The Influence of FX Risk on Credit Spreads

Transcription

1 he Influence of FX Risk on Credi Spreads Philippe Ehlers and Philipp Schönbucher Deparemen of Mahemaics, EH ürich March 24, his version December 24 Absrac We analyze he connecions beween he credi spreads ha he same credi risk commands in differen currencies. We show ha he empirically observed differences in hese credi spreads are mosly driven by he dependency beween he defaul risk of he obligor and he exchange rae. In our model here are wo differen channels o capure his dependence: Firs, he diffusions driving FX and defaul inensiies may be correlaed, and second, an addiional jump in he exchange rae may occur a he ime of defaul. he differences beween he defaul inensiies under he domesic and foreign pricing measures are analyzed and closed-form prices for a variey of securiies affeced by defaul risk and FX risk are given (including CDS). In he empirical par of he paper we find ha a purely diffusion-based correlaion beween he exchange rae and he defaul inensiy is no able o explain he observed differences beween JPY and USD CDS raes for a se of large Japanese obligors. he daa implies a significan addiional jump in he FX rae a defaul. 1 Inroducion In modern deb markes, many large debors issue deb in more han one currency, e.g. a large Japanese obligor may find i advanageous o issue deb in USD, or a European obligor in JPY. Furhermore, since he adven of liquid markes for credi defaul swaps (CDS) here are markes for credi proecion in currencies differen from he obligors home currency, even if he obligor has no issued bonds in ha currency. (here is Auhors Address: EH urich, D-MAH, Rämisr. 11, CH-892 urich, Swizerland. [email protected], ehlers, [email protected], his paper was presened a he RiskDay 24 in urich. Financial suppor by he Naional Cenre of Compeence in Research Financial Valuaion and Risk Managemen (NCCR FINRISK), Projec 5: Credi Risk is graefully acknowledged. he NCCR FIN- RISK is a research program suppored by he Swiss Naional Science Foundaion. We would like o hank Lombard Risk Sysems for generously making he ValuSpread CDS Hisorical Daabase ( available o us, and ABN Amro for providing us wih FX and ineresrae daa. All errors are our own. Commens and suggesions are very welcome. 1

2 demand for his in order o hedge loan exposures or OC derivaives ransacions.) In paricular, CDS proecion on many inernaional corporaions is now available in heir home currency and EUR and USD. Given his siuaion, i is naural o ask abou he correc relaive pricing of he credi risk in he differen currencies, i.e.: How should he credi spread be adjused (eiher on bonds or on CDS) if a differen currency is used? And: Wha informaion abou likely FX movemens in crisis evens can we imply from he relaive difference of CDS spreads in differen currencies? We aim o answer his quesion in wo seps. Firs, we analyse he connecions beween local and foreign currency credi spreads on a heoreical basis in an inensiy-based framework and highligh he effecs ha we can expec o encouner. We find ha he essenial feaure driving differences beween credi spreads in differen currencies is he dependency beween defaul risk and FX risk. If defaul risk and FX risk are independen (in a sense which will be made precise laer on), credi spreads in differen currencies should no differ. In order o capure empirically observed differences, we model dependency beween spreads in wo differen ways. Firs, here may be correlaion beween he diffusions driving FX and defaul inensiies, and second, an addiional jump in he exchange rae may occur a he ime of defaul, i.e. he defaul causes a devaluaion of he currency. We give closedform soluions for CDS raes and defaulable bond prices in a model which encompasses boh cases using an affine jump-diffusion (AJD) model. In he empirical par of he paper hese models are esimaed using a hisorical daabase of CDS raes in Japanese yen (JPY) and US dollars (USD) on a se of major Japanese corporae obligors. Besides being relaively clean, liquid and sandardised, CDS daa has he addiional advanage ha he recovery raes on USD-denominaed and JPYdenominaed CDS will be idenical by definiion (by documenaion, o be more precise). hus, spread differences in CDS raes canno be caused by he effecs of differen legal regimes and bond specificaion which may affec corporae bond daa. In he pure diffusion hypohesis (i.e. wihou jumps in he FX rae), we firs esimae he model parameers using USD CDS spreads and he JPY/USD rae, wihou using he JPY CDS spread. In he second sep, we hen calculae he JPY CDS spread which would hold if he model were correc and compare i o he empirically observed JPY CDS spreads. In all cases, we can srongly rejec he hypohesis ha he empirically observed JPY CDS spreads are noisy observaions of he model predicions, he prediced spread difference is only a small fracion of he observed spread difference. Consequenly, we rejec he pure diffusion model, here mus be jumps in he exchange rae a defaul. In many cases, he implied jump in FX raes a defaul of he obligors seems quie large and is difficul o explain unless we assume ha he obligor s defaul was caused by a major macroeconomic crisis. he echniques used in his paper are also useful for a number of oher applicaions like he pricing of counerpary risk and he pricing of sovereign defaul risk. Some of hese applicaions are also poined ou in he final secion, and we give prices for some more exoic FX-relaed credi derivaives. Sparked by he Asian crisis in he lae 199 s (and he Peso crisis earlier on) here is a large lieraure on sovereign defaul risk, banking crises and currency crises in emerging 2

3 markes 1. While some of he echniques used in he presen paper can also be applied o hese siuaions, his paper has a differen focus han he domesic deb of a sovereign obligor which is special because (a leas in heory) he sovereign could always repay ha deb by prining more domesic currency. Similarly, his paper also differs in focus from papers which empirically invesigae sovereign credi spreads, like e.g. Duffie e al. (23) and Singh (23). In his paper our main focus are mulinaional corporaions which face a quie differen siuaion from sovereigns. Here, he foreign sovereign ofen has negligible defaul risk compared o he corporae (unless i is an emerging marke), and he exchange rae floas freely. Furhermore, he available daa is significanly differen: CDS on large corporaions are rouinely quoed in he major currencies (USD, EUR, and also JPY, GBP, CHF). Despie his focus on corporae risk, he echniques of his paper can be used o back ou marke implied informaion abou he sovereign, in paricular values for he expeced currency devaluaion upon a defaul of he corporae. he same crieria apply o he CDS spreads of sovereigns in differen foreign (o he sovereign) currencies (e.g. USD and EUR-denominaed CDS on Brazil) which also fall wihin he scope of his paper. Finally, i is no difficul o exend he model o cover corporaes which are based in a counry wih non-negligible defaul risk. In a relaed paper Jankowisch and Pichler (23), he auhors address he quesion of he consrucion of corporae credi spread curves from corporae bond prices in differen currencies. In heir sample, he auhors find srong evidence agains he assumpion of independence of corporae bond credi spreads and exchange raes. Our paper differs from Jankowisch and Pichler (23) in several respecs: Firs, we provide a full heoreical model which is able o capure he sochasic dependency beween defaul inensiies and exchange raes and o replicae he observed spread differences. Second, we base our analysis on CDS which are significanly beer suied o he empirical and heoreical analysis of hese quesions as i avoids he issues caused by differences in recovery raes in differen currencies. Anoher relaed paper is Warnes and Acosa (22) who exend he classical Meron (1974) firm s value approach o incorporae deb in a foreign currency and provide closed-form soluions for deb prices under he assumpion of consan ineres-raes in boh counries. he res of he paper is srucured as follows: o se he sage for he res of he paper, we recapiulae in he nex secion he payoff mechanics of credi defaul swaps (CDS) and show ha he delivery opion in he proecion leg of he CDS makes he effecive recovery rae currency-independen. In secion 3 we hen se up he mahemaical background o he general FX model under defaul-risk which follows in secion 4. o provide a concree specificaion for he empirical esimaion in secion 5, we also specify an affine jumpdiffusion (AJD) version of he model in secions 3 and 4. Furhermore, secion 4 conains he presenaion of change-of-measure echniques ha apply o he valuaion of payoffs a sopping imes when he numeraire asse jumps a his sopping ime, an analysis of he he relaionship beween he defaul inensiies of he obligor under he domesic and he foreign pricing measures, and some concree pricing resuls for basic defaulable securiies (zero-coupon bonds and CDS) under a variey of recovery assumpions. As our focus is 1 See e.g. Kaminsky and Reinhar (1999), Reinhar (22), Bulow and Rogoff (1989) or oher papers lised on N. Roubini s hp:// 3

4 on he difference beween domesic and foreign CDS raes, we discuss in secion 4.3 he effecs which we expec o influence his quaniy. Secion 5 conains an empirical analysis of he AJD-model using hisorical CDS daa on a number of large Japanese obligors. We show ha here is a persisen, significan and raher large difference beween CDS raes in JPY and USD which canno be explained by a purely diffusion-based dependency beween defaul inensiy and FX rae alone. hus, we conclude ha he marke mus be pricing an implici devaluaion a defaul ino hese CDS spreads. Finally, in secion 6 he empirical resuls are discussed and i is shown how he echniques inroduced in secions 3 and 4 can be applied o oher defaul-sensiive FX derivaives. 2 CDS in Muliple Currencies Credi defaul swaps (CDS) are derivaive insrumens which allow he rading of payoffs coningen on he occurrence of a credi even. Single-name CDS are he mos imporan class of credi derivaives ransacions, Pael (23) finds ha in 23 hey accouned for around 72.5% of he enire credi derivaives marke in erms of noional ousanding which was equivalen o a noional of around bn USD. In many cases, he liquidiy of he CDS marke has surpassed he liquidiy of he marke for he bonds of he underlying obligor. his rading volume and liquidiy has been made possible by he sandardisaion of he documenaion for CDS ransacions which has been proposed by he Inernaional Swap Dealers Associaion (ISDA) (see hp:// and (ISDA) (1999)). In paricular when i comes o he analysis of he defaul risk of any given obligor in wo or more differen currencies (and hus in wo differen jurisdicions) his sandardisaion is essenial: Bonds in domesic and foreign currency are ypically issued in differen jurisdicions and herefore are governed by differen legal rules which has a significan impac on he resuling recovery raes of he bonds (see e.g. Davydenko and Franks (24)). CDSs referencing he obligor on he oher hand will be governed by he same sandardised ISDA documenaion even if hey are denominaed in differen currencies, in paricular hey will have he same recovery raes. hus, for he purposes of his paper we consider CDS o be more sandardised and more easily comparable han he underlying corporae bonds. We now presen a quick summary of he payoff mechanics of an ISDA-sandard CDS wih physical selemen in order o explain why he recovery rae of a CDS is ypically independen from he currency of is denominaion: Being an over-he-couner raded derivaive, a CDS is a conrac beween wo counerparies: he proecion buyer and he proecion seller. he proecion buyer makes he paymens of he fee leg of he CDS, he proecion seller pays he proecion leg. In order o define hese paymen sreams, he following daa is specified in every CDS: he noional amoun N, and he currency c of he noional amoun, he mauriy dae, he CDS rae s, he reference credi (i.e. he obligor whose credi risk is raded) he applicable (precise) definiion of he credi even, and 4

