Volatility Options: Hedging Effectiveness, Pricing, and. Model Error *

Transcription

1 Volailiy Opions: Hedging Effeciveness, Pricing, and Model Error * Dimiris Psychoyios ** and George Skiadopoulos *** Absrac Moivaed by he growing lieraure on volailiy opions and heir imminen inroducion in major exchanges, his paper addresses wo issues. Firs, we examine wheher volailiy opions are superior o sandard opions in erms of hedging volailiy risk. Second, we invesigae he comparaive pricing and hedging performance of various volailiy opion pricing models in he presence of model error. Mone Carlo simulaions wihin a sochasic volailiy seup are employed o address hese quesions. Alernaive dynamic hedging schemes are compared, and various opion-pricing models are considered. The resuls have imporan implicaions for he use of volailiy opions as hedging insrumens, and for he robusness of he volailiy opion pricing models. JEL Classificaion: G11, G12, G13. Keywords: Hedging Effeciveness, Model Error, Mone Carlo Simulaion, Sochasic Volailiy, Volailiy risk, Volailiy Opions. * We are paricularly graeful o Nicole Branger, Peer Carr, Jens Jackwerh, Iakovos Iliadis, and Sahis Tompaidis for many exensive discussions. We would like also o hank Iliana Anagnou, Charles Cao, Peros Dellaporas, Sephen Figlewski, Aposolos Refenes, Uwe Wysup, and he paricipans a he 2003 French Finance Associaion Meeing (Paris), he 2004 Bachelier World Congress (Chicago), he 2004 European Invesmen Review (London), he 2004 RISK Quan Congress Europe (London), and he AUEB, Universiy of Piraeus-ADEX, Universiy of Warwick seminars for helpful discussions and commens. Par of his paper was funded by he Financial Engineering Research Cenre and he Ahens Derivaives Exchange wihin he projec Volailiy Derivaives. Financial suppor from he Research Cenre of he Universiy of Piraeus is also graefully acknowledged. Previous versions of his paper have been circulaed under he ile How Useful are Volailiy Opions for Hedging Vega Risk?. Any remaining errors are our responsibiliy alone. ** Financial Engineering Research Cenre, Deparmen of Managemen Science and Technology Ahens Universiy of Economics and Business, dpsycho@aueb.gr *** Corresponding Auhor. Universiy of Piraeus, Deparmen of Banking and Financial Managemen, and Financial Opions Research Cenre, Warwick Business School, Universiy of Warwick, gskiado@unipi.gr

2 I. Inroducion The main sources of risk ha an invesor faces are price and volailiy risk (vega risk). Price risk is he invesor s exposure o changes in he asse price. Volailiy risk is he exposure o changes in volailiy. The laer ype of risk has been responsible for he collapse of major financial insiuions in he pas fifeen years (e.g. Barings Bank, Long Term Capial Managemen). To dae, he hedging of volailiy risk has been carried ou by using he exchange raded sandard fuures and plain-vanilla opions. However, hese insrumens are designed so as o deal wih price risk, primarily. A naural candidae o hedge volailiy risk is volailiy opions. These are insrumens whose payoff depends explicily on some measure of volailiy. The growing lieraure on volailiy opions has emerged afer he 1987 crash. Brenner and Galai (1989, 1993) firs suggesed opions wrien on a volailiy index ha would serve as he underlying asse. Towards his end, Whaley (1993) consruced VIX (currenly ermed VXO), a volailiy index based on he S&P 100 opion s implied volailiies raded in he Chicago Board of Exchange (CBOE). Ever since, oher implied volailiy indices have also been developed (e.g., VDAX in Germany, VXN in CBOE, VX1 and VX6 in France) and he properies of some of hem have been sudied (see e.g., Fleming e al. 1995, Moraux e al. 1999, Whaley 2000, Blair e al. 2001, Corrado and Miller 2003, and Simon 2003). Various models o price volailiy opions wrien on he insananeous volailiy have also been developed (see e.g., Whaley 1993, Grünbichler and Longsaff 1996, and Deemple and Osakwe 2000). These models differ in he specificaion of he assumed sochasic process, and he assumpions made abou he volailiy risk premium. In 2003, CBOE adoped a new mehodology o calculae he implied volailiy index, and i announced he immediae inroducion of volailiy opions in an organized exchange. However, o he bes of our knowledge, he hedging effeciveness of volailiy opions compared o ha of plain-vanilla opions has no ye been sudied. Jiang and Oomen (2001) 2

3 have examined he hedging performance only of volailiy fuures versus sandard opions; we commen furher on he relevance of heir sudy o ours in he concluding secion of he paper. This may be surprising given ha one of he main argumens for inroducing volailiy opions is based on heir use as hedging insrumens 1. Furhermore, he comparaive hedging and pricing performance of he exising volailiy opion pricing models in he presence of model error has araced very lile aenion 2 ; Daouk and Guo (2004) have focused on he pricing side and hey have invesigaed he impac of model error o he performance of only one (Grünbichler and Longsaff 1996) of he developed volailiy opion pricing models. This paper makes wo conribuions o he volailiy opions lieraure by exploring hese wo issues, respecively. Firs, i compares he hedging performance of volailiy versus sandard European opions. Second, i answers he following quesion: Assuming ha we know he rue daa generaing process of he underlying asse price and of volailiy, wha is he impac of using a mis-specified process on he hedging and pricing performance of he volailiy opion pricing models under scruiny? Undersanding he hedging performance of volailiy opions, as well as he comparaive pricing performance of various volailiy opion 1 Volailiy opions can also be used o speculae on he flucuaions of volailiy. Ineresingly, Dupire (1993), Derman e al. (1997), and Brien-Jones and Neuberger (2000) have shown ha volailiy rading/hedging can also be performed indirecly by using saic posiions in sandard European calls. For a review of he volailiy rading/hedging echniques, see also Carr and Madan (1998). However, ransacion coss may hamper he implemenaion of such sraegies. 2 Crouhy e al. (1998) define as model error eiher he mis-specificaion of he model, and/or he parameer misesimaion wihin any given model, and/or he incorrec implemenaion of any given model. The exising sudies on he impac of model error o he hedging effeciveness use as a arge opion eiher a sandard European opion (see e.g., Galai 1983, Figlewski 1989, and Carr and Wu 2002) or various exoic opions (see e.g., Hull and Suo 2002). 3

