A heuristic pricing and hedging framework for multi-currency fixed income desks
aa r X i v : . [ q -f i n . P R ] J a n A heuristic pricing and hedging framework formulti-currency fixed income desks
Eduard Gim´enez ∗ Alberto Elices † Giovanna Villani ‡ .January 9, 2017 Abstract
It is well known that traded foreign exchange forwards and crosscurrency swaps (CCS) cannot be priced applying overnight cash andcarry arguments as they imply absence of funding advantage of onecurrency to the other. This paper proposes a heuristic present valueconcept for multi-currency pricing and hedging which allows takinginto account the funding and therefore the collateral currency andits pricing impact. For uncollateralized operations, it provides morefunding optionality to achieve either cheaper or more connected fund-ing to the hedging instruments. When foreign exchange forwards getaligned with overnight cash and carry arguments, this method nat-urally converges to the well established OIS discounting where eachleg is funded in its own currency. A worked example compares thisapproach with a benchmark.
Before the financial crisis started in July 2007 with Bear Stearns default,interest rate desks would essentially use a unique interest rate curve for each ∗ Head of Model Development Group, Front Office, CaixaBank, Av. Diagonal, 621-629T.I. P13, Barcelona, Spain [email protected] . † Head of XVA Model Validation, Model Risk, Bank Santander, Av. Cantabria s/n,28660 Boadilla del Monte, Spain, [email protected] . ‡ Quantitative Analyst, Front Office, CaixaBank, Av. Diagonal, 621-629 T.I. P13,Barcelona, Spain, [email protected] . , this heuristic multi-currency pricingconverges to OIS cash flow discounting in the currency in which cash flowsare denominated. This heuristic framework is valid only for the most fre-quently traded products (e.g. swaps and cross currency swaps). Valuationof complex multi-currency exotic products such as callable swaps are beyondthe scope of this framework.When the paper assumes full collateralization of trades (see [1]), it alsoconsiders a funding structure between Front Office desks and the balancesheet of the bank at the overnight index-rate, provided that desks do nothave a consistent liquidity imbalance between borrowing and lending. Whendealing with uncollateralized pricing and even if these hypotheses are notfulfilled, the collateralized framework will always provide a reference or the-oretical price on top of which funding and credit value adjustments can beadded.Section 2 shows how the funding (and collateral) currency is chosen fortrading. The proposed decomposition method to price cross currency swapsand foreign exchange forwards is presented in sections 3 and 4. Section 5compares the proposed method with a benchmark using a worked example.Finally, section 6 concludes. The three basic structures which will be considered in this paper are thefloating rate note (FRN), the resettable or marked-to-market cross currencyswap (CCS) and the non-resettable cross currency swap (NCS). They willbe respectively denoted by
F RN s C t ,t N , CCS s Cd ,s Cf t ,t N and N CS s Cd ,s Cf t ,t N , where C d , C f and C are domestic, foreign or any given currency in which legs aredenominated, t and t N are the starting and ending dates of the structures, s C are the spreads added to the floating leg denominated in currency C and X i is the foreign exchange rate fixing at t i to convert 1 unit of domesticcurrency to foreign. According to this notation, a long position on FRNreceives the notional at t and pays it back at t N and in between a floatingleg of a given tenor frequency plus the spread s C is paid. A long position Overnight cash and carry arguments imply FX forwards calculated with OIS discountfactors in both currencies. This implicitly assumes that funding in both currencies issymmetric and there is not a funding advantage of one currency with respect to the other.
