MULTIPLE CURVE CONSTRUCTION

MULTIPLE CURVE CONSTRUCTION RICHARD WHITE 1. Introducton In the post-credt-crunch world, swaps are generally collateralzed under a ISDA Master Agreement Andersen and Pterbarg p266, wth collateral rates beng Fed Funds USD, Eona Euro, Sona GBP, etc.. Ths dvorces the forward Lbor rates from the dscount factors, meanng two curves are needed to prce swaps: a rsk-free dscount curve whch s used to dscount future cash flows, and a rsky ndex or forward curve from whch the forward Lbor rates are derved. Further to ths, there exsts tenor swaps whch pay a partcular Lbor tenor e.g. 3M Lbor every 3 months plus a spread, and receve a dfferent Lbor tenor e.g. 6M Lbor every sx months. Generally, a postve spread s pad on the leg of the shorter tenor, reflectng the greater rsk of the longer tenor. So for a partcular country a whole famly of curves must be created - ths could be expressed as a rsk-free curve e.g. OIS for USD, and a set of spread curves, one for each Lbor tenor. Each country would need ts own set of such curves. Uncollateralzed nterest rate dervatves have credt rsk, and proper treatment requres a credt valuaton adjustment CVA; somethng that we do not consder here. 2. Notaton P X,I t, T The pseudo dscount factor from tme T to t for currency X and ndex I - f the ndex superscrpt s absent we have the rsk-free curve. P X,I T P X,I 0, T s today s dscount curve. For example P $,3M T s the USD 3M Lbor curve. R X,I t, T 1 T t lnp X,I t, T s the yeld curve at tme t. L X,I t L X,I t, S X,I, T X,I E X,T t L X,I t, S X,I, T X,I ] s the forward ndex e.g. Lbor rate, where the expectaton s taken at tme t n the T forward measure.e. the numerare s P X t, T. The tmes S X,I and T X,I are the fxng effectve and maturty tmes of the ndex. t X 1... tx M s the set of fxed payment tmes t X,I 1... t X,I N s the set of floatng payment tmes τ X s the year fracton for the fxed payment at tme t X α X,I s the year fracton for the ndex rate L X,I N X an amount n currency X Date: Frst verson: September 26, 2011; ths verson March 30, 2012. Verson 1.0. 1

2 RICHARD WHITE f X,Y t the spot exchange rate at tme t - amount of currency X per unt of currency Y. So N X = f X,Y N Y and f X,Y = f Y,X 1 f X,Y t, T forward exchange rate seen at tme t for exchange at tme T When the currency and/or ndex s clear we wll suppress these superscrpts. 3. Instruments We now present how varous Interest Rate Dervatves IRD can be prced usng multple curves. The goal, of course, s to construct multple curves from market prces 1 of IRD. The companon document The Analytc Framework for Implyng Yeld Curves from Market Data dscusses ths n detal. Essentally, yeld curves are represented as splnes and the dscount factors trvally derved from these, wth nodes or knots at the maturtes of IRD. If the total number of nodes across all relevant curves equals the total number of IRD senstve to these curves.e. have prces that depend on the curves, then t s just a mult-dmensonal root-fndng excse to fnd the values of the nodes that prce the IRD back to the market exactly. 3.1. Interest Rate Swaps IRS. The forward ndex s related to the ndex curve pseudo dscount curve by 1 L X,I t = 1 P X,I t, S X,I α X,I P X,I t, T X,I 1 Ths allows us to wrte down the floatng leg reference ndex I n currency X as P V X,I float = N =1 α X,I 2 N P X,I t, S X,I = =1 P X,I t, T X,I 1 and fxed leg of a swap n currency X as M 3 P V X fxed = k =1 L X,I P X t, t X,I P X t, t X,I τ X P X t, t X Here the fact the the floatng leg depends on two curves s clear. Academc treatment normally has T X,I = S X,I X,I 1.e. spannng forward rates and T = t X,I.e. the floatng payment s on the same date as the maturty of the forward rate. Nether of these condtons necessarly holds n practce they wll dffer by a few days, so three sets of dates wll be needed for the floatng leg. The PV of a payer swap.e. pay fxed and receve floatng payments based on the ndex I, n currency X s N 4 P V X,I t = =1 P X,I t, S X,I P X,I t, T X,I 1 P X t, t X,I k M τ X P X t, t X 1 prce n a generc sense - n practce t could be swap rates, spreads, etc. that make an nstrument have zero PV =1

