A Penny Saved is a Penny Earned: Less Expensive Zero Coupon Bonds

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1 A Penny Saved is a Penny Earned: Less Expensive Zero Coupon Bonds Alessandro Gnoao Marino Grasselli Echard Plaen March 9, 018 arxiv: v [q-fin.pr] 8 Mar 018 Absrac In his paper we show how o hedge a zero coupon bond wih a smaller amoun of iniial capial han required by he classical ris neural paradigm, whose rivial hedging sraegy does no sugges o inves in he risy asses. Long daed zero coupon bonds we derive, inves firs primarily in risy securiies and when approaching more and more he mauriy dae hey increase also more and more he fracion invesed in fixed income. The convenional wisdom of financial planners suggesing invesor o inves in risy securiies when hey are young and mosly in fixed income when hey approach reiremen, is here made rigorous. The paper provides a srong warning for life insurers, pension fund managers and long erm invesors o ae he possibiliy of less expensive producs seriously o avoid he adverse consequences of he low ineres rae regimes ha many developed economies face. Key words: Forex, benchmar approach, benchmared ris minimizaion, sochasic volailiy, long erm securiies. JEL Classificaion: C6, C63, G1, G1, G13 1 Inroducion This paper aims o draw he aenion o a more general modeling world han available under he classical no-arbirage paradigm in finance. To explain he new approach and illusrae firs imporan consequences, he less expensive pricing of long daed zero coupon bonds will be demonsraed in his paper. Under he benchmar approach, see Plaen and Heah 010, many payoffs can be less expensively produced han curren heory and pracice sugges. The inuiive verbal advice of financial planners o inves in risy securiies when he invesor is young and mosly in fixed income when he/she approaches reiremen, is made rigorous. The long daed zero coupon bonds we derive, inves firs primarily in risy securiies and, when approaching more and more he mauriy dae, hey increase also more and more he fracion invesed in fixed income. These less expensive zero coupon bonds provide only a firs example ha is indicaive for he changes ha he new approach offers in a much wider modeling world han he classical one. Hisorically, Long 1990 was he firs who observed ha one can rewrie he ris neural pricing formula ino a pricing formula ha aes is expecaion under he real world probabiliy measure and employs he, so called, numéraire porfolio NP as numéraire. The benchmar approach assumes only Deparmen of Economics, Universiy of Verona. Via Canarane 4, 3719 Verona, Ialy. alessandro.gnoao@univr.i Diparimeno di Maemaica, Universià degli Sudi di Padova Ialy and Léonard de Vinci Pôle Universiaire, Research Cener, Finance Group, Paris La Défense, France. grassell@mah.unipd.i. Universiy of Technology, Sydney, Finance Discipline Group and School of Mahemaical and Physical Sciences, PO Box 13, Broadway, NSW, 007, Ausralia, and Universiy of Cape Town, Deparmen of Acuarial Science. echard.plaen@us.edu.au. 1

2 he exisence of he NP and no longer he raher resricive classical no-arbirage assumpions, which are equivalen o he exisence of an equivalen ris neural probabiliy measure. Under his much weaer assumpion, one can sill perform all essenial ass of valuaion and ris managemen. The only condiion imposed is ha he NP, which is he in he long run pahwise bes performing porfolio, remains finie in finie ime. Obviously, when his assumpion is violaed for a model, hen some economically meaningful arbirage mus exis causing he candidae for he NP o explode. In his case he respecive model maes no much heoreical and pracical sense. Noe ha when a finie NP exiss, various forms of classical arbirage may be presen in he mare; see e.g. Loewensein and Willard 000 and Heson e al. 007 for various examples in he lieraure on bubbles. The curren paper illusraes he divergence of he benchmar approach from he classical approach by focusing on he currency mare, which is one of he mos acive mares. We presen and calibrae a hybrid model describing he dynamics of a vecor of foreign exchange FX raes and he associaed ineres raes. We exend and unify he FX mulifacor sochasic volailiy models of De Col e al. 013 and Baldeaux e al. 015b by means of he general ransform formula presened in Grasselli 017. The resuling general model ha we develop allows for he simulaneous presence of muliple sochasic volailiy facors boh of square roo see Heson 1993 and 3/ ype see Heson 1997 and Plaen More explicily, he square roo of each CIR facor appears in boh he numeraor and he denominaor of he diffusion erms. Based on Baldeaux e al. 015b, we refer o his model as he 4/ model. Our specificaion for he volailiy process spans a large class of dynamics ranging from he 3/ o he Heson model. This means ha we can le mare daa dicae he relaive imporance of he wo sochasic volailiy effecs ha we consider. While he 4/ model migh appear as an involved choice, we will show in Subsecion 3.1 ha i naurally emerges e.g. in a simple Heson seing for a suiable choice of he ris premium. Moreover, such CIR facors can be freely combined in order o drive sochasic ineres raes. Therefore, he model is suiable for he valuaion of long-daed FX producs, for which ineres rae ris becomes a relevan ris facor, see he discussions in Gnoao and Grasselli 014. The framewor we propose is general o he exen ha for suiable parameer combinaions our model may no admi he exisence of an equivalen ris-neural probabiliy measure for some economies. In spie of his feaure, he problem of pricing and hedging coningen claims can always be solved under he more general benchmar approach of Plaen and Heah 010: he real world pricing formula see Plaen and Heah 010 and benchmared ris minimizaion for hedging see Du and Plaen 016 will be he ools we employ o solve boh problems. Despie of he richness of our framewor, i is possible o efficienly solve and implemen he pricing of plain vanilla insrumens via Fourier-based echniques, see Carr and Madan 1999 and Lewis 001. Semi analyical closed form soluions for producs such as European FX opions can be compued hans o he availabiliy of he exac formula for he join Fourier ransform of he model s sae variables, see Grasselli 017. We es our model on real daases of vanilla FX opions by performing several calibraion experimens. Our empirical resuls are wofold. On he one side, we confirm he empirical findings of Baldeaux e al. 015b on he violaion of he ris neural pricing paradigm. The appearence of such violaions may change over ime and across currencies. In fac, our muliple calibraion experimen seems o sugges he presence of regime swiches in raded FX-opion prices beween he sandard ris neural and he real world pricing approach. Such a feaure calls for a modelling framewor which is able o span boh valuaion principles, which is provided by he benchmar approach. On he oher side, we quanify he impac of such violaions in a simplifying ye clarifying siuaion, namely he pricing and hedging of zero coupon bonds in a deerminisic ineres rae seing. We find ha he impac is significan, mosly for long daed bonds, where he difference beween he price given by he real world pricing of he benchmar approach and he one provided by he puaive and empirically rejeced ris neural paradigm, may be very large abou 30% for mauriies around 0 years. This surprising resul can explain he effec ha has been inuiively exploied in financial planning, where one uses he exra growh presen in risy securiies o accumulae over long ime periods more wealh wih lile flucuaions a mauriy han by invesing over he enire period in fixed income, see e.g. Gourinchas and Parer 00. The possibiliy

