Interest Rates and The Credit Crunch: New Formulas and Market Models

Transcription

1 Interest Rates and The Credt Crunch: New Formulas and Market Models Fabo Mercuro QFR, Bloomberg Frst verson: 12 November 2008 Ths verson: 5 February 2009 Abstract We start by descrbng the major changes that occurred n the quotes of market rates after the 2007 subprme mortgage crss. We comment on ther lost analoges and consstences, and hnt on a possble, smple way to formally reconcle them. We then show how to prce nterest rate swaps under the new market practce of usng dfferent curves for generatng future LIBOR rates and for dscountng cash flows. Straghtforward modfcatons of the market formulas for caps and swaptons wll also be derved. Fnally, we wll ntroduce a new LIBOR market model, whch wll be based on modelng the jont evoluton of FRA rates and forward rates belongng to the dscount curve. We wll start by analyzng the basc lognormal case and then add stochastc volatlty. The dynamcs of FRA rates under dfferent measures wll be obtaned and closed form formulas for caplets and swaptons derved n the lognormal and Heston (1993) cases. 1 Introducton Before the credt crunch of 2007, the nterest rates quoted n the market showed typcal consstences that we learned on books. We knew that a floatng rate bond, where rates are set at the begnnng of ther applcaton perod and pad at the end, s always worth par at ncepton, rrespectvely of the length of the underlyng rate (as soon as the payment schedule s re-adjusted accordngly). For nstance, Hull (2002) rectes: The floatng-rate bond underlyng the swap pays LIBOR. As a result, the value of ths bond equals the swap Stmulatng dscussons wth Peter Carr, Bjorn Flesaker and Antono Castagna are gratefully acknowledged. The author also thanks Marco Banchett and Massmo Morn for ther helpful comments and Paola Moscon and Sabrna vorsk for proofreadng the artcle s frst draft. Needless to say, all errors are the author s responsblty. 1

2 2 prncpal. We also knew that a forward rate agreement (FRA) could be replcated by gong long a depost and sellng short another wth maturtes equal to the FRA s maturty and reset tme. These consstences between rates allowed the constructon of a well-defned zero-coupon curve, typcally usng bootstrappng technques n conjuncton wth nterpolaton methods. 1 fferences between smlar rates were present n the market, but generally regarded as neglgble. For nstance, depost rates and OIS (EONIA) rates for the same maturty would chase each other, but keepng a safety dstance (the bass) of a few bass ponts. Smlarly, swap rates wth the same maturty, but based on dfferent lengths for the underlyng floatng rates, would be quoted at a non-zero (but agan neglgble) spread. Then, August 2007 arrved, and our convctons became to weaver. The lqudty crss wdened the bass, so that market rates that were consstent wth each other suddenly revealed a degree of ncompatblty that worsened as tme passed by. For nstance, the forward rates mpled by two consecutve deposts became dfferent than the quoted FRA rates or the forward rates mpled by OIS (EONIA) quotes. Remarkably, ths dvergence n values does not create arbtrage opportuntes when credt or lqudty ssues are taken nto account. As an example, a swap rate based on semannual payments of the sx-month LIBOR rate can be dfferent (and hgher) than the same-maturty swap rate based on quarterly payments of the three-month LIBOR rate. These stylzed facts suggest that the consstent constructon of a yeld curve s possble only thanks to credt and lqudty theores justfyng the smultaneous exstence of dfferent values for same-tenor market rates. Morn (2008) s, to our knowledge, the frst to desgn a theoretcal framework that motvates the dvergence n value of such rates. To ths end, he ntroduces a stochastc default probablty and, assumng no lqudty rsk and that the rsk n the FRA contract exceeds that n the LIBOR rates, obtans patterns smlar to the market s. 2 However, whle watng for a combned credt-lqudty theory to be produced and become effectve, practtoners seem to agree on an emprcal approach, whch s based on the constructon of as many curves as possble rate lengths (e.g. 1m, 3m, 6m, 1y). Future cash flows are thus generated through the curves assocated to the underlyng rates and then dscounted by another curve, whch we term dscount curve. Assumng dfferent curves for dfferent rate lengths, however, mmedately nvaldates the classc prcng approaches, whch were bult on the cornerstone of a unque, and fully consstent, zero-coupon curve, used both n the generaton of future cash flows and n the calculaton of ther present value. Ths paper shows how to generalze the man (nterest rate) market models so as to account for the new market practce of usng multple curves for each sngle currency. The valuaton of nterest rate dervatves under dfferent curves for generatng future rates and for dscountng receved lttle attenton n the (non-credt related) fnancal lt- 1 The bootstrappng amed at nferrng the dscount factors (zero-coupon bond prces) for the market maturtes (pllars). Interpolaton methods were needed to obtan nterest rate values between two market pllars or outsde the quoted nterval. 2 We also hnt at a possble soluton n Secton 2.2. Compared to Morn, we consder smplfed assumptons on defaults, but allow the nterbank counterparty to change over tme.

3 3 erature, and manly concernng the valuaton of cross currency swaps, see Fruchard et al. (1995), Boenkost and Schmdt (2005) and Kjma et al. (2008). To our knowledge, Banchett (2008) s the frst to apply the methodology to the sngle currency case. In ths artcle, we start from the approach proposed by Kjma et al. (2008), and show how to extend accordngly the (sngle currency) LIBOR market model (LMM). Our extended verson of the LMM s based on the jont evoluton of FRA rates, namely of the fxed rates that gve zero value to the related forward rate agreements. 3 In the sngle-curve case, an FRA rate can be defned by the expectaton of the correspondng LIBOR rate under a gven forward measure, see e.g. Brgo and Mercuro (2006). In our mult-curve settng, an analogous defnton apples, but wth the complcaton that the LIBOR rate and the forward measure belong, n general, to dfferent curves. FRA rates thus become dfferent objects than the LIBOR rates they orgnate from, and as such can be modeled wth ther own dynamcs. In fact, FRA rates are martngales under the assocated forward measure for the dscount curve, but modelng ther jont evoluton s not equvalent to defnng ther nstantaneous covaraton structure. In ths artcle, we wll start by consderng the basc example of lognormal dynamcs and then ntroduce general stochastc volatlty processes. The dynamcs of FRA rates under non-canoncal measures wll be shown to be smlar to those n the classc LMM. The man dfference s gven by the drft rates that depend on the relevant forward rates for the dscount curve, rather then the other FRA rates n the consdered famly. A last remark s n order. Also when we prce nterest rate dervatves under credt rsk we eventually deal wth two curves, one for generatng cash flows and the other for dscountng, see e.g. the LMM of Schönbucher (2000). However, n ths artcle we do not want to model the yeld curve of a gven rsky ssuer or counterparty. We rather acknowledge that dstnct rates n the market account for dfferent credt or lqudty effects, and we start from ths stylzed fact to buld a new LMM consstent wth t. The artcle s organzed as follows. Secton 2 brefly descrbes the changes n the man nterest rate quotes occurred after August 2007, proposng a smple formal explanaton for ther dfferences. It also descrbes the market practce of buldng dfferent curves and motvates the approach we follow n the artcle. Secton 3 ntroduces the man defntons and notatons. Secton 4 shows how to value nterest rate swaps when future LIBOR rates are generated wth a correspondng yeld curve but dscounted wth another. Secton 5 extends the market Black formulas for caplets and swaptons to the double-curve case. Secton 6 ntroduces the extended lognormal LIBOR market model and derves the FRA and forward rates dynamcs under dfferent measures and the prcng formulas for caplets and swaptons. Secton 7 ntroduces stochastc volatlty and derves the dynamcs of rates and volatltes under generc forward and swap measures. Hnts on the dervaton of prcng formulas for caps and swaptons are then provded n the specfc case of the Wu and Zhang (2006) model. Secton 8 concludes the artcle. 3 These forward rate agreements are actually swaplets, n that, contrary to market FRAs, they pay at the end of the applcaton perod.