3n NCS exchanges notionals at t (receives 1 domestic and pays X foreign),does the reverse notional exchange at t N and in between, floating paymentsplus spreads are exchanged (foreign are received and domestic paid). CCSfollows the same convention but resets notionals on each payment date . N CS s Cd ,s Cf t ,t N = F RN s Cd t ,t N − X t · F RN s Cf t ,t N (1) N CS s Cd ,s Cf t ,t N = CCS s Cd ,s Cf t ,t N + N − X i =1 (cid:16) X t i − X t i − (cid:17) F RN s Cf t i ,t N (2)Equation (1) presents the decomposition of a non-resettable NCS into asum of two floating rate notes and equation (2) shows how a NCS can bedecomposed into a resettable CCS plus a sum of floating rate notes, where X t i denotes the foreign exchange fixing on t i .Consider the operator, V C d t [ C f { t = T } ], where the indicator function, C f { t = T } ,represents a cash flow payment in foreign currency C f at time T . Thisoperator is defined as the present value (at time t ) in domestic currency C d ,of a cash flow denominated in C f when it is funded in C d . Equation (3) showshow this operator is defined depending on whether the funding and cash flowcurrencies are the same. DF Ct,T is the OIS discount factor between t and T of currency C and X t,T is the forward exchange rate at time t to changeone unit of currency C d to C f at time T (these forward exchange rates areinterpolated according to the end of section 3). Therefore, the operator V issimply defined as OIS discounting when both the cash flow and the fundingcurrencies are the same and when they are different, the operator is definedas the cash flow value converted to the funding currency using the forwardforeign exchange and thereafter OIS discounted in that currency. Whenthis operator is applied to a linear combination of cash flows, the result isdefined as the linear combination of the operator applied to each cash flow.Cash flows are assumed to be deterministic and no probability measure isdefined. In this context, the operator only calculates present values usingOIS discounting and it will be assumed that before applying the operator V ,to FRN, XCS and CCS structures, all contingent index-rate fixings will have On top of the NCS floating exchanges, on each payment date, t i , a reverse notionalexchange from previous date is performed (pay 1 domestic and receive X i − foreign) alongwith an initial notional exchange on current date (receive 1 domestic and pay X i foreign).Domestic notional exchanges cancel each other and foreign compensate each other andonly the difference is exchanged (receive X i − − X i foreign). V Ct h C { t = T } i = DF Ct,T V C d t h C f { t = T } i = X − t,T DF C d t,T (3)Since current regulation enforces inter-bank trades to be liquidated throughClearing Counter Parties (CCP) such as London Clearing House or exchangecollateral in form of variation margin for over-the-counter structures and col-lateral interest payments are indexed to overnight rates such as EONIA, FedFunds, Sonia, etc, it is well accepted in the market place to use OIS dis-counting as the reference funding rate in each currency. In the ISDA MasterAgreement which defines the netting set in case of default for a given coun-terparty, the CSA (Credit Support Annex) specifies the currency in whichcollateral is exchanged for a set of products. In case of default, the deals as-sociated to each CSA net out in the CSA currency and the collateral balanceis taken to compensate for the loss. Therefore, the funding currency is spec-ified by the CSA as the set of deals associated with it exchange collateralin that currency and the funding rate is the OIS. Therefore, the heuristicmethod takes into account the currency of the collateral, by changing eachdeterministic cash flow from its own currency to the collateral currency us-ing the foreign exchange forward and thereafter by OIS discounting in thatcurrency according to the right side of equation (3). V C d t (cid:20) F RN s Cf t ,t N (cid:21) = DF C d t,t X t,t − N − X i =0 (cid:16) L C f t,t i + s C f (cid:17) τ C f i DF C d t,t i +1 X t,t i +1 − DF C d t,t N X t,t N (4) V C d t h F RN s Cd t ,t N i = DF C d t,t − N − X i =0 (cid:16) L C d t,t i + s C d (cid:17) τ C d i DF C d t,t i +1 − DF C d t,t N (5)Equations (4) and (5) show two examples of how the V operator is definedto calculate the present value expressed and funded in domestic units of anFRN denominated in foreign and domestic currencies. L Ct,t i is the forwardLibor index rate of currency C at time t of the period from t i to t i +1 and τ Ci is the year fraction from t i to t i +1 according to the conventions of currency C .See that the result of equation (4) will not be equal to V C f t h F RN s Cf t ,t N i X − t ,unless foreign exchange forwards get aligned with overnight cash and carryarguments as the funding currency is different.5he present values provided by the operator V are indeed a heuristicapproximation of the actual price. Convexity corrections or the effect ofcorrelation between Libor and foreign exchange rates are ignored. The errorsof this approximation will not have much impact, because the floating cashflows which do not net out in the derivative portfolio will be hedged andtherefore swapped into fixed deterministic cash flows which will be properlypriced according to the V operator. Four-curve method Heuristic method
Build FF and EO OIS curves forcing overnight index swaps (OIS) = 0.Build U3M & E6M estimation curves forcing most liquid IRS = 0 anddiscounting with FF and EO.Build USLIBOR & EURIBOR estimation curves for rest of tenors forc-ing tenor swaps = 0.Build USD collateralized EUR BA-SIS discount curve forcing CCS =0. Calculate FX forward curve withFF and EUR BASIS curves. Build FX forward curve from mkt(short term) and forcing USDfunded heuristic value of CCS = 0(medium and long term).Build EUR collateralized USD BA-SIS discount curve from unchangedFX forwards and EO OIS curve. Build EUR col. CCB curve forcingEUR funded heuristic value of CCS= 0 with unchanged FX forwards.