3 of less expensive producion coss for argeed long daed payoffs has major pracical implicaions for annuiies, life insurance conracs, pensions and many oher long erm conracs. The paper is srucured as follows: In Secion we inroduce he general muli-currency modeling framewor and recall some noions from he benchmar approach. Secion 3 moivaes and formally inroduces he 4/ model as a unifying framewor for sochaic volailiy models driven by he CIR process as he Heson-based model of De Col e al. 013 and he 3/-based model of Baldeaux e al. 015b. The 4/ model exends he Heson model and allows for he possibiliy of a failure of he ris neural paradigm. The analyical racabiliy of he 4/ model is demonsraed in Secion 4, which consiues a prerequisie for an efficien model calibraion, presened in Secion 5. A pracical consequence of he calibraion is analyzed in Secion 6, where we perform hedging in an incomplee mare seing wihou he exisence of a ris neural probabiliy measure. In paricular, in he absence of a ris neural probabiliy measure, we follow he concep of benchmared ris minimizaion of Du and Plaen 016. We show how o hedge long erm producs significanly less expensively han he classic ris neural paradigm permis. General Seup In his secion we presen he general modeling framewor of he benchmar approach for a foreign exchange FX mare. Subsecion.1 provides a general seup driven by a muli-dimensional diffusion process..1 Specificaion of he Currency Mare We use superscrips o reference differen currencies, and employ bold leers for vecors and subscrips for elemens hereof. Unless specified by a suiable superscrip, all expecaions are considered wih respec o he real world probabiliy measure P. We model he currency mare on a probabiliy space Ω, F T, P, where T < is a finie ime horizon. On his space we inroduce a filraion F 0 T o model he evoluion of informaion, saisfying he usual assumpions. The above filered probabiliy space suppors a sandard d-dimensional P-Brownian moion Z={Z = Z 1,..., Z d, 0 T } for modeling he raded uncerainy. The consan N denoes he number of currencies in he model, whereas d is he number of raded risy facors we employ. In each economy we posulae he exisence of a money mare accoun, i.e. he i-h money mare accoun,when denominaed in unis of he i-h currency, evolves according o he relaion db i = B i r i d, B i 0 = 1, 0 T ;.1 wih he R-valued, adaped i-h shor rae process r i = { r i, 0 T } {. We denoe by S i,j = S i,j, 0 T } he coninuous exchange rae process beween currency i and j. Here S i,j denoes he price of one uni of currency i in unis of currency j, meaning ha, e.g. for i = USD and j = EUR and S i,j = 0.9 we have, in line wih he sandard FORDOM convenion, ha he price of one USD is 0.9 EUR a ime. Le us follow Plaen and Heah 010 and Heah and Plaen 006 and inroduce a family of primary securiy accoun processes via B i,j = S i,j B j, 0 T for i j. Obviously, for i = j we have B i,i = B i. We ae he perspecive of a generic currency referenced wih superscrip i and inroduce he vecor of money mare accouns of he form B i = B i,1,..., B i,n, i = 1,..., N. Given his vecor of primary securiy accouns, an invesor may rade on hem. This is represened by inroducing a family of predicable B i -inegrable sochasic processes δ = { δ = δ 1,..., δ N, 0 T } for i = 1,..., N, called sraegies. Each δ j R denoes he number of unis ha an agen holds in he j-h primary securiy accoun a ime. Le us inroduce 3

4 he process V i,δ = { V i,δ, 0 T }, which describes he value process in i-h currency denominaion corresponding o he porfolio sraegy δ, i.e. The sraegy δ is said o be self-financing if V i,δ = dv i,δ = N δ j B i,j.. j=1 N δ j db i,j..3 j=1 In line wih Plaen and Heah 010, we assume limied liabiliy for all invesors. For his purpose, we inroduce V + as he se of all self-financing sraegies forming sricly posiive porfolios. For our purposes, we will be ineresed in a paricular sraegy δ V +, which yields he growh opimal porfolio GOP, which can be shown o be equivalen o he numéraire porfolio NP, and is defined as follows: Definiion.1. A soluion δ of he maximizaion problem [ V i,δ T log V i,δ 0 sup E δ V + ], for all i = 1,..., N and 0 T T is called a growh opimal porfolio sraegy. I has been shown in Plaen and Heah 010 ha he GOP value process is unique in an incomplee jump-diffusion mare seing. We summarize he discussion above in he following assumpions. Assumpion.1. We assume he exisence of he growh opimal porfolio GOP, and denoe by D i = { D i 0, +, 0 T }, i = 1,..., N he value of he GOP denominaed in he i-h currency. The dynamics of he GOP are given by dd i D i = ri d + π i, π i d + dz, D i 0 > 0.4 for [0, T ] and i = 1,... N, where for N, d N, he N-dimensional family of predicable, R d -valued sochasic processes π = { π i = π i 1,..., π i d, 0 T } represen he mare prices of ris wih respec o he i-h currency denominaion. The processes π i are assumed o be inegrable wih respec o he d-dimensional sandard Brownian moion Z. The GOP can be shown o be in many ways he bes performing porfolio. In paricular, in he long run is value ouperforms almos surely hose of any oher sricly posiive porfolio. Here we assume ha i remains finie in finie ime in all currency denominaions. If we were o consider a model where he GOP explodes in any of he currency denominaions, hen he model would allow an obvious form of economically meaningful arbirage, since one could generae in ha currency denominaion in finie ime unbounded wealh from finie iniial capial. Given he uniqueness of he GOP, all exchange raes S i,j can be uniquely deermined as raios of differen denominaions of he GOP in he respecive currencies. Assumpion.. The family of exchange rae processes S i,j = { S i,j, 0 T } is deermined by he raios for 0 T and i, j = 1,..., N. S i,j = Di D j Given Assumpion., i is immediae o compue via a direc applicaion of he Iô formula he dynamics of all exchange raes and all primary securiy accouns in all currency denominaions..5 4

