The Price-setting Behavior of Banks: An Analysis of Open-end Leverage Certificates on the German Market

Transcription

1 The Price-seing Behavior of Banks: An Analysis of Open-end Leverage Cerificaes on he German Marke Oliver Enrop +* Hendrik Scholz ++ Marco Wilkens +++ Working Paper Firs Version: Augus 10, 2006 This Version: January 15, Oliver Enrop, Caholic Universiy of Eichsae-Ingolsad, Ingolsad School of Managemen, Auf der Schanz 49, D Ingolsad, Auf der Schanz 49, D Ingolsad, Germany, phone: , fax: , oliver.enrop@ku-eichsae.de ++ Hendrik Scholz, Caholic Universiy of Eichsae-Ingolsad, Ingolsad School of Managemen, Auf der Schanz 49, D Ingolsad, Germany, phone: , fax: , hendrik.scholz@ku-eichsae.de. +++ Marco Wilkens, Caholic Universiy of Eichsae-Ingolsad, Ingolsad School of Managemen, Auf der Schanz 49, D Ingolsad, Germany, phone: , fax: , marco.wilkens@ku-eichsae.de. * Pars of his research were done while Oliver Enrop was visiing he School of Banking and Finance, Universiy of New Souh Wales. He hanks Terry Waler and he academic and adminisraive saff for heir hospialiy and suppor. We are graeful o paricipaions a he Ausralasian Banking and Finance Conference 2006, Sydney, for helpful commens and suggesions on an earlier draf of his paper.

2 The Price-seing Behavior of Banks: An Analysis of Open-end Leverage Cerificaes on he German Marke Absrac This paper presens he firs analysis of he pricing of exchange-raded open-end leverage cerificaes on he German reail marke. The major innovaions of hese cerificaes are wofold. Firs, he issuers announce ex-ane a price-seing formula according o which hey are willing o buy and sell he cerificaes over ime. In paricular, his formula is independen of he volailiy of he underlying. Second, he produc s lifeime is poenially endless. Our main findings can be summarized as follows: i) The price-seing formula srongly favors he issuers. ii) Issuers can hedge hese cerificaes easily wih a semi-saic superhedge using spo marke insrumens. iii) The price-seing formula confirms he main oucome of he life cycle hypohesis for srucured financial producs (e.g., Soimenov and Wilkens, 2005), insuring ha profis by issuers sysemaically increase in he course of produc lifeimes. iv) The value of he mispricing by consrucion depends on he volailiy of he underlying and he so-called funding rae spread. Compared wih hese facors, he influence of ineres raes and heir dynamics was found o be negligible. JEL: G13; G21; G24 Keywords: Srucured producs; Cerificaes; Hedging; German marke; Pricing

3 1 Inroducion In many counries since he mid-nineies, exchange-raded innovaive financial producs (IFPs), such as equiy-linked bonds and leverage producs, have become increasingly imporan in he reail marke (see Soimenov and Wilkens, 2005, for a curren overview of he German marke as one of he mos imporan markes for IFPs). Off-exchange rades of IFPs are usually seled by issuers, whereas exchange rades are primarily conduced by marke makers. As issuing banks normally handle he marke making by hemselves, hey de faco dominae no only he primary bu also he secondary marke. Since shor selling of IFPs is virually impossible, marke makers can sysemaically quoe prices ha do no mach fair heoreical values bu favor hemselves. In fac, he price-seing mechanism applied by issuers is generally kep hidden from invesors. Since IFPs are ofen complex, i is frequenly difficul for privae invesors o calculae heir fair values and, hence, o evaluae he fairness of he quoes. Due o his inransparency, a large body of empirical work has been carried ou o analyze he price-seing behavior of issuers by comparing quoed prices and heoreical fair values. Chen and Kensinger (1990), Chen and Sears (1990), Baubonis e al. (1993), and Bene e al. (2006) repor significan deviaions for equiy-linked producs on he US marke. Brown and Davis (2004) recenly deeced significan price deviaions for endowmen warrans on he Ausralian marke. An analogue resul was found for he Swiss marke by Wasserfallen and Schenk (1996), Burh e al. (2001), and Grünbichler and Wohlwend (2005), and for he German marke by Wilkens e al. (2003), Soimenov and Wilkens (2005), and Baule e al. (2006). All hese empirical sudies reveal he pricing behavior of issuers: A issuance, hey regularly sell IFPs for heir heoreical value plus a posiive premium, and laer on hey buy hem back paying he heoreical value plus a decreased premium. As a resul, issuers gain by diminishing overpricing in he course of produc lifeimes. Wilkens e al. (2003) and Soimenov and Wilkens (2005) analyze his behavior in deail for srucured financial producs and subsume decreasing premiums over ime under life cycle hypohesis. In recen years, several banks have issued leverage producs as a new ype of IFP. Alhough hey were no issued unil Ocober 2001, his marke segmen now replaces a subsanial porion of he classical warran marke in Germany. The main characerisics of he firs generaion of leverage producs are equivalen o hose of one-sided barrier opions. Muck (2006a, 2006b) and Wilkens and Soimenov (2006) analyze he pricing of hese cerificaes similarly o he above-menioned sudies. Muck (2006a) and Wilkens and Soimenov (2006) 1

4 repor clear posiive premiums ha favor issuers. In conras, Muck (2006b) finds ha jump risk a leas parially explains hese premiums. Looking a he whole daa se conaining seven issuers, Muck (2006a) could no confirm decreasing premiums over he produc lifeime, while he finds only weak evidence supporing he life cycle hypohesis on an individual issuer level. Wilkens and Soimenov (2006) refrain from pursuing he life cycle hypohesis because he knock-ou characerisic of leverage cerificaes yields sochasic lifeimes. This paper is he firs analyzing a new generaion of leverage producs on he German reail marke, namely open-end leverage cerificaes. 1 In Ocober 2002, banks sared issuing his generaion, beginning wih 84 cerificaes in 2002 and reaching 14,030 during he firs hree quarers of Compared o financial producs analyzed by he sudies menioned above, and in paricular in conras o he firs generaion of leverage cerificaes discussed in Muck (2006a, 2006b) and Wilkens and Soimenov (2006), his new generaion exhibis wo main innovaive feaures: i) Issuers announce ex-ane a relaively simple price-seing formula, according o which hey are willing o sell and repurchase hese cerificaes over ime. ii) Open-end leverage cerificaes do no have a fixed produc mauriy, bu a poenially perpeual lifeime. Feaure i) removes he arbirariness of he issuers quoes for IFPs, from he invesors poin of view; his arbirariness is normally presen and only slighly limied hrough compeiion across issuing banks. To he bes of our knowledge his sudy is he firs ha does no have o rely on quoes colleced on he primary and secondary marke for analyzing he price-seing behavior of issuers. Since we focus direcly on he price-seing formula used by issuers, we are able o fill a conspicuous gap in he presen empirical lieraure. The paper is organized as follows: Secion 2 describes he consrucion of open-end leverage cerificaes and he price-seing formula applied by banks. Secion 3 analyzes a semi-saic superhedge of open-end leverage cerificaes using spo marke insrumens. Based on his superhedge we find he price-seing formula srongly favors issuers. Addiionally, we show ha he life cycle hypohesis clearly holds for open-end leverage cerificaes a finding ha could no be shown for he firs generaion of leverage cerificaes. Secion 4 presens a comparaive saic analysis of he value of mispricing from he issuers poin of view. This valuaion is based on a Black and Scholes (1973) and Meron (1973) world wih sochasic 1 These cerificaes are sold under differen names, see Table 3. 2