4 4 2 Credt-crunch nterest-rate quotes An mmedate consequence of the 2007 credt crunch was the dvergence of rates that untl then closely chased each other, ether because related to the same tme nterval or because mpled by other market quotes. Rates related to the same tme nterval are, for nstance, depost and OIS rates wth the same maturty. Another example s gven by swap rates wth the same maturty, but dfferent floatng legs (n terms of payment frequency and length of the pad rate). Rates mpled by other market quotes are, for nstance, FRA rates, whch we learnt to be equal to the forward rate mpled by two related deposts. All these rates, whch were so closely nterconnected, suddenly became dfferent objects, each one ncorporatng ts own lqudty or credt premum. 4 Hstorcal values of some relevant rates are shown n Fgures 1 and 2. In Fgure 1 we compare the last values of one-month EONIA rates and one-month depost rates, from November 14th, 2005 to November 12, We can see that the bass was well below ten bp untl August 2007, but snce then started movng erratcally around dfferent levels. In Fgure 2 we compare the last values of two two-year swap rates, the frst payng quarterly the three-month LIBOR rate, the second payng semannually the sx-month LIBOR rate, from November 14th, 2005 to November 12, Agan, we can notce the change n behavor occurred n August In Fgure 3 we compare the last values of 3x6 EONIA forward rates and 3x6 FRA rates, from November 14th, 2005 to November 12, Once agan, these rates have been rather algned untl August 2007, but dverged heavly thereafter. 2.1 vergence between FRA rates and forward rates mpled by deposts The closng values of the three-month and sx-month deposts on November 12, 2008 were, respectvely, 4.286% and 4.345%. Assumng, for smplcty, 30/360 as day-count conventon (the actual one for the EUR LIBOR rate s ACT/360), the mpled three-month forward rate n three months s 4.357%, whereas the value of the correspondng FRA rate was 1.5% lower, quoted at 2.85%. Surprsngly enough, these values do not necessarly lead to arbtrage opportuntes. In fact, let us denote the FRA rate and the forward rate mpled by the two deposts wth maturty T 1 and T 2 by F X and F, respectvely, and assume that F > F X. One may then be tempted to mplement the followng strategy (τ 1,2 s the year fracton for (T 1, T 2 ]): a) Buy (1 + τ 1,2 F ) bonds wth maturty T 2, payng (1 + τ 1,2 F )(0, T 2 ) = (0, T 1 ) 4 Futures rates are less straghtforward to compare because of ther fxed IMM maturtes and ther mplct convexty correcton. Ther values, however, tend to be rather close to the correspondng FRA rates, not dsplayng the large dscrepances observed wth other rates.

5 5 Fgure 1: Euro 1m EONIA rates vs 1m depost rates, from 14 Nov 2005 to 12 Nov Source: Bloomberg. dollars, where (0, T ) denotes the tme-0 bond prce for maturty T ; b) Sell 1 bond wth maturty T 1, recevng (0, T 1 ) dollars; c) Enter a (payer) FRA, payng out at tme T 1 τ 1,2 (L(T 1, T 2 ) F X ) 1 + τ 1,2 L(T 1, T 2 ) where L(T 1, T 2 ) s the LIBOR rate set at T 1 for maturty T 2. The value of ths strategy at the current tme s zero. At tme T 1, b) plus c) yeld τ 1,2 (L(T 1, T 2 ) F X ) 1 + τ 1,2 L(T 1, T 2 ) 1 = 1 + τ 1,2F X 1 + τ 1,2 L(T 1, T 2 ), whch s negatve f rates are assumed to be postve. To pay ths resdual debt, we sell the 1 + τ 1,2 F bonds wth maturty T 2, remanng wth 1 + τ 1,2 F 1 + τ 1,2 L(T 1, T 2 ) 1 + τ 1,2 F X 1 + τ 1,2 L(T 1, T 2 ) = τ 1,2(F F X ) 1 + τ 1,2 L(T 1, T 2 ) > 0 n cash at T 1, whch s equvalent to τ 1,2 (F F X ) receved at 2. Ths s clearly an arbtrage, snce a zero nvestment today produces a (stochastc but) postve gan at tme T 1 or, equvalently, a determnstc postve gan at T 2 (wth no ntermedate net cash

6 6 Fgure 2: Euro 2y swap rates (3m vs 6m), from 14 Nov 2005 to 12 Nov Source: Bloomberg. flows). However, there are two ssues that, n the current market envronment, can not be neglected any more (we assume that the FRA s default-free): ) Possblty of default before T 2 of the counterparty we lent money to; ) Possblty of lqudty crunch at tmes 0 or T 1. If ether events occur, we can end up wth a loss at fnal tme T 2 that may outvalue the postve gan τ 1,2 (F F X ). 5 Therefore, we can conclude that the strategy above does not necessarly consttute an arbtrage opportunty. The forward rates F and F X are n fact allowed to dverge, and ther dfference can be seen as representatve of the market estmate of future credt and lqudty ssues. 2.2 Explanng the dfference n value of smlar rates The dfference n value between formerly equvalent rates can be explaned by means of a smple credt model, whch s based on assumng that the generc nterbank counterparty s subject to default rsk. 6 To ths end, let us denote by τ t the default tme of the generc 5 Even assumng we can sell back at T 1 the T 2 -bonds to the counterparty we ntally lent money to, default stll plays aganst us. 6 Morn (2008) develops a smlar approach wth stochastc probablty of default. In addton to ours, he consders blateral default rsk. Hs nterbank counterparty s, however, kept the same, and hs defnton of FRA contract s dfferent than that used by the market.