EUR col : use estimation curves,discount with EO and USD BASIS.
US col : use estimation curves, dis-count with FF and EUR BASIS. Apply heuristic decomposition andvaluation funded in collateral cur-rency using derived CCB curve(EUR col) or mkt one (USD col).
EUR col : estim. EUR/USD FXfwd with EO & USD BASIS.
US col : estim. EUR/USD FX fwdwith FF & EUR BASIS. Estimate EUR/USD FX fwd byheuristic interpolation in FX fwdcurve (see end of section 3).Table 1: Comparison of four-curve (benchmark) and heuristic approaches.Table 1 compares the steps for valuation using the heuristic method andwhat will be considered the benchmark: the four-curve approach. It is de-scribed for the particular case of EUR and USD currencies but can be appliedto any pair of currencies without loss of generality (for these pair of curren-cies, the market currency swap is collateralized in USD). See that the firstthree steps are common and they are all carried out through a bootstrap-6ing process. The first one constructs the OIS curves (“FF”, Fed Funds forUSD and “EO”, EONIA, for EUR) by forcing the overnight index swaps tobe equal to zero. The second obtains the estimation curves associated withthe most liquid tenors (EURIBOR6M, “E6M” in EUR and USLIBOR3M,“U3M” in USD) by forcing fixed versus floating interest rate swaps to beequal to zero and the third builds the rest of tenors of the estimation curvesby forcing tenor swaps (floating versus floating such as EURIBOR3M versusEURIBOR6M) to be equal to zero (see [3] for more details).The fourth step of the four-curve approach replaces the discount curve ofthe currency different from the collateral currency of the market CCS by abasis discount curve so that CCS are equal to zero (using the estimation andthe OIS discount curve for the funding or collateral currency). Foreign ex-change forwards are then calculated from the basis and OIS discount curves.In the heuristic approach, the foreign exchange forward curve is build outof swap point market quotes for the short term and the rest of the curve isestimated by a bootstrapping method forcing to zero the heuristic valuationof the cash flows associated with market cross currency swaps funded in itscollateral currency according to equations (4) and (5). The interpolation ofFX forwards in between two consecutive maturities is carried out accordingto the end of section 3. For the purpose of valuation of the decomposition insection 4, this foreign exchange forward curve does not need to be very exactas the cash flows involved are small.The fifth step calculates the basis discount curve of the four-curve ap-proach when collateral is not exchanged in the currency of the market CCS.This basis curve is calculated assuming that foreign exchange forwards areconstant irrespective of the currency in which collateral is exchanged. As theyare calculated from the basis and OIS discount curves, the basis curve canbe derived from the foreign exchange forward and the OIS curve of the cur-rency in which collateral is exchanged. For the heuristic approach, the crosscurrency basis spread (CCB) curve quoted by the market assumes collateral-ization in the currency defined in CCS contracts (USD for CCS exchangingEUR and USD cash flows). However, if collateralization is carried out in theother currency, it is necessary to derive an alternative EUR collateralizedcross currency basis (CCB) curve which will be used in the decompositionof section 3. This curve is derived by finding the cross currency basis spreadto force the heuristic valuation to be equal to zero of CCS for all maturi-ties funded in the other currency from the market convention according toequations (4) and (5). The difference between cross currency basis spreads7ssuming collateralization in either the currency of the CCS or the other oneis usually or the order of a few basis points.The sixth step shows the valuation with the four-curve approach depend-ing on the funding currency. Estimation curves are used irrespective of thefunding currency, the OIS