4 pricing models will faciliae he inroducion of volailiy opions in organized exchanges, and heir use by invesors 3. To address our research quesions, Mone Carlo (MC) simulaions under a sochasic volailiy seup are employed. MC simulaion has been used in he lieraure exensively o invesigae he pricing and hedging performance of various models, as well as he impac of model error (see e.g., Hull and Whie 1987, Figlewski 1989, Jiang and Oomen 2001, Carr and Wu 2002, Daouk and Guo 2004). This is because i enables he selecion of he daa generaing process, and he conrol of he values of is parameers. Comparaive analysis for various parameer values is also possible. Moreover, in our case he use of MC simulaion is dicaed by he lack of daa on volailiy opions; volailiy opions are no raded ye. Alernaive mehods such as hisorical simulaion (Green and Figlewski 1999), or calibraion of he pricing model o marke daa (see e.g., Backshi e al. 1997, Dumas e al. 1998, and Hull and Suo 2002) ha have been used o answer similar quesions canno be followed. Following Hull and Suo (2002), he sochasic volailiy seup has been adoped as he rue daa generaing process. This is a legiimae assumpion since here is broad empirical evidence ha volailiy is sochasic. Moreover, his seup is preferred o a more complex one ha also includes oher sources of risk, e.g., jumps and sochasic ineres raes. Backshi e al. 3 Surprisingly, he rading of volailiy derivaives in exchanges has no ye been insiued. The only aemp o inroduce conracs on volailiy in an organized marke was underaken by he German Exchange in 1997; ha was a volailiy fuure (VOLAX) on he German implied volailiy index VDAX. However, he rading of VOLAX ceased in An anecdoal explanaion ha is offered by praciioners for he failure of VOLAX, as well as for he delay in inroducing volailiy opions, is ha marke makers are neiher familiar wih he models ha have been developed o price volailiy fuures and opions, nor wih heir use for hedging purposes. In accordance wih his claim, Whaley (1998) also saes In summary, I believe ha volailiy derivaives are a viable exchange-raded produc I also believe ha he conracs have no been successful largely because poenial marke makers have no sepped forward. The reason is fear. 4

5 (1997) examine such a general model and hey conclude aking sochasic volailiy ino accoun is of firs-order imporance in improving upon he BS formula (pp ). We assume ha a shor posiion in a sandard European sock call opion is o be hedged (arge opion). A naural way o hedge boh price and volailiy risk is o hedge he arge opion wih he sock and anoher opion (dela-vega sraegy, see e.g., Hull and Whie 1987 and Sco 1991). Two alernaive dynamic dela-vega neural hedging schemes are considered. These differ on he ype of opion employed as he hedging insrumen; sandard and volailiy European call opions are used as opion hedging insrumens, respecively. Nex, he sock and volailiy price series are joinly generaed. We inroduce he hedging error by rebalancing he posiion in he hedging insrumen discreely. Then, he performance of he wo hedging sraegies is assessed on he simulaed daa across differen srikes and mauriies of he arge opion, and for various values of correlaion beween he underlying asse and is volailiy. Two differen rebalancing frequencies (daily and weekly) are used. This will enable us o invesigae he effec of he conrac specificaions, correlaion, and he rebalancing frequency on he hedging effeciveness of he wo sraegies. To implemen he wo hedging schemes, he hedge raios are calculaed by employing separaely he Black-Scholes (BS, 1973) and Heson (1993) models in he case of (arge and insrumen) sandard opions. In he case of volailiy opions, Whaley (1993), Grünbichler and Longsaff (1997), and Deemple and Osakwe (2000) models are used. The use of various opion pricing models will shed ligh on he robusness of he volailiy opion pricing models for hedging and pricing purposes in he presence of model error. In erms of he hedging effeciveness, we find ha volailiy opions are no superior hedging vehicles o sandard European opions in he conex of he examined hedging schemes. However, he difference in he performance of he wo hedging schemes depends on he characerisics of he arge opion, and on wheher a roll-over sraegy in volailiy opions is carried ou. In erms of he effec of he model error, we find ha he seemingly 5

6 wors mis-specified volailiy opion-pricing model can be reliably used for pricing and hedging purposes. The remainder of he paper is srucured as follows. In he nex Secion, he employed opion pricing models are described. In Secion 3, he simulaion seup and he measure of hedging effeciveness are inroduced. Secion 4 presens and discusses he resuls on he hedging performance of he wo hedging sraegies. Secion 5 presens and discusses he resuls on he robusness of he volailiy opion pricing models for hedging and pricing purposes in he presence of model error. The las Secion provides a brief summary, he implicaions of he resuls are addressed, and opics for fuure research are suggesed. II. Descripion of he Models In his secion, he Black-Scholes (1973), Heson (1993), Whaley (1993), Grünbichler and Longsaff (1996), and Deemple and Osakwe (2000) opion pricing models ha will be used in he remainder of he paper are briefly described. The firs wo models are used for he pricing/hedging of sandard European opions. The remaining hree are used for he pricing/hedging of volailiy opions. Only he European call pricing formulae are provided for he purposes of our analysis. A. Sandard Opions: Black-Scholes (1973) Le he sock price S of a non-dividend paying sock follow a Geomeric Brownian Moion Process (GBMP), i.e.: [ ) ds = rs d + VS dw, 0, +, (1) where V is he volailiy of he asse price, and W is a Wiener process under he risk-adjused measure Q. The Black and Scholes (BS) price C( S, KT, ) of a European call opion is given by: 6

7 ( ) 1 ( ) ( ) rt CS (, KT, ) = SN d e KN d2 (2) where T- is he ime o mauriy, K is he srike price, N( ) is he cumulaive sandard normal probabiliy disribuion funcion, and d = S 1 2 ln ( ) ( ) ( ) + r+ V T K 2 V T, d 1 2 = d1 V T. (3) B. Sandard Opions: Heson (1993) Heson (1993) assumes ha he sock price follows a GBMP and he variance υ evolves as a Mean Revering Square Roo Process (MRSRP), i.e.: ds = rs d + υ S dw (4) 1 ( ) 2 dυ = β m υ d + σ υ dw (5) where W1 and W are wo correlaed Wiener processes under he risk-adjused measure Q 2 and σ is he volailiy of he variance. The parameers β and m are he rae of mean reversion and he long-run mean of volailiy, respecively, ha incorporae he marke price of volailiy risk h. Heson assumes ha h is proporional o he curren level of variance, i.e. [ ) h = ςυ, where ζ 0, + is a consan parameer. Therefore, under he risk-adjused measure Q, β=λ+ζ, and m = λµ λ + ζ, where λ and µ are he rae of mean reversion and he long-run mean of he equivalen MRSRP under he physical probabiliy measure P, respecively. Then, he price a ime of a European call opion is given by: 1 ( ) rt CS (, υ, ) = SP Ke P2, (6) where P j (j=1,2) is defined in Heson (1993, page 331, equaions 16, 17, and 18). 7

