy obtined in the first step 20 We thnk the Deutsche Boerse AG for kindly providing the dt 21 Cll options with expirtion dtes less thn 30 dys yielded implusible estimtes or ggrvted convergence problems in the model estimtions 22 The descriptive sttistics of Tble 1 re bsed on the cll options trded t the Eurex; this explins the difference in comprison to the mrket sttistics for ll options reported by the EUREX vilble t 23 The estimtion ws crried out using the proc model procedure in SAS 91 We lso tried estimtions on weekly bsis Overll, the results on weekly bsis re very similr to those obtined from the dy-to-dy estimtions 24 The following definitions were used for the weights Let rtio = strike price/current stock price If rtio > 1 then the weight in the estimtion is given s 2 (1 Φ ((rtio 1)/03)) For exmple, if rtio = 13 then weight = 0317, if rtio = 16 then weight =

14 of the nlysis (refer to eq 31) These estimtions re bsed on the observed trnsctions of cll options on the DJ Eurostoxx 50 index It cn be stted tht the overll pttern of µ m nd σ m over time ppers to be quite plusible We observe n increse of µ m nd σ m in the second hlf of 2002, but the estimtes grdully decline fterwrds We lso find strong correltion between µ m nd σm 2 Tble 2 contins the loction nd dispersion sttistics for the estimted implicit bets using the obtined µ s nd σ s for ech cll from the second step estimtion Agin, the estimtion is crried out for ech dy with observtion weights for outof-money clls s described bove nd the restriction tht µ m + 05σm 2 > r f is used Generlly, there re 253 to 255 trding dys per yer Tble 2 lso displys the number of dys of the yer for which the prmeter estimtes re obtined in the second step Missing estimtes of µ s nd σ s for some dys re minly the result of non-convergence in the model estimtions, nd sometimes due to n insufficient number of observtions Furthermore, estimtes of µ s nd σ s tht give ρ > 1 (refer to eq 35) re set to missing Tble 2 shows tht the yerly verges of estimted bets for the underlying ssets re in plusible rnges The computed 95% confidence intervls (CI) for expected vlues of bets show tht in ll cses expected vlues of bets re significntly different from zero The implicit bet estimtes for the technology-compny Noki corportion re higher thn the implicit bet estimtes for EON AG, big utility compny, gin confirming the plusibility of the results Another interesting result is tht implicit bets show some considerble vrition over time For instnce, Deutsche Telekom AG hd implicit bets lrger thn one over the yers 2000 to 2002, but fterwrds hd lower implicit bets, development which cn lso be observed for SAP Accordingly, n nlysis bsed on the ssumption of time-invrince on bets might provide misleding evidence 6 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 (1976) 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 To illustrte the pplicbility of this new pproch, we provide evidence on implicit bet estimtes using dt on cll options from the EUREX We find tht most of implicit bets re in plusible rnge, nd 10

15 the dispersion of bets within yers ppers to be resonble The estimtion results highlight tht bet vlues chnge over the yers, which implies tht the results from the conventionl regressive nlysis using historicl dt to obtin bets might be misleding if time-invrince of bet is ssumed This issue will be n interesting venue for future reserch 11

16 Appendix A: The Lognorml Distribution The definition of density of norml distribution is f(x) = 1 (x µ)2 e 2σ 2 (36) 2πσ 2 Φ( ) is the stndrd norml distribution (µ = 0 nd σ = 1) A vrite is lognormlly distributed if its nturl logrithm is normlly distributed The definition of bivrite norml distribution is 1 f(x, y) = e 1 2(1 ρ 2 ) 2π σxσ 2 y(1 2 ρ 2 ) (x µx) 2 σ 2 x «2ρ (x µ x)(y µy) + (y µ y) 2 σxσy σy 2 (37) 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 (1), (2), (3) nd (4), the simplifictions of the following specil integrls re required: ( e cx f(x) dx = e cµ+ 1 2 (cσ)2 Φ e cx 2 f(x 1, x 2 ) dx 1 dx 2 = e cµ (cσ 2) 2 Φ ) +µ+cσ 2 σ ( ) +µ 1 +cρ σ 1 σ 2 σ 1 ( ) e x 1 e cx 2 f(x 1, x 2 ) dx 1 dx 2 = e µ σ2 1 +cµ (cσ 2) 2 +cρ σ 1 σ2 +µ1 +σ1 Φ 2+cρ σ 1σ 2 σ 1 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 f(x 2 x 1 ) = f(x 1, x 2 ) f(x 1 ) If we pply this definition to the bivrite norml distribution, we get (38) (39) (40) f(x 2 x 1 ) = σ 2 2 σ (x 1 1 µ 1 ) )) 1 e (x2 ( µ2+ρ 2πσ 2 2 (1 ρ 2 ) 2σ 2 2(1 ρ2 ) (41) Note tht the conditionl density of the bivrite norml distribution equls the density of the norml distribution with the prmeters µ x2 x 1 = µ 2 + ρ σ 2 σ 1 (x 1 µ 1 ) und (42) σ 2 x 2 x 1 = σ 2 2(1 ρ 2 ) (43) 12