5 Lemma.1. The exchange rae S i,j under he real world probabiliy measure P evolves according o he dynamics ds i,j S i,j = ri r j d + π i π j, π i d + dz, S i,j 0 = s i,j > 0,.6 and he generic j-h primary securiy accoun B i,j, in i-h currency denominaion and under he real world probabiliy measure P, evolves according o he dynamics db i,j B i,j = ri d + π i π j, π i d + dz, B i,j 0 = b i,j,.7 for i, j = 1,..., N and [0, T ].. The Benchmar Approach In he presen paper we evaluae coningen claims under he benchmar approach of Plaen and Heah 010. Under his approach, price processes denominaed in erms of he GOP are called benchmared price processes. More precisely, for i, j = 1,... N, le us inroduce he benchmared price process { } ˆB j = ˆB j := Bi,j D i, 0 T. We call ˆBj he benchmared j-h primary securiy accoun. Noe ha ˆB j does no depend on he index i of he currency denominaion we sared from. Given.7 and.4, upon an applicaion of he Iô formula, i is immediae o conclude ha all benchmared price processes ˆB j form P-local maringales. Even more, hey are non-negaive P-local maringales. Hence, due o Faou s lemma, hey are also P-supermaringales. Analogously, we also have ha benchmared non-negaive porfolio values ˆV δ := V i,δ /D i form P-supermaringales. Besides he exclusion of forms of economically meaningful arbirage, which are equivalen o he explosion of he GOP, forms of classical arbirage ha are excluded under classical no-arbirage assumpions may exis in our model, see e.g. Loewensein and Willard 000. Le us now inroduce for he i-h currency denominaion he Radon-Niodym derivaive process, denoed by Λ i = { Λ i, 0 T }, by seing Λ i = ˆB i, i = 1,..., N..8 ˆB i 0 This is he ris neural densiy for he puaive ris-neural measure Q i of he i-h currency denominaion. I arises e.g. when we consider replicable claims and assume he exisence of an equivalen ris neural probabiliy measure Q i. As each Λ i equals he corresponding benchmared savings accoun ˆB i up o a consan facor, i is clear ha Λ i is a P-local maringale, for i = 1,..., N. The classical assumpion in he foreign exchange lieraure ha here exiss an equivalen ris-neural probabiliy measure for each currency denominaion, corresponds o he requiremen ha each process Λ i is a rue maringale for i = 1,..., N. Such a requiremen is raher srong and may be empirically rejeced, see e.g. Heah and Plaen 006, he findings in Baldeaux e al. 015b and he necessary and sufficien condiions of Hulley and Ruf 015. Hence, in he presen paper, we shall allow each Λ i o be eiher a rue maringale or a sric local maringale. To wor in such a generalized seing requires a more general pricing concep han he one provided under he classical ris neural paradigm. In he following, we will employ he noion of real world pricing: a price process V i = { V i, 0 T }, here denominaed in i-h currency, is said o be fair if, when expressed in unis of he GOP D i, forms a P-maringale, his means is benchmared value forms a rue P-maringale, see Definiion 9.1. in Plaen and Heah 010. For a fixed mauriy 5

6 T [0, T ], we le HT = VT be an F T -measurable non-negaive benchmared coningen claim, such ha when expressed in unis of he i-h currency denominaion as H i T = D i T HT we have [ H i ] T E [HT F ] = E D i T F < for all 0 T T, i = 1,..., N. The benchmared fair price ˆV = V i /D i of his coningen claim is he minimal possible price and given by he following condiional expecaion under he real-world probabiliy measure P: ] ˆV = E [ĤT F,.9 which is nown in he lieraure as real-world pricing formula, see Corollary in Plaen and Heah 010. Noe ha benchmared ris minimisaion, described in Du and Plaen 016, gives.9 generally. In case Λ i is a rue maringale we obain, by changing in.9 from he real world probabiliy measure P o he equivalen ris neural probabiliy measure Q i, he ris neural pricing formula [ B V i i B i T D i ] = E B i T B i D i T Hi T F [ Λ i T B i ] [ = E Λ i B i T Hi T B i ].10 F = E Qi B i T Hi T F. This shows ha in his case he real-world pricing formula generalises he classical ris-neural valuaion formula and he Radon-Niodym derivaive for he respecive ris neural probabiliy measure is given by.8. In general, due o he supermaringale propery of benchmared price processes in he case when Λ i is a sric supermaringale, a formally obained ris neural price is greaer han or equal o he real world price, see Du and Plaen 016. Laer on we consider he pricing of zero coupon bonds. According o.9, he minimal possible price P i, T a ime of a zero coupon bond, denominaed in he i-h currency and paying a mauriy T one uni of he i-h currency, is given by he formula [ ] P i, T = D i 1 E D i T F..11 All benchmared non-negaive price processes are supermaringales. Therefore, as already menioned, he fair price process of a coningen claim is he minimal possible price process. There may exis more expensive price processes ha deliver he same payoff. As we will show laer on, over long periods of ime a formally obained ris neural price can be significanly more expensive han he fair one. This paper focuses on he emerging possibiliy o poenially produce long daed zero coupon bonds less expensively han suggesed under he classical paradigm. This permis, e.g. he less expensive producion of conracs involving long daed zero coupon bonds as building blocs as in he case for pensions, annuiies and life insurances. 3 The 4/ Model To demonsrae he fac ha in realiy here may exis hedgeable long daed zero coupon bond price processes ha are significanly less expensive han respecive ris neural price processes, we need some model ha could capure his phenomenon when i is presen in he mare. In his secion we provide such model, called he 4/ model. Below we describe he 4/ model ha unifies naurally several wellnown models. In Subsecion 3.3 we sae he precise condiions under which he crucial maringale propery of he benchmared savings accoun fails for he 4/ model. 6