so the par-swap rate s gven by 5 k = MULTIPLE CURVE CONSTRUCTION 3 N =1 P X,I t,s X,I P X,I t,t X,I 1 M =1 τ X P X t, t X P X t, t X,I The market wll gve a set of swap rates for dfferent length swaps typcally 2 to 30 years. Wthout makng some further assumptons e.g. constant spread, t s not possble to back out both the dscount and ndex curves just from the set of swaps. However, f the dscount curve were extraneously gven, then a ndex curve wth the same number of nodes as swap rates s easly found 2. 3.2. OIS Swaps. In the USD market, from one week to twelve months, an OIS swap has a sngle floatng payment equal to the geometrc average of the Fed Funds overnght rate. That s, N 6 float leg = 1 + r N year =1 where r s the overnght rate on day, N s the number of days n the perod, and N year s the number of days n a year. Ths can be approxmated by a contnuously compounded rate t1 7 float leg = exp r s ds wth t 0 beng the trade date normally 1 or 2 days from now and t 1 beng the payment date. The fxed leg s just 1 + αk where α s the year fracton. Ths means that the PV of ths swap s 8 P V =E =E e R t 1 0 rsds R t1 e t 0 ] t r sds 0 1 + αk e R t 0 0 rsds] 1 + αke e R t 1 0 rsds] where the rsk-free rate used for dscountng s dentcal to the overnght rate - the OIS curve s the rsk-free curve for USD and other markets where smlar nstruments exst. Ths can be wrtten n terms of zero coupon bounds, P 0, t as 9 P V = P 0, t 0 1 + αkp 0, t 1 whch for zero PV gves 10 k = 1 α P 0, t0 P 0, t 1 1 whch of course s exactly the form for Lbor rates. Therefore, out to one year we can prce OIS swaps usng a sngle rsk-free curve, and by nverson we can construct the OIS curve to one year from the OIS swap rate. 2 The reverse does not necessarly hold, as the payment nettng means there s only weak senstvty of the swap rate or equvalently the PV to the dscount curve.

4 RICHARD WHITE For two to ten years, OIS swaps have annual coupons and can be handled exactly lke ordnary fxed-float IRS where the ndex and dscount curves are both the OIS rskfree curve. Beyond 10 years n the USD market we must consder bass swaps see next secton. In the Euro and GBP markets, EONIA and SONIA swaps go out to 30 years, whch allows constructon of full rsk-free curves n these currences. The JPY market only has OIS swaps to 3 years, and cross-currency swaps CCS are needed to construct the full OIS curve see secton 3.5. Ths s also the case n other OECD countres. 3.3. Bass Swap n a Sngle Currency. A bass swap exchanges payments based on one ndex e.g. Fed-Funds for another e.g. 3m-Lbor on the same notonal amount. A tenor swap exchanges payments based on dfferent Lbor tenors, e.g. 3 month Lbor pad quarterly for 6 month Lbor pad sem-annually. If legs of the swap A & B have reference rates, and L X,B, the PV of the recever of leg A s 11 N A P V X = =1 N A = =1 N B =1 α X,A P X,A t, S X,A P X t, t X,A P X,A t, T X,A 1 P X,B t, S X,B P X,B t, T X,B N B =1 α X,B L X,B + sp X t, t X,B P X t, t X,A 1 + αx,b s P X t, t X,B where s s the spread, whch n ths set up can take postve or negatve values. It s clear that the PV depends on the value of three curves. Makng the PV zero, the spread becomes 12 s = NA =1 αx,a P X t, t X,A N B NB =1 αx,b =1 αx,b P X t, t X,B L X,B P X t, t X,B If two of the curves are already known, these spreads can be used to construct the thrd. For example, f the dscount curve from OIS swaps and the 3M Lbor curve from Lbor rates, FRAs and IRS are known, the 6M Lbor curve can be found from 3M-6M tenor swap spreads. The nterestng thng for the USD market s usng the US Fed Funds spreads - the USD Bass Swap Fed Funds versus USD three-month Lbor - to extend the OIS curve to 30 years. If the Fed Funds leg were just the geometrc average of the FF effectve rate as s the case for OIS, then there would be just two curves nvolved - the OIS curve for dscountng and calculatng the FF-based floatng payments and the 3M Lbor curve. Snce these are also the two curves nvolved n prcng USD IRS, the two sets of market nformaton US Fed Funds spreads and USD swap rates can be used to smultaneously construct the OIS and 3M Lbor curves out to 30 years.