7 7 Fgure 3: 3x6 EONIA forward rates vs 3x6 FRA rates, from 14 Nov 2005 to 12 Nov Source: Bloomberg. nterbank counterparty at tme t, where the subscrpt t ndcates that the random varable τ t can be dfferent at dfferent tmes. Assumng ndependence between default and nterest rates and denotng by R the (assumed constant) recovery rate, the value at tme t of a depost startng at that tme and wth maturty T s (t, T ) = E [e ( T ) ] t r(u) du R+(1 R)1 {τt>t } Ft = RP (t, T )+(1 R)P (t, T )E [ ] 1 {τt>t } F t, where E denotes expectaton under the rsk-neutral measure, r the default-free nstantaneous nterest rate, P (t, T ) the prce of a default-free zero coupon bond at tme t for maturty T and F t s the nformaton avalable n the market at tme t. 7 Settng Q(t, T ) := E [ 1 {τt>t } F t ], the LIBOR rate L(T 1, T 2 ), whch s the smple nterest earned by the depost (T 1, T 2 ), s gven by L(T 1, T 2 ) = 1 [ ] 1 τ 1,2 (T 1, T 2 ) 1 = 1 [ ] 1 1 τ 1,2 P (T 1, T 2 ) R + (1 R)Q(T 1, T 2 ) 1. 7 We also refer to the next secton for all defntons and notatons.

8 8 Assumng that the above FRA has no counterparty rsk, ts tme-0 value can be wrtten as [ 0 = E e T 1 0 r(u) du τ ] 1,2(L(T 1, T 2 ) F X ) 1 + τ 1,2 L(T 1, T 2 ) [ ( )] = E e T 1 0 r(u) du 1 + τ 1,2 F X τ 1,2 L(T 1, T 2 ) [ = E e ( T 1 0 r(u) du 1 (1 + τ 1,2 F X )P (T 1, T 2 )(R + (1 R)Q(T 1, T 2 )) )] = P (0, T 1 ) (1 + τ 1,2 F X )P (0, T 2 ) ( R + (1 R)E [ Q(T 1, T 2 ) ]) whch yelds the value of the FRA rate F X : F X = 1 [ ] P (0, T1 ) 1 τ 1,2 P (0, T 2 ) R + (1 R)E [ Q(T 1, T 2 ) ] 1. Snce then so that 0 R 1, 0 < Q(T 1, T 2 ) < 1, 0 < R + (1 R)E [ Q(T 1, T 2 ) ] < 1 F X > 1 [ ] P (0, T1 ) τ 1,2 P (0, T 2 ) 1. (1) Therefore, the FRA rate F X s larger than the forward rate mpled by the default-free bonds P (0, T 1 ) and P (0, T 2 ). If the OIS (EONIA) swap curve s elected to be the rsk-free curve, whch s reasonable snce the credt rsk n an overnght rate s deemed to be neglgble even n ths new market stuaton, then (1) explans that the FRA rate F X can be (arbtrarly) hgher than the correspondng forward OIS rate f the default rsk mplct n the LIBOR rate s taken nto account. Smlarly, the forward rate mpled by the two deposts (0, T 1 ) and (0, T 2 ),.e. F = 1 [ ] (0, T1 ) τ 1,2 (0, T 2 ) 1 = 1 [ ] R + (1 R)Q(0, T1 ) P (0, T 1 ) τ 1,2 R + (1 R)Q(0, T 2 ) P (0, T 2 ) 1 wll be larger than the FRA rate F X f R + (1 R)Q(0, T 1 ) R + (1 R)Q(0, T 2 ) > 1 R + (1 R)E [ Q(T 1, T 2 ) ]. Ths happens, for nstance, when R < 1 and the market expectaton for the future credt premum from T 1 to T 2 (nversely proportonal to Q(T 1, T 2 )) s low compared to the value mpled by the spot quanttes Q(0, T 1 ) and Q(0, T 2 ). 8 8 Even though the quanttes Q(T 1, T 2 ) and Q(0, T ), = 1, 2, refer to dfferent default tmes τ 0 and τ T1, they can not be regarded as completely unrelated to each other, snce they both depend on the credt worthness of the generc nterbank counterparty from T 1 to T 2.

9 9 Further degrees of freedom to be calbrated to market quotes can be added by also modelng lqudty rsk. 9 A thorough and sensble treatment of lqudty effects, s however beyond the scope of ths work. 2.3 Usng multple curves The analyss just performed s meant to provde a smple theoretcal justfcaton for the current dvergence of market rates that refer to the same tme nterval. Such rates, n fact, become compatble wth each other as soon as credt and lqudty rsks are taken nto account. However, nstead of explctly modelng credt and lqudty effects, practtoners seem to deal wth the above dscrepances by segmentng market rates, labelng them dfferently accordng to ther applcaton perod. Ths results n the constructon of dfferent zero-coupon curves, one for each possble rate length consdered. One of ths curves, or any verson obtaned by mxng nhomogeneous rates, s then elected to act as the dscount curve. As far as dervatves prcng s concerned, however, t s stll not clear how to account for these new market features and practce. When prcng nterest rate dervatves wth a gven model, the usual frst step s the model calbraton to the term structure of market rates. Ths task, before August 2007, was straghtforward to accomplsh thanks to the exstence of a unque, well defned yeld curve. When dealng wth multple curves, however, not only the calbraton to market rates but also the modelng of ther evoluton becomes a non-trval task. To ths end, one may dentfy two possble solutons: ) Modelng default-free rates n conjuncton wth default tmes τ t and/or lqudty effects. ) Modelng the jont, but dstnct, evoluton of rates that apples to the same nterval. The former choce s consstent wth the above procedure to justfy the smultaneous exstence of formerly equvalent rates. However, devsng a sensble model for the evoluton of default tmes may not be so obvous. Notce, n fact, that the standard theores on credt rsk do not mmedately apply here, snce the default tme does not refer to a sngle credt entty, but t s representatve of a generc sector, the nterbank one. The random varable τ t, therefore, does not change over tme because the credt worthness of the reference entty evolves stochastcally, but because the counterparty s generc and a new default tme τ t s generated at each tme t to assess the credt premum n the LIBOR rate at that tme. In ths artcle, we prefer to follow the latter approach and apply a logc smlar to that used n the yeld curves constructon. In fact, gven that practtoners buld dfferent curves for dfferent tenors, t s qute reasonable to ntroduce an nterest rate model where such curves are modeled jontly but dstnctly. To ths end, we wll model forward rates wth a gven tenor n conjuncton wth those mpled by the dscount curve. Ths wll be acheved n the sprt of Kjma et al (2008). The forward (or growth ) curve assocated to a gven rate tenor can be constructed wth standard bootstrappng technques. The man dfference wth the methodology fol- 9 Lqudty effects are modeled, among others, by Cetn et al. (2006) and Acerb and Scandolo (2007).