17 We next prove eqution (38) e cx 1 σ 2π (x µ) 2 e 2σ 2 dx = = = = 1 σ 2π e 1 σ 2π e 1 σ 2π e 1 σ 2π e = e cµ+ 1 2 (cσ)2 x 2 2xµ 2cσ 2 x+µ 2 2σ 2 dx x 2 2x(µ+cσ 2 )+µ 2 2σ 2 dx x 2 2x(µ+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 1 σ 2π e (x (µ+cσ 2 )) 2 2σ 2 dx Eqution (38) follows with 1 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 f(x 1, x 2 ) dx 1 dx 2 = e cx 2 f(x 2 x 1 ) dx 2 f(x 1 ) dx 1 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 f(x 2 x 1 ) dx 2 f(x 1 ) dx 1 = [e c µ 2 +ρ σ 2 (x σ 1 µ 1 ) c2 (σ2 2(1 ρ2 )) ] f(x 1 ) dx 1 = e cµ 2 cρ σ 2 σ 1 µ c2 σ c2 σ 2 2 ρ2 e cρ σ 2 σ 1 x 1 f(x 1 ) dx 1 If we define the helping vrible c := cρ σ 2 σ 1, 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 1 f(x 1, x 2 ) dx 1 dx 2 = e cx 2 f(x 2 x 1 ) dx 2 e x 1 f(x 1 ) dx 1 = e cµ 2 cρ σ 2 σ 1 µ c2 σ c2 σ 2 2 ρ2 e cρ σ 2 σ 1 x 1 +x 1 f(x 1 ) dx 1 If we define the helping vrible c := cρ σ 2 σ 1 + 1, we rrive t the desired result (40) fter repeted ppliction of eqution (38) nd shortening of the terms in exponents 13

18 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 rs K, 0 f(r s ) dr s = p( X s ) e r s f(r s ) dr s K f(r s ) dr s ln(k/p( Xs)) ln(k/p( Xs)) = p( X ( ) ( ) s )e µ s+ 1 2 σ2 s Φ ln(p( Xs)/K)+µ s+σs 2 σ s KΦ ln(p( 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 ] = mx(p( X s ) e r s K, 0) e r m f(r s, r m ) dr m dr s = (p( X s ) e rs K) e rm f(r s, r m ) dr m dr s ln(k/p( Xs)) = p( X s ) e r s e r m f(r s, r m ) dr m dr s K e r m f(r s, r m ) dr m dr s ln(k/p( Xs)) = p( X s ) e µs+ 1 2 σ2 s+µ m+ 1 2 σ2 m+ρ σ sσm Φ ( ) K e µm+ 1 2 σ2 ln(k/p( m Φ X s ))+µ s +ρ σ s σ m σ s ln(k/p( Xs)) ( ) ln(k/p( X s ))+µ s +σs 2+ρ σ sσ m σ s (45) Following the decomposition theorem, we rrive t the covrince (8) with (44) und (45), fter elementry conversions Appendix C: Tbles nd Figures 14

19 Figure 1: Estimtes of µm, σm for the EuroStoXX50, yers σm µm (For dt description nd sources, see Section 5) 15

20 Tble 1: Description of Cll Options nd the Nmes of Underlying Assets Cll Option On SECU DJ EURO OES trnsctions STOXX 50 INDEX contrcts [million] volu in e [billion] ALLIANZ AG ALV trnsctions contrcts [million] volu in e [billion] DEUTSCHE DBK trnsctions BANK AG contrcts [million] volu in e [billion] DAIMLER DCX trnsctions CHRYSLER AG contrcts [million] volu in e [billion] DEUTSCHE DTE trnsctions TELEKOM AG contrcts [million] volu in e [billion] EON AG EOA trnsctions contrcts [million] volu in e [billion] MUENCHNER MUV trnsctions RUECKVERS AG contrcts [million] volu in e [billion] NOKIA CORP NOA trnsctions contrcts [million] volu in e [billion] SAP AG SAP trnsctions contrcts [million] volu in e [billion] SIEMENS AG SIE trnsctions contrcts [million] volu in e [billion]

21 Tble 2: Estimtion Results for Implicit Bets SECU ALV men Bet % CI [038,054] [050,067] [087,103] [140,156] [091,110] dys DBK men Bet % CI [072,090] [087,106] [100,110] [093,108] [083,101] dys DCX men Bet % CI [110,123] [096,114] [126,133] [113,123] [120,131] dys DTE men Bet % CI [204,220] [163,190] [139,155] [049,064] [105,120] dys EOA men Bet % CI [078,097] [049,069] [066,076] [066,079] [059,078] dys MUV men Bet % CI [079,110] [040,062] [091,109] [139,156] [079,099] dys NOA men Bet % CI [179,206] [221,254] [162,183] [093,110] [158,170] dys SAP men Bet % CI [113,149] [099,122] [069,088] [040,058] dys SIE men Bet % CI [123,144] [134,154] [117,130] [087,101] [084,099] dys

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