5 he se of deliverable obligaions. I is imporan o noe ha boh he definiion of he credi even and he lis of he deliverable obligaions usually do no depend on he currency of he CDS. he ISDAdefiniion of a credi even includes bankrupcy, failure o make due paymens on bonds or loans ( failure o pay ), repudiaion or moraorium, cross-acceleraion, obligaion defaul, disressed resrucuring and credi evens upon mergers. hese evens apply globally o he reference obligor and are in mos cases objecively verifiable and independen from local legal rules. he se of deliverable obligaions conains mos bonds and loans issued by he reference credi irrespecive of heir currency, excluding special cases such as subordinaed bonds or bonds wih unusual mauriy daes bu including all major bond issues. he Fee Leg. he proecion buyer makes fee paymens o he proecion seller a regular inervals unil he CDS maures or unil a credi even occurs. he fee paymens are made in he currency of he CDS and are calculaed as daycoun fracion CDS rae Noional = s N. he Proecion Leg. A he credi even, he proecion buyer chooses a porfolio from he se of deliverable obligaions such ha he oal noional amoun of he porfolio is N. If any obligaions in he porfolio are denominaed in oher currencies han he CDS reference currency c, hen he noional amoun of hese obligaions is convered ino he reference currency using he acual exchange rae of he day. he proecion buyer hen delivers his porfolio o he proecion seller who has o pay he full noional N for i. Clearly, alhough hey will rade a a significan discoun o par, no all deliverable obligaions will rade a he same price. he proecion buyer has a delivery opion: he proecion buyer will choose o deliver hose bonds which rade a he highes discoun o heir par value, he cheapes-o-deliver bonds. Ineresingly, he choice of he cheapeso-deliver bond is independen of he currency in which ha bond is denominaed, i only depends on he relaive discoun of he bond o is par value. o illusrae his le us assume he bond ha we wan o deliver rades a a discoun of q, i.e. a a price of (1 q) per uni 1. of noional in is currency c c. If one uni of c is worh X unis of c a he ime of he credi even, hen in order o reach a porfolio of noional N in currency c, we have o buy a porfolio of noional N/X in he bond s currency c. hus he delivery porfolio coss (1 q) X N/X = (1 q) N in he CDS s currency. his porfolio is pu o he proecion seller for a paymen of N in c which yields a ne value of he proecion paymen of N q in c. his value does no depend on he exchange rae X any more. hus, in order o maximise he value of he proecion paymen, he proecion buyer will choose o deliver a porfolio of hose bonds which have he lowes (1 q), irrespecive of he exchange rae for he currency of denominaion of his bond. In paricular, he effecive recovery rae for he CDS will be independen from he currency in which he CDS is denominaed. his does no mean ha CDS which are denominaed in differen currencies are idenical. Consider wo CDS on he same reference credi wih he same deliverable obligaions bu denominaed in differen currencies c (wih noional N), and c (wih noional Ñ). A defaul, he firs CDS will pay off N q in currency c, and he oher will pay off Ñ q in currency c. hus, he amoun of proecion ha he c-cds provides in currency c 5

6 depends on he exchange rae a he ime of defaul. his is he relaionship which we are going o explore in his paper. 3 Mahemaical Framework 3.1 Se-Up Our model is se in a filered probabiliy space (Ω, F, F, Q) wih finie ime horizon <. he filraion F = (F ), is assumed o saisfy he usual condiions, F = F and is generaed by an N-dimensional Brownian moion (BM) W and a K- dimensional purely disconinuous process J wih jumps J = (, 1 K. We represen J using is associaed jump measure µ(d, dz) on, as J := z µ(dz, ds); he compensaor measure of µ(dz, d) under Q is denoed wih ν(dz, d). We assume (Ω, F, F, Q) has he predicable represenaion propery wih respec o W and µ ν. he ranspose of a marix M is denoed by M, and if x is a vecor, hen diag(x) is a diagonal marix wih he elemens of x on is diagonal. Sandard arihmeical funcions, inegrals and comparisons of vecors are mean componenwise, excep in he case of muliplicaions we will use marix muliplicaions. l denoes he Lebesgue measure on R. Regarding ime poins we always assume. Assumpion 1. (i) N := µ(dz, ds) is a couning process, i.e. N < Q-a.s. for all. We denoe he ime of he firs jump by he sopping ime τ := inf{; N > } (1) (wih he convenion inf = ). (ii) here exis a (nonnegaive) F W -adaped (hence predicable) processes λ and an F W - adaped (predicable) funcion 2 F on, such ha he predicable compensaor ν of µ saisfies 1 { τ} ν(dz, d) = 1 { τ} df (z)λ d and F is a disribuion funcion on. From his assumpion follows ha τ is a oally inaccessible sopping ime, and τ λ s ds is he predicable compensaor of 1 {<τ}, i.e. 1 { τ} λ is he predicable inensiy of 1 {<τ}. Noe ha we do no make any assumpion abou he form of he predicable compensaor afer τ apar from ν(dz, ds) < Q-a.s. for all.3 We define a family of condiional Laplace ransforms L : R K + R + indexed by, L(; ) : u 1 { τ} (1 z) u df (z) := ( K ) 1 { τ} (1 z k ) u k df (z) 2 See e.g. Jacod and Shiryaev (1988) for he definiion of predicable funcions. 3 his is a consequence of N < Q-a.s. k=1 6

7 suppressing he dependence on ω Ω. Inuiively, L is he Laplace ransform of he jump log(1 J ) condiional on he even {τ = }. Someimes we will use a version of L exended o all values u R K wih L(; u) < Q l-a.s. Furher we define a coninuous ime-homogeneous N-dimensional Markov process Y as he unique srong soluion of a sochasic differenial equaion (SDE) dy = γ(y )d + σ(y )dw, Y () = Y (2) wih affine funcions γ( ) and σσ ( ). 4 he process Y is called an affine diffusion (AD). We say, he process W + J is a jump diffusion (JD); and, if Y is an AD, λ is affine in Y and L(; u) is exponenially affine 5, in Y for every u hen we call Y + J an affine jump diffusion (AJD). 3.2 Hypohesis H and is Implicaions In he se-up we only assumed F = F W F J. In addiion we assume ha hypohesis H: Every F W -maringale is an F-maringale. is saisfied. See Jeanblanc and Rukowski (2) for deails on hypohesis H. In presence of hypohesis H he compuaion of he condiional expecaion of a defaulable claim, i.e. an F -measurable random variable wih 1 {τ } =, can be reduced o a condiional expecaion of a relaed F W -measurable random variable. he following lemma gives he precise resul. Lemma 2. Le g be F W -measurable wih E g <. If hypohesis H holds, hen E 1 { <τ} g F = 1{<τ} E e R λ sds g F. (3) I should be noed ha in paricular E e R λ sds g F = E e R λ sds g F W by hypohesis H. Wih Fubinis heorem we immediaely derive he corollary below. Corollary 3. Le hypohesis H hold and g be F W -adaped wih E g s ds <. hen E 1 {s τ} g s ds F = 1 {<τ} E e R s λudu g s F ds. And he propery of he predicable compensaor yields he following resul. Corollary 4. Le H hold and f be an F W -adaped (hence predicable) funcion wih E 1 { τ} f(z, ) df (z)λ d <. (4) 4 Indeed as shown by Duffie and Kan (1996), under ligh echnical assumpions given in secion 3.4, SDE (2) admis a unique srong soluion on (Ω, F, Q). 5 I.e. here exis real funcions α and β such ha L(; u) = e α(u)+β(u) Y. 7

8 Define G := f(z, )df (z). hen E 1 {s τ} f(z, s)µ(dz, ds) F = 1 {<τ} E e R s λudu G s λ s F ds. he proofs can be found in he appendix. 3.3 Jump Diffusions under Change of Measure he disribuional properies of he processes Y and J under differen probabiliy measures will be a he hear of his aricle. We give a general form of Girsanov s heorem which is valid for he probabiliy space under consideraion. For he proof see e.g. Jacod and Shiryaev (1988). heorem 5 (Girsanov s heorem for JD). Le L be an F-maringale under Q wih dl ( ) = φ() dw + Φ(z, ) 1 (µ ν)(dz, d), L = 1 L for a predicable process φ and a predicable funcion Φ. hen, he probabiliy measure Q on (Ω, F), defined by d Q dq = L, (5) F is absoluely coninuous wr. Q ( Q Q) and i holds ha: (i) he process W = W φ(s)ds is a BM under Q. (ii) he predicable compensaor ν of µ under Q saisfies 1 { τ} ν(dz, d) = 1 { τ} d F (z) λ d, where λ() = λ() Φ(z, )df (z) and F is a disribuion on for all, wih { Φ(z,) d F (z) = R Φ(z,)dF(z)dF (z) if Φ(z, )df (z) >, df (z) oherwise. Noe ha by (ii), 1 { τ} λ is he predicable inensiy of 1 {<τ} under Q. he following corollary is a sraighforward implicaion of Girsanov s heorem. Corollary 6. Le Q Q. hen de Q dq, N τ = Q-a.s. for all, if and only if 1 { τ} λ() = 1{ τ} λ() Q l-a.e.. (6) I.e. he indicaor 1 {<τ} has he same inensiy under wo equivalen measures Q Q if de Q dq, N τ = a.s. However, his does no imply ha ν = ν. 8

9 Proof of Corollary 6. Le L := de Q dq be as in heorem 5. If Q Q, hen L > Q-a.s. for all. By heorem 5 he predicable covariaion of L and 1 {<τ} is given by L, 1 { <τ} = L, N τ = τ ( ) τ L s Φ(z, s) 1 ν(dz, ds) = L s ( λ λ)(s)ds. hence 1 { τ} L ( λ λ) = Q l-a.e. if and only if L, N τ = Q-a.s for all. and he claim follows. In he sequel of he paper we use he abbreviaion Ẽ := EL when aking expecaions under Q. he following lemma saes, how he expeced value of sochasic inegrals wr. µ acs under a measure Q Q. Lemma 7 (Sochasic Inegrals wr. µ). Le L and Q be as in heorem 5 and f be a predicable funcion such ha f(z, s) µ(dz, ds) is Q-inegrable. 6 Define h := f(z, s)µ(dz, ds). If E h2 d L <, 7 hen E Proof. Appendix. L s Φ(z, s)f(z, s)µ(dz, ds) F = L Ẽ f(z, s)µ(dz, ds) F Inuiively, if L + L is a facor of he inegrand, hen i can be facored ou of he sochasic inegral when expecaions are aken. Nex we esablish a similar connecion for expecaions of inegrals wih respec o ime where we are allowed o ake L s ou of he inegral and perform he change of measure. Lemma 8 (Sochasic Inegrals wr. l). Le L and Q be as in heorem 5 and g be a predicable process wih E ( g s ds) 2 d L <. hen Proof. Appendix. E L s g s ds F = L Ẽ g s ds F. 3.4 Classificaion and Properies of he AD Y here is a large lieraure on AD in he conex of pricing defaul free and defaulable bonds. he main feaure of ADs is (see Duffie and Kan (1996)) ha under echnical condiions, for any affine funcion f : R N R here exis deerminisic funcions A, B such ha E e R f(ys)ds F = e A( )+B( ) Y, 6 I.e. L f(z, s) µ(dz, ds) is Q-inegrable. 7 his condiion can be replaced by any condiion ensuring ha h s dl s is a maringale. 9