8 C. Volailiy Opions: Whaley (1993) Whaley uses he Black s (1976) fuures model o price volailiy opions wrien on an implied volailiy index; he considers volailiy fuures opions wih a zero cos-of-carry. Hence, he assumes implicily ha he volailiy index is a raded asse ha follows a GBMP. The value C( V, KT, ) of a European volailiy call opion on he volailiy index a ime is given by: ( ) ( ) ( ) rt CV (, KT, ) = e VN d KN d, where V is he value of he volailiy index a ime, and 1 2 (7) V 2 + σ ( T ) ( T ) ln 0.5 K d1 = and d2 = d1 σ T. σ (8) where σ is he volailiy of he volailiy index reurns. D. Volailiy Opions: Grünbichler and Longsaff (1996) Grünbichler and Longsaff (GL) derive a closed form expression o price European volailiy opions. The underlying asse is he insananeous volailiy of he reurns of a sock index. They regard volailiy as a non-radable asse; he marke price of volailiy risk has o be inroduced. Following Heson (1993) hey assume ha he expeced volailiy risk premium h is proporional o he curren level of volailiy. GL model he volailiy (and no he variance as in Heson s case) as a MRSRP, i.e.: ( ) σ, [ 0, ) dv = β m V d + V dw + (9) where W is he Brownian moion under he risk-adjused probabiliy measure Q. Then, he value of a European volailiy call opion CV (, KT, ) a ime is given by: ( ) β ( ) ( ) ( ( ) β ( ) ( ) ( γ ; ν, θ) ( γ ν θ) rt T CV (, KT, ) = e e V X K; + 4, + rt T e m 1 e X γ K; ν + 2, θ rt e K X K ) (10) 8

9 where X( ) is he cumulaive disribuion funcion of he complemenary non-cenral chi- 2 squared disribuion χ ( ) ν θ, wih ν degrees of freedom and non-cenraliy parameer θ (see Chaper 28 of Johnson and Koz 1970). The parameers γ, θ, and v are defined in Grünbichler and Longsaff (1996, page 989, equaion 6). E. Volailiy Opions: Deemple and Osakwe (2000) Deemple and Osakwe (DO) provide an analyic pricing formulae for European volailiy opions assuming ha he marke price of volailiy risk is zero, and ha volailiy follows a Mean Revering Logarihmic process (MRLP), i.e.: ( ) dlnv = λ µ ln V d+ σdw, (11) Then, he value of a European volailiy call opion is given by: 1 2 rt ( ) µ ( 1 φt ) + a φ T T ( ) ( 2 CV (, KT, ) = e V e N dt + at KN dt, (12) where φ Τ, α Τ, and d Τ are defined in Deemple and Osakwe (2000, page 25). ) III. Assessing he Hedging Effeciveness We invesigae wheher he hedging performance of volailiy opions is superior o ha of sandard plain vanilla opions in a sochasic volailiy environmen. Towards his end, he performance of various hedging sraegies is assessed under a sochasic volailiy simulaion seup. A. Generaing he Daa: The Simulaion Seup A join Mone Carlo simulaion of he sock price and volailiy generaes he daa. We assume ha he asse price S follows a GBMP, and he volailiy V follows a MRSRP (i.e. equaions 1 and 9). The MRSRP has been widely used o model he sochasic evoluion of volailiy over ime (see among ohers, Hull and Whie 1988, Heson 1993, Ball and Roma 9

10 1994, Grünbichler and Longsaff 1996, and Psychoyios e al for a review on he coninuous ime sochasic volailiy processes). This is because i is consisen wih he empirical evidence ha volailiy follows a mean-revering process (see e.g., Sco 1987, Merville and Piepea 1989, and Sheikh 1993), and i precludes volailiy from aking negaive values. Furhermore, a zero volailiy risk premium is assumed, i.e. ζ=0. Hence, equaions (1) and (9) are re-wrien as ds rd V dw 1,, S = + (13) ( ) σ [ ) dv = λ µ V d + VdW2,, 0, +. (14) The Brownian moions in he wo processes are assumed o be correlaed wih insananeous correlaion coefficien ρ, i.e. Cor( dw, dw ) = ρ (15) 1, 2, The assumpion of a zero volailiy risk premium is necessary so as o include Whaley (1993) and DO in he lis of models o be compared subsequenly. On he oher hand, his assumpion is in conras wih he empirical evidence ha suggess ha he volailiy risk premium is non-zero and ime varying (see e.g., Backshi and Kappadia 2003). However, he assumpion of a zero volailiy risk premium does no impose any limiaions on our subsequen analysis. We commen furher on his in Secion 4.4. Moreover, such an assumpion is no unrealisic wihin our conex. Whaley (1993) argues ha risk neural valuaion is valid given ha he (implied) volailiy index can be replicaed by forming an opion rading sraegy in line wih he seps necessary o consruc he index. Finally, he non-zero correlaion beween he asse price and volailiy is also empirically documened. Typically i has been found o be negaive; his has been ermed as leverage effec (see e.g., Figlewski and Wang 2000, for a deailed review and an empirical analysis). 10