5 ineres raes. In his conex we discuss main facors influencing he heoreical value of open-end leverage cerificaes, in paricular he volailiy of he underlying and he so-called funding rae spread se by he issuer. Secion 5 discusses he impac of differing produc feaures on our main findings. Secion 6 concludes. 2 Main characerisics of open-end leverage cerificaes Open-end leverage cerificaes are issued in wo basic forms: as long cerificaes which benefi from he increasing prices of he underlying, and as shor cerificaes which profi from decreasing prices. We focus on open-end long cerificaes (OELCs), which are more imporan considering he number of issues in Germany from Ocober 2002 o Sepember 2006 (see Table 1). However, he analysis can be ransferred o open-end shor cerificaes sraighforwardly. 2 Table 2 repors he number of issues in relaion o he specific underlying. Obviously socks and sock indices are he mos common underlyings. Inser Table 1 abou here Inser Table 2 abou here Depending on he issuer of OELCs, we find slighly varying produc feaures in pracice. However, he main characerisics of hese cerificaes mach closely. An OELC is based on an underlying S and has a srike X, and a knock-ou barrier B. To keep he illusraion general and as inuiive as possible, he following analysis is based on a sylized definiion of OELCs ha is closely relaed o cerificaes of HSBC Trinkaus & Burkhard. 3 We analyze he influence of differing produc feaures in Secion 5. As already poined ou, OELCs do no have a fixed produc mauriy, bu a poenially perpeual lifeime. However, hey become due when he price of he underlying his or falls below he barrier for he firs ime. This firs passage ime τ is given by 2 3 We discuss briefly he price-seing formula issuers apply for open-end shor cerificaes in Secion 3, Foonoe 9. We choose his issuer, as he explicily declares a consan funding rae spread and a consan relaive difference beween barrier and srike over ime (which will be defined more precisely laer) in sales prospecs (see HSBC Trinkaus & Burkhard, 2006). Furhermore, OELCs of his issuer show daily changing srikes and barriers as well. 3

6 τ = inf{: S B }, (1) where S denoes he price of he underlying and B he barrier in. In he case of a knock-ou in, he invesor receives a selemen amoun (rebae) P, which is he difference beween he price of he underlying S and he srike X : 4 P = S X. (2) In pracice HSBC Trinkaus & Burkhard, for example, deermine he selemen amoun P of OELCs wihin one hour following he knock-ou based on he prices hey ge from erminaing heir hedging insrumens (see HSBC Trinkaus & Burkhard, 2006). Therefore he exac price of he underlying S a knock-ou in is no necessarily relevan for he selemen amoun. Iner alia his characerisic ransfers he liquidiy risk in conex wih he hedging insrumens o he invesor. However, we will absrac from his in our analysis. Barrier and srike are no consan over ime. The iniial srike X 0 increases over ime, relaed o he so-called funding rae. This funding rae consiss of a (variable) shor-erm money marke ineres rae r such as EONIA (Euro OverNigh Index Average) and a funding rae spread z > 0. The barrier is designed o permanenly exceed he srike by he facor a > 0: B = (1 + a) X. 5 Assuming r and z are coninuously compounded we have: X = X 0 exp (r ś + z) ds = X 0 exp r ś ds + z, (3) 0 0 B = (1 + a) X = (1 + a) X 0 exp r ś ds + z. (4) 0 Subsiuing (3) ino (2), he selemen amoun in he case of a knock-ou in is given by: 4 5 The conversion raio is ofen no one, hus he invesor receives a par or a muliple of his difference. As a rule, facor a is large enough o ensure ha, in general, he price of he underlying is higher han he srike, even in he case of an occasional illiquidiy of he underlying afer he knock-ou even. Changing marke condiions, in paricular clearly increasing volailiies, could cause issuers o increase he facor a o resore a close-o-zero probabiliy of a negaive selemen amoun (see also Secion 5). Only in he mos unlikely even of surprisingly high illiquidiy of he underlying or large negaive jumps of he underlying price no only beneah he barrier bu also beneah he srike could he selemen amoun according o (2) be negaive. Since, in general, he selemen amoun of real cerificaes is defined as non-negaive, issuers ake his mos unlikely, pracically negligible risk. 4

7 P = S X 0 exp r ś ds + z. (5) 0 Equaion (5) no only describes he selemen amoun, i is also used by he issuer as a priceseing formula for he secondary marke. 6 A any ime during he lifeime of a cerificae, he issuer is willing o sell or buy back he cerificae for a price according o (5). 7 Hence, he price of he cerificae in only depends on he sock price in, he iniial srike, pas money marke raes, and he funding rae spread. We emphasize ha Equaion (5) is independen of he knock-ou barrier and he volailiy of he underlying. 3 Semi-saic superhedge and life cycle hypohesis Wha is he inuiion behind he price-seing formula (5)? When he issuer sells he cerificae for P 0 = S 0 X 0 in = 0, he can a he same ime purchase he underlying for S 0 and issue revolving shor-erm deb for he noional amoun X 0 which comes ou o a oal paymen of zero. As banks can refinance hemselves a shor-erm money marke ineres raes r, such as EONIA in he iner-bank marke, 8 he value of his hedge posiion, he leveraged underlying, LU, in is given by: LU = S X 0 exp r ś ds, (6) 0 where we assume ha ineres rae paymens are accrued. A decomposiion of he priceseing formula for OELCs according o (5) clarifies he relaion beween his formula and he leveraged underlying: P = S X 0 exp r ś ds X 0 exp r ś ds + z exp r ś ds. (7) leveraged underlying LU profi poenial PP Invesors can ofen also exercise cerificaes daily (e.g., Goldman Sachs, 2006) or monhly (e.g., HSBC Trinkaus & Burkhard, 2006). In ha case hey receive a selemen amoun according o he paricular priceseing formula based on he closing-price of he underlying. In he following we absrac from bid-ask spreads exising in pracice. Addiionally, he issuer can use he purchased underlying o collaeralize oher deb which reduces his ineres rae paymens. In his respec, he can refund he underlying a a ne ineres rae ha is even lower han r. 5

