Edu-Risk International Financial Risk Management & Training Justin Clarke justin.clarke@edurisk.ie

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1 Edu-Risk Iteratioal Fiacial Risk Maagemet & Traiig Justi Clarke usti.clarke@edurisk.ie Swap Discoutig & Pricig Usig the OIS Itroductio Sice August 007 ad the start of the fiacial crisis, swap pricig has udergoe a sigificat revolutio. Prior to that date, swap pricig was thought to be fully uderstood ad pricig models were cosidered adequate. Market participats recogised that there were some simplificatios that were made i the pricig of swaps such as accoutig for couterparty risk, but these were mior i ature ad had limited impact o pricig at the iterbak level ad were geerally igored. I the moths subsequet to August 007, teor basis ad cross currecy basis was observed i swap pricig ad accouted for by baks. However, as the fiacial crisis ad its effects persisted, market players started takig ito accout a wide rage of other pricig aomalies arisig primarily from credit ad liquidity risk related issues i trasactios. Differet baks approached the issue i differet ways, but sice mid 00, a commo approach to swap pricig has emerged. The core issue that has ow uified the approach to swap pricig has bee the adoptio of the OIS curve for swap discoutig where swaps are uder daily margiig agreemets with daily collateral calls. Adoptio of the OIS curve has compelled baks to reassess their approach to the followig iterrelated issues: Due to the problems experieced i a umber of baks, the market became acutely aware of credit ad liquidity risks associated with short-term iterbak ledig. High liquidity ad credit premiums started beig paid o iterbak Libor fudig. Libor rates of differet teors displayed differet proportios of credit/liquidity spread which icreased with term. As a example, this meat that rollig moth Libor rates for a loger term (e.g. for 6 moths) resulted i a differet spread beig paid tha o a sigle 6-moth roll. The et result of this termdepedet credit spread was to create sigificat basis spreads betwee swaps with differet floatig leg reset teors. The spread o a 6-moth resettig swap was much greater tha the spread for a -moth resettig swap. Furthermore, spreads opeed up betwee the various Libor rates ad the equivalet term Treasury rate, the so-called TED spread. Pricig swaps where collateral is placed i a differet currecy. Libor curve costructio. Pricig of o-collateralised swaps. Specificatio of cross currecy swap curves. I this documet, we will examie the ustificatio for adoptig the OIS curve ad also touch o the issues eumerated above. Market Dyamics Leadig To OIS Discoutig Prior to 007, the swap market priced ad traded off a very simple set of yield curves which could be applied to swaps across a wide rage of product features, couterparts ad credit mitigatio arragemets. The issues that have cotributed to the chage i approach are all iter-liked to some extet ad are: Figure : Graph showig the spread betwee USD Libor rates of differet teor ad the OIS Rate which is a good illustratio of teor spreads. The market became more sesitive to the pricig of couterparty default i swap trasactios. Prior to the crisis, the market was aware of the eed to price-i the cost of couterparty default, but the calculatio ad applicatio of such adustmets was icosistetly priced ad applied across the market. The cocept of Credit value Adustmets (CVA) is ow widely accepted ad applied across the markets. A cosistet approach has developed ad the wider acceptace of bilateral CVA adustmets has ow meat baks have a method of comparig their assessmets ad approach to CVAs tha existed i the past. I additio to a more cosistet defiitio of CVA, baks have also take ito accout the ature of credit mitigatio arragemets (such as allowed for i ISDA Credit Support Aexes

2 CSAs). The terms of the CSA modifies the effective couterparty credit risk exposure of a trasactio or group of trasactios ad therefore would have a maor impact o deal valuatio. The market came to a realisatio that all term iterbak fixig rates (Libor) icluded some elemet of credit or liquidity premium. The oly curve which is almost free from such effects was the OIS curve where fixigs were based o cetral bak overight collateralised accommodatio rates ad therefore did ot suffer from iterbak credit risk. The OIS curve is therefore recogised as beig the oly yield curve ow which is practically free from basis effects due to credit or liquidity issues (provided the curve is built from the prices of OIS swaps subect to overight margiig). The bulk of iter-bak professioal swap dealig is based o stadard ISDA agreemets which allow for ettig of all deals traded uder those ISDAs i the evet of default. I additio a sigificat portio of the trades covered by a ISDA agreemet are subect to regular collateral calls (daily i may cases) ad iter couterparty settlemet of outstadig MTMs. Collateral balaces geerally attract iterest at the relevat overight accommodatio rate, such as the Fed Fuds Rate, SONIA, EONIA or JIBAR rate. We will call this the rate