discount curve is used for the currency in whichthe operation is funded (the collateral currency) and the basis discount curveis used for the other currency. The heuristic method applies the decomposi-tion of section 4 using the market cross currency basis curve if the collateralexchanged is the one specified in CCS contracts or the derived cross currencybasis curve if collateral is exchanged in the other currency.Finally, the last step shows the estimation of the foreign exchange for-wards. For the four-curve approach, they are calculated using the OIS andbasis discount curves used for valuation (see previous steps) and the heuristicmethod just interpolates them in the previously calculated foreign exchangeforward curve. DF USBASISt,t i = X − t,t i DF EOt,t i X t ⇒ X t,t i = X t DF EOt,t i DF USBASISt,t i (6) DF EURBASISt,t i = X t,t i DF F Ft,t i X − t ⇒ X t,t i = X t DF EURBASISt,t i DF F Ft,t i (7)Equations (6) and (7) show the equivalence of a discount factor of a basiscurve and the valuation of the corresponding cash flow with the heuristicapproach. The left hand side of equation (6) represents the present valueof a USD cash flow funded in EUR (that is why it is discounted with theUSD basis curve). The right side shows the USD cash flow converted toEUR through the FX forward and discounted with the EONIA OIS curveto fund it in EUR and converted again back to USD with the foreign ex-change spot rate. A similar argument follows from equation (7) when thefunding currency is USD. See that foreign exchange forwards must be thesame from both equations and both involve the two discount curves usedfor valuation depending on the funding (collateral) currency. When marketforeign exchange forwards match those obtained applying overnight cash andcarry arguments (discount factors calculated with OIS curves), the asymme-try of funding advantage disappears and basis curves collapse into OIS. Inthis situation the heuristic method converges into OIS discounting of eachcash flow in the currency in which it is denominated.8n this context, the four-curve and heuristic methods are rather equivalentto each other. Although the four-curve method provides a more generalframework for exotic product valuation, the end of section 4 shows that forthe most frequently traded products (swaps and currency swaps) the heuristicvaluation provides more funding flexibility for uncollateralied operations (e.g.trades with non-financial institutions and project financing). This section presents how to calculate the market spread of a forward startingresettable currency swap which makes its present value equal to zero. Thecalculation is carried out in two steps. In the first step, spreads of forwardstart CCS whose payment dates belong to the current market schedule (dateson exact multiples of the considered frequency tenor) are calculated through abootstrapping method. The second step estimates the spreads of customizedforward start CCS whose product payment dates are in between that marketschedule.Consider t mkti the market schedule of payments considered at present time, t = t mkt , for a given tenor frequency (e.g. 3 months, 6 months, etc) withyear fractions, τ mkti , corresponding to periods from t mkti to t mkti +1 . The function p ( T ) returns the period number ending at or beyond T within the marketshedule ( t mktp ( t mkti ) = t mkti ). p ( t N ) P i =1 s mktN τ mkti − DF C d t,t i = p ( t M ) P i =1 s mktM τ mkti − DF C d t,t i + p ( t N ) P i = p ( t M )+1 s mktMN τ mkti − DF C d t,t i (8)The first step calculates the spread, s mktMN , of forward starting CCS withinmarket schedule applying the bootrapping equation (8). This equation showsthe relation among the spreads of market spot and forward starting CCS. Aquoted spot starting CCS expiring at t N , CCS s Cd = s mktN ,s Cf =0 t,t N , can be expressedas the composition of a spot starting CCS expiring at t M , CCS s Cd = s mktM ,s Cf =0 t,t M ,and a forward starting CCS, CCS s Cd = s mktMN ,s Cf =0 t M ,t N , starting at t M and