10 where A, B are deermined by a se of ordinary differenial equaions (ODEs) involving γ, σσ and f. (Noe ha E e R f(ys)ds F = E e R f(ys)ds F W by hypohesis H.) When one wans o parameerize he funcions γ and σσ in (2), i has o be aken ino accoun ha σ is a square roo of an affine marix funcion (σσ (Y )), hence i may no be well-defined for all values of Y. In he ligh of his problem Dai and Singleon (2) inroduced a paramerizaion under which he admissibiliy of he marix σσ can be checked easily. hey call a pair (γ, σσ ) admissible if (2) admis a unique srong soluion. 8 Furhermore heir paramerizaion allows for a simple classificaion of ADs. We presen a (very sligh) exension of heir parameerizaion 9. For m {, 1,..., N} fixed, A m (N) is he class of admissible N-dimensional ADs wih σ depending on exacly m componens of Y. Consider he following parameerized version of he SDE (2) dy = (Θ KY )d + S dw, Y () = Y, (7) where S is a diagonal marix wih S ii = a i + m j=1 b ijy j for a R N and b R N N, Y, Θ R N and K R N N. Now, if he following condiions are saisfied hen (γ, σ) is admissible and he soluion Y of (7) belongs o A m (N): b, and for all 1 i j m we have : Y i, Θ i, a i =, K ij, b ii = 1, b ij =. and for all m < k N and 1 i m we have Θ k =, a k {, 1} b ik = K ik =. hen he firs m componens Y 1,..., Y m of Y are nonnegaive and σ(y ) depends only on hese componens. Y is called a canonical represenaive of he class A m (N) which is formed by all regular affine ransforms of Y (i.e. all processes = η + θy, where η R N, θ R N N inverible). If furhermore: (i) K ii > for 1 i N, (ii) he inequaliies concerning Y hold sricly, (iii) Θ i > 1, 1 i m and a 2 i + N j=1 b ij > for all 1 i N, hen (Y 1,..., Y m ) remains sricly posiive and Y is non-explosive Q-a.s. on,, and F W = F Y. In he sequel of he paper we will always consider his case. Proposiion 1 (Quadraic Variaion). Le Y be a canonical represenaive of A m (N) and α R and β R N. hen he quadraic variaion of α + β Y saisfies d d α + β Y = v(β) + w(β) Y where v and w in marix noaion are given by v(β) = a diag(β)β = β diag(a)β and w(β) = b diag(β)β. 8 In paricular, a real marix σ wih σ σ = σσ (Y ) is Q-a.s. definable for all,. 9 A commen in Ai-Sahalia and Kimmel (22) shows ha he Dai and Singleon (2) paramerizaion does no include all ADs. 1

11 Proof. We have d d β Y = β Sβ = N i=1 β 2 i ( N ) a i + b ij Y j = j=1 N N βi 2 a i + βi 2 b ijy j. i=1 i,j=1 We observe ha v( ) and w( ) is R m + {} N m -valued. Furher noe he rules v(β 1 +β 2 ) = v(β 1 ) + v(β 2 ) + 2β 1 diag(a)β 2 w(β 1 +β 2 ) = w(β 1 ) + w(β 2 ) + 2b diag(β 1 )β 2 and ha he parial derivaives of v and w wr. β in marix noaion are given by v(β) β = 2a diag(β) = 2β diag(a) and w(β) β = 2b diag(β). he following lemmaa for he calculaion of exended ransforms are well-known (see e.g. Duffie and Kan (1996) or Dai and Singleon (2)), in our paramerizaion hey are: Lemma 9. Le Y be he canonical represenaive of A m (N) saisfying (7), α R and β R N, and A : R R and B : R R N solve he Riccai ODEs x A(x) = α + Θ B(x) + 1 ( ) 2 v B(x), x B(x) = β K B(x) + 1 ( ) 2 w B(x) wih iniial condiions A() = and B() =. If here exiss B < wih B B on, and E e 1 R 2 w(b ) Y sds <, hen E e R α+β Y sds F = e A( )+B( ) Y (8) he following lemma generalizes he resul of lemma 9. Lemma 1. Le he assumpions of lemma 9 be saisfied, ζ and ξ R m + {} N m and A : R R and B : R R N solve he ODEs A(x) = Θ B(x) + B(x) diag(a)b(x) x B(x) ( ) = K B(x) + b diag B(x) B(x). (9) x wih iniial condiions A() = ζ and B() = ξ. If E e 1 R ) Yd 2 (B w + B A +B Y <, where A = min A(), B = min B(), hen E e R α+β Y sds (ζ +ξ Y ) F = ( ) A( )+B( ) Y e A( )+B( ) Y. 11

12 he proofs of lemma 9 and 1 can be found in he appendix. We say (α, β) is Q-regular if α, β and he parameers (Θ, K, a, b) governing he dynamics of Y under Q saisfy he condiions of lemma 9, and if he condiions of lemma 1 are saisfied, we say (α, β, ζ, ξ) is Q-regular. Remark 11. (i) he following relaionship is useful for applicaions and follows immediaely from lemma 1. If (α, β, ζ = α, ξ = β) is Q-regular, hen E e R α+β Y sds (α+β Y ) F = ea( )+B( ) Y i.e. A and B are he derivaives of A and B wih respec o ime. (ii) he resul of of lemma 1 can be generalized o values of ζ R and ξ R N, bu hen one has o impose a condiion which is more difficul o check han E exp{ 1 w( ) B Y + B 2 A +B d} <. Y Duffie and Singleon (1999) direcly calculae expecaions of he form E e R α+β Y sds+γ Y (ζ +ξ Y ) using differeniaion hrough he inegral. We believe we have found a more naural way o look a his, namely we choose φ = S γ and Φ = 1 in Girsanov s heorem 5. hen L = e γ (Y Y ) R Θ KYsds 1 R 2 v(γ)+w(γ) Y sds and (see also proposiion 3) Y is also an AD under Q. hus we can always find a measure Q Q under which Y remains an AD and α, β such ha E e R α+β Y sds+γ Y (ζ +ξ Y ) F = e γ Y Ẽ e R eα+e β Y sds (ζ +ξ Y ) F, hence lemma 9 or 1 apply again. In he sequel of his paper he circuious deour described above o find α, β in order o derive he respecive expecaions will never be necessary. he maringale L, raher han he facor e γ Y, will have an imporan financial inerpreaion. 4 Fixed Income Securiies in Differen Currencies In his secion we discuss how o deermine he prices of defaul-sensiive foreign currency insrumens such as bonds and CDSs denominaed in differen currencies. I is wellknown ha wih he change of numeraire mehod he problem of pricing foreign currency claims a fixed imes can be reduced o a relaed domesic currency pricing problem. he recovery of defaulable bonds or he proecion paymens of CDSs, however, are payoffs a sopping imes which require a modificaion of hese echniques, in paricular when he value process of he numeraire (and hus he densiy of he change of measure) is disconinuous a he sopping ime. he general form of his modificaion is given in lemma 7, here we apply his modificaion o he case of foreign currency claims payable a defaul and ransform hese pricing problems ino an equivalen domesic currency pricing problem, which is hen solved easily using he AJD-specificaion of he model. All noaion from secion 3 is carried over o his par of he paper. 12

13 4.1 Defaul Inensiies under Domesic and Foreign Maringale Measures he model for foreign exchange risk in he presence of defaul risk is se up as follows: Assumpion 12 (he Defaulable FX Model). (i) he marke is modelled by he filered probabiliy space (Ω, F, F, Q) defined in 3.1, where Q is a domesic spo maringale measure (DSMM) 1, and F is he informaion of he marke o which all processes are adaped. (ii) he ime of defaul of he obligor is he sopping ime τ defined in (1). i.e. he ime of he firs jump of N(). Given defaul, he severiy of defaul is characerized by he realizaion of he marker z τ := J τ of he marked poin process. (iii) he exchange (FX) rae beween foreign currency c f and domesic currency c d is denoed wih X. 11 r i are he shor-erm ineres raes and b i () := e R r i(s)ds insananeous bank accouns in he respecive currencies c i, i = d, f. We someimes wrie Q d insead of Q when we find i necessary o emphasize he fac ha Q is he domesic SMM, or λ d insead of λ for he defaul inensiy under Q d. 12 Q d does no need o be unique, we only assume i is he pricing measure chosen by he marke. From assumpion 12 (i) and (iii) i follows direcly ha X saisfies a SDE of he form dx X = (r d r f )()d + φ X () dw δ(z, )(µ ν)(dz, d) (1) where φ X is predicable process, and δ(, ) 1 is a predicable funcion. o see his noe b ha X f () is he discouned value in c b d () d of a foreign bank accoun and hence needs o be a Q-local maringale. Furhermore our probabiliy space has he predicable represenaion propery and X mus be a nonnegaive process. Regarding he drif erm in (1) we say X saisfies he FX drif resricion under Q. he dependency beween defauls and he movemens of he exchange rae X has imporan consequences for he dynamics of he model under he pricing measures ha we will inroduce in he following. Here, equaion (1) has wo implicaions. Firs, FX rae and defaul inensiy may be condiionally correlaed, if λ, X is no idenically zero. If for example we have posiive local correlaion, an increasing (decreasing) FX rae will indicae a rise (lowering) in he defaul inensiy. Second, here is a direc jump-influence from he defaul even iself on he exchange rae which is capured in he funcion δ via dn, X = X δ(z, )µ(dz, d) or X τ = X τ δ(z τ, ). A defaul τ, he foreign currency c f is devaluaed (relaive o c d ) in a jump of a fracion δ(z τ, τ) of he pre-defaul value of X. As defauls are he only jumps in his model, here is no disconinuiy in X before τ. 1 I.e. every raded asse (denominaed in domesic currency) is a Q-local maringale. 11 I.e. X is he value a ime of one uni of c f, expressed in unis of c d. 12 We also someimes call λ iself defaul inensiy, omiing he indicaor funcion in he mahemaically correc expression 1 { τ} λ(). 13