the FED rate for coveiece, irrespective of its currecy. Move to the OIS The last-metioed poit i the list above has had sigificat impact o the pricig of swaps ad i Jue 00, the Lodo Clearig House (LCH.Clearet) aouced that : LCH.Clearet Ltd (LCH.Clearet), which operates the world s leadig iterest rate swap (IRS) clearig service, SwapClear, is to begi usig the overight idex swap (OIS) rate curves to discout its $8 trillio IRS portfolio. Previously, i lie with market practice, the portfolio was discouted usig LIBOR. After extesive cosultatio with market participats, LCH.Clearet has decided to move to OIS to esure the most accurate valuatio of its portfolio for risk maagemet purposes. The ratioale behid LCH s decisio to move to OIS discoutig for collateralised ad daily margied trades was prompted maily by the fact that most of the swaps clearig through their system were subect to the (ow) stadard CSA requirig daily collateral calls o the swap MTMs. The collateral placed would ear iterest based o the prevailig O/N FED rate. The core reaso for movig to the OIS curve for discoutig is the fact that the collateral ears iterest at the FED rate. To uderstad why the terms of iterest accrual o the collateral traslates ito the selectio of the discoutig curve we ca use the argumet i the followig sectio. Ratioale for OIS Discoutig Assume that a certai swap (S) has a MTM value of V(S), we ca argue that V(S) is simply the preset value of all the expected cash flows of the Swap. Assume the swap has fixed flows (c fixed ) util maturity ad m floatig flows (c float )to maturity Usig the otatio d i as the discout factor to time t i, we have the followig: V ( S) c d c d m fixed, i i i float, (Note that the c fixed ad the c float are ormally of differet sig as oe would pay fixed/float ad receive float/fixed. Also, the argumet is preseted for a fixed/float swap, but it could exted to ay swap).the collateral C placed by the couterparty would be equal to: C mi( V( S),0) The above equatio states that the collateral will oly be placed if the swap couterpart is out of the moey. If the swap is i the moey, o collateral would be placed, but the other party would be out of the moey ad they would place the collateral. I all cases, collateral placed by either party will equal V(S). The rate eared o collateral (equivalet to the FED rate) is a itegral part of the OIS curve ad lies o the OIS curve because this rate represets the floatig rate idex rate for OIS trades. Therefore if the collateral placed o a swap was ivested at the OIS rate, the couterpart would be idifferet to earig the FED or OIS rate because of the ability to swap directly betwee the FED rate ad the term OIS rate usig a OIS. Assume therefore that sufficiet of the collateral was ivested at the same set of cash flow dates as the swap flows usig a series of OIS trades to esure that the proceeds of the OIS trades exactly match the swap s expected cash flows. Assume that theoretically, this process was followed each time there was a collateral call. I order for there to be o arbitrage possible, there should be o residual amout of collateral left over uivested after each ew set of OIS trades are executed. If there was some uivested collateral, the swap couterparty could make a arbitrage retur. Alteratively, all the collateral could be ivested at OIS rates to the swap cash flow dates ad if the OIS ad swap cash flows did ot match exactly at the cash flow dates the a arbitrage free profit would result. LCH.Clearet curretly clears USD7 Trillio of derivatives.

3 For the o arbitrage coditio to apply, the followig must hold: m c fixed, id i c float, d V ( s) i c d m fixed, i OIS, i i c float, d OIS, Where d OIS,i is the discout factor read off the OIS curve to time t i. Equatio simply states that the collateral posted (equal to V(s) as see i equatio ) should equal the future expected cash flows of the swap discouted by the discout curve ad also whe discouted by the OIS curve. The oly coditio that geerally satisfies equatio would be if d i = d OIS,i for all i. Hece oe ca the say that the correct curve to use for swap discoutig is the OIS curve. Creatig the OIS The OIS curve is geerated directly off quoted OIS rates. OIS rates quoted i the market are simple iterest rates for OIS trades of less tha year maturity ad aual iterest paymets for trades of greater tha year maturity. OIS trades are quoted for regular itervals i the 0 to year rage ad aually thereafter. Sice the adoptio of OIS discoutig i July 00, there has bee a resurgece i the OIS markets. The OIS market has become icreasigly more liquid, ad quotes are made ow for loger dated OIS trades. I practice, OIS curve bootstrappig is ormally split ito two regimes. The first is the short ed of the curve (possibly out to moths) ad the other is for loger periods. I the short-dated regime, the curve ca be costructed usig flat expectatios of FED rates up to the ext (or subsequet) Moetary Policy Committee (MPC) meetig, where FED rates are set. The assumptio is that the FED rate will remai (early) costat util the ext MPC. I additio, if