expiringat t N . Equation (8) only considers the spread payments because the rest9f cash flows are the same between the spot CCS expirying at t N and thecomposition of spot and forward start CCS. This equation assumes that thefixed spreads are set on the domestic currency leg (this case corresponds toa domestic currency different from USD). The solution of the first step isthe calculation of the unobserved spread, s mktMN , from s mktN and s mktM which arequoted in the market .Similarly, t prdi , is the schedule of payments of the customized productstarting at t U and ending at t V and the function q ( T ) returns the periodnumber within the product schedule ending at or beyond T . The nearesttimes of the market schedule before and after a given time, t H , are respec-tively denoted by t mktH and t mktH . V C d t (cid:20) F RN s Cd = s OISUV t U ,t V (cid:21) − X − t q ( t V ) X j =1 X OISt,t prdj − V C f t (cid:20) F RN s Cf =0 t prdj − ,t prdj (cid:21) = 0 (9) X OISt,T = X t DF C d t,T DF C f t,T (10)The second step starts considering the customized forward start CCS(from t U to t V ) and the two contiguous CCS with dates in the market schedulewhich have the same time duration and start just before and after the startdate of the customized CCS (from t mktU to t mktV and from t mktU to t mktV ). Forthese three structures, the spreads forcing their zero value are calculatedassuming OIS discounting of every cash flow in its own currency. For thecustomized CCS, this implies that s OISUV is derived to satisfy equation (9) .Similar equations can be written for the contiguous CCS (replacing t U by t U , t V by t V , t prdj by t mktj and s OISUV by s OISUV ). The foreign exchange forward, X OISt,T , is approximated by equation (10) assuming overnight cash and carryarguments based on OIS curves. The solution would give the spreads s OISUV , s OISUV and s OISUV , which do not include the cross currency basis spread yet. e UV = s mktUV − s OISUV e UV = s mktUV − s OISUV (11) It is assumed that the frequency of the floating legs is 3 months. If the floating legfrequency is different from 3 months in either leg, a synthetic market quote for the spotCCS should be obtained introducing the quotes of tenor swaps. See that in equation (9), the resettable leg of the CCS has been expressed as the sumof forward starting one period FRN structures. UV = t mktU − t U t mktU − t mktU e UV + t U − t mktU t mktU − t mktU e UV (12)The second step follows from equation (11) which calculates the error ofthe approximation of the spreads of the two contiguous CCS, s OISUV and s OISUV ,obtained from equation (9). Finally, the error made by the heuristic equation(9) for the customized forward starting CCS spread, s OISUV , is interpolated be-tween the errors of the two contiguous CCS according to equation (12). Thisinterpolation has good accuracy as it involves the pricing error between twowell-calculated market points. Therefore, the accuracy is not significantlylost by the error interpolation because it is indeed small. The spread, s OISUV ,would be correctly calculated and no correction would be needed when theFX forward is obtained from overnight cash and carry arguments. s mktUV = s OISUV + e UV (13)Equation (13) finally obtains the customized forward CCS spread byadding to the approximated spread, s OISUV , the error, e UV , which is obtainedthrough linear interpolation between two accurately-estimated errors.The interpolation of forward foreign exchange rates on a particular dateis carried out similarly. They are priced according to the usual cash andcarry argument applied to OIS curves according to equation (10). The errorsbetween the actual market quotes and this approximation are calculated asin equation (11) for the foreign exchange quoted dates previous and followingthe date to interpolate. The error on the interpolation date is approximatedwith equation (12). Finally, the interpolated error is added to the X OIS estimation of the foreign exchange forward as in equation (13).