8 The price of he OELC in equals he value of he leveraged underlying minus a erm PP which is posiive by definiion a any > 0. Accordingly, he value of he leveraged underlying is always higher han he quoed price of he OELC. Hence, buying he leveraged underlying in = 0 represens a semi-saic superhedging sraegy which has o be erminaed by he issuer in he case of a knock-ou or a repurchase. 9 If he invesor reurns he cerificae in or, alernaively, if he OELC is knocked ou in, he bank can sell he leveraged underlying and sele he invesor s claim, which yields a bank s posiive (gross 10 ) profi PP. Therefore, we denoe PP as banks profi poenial. The erm poenial is used because he poin in ime of a knock-ou or repurchase is ex-ane unknown. As he price according o (7) always develops more poorly han he superhedge, he price-seing formula clearly favors he issuer. Noe ha he semi-saic superhedge is based on spo marke insrumens and can easily be implemened, as i works wihou poenially illiquid derivaives. The issuer s profi poenial PP from OELCs equals zero a issuance and increases over ime. This finding is clearly in line wih he main oucome of he life cycle hypohesis for IFPs, quoing sysemaically rising gains for issuers in he course of produc lifeimes. However, he above-menioned sudies dealing wih he firs generaion of leverage cerificaes could no confirm he life cycle hypohesis. Addiionally, in conras o papers analyzing he life cycle hypohesis for oher IFPs, our resul can be derived analyically based on he price-seing formula raher han indirecly by observing quoed prices and calculaing fair heoreical values relying on opion valuaion models. Hence, in conras o earlier sudies our conclusions are no subjec o model risk The decomposed price-seing formula for open-end shor cerificaes is: shor P = X 0 exp r ś ds S X 0 exp r ś ds exp r ś ds z. Obviously, a semi-saic superhedging sraegy for open-end shor cerificaes conains a shor posiion in he underlying S and an invesmen of X 0 in a shor-erm money marke accoun. The coss of srucuring, disribuion ec. have o be deduced from he gross profi (poenial) o ge o he ne profi (poenial). As long as here is no danger of confusion, he addiion gross will be abandoned in he following. 6

9 4 Valuaion from he bank s perspecive 4.1 Valuaion algorihm Wha are he fair heoreical values of OELCs? Alhough we do no have o apply valuaion models o confirm he life cycle hypohesis for OELCs in Secion 3, we sill need o derive a valuaion algorihm o compue he fair heoreical values of OELCs and he value of issuers mispricing. Based on his algorihm and he laer comparaive saic analysis, invesors and issuers can assess how aracive OELCs are depending on differing underlyings, produc feaures, and marke condiions. Moreover, comparing OELCs and oher IFPs, in erms of he differences beween marke prices and fair heoreical values, reveals he comparaively exraordinary gains for banks issuing OELCs. All sudies menioned in Secion 1 analyzing quoes of IFPs assume arbirage-free markes and apply risk-neural valuaion echniques o calculae fair heoreical values of financial producs. In conras, i is eviden ha IFPs on he whole, and OELCs in paricular, offer arbirage opporuniies o banks as already discussed in Secions 1 and 3. This seeming conflic can be resolved by a marke segmenaion hypohesis similar o ha of Jarrow and van Devener (1998) in he conex of credi cards loans and demand deposis: There are a number of banks wih access o capial markes, whereas his access is limied for individual invesors for several reasons, such as legal resricions, large enry barriers or excessive ransacion coss. I can be assumed ha markes are (nearly) arbirage-free for issuers when hey hedge such IFPs, bu privae invesors canno buy he replicaing porfolio or he payoff profile of IFPs, a leas no wihou addiional coss. Addiionally, hey canno ake shorposiions in IFPs o benefi from unfair quoes, i.e. o make arbirage gains. Hence, when riskneural valuaion echniques are applied, he resuling values are fair heoreical values for banks and lower boundaries for he value ou of he perspecive of privae invesors. The following analysis calculaes he value of OELCs from he bank s poin of view. Assumpion 1: For banks, capial markes are arbirage-free, fricionless, and complee. Under mild regulariy condiions his ensures he exisence of a unique equivalen risk-neural measure Q (Harrison and Pliska, 1981, Heah e al., 1992). Furher, we assume ha he underlying of he OELC is a sock (index) following a geomeric Brownian moion and allow he defaul-free shor rae o be sochasic (Meron, 1973). For simpliciy, we assume he sock o be non-dividend paying. However, his could easily be relaxed. 7

10 Assumpion 2: The defaul-free shor rae r is sochasic. Under Q, he sock price saisfies he equaion ds = r S d + σ S dw, (8) where σ > 0 denoes he consan volailiy and W a sandard Brownian moion. Sudying leverage cerificaes of he firs generaion in he German marke wih finie lifeimes and wihou an openly communicaed price-seing formula, Muck (2006b) allows for sochasic volailiy and jumps. While i urns ou ha sochasic volailiy has only a marginal effec on heoreical values of hese leverage cerificaes, jump risk exercises a subsanial impac. The reason for his is ha he ype of leverage cerificaes analyzed in Muck (2006b) exhibis ime-invarian and coinciding barrier and srike which, in he case of a knock-ou even, yields a selemen amoun of zero. Therefore, possible negaive jumps do no necessarily harm invesors severely because if he sock price jumps beneah he barrier (and he srike) which yields a knock-ou of he cerificae, is value becomes zero regardless of he amoun he srike is undersho. In conras, posiive jumps always fully benefi invesors. Since he issuer has o bear his negaive jump risk, he feaure barrier equals srike has a posiive impac on he heoreical value of leverage cerificaes in he presence of jump risk, as Muck (2006b) shows. However, he innovaive OELCs sudied in his paper exhibi a barrier ha permanenly exceeds he srike by a facor a, which is usually large enough o make i exremely unlikely ha a sock price will undershoo no only he barrier bu also he srike. Therefore, he srong posiive impac of jump risk on heoreical values of leverage cerificaes analyzed in Muck (2006b) does no ransfer o he OELCs considered here. Consequenly, we ignore possible jump risk. For simpliciy, we disregard credi spreads in he money marke and assume he bank s shorerm refinance rae r equals he defaul-free shor rae r. Addiionally, we assume he issuer is defaul-free. 11 Assumpion 3: The issuer of he cerificae is defaul-free. The bank s shor-erm refinance rae r equals he defaul-free shor rae r. 11 The influence of he issuer s credi risk could, for example, be analyzed along he lines of Hull and Whie (1995) or Baule e al. (2006). 8