14 Of course no only X and λ bu also r d, r f, λ, X may show muual correlaion and r d, r f migh also jump a τ. However, in his paper we mainly focus on he dependence of FX and credi risk, and we will no rea hese addiional feaures in deail. For echnical reasons, we make he following assumpion. Assumpion 13 (FX Maringale Propery). he discouned value in c d of a foreign bank accoun L := X b f () X b d () is uniformly inegrable, i.e. a rue maringale under Q, and L > Q-a.s. Saring from a DSMM Q, he following pricing measure is commonly used o analyze FXrelaed insrumens when assumpion 13 is saisfied (see Musiela and Rukowski (1997)). Definiion 1 (FSMM). he equivalen measure Q f Q d on (Ω, F) defined by dq f dq d := L = X b f () F X b d () is called he foreign spo maringale measure (FSMM) induced by Q d and X. 13 he defaul inensiy under Q f is denoed wih λ f. he FSMM is useful for pricing foreign currency payoffs because he price in c d a ime of an insrumen ha pays unis of c f a a fixed ime (i.e. X unis of c d ) is p() = E Q d e R rd(s)ds X F = X E Q f e R r f (s)ds F, (11) i.e. E Q f e R r f (s)ds F is he price of his insrumen in c f. hus, in order o price a foreign currency coningen claim a fixed imes, only he disribuional properies of foreign ineres raes and he considered claim under he FSMM are needed. If he FX rae X (and so he Radon-Nikodym densiy dq f dq d ) jumps a defaul, hen, in general, he domesic and he foreign defaul inensiy do no coincide, and hus, defauls occur wih differen probabiliies under he FSMM and under he DSMM. As was already shown in corollary 6, λ f = λ d if and only if N, L τ. Here, given he paricular naure of our numeraire asse, more can be said abou he link beween he devaluaion fracion of he currency and he wo defaul inensiies: Proposiion 2 (FSMM Defaul Inensiy). Le assumpions 12 and 13 hold. Define he locally expeced devaluaion fracion δ() a ime (under Q d, condiional on defaul occurring a τ = ) δ() := 1 { τ} δ(z, )df (z). (12) hen, he defaul inensiy under he foreign spo maringale measure (FSMM) equals λ f () = ( 1 δ() ) λ d (). 13 We emphasize ha every DSMM induces anoher associaed FSMM. If Q d is unique, hen so is Q f. 14

15 Proof. Subsiue (1) in heorem 5. Inuiively, he adjusmen facor beween local and foreign defaul inensiy is equal o he locally expeced devaluaion of he FX rae X, if a defaul were o happen a ime : λ f () λ d () = 1 δ() = EQ X() τ =, F E Q X() τ >, F. o specify a racable model ha can be esimaed saisically we reurn o an AJD framework and use he canonical represenaive Y A m (N) defined in (7) as an observable background process driving he coninuous dynamics of he oher variables in he marke. Assumpion 14 (AJD Economy). Le Y be as in (7) wih admissible parameers. (i) Defaul inensiy and ineres raes are affine in Y, i.e. here exis α i R, β i R N, i = d, f and α λ, β λ R m + {} N m 14 such ha r i = α i + β i Y, i = d, f and λ = α λ + β λ Y. (ii) here exis γ R N and x R K wih L(; x) < Q-a.s. for all such ha in he FX dynamics (1) φ X () = S γ, and δ(z, ) = δ(z) = 1 (1 z) x. Occasionally, we prefer o wrie α λd and β λd insead of α λ and β λ. Under assumpion 14 he covariaion beween inensiy and exchange rae λ, X is compleely deermined by he inner produc γ Sβ λ because dλ, X = X γ Sβ λ d. As we did no make any assumpions on he condiional disribuion F (z) of he jump severiy, he assumpion regarding δ(, ) is no resricive and was only chosen in order o express δ() = 1 L(; x) wih he Laplace ransform L(; x). he following proposiion provides he relaions beween he domesic and he foreign spo maringale measure in an AJD framework. Proposiion 3 (AJD FX Rae). Le assumpion 14 be saisfied. hen (i) Y saisfies he SDE dy = (Θ f K f Y )d + S dw f, Y () = Y, where Θ f = Θ + diag(γ)a, K f = K diag(γ)b and W f is a BM under Q f, i.e. Y is also an AD under Q f. (ii) he defaul inensiy under Q f is given by λ f () = L(; x)λ(), (iii) and he sochasic Laplace ransform of he defaul severiy disribuion under Q f saisfies L(; u+x) L f (; u) =. L(; x) 14 his ensures nonnegaiviy of λ. 15

16 (iv) λ f is an AD under Q f, if and only if L(; x) = L(x) < is ime-invarian. 15 In his case we define consan coefficiens α λf := L(x)α λ and β λf := L(x)β λ wih λ f = α λf + β λ f Y, and we call X an affine FX rae. Proof. Apply Girsanov s heorem 5 o our AJD framework. Mean reversion speed and level of Y (and hus of λ, r d, r f ) ransform under he changes of measure considered in proposiion 3, whereas he parameers a and b governing he volailiy of Y are clearly invarian. Noe ha he parameer resricions for canonical ADs given in secion 3.4 are auomaically saisfied for K f and he m firs componens of Θ f, bu in general no for he N m las componens of Θ f. Of course (Θ f, K f, a, b) is an admissible parameer se, bu i does no necessarily belong o a canonical represenaive of A m (N). Furher he inequaliies K fii >, i = 1,..., N may be violaed, i.e. we migh lose mean reversion under he FSMM, which would no be economically meaningful. In applicaions one should check ha hese inequaliies hold. Remark 15. (i) In general i is difficul o check he validiy of assumpion 13. In he AJD case however, he maringale propery of L is ensured if he following condiion holds (see e.g. Lepingle and Memin (1978)). Le X be as in proposiion 3, and E exp{ 1 ( w(γ) Y d + (x log(1 z) (1 z) x 1)dF (z) ) λ()d} < 2 For x, i.e. if he foreign currency can only be devaluaed a defaul, i suffices o check he Novikov condiion E e 1 R 2 w(γ) Y d <. (ii) Proposiion 3 remains rue for all sufficienly regular 16 ineres raes. 4.2 Basic Defaul-Free and Defaul-Sensiive Insrumens We urn o he pricing of domesic and foreign securiies which may be sensiive o boh, ime and severiy of defaul. Basically one has o disinguish wo ypes of defaul-sensiive insrumens. Firs, payoffs upon survival unil a fixed mauriy (e.g. a defaulable bond wih zero recovery), and second, insrumens wih paymens ha become due a defaul (e.g. recovery paymens or proecion paymens in CDS). As hypohesis H underlies our probabiliy space, we can always reduce he pricing problem of a defaulable claim o a relaed defaul-free pricing problem, following he ools provided in secion 3.2. Bu firs we address he well-known problem of pricing defaul-free zero coupon bonds (CBs) Defaul-Free ero Coupon Bonds If he ineres rae coefficiens (α i, β i ) are Q i -regular for i = d, f, hen, according o (11) wih = 1 and lemma 9, domesic and foreign defaul-free CB prices exis for all 15 F = cons. is a sufficien bu no a necessary condiion for his. 16 E.g. b i () < Q-a.s. for i = d, f. 16

17 and are given (in heir respecive payoff currency) by B i (, ) = E Q i e R r i(s)ds F = e A i( )+B i ( ) Y, where A i, B i solve (8) wih ( α i, β i, Θ i, K i, a, b) for i = d, f. In he sequel of he aricle, we will always assume ha defaul free CB prices exis. If we do no allow for negaive ineres raes, i.e. α i and β i R m {} N m for i = d, f, hen he exisence of CB prices is immediae. However, in ha case λ can only have nonnegaive correlaion wih r i (because hen β λ Sβ i ), which is no always a desirable propery (see e.g. Duffee (1998) for empirical evidence of negaive correlaion). Assumpion 16 (Recoveries). Generally, we model he loss given defaul (of a c i - bond) wih a predicable, 1-valued funcion q i (z, ) 17 which capures he dependency of recovery on he defaul severiy marker z. In he AJD framework we assume ha he recoveries can be wrien as 1 q i (z) := (1 z) u i for some fixed u i R K +. We also assume L i(; u) = L i (u) for all. In combinaion wih assumpion 14 (ii) on he devaluaion fracion, he assumpion on he funcional form of q(z, ) is resricive, bu i preserves he affine srucure. Despie his, using a higher-dimensional marker space we sill have a large degree of flexibiliy in he modelling of he correlaion beween loss given defaul q and FX devaluaion δ Defaulable ero Coupon Bonds We presen he prices of CB under a variey of recovery assumpions ha have been proposed in he lieraure: ero Recovery. A c i -CB wih zero recovery (R) pays 1 { >τ} unis of c i a is mauriy, i = d, f. By equaion (11) wih = 1 { >τ}, he price of domesic and foreign CB wih R is B i (, ) := E Q i e R ri(s)ds 1 { >τ} F, i = d, f in he respecive payoff currency. Using hypohesis H and lemma 2, he survival funcion 1 { >τ} can be replaced by 1 {>τ} e R λ i (s)ds in he above expecaion. hen lemma 9 yields for he AJD seup B i (, ) = 1 {>τ} E Q i e R (r i+λ i )(s)ds F = 1 {>τ} e A i( )+B i ( ) Y, (13) where A i and B i solve (8) wih ( α i, β i, Θ i, K i, a, b) where α i := α i + α λi and β i = β i + β λi, i = d, f. Noe ha no only are he payoffs in differen currencies discouned wih differen ineres raes, bu also wih differen defaul inensiies. Any posiive recovery is a paymen a a sopping ime, and hus, is value is equal o he expecaion of a sochasic inegral wr. he jump measure µ. his renders he pricing problem a bi more complicaed. We consider he following recovery assumpions. 17 he recovery rae 1 q of a foreign-issued bond is in general no equal o ha of a domesic bond because he respecive bankrupcy cours use differen legal rules. 17