11 B. The Hedging Scenario and he Hedging Error Meric Assume ha a financial insiuion sells a sandard European call opion wih τ 1 days o mauriy (arge porfolio). Acceping he sochasic naure of volailiy, he aim is o delavega hedge he arge porfolio wih an insrumen porfolio ha is composed of he underlying asse and eiher a sandard opion, or a volailiy opion. To make his concree, le T and I be he prices of he arge and insrumen porfolios a ime, respecively. Then, he price P of he porfolio formed by he arge and he insrumen porfolio (exended porfolio) is given by P = T + I = T+n S +n i (16) 1, 2, where i and S are he prices of he insrumen opion and he sock price a ime, respecively. Noe ha he weighs n 1 and n 2 are funcions of ime, since he dela-vega neuraliy requires (coninuous) rebalancing. In paricular, a coninuous dela-vega neural hedge requires he following equaions o hold and P T i = + n 2, 0 V V V = (17) P = T + n + n i = S S ds 1, 2, 0 (18) Equaions (17) and (18) express he condiions for vega and dela neuraliy, respecively. However, in pracice he presence of ransacion coss does no allow coninuous rebalancing; his inroduces a hedging error. Assuming a rebalancing frequency, he price of he exended porfolio a ime + is given by P = T + I = T +n S +n i (19) , + 2, + Noice ha he weighs in equaion (19) are sill funcions of ime because he hedge has no been adjused since ime (see also Boyle and Emanuel 1980 for a similar seup). Assuming ha here are no arbirage opporuniies, a riskless exended porfolio (i.e. a perfec hedge is 11

12 achieved) should earn he risk-free rae, i.e. P = r P. Following Figlewski (1989) and Sco (1991), he hedging error HE(+ ) a ime + is defined as HE( + ) = P rp = ( T T ) + n ( S S ) + + 1, + n ( i i )- r[t+n S +n i ] 2, + 1, 2, (20) HE() is calculaed and recorded for each dae and for each hedging scheme. A commonly used meric o assess he hedging effeciveness is he Toal Dollar Hedging Error (TDHE, see Figlewski 1989, and Backshi e al. 1997). The TDHE is defined as: M M l ( ) ( 1 ) (21) TDHE = HE l + r l= 1 where M is he number of rebalancing daes. Therefore, he TDHE is he sum of hedging errors across he rebalancing daes compounded up o he expiry dae (aggregae hedging error). Consequenly, he hedging scheme wih he lowes TDHE should be chosen among alernaive compeing schemes. However, wihin a simulaion seup he calculaed hedging error is condiional on he generaed asse and volailiy pahs. To eliminae his dependence, we calculae HE(+, k) by running K join MC simulaion runs of he asse and volailiy prices (k=1,2,,k). Then, he TDHE is calculaed for each simulaion run. Finally, he uncondiional TDHE (UTDHE) is calculaed as he average of he absolue values of he TDHE across he K simulaion runs, i.e.: K K M 1 1 UTDHE( ) = TDHE( k) = HE( l, k) 1+ r K K M l ( ) (22) k= 1 k= 1 l= 1 The absolue values are used so as o avoid offseing he posiive wih he negaive signed TDHEs and hence disoring he magniude of he chosen measure of hedging error. The UTDHE can be inerpreed as a measure of he dispersion of he disribuion of he TDHEs. The variabiliy of he hedging error has been commonly used in he lieraure as a meric o assess he hedging performance (see e.g., Hull and Whie 1987, Figlewski 1989, Green and Figlewski 1999). The mos effecive hedging scheme is he one wih he lowes UTDHE; he 12

13 idea is analogous o he approach followed o calculae a sandard opimal hedge raio, i.e. o minimize he variance in he changes of he price of an exended porfolio. C. Hedging Schemes and Opion Pricing Models The described simulaion seup is applied o wo differen hedging schemes: Hedging Scheme 1 (HS1): Αn insrumen porfolio consising of n 1 shares, and n 2 a-hemoney sandard European call opions wih τ 2 days o mauriy (τ 1 <τ 2 ), and Hedging Scheme 2, (HS2): An insrumen porfolio consising of n 1 shares, and n 2 a-hemoney European volailiy call opions wih τ 3 days o mauriy ( τ 3 τ 1 < τ 2 ). Hence, a rollover sraegy wih volailiy opions may be considered. Wihin he above wo schemes, five models are employed o calculae he hedge raios (equaions 17 and 18). These are: Black-Scholes (BS, 1973), Heson (He, 1993), Whaley (1993), Grünbichler and Longsaff (GL, 1996), and Deemple and Osakwe (DO, 2000) models 4 5. The observed prices of he arge and he insrumen opions are assumed o be generaed by he Heson (1993) and Grünbichler and Longsaff (1996) models in he cases of he plain-vanilla and volailiy opions, respecively. These are he models ha are closes o he assumed daa generaing sock/volailiy processes. Then, he resuling hedging error is calculaed (equaion 20). Hence, he resuls will be a join es of he hedging scheme and of he model employed. In oal, he hedging performance of eigh combinaions is compared. These are he following: 4 The dela-vega neural sraegy is inconsisen wih he BS assumpion of consan volailiy. However, i is widely used in pracice by praciioners so as o accommodae he non-consancy of volailiy wihin he BS model. 5 There exis no closed-form formulae o calculae he GL hedge raios; hese are calculaed numerically. The hedge raios in he oher models are calculaed by using he formulae ha are provided in he respecive papers. The simulaed sock price/ insananeous volailiy are used as inpus. 13

14 HS 1, BS-BS: In Hedging Scheme 1, BS model is used o calculae he Greeks of he arge and of he insrumen porfolio. HS 1, He-He: In Hedging Scheme 1, Heson model is used o calculae he Greeks of he arge and of he insrumen porfolio. HS 2, BS-DO: In Hedging Scheme 2, BS and DO models are used o calculae he Greeks of he arge and of he insrumen porfolio, respecively. HS 2, He-DO: In Hedging Scheme 2, Heson and DO models are used o calculae he Greeks of he arge and of he insrumen porfolio, respecively. HS 2, BS-GL: In Hedging Scheme 2, BS and GL models are used o calculae he Greeks of he arge and of he insrumen porfolio, respecively. HS 2, He-GL: In Hedging Scheme 2, Heson and GL models are used o calculae he Greeks of he arge and of he insrumen porfolio, respecively. HS 2, BS-W: In Hedging Scheme 2, BS and Whaley models are used o calculae he Greeks of he arge and of he insrumen porfolio, respecively. HS 2, He-W: In Hedging Scheme 2, Heson and Whaley models are used o calculae he Greeks of he arge and of he insrumen porfolio, respecively 6. D. The Implemenaion The join MC simulaions in equaions (13) and (14) are performed by adoping he Grünbichler and Longsaff (1996) parameers values µ=0.15 and λ=4. The iniial asse price 6 Heson (1993) models he sochasic evoluion of he variance. In our simulaion framework, he volailiy is simulaed. This raises wo issues. Firs, o implemen Heson s model ha requires he variance as an inpu he simulaed volailiy has o be squared. Second, Heson defines vega as he firs derivaive of he opion price wih respec o he variance. In our case, vega is defined as he firs derivaive of he opion price wih respec o volailiy. Hence, he chain rule is applied o calculae Heson s vega using he laer definiion, i.e.: 2 ( ) dc dc dvar dc d V dc = = = 2V dv dvar dv dvar dv dvar where C is he call opion price, and V, Var denoe he volailiy and he variance, respecively. 14