This section prices a customized resettable currency swap through a decom-position procedure in which the currency swap is expressed as the sum ofa forward starting market currency swap (whose zero value spread is calcu-lated according to section 3) plus some additional small contributions pricedaccording to section 2. This will allow flexibility to choose the currency fromwhich funding and hedging are carried out. Pricing inaccuracies in thesesmall contributions do not usually have a significant impact in pricing (it is11ike a fine tuning) as the big contributions have been expressed in terms ofmarket quotes.Consider a customized CCS starting at t U and ending at t V with cus-tomized fixed spreads in both legs s C d UV and s C f UV . The swap is valued attime t = t L , in between the two product fixing dates t prdL and t prdL where t prdL < t < t prdL ( t q ( t L ) = t L and t q ( t L ) = t L ). CCS s Cd = s CdUV ,s Cf = s CfUV t U ,t V = CCS s Cd = s mktLV ,s Cf =0 t prdL ,t V − X t prdL F RN s Cf = s CfUV t prdL ,t prdL + F RN s Cd = s CdUV t prdL ,t prdL + q ( t V ) P j = q (cid:16) t prdL (cid:17) +1 (cid:20)(cid:16) s C d UV − s mktLV (cid:17) C d { t = t prdj } + s C f UV C f { t = t prdj } (cid:21) (14)Equation (14) shows the currency swap decomposition. The first termof the right hand side is a forward starting resettable currency swap whosespread, s mktLV , is obtained according to equation (13) using the method of sec-tion 3. The second and third terms involving FRN structures represent theswap structure of the already started hub period (exchange of notionals andfloating payments at the end of the period) and the sum includes the fixedspread, s C f UV , of the foreign curve and the fixed payments of the domestic leg,equal to the difference between the customized and market spreads, s C d UV and s mktLV . No valuation has been performed yet, only a payoff decomposition. Todo the pricing, three terms are considered: the CCS, the two FRN struc-tures and the last sum of cash flows. The only multi-currency piece of thedecomposition is the forward starting CCS whose valuation is zero ( s mktLV iscalculated to satisfy this condition). The rest of the pieces involve single cur-rency fixed cash flows. The two FRN cash flows CF F RN in equation (15) ,must be priced jointly applying either the operator V C d t [ · ] or V C f t [ · ] assum-ing a common funding currency, either domestic or foreign (see that equation(15) has replaced contingent index fixings by their forward values before ap-plying the operator). This common funding avoids the inconsistency betweenthe market forward foreign exchange rate, X t,t prdL and the estimated foreignexchange rate, X OISt,t prdL , of equation (10) using OIS discount factors . Equa- The FRN cash flows in equation (15) change signs with respect to equation (14)because the notation for a long position of FRN was defined returning the notional andpaying the floating rate at expiry. This inconsistency could allow for an arbitrage, as these cash flows are big enough CF F RN = CF C f C f { t = t prd ¯ L } − CF C d C d { t = t prd ¯ L } CF C d = [1 + ( L C d t,t prdL + s C d UV ) τ prdq ( t prdL ) ] CF C f = X t,t prdL [1 + ( L C f t,t prdL + s C f UV ) τ prdq ( t prdL ) ] (15) V C d t [ CF F RN ] = ( CF C f X − t,t prd ¯ L − CF C d ) DF C d t,t prd ¯ L V C f t [ CF F RN ] = ( CF C f − CF C d X t,t prd ¯ L ) DF C f t,t prd ¯ L (16)Similarly, the last spread sum of equation (14) should be jointly fundedin the collateral currency and for uncollateralized operations, the currencyin which it is decided to fund them. Independently of the funding currencychosen for valuation, every price must be converted to the common currencyin which pricing is carried out with the foreign exchange spot. This heuristicmethod allows choosing the funding currency of these additional cash flowswhich do not belong to the cross currency swap.Non-resettable currency swaps can be priced according to the decompo-sition of equation (2). The first term on the right hand side is a customizedresettable cross currency swap which can already be priced. The second termis a sum of floating rate notes which can be priced applying operator V as-suming funding in the currency of the collateral or if the operation is notcollateralized, the cheapest or most convenient currency to fund it. Thesediscounted cash flows must be converted to the currency in which pricing isaccomplished using the corresponding foreign exchange spot.As it has already been mentioned, if collateralized operations are consid-ered, a common funding must be chosen equal to the collateral currency toallow for