10 10 lowed n the pre-credt-crunch stuaton s that now only the market quotes correspondng to the gven tenor are employed n the strppng procedure. For nstance, the three-month curve can be constructed by bootstrappng zero-coupon rates from the market quotes of the three-month depost, the futures (or 3m FRAs) for the man maturtes and the lqud swaps (vs 3m). The dscount curve, nstead, can be selected n several dfferent ways, dependng on the contract to prce. For nstance, n absence of counterparty rsk or n case of collateralzed dervatves, t can be deemed to be the classc rsk-neutral curve, whose best proxy s the OIS swap curve, obtaned by sutably nterpolatng and extrapolatng OIS swap quotes. 10 For a contract sgned wth a generc nterbank counterparty wthout collateral, the dscount curve should reflect the fact that future cash flows are at rsk and, as such, must be dscounted at LIBOR, whch s the rate reflectng the credt rsk of the nterbank sector. In such a case, therefore, the dscount curve may be bootstrapped (and extrapolated) from the quoted depost rates. In general, the dscount curve can be selected as the yeld curve assocated the counterparty n queston. 11 In the followng, we wll assume that future cash flows are all dscounted wth the same dscount curve. The extenson to a more general case nvolves a heaver notaton and here neglected for smplcty. 3 Basc defntons and notaton Let us assume that, n a sngle currency economy, we have selected N dfferent nterest-rate lengths δ 1,..., δ N and constructed the correspondng yeld curves. The curve assocated to length δ wll be shortly referred to as curve. 12 We denote by P (t, T ) the assocated dscount factor (equvalently, zero-coupon bond prce) at tme t for maturty T. We also assume we are gven a curve for dscountng future cash flows. We denote by P (t, T ) the curve- dscount factor at tme t for maturty T. We wll consder the tme structures {T0, T1,...}, where the superscrpt denotes the curve t belongs to, and {T0 S, T1 S,...}, whch ncludes the payment tmes of a swap s fxed leg. Forward rates can be defned for each gven curve. Precsely, for each curve x {1, 2,..., N, }, the (smply-compounded) forward rate prevalng at tme t and appled to the future tme nterval [T, S] s defned by [ ] 1 Px (t, T ) F x (t; T, S) := τ x (T, S) P x (t, S) 1, (2) 10 Notce that OIS rates carry the credt rsk of an overnght rate, whch may be regarded as neglgble n most stuatons. 11 A detaled descrpton of a possble methodology for constructng forward and dscount curves s outlned n Ametrano and Banchett (2008). In general, bootstrappng multple curves, for the same currency, nvolves plenty of techncaltes and subjectve choces. 12 In the next secton, we wll hnt at a possble bootstrap methodology.

11 11 where τ x (T, S) s the year fracton for the nterval [T, S] under the conventon of curve x. 13 Gven the tmes t Tk 1 < T k and the curve x {1,..., N, }, we wll make use of the followng short-hand notaton: Fk x (t) := F x (t; Tk 1, Tk) = 1 [ Px (t, Tk 1 ) ] τk x P x (t, Tk ) 1 (3) where τ x k s the year fracton for the nterval [T k 1, T k ] for curve x, namely τ x k := τ x(t k 1, T k ). As n Kjma et al (2008), the prcng measures we wll consder are those assocated to the dscount curve. To denote these measures we wll adopt the notaton Q z x, where the subscrpt x (manly ) dentfes the underlyng yeld curve, and the superscrpt z defnes the measure n queston. More precsely, we denote by: Q T the T -forward measure, whose numerare s the zero-coupon bond P (, T ). Q T the spot LIBOR measure assocated to tmes T = {T 0,..., TM }, whose numerare s the dscretely-rebalanced bank account B T : B T (t) = P (t, T m) m j=0 P (T j 1, T j ), T m 1 < t T m, m = 1,..., M. Q c,d the forward swap measure defned by the tme structure {T S c, T S c+1,..., T S d }, whose numerare s the annuty where τ S j := τ (T S j 1, T S j ). C c,d (t) = τ S j P (t, T S j ), The expectaton under the generc measure Q z x wll be denoted by E z x, where agan the ndces x and z dentfy, respectvely, the underlyng yeld curve and the measure n queston. The nformaton avalable n the market at each tme t wll be descrbed by the fltraton F t. 4 The valuaton of nterest rate swaps In ths secton, we show how to value lnear nterest rate dervatves under our assumpton of dstnct forward and dscount curves. To ths end, let us consder a set of tmes T a,..., T b compatble wth curve, 14 and an nterest rate swap where the floatng leg pays at each 13 In practce, for curves = 1,..., N, we wll consder only ntervals where S = T +δ, whereas for curve the nterval [S, T ] may be totally arbtrary. 14 For nstance, f denotes the three-month curve, then the tmes Tk must be three-month spaced.

12 12 tme Tk the LIBOR rate of curve set at the prevous tme T k 1, k = a + 1,..., b. In formulas, the tme-tk payoff of the floatng leg s FL(T k; T k 1, T k) = τ kf k(t k 1) = 1 P (Tk 1, T 1. (4) k ) The tme-t value, FL(t; Tk 1, T k ), of such a payoff can be obtaned by takng the dscounted expectaton under the forward measure Q T k :15 FL(t; Tk 1, Tk) = τkp (t, Tk)E T k [ ] F k (Tk 1) F t. efnng the tme-t FRA rate as the fxed rate to be exchanged at tme Tk payment (4) so that the swap has zero value at tme t, 16.e for the floatng we can wrte L k(t) := FRA(t; Tk 1, Tk) = E T k [ ] F k (Tk 1) F t, FL(t; T k 1, T k) = τ kp (t, T k)l k(t). (5) In the classc sngle curve valuaton ( ), the forward rate Fk s a martngale under the assocated Tk -forward measure (concdng wth QT k ), so that the expected value L k (t) concdes wth the current forward rate: L k(t) = F k(t). Accordngly, as s well known, the present value of each payment n the swap s floatng leg can be smplfed as follows: FL(t; T k 1, T k) = τ kp (t, T k)l k(t) = τ kp (t, T k)f k(t) = P (t, T k 1) P (t, T k), whch leads to the classc result that the LIBOR rate set at tme Tk 1 and pad at tme Tk can be replcated by a long poston n a zero-coupon bond exprng at tme T k 1 and a short poston n another bond wth maturty Tk. In the stuaton we are dealng wth, however, curves and are dfferent, n general. The forward rate Fk s not a martngale under the forward measure QT k, and the FRA rate L k (t) s dfferent from F k (t). Therefore, the present value of a future LIBOR rate s no longer obtaned by dscountng the correspondng forward rate, but by dscountng the correspondng FRA rate. 15 For most swaps, thanks to the presence of collaterals or nettng clauses, curve can be assumed to be the rsk-free one (as obtaned from OIS swap rates). 16 Ths FRA rate s slghtly dfferent than that defned by the market, see Secton 2.2. Ths slght abuse of termnology s justfed by the defnton that apples when payments occur at the end of the applcaton perod (lke n ths case).