15 and he iniial volailiy are se equal o 100 and 0.1, respecively, r=5%, σ 2 =0.133, and τ 2 =100. 1,000 simulaion runs are performed o calculae he UTDHE. Equaion (22) shows ha he UTDHE is a funcion of he rebalancing frequency. Therefore, he hedging effeciveness of each one of he eigh combinaions is examined by calculaing he UTDHE for wo differen rebalancing frequencies: daily and every five days. Under he assumpion ha volailiy follows a mean revering process, he hedging effeciveness of he volailiy opion decreases as he ime o mauriy increases (see Grünbichler and Longsaff 1996, and Deemple and Osawke 2000). Therefore, shor mauriy volailiy opions should be preferred as hedging vehicles. To es his hypohesis, we considered wo differen hedging sraegies wihin HS2. A roll-over in volailiy opions sraegy wih N shor mauriy τ 3 volailiy opions (τ 3 << τ 1 ), so as N τ 3 = τ1, and a no-rollover sraegy wih only one volailiy opion wih τ 1 ime-o-mauriy. Moreover, he hedging performance of he insrumen opions may depend on he srike price and he ime-o-mauriy of he arge opion (see e.g., Hull and Whie 1987, and Backshi e al. 1997). Hence, regarding he srike price, hree arge opions are hedged separaely: an in-he-money (ITM), an a-he-money (ATM), and an ou-of-he-money (OTM) call opion (Κ=90, 100, and 110, respecively). For each srike price, arge opions of shor, inermediae, and long mauriies are sudied separaely (τ 1 =20, 40, and 80, respecively). In he case of he roll-over sraegy, he ime-o-mauriy of he volailiy opion is τ 3 =30, and he roll-over is performed en days prior o i s expiry dae. Obviously, in he case of he shor mauriy arge opion, here is no roll-over. In he inermediae and he long mauriy, here are one and hree roll-over poins, respecively. Finally, i may be he case ha he hedging performance also depends on he value of correlaion beween he asse price and he volailiy. For example, Nandi (1998) finds ha he Heson (1993) model performs beer han he BS model in erms of hedging only in he case where he correlaion is non-zero. Hence, for each srike price and ime-o-mauriy, he 15

16 join MC simulaions in equaions (13) and (14) are performed for differen values of ρ=-0.5, 0, The resuls are no sensiive o he choice of he correlaion value. Therefore, hey are repored only for he case of ρ=-0.5 ha is consisen wih he empirically documened leverage effec. Cholesky decomposiion is applied o creae he correlaed random numbers for he purposes of MC simulaion. IV. Hedging Effeciveness: Resuls and Discussion In his Secion, he hedging effeciveness of volailiy versus plain vanilla opions is examined. Table 1 shows he UTDHE for HS1 and HS2 across differen rebalancing frequencies, various imes-o-mauriy, and moneyness levels of he arge opion. The resuls are also repored for he roll-over versus he no roll-over of he volailiy opion case; in he shor mauriy no roll-over is performed. (INSERT TABLE 1 HERE) We can see ha in general, he hedging performance of HS1 is superior o ha of HS2, i.e. he disribuion of he TDHE has a smaller dispersion under HS1 han HS2. HS1 ouperforms HS2 in boh he roll-over and he no roll-over of he volailiy opion case. In paricular, HS1 He-He has he lowes UTDHE for all mauriies and moneyness levels. Nex, he resuls on he hedging effeciveness are discussed by focusing on he change in he difference in he hedging performance of he wo hedging schemes across each one of he following four crieria: he rebalancing frequency, he applicaion of roll-over versus noroll-over, he ime-o-mauriy of he arge opion, and he srike price of he arge opion. The analysis is ceeris paribus. A. Hedging Effeciveness: Rebalancing Frequency and Roll-Over Effec Table 1 shows ha he difference beween HS1 and HS2 increases in he case of he weekly rebalancing. In he shor and inermediae mauriies and for all moneyness levels, he 16

17 hedging error is less under daily han weekly rebalancing. Ineresingly, for some combinaions in he case of he long mauriy arge opions, he hedging error is slighly greaer under daily han weekly rebalancing; he resul may seem counerinuiive. Boyle and Emanuel (1980) have showed ha he hedging error decreases as he rebalancing frequency increases. However, heir sudy focuses on he effec of he rebalancing frequency on he dela hedging error under he Black-Scholes assumpions. To he bes of our knowledge, here is no similar sudy in he conex of dela-vega hedging sraegies in a sochasic volailiy seup. Inuiively, our findings may be explained as follows. The hedging error includes he dela and vega hedging errors among oher sources of error. Given he assumed mean-revering naure of volailiy, he vega hedging error erm is expeced o be smaller under weekly rebalancing in he case of he long-mauriy arge opion. This is because here is a greaer probabiliy for he long-run mean of volailiy o be reached, and hence volailiy does no vary any longer. Regarding he roll-over effec, Table 1 shows ha HS1 dominaes HS2 in boh he noroll-over and he roll-over sraegies. Ineresingly, in he roll-over case, he difference beween HS1 and HS2 has decreased, in general. This finding is expeced; long-daed volailiy opions should no be used o hedge volailiy since heir prices are less sensiive o changes in volailiy han he shor-erm ones (see Grünbichler and Longsaff 1996, and Deemple and Osakwe 2000). The resul implies ha he hedging effeciveness of volailiy opions increases by following a roll-over sraegy wih shor mauriy volailiy opions. B. Hedging Effeciveness: Mauriy and Srike Effec To compare he hedging effeciveness across various mauriies and srikes, he UTDHE calculaed from equaion (22) needs o be sandardized. The choice of he numeraire o be used for he sandardisaion depends on he ype of risk o be hedged. In he conex of comparing alernaive dela hedging sraegies, he resuling hedging error is usually divided 17