11 Clearly, heoreical fair OELC values depend on he holding period T of invesors, since he realized profi poenial of he bank increases wih he lengh of ime invesors hold hese cerificaes. The above-menioned sudies deal wih IFPs exhibiing a fixed mauriy. Furhermore, i is implicily assumed ha invesors hold hese producs unil mauriy, alhough in general hey have he opporuniy o sell hem back o he issuer a any ime. To obain comparable resuls, we assume ha invesors plan o hold OELCs for a cerain ime period T. Therefore, a paymen prior o he expiraion of his holding period only occurs in he case of a knock-ou. To show he impac of differen assumed holding periods on heoreical values, we laer repor he resuls for various choices of T in a comparaive saic analysis und discuss furher exensions in Secion 6. Assumpion 4: Invesors plan o hold he OELC for a finie holding period T. According o he risk-neural valuaion echnique, he presen value of a securiy resuls from he expeced value of he discouned payoffs. Taking a possible knock-ou ino accoun, he poin in ime when he invesor receives a paymen according o he price-seing formula is τ T = min(τ, T) given a planned holding period T. Based on Equaion (7), oday s fair heoreical value OELC 0T of open-end long cerificaes can be calculaed as: T OELC 0 = E Q τ T exp r s ds P τ T 0 = E Q τ exp T τ T τ T r s ds S τ T X 0 exp r s ds X 0 exp r s ds + z τ T τ T exp r s ds = S 0 X 0 X 0 (E Q (exp(z τ T ) 1) = S 0 X 0 X 0 (exp(z T) (1 Q(τ T)) + E Q (1 {τ T} exp(z τ)) 1) (9) LU 0 VPP 0 T where E Q ( ) denoes expecaion wih respec o Q. In he Appendix we derive closed-form soluions for he risk-neural cumulaive knock-ou probabiliy Q(τ ) and he expression E Q (1 {τ } exp(z τ)) for every > 0. We have Q(τ ) = N(h 1 ()) + 2 (σ 2 / 2 + z) B 0 S 0 σ 2 N(h 2 ()), (10) 9

12 E Q (1 {τ } exp(z τ)) = 2 z B 0 S 0 σ 2 N(h 3 ()) + 1 B 0 S 0 N(h 4 ()) wih h 1 () = ln(b 0 / S 0 ) + (σ 2 / 2 + z), h 2 () = ln(b 0 / S 0 ) (σ 2 / 2 + z), σ σ h 3 () = ln(b 0 / S 0 ) + (σ 2 / 2 z), and h 4 () = ln(b 0 / S 0 ) (σ 2 / 2 z), σ σ where N( ) denoes he sandard Gaussian cumulaive disribuion funcion. Since we have σ 2 / 2 + z > 0, Q(τ ) converges o 1 for large. The valuaion of OELCs according o (9) discloses ha heir values do no depend on he shor rae and is dynamics. This is a naural resul, since he shor rae eners boh he drif of he sock price process (8) and he imevarying barrier (4). The las row in (9) allows for an economic inerpreaion of he heoreical value of OELCs. Today s cerificae value OELC 0 T consiss of he value of he leveraged underlying LU 0 (= P 0 = S 0 X 0 ) minus oday s heoreical value of he profi poenial VPP 0T given he holding period T. In oher words, VPP 0 T represens oday s value of he difference beween he semisaic superhedging sraegy and he price of he OELC. Clearly, banks are mos ineresed in increasing his difference, as i presens he value of heir arbirage gains. The essenial arge analyzed in earlier sudies is he relaive price deviaion beween he price se by he issuer and he fair heoreical value of he IFP. This relaive price deviaion can be inerpreed as issuers percenage profi under he sandard assumpion of invesors being invesed ino he produc unil mauriy. An analogue proceeding is possible in he conex of he OELCs analyzed here: RPD T 0 = P T 0 OELC 0 P = VPP T 0 0 P, (11) 0 where he numeraor denoes he absolue price deviaion beween he quoed price P 0 and he value of he cerificae OELC T 0. In conras o oher sudies, here we relae his difference o he curren price P 0 of he OELC and no o is heoreical value. This means ha according o (11), he relaive price deviaion is relaed o he price of he hedging insrumens in = 0. In he case of OELCs, he value of his hedge posiion exacly equals he curren price, whereas 10

13 he value of common hedge posiions of classical IFPs (e.g., discoun cerificaes) maches he heoreical value of he produc. Wih his in view, we relae he price deviaion o he value of he hedge posiion, like in oher empirical sudies. 4.2 Comparaive saic analysis In his Secion, we analyze he impac of differen produc designs and marke condiions on he heoreical values of OELCs. From he perspecive of banks or invesors, his shows which produc design is especially profiable or disadvanageous depending on marke condiions. As a saring poin, we examine a noional OELC on he German blue-chip sock index DAX. Since he DAX is a performance index, no adjusmens for dividends are needed. The main characerisics of he cerificae mach real OELCs offered by HSBC Trinkaus & Burkhard. Iniially, he cerificae has a srike X 0 of 5, and a barrier exceeding he srike by a = 1.5 %. Thus a issuance, he barrier amouns o B 0 = 5, According o he price-seing formula (5) he cerificae s srike is coninuously compounded based on he funding rae r + z = r %. For he curren DAX of S 0 = 5,700.00, he price of he cerificae a issuance is P 0 = Given a consan shor rae of r = 3 %, he lef ordinae of Figure 1 shows he issuer s profi poenial PP. Iniially, he price of he cerificae and he value of he leveraged underlying mach a Consequenly, he profi poenial and is value are zero. The black-labeled line shows he profi poenial PP almos linearly increasing in. For example, given a oneyear holding period, he profi poenial PP 1 already reaches Tha is exacly he amoun he issuer gains if he cerificae has no been knocked-ou before and he invesor sill holds he cerificae in one year. Wih respec o he curren price of he cerificae, he relaive profi poenial is as huge as % (= PP 1 / P 0 = / ) much more han for oher IFPs analyzed in he sudies menioned above. Inser Figure 1 abou here The grey ploed line in Figure 1 shows, a he same ime, he oday s value of he profi poenial VPP T 0 (lef ordinae) and he relaive price deviaion RPD T 0 (righ ordinae). Due o he possibiliy of an early knock-ou, boh increase much more flaly han he profi poenial. Obviously, he risk-neural probabiliy of a premaure knock-ou of he cerificae has a srong influence on he value of he profi poenial VPP T 0. Besides he raio of he iniial barrier and 11