18 Recovery of Par. (See e.g. Duffie (1998).) In addiion o he survival payoff of 1 { >τ} unis of c i a mauriy, a defaulable CB pays 1 q i (z τ, τ) unis of c i a he defaul ime τ if τ in he recovery of par seing (RP). his (he recovery paymen) can be wrien as ( 1 qi (z, ) ) 1 { τ} µ(dz, d). he no-arbirage price of he domesic bond is hus B RP d (, ) = B d (, ) + E Q d e ( R s r d(v)dv 1 q d (z, s) ) 1 {s τ} µ(dz, ds) F on he se { < τ}. he valuaion of he foreign recovery paymen is slighly more involved. In unis of c d, he foreign recovery paymen is ( )( X 1 δ(z, ) 1 qf (z, ) ) 1 { τ} µ(dz, d). he sandard formula (11) o eliminae X by changing o he FSMM is no direcly applicable here because he paymen akes place a a sopping ime. We use lemma 7 insead, again on { < τ}. Assume ha L and f(z, ) := e R r f (v)dv (1 q f (z, ))1 { τ} saisfy he condiions of lemma hen E Q d e R s ( )( rd(v)dv X s 1 δ(z, s) 1 qf (z, s) ) 1 {s τ} µ(dz, ds) F = X E Q d = X E Q f L s ( ) R ( 1 δ(z, s) e s r f (v)dv 1 q f (z, s) ) 1 {s τ} µ(dz, ds) L F ( r f (v)dv 1 q f (z, s) ) 1 {s τ} µ(dz, ds) F. (14) e R s he imporan difference o a naive applicaion of (11) is, ha no jus he predicable par of he exchange rae X, bu also he jump erm 1 δ(z, s) is removed in he change of measure. Furher noe ha pricing equaion (14) remains rue for all regular 19 payoff funcions p(z, ) insead of 1 q f. Hence (14) is an equivalen o (11) for he valuaion of foreign currency paymens a defaul. hen, by corollary 4 and lemma 1 on { < τ} B RP i (, ) = B i (, ) + E Q i e R s ri(v)dv (1 z) u i 1 {s τ} µ(dz, ds) = B i (, ) + L i (u i ) = B i (, ) + L i (u i ) E Q i e R s (r i+λ i )(v)dv λ i (s) F ds F ( Ai (s )+B i (s ) Y ) e A i (s )+B i (s ) Y ds where A i, B i solve (9) wih ( α i, β i, ζ i := α λi, ξ i := β λi, Θ i, K i, a, b). Noe ha by remark 11 he exisence of E Q i e R s (r i+λ i )(v)dv λ i (s) F is immediae if ri. 18 If r f, hen he process h = lemma 7 are already saisfied, when L is a square-inegrable maringale. 19 Again, he condiions of lemma 7 mus be saisfied. f(z, s)µ(dz, ds) is bounded by 1. In his case he condiions of 18

19 Remark 17. A special case in he RP seing is obained when u i =. Such a CB pays 1 uni of c i a τ 2 and we denoe is price by B i (, ). Muliple Defauls. (See e.g. Schönbucher (1998) or Duffie e al. (23).) A muliple defaul (MD) CB wih mauriy has he payoff p MD i ( ) unis of c i a which solves he SDE dp MD i () ( ) = q i (z, )µ(dz, d), p MD i () = 1. p MD i An MD bond can defaul more han once and q i is he loss fracion a a defaul. he value p MD i () can be seen as he remaining promised payoff a ime. In order o derive closed form soluions of CB prices in he MD seing, knowledge of he compensaor ν afer τ is needed. ha is he disribuional properies of he sopping imes τ j := inf{ > τ j 1 ; N > N τj 1 } (i.e. he ime of he jh defaul) and he condiional disribuion funcion of he jh jump size F τj (z) (given he jh defaul occurs). Hence addiional assumpions have o made. Assumpion 18 (MD). he predicable compensaor of µ under Q d is of he form ν(dz, d) = df (z)λ()d, where F is a ime-invarian disribuion funcion on and he loss fracions a each defaul are q i (z) = 1 (1 z) u i. Under assumpion 18, N is a Cox process and hus B MD i (, ) = E Q i e R ri(s)ds p MD i ( ) F where A MD i L i (u i )α λi and B MD i and β MD i = p MD i () e AMD i ( )+Bi MD ( ) Y, (15) solve (8) wih ( α MD i, β MD i, Θ i, K i, a, b), where α MD := β i + L i (u i )β λi, i = d, f. (A proof can be found in he appendix.) i := α i + Recovery of reasury. (See e.g. Jarrow and urnbull (1995).) If a defaul occurs before mauriy (τ ), a CB holder receives under recovery of reasury (R) ( 1 qi (z, ) ) 1 { τ} µ(dz, d) (16) defaul-free CBs wih he same mauriy and par value 1c i. By a simple hedging argumen, he value of he recovery is equal o ha of receiving (1 q i (z τ, τ))1 {τ< } unis of c i a ime. Using he domesic and foreign -forward measures dp i dq i := L i () := B i(, ) for i = d, f, F B i (, )b i () 2 his is also rue for arbirary u i if F () = 1 Q-a.s. for all 19

20 we can always reduce he R case o a relaed RP case 21, namely on { < τ} B R i (, ) = B i (, ) + E Q i = B i (, ) + B i (, ) E P i e ( R s r i(v)dv 1 q i (z, s) ) B i (s, )1 {s τ} µ(dz, d) F ( 1 qi (z, s) ) 1 {s τ} µ(dz, d) F. (17) If we posulae again 1 q i = (1 z) u i, hen his leads o a AJD pricing problem wih deerminisic bu ime-dependen coefficiens (Θ i (), K i (), a, b). We will no furher rea he ime-heerogeneous case in his paper Credi Defaul Swaps he cash flows involved in a CDS conrac were already described in secion 2. Afer defaul a CDS conrac is unwound, hus we always assume ha defaul has no ye occurred ( < τ) in order o avoid rivialiies. We also assume he amoun of noional insured is always 1 uni of c i, i = d, f. For a conrac wih mauriy enered a, he fee leg hen consiss of paymens s i (, ) ( j j 1 ) in c i a quarerly daes j. We approximae his paymen sream by an inegral. he Value of he Fee Sream (in is payoff currency) is hus given by V fee i (, ) = s i (, ) B i (, s)ds = s i (, ) E Q i e R s (r i+λ i )(v)dv F ds. he fee sream of a CDS can be inerpreed as a defaulable coupon bond wih coninuously paid coupon s i (, )ds and par value zero. Value of Proecion Leg. he proecion paymen of a CDS akes place if and only if τ. In his case i is made a he ime of defaul τ, and is size is 1 B cd (τ) unis of c i, where B cd is he price of he cheapes-o-deliver bond. As argued in secion 2, he cheapes-o-deliver bond will usually be a coupon-bearing bond and he only relevan quaniy is is relaive discoun o par value in he currency in which i was issued. We model his using he RP seup which is he mos appropriae choice for CDS recovery modelling (see e.g. Houweling and Vors (25)). In his case, we can wrie q(z τ, τ) = 1 B cd (τ) = 1 (1 z τ ) u cd. If e.g. Li (; u cd ) = L i (u cd ) is deerminisic and imeindependen, hen he value of he proecion leg saisfies V pro i (, ) = ( 1 L i (u cd ) ) E Q i e R s (r i+λ i )(v)dv λ i (s) F ds. (18) he fair CDS rae s i (, ) is obained when he value of fee and proecion leg are equal: s i (, ) = V pro i (, ) B i (, s)ds. (19) 21 Firs (14) shows ha (17) is also valid for i = f. hen lemma 7 wih L i applies again because L i is coninuous a τ. Moreover heorem 5 yields ha Y has he coefficiens Θ i () := Θ i + diag ( B i ( ) ) a and K i () := K i diag ( B i ( ) ) b under he respecive measure P i. 2

21 I is no hard o see ha lim s i (, ) = ( 1 L i (; u cd ) ) λ i () Q i -a.s. (2) his limiing propery remains rue when L i (; u cd ) is a righ-coninuous sochasic process. 4.3 he relaionship beween domesic and foreign CDS raes In his subsecion we wan o achieve some inuiion regarding he relaionships beween domesic and foreign CDS raes s d and s f. We will idenify he hree componens of credi risk (defaul inensiy risk, defaul even risk and defaul severiy risk) and he defaul-free erm srucures of ineres raes as he driving facors of his relaion. For simpliciy we assume ha he expeced recovery raes remain consan over ime as in (18) for boh currencies and wrie q i := 1 L i (u cd ), i = d, f. Noe ha if defaul occurs, here will be one unique recovery rae o boh CDS due o he proecion buyer s delivery opion discussed on secion 2. he quaniies L i (u cd ), however, may differ from each oher because hey are he expecaions of his recovery rae under differen measures. Mahemaically his becomes clear from he definiion of F in (ii) of Girsanov s heorem 5 or from (iii) in proposiion 3. From an economic poin of view his phenomenon is explained by he possible dependence of recovery and devaluaion fracion of he foreign currency a defaul. E.g. if recovery and devaluaion show negaive correlaion, hen he proecion buyer of a foreign CDS is doubly punished. In a ligh defaul scenario, i.e. when recovery is large, he LGD payed o he proecion buyer will be small. In he case of a severe defaul, i.e. when recovery is small, hen he LGD amoun in foreign currency will be large, bu is value in domesic currency will ypically be reduced by a high devaluaion fracion. Clearly, when he recovery rae (or/and he devaluaion fracion) is consan, hen q d = q f := q. 22 In he sequel we assume such a q exiss bu one could easily include he consideraions concerning recovery risk (=defaul severiy risk) in he subsequen discussion. he value of he proecion leg in (18) was expressed as an inegral over securiies wih a payoff in unis of defaulable CBs. he following probabiliy measures are ofen used in his siuaion (see Schönbucher 1999, 24). P d and P f dp i dq i := L i () := B i(, ) F B i (, )b i (), i = d, f are called he domesic and he foreign -survival measure. 23 Furher we define he erm srucures of defaul inensiies λ i (, s) := E Ps i λi (s) F and defaulable weighs w i (s;, ) := B i(,s). hen (19) simplifies o R B i (,s)ds s i (, ) = q w i (s;, ) λ i (, s)ds. (21) 22 his is also rue when recovery and devaluaion are independen, given defaul occurs. 23 We assume L i is a Q i -maringale. 21

22 he weighs w i are proporional o he defaulable CB price for he corresponding mauriies. Overall, he price curve of defaulable CBs will be downward-sloping. he currency wih he higher level of defaulable ineres raes will have a sronger downward slope in he defaulable CB price B i (, s) and hus i will have a higher weigh on early (small s) values of λ i (, s), compared o he currency wih a lower level of defaulable ineres raes. In many cases he fac ha we have differen weighs w d and w f will only have a small influence on he differences in CDS raes, because he weighs w i will no differ by much, and he erm-srucure of defaul inensiies will be quie fla (i.e. λ i (, s) is close o a consan funcion of s). For fla erm srucures of λ i (, s) he weighing has no influence a all; and if he slope is small, he influence of slighly varying weighs will also be small. In hese cases we may argue ha s i () qλ i (). I remains o analyze he difference beween he domesic and foreign erm srucure of defaul inensiies. Assume ha X is an affine FX rae. hen δ := 1 L(x), he expeced devaluaion fracion of X a defaul (under Q d and given defaul occurs), is a consan and λ f () = (1 δ)λ d (). Furher, chaining densiies we can define he P d -maringale L df () := dp f dp = L X () L f () d F L d () = X B f (, )/B d (, ) X B f (, )/B d (, ). I should be noed ha when domesic and foreign CB price do no differ by much, hen L df is almos proporional o he FX rae X. In any case s L λ f (, s) = (1 δ) E Ps df (s) d L s df () λ d(s) F. hus, differen erm srucures of defaul inensiies may arise for wo reasons. Firs, when domesic and foreign defaul inensiies are no equal, i.e. when here is a non-zero expeced devaluaion fracion δ. Second, when L s df and he domesic defaul inensiy λ d (s) are correlaed under he domesic s-survival measure P s d. Overall we have idenified four drivers of he difference beween CDS raes in domesic and foreign currency. Firs, foreign and domesic defaul inensiies are no equal, when he FX rae is subjec o defaul even risk, i.e. when X jumps a defaul. 24 Second, foreign and domesic erm srucure of defaul inensiies do no coincide (nor are hey proporional) when here is covariance beween λ d and he P s d -maringale Ls df under he domesic survival measure P s d a any ime s. I can be shown ha hese erm srucures are equal, up o he muliplicaive consan (1 δ), when he defaul-free ineres raes r d, r f are independen of λ and λ, X =, i.e. when X is no subjec o defaul inensiy risk. 25 hird, he expeced LGD q i may differ beween he pricing measures Q d and Q f when devaluaion is no independen of defaul severiy risk. Fourh, he slopes of domesic and foreign erm srucure of defaul free ineres raes deermine (via heir impac on he weighs w i ) how he respecive erm srucures of defaul inensiies mus be weighed in order o derive he CDS rae in each currency. 24 Unless he expeced devaluaion fracion is equal o zero. 25 Precisely, we mean hey have orhogonal volailiies (λ, X = ). 22