18 by he iniial premium of he arge opion (see e.g., Figlewski and Green 1999, Carr and Wu 2002, and Bakshi and Kappadia 2003). For he purposes of our analysis, we choose he numeraire o be he Black-Scholes vega of he arge opion a he iniiaion dae of he hedging sraegy. Hence, he sandardised hedging errors can be compared across various arge opions ha have he same exposure o volailiy risk 7. Table 2 shows he vegasandardised UTDHE for HS1 and HS2 across differen rebalancing frequencies, and various imes-o-mauriy and moneyness levels of he arge opion. The resuls are also repored for he roll-over versus he no roll-over of he volailiy opion case. (INSERT TABLE 2 HERE) HS1 dominaes HS2 for all mauriies. The effec of he ime-o-mauriy of he arge opion on he sandardized UTDHE depends on he moneyness of he arge opion. In paricular, he sandardized UTDHE increases in he cases of ITM and OTM arge opions. On he oher hand, he sandardized UTDHE decreases in he case of ATM opions. The resuls hold for boh HS1 and HS2. The effec of he ime-o-mauriy of he arge opion on he difference beween he hedging performance of HS1 and HS2 depends on he moneyness of he arge opion, as well. In general, as he ime-o-mauriy increases, he difference in he performance of he wo hedging schemes decreases for ITM and OTM arge opions, and i increases for ATM arge opions. The resuls hold for boh he roll-over and he no roll-over sraegies. Finally, as far as he srike price of he arge opion is concerned, HS1 dominaes HS2 for all moneyness levels. We can see ha he difference in he hedging performance of 7 Given ha our analysis focuses on hedging vega risk, i is no appropriae o sandardise he hedging error by using he opion premium as a numeraire. This is because he vega hedging error is expeced o increase monoonically wih he vega. However, he relaionship beween he vega hedging error and he premium is no monoonic. On he oher hand, in he conex of dela hedging, he dela hedging error is monoonically relaed o boh he opion premium and he dela; eiher of hem can be used as numeraire for he purposes of comparing dela hedging errors across various arge posiions. 18

19 he wo schemes depends on he moneyness level. The difference is maximised for ITM arge opions, and i is minimised for ATM arge opions. Regarding he performance of each individual scheme, boh HS1 and HS2 perform bes for ATM arge opions and worse for ITM arge opions. Focusing on HS2, we can see ha i performs bes when ATM opions are o be hedged; hese happen o be he mos liquid insrumens in he derivaives exchanges. The resuls hold for boh he roll-over and he no-roll-over sraegies. C. Assessing he Differences in he Hedging Effeciveness So far, we have commened on he differences beween he UTDHE across he various hedging schemes. The UTDHE measures he dispersion of he disribuion of he TDHE. To ge a feel of wheher he TDHE differs across he hedging schemes under scruiny, we consruc 95% confidence inervals from he disribuion of he TDHE. The inersecion of any wo of he consruced 95% confidence inervals may sugges ha he corresponding values of he TDHE are no significanly differen in a saisical sense 8. Tables 3 and 4 show he UTDHE and (wihin he brackes) he lower and upper bounds of he 95% confidence inervals of he TDHE in he case of daily and weekly rebalancing, respecively. Resuls are repored for he roll-over and no-roll-over case. (INSERT TABLES 3 AND 4 HERE) The confidence inervals es suggess ha he difference beween HS1 and HS2 becomes sronger as he ime-o-mauriy of he arge opion increases; his holds regardless of he rebalancing frequency. Ineresingly, his relaionship is more eviden in he no-roll- 8 The sandard saisical mehods ha assess he saisical significance of wo means (e.g. he one sample -es or he independen-samples es) canno be used in our case because hey are based on he assumpion ha he populaion follows a normal disribuion. However, applicaion of Jarque-Bera ess showed ha he TDHE is no normally disribued. 19

20 over sraegy; hese findings are in accordance wih he discussion in Secion IV.B where he differences beween HS1 and HS2 are repored o be greaer for he no-roll-over case. In paricular, we can see ha he differences in he TDHE beween HS1 and HS2 are no significan for he shor mauriy opions; his holds for all moneyness levels. In he case of he inermediae mauriy opions, he resuls depend on he moneyness level. More specifically, in he case of ITM opions he differences beween HS1 and HS2 are no significan. In he case of ATM opions, he differences beween HS1 He-He and HS2 are significan. On he oher hand, he differences beween he HS1 BS-BS and HS2 are no significan. Finally, considering OTM opions, he differences are in general significan; some excepions occur comparing he differences of HS1 BS-BS wih HS2 He-GL and HS2 He- DO. In he case of he long mauriy opions, he differences beween HS1 and HS2 are significan for ATM and OTM opions. D. The Effec of a Non-Zero Volailiy Risk Premium A his poin, we should examine wheher our resuls are robus o he assumpion of a zero volailiy risk premium. The exisence of a non-negaive volailiy risk premium may affec he hedging effeciveness of volailiy opions (provided ha he respecive model admis he exisence of a volailiy risk premium). For insance, Henderson e al. (2003) showed ha in a quie general sochasic volailiy seing, he prices of sandard calls and pus are a decreasing funcion of he marke price of risk of volailiy. There may be an analogous resul o be proved for he case of volailiy opions. Therefore, we invesigaed he comparaive performance of combinaions ha admi he exisence of a non-zero volailiy risk premium, i.e. HS1 He-He versus HS2 He-GL. To his end, he same analysis was performed by choosing ζ=-1. This choice is consisen wih he average esimae of ζ over he period in Guo (1998); he esimaed he implici volailiy risk premium from Heson s model using currency opion prices. Hence, his resuls are direcly applicable o implemening 20

21 Heson and GL models in our seup. We found ha HS1 sill ouperforms HS2. Hence, he previously repored resuls are robus o he choice of a zero value for he volailiy risk premium. This is in line wih he resuls in Daouk and Guo (2004) who also found ha he effec of model error on he pricing performance of he GL model is no affeced by he choice of he volailiy risk premium. V. Volailiy Opions and he Effec of Model Error In his Secion, he focus is on HS2, and he effec of model error on he hedging and pricing performance of he various volailiy opion pricing models is sudied. The He, GL models are used as benchmarks. This is because among he opion models under scruiny, hey assume a volailiy process (MRSRP) ha is closes o our assumed (possibly correlaed sock and volailiy) daa generaing processes. A. Model Error and he Hedging Performance Given ha he He-GL is he rue combinaion, he model error may arise from any of he hree following alernaive scenarios: (a) he hedger uses he rue model o calculae he hedge raio of he volailiy opion, and a false model o calculae he hedge raio of he arge opion. In his case, we compare He-GL wih BS-GL. He-GL ouperforms BS-GL only in he inermediae mauriy roll-over case, irrespecively of he moneyness level. The ou performance of his combinaion also appears in he OTM inermediae and long mauriy opions for boh he roll-over and no-roll-over case. (b) The hedger uses he false model o calculae he hedge raio of he volailiy opion and he rue model o calculae he hedge raio of he arge opion. In his case, we compare He-DO and He-W wih He-GL. He-GL is superior o he He-W in almos all cases. On he oher hand, he performance of he He-GL relaive o He-DO depends on wheher roll-over is performed. In he case of he no-roll-over sraegy, He-DO performs bes, while in he case of he roll-over sraegy He-GL performs 21