valuation agreement with the counterparty with which collateral isexchanged. However, if an uncollateralized transaction is considered, the de-composition of equation (14) allows for more flexibility to fund the remainingpieces apart from the currency swap depending on trading convenience andpreferences. A typical example happens when a European bank hedges crosscurrency exposure between USD and EUR with other European institutions (they involve a Libor rate instead of a spread) and the term between their fixing andpayment is very liquid (just 3 months). This section presents a worked example comparing the two methods with thepotential advantage of the heuristic method. The example illustrates the casein which a European institution hedges currency exposure between EUR andUSD with other banks under a CSA agreement that exchanges collateral inEUR. The four-curve method funds everything in domestic currency (EUR)and the heuristic method funds the currency exposure in EUR but the rest inUSD. The comparison will be done for a resettable (CCS) and a non-resetable(NCS) currency swap, whose legs are denominated in USD and EUR, spreadsadded to the EUR floating leg of -5.75 basis points and notional amount of100 million EUR. Market data has been taken on January 29th 2014, withspot and forward foreign exchange rates in USD per EUR of { Spot: 1.3533,1y: 1.3543, 2y: 1.3610, 3y: 1.3741, 4y: 1.3928, 5y: 1.4143, 7y: 1.4589, 10y:1.5145 } , and a cross currency basis spread curve in basis points added to theEUR floating leg and collateralized in EUR of { } with zero spreadon the USD floating leg.The cross currency quotes are provided by an institution with a CSAin EUR . Five curves are considered: EONIA (“EO”), 3 month Euribor The sensitivity propagation method used by the system in which the four-curvemethod is implemented can only recalibrate the set of curves chosen for valuation. If
Mat
EO E3M FF U3M CCB EO E3M FF U3M CCB NPV e e Mat
EO E3M FF U3M CCB EO E3M FF U3M CCB -1 1 0 0 0 0 0 -1 1 0 -2 2 0 0 0 0 0 -2 2 0
15 -15 0 0 97 0 0 14 -14 96
NPV e -58 e -120.34Table 3: 10 year NCS NPV and deltas (thousand EUR) for four-curve andheuristic methods moving each curve by one basis point.Tables 2 and 3 show NPV (net present value) and delta sensitivities inthousand EUR of each curve for 10 year CCS and NCS under rate movementsof one basis point of the four-curve and heuristic methods. The CCS has zeroNPV (net present value) for both pricing methods as they are calibrated tomarket. The NPV difference for the NCS between both methods (see table 3)is 62.26 thousand which reflects the pricing difference of using USD insteadof EUR funding for the non-cross currency risk. For the position shown intable 3 the heuristic method provides a more negative NPV. This means thatEUR funding providing the price given by the four-curve method would be valuation considers EUR collateral, then market CCS are valued with those set of curveswhen recalibration is carried out and therefore sensitivities are provided as if collateralizedin EUR. curves as theyare funded and discounted with “FF” curve. See that sensitivities to “EO”and “E3M” curves are zero as every EUR cash flow is incorporated into theCCS (the sensitivity appears in the CCB curve). The four-curve method onlyprovides sensitivities to EUR curves with the same sign as the correspondingUSD sensitivities of the proposed method . The sensitivities are of the sameorder for both “EO” and “FF” curves as they have similar interest rate levels.∆ $ = ∂ ( V $ X − t ) ∂ ( X − t ) = − ∆ e X t + V $ ∆ e = ∂V $ ∂X t = − ∆ $ X − t + V $ X − t (17) Rising “U3M” curve increases USD paid cash flows and so the biggest sensitivity isnegative as more has to be paid. Rising “FF” curve decreases USD payments providing apositive sign for the biggest sensitivity. Rising “EO” curve reduces value of EUR leg of the calibrated CCS, which has tobe compensated by an increase of USD basis discount curve after propagation (marketCCS re-calibration) providing a positive sensitivity (the same sign of the “FF” curve forthe heuristic). Rising “E3M” curve lowers USD basis discount curve after propagation,yielding a negative sensitivity (opposite sign of “FF” curve for heuristic). $ USD) and the US (∆ e EUR), where V $ is the price in USD and X t is thespot foreign exchange rate (USD per unit of EUR). Pricing systems based inEurope usually report ∆ e − V $ X − t as FX delta, because the deal premium isalready in EUR and this amount has to be subtracted from the total amountof Euros to cancel, ∆ e , in order to be delta hedged. See that this cancellingEUR transaction, ∆ e − V $ X − t = − ∆ $ X − t , is carried out against a quantityof USD, − ∆ $ , which cancels the open US sensitivity, ∆ $ . For a US investor,the delta reported by the system would be ∆ $ − V $ .Four-curve HeuristicFX Delta CCS NCS CCS NCS∆ e EUR 39 -1,380 -37 -2,623∆ $ USD 26 1,884 52 3,417Table 4: 10 year foreign exchange deltas (thousands) of CCS and NCS forfour-curve and proposed methods.Table 4 shows the foreign exchange deltas of CCS and NCS for four-curveand heuristic methods according to equation (17). Values corresponding to∆ e are in thousand EUR and to ∆ $ in thousand USD. To understand thedifferences with respect to the two products and methods, see that for theheuristic method under the assumption that terms not belonging to the CCS(the foreign exchange delta is not affected by the funding currency of theCCS) are funded in USD, the foreign exchange deltas are given by equation(18), where ∆ $ CCS,USD stands for ∆ $ for the resettable currency swap whoseadditional cash flows of the decomposition are funded in USD. See that forthe CCS, only the first 3 month period contributes . For the NCS, the whole10 year period contributes . According to the decomposition of equation (14) the terms which affect the foreignexchange delta of equation (17) are the FRN denominated in foreign currency (USD), − X t F RN s $ =0 t ,t +3 m and the sum of foreign spreads, s C f UV , which are zero in this example. According to the decomposition of equation (2) the CCS contributes with the first pe-riod, − X t F RN s $ =0 t ,t +3 m and from the sum, only the term − X t F RN s $ =0 t ,t +10 y contributesas the foreign exchange rate has already been fixed. These two terms together are equiv-alent to − X t F RN s $ =0 t ,t +10 y . $ CCS,USD = V $ t h − X t F RN s $ =0 t ,t +3 m i ∆ $ NCS,USD = V $ t h − X t F RN s $ =0 t ,t +10 y i (18)Foreign exchange deltas of CCS are much smaller than those of NCSbecause for the CCS only a period of 3 months contributes, whereas for theNCS, the whole 10 year period contributes. CCS EUR deltas (∆ e ) changesigns between four-curve and proposed methods (39 versus -37), becauseboth CCS premium and delta are small and comparable (see equation (17)to relate EUR and USD deltas).Equation (19) shows the foreign exchange deltas when the heuristic methodfunds the remaining terms apart from the CCS component in EUR. See thatthe operator V e t [ · ] is used to represent that cash flows are converted with for-eign exchange forwards to EUR and thereafter discounted with curve “EO”.If this heuristic valuation is differentiated with respect to X − t , the X − t fac-tors of the foreign exchange forwards will disappear which is equivalent tomultiply deltas by X t as shown in equation (19). If these deltas are calcu-lated with the heuristic method as shown in equation (19) in thousand USD,see that they are similar to those calculated by the four-curve method (seesecond row, left side of table 4).∆ $ CCS,EUR = X t V e t [ − X t F RN s $ =0 t ,t +3 m ] = 26∆ $ NCS,EUR = X t V e t [ − X t F RN s $ =0 t ,t +10 y ] = 1 ,
906 (19)See that foreign exchange deltas differ depending on the funding currencychosen by the heuristic method mainly because the valuation of the last no-tional payment of the FRN may differ depending on whether it is discountedwith “FF” or converted to EUR and discounted with curve “EO”. As EURfunding has advantage compared to USD, the EUR funded delta is lower.
A heuristic present value concept for multi-currency pricing and hedging hasbeen proposed which naturally allows choosing the funding and thereforethe valuation collateral currency. This concept is applied to the valuationof cross currency swaps by decomposing them into a major component withthe cross currency basis risk plus minor residual components. This method18as been compared with the four-curve method which is the current bench-mark. For collateralized operations with funding managed in the collateralcurrency both methods are rather equivalent. For uncollateralized opera-tions, the heuristic method allows more optionality to choose the funding ofthe components without cross currency basis risk to achieve either a cheaperfunding or more connected one to the hedging products. Although the four-curve method provides a more general framework for exotic valuation, theheuristic method provides more funding flexibility for the most frequentlytraded products. The heuristic method converges to OIS discounting in eachcurrency when foreign exchange forwards follow overnight cash and carryarguments.
Acknowledgement : the authors want to thank E. Cuesta and G. Mon-tesinos for their support, feedback and clarifying discussions.