7 3.1 The 4/ Model as a Unifying Framewor In his subsecion, we provide a specificaion of he volailiy dynamics of he exchange raes. In paricular, we consider, for simpliciy, wo currencies wih D 1 denoing he GOP in domesic currency and D denoing he GOP in foreign currency. For example, S 1, = D 1 /D can follow a sochasic volailiy model of Heson ype see Heson 1993, where ds 1, S 1, = r 1 r d + V dz + λd, S 1, 0 = s 1, > 0, dv =κθ V d + σv 1/ ρdz + 1 ρ dz, V 0 = v > Here Z = { Z, 0 T } is a P-Brownian moion independen of Z, κ > 0, θ > 0, ρ [ 1, 1] wih he predicable processes r 1, r and λ. In he remainder of he paper, we will repeaedly employ he following erminology: Heson ype model, 3/ ype model, 4/ ype model. Le us now clarify he respecive models. In he following, we consider a, b R, and le D i = { D i, 0 T } denoe a generic place-holder for a GOP process saisfying a scalar diffusive sochasic differenial equaion SDE. Moreover, le V = { V, 0 T } be a square roo process as given in 3.1. A model is said o be of Heson ype resp. of 3/ ype, resp. of 4/ ype if he diffusion coefficien in he dynamics of he GOP D i is proporial o a V resp. b/ V, resp. a V + b/ V. The quesion we would lie o address is he following: Given a specificaion of he mare price of ris process λ for he domesic currency denominaion of securiies, wha are he associaed dynamics of he domesic and foreign specificaions of he GOP? Lemma 3.1. Consider a wo-currency model, where he exchange rae model for S 1, is of Heson ype 3.1. The following saemens hold rue: 1. If λ = a V, a R, hen he GOP denominaions D 1 and D follow boh Heson-ype models.. If λ = b, b R, hen he GOP denominaion D 1 follows a 3/ model, whereas D follows a V 4/ model. 3. If λ = a V + b, a, b R, hen he GOP denominaions D 1 and D follow boh 4/ ype V models. The proof for his resul is given in Appendix A. Noe ha if we had sared in 3.1 wih he volailiy 1/ V, hen for λ = b we would have V always fallen ino he world of 4/ ype models. Lemma 3.1 highlighs an ineresing inerplay beween several well-nown financial models. I shows ha he 4/ model arises naurally from a sandard Heson model when he mare price of ris belongs o he essenially affine class see Duffee 00. Furhermore, i demonsraes ha he 4/ model provides a general framewor ha ness oher popular model choices. Differen specificaions of he mare price of ris do no only impac on he shape of he GOP dynamics. In fac, depending on he calibraed values of he model parameers, we may incur siuaions where classical ris neural pricing is no longer possible because an equivalen ris neural probabiliy measure does no exis. To see his, we observe ha from a direc inspecion of he dynamics in 3.1 in he 7

8 Heson model seing, i is emping o define he following wo coninuous processes Z Q1 := Z + Z Q := Z λsds λs V s ds, which, if he assumpions of he Girsanov heorem were in boh cases fulfilled, would hen be Q 1 - resp. Q - Brownian moions. Le us assume ha under he real world probabiliy measure P he Feller condiion see Karazas and Shreve 1991, Secion 5.5 is fulfilled by he parameers of he volailiy process V, i.e. we have κθ σ 0, so ha he square roo process V remains sricly posiive P-a.s. for all [0, T ]. The following lemma shows ha, depending on he specificaion of λ, i is possible o obain a variance process V under he puaive ris neural measure ha may no saisfy he Feller condiion, implying ha he puaive ris neural measure may fail o be equivalen o he real world probabiliy measure. Lemma 3.. Consider a wo-currency model, where he dynamics of he exchange rae S 1, is of he Heson ype 3.1, such ha he variance process V fulfills he Feller condiion, i.e. κθ σ 0. Le Z Q1, Z Q, as in 3., be he candidae Brownian moions under he puaive ris neural measures Q 1 and Q, respecively. The following holds: 1. For he puaive ris neural measure Q 1 we ge: a If λ = a V, a R, hen he drif of he variance process V under Q 1 is κ θ V σρav and he Feller condiion is always saisfied under Q 1, which is hen a rue equivalen maringale measure. b If λ = a V + b, a, b R, hen he drif of he variance process V under Q 1 equals V κ θ V σ V ρ a V + b V 3. and he Feller condiion may be violaed, implying ha Q 1 may no be equivalen o P. c If λ = b, b R, hen he drif of he variance process V under Q 1 is V κ θ V σρb and he Feller condiion may be violaed, implying ha Q 1 may no be equivalen o P.. For he puaive ris neural measure Q we have a If λ = a V, a R, hen he drif of he variance process V under Q equals κ θ V σρa 1V and he Feller condiion is always saisfied under Q, which is hen a rue equivalen maringale measure. b If λ = a V + b, a, b R, hen he drif of he variance process V under Q equals V κ θ V σ V ρ a 1 V + b V and he Feller condiion may be violaed, implying ha Q would be in such case no equivalen o P. 8

9 c If λ = b, b R, hen he drif of he variance process V under Q is V κ θ V σρb + σρv and he Feller condiion may be violaed, implying ha Q would be in such case no equivalen o P. The proof of hese saemens is sraighforward using our previous noaion and relaionships and, herefore, omied. 3. Formal Presenaion of he 4/ Model To provide in he case of more han wo currencies a concree specificaion of he mare prices of ris, we proceed o inroduce he R d -valued nonnegaive sochasic process, called he volailiy facor process, ha we denoe by V = { V = V 1,..., V d, 0 T }. The -h componen V of he vecor process V is assumed o solve he SDE dv =κ θ V d + σ V 1/ dw, V 0 = v > 0, 3.3 for [0, T ], where he parameers κ > 0, θ > 0, σ > 0, are admissible in he sense of Duffie e al. 003, = 1,..., d. In addiion, o avoid zero volailiy facors, we impose he following assumpion. Assumpion 3.1. For every = 1,..., d, he parameers in 3.3 saisfy he relaion κ θ σ We also allow for non-zero correlaion beween asses and heir volailiies via he following condiion: Assumpion 3.. The Brownian moions Z and W have a covariaion saisfying d W, Z l d where δ l denoes he Dirac dela funcion for he indices and l. = δ l ρ,, l = 1,..., d, 3.5 We hen proceed o provide a general specificaion for he family of mare prices of ris. Assumpion 3.3. We assume ha he i-h mare price of ris vecor π i is a projecion of he common volailiy facor V, along a direcion paramerized by a consan vecor a i R d and a projecion of he invered elemens of V along anoher direcion paramerized by b i R d, according o he following relaions π i = Diag 1/ V a i + Diag 1/ V b i, i = 1,..., N, 3.6 where Diag 1/ u denoes he diagonal marix whose diagonal enries are he respecive square roos of he componens of he vecor u R d. The family of shor-rae processes r i, i = 1,..., N is assumed o be given in he form r i = h i + H i, V + G i, V 1, 3.7 where V 1 is a vecor whose componens are he inverses of hose of V. Under Assumpion 3.3, we can express he dynamics of he GOP as dd i D i = r i + a i DiagV a i + b i Diag 1 V b i + a i b i d + a i Diag 1/ V dz + b i Diag 1/ V dz. 9