13 13 The net present value of the swap s floatng leg s smply gven by summng the values (5) of sngle payments: FL(t; T a,..., T b) = b k=a+1 FL(t; T k 1, T k) = b k=a+1 τ kp (t, T k)l k(t), (6) whch, for the reasons just explaned, wll be dfferent n general than P (t, Ta) P (t, Tb ) or P (t, Ta) P (t, Tb ). Let us then consder the swap s fxed leg and denote by K the fxed rate pad on the fxed leg s dates Tc S,..., Td S. The present value of these payments s mmedately obtaned by dscountng them wth the dscount curve : τ S j KP (t, T S j ) = K τ S j P (t, T S j ), where we remember that τ S j = τ (T S j 1, T S j ). Therefore, the nterest rate swap value, to the fxed-rate payer, s gven by IRS(t, K; T a,..., T b, T S c,..., T S c ) = b k=a+1 τ kp (t, T k)l k(t) K τ S j P (t, T S j ). We can then calculate the correspondng forward swap rate as the fxed rate K that makes the IRS value equal to zero at tme t. We get: S a,b,c,d(t) = b k=a+1 τ k P (t, T k )L k (t) τ S j P (t, T S j ). (7) Ths s the forward swap rate of an nterest rate swap where cash flows are generated through curve and dscounted wth curve. In the partcular case of a spot-startng swap, wth payment tmes for the floatng and fxed legs gven, respectvely, by T1,..., Tb and T 1 S,..., Td S, wth T b = T d S, the swap rate becomes: b S0,b,0,d(0) k=1 = τ k P (0, Tk )L k (0) d j=1 τ j SP (0, Tj S), (8) where L 1 (0) s the constant frst floatng payment (known at tme 0). As already notced by Kjma et at. (2008), nether leg of a spot-startng swap needs be worth par (when a fcttous exchange of notonals s ntroduced at maturty). However, ths s not a problem, snce the only requrement for quoted spot-startng swaps s that ther net present value must be equal to zero. Remark 1 As tradtonally done n any bootstrappng algorthm, equaton (8) can be used to nfer the expected rates L k mpled by the market quotes of spot-startng swaps, whch by

14 14 defnton have zero value. The bootstrapped L k can then be used, n conjuncton wth any nterpolaton tool, to prce other swaps based on curve. As already notced by Boenkost and Schmdt (2005) and by Kjma et al. (2008), these other swaps wll have dfferent values, n general, than those obtaned through classc bootstrappng methods appled to swap rates S 0,d (0) = 1 P (0, Td S ) d j=1 τ j SP (0, Tj S). However, ths s perfectly reasonable snce we are here usng an alternatve, and more general, approach. 5 The prcng of caplets and swaptons Smlarly to what we just dd for nterest rate swaps, the purpose of ths secton s to derve prcng formulas for optons on the man nterest rates, whch wll result n modfcatons of the correspondng Black-lke formulas governed by our double-curve paradgm. As s well known, the formal justfcatons for the use of Black-lke formulas for caps and swaptons come, respectvely, from the lognormal LMM of Brace et al. (1997) and Mltersen et al. (1997) and the lognormal swap model of Jamshdan (1997). 17 To be able to adapt such formulas to our double-curve case, we wll have to reformulate accordngly the correspondng market models. Agan, the choce of the dscount curve depends on the credt worthness of the counterparty and on the possble presence of a collateral mtgatng the credt rsk exposure. 5.1 Market formula for caplets We frst consder the case of a caplet payng out at tme T k τ k[f k(t k 1) K] +. (9) To prce such payoff n the basc sngle-curve case, one notces that the forward rate F k s a martngale under the Tk -forward measure QT k caplet prce for curve, and then calculates the tme-t Cplt(t, K; Tk 1, Tk) = τkp (t, Tk)E T k { } [F k (Tk 1) K] + F t accordng to the chosen dynamcs. For nstance, the classc choce of a drftless geometrc Brownan moton 18 dfk(t) = σ k Fk(t) dz k (t), t Tk 1, 17 It s worth mentonng that the frst proof that Black-lke formulas for caps and swaptons are arbtrage free s due to Jamshdan (1996). 18 We wll use the symbol d to denote dfferentals as opposed to d, whch nstead denotes the ndex of the fnal date n the swap s fxed leg.

15 15 where σ k s a constant and Z k s a Q T k -Brownan moton, leads to Black s prcng formula: where Cplt(t, K; T k 1, T k) = τ kp (t, T k) Bl ( K, F k(t), σ k T k 1 t) (10) ( ) ( ) ln(f/k) + v 2 /2 ln(f/k) v 2 /2 Bl(K, F, v) = F Φ KΦ, v v and Φ denotes the standard normal dstrbuton functon. In our double-curve settng, the caplet valuaton requres more attenton. In fact, snce the prcng measure s now the forward measure Q T k t becomes for curve, the caplet prce at tme Cplt(t, K; Tk 1, Tk) = τkp (t, Tk)E T k { } [F k (Tk 1) K] + F t. As already explaned n the IRS case, the problem wth ths new expectaton s that the forward rate Fk s not, n general, a martngale under QT k. A possble way to value t s to model the dynamcs of Fk under ts own measure QT k and then to model the Radon- Nkodym dervatve dq T k /dq T k that defnes the measure change from QT k to Q T k. Ths s the approach proposed by Banchett (2008), who uses a foregn-currency analogy and derves a quanto-lke correcton for the drft of Fk. Here, nstead, we take a dfferent route. Our dea s to follow a conceptually smlar approach as n the classc LMM. There, the trck was to replace the LIBOR rate enterng the caplet payoff wth the equvalent forward rate, snce the latter has better dynamcs (a martngale) under the reference prcng measure. Here, we make a step forward, and replace the forward rate wth ts condtonal expected value (the FRA rate). The purpose s the same as before, namely to ntroduce an underlyng asset whose dynamcs s easer to model. Snce L k(t) = E T k [ ] F k (Tk 1) F t, at the reset tme T k 1 the two rates F k and L k concdes: L k(t k 1) = F k(t k 1). We can, therefore, replace the payoff (9) wth τ k[l k(t k 1) K] + (11) and vew the caplet as a call opton no more on Fk (T k 1 ) but on L k (T k 1 ). Ths leads to: Cplt(t, K; Tk 1, Tk) = τkp (t, Tk)E T k { } [L k (Tk 1) K] + F t. (12) The FRA rate L k (t) s, by defnton, a martngale under the measure QT k. If we smartly choose the dynamcs of such a rate, we can value the last expectaton analytcally and