14 he price of he underlying a issuance (B 0 / S 0 ), he essenial facors deermining he riskneural knock-ou probabiliy are he volailiy of he underlying σ and he funding rae spread z (see Equaion (10)). 12 Figure 2 shows he cumulaive risk-neural knock-ou probabiliy of he noional cerificae depending on ime and volailiy σ. Given a volailiy of σ = 0, he exemplary cerificae is going o be knocked-ou a ime τ = 2.98 (= ln(s 0 / B 0 ) / z) because in he risk-neural world he barrier grows faser han he underlying due o he funding rae spread z. 13 Consequenly, here is a knock-ou probabiliy of zero for any momen before. A firs, higher volailiies increase he knock-ou probabiliy quie seeply, bu i flaens laer on. I converges o 1 for large. Inser Figure 2 abou here Figure 3 shows he impac of he volailiy on he value of he profi poenial VPP T 0 for a planned holding period of T = 1 and for various choices of he funding rae spread z of 1.5 %, 2.5 %, and 3.5 %, and curren DAX values S 0 of 5, and 6, Again, given a volailiy of zero, he risk-neural knock-ou probabiliy of he cerificae is zero wihin he firs year for all hree choices of he funding rae spread z. Hence, he high profi poenial PP 1 will definiely be realized by he bank. The value of he profi poenial VPP T 0 decreases wih increasing volailiy. Consequenly, a higher volailiy is disadvanageous for he bank, as i increases he value of he cerificae. This is caused by he fac ha he invesor is more likely o be forced ou of he cerificae earlier by a higher volailiy since i causes a higher probabiliy of premaure knock-ou, which implies a lower value of he profi poenial VPP 0T. A higher curren DAX of 6, causes a higher value of he profi poenial, as i increases he difference beween underlying and barrier and hus lowers he probabiliy of a knock-ou. Inser Figure 3 abou here The sensiiviies of he risk-neural knock-ou probabiliy can be derived analyically. The posiive dependence of he risk-neural knock-ou probabiliy on he volailiy of he underlying holds for ln(b 0 / S 0 ) + z < 0. This condiion is equivalen o he probabiliy of a knock-ou unil ime converges o zero, given σ 0. Given a volailiy of zero, he price of he underlying in is S = S 0 exp r s ds while he barrier is B = B 0 0 exp r s ds + z. This yields he non-sochasic knock-ou ime τ = ln(s 0 / B 0 ) / z. 0 12

15 In pracice, banks issue various cerificaes on he same underlying wih differen srikes. For example, when BNP Paribas enered he marke for OELCs in June 2006, hey a once offered 25 cerificaes on he DAX wih a wide range of predominanly equidisan srikes (BNP Paribas, 2006). Figure 4 shows he relaive profi poenial (PP / P 0 ) and he relaive price T deviaion RPD 0 for he noional OELC for differen srikes as well as various choices of he funding rae spread z of 1.5 %, 2.5 %, and 3.5 % and, again, a holding period of T = 1 and a shor rae of consanly 3 %. Clearly, he relaive profi poenial (black lines) increases wih 1 higher srikes. On he oher hand, he grey ploed lines of he relaive price deviaions RPD 0 rise a firs, bu reach a peak a abou 7 %, 11 %, and 15 %, respecively. Finally, he relaive price deviaions decrease o zero a an iniial srike of 5, Given his srike, he barrier jus maches he curren DAX; B 0 = (1 + a) X 0 = , = 5, = S 0. Therefore, he cerificae is insanly knocked-ou and hus he issuer canno creae any profi. The difference beween he relaive profi poenial and he relaive price deviaion is again mainly deermined by he knock-ou probabiliies. Cerificaes wih a higher srike creae a higher relaive profi poenial for he issuer over ime. However, he knock-ou probabiliy increases as well, especially a iniial srikes higher han abou X 0 = 4,000 which couneracs he simulaneous increase in he relaive profi poenial. Since he profi poenial is zero for an iniial srike of X 0 = 0 and for an iniial srike of X 0 = S 0 / (1 + a), an opimal iniial srike exiss yielding a maximum relaive price deviaion. To maximize heir (relaive) gains, issuers could prefer o issue OELCs exhibiing his mos profiable iniial srike. In conras, banks regularly offer a variey of srikes (see, e.g., he issues of BNP Paribas in June 2006). This is because banks realize profis over ime regardless of he srikes of OELCs. Therefore, hey offer various srikes o arac a large number of invesors. Since a higher srike yields a lower price according o (2) and a higher leverage of he cerificae, cerificaes wih differen srikes can be aracive for invesors preferring differen risk-profiles. Inser Figure 4 abou here 13

16 5 Impac of differing produc feaures The open-end leverage long cerificaes analyzed in he previous secions are closely relaed o he produc design by HSBC Trinkaus & Burkhard, bu exis in very similar forms for oher issuers. Table 3 provides an overview of he characerisics of OELCs on he DAX for differen banks. Whereas all banks adjus he srike daily, mos rely on monhly adjusmens of he barrier. This ceeris paribus leads o slighly lower knock-ou probabiliies. Addiionally, i suggess ha ineres raes influence knock-ou probabiliies, as he shor rae no longer vanishes in he formula for he knock-ou probabiliies (see he derivaion of he knock-ou probabiliy in he Appendix). However, he resuling effec on he value of OELCs should be negligible. Furhermore, facor a, he relaive difference beween he srike and he barrier, need no be fixed over he produc s lifeime. Some banks sae hey migh change i in exraordinary marke condiions such as srongly increasing volailiies (see, e.g., Sal. Oppenheim, 2006). Higher volailiies resul in higher knock-ou probabiliies and a higher probabiliy ha he difference beween he price of he underlying and he srike is non-posiive when he invesors claim is seled following a knock-ou. This would be unfavorable for issuers. Enhancing a increases he probabiliy of a posiive difference bu simulaneously decreases he value of he banks profi poenial. Some banks addiionally sae in heir produc brochures ha hey migh vary he funding rae spread z over he produc s lifeime as well. In he conex of a possible enhancemen of a, increasing he funding rae spread z a he same ime could allow issuers o keep he value of heir profi poenial sable. Inser Table 3 abou here How are our findings affeced by non-fixed relaive differences beween srike and barrier and funding rae spreads over ime? As he value of he superhedge porfolio is a any ime > 0 above he price-seing formula (see (7)), as long as z is posiive, regardless of a, he price seing formula favors he issuer in any siuaion, even if z is sochasic. Analog consideraions hold for he life cycle hypohesis since he profi poenial increases over ime, even for a sochasic z. However, he valuaion model had o be modified. 14