10 Here we suppress he explici formulaion of he dependence of r i on V. Consequenly, he dynamics of he exchange rae S i,j is given by he SDE ds i,j S i,j = r i r j + a i b i a i b j a j b i + a i DiagV a i a j + b i Diag 1 V b i b j d + a i a j Diag 1/ V + b i b j Diag 1/ V dz. Noice ha he dynamics of he exchange raes are fully funcionally symmeric w.r.. he consrucion of produc/raios hereof. 3.3 Sric Local Maringaliy In Secion 3.1, we observed ha, for a sufficienly general specificaion of he mare price of ris for a given currency, we can have a siuaion where an equivalen ris neural probabiliy measure may fail o exis, due o he behavior of he variance process near zero under he puaive ris neural measure. In his subsecion, we invesigae he condiions under which he i-h benchmared savings accoun, ˆB i = Bi D i, is a sric P-local maringale, i = 1,..., N. As observed in Secion., ˆBi, afer normalizaion o one a he iniial ime, corresponds o he Radon-Niodym derivaive for he puaive ris neural measure of he i-h currency denominaion. Should ˆB i be a sric P-local maringale, we noe ha classical ris neural pricing is no applicable. However, real world pricing in line wih.9 is sill applicable, see Plaen and Heah 010, and provides he minimal possible price. Given.7 and.4, he dynamics of ˆB i are given by he SDE 3.8 d ˆB i = ˆB i a i DiagV 1/ dz + b i DiagV 1/ dz. 3.9 Upon inegraion of he above SDE we obain [ ] E ˆBi = ˆB i 0 d E [ ξ i ], where we define he exponenial local maringale process ξ i = { ξ i, 0} via =1 ξ i := exp { ρ a i V s 1/ + b i V s 1/ dw s 1 ρ 0 0 } a i V s 1/ + b i V s 1/ ds. The puaive change of measure wih respec o he i-h currency denominaion involves 3.10 d W = dw + ρ a i V 1/ + b i V 1/ d, where under classical assumpions W should be a Wiener process under he puaive ris neural measure Q i. Under his measure he process V would hen solve he SDE dv = κ θ V d ρ σ a i V + b i d + σ V 1 d W = κ θ ρ σ b i d κ 1 + ρ σ a i κ V d + σ V 1 d W Under P, he process V does no reach 0 if he Feller condiion is saisfied, i.e. κ θ σ, 10

11 while under he puaive ris neural measure he process V would no reach 0 if he corresponding Feller condiion would be saisfied, ha is κ θ σ + ρ σ b i. Therefore, he process V would have a differen behavior a 0 under he wo measures, provided ha σ κ θ < σ + ρ σ b i. 3.1 In his case he puaive ris neural measure would no be an equivalen probabiliy measure and classical ris neural pricing would no be well-founded. In order o ge an inuiion of wha is he ypical pah behavior when dealing wih rue and sric local maringales, we simulae some pahs of he Radon-Niodym derivaive for he puaive ris neural measure of he i-h currency denominaion ˆB i = Bi D i according o he corresponding SDE 3.9, ogeher wih he respecive quadraic variaion processes, for ime horizon = 10 years. In his illusraion we consider a one facor specificaion of he 4/ model i.e. d = 1 and fix he parameers as follows: κ = ; θ = ; σ = ; V 0 = ; ρ = ; a = We le he parameer b range in he inerval [ 0.4, 0.4] in order o generae siuaions in which he process ˆB i is a rue maringale b posiive or a sric local maringale b negaive. We see in Figure 1 ha he quadraic variaion of he sric local maringale process almos explodes from ime o ime and increases hrough hese upward jumps visually much faser han in he case corresponding o he rue maringale process, in line wih he well-nown unbounded expeced quadraic variaion process for square inegrable sric local maringales, see e.g. Plaen and Heah 010. To conclude his secion, le us observe ha we can compue he prices of zero coupon bonds for all currency denominaions, meaning ha i is a priori possible o devise a model for long-daed FX producs, in he spiri of Gnoao and Grasselli 014, where a join calibraion o FX surfaces and yield curves is performed. Depending on he parameer values, our general framewor may be inerpreed boh from he poin of view of real world pricing and classical ris-neural valuaion, respecively: Should mare daa imply he exisence of a ris-neural probabiliy measure for he i-h currency denominaion, hen i would be possible o equivalenly employ he i-h money mare accoun as numéraire. In he oher case, i.e. when ris neural pricing is no possible for he i-h currency denominaion due o he sric local maringale propery of he i-h benchmared money-mare accoun, hen discouning should be performed via he GOP. 4 Valuaion of Derivaives In he presen secion we solve he valuaion problem for various coningen claims. The general valuaion ool will be given by he real-world pricing formula.9. Subsecion 4.1 concenraes on plain vanilla European FX opions, for which a semi-closed form valuaion is available by means of Fourier echniques. These require he nowledge of he characerisic funcion of he log-underlying, given below in Theorem 4.1, which provides as a by-produc a closed form valuaion formula for benchmared zero-coupon bonds. 4.1 FX opions We firs provide he calculaion of he discouned condiional Fourier/Laplace ransform of x i,j := lns i,j, which will be useful for opion pricing purposes. Le us consider a European call opion CS i,j, K i,j, τ a ime, i, j = 1,..., N, i j, on a generic exchange rae process S i,j wih srie K i,j, mauriy T = + τ and face value equal o one uni of he foreign currency. We denoe via Y i = logd i he logarihm of he GOP in i-h currency denominaion. Hence he log-exchange 11