22 bes. (c) The hedger uses he false model o calculae he hedge raios of boh he volailiy and he arge opion. In his case, we compare BS-DO, and BS-W wih He-GL. In he case of he no roll-over sraegy, BS-DO, and BS-W perform beer han He-GL. In he case of he roll-over sraegy, He-GL performs, in general, beer han BS-DO, and BS-W 9 Overall, Table 1 shows ha he combinaions ha use simpler models such as BS and Whaley s seem o perform equally well wih he benchmark combinaion He-GL. This evidence is corroboraed by he confidence inervals ess presened in Tables 3 and 4 where he differences in he hedging error across he various models appear o be saisically insignifican. These findings have an imporan implicaion: here is no need o resor o complex volailiy opion pricing models in order o hedge a arge opion wih a volailiy opion. I suffices o use Whaley s model ha is based on simpler assumpions. This resul is analogous o he empirical findings of Sco (1991) and Backshi e al. (1997) who compared he dela-vega hedging performance of BS versus more complex models (sochasic volailiy and jump models, respecively). In a broader sudy, Dumas e al. (1998) reached similar resuls and hey sae, Simpler is beer. B. Model Error and he Pricing Performance The effec of model error on he pricing performance of he various volailiy opion pricing models is examined by evaluaing he pricing performance of he DO and Whaley models relaive o he rue GL model. The square roo of he uncondiional mean square pricing error is used as a measure of he pricing error (rue GL price minus he price of each one of he oher wo models). This is calculaed as follows. For each model and for any given simulaed volailiy pah, he square pricing errors are calculaed a each rebalancing poin. Nex, he mean square pricing error is calculaed across he rebalancing poins. This exercise is repeaed for each one of he 1,000 volailiy 9 Some excepions occur in he case of he OTM arge opions. 22

23 simulaion runs. Finally, he uncondiional mean square pricing error is obained as he average of he 1,000 mean square pricing errors. Three differen correlaion values beween he sock price and volailiy (ρ=-0.5, 0, +0.5) are used o perform he join MC simulaion. The pricing performance of each model is assessed for a volailiy opion wih hree separae moneyness levels (ITM, ATM, OTM), and hree separae mauriies (shor, inermediae, long). Table 5 shows he square roo uncondiional mean square pricing error of he DO and Whaley models for differen mauriies and moneyness levels across he hree correlaion values. We can see ha he pricing performance of he Whaley model is superior o ha of he DO. This holds regardless of he opion moneyness/mauriy, as well as of he correlaion values. Despie he fac ha he DO model relies on a volailiy process ha is closer o he real one, i performs worse han Whaley s model ha is based on a seemingly more unrealisic process. This is in accordance wih he resuls from he analysis of he hedging performance where we found ha he simpler is beer. (INSERT TABLE 5 HERE) Focusing on he performance of each individual model, regarding he moneyness dimension, we can see ha Whaley s model performs beer for OTM opions; he DO model performs equally worse for all moneyness levels. Regarding he mauriy dimension, Whaley s model performs beer for shor mauriy opions. On he oher hand, he DO model performs beer for long mauriy opions. VI. Conclusions This paper makes wo conribuions o he rapidly evolving volailiy opions lieraure by addressing he following wo quesions: (a) Are volailiy opions superior o sandard opions in erms of hedging volailiy risk?, and (b) Are he volailiy opion pricing models robus for hedging and pricing purposes in he presence of model risk? 23

24 To his end, a join Mone Carlo simulaion of he sock price and volailiy in a sochasic volailiy seup has been employed. Firs, wo alernaive dynamic dela-vega wih discree rebalancing hedging schemes were consruced o assess he hedging performance of plain vanilla opions versus volailiy opions. A shor sandard European call is assumed o be he opion o be hedged (arge opion). A sandard European call and a European volailiy call opion are he alernaive hedging insrumens. Black Scholes (1973), Heson (1993), Whaley (1993), Grünbichler and Longsaff (1996), and Deemple and Osakwe (2000) models have been used o hedge he sandard and volailiy opions. Then, he robusness of he hedging and pricing effeciveness of he volailiy opion pricing models in he presence of model error was invesigaed. The Grünbichler-Longsaff and Heson models were assumed o be he rue models; his is consisen wih he choice of he volailiy daa generaing process used o simulae he daa. Our wo research quesions have been examined for various expiry daes and srikes of he arge opion, as well as for alernaive correlaion values beween he sock price and volailiy, and for differen rebalancing frequencies. Roll-over and no-roll over sraegies in he volailiy opion were also considered. In erms of he hedging effeciveness, we found ha he hedging scheme ha uses volailiy opions as hedging insrumens is no superior o he one ha uses sandard opions. In erms of he impac of model error o he hedging performance, combinaions ha use simpler models such as Black-Scholes and Whaley s seem o perform equally well wih he benchmark combinaion (Grünbichler-Longsaff and Heson). Regarding he impac of model error o he pricing performance, Whaley s model performs beer han Deemple-Osakwe s model. This sudy has a leas five imporan implicaions for he use of volailiy opions and heir pricing models. Firs, volailiy opions are no beer hedging vehicles han plain-vanilla opions for he purposes of hedging sandard opions. This finding exends he conclusions in Jiang and Oomen (2001) who examined he hedging effeciveness of volailiy fuures versus 24