23 5 Empirical Resuls Here we focus on he firs wo drivers. here is heoreical and empirical evidence ha he correlaion beween defaul-free ineres-raes and defaul inensiies has only a very small effec on CDS raes (see e.g. Houweling and Vors (25) or Schönbucher (22)). Regarding he relaive prices of CDS, his effec is likely o be even smaller here, so we feel jusified in ignoring his effec. Furhermore, we will also assume ha he correlaion beween recovery rae and devaluaion fracion is no significan. Assumpion 19 (Empirical Esimaion Seup). (i) We assume ha domesic and foreign ineres-raes have zero covariaion wih exchange rae and defaul inensiy: r d, X = = r d, λ and r f, X = = r f, λ. (ii) he cheapes-o-deliver bond of a CDS has a consan LGD rae q and, a defaul, he FX rae is devaluaed by a consan fracion δ. By virue of assumpion 19 (i) we do no have o model an affine model for he defaul-free ineres-raes any more and we direcly use he curren 1M Libor raes as approximaion for he shor rae in he FX drif (23) and he curren erm-srucure of ineres-raes o discoun fuure cash-flows. As daa source for CDS quoes we use he ValuSpread CDS daabase by Lombard Risk Sysems Ld., a provider of a daa pooling service for he CDS marke. 26 he daa sars in he hird quarer of A he beginning of he period he se of CDSs on he same reference eniy which are available in more han one currency is relaively sparse and he daa frequency is only monhly bu from 22 onwards he daa qualiy improves significanly boh in frequency (weekly, hen daily from 23) and in he number of obligors wih CDS raes in boh JPY and USD. From he available se of Japanese reference names we seleced he 25 obligors wih he larges number of simulaneous daa poins in boh JPY and USD. For defaul-free ineres-raes we used JPY and USD ermsrucures of ineres-raes based upon Bloomberg swap and money-marke daa. he JPY/USD exchange rae daa was also aken from Bloomberg. he snapshos in figure 5 ac as a good example for he magniude of he spread beween domesic and foreign CDS raes in our daase. JPY CDS raes on many large Japanese reference eniies rade ypically around 2 percen lower han heir US$ equivalen. 5.1 A Simple Esimaor for he Devaluaion Fracion As a firs sep, we esed for he presence of significan differences beween JPY and USD CDS raes used a simple saisic based upon he limiing propery (2), which saes ha 26 A regular (daily) inervals, Lombard Risk Sysems collecs CDS quoes on a large number of reference names from a se of major marke makers and dealers. he submied daa is cleaned and averaged and hen added o he ValuSpread CDS daa base and disribued back o he original daa providers and o oher subscribers of he service who use his daa o mark heir books and for various risk managemen purposes. 23

24 6 5 4 US$ CDS JPY CDS Spread 4 3 US$ CDS JPY CDS Spread Figure 1: 5y CDS raes on Sony (lef) and okyo Elecric Power US$ and JPY. he mean spread amouns 5.4 bp for Sony and 3.6 bp for okyo Elecric Power. CDS raes for shor mauriies are approximaely equal o LGD imes defaul inensiy. Remembering λ f = (1 δ)λ d (proposiion 2), he following (approximae) esimaor for δ is sraighforward. δ k := 1 n n i=1 { 1 s } Y ( i, i +k), k >. (22) s $ ( i, i +k) Apar from he limi argumen (2), we will see anoher argumen why his esimaor can be inerpreed as an implied devaluaion fracion in equaion (28). We give he esimaes 1 year CDS 5 year CDS icker (Company) δ1 % p-value % δ5 % p-value % ASAGLA (Asahi Glass Company, Limied) BO1 (Bank of okyo-misubishi, Ld.) EJRAIL (Eas Japan Railway Company) FUJIS1 (Fujisu Ld) HIACH (Hiachi, Ld.) HONDA (Honda Moor Co., Ld.) MASEL (Masushia Elecric Indusrial Co., Ld.) MICO1 (Misubishi Corp) MISCO (Misui & Co., Ld.) NECORP (NEC Corpoarion) NIPOIL/NIPOIL1 (Nippon Oil Corporaion) NIPSL (Nippon Seel Corporaion) N (Nippon elegraph & elephone Corporaion) NDCM (N DoCoMo Inc.) ORIX (Orix Corporaion) SHARP (Sharp Corporaion) SNE (Sony Corporaion) SUMIBK2 (Sumiomo Misui Banking Corporaion) SUM (Sumiomo Corporaion) AKFUJ (akefuji Corporaion) OKELP (okyo Elecric Power Co., Inc.) OKIO (okio Marine and Fire Insurance Company Limied) OSH (oshiba Corporaion) OYOA1 (oyoa Moor Corporaion) YAMAHA (Yamaha Moor Co., Ld.) able 1: Esimaes for he devaluaion fracion implied by he daa. δ k for he mauriies k = 1 years and k = 5 years in able 1. he 1Y mauriy is he shores mauriy available o us, i is repored in he firs pair of columns. We also added he 5Y mauriy because marke liquidiy is usually concenraed a his poin of he CDS erm srucure. Using a -es on 1 s Y /s $ we esed he hypohesis wheher 24

25 he deviaions of he CDS raio s Y /s $ from 1 is jus noise in he daa. In all cases his hypohesis could be clearly rejeced: We are facing a sysemaic feaure of he daa. Ineresingly he devaluaion fracions implied by he daa is ypically higher for he 5 CDS raes han for he 1 year CDS raes. 5.2 A Correlaion Model he differences beween JPY and USD CDS raes repored in able 1 need no necessarily be caused by an implied devaluaion of he JPY a defaul (i.e. a δ > ). As seen above, a difference beween CDS raes in differen currencies can also arise when defaul inensiies and FX rae are correlaed via heir diffusion componens. hus, we now wan o invesigae wheher i is possible o reproduce he observed differences in CDS raes wihou assuming a devaluaion a defaul, bu only using he dependency beween defauls and exchange rae X ha is generaed in a purely diffusion-based seup. We build up a concree model for he observable background driving process Y using he classificaion of ADs ino he families A m (N). Because assumpion 19 relieves us from modelling defaul-free ineres-raes, we are lef wih he ask of modelling he correlaion srucure of wo variables: λ and he diffusion par of X, hence we need a dimension of a leas N 2. Second we have o choose he number m N of componens of Y ha we wan o remain posiive. In our case λ > is desirable, hence we choose m 1. hus, an A 1 (2) or an A 2 (2) model is appropriae. We chose Y A 1 (2) because for fixed N, he number of parameer resricions is increasing in m (see 3.4), and if Y A 2 (2), hen he wo componens driving he sochasic volailiy of Y, and hus of λ and X, can only have nonnegaive correlaion. (See Dai and Singleon (2)) hen λ is a CIR process up o a posiive addiive consan. We resric his consan and he marix b defined in (7) o zero. his yields he following model under he DSMM Q d. dλ = κ(θ λ)d + σ λ dw 1 dx X = (r d r f )d + γ 1 λ dw 1 + γ 2 dw 2 δ(dn λd) (23) wih κ, θ, σ, γ 2 >, γ 1 R and δ < 1 and (W 1, W 2 ) a sandard BM under Q d. he insananeous correlaion of defaul inensiy and log FX rae, ρ(log X c, λ) := d ( ) dlog Xc, λ 1 = sgn(γ 1 ) 1 + γ2 2 2, d log d Xc d λ γ1 2λ d is essenially conrolled by he raio γ 1 /γ 2, bu also depends on he curren level of he sochasic process λ. Imporanly is sign depends only on he sign of γ 1. Furher we have o link he DSMM o he physical measure P, under which he daa was generaed. For racabiliy we assume ha he sae price densiy is of he form dp dq := L wih F dl L = φ 1 λ dw 1 + φ 2 dw 2 Φ(dN λd) for φ 1, φ 2 R and Φ < 1. hen Y is also an AD A 1 (2) under P. 25

26 In order o esimae he parameers of an A m (N) model a large number of relaively demanding esimaion mehodologies have been proposed for non-gaussian AD models (Singleon (21), Ai-Sahalia (22), Gallan and auchen (1996)). Given he simpliciy of our model and he high frequency of our daa (daily for mos of he daase) we used a simple quasi maximum likelihood (QML) esimaor by approximaing (23) wih is Euler discreisaion. hen he parameer esimaion reduces o a linear regression problem in ransformed variables. Noe ha for every volailiy esimaor of he above model he value of δ plays no role as long as defaul has no ye occurred. As he inensiy λ is no direcly quoed in he markes, we need o find a proxy for i. Here, he limiing propery (2) suggess ha CDS raes wih a shor mauriy are approximaely proporional o λ, and equaion (21) ells us ha also for longer imes o mauriy, CDS raes are proporional o a weighed average of forward hazard raes. herefore, we decided o use CDS raes as approximaion for qλ d (). Ideally, we would have liked o use 1Y CDS raes, bu he daa for he 5Y mauriy urned ou o be cleaner and more liquid, so we used 5Y USD CDS raes, s d (, + 5). If q is unknown, hen γ 1 can only be esimaed up o a posiive consan. herefore, we give QML esimaes and 95%-confidence inervals for γ 1 in able 2. However, he qσ icker (Company) ( γ 1 qσ ) 95%-CI ρ(log X c, λ) (mean, %) ASAGLA (Asahi Glass Company Ld.) , BO1 (Bank of okyo-misubishi Ld.) , EJRAIL (Eas Japan Railway Co.) , FUJIS1 (Fujisu Ld) , HIACH (Hiachi Ld.) , HONDA (Honda Moor Co. Ld.) , MASEL (Masushia Elecric Indusrial Co. Ld.) , MICO1 (Misubishi Corp.) , MISCO (Misui & Co. Ld.) , NECORP (NEC Corp.) , NIPOIL1/NIPOIL (Nippon Oil Corp.) , NIPSL (Nippon Seel Corp.) , N (Nippon elegraph & elephone Corp.) , NDCM (N DoCoMo Inc.) , ORIX (Orix Corp.) , SHARP (Sharp Corp.) , SNE (Sony Corp.) , SUMIBK2 (Sumiomo Misui Banking Corp.) , SUM (Sumiomo Corp.) , AKFUJ (akefuji Corp.).7.48, OKELP (okyo Elecric Power Co. Inc.) , OKIO (okyo Marine & Fire Insurance Co. Ld.) , OSH (oshiba Corp.) , OYOA1 (oyoa Moor Corp.) , YAMAHA (Yamaha Moor Co., Ld.) , able 2: QLM esimaes of he Correlaion Parameer and Averaged Correlaion level of correlaion seems o be raher low: For only four companies, MISCO, NIPSL, SHARP and SUM, he null hypohesis H : γ 1 = can be rejeced on he 95% level. o provide a quaniy which is more inuiively undersandable, we also compued he average insananeous correlaion funcion ρ(log X c, λ) by combining he esimaes for γ 1 qσ wih he esimaes of γ 2 and σ. his quaniy can be viewed as he average correlaion beween defaul inensiy and exchange rae over our sample period. 26