12 rae may be wrien as x i,j = logs i,j = Y i Y j. Le us inroduce he following condiional expecaion [ ] φ i,j,t z = 1 Di E i,j D i T eizx T F [ ] 4.1 = e Y i E e Y i T +izy i T Y j T F for i = 1. For z = u R we will use he erminology of a discouned characerisic funcion, whereas for z C when he expecaion exiss, he funcion φ i,j,t will be called a generalized discouned characerisic funcion. If we denoe by Ψ,T z he join condiional generalized characerisic funcion of he vecor of GOP denominaions Y T = Y 1 T,..., Y N T, ha is ] Ψ,T ζ := E [e i ζ,y T F, ζ C N, 4. hen we have φ i,j,t z = Di Ψ,T ζ, 4.3 for ζ being a vecor wih ζ i = z + i, ζ j = z and all oher enries being equal o zero. Now, from he real-world pricing formula.9, he ime price of a call opion can be wrien as he following expeced value: [ CS i,j, K i,j, τ = D i 1 E S i,j D i T K i,j ] + T F. Following Lewis 001, we now ha opion prices may be inerpreed as a convoluion of he payoff and he probabiliy densiy funcion of he log-underlying. As a consequence, he pricing of a derivaive may be solved in Fourier space by relying on he Plancherel/Parseval ideniy, see Lewis 001, where we have for f, g L R, C fxgxdx = 1 ˆfuĝudu π for u R and ˆf, ĝ denoing he Fourier ransforms of f, g, respecively. Applying he reasoning above in an opion pricing seing requires some addiional care. In fac, mos payoff funcions do no admi a Fourier ransform in he classical sense. For example, i is well-nown ha for he call opion one has Φz = R e izx e x K i,j + dx = K i,j iz+1 zz i, provided we le z C wih Imz > 1, meaning ha Φz is he Fourier ransform of he payoff funcion in he generalized sense. Such resricions mus be coupled wih hose ha idenify he domain where he generalized characerisic funcion of he log-price is well defined. The reasoning we jus repored is developed in Theorem 3. in Lewis 001, where he following general formula is presened here we wrie φ i,j for φ i,j,t in order o simplify noaion: CS i,j, K i,j, τ = 1 φ i,j zφzdz, 4.4 π Z wih Z denoing he line in he complex plane, parallel o he real axis, where he inegraion is performed. The aricle Carr and Madan 1999 followed a differen procedure by inroducing he concep of a dampened opion price. However, as Lewis 001 and Lee 004 poin ou, his alernaive approach is jus a paricular case of he firs one. In Lee 004, he Fourier represenaion of opion prices is exended o he case where ineres raes are sochasic. Moreover, he shifing of conours, pioneered 1

13 by Lewis 001, is employed o prove Theorem 5.1 in Lee 004. There he following general opion pricing formula is presened: CS i,j,k i,j, τ = R S i,j, K i,j, α + 1 π iα 0 iα Re e izi,j φi,j z i dz. zz i 4.5 Here i,j = logk i,j, α denoes he conour of inegraion and he erm coming from he applicaion of he residue heorem is given by φ i,j i K i,j φ i,j 0, if α < 1 R S i,j, K i,j, α φ i,j i Ki,j φi,j 0, if α = 1 = φ i,j i if 1 < α < φi,j i if α = 0 0 if α > 0. The following heorem provides he explici compuaion of he generalized discouned characerisic funcion. Theorem 4.1. The join condiional generalized characerisic funcion Ψ,T in 4. is given by { N Ψ,T ζ = exp iζ i Y i + h i T } =1 [ d N 1 ρ exp T β, V e + iζ i a i b i + iζ i κ ρ σ V iζ i ρ a i σ m +1 V κ θ 1 σ κ θ T A V coh N iζ i iζ j a i b j + aj bi j=1 iζ i ρ b i σ b i θ a i logv σ λ + K 1 A T 1 1 F 1 + m α + κ θ, m + 1, σ ]} + m α + κ θ σ +κ V Γ 1 + m α + κ θ σ Γm + 1 β, V, 4λ + K 4.7 where κθ σ m = σ A = κ + µ σ, + σ ν, A x β, x = σ, sinh A T K = 1 A A T σ coh + κ, 13

14 and α = ρ σ λ = ρ σ Le be given he funcions N iζ i b i 4.8 N iζ i a i 4.9 N µ = iζ i H i + iζi ai + 1 N ρ iζ i iζ j a i a j + iζi ρ a i j=1 N ν = iζ i G i + iζi bi + 1 N ρ iζ i iζ j b i b j j=1 iζi ρ b i κ θ σ. σ f 1 Imζ := κ + σ f Imζ := N [ Imζ i H i Imζi a i + 1 ρ N Imζ i Imζ j a i a j Imζi ρ a i j=1 κ θ σ + σ + 1 ρ N j=1 f 3 Imζ := κ θ + σ + f Imζ f 4 Imζ := ρ σ σ N [ Imζ i G i Imζi b i Imζ i Imζ j b i b j + Imζi ρ b i σ N f Imζ i a i + Imζ + κ, in conjuncion wih he following condiions i f 1 Imζ > 0, = 1,..., d; σ κ σ κ σ 4.10 κ θ σ 4.11 ii f Imζ 0, = 1,..., d; iii f 3 Imζ > 0, = 1,..., d; iv f 4 Imζ 0, = 1,..., d. The ransform formula 4.7 is well defined for all [0, T ] when he complex vecor iζ belongs o he srip D,+ = A,+ ir N C N, where he convergence se A,+ R N is given by A,+ := { Imζ R N f l Imζ, l = 1,..., 4 saisfying i-iv } 14