16 16 obtan a closed-form formula for the caplet prce. For nstance, the obvous choce of a drftless geometrc Brownan moton dl k(t) = v k L k(t) dz k (t), t T k 1 where v k s a constant and Z k s now a Q T k -Brownan moton, leads to the followng prcng formula: Cplt(t, K; Tk 1, Tk) = τkp (t, Tk) Bl ( K, L k(t), v k Tk 1 t). Therefore, under lognormal dynamcs for the rate L k, the caplet prce s agan gven by Black s formula wth an mpled volatlty v k. The dfferences wth respect to the classc formula (10) are gven by the underlyng rate, whch here s the FRA rate L k, and by the dscount factor, whch here belongs to curve. 5.2 Market formula for swaptons The other plan-vanlla opton n the nterest rate market s the European swapton. A payer swapton gves the rght to enter at tme Ta = Tc S an IRS wth payment tmes for the floatng and fxed legs gven, respectvely, by Ta+1,..., Tb and T c+1, S..., Td S, wth T b = T d S and where the fxed rate s K. Its payoff at tme Ta = Tc S s therefore [ S a,b,c,d (T a) K ] + d τ S j P (T S c, T S j ), (13) where, see (7), S a,b,c,d(t) = b k=a+1 τ k P (t, T k )L k (t) τ S j P (t, T S j ). Settng C c,d (t) = τ S j P (t, T S j ) the payoff (13) s convenently prced under the swap measure Q c,d, whose assocated numerare s the annuty C c,d (t). In fact, we get: PS(t, K; Ta+1,..., Tb, Tc+1, S..., Td S ) {[ S = τj S P (t, Tj S ) E Qc,d a,b,c,d (Ta) K ] + τ j S P (Tc S, T S } j ) C c,d (T F t c S ) { [ = τj S P (t, Tj S ) E Qc,d S a,b,c,d (Ta) K ] + } Ft so that, also n our mult-curve paradgm, prcng a swapton s equvalent to prcng an opton on the underlyng swap rate. (14)

17 17 As n the sngle-curve case, the forward swap rate Sa,b,c,d (t) s a martngale under the swap measure Q c,d. In fact, by (6), S a,b,c,d (t) s equal to a tradable asset (the floatng leg of the swap) dvded by the numerare C c,d (t): S a,b,c,d(t) = b k=a+1 τ k P (t, T k )L k (t) τ S j P (t, T S j ) = FL(t; T a,..., T b ) C c,d (t). (15) Assumng that the swap rate Sa,b,c,d Brownan moton: evolves, under Qc,d, accordng to a drftless geometrc ds a,b,c,d(t) = ν a,b,c,d S a,b,c,d(t) dz a,b,c,d (t), t T a where ν a,b,c,d s a constant and Z a,b,c,d s a Q c,d -Brownan moton, the expectaton n (14) can be explctly calculated as n the caplet case, leadng to the generalzed Black formula: PS(t, K; T a+1,..., T b, T S c+1,..., T S d ) = τ S j P (t, T S j ) Bl ( K, S a,b,c,d(t), ν a,b,c,d T a t ). Therefore, the double-curve swapton prce s stll gven by a Black-lke formula, wth the only dfferences wth respect to the basc case that dscountng s done through curve and that the swap rate Sa,b,c,d (t) has a more general defnton. After havng derved market formulas for caps and swaptons under dstnct dscount and forward curves, we are now ready to extend the basc LMMs. We start by consderng the fundamental case of lognormal dynamcs, and then ntroduce stochastc volatlty n a rather general fashon. 6 The double-curve lognormal LMM In the classc (sngle-curve) LMM, one models the jont evoluton of a set of consecutve forward LIBOR rates under a common prcng measure, typcally some termnal forward measure or the spot LIBOR measure correspondng to the set of tmes defnng the famly of forward rates. enotng by T = {T0,..., TM } the tmes n queston, one then jontly models rates Fk, k = 1,..., M, under the forward measure QT M or under the spot LIBOR measure Q T. Usng measure change technques, one fnally derves prcng formulas for the man calbraton nstruments (caps and swaptons) ether n closed form or through effcent approxmatons. To extend the LMM to the mult-curve case, we frst need to dentfy the rates we need to model. The prevous secton suggests that the FRA rates L k are convenent rates to model as soon as we have to prce a payoff, lke that of a caplet, whch depends on LIBOR

18 18 rates belongng to the same curve. Moreover, n case of a swap-rate dependent payoff, we notce we can wrte S a,b,c,d(t) = b k=a+1 τ k P (t, T k )L k (t) τ S j P (t, T S j ) = b k=a+1 ω k (t)l k(t), (16) where the weghts ω k are defned by ω k (t) := τk P (t, Tk ) d τ j SP (17) (t, Tj S ). Characterzng the forward swap rate Sa,b,c,d (t) as a lnear combnaton of FRA rates L k (t) gves another argument supportng the modelng of FRA rates as fundamental brcks to generate sensble future payoffs n the prcng of nterest rate dervatves. Notce, also the consstency wth the standard sngle-curve case, where the forward LIBOR rates Fk (t) and the FRA rates L k (t) concde by defnton. However, there s a major dfference wth respect to the sngle-curve case, namely that forward rates belongng to the dscount curve need to be modeled too. In fact, as s evdent from equaton (16), future swap rates also depend on future dscount factors whch, unless we unrealstcally assume a determnstc dscount curve, wll evolve stochastcally over tme. Moreover, we wll show below that the dynamcs of FRA rates under typcal prcng measures depend on forward rates of curve, so that also path-dependent payoffs on LIBOR rates wll depend on the dynamcs of the dscount curve. 6.1 The model dynamcs The LMM was ntroduced n the fnancal lterature by Brace et al. (1997) and Mltersen et al. (1997) by assumng that forward LIBOR rates have a lognormal-type dffuson coeffcent. 19 Here, we extend ther approach to the case where the curve used for dscountng s dfferent than that used to generate the relevant future rates. For smplcty, we stck to the case where these rates belong to the same curve. Let us consder a set of tmes T = {0 < T0,..., TM }, whch we assume to be compatble wth curve. We assume that each rate L k (t) evolves, under ts canoncal forward measure Q T k, as a drftless geometrc Brownan moton: dl k(t) = σ k (t)l k(t) dz k (t), t T k 1 (18) where the nstantaneous volatlty σ k (t) s determnstc and Z k s the k-th component of an M-dmensonal Q T k -Brownan moton Z wth nstantaneous correlaton matrx (ρ k,j) k,j=1,...,m, namely dz k (t) dz j (t) = ρ k,j dt. 19 Ths mples that each forward LIBOR rate evolves accordng to a geometrc Brownan moton under ts assocated forward measure.