27 5.3 Evidence for Devaluaion a Defaul As we oulined in 4.3, γ 1 if and only if X, λ, which implies differen erm srucures of defaul inensiies in USD and JPY. We now aemp o answer he quesion wheher his purely correlaion based dependence is able o explain he whole spread beween USD and JPY CDS raes, i.e. he quesion wheher we can we can make our life much easier and se he jump ineracion parameer δ o zero? Formally, our null hypohesis is: he difference beween JPY and USD CDS spreads is caused by a model like (23) wih no devaluaion a defaul, i.e. δ =. In order o evaluae he Null, we mus firs compleely specify he model (23). Given δ =, and for a given value of he loss given defaul q (which is also provided in our CDS daabase), we are able o compue QML-esimaes for all parameers in (23) including he marke price of risk φ 1, φ 2. In he esimaes of hese parameers only USD CDS raes (and no JPY CDS raes) are used besides he exchange rae and ineres-rae daa. Nex, we ry o find ou wheher here are any combinaions of λ and he defaul-free erm srucures for which he resuling raio of JPY CDS spreads o USD CDS spreads is of a similar order of magniude as he one observed in able 1. We do no need o consider he curren value of X as i only eners he CDS spreads hrough is dependency wih λ under he differen measures (see e.g. equaion (21)). For his, we ook he mean erm srucures of defaul-free ineres raes over he sample period in US$ and JPY. We also considered he erm srucure which urned ou o be mos favorable (disfavorable) o our null hypohesis: he seepes (flaes) defaul-free USD erm srucure of ineresraes of his period, ogeher wih he JPY erm srucure of he same day. We firs used λ( i ) 1 q s $( i, i +5) as an approximaion for he defaul inensiy a i and hen calculaed a each i he corresponding 5Y USD and JPY CDS raes (21). he resuling spread beween USD and JPY CDS raes was in no case even close o he spread empirically observed in he daa: Even wih he seepes erm srucure of USD-ineres-raes we ypically found a heoreical relaive spread of less han 5%, (wih he mean erm srucure one of less han 3% and wih he flaes less han 2%): We applied he esimaor (22) o he resuling heoreical 5Y CDS raes and also calculaed { min i 1 s Y ( i, i +5) } s $ ( i, i +5). Wih he mean and he seepes erm srucures of ineres raes in his period we reached he values repored in able 3. (We considered only he ickers mean erm srucure seepes erm srucure icker (Company) δ5 95%-CI δ 5,max δ5 95%-CI δ 5,max BO1 (Bank of okyo-misubishi Ld.).9 1.8, , EJRAIL (Eas Japan Railway Co.)..7, , HONDA (Honda Moor Co. Ld.).3.3, , MICO1 (Misubishi Corp.) , , MISCO (Misui & Co. Ld.).9.5, , N (Nippon elegraph & elephone Corp.) , , NDCM (N DoCoMo Inc.) , , SHARP (Sharp Corp.).7.4, , SUM (Sumiomo Corp.) 1.3.4, , AKFUJ (akefuji Corp.) , , OKIO (okyo Marine & Fire Insurance Co. Ld.).9.8, , OYOA1 (oyoa Moor Corp.) , , YAMAHA (Yamaha Moor Co., Ld.) , , able 3: Implied devaluaion fracions % in he pure-diffusion seup. which showed posiive mean reversion κ under Q.) Comparing hese resuls o he relaive 27

28 spreads repored in able 1 we rejec he null hypohesis of δ =. On he oher hand, for any given relaive spread of USD and JPY CDS raes (and for any given erm srucures of defaul-free ineres-raes) we can find a nonzero value of δ such ha his relaive spread is reproduced. So we canno rejec he hypohesis ha he devaluaion fracion δ is no zero. 6 Applicaions and Exensions 6.1 Discussion of he Empirical Resuls he spread beween domesic and foreign CDS raes implies a currency devaluaion of beween 15 and 25% a a defaul of one of he larger obligors in our sample. his may seem raher large, and i may be (a leas parially) caused by marke imperfecions. Neverheless, he size and persisence of he effec (i did no change significanly alhough he marke liquidiy has muliplied in our sample) indicaes ha here is a leas o some exen a real foundaion o i. For example, a defaul of a large Japanese firm will mos likely happen in a serious recession scenario, or i will be an indicaor of severe fundamenal problems in he economy (e.g. a bank defaul would be a srong indicaor ha he Japanese bad-loans problem is more serious han expeced). Furhermore, we would like o poin ou ha we excluded some sources of dependency in our seup, in paricular he possibiliy of join jumps of FX rae and defaul inensiy before defaul, and he dependency across obligors. In paricular he laer effec should be invesigaed furher: A macro-economic effec of a defaul (e.g. FX devaluaion) should be paricularly large if he defaul iself was caused by a sysemaic, macro-economic reason. Bu hen he defaul is also more likely o affec oher obligors. hus, he implied FX devaluaion a defaul may even give us informaion on he dependency beween he obligors and he macro economy if all variables (defauls and FX rae) are driven by he same macro-variables. We would have expeced o see significanly differen resuls for firms ha are acive in foreign rade as exporers, as imporers, and firms ha mosly service he domesic marke. Making his disincion for large Japanese corporaions is raher difficul and from a qualiaive inspecion of our resuls we did no see any such sysemaic connecion. o he mehodology of our sudy i was irrelevan ha he reference obligors were Japanese companies. Wha we needed was ha X was he exchange rae beween he CDScurrencies, bu no ha X had any connecion o he reference credi. hus, a similar sudy can also be performed using reference credis ha are no incorporaed in he foreign counry. A paricularly ineresing case arises if he reference credi of he CDS is a sovereign iself. Usually, his sovereign will no be a G7 counry, so (unless his is explicily saed in he erm shee) he CDS will no be denominaed in is currency. Neverheless, many developing counries have issued deb in muliple currencies, e.g. USD, EUR, and JPY, and CDS on hese sovereigns can also be raded in hese currencies. he relaive spreads of hese CDS will hen allow us o make saemens abou he implied effec of a sovereign defaul of e.g. Brazil on he EUR/USD exchange rae in much he 28

29 same way as we made saemens abou he JPY/USD exchange rae upon defaul of a corporae reference credi. In his case, i will no be clear which currency we would expec o devalue, i.e. ˆδ may also be negaive. 6.2 Oher Defaul-Sensiive FX Derivaives We used CDS wih denominaion in differen currencies o illusrae he key ideas and mehods which mus be used o analyse credi risk in muliple currencies, bu here are also a number of oher siuaions in which he mehods of his paper can be used, for example some exoic credi derivaives and he analysis of counerpary credi risk in OC derivaives ransacions. In his secion we presen some of hese connecions. As a firs applicaion, we define defaul-sensiive equivalens for he mos common FX derivaives such as FX swaps and FX forwards. hese derivaives behave like heir defaulfree equivalens excep ha hey are cancelled a defaul. I will urn ou ha some of hese insrumens have a naural connecion o he problem of pricing CDSs in differen currencies. We define he defaulable FX forward rae X as X(, ) = X() B f(, ) B d (, ). (24) his is he forward exchange rae o be used in a FX forward conrac which is cancelled a defaul (i.e. if a defaul occurs before he selemen dae ). Using he defaulable zero coupon bond prices given in secion 4.2.2, he defaulable FX forward raes can be given in closed-form. In a defaulable currency swap one side of he swap pays a sream of he defaulable currency swap rae x(, ) in c d, and he oher side of he swap pays a sream of 1 in c f, and boh paymen sreams sop a defaul, or if no defaul occurs before a he mauriy dae. (In conras o a classical currency swap we assume no exchange of principal a mauriy.) As boh legs of he currency swap mus have he same value, we reach he following represenaion for x(, ): and hus x(, ) B d (, s)ds = X() B f (, s)ds = X(, s)b d (, s)ds (25) B f (, s)ds x(, ) = X() B d (, s)ds = w d (s;, )X(, s)ds. hus, we can view he rae x as a weighed average of he defaulable FX-forward raes X(, s) over he life of he swap, or as a survival-coningen exchange rae for paymen sreams. Alernaively, i can be viewed as he relaive price of a uni payoff sream (an annuiy) in foreign-currency annuiy B d (, s)ds. B f (, s)ds, measured in unis of he domesic defaulable he defaulable currency swap in (25) allows us o ransform any CDS fee sream in c f ino a corresponding fee sream in c d. his leads naurally o he following insrumen: 29