15 Moreover, for iζ D, = A, ir N wih A, := { Imζ R N f l Imζ, l = 1,..., 3 saisfying i-iii and f 4 Imζ < 0 for some } A,+ he ransform is well defined unil he maximal ime given by 1 A = min log 1 s.. f 4 N Imζ<0 A κ + σ ρ Imζi a i A Proof. See Appendix B. The general ransform formula above is a powerful ool, however, checing he validiy of 4.7, may no be very pracical in a calibraion seing. For his reason, we provide a simple, ye handy, crierion. Recall ha we inroduced in.11 he price P i, T a ime [0, T ], 0 T T of a zero coupon bond for one uni of he i-h currency o be paid a T, i = 1,..., N, via he following condiional expecaion [ ] P i, T := D i 1 E D i T F = φ i,j,t The crierion is provided by he nex lemma. Lemma 4.1. Le 1 < α < 0 and z C wih z = u + iα. Assume P i, T P j, T <, hen [ D i 1 E S i,j D i T ] α T F <, moreover, he discouned characerisic funcion φ i,j z admis an analyic exension o he srip Proof. See Appendix C. Z = {z C z = u + iα, α 1, 0}. Given he resul in Lemma 4.1, we shall proceed o calibrae he model by employing he generalized Carr-Madan formula of Lee, i.e. 4.5, by seing R S i,j, K i,j, α = φ i,j i. 5 Model Calibraion o FX Triangles In line wih De Col e al. 013, Gnoao and Grasselli 014 and Baldeaux e al. 015b, we perform a join calibraion o a riangle of FX implied volailiy surfaces. More specifically, we consider he daa se employed in Gnoao and Grasselli 014, feauring implied volailiy surfaces for EURUSD, USDJPY and EURJPY as of July nd 010. We choose such a dae so as o obain a calibraion ha can be approximaely compared wih he one of De Col e al. 013, based on daa as of July 3 rd 010. We perform our calibraion o opions wih expiry daes ranging from one up o 18 monhs and moneyness ranging from 15 dela pu up o 15 dela call, and we consider a oal of 16 conracs. The model we consider for he calibraion is he full 4/ sochasic volailiy model, i.e. boh he Heson and he 3/ effecs are simulaneously considered. As we calibrae opions wih mauriy up o 18 monhs, we do no consider sochasic ineres raes due o he limied ineres rae ris, see Gnoao and Grasselli 014. In line wih he references above, we choose he following penaly funcion σ imp i,m i,model σimp, i 15

16 where σ imp i,m is he i-h observed mare volailiy and σimp i,model is he i-h model-derived implied volailiy. For each opion conrac, σ imp i,model is consruced along he following seps: firs, given a se of model parameers, 4.5 for 1 < α < 0 is employed so as o obain he corresponding model derived price, secondly, he obained price is convered ino σ imp i,model, via a sandard implied volailiy solver. As far as he implemenaion of 4.5 is concerned, we approximaed he inegral via a 4096-poin FFT rouine, wih grid spacing equal o 0.1, so ha he improper inegral is runcaed a he poin e 409. The corresponding srie range is hen given by [ e , e ] and Simpson s rule weighs are inroduced for increased accuracy, see Carr and Madan The FFT reurns hen a vecor of opion prices for a fixed grid of sries. Opion prices for he sries of ineres are obained via a linear inerpolaion. We assume ha he model is driven by wo square roo facors. The parameers we need o calibrae are given by hose appearing in he dynamics of each square roo process, i.e. κ, θ, σ, V 0, = 1,, coupled wih a wo-dimensional vecor of correlaions and six wo-dimensional vecors of projecions for each currency area, i.e. a i, b i, i = 1,, 3, meaning ha we proceed o esimae a oal of parameers. Clearly, in order o preven insabiliy and over-paramerizaion issues, simplified versions of he model may be considered. The resul of he calibraion is presened in Figure a for July nd, 010, while he corresponding parameers are repored in Table 1. We obain a good fi over all hree surfaces we consider, in line wih De Col e al This shows ha a saisfacory calibraion of he model can be achieved. I allows us o perform he following analysis, which consiues an ineresing empirical resul of he curren paper. Given he se of parameers we obain from he calibraion, we can ry o analyze wheher mare daa of FX opions are supporing he common use of classical ris neural pricing. Our approach is so flexible, ha we can, in he seing of a single model, span boh he ris-neural valuaion and he pricing under he real-world measure. Such an analysis is summarized in Table. We consider differen measures for pricing: he real world probabiliy measure P, and he puaive ris neural measures Q usd, Q eur and Q jpy. For each measure we compue he corresponding Feller condiion for each square roo process. Under he real world probabiliy measure P, we observe ha he Feller condiion κ θ σ is saisfied by boh V 1 and V. We nex proceed o perform he same analysis under he wo puaive ris neural measures Q usd and Q jpy, respecively. We observe ha for he firs one we sill have ha boh processes do never reach zero, whereas for Q jpy we have ha he Feller condiion is no saisfied by he second componen. As discussed in Secion 3.3, if a leas one of he square roo processes has a differen behavior under he puaive ris-neural measure, hen we have ha classical ris-neural pricing is no well founded. In summary, we have a siuaion where mare daa sugges ha for he USD currency denominaion ris-neural pricing is poenially applicable, while in he JPY denominaion i is no heoreically founded. We also perform a second calibraion experimen. The srucure of he sample of he daase is he same as in he previous case and mare daa were provided as of Feb 3 rd, March 3 rd, April nd, May nd and June nd, 015. Essenially, we are aing he perspecive of a derivaive des following he mare pracice ha involves a periodic model re-calibraion across differen rading daes. Such analysis allows us o provide some firs evidence regarding he sabiliy of he parameer esimaes we obain. By looing a Table 5, we observe a saisfacory sabiliy of he calibraed parameers. A relevan change in he esimaes is observed only beween he February and March calibraion. The qualiy of he fi is comparable wih he one obained in our firs calibraion and he above menioned papers of Baldeaux e al. 015b and De Col e al Calibraed parameer values are lised in Table 5, whereas he Feller condiion under all measures is repored in Table 6. We observe in his case a violaion of he Feller condiion for he second facor under he Q usd puaive ris neural measure, whereas for he Q jpy measure he condiion is passed. For Q eur, insead, we observe ha he condiion is iniially passed and hen, saring from April nd, we have repeaed violaions. The overall resuls of our analysis allow us o sugges ha mares are subjec o wha we may erm as regime swiches in pricing beween he classical ris neural and he more general real world pricing approach. Such a feaure would clearly provide a srong moivaion for he inroducion 16