19 19 In a double-curve settng, we also need to model the evoluton of rates Fk (t) = F (t; Tk 1, Tk) = 1 [ P (t, Tk 1 ) ] τk P (t, Tk ) 1 τk = τ (Tk 1, Tk) To ths end, we assume that the dynamcs of each rate F h measure Q T h under the assocated forward s gven by: df h (t) = σ h (t)f h (t) dz h (t), t T h 1 (19) where the nstantaneous volatlty σh (t) s determnstc and Z h s the h-th component of an M-dmensonal Q T h -Brownan moton Z whose correlatons are dzk (t) dzh (t) = ρ, k,h dt dz k (t) dzh (t) = ρ, k,h dt Clearly, correlatons ρ = (ρ k,j ) k,j=1,...,m, ρ, = (ρ, k,h ) k,h=1,...,m and ρ, = (ρ, k,h ) k,h=1,...,m must be chosen so as to ensure that the global matrx [ ] ρ ρ, R := ( ) ρ, s postve (sem)defnte. Remark 2 In some stuatons, t may be more realstc to resort to an alternatve approach and model ether curve or jontly wth the spread between them, see e.g. Kjma (2008) or Schönbucher (2000). Ths happens, for nstance, when one curve s above the other and there are sound fnancal reasons why the spread should be preserved postve n the future. In such a case, one can assume that each spread X k (t) := L k (t) F k (t) evolves under the correspondng forward measure Q T k, accordng to some ρ, dx k (t) = σ X k (t, X k (t)) dz X k (t), whose soluton s postvely dstrbuted. Stckng to (18), the dynamcs (19) of forward rates Fk must then be replaced wth df k (t) = dl k(t) ± dx k (t), where the sgn ± depends on the relatve poston of curves and. The analyss that follows can be equvalently appled to the new dynamcs of rates F k The calculatons are essentally the same. Ther length depends on the chosen volatlty functon σ X k.

20 ynamcs under a general forward measure To derve the dynamcs of the FRA rate L k (t) under the forward measure QT j we start from (18) and perform a change of measure from Q T k to QT j, whose assocated numerares are the curve- zero-coupon bonds wth maturtes Tk and T j, respectvely. To ths end, we apply the change-of-numerare formula relatng the drfts of a gven process under two measures wth known numerares, see for nstance Brgo and Mercuro (2006). The drft of L k (t) under QT j s then equal to rft(l k; Q T j ) = d L k, ln(p (, Tk )/P (, Tj )) t, dt where X, Y t denotes the nstantaneous covaraton between processes X and Y at tme t. Let us frst consder the case j < k. The log of the rato of the two numerares can be wrtten as ( [ k ]) ln(p (t, Tk)/P (t, Tj )) = ln 1/ (1 + τh Fh (t)) from whch we get: = k h=j+1 h=j+1 ln ( 1 + τ h F h (t) ) rft(l k; Q T j ) = d L k, ln(p (, T k )/P (, T j )) t dt = k h=j+1 τ h d L k, F h t 1 + τh F h (t). dt = k h=j+1 d L k, ln ( 1 + τh F h dt ) t In the standard LMM, the drft term of L k under QT j depends on the nstantaneous covaratons between forward rates Fk and F h, h = j + 1,..., k. The ntal assumptons on the jont dynamcs of forward rates are therefore suffcent to determne such a drft term. Here, however, the stuaton s dfferent snce rates L k and F h belong, n general, to dfferent curves, and to calculate the nstantaneous covaratons n the drft term, we also need the dynamcs of rates Fh. Under (19), we thus obtan rft(l k; Q T j ) = σ k(t)l k(t) k h=j+1 ρ, k,h τ h σ h (t)f h (t) 1 + τ h F h (t). The dervaton of the drft rate n the case j > k s perfectly analogous.

21 21 As to forward rates Fk, ther QT j -dynamcs are equvalent to those we obtan n the classc sngle-curve case, see Brgo and Mercuro (2006), snce these probablty measures and rates are assocated to the same curve. The jont evoluton of all FRA rates L 1,..., L M and forward rates F 1,..., FM under a common forward measure s then summarzed n the followng. Proposton 3 The dynamcs of L k and F k cases j < k, j = k and j > k are, respectvely, [ k dl k (t) = σ k(t)l k (t) j < k, t Tj h=j+1 : [ k df k (t) = σ k (t)f k (t) { dl k (t) = σ k(t)l k (t) dzj k (t) j = k, t T k 1 : j > k, t T k 1 : dfk (t) = σ k (t)f k under the forward measure Q T j h=j+1 (t) dzj, k [ dl k (t) = σ k(t)l k (t) [ dfk (t) = σ k (t)f k (t) h=k+1 ρ, k,h τ h σ h (t)f h (t) 1 + τh F h (t) dt + dz j k (t) n the three ρ, k,h τ h σ h (t)f h (t) ] 1 + τh F h (t) dt + dz j, k (t) (t) j ρ, k,h τ h σ h (t)f h (t) ] 1 + τh F h (t) dt + dz j k (t) j ρ, k,h τ h σ h (t)f h (t) ] 1 + τh F h (t) dt + dz j, k (t) h=k+1 where Z j k and Zj, k are the k-th components of M-dmensonal Q T j -Brownan motons Zj and Z j, wth correlaton matrx R. Remark 4 Followng the same arguments used n the standard sngle-curve case, we can easly prove that the SEs (20) for the FRA rates all admt a unque strong soluton f the coeffcents σh are bounded. When curves and concde, we have already notced that the FRA rates L k concde wth the correspondng Fk. As a further santy check, we can also see that the FRA rate dynamcs reduce to those of the correspondng forward rates snce ρ, k,h ρ k,h τh τ h σh (t) σ h(t) Fh (t) F h (t) for each h, k. The extended dynamcs (20) may rase some concern on numercal ssues. In fact, havng doubled the number of rates to smulate, the computatonal burden of the lognormal ] (20)

22 22 LMM (20) s doubled wth respect to that of the sngle-curve case, snce the SEs for the homologues L k and F k share the same structure. However, some smart selecton of the correlatons between rates can reduce the smulaton tme. For nstance, assumng that ρ, k,h = ρ, k,h for each h, k, leads to the same drft rates for L k and the correspondng F k, thus halvenng the number of drfts to be calculated at each smulaton tme. Ths gves a valuable advantage snce t s well known that the drft calculatons n a LMM are extremely tme consumng. 6.3 ynamcs under the spot LIBOR measure Another measure commonly used for modelng the jont evoluton of the gven rates and for prcng related dervatves s the spot LIBOR measure Q T assocated to tmes T = {T0,..., TM }, whose numerare s the dscretely-rebalanced bank account BT B T (t) = P (t, T β(t) 1 ) β(t) 1 j=0 P (T j 1, T j ), where β(t) = m f T m 2 < t T m 1, m 1, so that t (T β(t) 2, T β(t) 1 ]. Applcaton of the change-of numerare technque, mmedately leads to the followng. Proposton 5 The dynamcs of FRA and forward rates under the spot LIBOR measure Q T are gven by: k ρ, dl k(t) = σ k (t)l k,h k(t) τ h σ h (t)f h (t) 1 + τh F h (t) dt + σ k (t)l k(t) dzk(t) d df k (t) = σ k (t)f k (t) k ρ, k,h τ h σ h (t)f h (t) 1 + τh F h (t) dt + σk (t)fk (t) dz d, k (t) where Z d = {Z1, d..., ZM d } and Zd, = {Z d, 1,..., Z d, M } are M-dmensonal QT -Brownan motons wth correlaton matrx R. 6.4 Prcng caplets n the lognormal LMM The prcng of caplets n the LMM s straghtforward and follows from the same arguments of Secton 5. We get: where Cplt(t, K; T k 1, T k) = τ kp (t, T k) Bl(K, L k(t), v k (t)) v k (t) := T k 1 t σ k (u) 2 du As expected, thanks to the lognormalty assumpton, ths formula for caplets (and hence caps) s analogous to that obtaned n the basc lognormal LMM. We just have to replace forward rates wth FRA rates and use the dscount factors comng from curve. (21)