17 6 Conclusion This paper presens he firs analysis of he pricing and valuaion of open-end leverage cerificaes on he German reail marke. In conras o earlier sudies focusing on he priceseing behavior of banks issuing IFPs, we do no have o rely on prices colleced from primary and secondary markes since issuers communicae heir price-seing formulas for open-end leverage cerificaes. By applying a semi-saic superhedge based on spo marke insrumens, i urned ou ha he issuers price-seing formula srongly favors hemselves. This mispricing by consrucion does no cause arbirage aciviies of oher marke paricipans, because shor posiions in he cerificaes are pracically impossible. Furhermore, our findings clearly confirm he life cycle hypohesis for leverage producs for he firs ime, meaning sysemaically increasing profis for issuers over he produc s lifeime. Applying sandard valuaion echniques and assuming fixed planned holding periods, we deermine he value of he issuers mispricing of OELCs. Given a holding period of one year and assuming realisic parameers for he DAX, respecive open-end leverage long cerificaes are regularly sold a (leas) abou 5 o 10 % above heir heoreical values. Clearly, his analysis depends on he assumed behavior of invesors. This could be derived from empirical daa abou he buying and selling decisions of invesors. However, due o a lack of daa a presen, his challenging analysis will be he opic of a subsequen sudy. Moreover, issuers can easily gain ex-pos realized poenial profis of abou 20 o 30 % relaed o he iniial price, if he DAX increases o a cerain level over ime, so ha value and price of he cerificae rise as well and neiher a knock-ou nor a repurchase occurs. Therefore a posiive price developmen of he underlying of open-end long leverage cerificaes can be a he same ime beneficial o issuers and invesors. Furhermore, hese relaively high profis over ime are also possible for issuers, if he underlying develops in a negaive direcion which may cause cerificaes o be premaurely knocked-ou. I is only necessary for cerificaes ha were knocked-ou o be subsiued hrough new ones for he oal volume of ousanding open-end cerificaes o remain roughly consan. By purchasing open-end leverage cerificaes, invesors paricipae o a disproporionaely high exen in changes of he underlying. However, banks and insiuional invesors can also aain equal or similar payoff profiles a lower coss using forwards or fuures conracs raded on he EUREX (European Exchange) or over-he-couner. Privae invesors, however, ofen have no access o hese markes. Hence he exisence of open-end leverage cerificaes is 15

18 jusified by incomplee capial markes or imperfecions, such as marke access barriers, ransacion coss, and informaion asymmeries. Privae invesors ineresed in he payoff profiles of open-end leverage cerificaes herefore normally rely on purchasing hose cerificaes from banks. Since banks regularly apply he unfair price-seing formula discussed in his paper, hey produce an enormous profi poenial due o he funding rae spread included in he price. However, in deermining he ne profi of issuers, an adequae paymen for he issuer s service o reail cosumers should be incorporaed; his should a leas cover he cos of srucuring, disribuion ec. Considering he simple semi-saic superhedging sraegy shown above, he idenified profi poenials for issuers are sill noeworhy. However, in he fuure we do expec decreasing issuers profis due o he rising compeiion in his segmen of he German reail marke, which will probably be refleced in lower funding rae spreads. 16

19 Appendix The derivaions of he cumulaive risk-neural knock-probabiliy Q(τ ) and he expression E Q (1 {τ } exp(z τ)) are based on he following wo lemmas: Lemma 1 (e.g., Bielecki and Rukowski, 2002, p. 67): For every > 0 le he sochasic process Y be given by Y = y 0 + ν + σ W for some consans y 0 > 0, ν, σ > 0 and a sandard Brownian moion W under he probabiliy measure Q. The sopping ime τ is defined by τ = inf{ : Y 0}. For any s > 0, we have Q(τ s) = N(h 1 (s)) + exp( 2 ν σ 2 y 0 ) N(h 2 (s)) where N( ) sands for he sandard Gaussian cumulaive disribuion funcion, and h 1 (s) = y 0 ν s σ s, h 2(s) = y 0 + ν s σ s. Lemma 2 (Bielecki and Rukowski, 2002, p. 74): Le a, b, c be consans wih b < 0 and c 2 > 2 a. For every y > 0, we have y 0 exp(a x) dn b c x x = d + c 2 d g(y) + d c 2 d f(y), where d = c 2 2 a, g(y) = exp(b (c d)) N b d y y and f(y) = exp(b (c + d)) N b + d y y. Derivaion of Q(τ ) Le he process S, 0, saisfy ds = r S d + σ S dw under he risk-neural probabiliy measure Q, where r denoes he shor rae in, σ > 0 is a consan and W denoes a sandard Brownian moion. By definiion we have S = S 0 exp r s ds σ 2 / 2 + σ W. Le he barrier in be given by B = B 0 exp r s ds + z 0 0 for some consan z and 0 < B 0 < S 0. The firs passage ime is defined by τ = inf{: S B }. 17