MULTIPLE CURVE CONSTRUCTION 5 Unfortunately, the FF bass swap cash flows are calculated from an arthmetc average of the effectve rate. The payment can be wrtten as α N 13 r N =1 where α s the year fracton and the ndex runs of the days n the perod. The expected PV of ths payment can be approxmated by 14 L $,F F = E e R t t+1 ] +1 0 r sds r s ds wth r s beng the rsk-free short rate. If r s s hgh n the observaton perod between t and t +1 then the payment at t +1 wll be hgh, but the dscountng wll be greater lowerng the overall effect on the PV of the payment the same argument holds f r s s low n that perod. The smplest approach s pretend the the averagng s geometrc and proceed as for OIS. A more correct approach would be to express equaton 14 n terms of dscount factors and a convexty adjustment. 3.4. Forward FX rates. The forward FX rate s related to the spot and the dscount curves n each currency by 15 f X,Y t, T = f X,Y t P Y t, T P X t, T These nstruments are lqud n a range of currency pars out to a few years, allowng the short end of a dscount curve n one currency to be found from the short end n another 3. 3.5. Cross-Currency Swaps CCS. A CCS s effectvely an exchange of a FRN wth notonal N X n currency X for one wth notonal N Y n currency Y. On the trade date typcally a couple of days forward the notonal amounts are exchanged. Coupon payments are then made at defned dates, wth the reverse exchange of notonals at maturty. 3.5.1. Fxed-Fxed CCS. For a fxed-for-fxed CCS the PV from a currency X nvestor s 16 N X P V =N x P X 0, T + P X 0, T + k X τ X P X 0, t X =1 N Y N Y P X 0, t 0 f X,Y 0, t 0 + P X 0, T f X,Y 0, T + k Y τ Y P X 0, t Y f X,Y 0, t Y Here coupon payments can be at dfferent tmes on each leg, but the ntal and fnal reverse exchange of notonal are at tmes t 0 and T. The NPV of the ntal exchange s 17 NP V = N x P X 0, t 0 + N Y P X 0, t 0 f X,Y 0, t 0 3 Ths doesn t requre root fndng - the node values can just be read off. t =1

6 RICHARD WHITE whch can be made zero by settng N Y = N X f X,Y 0, t 0. It can be the case that N Y = N X f X,Y 0 whch means there could be a small ntal payment f the two day forward FX rate dffers from spot. Assumng the former condton holds, we rewrte the PV as takng N X = 1 18 N X N Y P V = P X 0, T + k X τ X P X 0, t X f X,Y 0 P Y 0, T + k Y τ Y P Y 0, t Y =1 If k X s set to make the X currency payments PV to zero.e. k X = P X 0,t 0 P X 0,T P, NX =1 τ XP X 0,t X then the PV of the whole swap becomes N Y 19 P V = P X 0, t 0 f X,Y 0 P Y 0, T + k Y τ Y P Y 0, t Y So for a far swap we must set 20 k Y = f X,Y 0P Y 0, T P X 0, t 0 NY =1 τ Y P Y 0, t Y If a set of fxed coupons k Y are gven then the currency Y dscount curve can be constructed. 3.6. Float-Float CCS. Ths exchanges floatng payments based on an ndex n one currency for floatng payments based on an ndex n a dfferent currency. Assumng the same treatment of notonal payments are above.e. they net to zero, then PV of ths swap from the pont of vew of the X nvestor s 21 P V =P X 0, T + The spread s for a far swap s 22 N X =1 α X,A =1 P X 0, t X,A =1 =1 N Y f X,Y 0 P Y 0, T + α Y,B L Y,B + sp Y 0, t Y,B s = P X 0, T f X,Y 0P Y 0, T + N X =1 αx,a NY =1 αy,b P X 0, t X,A f X,Y 0 N Y P Y 0, t Y,B =1 αy,b The market values PV or spread depends on four curves - the dscount curve n each currency and the ndex curve n each currency. As a concrete example, assume we have already constructed the USD dscount curve and 3M Lbor curve from OIS swaps, Fed Funds bass swaps, Lbor rates, FRAs and swaps. The short end of the JPY dscount curve can be covered by OIS swaps whch go out to 3 years n JPY, and the short end of the Lbor curve by Lbor rate and FRA out to 2 years. However there s no OIS market beyond 3 years, and ths s where USDJPY CCS spreads swappng 3M USD Lbor for 3M JPY Lbor and JPY IRS swappng fxed for 3M Lbor n the JPY domestc market come n. Usng these together wth L Y,B P Y 0, t Y,B

MULTIPLE CURVE CONSTRUCTION 7 the known USD curves we can construct the JPY dscount curve and 3M Lbor curve out to 30 years. There s an addtonal complcaton to the above story: The lqud USDJPY CCS market s n 3M Lbor, but the man JPY IRS market s n 6M Lbor. However there s a JPY 3M/6M tenor swap. So we wll need to ft 3 curves smultaneously usng the USD curves together wth OIS swaps, FRA 6M, IRS 6M tenor, USDJPY CCS and 3M/6M JPY tenor swaps. 4. Spread Representaton Varatons of the USD 3M Lbor curve from the dscount OIS curve wll generally be small. If we treat the dscount curve n each currency as the fundamental curve, then all of the varous ndex curves can be represented as spread curves over ths,.e. 23 P X,I t, T = Q X,I t, T P X t, T or, n terms of the yeld curves 24 R X,I t, T = S X,I t, T + R X t, T where S X,I t, T = 1 T t lnqx,i t, T. The spread curves could then be endowed wth far fewer nodes than the fundamental curve, whle the ndex curves stll dsplay complex structure nherted from the fundamental curve. E-mal address: rchard@opengamma.com