25 plain-vanilla opions; hey found ha he laer perform beer han he former. However, his does no invalidae he imminen inroducion of volailiy opions (and volailiy fuures) in various exchanges. Volailiy opions may be proved o be very useful for volailiy rading and for hedging oher ypes of opions, e.g., exoic opions. The liquidiy and he ransacion coss will be criical facors for he success of his emerging new marke, as well. Second, in he case ha an invesor chooses he volailiy opions as hedging insrumens, hese should be used o hedge a-he-money and ou-of-he-money raher han in-he-money arge opions. This feaure may encourage he use of volailiy opions for hedging purposes given ha mos of he opions rading aciviy is concenraed on a-he-money opions. Third, he hedging performance of volailiy opions increases as heir rebalancing frequency increases. Fourh, he roll-over sraegy wih volailiy opions should be preferred since i performs beer han he no-roll-over sraegy. Finally, despie he fac ha Whaley s model is he wors misspecified model wihin our simulaion framework, i can be reliably used o hedge sandard opions wih volailiy opions, and o price volailiy opions. This is in accordance wih he resuls from previous sudies in he model error (sandard opions) lieraure ha found ha increasing he complexiy of he opion pricing model does no necessarily improve i s pricing and hedging performance (see e.g., Backshi e al. 1997, Dumas e al. 1998). The simulaion seup presened creaes hree srands for fuure research. Firs, he abiliy of volailiy opions o hedge exoic opions (e.g., barrier opions) saisfacorily should be explored. Second, he hedging effeciveness of opions wrien on alernaive measures of volailiy should be invesigaed. For example, Brenner e al. (2001) sugges he inroducion of opions on sraddles. Finally, i may be worh sudying he hedging effeciveness of volailiy opions and he impac of model error for alernaive daa generaing processes (see e.g., Carr and Wu 2002, and Daouk and Guo 2004). In he ineress of breviy, hese exensions are bes lef for fuure research. 25

26 References Bakshi, G., Gao, C., and Chen, Z., Empirical Performance of Alernaive Opion Pricing Models. Journal of Finance 52, Backshi, G., and Kapadia, N., Dela-Hedged Gains and he Negaive Marke Volailiy Risk Premium. Review of Financial Sudies 16, Ball, C., and Roma, A., Sochasic Volailiy Opion Pricing. Journal of Financial and Quaniaive Analysis 29, Black, F., The Pricing of Commodiy Conracs. Journal of Financial Economics 3, Black, F., and Scholes M The Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy 81, Blair, B.J., Poon, S., and Taylor, S.J., Forecasing S&P 100 Volailiy: he Incremenal Informaion Conen of Implied Volailiies and High--Frequency Index Reurns. Journal of Economerics 105, Boyle, P., and Emanuel, D., Discreely Adjused Opion Hedges. Journal of Financial Economics 8, Brenner, M., and Galai, D., New Financial Insrumens for Hedging Changes in Volailiy. Financial Analyss Journal July-Augus, Brenner, M., and Galai, D., Hedging Volailiy in Foreign Currencies. Journal of Derivaives, 1, Brenner, M.,, Ou, E., and Zhang, J., Hedging Volailiy Risk. Working Paper, New York Universiy. Brien-Jones, M. and Neuberger, A., Opion Prices, Implied Price Processes, and Sochasic Volailiy. Journal of Finance 55, Carr, P., and Madan, D Towards a Theory of Volailiy Trading, in: Jarrow, R., (Ed.), Volailiy. Risk Books, pp

27 Carr, P., and Wu, L., Saic Hedging of Sandard Opions. Working Paper. New York Universiy. Corrado, C.J., and Miller, T., The Forecas Qualiy of CBOE Implied Volailiy Indexes. Working Paper. Washingon Universiy, Olin School of Business. Crouhy, M., Galai, D., and Mark, R., Model Risk, Journal of Financial Engineering 7, Daouk, H. and Guo, J.Q., Swiching Asymmeric GARCH and Opions on a Volailiy Index. Journal of Fuures Markes 24, Derman, E., Kani, I., and Kamal, M., Trading and Hedging Local Volailiy. Journal of Financial Engineering 6, Deemple, J., and Osakwe, C., The Valuaion of Volailiy Opions. European Finance Review 4, Dumas, B., Fleming, J., Whaley, R. E., Implied Volailiy Funcions: Empirical Tess. Journal of Finance 53, Dupire, B., Model Ar. Risk 6, Figlewski, S Opion Arbirage in Imperfec Markes. Journal of Finance 44, Figlewski, S., and Wang, X., Is he Leverage Effec a Leverage Effec? Working Paper. New York Universiy, Sern School of Business. Fleming, J., Osdiek, B., and Whaley, R. E., Predicing Sock Marke Volailiy: A New Measure. Journal of Fuures Markes 15, Galai, D., The Componens of he Reurn from Hedging Opions Agains Socks. Journal of Business 56, Green, T. C., and Figlewski S., Marke Risk and Model Risk for a Financial Insiuion Wriing Opions. Journal of Finance 54,

28 Grünbichler, A., and Longsaff, F., Valuing Fuures and Opions on Volailiy. Journal of Banking and Finance 20, Guo, D., The Risk Premium of Volailiy Implici in Currency Opions. Journal of Business and Economic Saisics 16, Henderson, V., Hobson, D., Howison, S., and Kluge, T., A Comparison of Opion Prices under Differen Pricing Measures in a Sochasic Volailiy Model wih Correlaion. Working Paper. Princeon Universiy. Heson, S., A Closed-Form Soluion for Opions wih Sochasic Volailiy wih Applicaions o Bond and Currency Opions. Review of Financial Sudies 6, Hull, J. C., and Whie, A., Hedging he Risks from Wriing Foreign Currency Opions. Journal of Inernaional Money and Finance 6, Hull, J. C., and Whie, A., An Analysis of he Bias in Opion Pricing Caused by Sochasic Volailiy. Advances in Fuures and Opions Research 3, Hull, J. C., and Suo, W., A Mehodology for Assessing Model Risk and is Applicaions o he Implied Volailiy Funcion Model. Journal of Financial and Quaniaive Analysis 37, Jiang, G., and Oomen, R., Hedging Derivaives Risks: A Simulaion Sudy. Working Paper. Universiy of Warwick. Johnson, N. L., and Koz, S., Coninuous Univariae Disribuions, Vols 1 and 2, John Wiley, New York. Merville, L. J., and Piepea, D. R., Sock-Price Volailiy, Mean-Revering Diffusion, and Noise. Journal of Financial Economics 24, Moraux, F., Navae, P., and Villa, C., The Predicive Power of he French Marke Volailiy Index: A Muli Horizons Sudy. European Finance Review 2,