17 of models ha are able o accomodae boh valuaion framewors, lie he 4/ ype specificaion ha we propose. 6 Pricing and Hedging of Long-Daed Zero Coupon Bonds In he previous secion we obained a prooypical calibraion o mare daa ha shows he coexisence of pricing under he ris-neural and he more general benchmar approach. In he presen secion, we ae he calibraed values as given and consider he problem of hedging coningen claims under he 4/ model. We sar in Subsecion 6.1 by providing he necessary bacground on quadraic hedging and, in paricular, on benchmared ris minimizaion. We resric our aenion o a very simple coningen claim, namely a zero coupon bond, which is a core building bloc of annuiies and many oher insurance producs. Such an experimen is simple and ye exremely powerful in showing how he benchmar approach allows for he hedging of coningen claims for a lower cos. The consrucion of he hedging scheme requires a maringale represenaion for he claim under consideraion, where he iniial price represens he saring poin of he value of he sraegy. In Subsecion 6. we analyze he valuaion formula for a zero coupon bond and explicily highligh he consequences of he failure of he maringale propery from an analyical poin of view. Finally, in Subsecion 6.3 we explicily compue he dynamic hedging scheme. 6.1 Benchmared Ris Minimizaion The 4/ model is a sochasic volailiy model. Hence, due o he presence of he insananeous volailiy uncerainy, we have an example of an incomplee mare. Hedging claims in an incomplee mare seing is a non-rivial as, which may be performed by means of differen possible crieria. Incompleeness means ha, in general, i is no possible o consruc a self-financing rading sraegy ha delivers a mauriy T he final payoff HT almos surely. According o he survey paper of Schweizer 001, in an incomplee mare seing one may relax he requiremens on he family of possible rading sraegies in eiher wo direcions: 1. A firs possibiliy is o relax he requiremen ha he erminal value process of he sraegy reaches he final payoff HT a.s., and hence one is induced o minimize he expeced quadraic hedging error a mauriy over a suiable se of self-financing rading sraegies. This firs approach is called mean-variance hedging, and is presened in Bouleau and Lamberon 1989, Duffie and Richardson 1991 and Schweizer Alernaively, one may relax he self-financing requiremen and insis on aaining HT a.s. a mauriy T. In his second case one is induced o minimize a quadraic funcion of he cos process of he non-self-financing rading sraegy. This second approach, referred o as local ris minimizaion, was inroduced in Föllmer and Sondermann 1986 and hen generalized in Schweizer 1991 in a general semimaringale seing assuming he exisence of a minimal equivalen maringale measure; see also Møller 001. In his paper we adop some ype of local ris minimizaion. I is imporan o sress ha we need o consider a generalized concep of local ris minimizaion in he conex of he benchmar approach. A firs generalizaion in his sense was provided by Biagini e al In he curren paper, we adop he more general approach of Du and Plaen 016, nown as benchmared ris minimizaion, which does no require second momen condiions. In Secion we inroduced he noion of a self-financing rading sraegy. We need o generalize our noaion, since we will consider, in general, also non-self-financing rading sraegies. To his end, in line wih Definiion.1 of Du and Plaen 016,{ we will call a dynamic rading sraegy, iniiaed a = 0, an R N+1 -valued process of he form v = v = η, ϑ 1,..., ϑ N }, 0 T T for a predicable B-inegrable process ϑ = { ϑ = ϑ 1,..., ϑ N, 0 T T }, which describes he unis invesed in he benchmared primary securiy accouns B and forms he self-financing par of he 17

18 associaed porfolio. The corresponding benchmared value process is hen V v = ϑ B+η, where η = { η, 0 T T } wih η0 = 0 moniors he non-self-financing par, so ha we can wrie V v = V v ϑs d Bs + η. The process η moniors he inflow/ouflow of exra capial and hence measures he cos of he sraegy, see Corollary 4. in Du and Plaen 016. When he monioring process is a local maringale, we say ha he sraegy v is mean-self-financing, see Definiion 4.4 in Du and Plaen 016. Moreover, when he monioring par η is orhogonal o he primary securiy accouns, in he sense ha η B forms a vecor local maringale, we say ha he rading sraegy v has an orhogonal benchmared profi and loss, see Definiion 5.1 in Du and Plaen 016. Given VĤT, he se of all mean-self-financing rading sraegies which deliver he final benchmared payoff ĤT P-a.s., wih an orhogonal benchmared profi and loss, see Definiion 5. in Du and Plaen 016, we say ha a sraegy ṽ VĤT is benchmared ris minimizing if, for all sraegies v VĤT, we have ha Vṽ V v P-a.s. for every 0 T T, see Definiion 5.3 in Du and Plaen 016. The pracical applicaion of he concep of benchmared ris minimizaion necessiaes he availabiliy of maringale represenaions for benchmared coningen claims, which will be given under he real-world probabiliy measure P. In he racable Marovian seing of he 4/ model, such represenaion can be explicily obained, so ha one represens a benchmared coningen claim ĤT in he form ] ĤT = E [ĤT F + ϑs d Bs + ηt η, 6.1 see Equaion 6.1 in Du and Plaen 016, and here exiss a benchmared ris minimizing sraegy v for ĤT. In summary, o deermine he sraegy, one has firs o compue he condiional expecaion in 6.1, which will be sraighforward in our case due o he analyical racabiliy of he 4/ model. In a second sep, an applicaion of he Iô formula allows for he idenificaion of he possible holdings ϑ in he self-financing par of he price process V v. Finally, he monioring par is given by η = V v ϑ B, which when muliplied wih B needs o have zero drif. In he nex subsecions, we carry ou his procedure for a zero-coupon bond in he Japanese currency denominaion, where, based on he calibraion resuls discussed in Secion 5, we observed a failure of he ris neural paradigm. Noice ha he calibraion of Secion 5 refers o a model where he shor rae is assumed o be consan. We do recognize ha his represens a simplificaion. Noe, insead one could also use as payoff one uni of he saving accouns o demonsrae he consequences of he failure of he ris neural approach. However, aing he shor rae consan allows us o illusrae easily he impac of he violaion of he ris neural paradigm on zero coupon bonds. Inroducing a sochasic shor rae can be easily achieved by allowing for non-zero projecion vecors H i, G i in 3.7. For an example of a hedging scheme in he presence of sochasic ineres raes we refer o Baldeaux e al. 015a. 6. Pricing of a Long-Daed Zero Coupon Bond In his subsecion we firs quanify he impac of he violaion of he classical ris neural assumpion on he pricing of a zero coupon bond wrien on JPY as domesic currency when assuming a consan ineres rae r JP Y. If ris neural valuaion were possible in his mare, hen pricing as well as hedging of his elemenary securiy would be rivial: I would consis in eeping he JPY amoun exp r JP Y T in he domesic ban accoun B JP Y and nohing in any oher asse. However, for he calibraed mare we showed ha under he 4/ model he ris neural probabiliy measure mos liely does no exis for he JPY currency denominaion, so ha ris neural pricing is no allowed. As a consequence, we price he zero coupon bond under he real world probabiliy measure using he real world pricing formula.11. Recall ha in our sochasic volailiy conex he mare is incomplee, so ha perfec hedging is no possible. As already indicaed, we adop benchmared ris 18