23 Prcng swaptons n the lognormal LMM An analytcal approxmaton for the mpled volatlty of swaptons can be derved also n our mult-curve settng. To ths end, remember (16) and (17): S a,b,c,d(t) = ω k (t) = b k=a+1 ω k (t)l k(t), τ k P (t, T k ) τ S j P (t, T S j ). The forward swap rate Sa,b,c,d (t) can be wrtten as a lnear combnaton of FRA rates L k (t). Contrary to the sngle curve case, the weghts are not a functon of the FRA rates only, snce they also depend on dscount factors calculated on curve. Therefore we can not wrte that, under the swap measure Q c,d, the swap rate S a,b,c,d (t) satsfes the S..E. ds a,b,c,d(t) = b k=a+1 S a,b,c,d (t) L k (t) σ k (t)l k(t) dz c,d k (t). However, we can resort to a standard approxmaton technque and freeze the weghts ω k at ther tme-zero value. Ths leads to the approxmaton whch enables us to wrte ds a,b,c,d(t) S a,b,c,d(t) b k=a+1 b k=a+1 ω k (0)L k(t), ω k (0)σ k (t)l k(t) dz c,d k (t). (22) Notce, n fact, that by freezng the weghts, we are also freezng the dependence of S a,b,c,d (t) on forward rates F h. To obtan a closed equaton of type ds a,b,c,d(t) = S a,b,c,d(t)v a,b,c,d (t) dz a,b,c,d (t), (23) we equate the nstantaneous quadratc varatons of (22) and (23) [ va,b,c,d (t)s a,b,c,d(t) ] 2 dt = b h,k=a+1 ω h (0)ω k (0)σ h (t)σ k (t)l h(t)l k(t)ρ h,k dt. (24) Freezng FRA and swap rates at ther tme-zero value, we obtan ths (approxmated) Q c,d -dynamcs for the swap rate S a,b,c,d (t): b dsa,b,c,d(t) = S a,b,c,d(t) h,k=a+1 ω h(0)ω k (0)σ h (t)σ k (t)l h (0)L k (0)ρ h,k dz (Sa,b,c,d a,b,c,d (t). (0))2

24 24 Ths mmedately leads to the followng (payer) swapton prce at tme 0: PS(0, K; T a+1,..., T b, T S c+1,..., T S d ) = τ S j P (0, T S j ) Bl ( K, S a,b,c,d(0), V a,b,c,d ), where the swapton mpled volatlty (multpled by T a) s gven by V a,b,c,d = b h,k=a+1 ω h (0)ω k (0)L h (0)L k (0)ρ h,k (S a,b,c,d (0))2 T a 0 σ h (t)σ k (t) dt. (25) Agan, ths formula s analogous n structure to that obtaned n the classc lognormal LMM, see Brgo and Mercuro (2006). The dfference here s that the swapton volatlty depends both on curves and, snce weghts ω s belong to curve, whereas the FRA and swap rates are calculated wth both curves. A better approxmaton for lognormal LMM swapton volatltes can be derved by assumng that each Tj S belongs to T = {T0,..., TM }, so that, for each j, there exsts an ndex j such that Tj S = T j. In ths case, we can wrte: ω k (t) = = τ k P (t, T k ) τ S j P (t, T S j ) = k τk h=a+1 τ S j τ h F h (t) j a=c τ h F h (t) τk P (t, Tk ) P (t, Ta) τ P (t, T j S j ) P (t, Ta) =: f(f a+1(t),..., F b (t)) where the last equalty defnes the functon f and where the subscrpts of rates Fh (t) range from a + 1 to b snce Td S = T b (namely d = b). Under the swap measure Q c,d, the swap rate S a,b,c,d (t) then satsfes the S..E. ds a,b,c,d(t) = b k=a+1 S a,b,c,d (t) L k (t) σ k (t)l k(t) dz c,d k (t) + b k=a+1 Sa,b,c,d (t) Fk (t) σ k (t)fk (t) dz c,d, k (t), where {Z c,d 1,..., Z c,d M } and {Zc,d, 1,..., Z c,d, M } are M-dmensonal Qc,d -Brownan motons wth correlaton matrx R. Matchng nstantaneous quadratc varatons as n (24) and freezng stochastc quanttes at ther tme-zero value, we can fnally obtan another, more accurate, approxmaton for the mpled swapton volatlty n the lognormal LMM, whch we here omt for brevty.

25 The termnal correlaton between FRA rates Assume we are nterested to calculate, at tme 0, the termnal correlaton between the FRA rates L k and L h at tme T j, wth j k 1 < h, under the forward measure Q T r, wth r j: Corr T ( r L k (Tj ), L h(tj ) ) = E T r E T r [ (L k (T j ))2] [ E T r [ L k (T j )L h (T j ) ] E T r ( L k (T j ))] 2 E T r [ L k (Tj ) ] E T [ r L h (Tj ) ] [ (L h (T j ))2] [ E T r ( L h (T j ))] 2 Mmckng the dervaton of the approxmaton formula n the sngle-curve lognormal LMM, see Brgo and Mercuro (2006), we frst recall the dynamcs of L x, x = k, h, under Q T r : where dl x(t) = µ x (t)l x(t) dt + σ x (t)l x(t) dz x (t) x ρ, x,l σ x (t) τ l σl (t)fl (t) f r < x 1 + τ l=j+1 l Fl (t) µ x (t) := 0 f r = x j ρ, x,l σ x (t) τ l σl (t)fl (t) f r > x 1 + τl Fl (t) l=x+1 Then, n the drft rate µ x (t), we freeze the forward rates Fl (t) at ther tme-0 value to obtan: dl x(t) = ν x (t)l x(t) dt + σ x (t)l x(t) dz x (t) where x ρ, x,l σ x (t) τ l σl (t)fl (0) f r < x 1 + τ l=j+1 l Fl (0) ν x (t) := 0 f r = x j ρ, x,l σ x (t) τ l σl (t)fl (0) f r > x 1 + τl Fl (0) l=x+1 Snce ν x s determnstc, L x follows (approxmately) a geometrc Brownan moton. The expectatons (26) are thus straghtforward to calculate. We get: E T [ r L x (Tj ) ] { T } = L j x(0) exp ν x (t) dt, x = k, h 0 [( L x (Tj ) ) { T 2] = (L x (0)) 2 j exp E T r E T r [ L k (T j )L h(t j ) ] = L k(0)l h(0) exp { T j 0 } [2ν x (t) + (σ x (t)) 2 ] dt, x = k, h 0 } [ν k (t) + ν h (t) + ρ k,h σ k (t)σ h (t)] dt (26)