30 A Quano CDS is a credi-defaul swap, which has a proecion paymen in one currency (e.g. c f ), bu he fee paymen is made in anoher currency (e.g. c d ) a he rae s quano. Using he defaulable currency swap o ransform he fee sreams we reach: s quano (, ) = s f (, ) x(, ). (26) A Defaul-Coningen FX Forward is a conrac o exchange 1 uni of foreign currency for X τ (, ) unis of domesic currency a he ime of defaul τ, if and only if defaul occurs before is mauriy. his insrumen may seem a bi arificial bu we believe here should be ineres in i. Ofen, invesors in foreign companies have secured srong collaeralisaion of heir loans. Upon a credi even, hese invesors will have o liquidae he collaeral and conver a significan amoun of cash back ino heir domesic currency. Such invesors migh be ineresed in a defaul-coningen FX forward, i.e. a FX forward conrac which allows he invesor o exchange if (and only if) a credi even has occurred. Assuming consan recovery raes 1 q, we can model his conrac as a porfolio of wo CDS conracs, one of which is shor proecion in foreign currency wih noional 1/q (hus paying one uni of c f a defaul), and one ha is long proecion in domesic currency wih noional X τ /q (which pays one uni of c d a defaul). We can conver he fees of he c f CDS ino domesic currency using he quano CDS inroduced above o reach a ne value in c d for he defaul-coningen FX forward of 1 ( ) s f (, ) x(, ) X τ (, ) s d (, ) B d (, s)ds, q i.e. he fee in domesic currency ha has o be paid for his conrac is 1 q (s f x X τ s d ). Finally, for he conrac o be fair (i.e. for he fee o be zero), we need o se he defaulconingen FX forward rae o X τ (, ) = x(, ) s f(, ) s d (, ). (27) he Raio of Domesic o Foreign CDS Raes: From (27) we reach immediaely s f (, ) s d (, ) = Xτ (, ) x(, ). (28) he raio of he foreign o he domesic CDS rae is he raio of wo exchange raes: he exchange rae X τ (, ) ha applies a he ime of defaul only if a defaul occurs over, o he exchange rae x(, ) ha applies over, only for he ime he obligor survives. References Yacine Ai-Sahalia. Maximum-likelihood esimaion of discreely-sampled diffusions: A closed-form approximaion approach. Economerica, 7: , 22. 3

31 Yacine Ai-Sahalia and Rober Kimmel. Esimaing affine mulifacor erm srucure models using closed-form likelihood expansions. Working paper, Deparmen of Economics, Princeon Universiy, Ocober 22. Jeremy Bulow and Kenneh Rogoff. A consan reconracing model of sovereign deb. Journal of Poliical Economy, 97(1): , February Qiang Dai and Kenneh Singleon. Specificaion analysis of affine erm srucure models. Journal of Finance, LV: , 2. Sergei A. Davydenko and Julian R. Franks. Do bankrupcy codes maer? A sudy of defauls in France, Germany, and he UK. Working paper, London Business School, June 24. Gregory R. Duffee. he relaion beween reasury yields and corporae bond yield spreads. he Journal of Finance, 53(6): , Darrell Duffie. Defaulable erm srucure models wih fracional recovery of par. Working paper, Graduae School of Business, Sanford Universiy, Darrell Duffie and Rui Kan. A yield-facor model of ineres raes. Mahemaical Finance, 6(4):379 46, Darrell Duffie, Lasse Pedersen, and Ken Singleon. Modeling sovereign yield spreads: A case sudy of Russian deb. he Journal of Finance, 58: , 23. Darrell Duffie and Kenneh J. Singleon. Modeling erm srucures of defaulable bonds. he Review of Financial Sudies, 12(4):687 72, A. R. Gallan and G. auchen. Which momens o mach? Economeric heory, 12: , Parick Houweling and on Vors. Pricing defaul swaps: Empirical evidence. Journal of Inernaional Money and Finance, forhcoming, 25. Inernaional Swap Dealers Associaion (ISDA) isda credi derivaives definiions. echnical repor, Jean Jacod and Alber N. Shiryaev. Limi heorems for Sochasic Processes. Springer, Berlin, Heidelberg, New York, Rainer Jankowisch and Sefan Pichler. Currency dependence of corporae credi spreads. Working paper, Deparmen of Finance, Vienna Universiy of echnology, March 23. Rober A. Jarrow and Suar M. urnbull. Pricing derivaives on financial securiies subjec o credi risk. Journal of Finance, 5:53 85, Monique Jeanblanc and Marek Rukowski. Mahemaical Finance - Bachelier Congress 2, chaper Defaul risk and hazard processes, pages Springer-Verlag, Berlin, 2. 31

32 Graciela L. Kaminsky and Carmen M. Reinhar. he win crises: he causes of banking and balance-of-paymens problems. he American Economic Review, 89(3):473 5, June Ioannis Karazas and Seve E. Shreve. Brownian Moion and Sochasic Calculus. Springer, Berlin, Heidelberg, New York, Dominique Lepingle and Jean Memin. Sur l inegrabilie uniforme des maringales exponenielles.. Wahrsch. Verw. Gebiee, 42:175 23, Rober C. Meron. On he pricing of corporae deb: he risk srucure of ineres raes. Journal of Finance, 29:449 47, Marek Musiela and Marek Rukowski. Maringale Mehods in Financial Modelling, volume 36 of Applicaions of Mahemaics. Springer, Berlin, Heidelberg, New York, Navroz Pael. Flow business booms. Risk Magazine, (2):2 23, February 23. Carmen M. Reinhar. Defaul, currency crises, and sovereign credi raings. Working paper, Universiy of Maryland and Inernaional Moneary Fund, 22. Philipp Schönbucher. A libor marke model wih defaul risk. Working paper, Universiy of Bonn, Philipp Schönbucher. A measure of survival. Risk Magazine, 17(8), Augus 24. Philipp J. Schönbucher. erm srucure modelling of defaulable bonds. he Review of Derivaives Sudies, Special Issue: Credi Risk, 2(2/3): , Philipp J. Schönbucher. A ree implemenaion of a credi spread model for credi derivaives. Journal of Compuaional Finance, 6(2), 22. Manmohan Singh. Are credi defaul swap spreads high in emerging markes? An alernaive mehodology for proxying recovery value. Working Paper WP 3/242, Inernaional Moneary Fund, December 23. Kenneh Singleon. Esimaion of affine asse pricing models using he empirical characerisic funcion. Journal of Economerics, 12(1): , 21. Ignacio Warnes and Gabriel Acosa. Valuaion of inernaional corporae deb issues. Working paper, U orcuao Di ella and U de General Sarmieno, January 22. A Appendix Proof of Lemma 2. M := E e R λ(s)ds g F,, is a uniformly inegrable maringale. Under hypohesis H (see definiion H 3 in Jeanblanc and Rukowski (2)) we have ha E e R λ(s)ds g F = E e R λ(s)ds g F W because e R λ(s)ds g is 32

33 F W -measurable and hus he srong predicable represenaion propery (see e.g Karazas and Shreve (1991)) implies ha here exiss a F W -adaped process φ M such ha M = M + φ M (s)dw s. (29) Y := 1 {<τ} e R λ(s)ds is a local maringale (dy = Y ( dn +λ()d)) wih Y = 1 { <τ} e R λ(s)ds. MY is also a local maringale because dm, Y = dm c, Y c + M Y =. Noe ha his follows only from (29), i.e. from hypohesis H. Furher E g F is uniformly inegrable and M Y E g F (Jensen s inequaliy for cond. expecaions). hus MY is uniformly inegrable, i.e. i is a maringale and we have 1 {<τ} E e R λ(s)ds g F = M Y = E M Y F = E 1 { <τ} g F. Proof of corollary 3. E 1 {s τ} g s ds < guaranees ha 1 { τ} g is Q l-inegrable. Hence Fubini s heorem applies and E 1 { τ} g < l-a.e. For every F F E 1 F 1 {s τ} g s ds = E 1 {s τ} g s 1 F ds = E 1 F E 1 {s τ} g s F ds = E 1 F E 1 {s τ} g s F ds = E 1 F E 1 {s<τ} g s F ds. In he las equaliy we used ha E 1 { τ} g = E 1{<τ} g Q l-a.e. because 1{ τ} = 1 {<τ} l-a.e. ogeher wih lemma 2 his proves he claim. Proof of corollary 4. By a variaion of heorem 1.8 of chaper II in Jacod and Shiryaev (1988) (4) ensures 1 27 {s τ}f(z, s)(µ ν)(dz, ds) is a maringale. hus, for all F F ( ) E 1 F 1 {s τ} f(z, s)µ(dz, ds) = E 1 F 1 {s τ} s)df s (z) λ s ds < f(z, By Fubini s heorem E f(z, ) df (z) 1 { τ} λ < l-a.e. and we can proceed as in he proof of lemma 3. Proof of Lemma 7. We omi he argumen of f and Φ. Recall L = L (Φ 1)µ(dz, d). By Iô s lemma s d (L s ) fµ(dz, du) = L s + L s = L s ( s fµ(dz, ds) + fµ(dz, ds) ( s fφµ(dz, ds) + 27 f(z, s) := 1{s>} 1 F f(z, s) is predicable for fixed and F F. ) fµ(dz, du) dl s ) fµ(dz, du) dl s. 33

34 Condiion E h2 d L < ensures ha he sochasic inegral wr. L, h s dl s, is a rue maringale, hence has zero expecaion. he argumens hold rue when f is replaced by f := 1 (, 1 A f for arbirary A F, his proves he claim. Proof of Lemma 8. From Iô s lemma i follows s ) ( s ) d (L s g u du = L s g s ds + g u du dl s Condiion E ( g s ds) 2 d L < guaranees ha ( s g udu) dl s is a rue maringale. his remains valid for g := 1 (, 1 A g for arbirary A F insead of g and he claim follows. Proof of lemma 9. he boundedness of B (and hence A) guaranees ha A and B are C 1 -funcions. Hence M() = e R α+β Y sds+a( )+B( ) Y,,, is differeniable in and i can be checked easily via Iô s lemma ha M solves he SDE dm = B( ) S dw M if A and B solve he ODE (8), hence M is a local maringale. By E e 1 R 2 w(b ) Y sds < (Novikov s condiion) M is a maringale and he boundary condiions A() = and B() = imply he erminal value M = e R α+β Y sds. Proof of lemma 1. If ξ =, hen lemma 9 applies. Hence we consider only he case where ξ has a leas one posiive componen. Noe ha (9) is linear in B wih bounded coefficiens as long as B is bounded, i.e. B is also bounded. Moreover B, for a leas one componen min B i () > and A. hus M() := e R α+β Y sds+a( )+B( ) Y ( A( )+B( ) Y ), is differeniable in and Q-a.s. posiive for all,. By Iô s lemma ( ) dm() M() = B( ) B( ) + S dw A( ) + B( ) Y i.e. M is a local maringale. Novikov s crierion E e 1 R ) Yd (B 2 w + B A +B Y < guaranees ha M is also a maringale and by he iniial condiion A() = ζ and B() = ξ is erminal value is M = e R ( ) α+β Y sds ζ +ξ Y. Proof of formula (15). By virue of (11) i suffices o proof he domesic formula. We firs define q d := 1 L d (u d ) and choose an F W -measurable g wih E Q d g <. hen he proof goes much in line wih ha of lemma 2. Define he Q d -maringale M := E Q d e bq R d λd(s)ds g F. I can be checked easily ha Y := p MD d ()e bq dr λd(s)ds is a Q d - local maringale. Due o assumpion of hypohesis H we have M = E e R λ(s)ds g F W 34

35 and hus MY is also a Q d -local maringale. Furher M Y E Q d g F, hence MY is a rue Q d -maringale and p MD d () E Q d e bq R d λ d (s)ds g F = M Y = E Q d M Y F = E Q d p MD d ( ) g F. We se g := e R r d (s)ds, hen lemma 9 yields he claim. 35