20 We have {S B } = S 0 exp r s ds σ 2 / 2 + σ W B 0 exp r s ds + z 0 0 = {S 0 exp( σ 2 / 2 + σ W ) B 0 exp(z )} = {ln(s 0 / B 0 ) + ( σ 2 / 2 z) + σ W 0}. By applying Lemma 1 o y 0 = ln(s 0 / B 0 ) and ν = σ 2 / 2 z we obain for every > 0 Q(τ ) = N(h 1 ()) + h 1 () = ln(b 0 / S 0 ) + (σ 2 / 2 + z) σ B 0 S 0 2 (σ 2 / 2 + z) σ 2 N(h 2 ()),, h 2 () = ln(b 0 / S 0 ) (σ 2 / 2 + z). σ Derivaion of E Q (1 {τ } exp(z τ)) Based on he above represenaion of Q(τ ) we can conclude E Q (1 {τ } exp(z τ)) = exp(z x) dq(τ x) 0 = exp(z x) dn(h 1 (x)) + 0 B 0 S 0 2 (σ 2 / 2 + z) σ 2 0 exp(z x) dn(h 2 (x)). By applying Lemma 2 o each summand in he above equaion separaely wih y =, a = z, b = ln(b 0 / S 0 ), c = (σ / 2 + z / σ) and = (σ / 2 + z / σ), respecively, we obain for σ 2 2 z σ ( c 2 2 a > 0) afer rearranging and collecing erms: E Q (1 {τ } exp(z τ)) = 2 z B 0 S 0 B 0 σ 2 N(h 3 (s)) + S 0 1 N(h 4 (s)), h 3 () = ln(b 0 / S 0 ) + (σ 2 / 2 z), h 4 () = ln(b 0 / S 0 ) (σ 2 / 2 z). σ σ The same formula holds for σ 2 = 2 z as E Q (1 {τ } exp(z τ)) is a coninuous bounded funcion of σ. 18

21 References BNP Paribas (2006): Open-End Long Warrans on he DAX Performance Index (in German). Sales Prospec, Frankfur/Main and Paris, June 22, Baubonis, C., Gasineau, G., Purcell, D. (1993): The Bankers s Guide o Equiy-Linked Cerificaes of Deposi. The Journal of Derivaives 1 (Winer), Baule, R., Enrop, O., Wilkens, M. (2006): Wha do Banks Earn wih Their Own Credi Risk? Evidence from he German Secondary Marke for Srucured Financial Producs. Working Paper, Universiy of Goeingen and Caholic Universiy of Eichsae-Ingolsad, Augus Bene, B.A., Giannei, A., Pissaris, S. (2006): Gains from srucured produc markes: The case of reverse-exchangeable securiies (RES). Journal of Banking and Finance 30, Black, F., Scholes, M. (1973): The Pricing of Opions and Corporae Liabiliies. Journal of Poliical Economy 81, Bielecki, T.R., Rukowski, M. (2002): Credi Risk: Modeling, Valuaion and Hedging. Berlin e al. Brown, C., Davis, K. (2004): Dividend Proecion a a Price. The Journal of Derivaives 12 (Winer), Burh, S., Kraus, T., Wohlwend, H. (2001): The Pricing of Srucured Producs in he Swiss Marke. The Journal of Derivaives 9 (Winer), Chen, A.H., Kensinger, J.W. (1990): An Analysis of Marke-Index Cerificaes of Deposi. Journal of Financial Services Research 4 (2), Chen, K.C., Sears, R.S. (1990): Pricing he SPIN. Financial Managemen 19 (Summer), Goldman Sachs (2006): Mini Fuure Turbo Warrans on Indices, Commodiies, Exchange Raes or Raher Fuure Conracs (in German). Sales Prospec, Frankfur/Main, Augus 1, Grünbichler, A., Wohlwend, H. (2005): The Valuaion of Srucured Producs: Empirical Findings for he Swiss Marke. Financial Markes and Porfolio Managemen 19, Harrison, J.M., Pliska, S.R. (1981): Maringales and Sochasic Inegrals in he Theory of Coninuous Trading. Sochasic Processes and Their Applicaions 11, Heah, D., Jarrow, R., Moron, A. (1992): Bond Pricing and he Term Srucure of Ineres Raes: A New Mehodology for Coningen Claim Valuaion. Economerica 60, HSBC Trinkhaus & Burkhard (2006): DAX Mini Fuure Cerificaes (in German). Sales Prospec, Duesseldorf, Augus 14, Hull, J.C., Whie, A. (1995): The impac of defaul risk on he prices of opions and oher derivaive securiies. Journal of Banking and Finance 19, Jarrow, R.A., van Devener, D.R. (1998): The arbirage-free valuaion and hedging of demand deposis and credi card loans. Journal of Banking and Finance 22, Meron, R.C. (1973): Theory of raional opion pricing. Bell Journal of Economics and Managemen Science 4, Muck, M. (2006a): Where Should You Buy Your Opions? The Pricing of Exchange-Traded Cerificaes and OTC Derivaives in Germany. The Journal of Derivaives 14 (Fall), Muck, M. (2006b): Pricing Turbo Cerificaes in he Presence of Sochasic Jumps, Ineres Raes, and Volailiy. Working Paper, WHU Oo Beisheim Graduae School of Managemen, Vallendar, January 24, 2006, forhcoming in: Die Beriebswirschaf. Sal. Oppenheim (2006): Index Turbo Open-End Warran (in German). Sales Prospec, Frankfur/Main, Augus 16, Soimenov, P.A., Wilkens, S. (2005): Are srucured producs fairly priced? An analysis of he German marke for equiy-linked insrumens. Journal of Banking and Finance 29,

22 Wasserfallen, W., Schenk, C. (1996): Porfolio Insurance for he Small Invesor in Swizerland. The Journal of Derivaives 3 (Spring), Wilkens, S., Erner, C., Röder, K. (2003): The Pricing of Srucured Producs in Germany. The Journal of Derivaives 11 (Fall), Wilkens, S., Soimenov, P.A. (2006): The Pricing of Leverage Producs: An Empirical Invesigaion of he German Marke for Long and Shor Sock Index Cerificaes. Working Paper, Ruhr Graduae School in Economics, Essen, and Frankfur/Main, May 8, 2006, forhcoming in: Journal of Banking and Finance. 20

23 Table 1: Number of open-end leverage cerificaes issued by banks on he German marke from 2002 o Sepember 2006 ABN Amro Bank BNP Paribas Ciigroup Commerzbank Deusche Bank Dresdner Bank Goldman Sachs HSBC Trinkaus & Burkhard Lang & Schwarz Raiffeisen Cenrobank Sal. Oppenheim Sociéé Générale all /2006 Long all Long all Long all Long all Long all Shor Shor Shor Shor Shor ,455 1,489 2, , ,340 1,814 2,972 2,412 3, , , ,379 2, , , ,034 1,720 1,164 1, , ,286 3,029 4,881 4,913 7,687 8,594 14, ,852 2,774 5,436 The able shows he number of open-end leverage cerificaes issued on he German marke from 2002 o Sepember 2006, lised according o issuers. Source: Deriva GmbH Financial IT and Consuling. 21

24 Table 2: Underlyings of open-end leverage cerificaes issued by banks on he German marke from 2002 o Sepember /2006 Underlying Long all Long all Long all Long all Long all Shor Shor Shor Shor Shor Socks Sock indices Exchange raes Precious meals Ohers all ,828 2,731 3,018 4,196 5,385 8, ,178 2, , , ,672 1,649 3, , , ,286 3,029 4,881 4,913 7,687 8,594 14, ,852 2,774 5,436 The able shows he number of open-end leverage cerificaes issued on he German marke from 2002 o Sepember 2006, lised according o he respecive underlying. Ohers conains open-end leverage cerificaes on, e.g., commodiies, ineres raes fuures, and foods such as cacao, orange juice, coffee, and sugar. Source: Deriva GmbH Financial IT and Consuling. 22

25 Table 3: Specificaion of open-end long leverage cerificaes on he DAX, lised according o he issuing bank in Augus 2006 Issuer ABN Amro BNP Paribas Ciigroup Commerzbank Deusche Bank Dresdner Bank Goldman Sachs HSBC Trinkaus & Burkhard Lang & Schwarz Sal. Oppenheim Sociéé Générale Produc name Mini Fuure Cerificae; DAX Index Mini Long Cerificae Open-end Turbo Long Warran Open-end Sop Loss Bull Turbo; Open-end Turbo Sop Loss Knock-ou Warran Unlimied Turbo Bull Cerificae Wave XXL; Call Wave XXL Knock-ou Warran Call Open-end Knock-ou Warran Mini Fuure Turbo Warran Reference ineres rae money marke rae 1-monh EURIBOR 1-monh EURIBOR 1-monh EURIBOR EONIA EONIA EUR LIBOR Overnigh Mini Fuure Cerificae EONIA 1.5 % Open-end Turbo Call Turbo Open-end Warran Open-end Turbo Long Knock-ou Warran 1-monh EURIBOR 1-monh EURIBOR EUR LIBOR Overnigh Funding rae spread z a issuance: 3.0 %, possible changes over ime, max 3.0 % a issuance: 2.5 % possible changes over ime, min 0 %, max 5.0 % currenly abou 2.0 %, possible changes over ime currenly abou 3.0 %, possible changes over ime a issuance: 3.75 %, possible changes over ime a issuance: 1.5 %, possible changes over ime a issuance: 2.0 %, possible changes over ime, max 4.0 % currenly 1.5 %, possible changes over ime a issuance: 2.0 %, possible changes over ime 2.5 % Facor a a issuance: 1.5 %, possible changes over ime, min 1.5 %, max 5.0 % normally: 1.5 %, for cerain cerificaes: 2.0 % or 3.0 % a issuance: abou 1.5 %, possible changes over ime currenly abou 1,5 %, possible changes over ime currenly 2.0%, possible changes over ime, min 2.0 %, max 10.0 % currenly abou 2.0 %, possible changes over ime, min 2.0 %, max 10.0 % a issuance: 2.0 %, possible changes over ime, max 5.0 % 1.5 % (older cerificaes: 3.0 %) Adjusmen Barrier Srike monhly daily monhly monhly monhly monhly monhly daily daily daily daily daily daily daily daily daily abou 1.75 % monhly daily a issuance: 3.0 %, possible changes over ime a issuance: 2.0 %, possible changes over ime, max 7.0 % The able shows he specificaion of open-end leverage producs on he DAX, lised according o he issuer. German produc names are ranslaed ino English. The daa were colleced from inerne-published documens such as produc brochures and sales prospecuses of issuers on Augus 20, daily monhly daily daily 23

26 Figure 1: Profi poenial of banks and is value, and relaive price deviaion of an open-end long leverage cerificae, depending on he poin in ime and he invesor s holding period T Profi poenial and is value PP VPP 0 T 24 % 21 % Relaive price deviaion 18 % 15 % 12 % 9 % 20 RPD 0 T 6 % 3 % and T, respecively 0 % For a noional open-end long leverage cerificae on he DAX, his figure shows he profi poenial PP (see (7)) as well as is value VPP 0 T (see (9)), and he relaive price deviaion RPD 0 T (see (11)) as a funcion of he poin in ime and he invesor s holding period T, respecively. The lef ordinae shows he profi poenial and is value, he righ ordinae he relaive price deviaion. The main feaures of he open-end leverage cerificae on he DAX are: iniial srike X 0 = 5,370.00, iniial barrier B 0 = 5,450.55, relaive difference beween barrier and srike a = 1.5 %, and funding rae spread z = 1.5 %. The shor rae is consanly r = 3 % and he parameers of he DAX are: S 0 = 5,700.00, σ = 20 %. 24

27 Figure 2: Cumulaive risk-neural knock-ou probabiliy of an open-end long leverage cerificae, depending on he poin in ime and he volailiy σ Knock-ou probabiliy 80 % 60 % 40 % 20 % 0% Time % 40 % 30 % Volailiy σ 20 % 10 % 0% For a noional open-end long leverage cerificae on he DAX, his figure shows he risk-neural knock-ou probabiliy (see (10)) as a funcion of he poin in ime and he volailiy σ of he DAX. The main feaures of he open-end leverage cerificae on he DAX are: iniial srike X 0 = 5,370.00, iniial barrier B 0 = 5,450.55, relaive difference beween barrier and srike a = 1,5 %, and funding rae spread z = 1.5 %. The curren DAX is S 0 = 5,

28 Figure 3: Value of he banks profi poenial from an open-end long leverage cerificae, depending on he volailiy σ of he underlying for differen funding rae spreads z 200 z = 3.5 % Value of he profi poenial z = 2.5 % z = 1.5 % Curren DAX S 0 = 5, % 1.5% S 0 = 6, % 5 % 10 % 15 % 20 % 25 % Volailiy σ For a noional open-end long leverage cerificae on he DAX, his figure shows he value of he poenial profi VPP 0 T (see (9)) as a funcion of he volailiy σ of he DAX for funding rae spreads of z = 1.5 %, z = 2.5 %, and z = 3.5 %. The oher main feaures of he open-end leverage cerificae on he DAX are: iniial srike X 0 = 5,370.00, iniial barrier B 0 = 5,450.55, and relaive difference beween barrier and srike a = 1,5 %. The planned holding period of he invesor is T = 1, he curren DAX is S 0 = 5, and S 0 = 6,000.00, respecively. 26