there is geeral market certaity about future MPC views o rates, this expectatio ca persist beyod the ext MPC. I practice, daily discout factors are compouded to produce OIS rates i this area of the curve. Short-dated OIS rates are quoted i the market i this part of the curve, but care must be take to iterpolate these rates due to the step-fuctio characteristics of FED rates aroud the times of MPC meetigs. I some markets OISs are quoted to MPC meetig dates, ad these ca be used to back out daily discout factors betwee each MPC date, ad these discout factors ca be used to bootstrap the short portio of the curve. C Due to the greater ucertaity of MPC actios over loger horizos, loger dated OIS rates do ot suffer from the same step-fuctio behaviour of shorter OIS curves. Here a traditioal bootstrappig ad iterpolatio approach ca be followed to costruct the curve i this area. Bootstrappig Libor s i a OIS World I the past, the ormal method of bootstrappig a swap curve relied o the fact that the iterest geeratio curve (hereiafter called the Libor ) ad the discout curve were idetical. With idetical discoutig ad Libor curves, a sigle set of swap rates would be sufficiet to bootstrap the curve where the discout curve ad Libor curve are bootstrapped i tadem With the adoptio of OIS discoutig methodology, a differet method of bootstrappig becomes ecessary. I effect, the ukow Libor curve must be bootstrapped relative to the kow discout curve. The approach take i the bootstrappig process follows the same approach as the traditioal bootstrappig methodology except that the discout curve is kow i advace. I essece the Libor curve is costructed where the obective is to esure that the swap prices to par whe the flows geerated off the Libor curve are discouted usig the OIS curve. Appedix A demostrates oe possible method of bootstrappig a Libor curve off a OIS curve. Libor s Libor curves eed to be bootstrapped for each teor ecoutered i the swap world. I practice, a set of - moth, -moth ad 6-moth Libor curves are produced for each currecy. All of these curves would be bootstrapped relative to the OIS curve. The cocept of bootstrappig a iterest geeratig curve relative to a already kow discout curve (or vice versa) would become a stadard process i a OIS discoutig world. Traditioally, -Moth Short term iterest rate futures such as Eurodollar futures are used i the -moth to -year portio of the curve, however, these futures caot be used i the costructio of the -moth or 6-moth curves. For these curves, s- s basis swaps are used for the - moth curve ad 6-moth FRAs ca be used for the 6- moth curve. I order to bootstrap a Libor curve usig OIS discoutig, all previous issues that existed whe the Libor curve was used for discoutig ad iterest geeratio still exist. These issues iclude:

4 Selectio of curve istrumets. Usage of covexity adustmets to Future prices. Geeratio of smooth forward curves. Iterpolatio of the curve durig costructio. Geeratio of a OIS Whe Collateral is Posted i a Differet Currecy I may cases, a commo CSA agreemet may cover a diverse set of trades makig up a couterparty s portfolio. Most ofte tha ot, the CSA would assume that collateral is placed i a sigle currecy, irrespective of the currecy of the uderlyig trasactios covered by the CSA. May CSAs allow the couterparty to place collateral i a rage of currecies, ad the couterpart could decide what currecy to place. A commo example would be where USD collateral is placed o a world-wide portfolio of trade of may differet currecies. I the discussio below, we assume that we are obliged to place USD collateral o all the trades covered uder a CSA. I this case, it would ot be sufficiet to use the home currecy OIS curve to discout trades i a o-collateral currecy. The reaso for the home currecy OIS curve beig iappropriate is because o could ot satisfy the the o-arbitrage argumet preseted earlier relatig to the reivestmet of the collateral to the cash flow dates of the trade. I order to fulfil the o-arbitrage argumet, oe would follow a process where firstly the USD collateral is ivested out to the cash flow date usig the USD OIS rate. Secodly, the forward USD cash flows resultig from step would be coverted to the home currecy at the forward exchage rate (the cross currecy basis swap rate). The ew effective discoutig curve for a trade where collateral calls are i a differet currecy would therefore be a equivalet forward implied currecy discout curve as described above. A importat feature of may CSA agreemets is that they allow for collateral to be placed i a selectio of differet currecies at the collateral placer s discretio. With the sigificat basis spreads betwee fudig i differet currecies, the party placig the collateral ca select the cheapest to deliver currecy ad place that. This feature has some sigificat impact o deal pricig because it itroduces a optioality elemet i the pricig of collateral ad the use of a appropriate discout curve. Discoutig Of Cross Currecy Swaps A questio exercisig may market participats is the method for determiig the discout curve for cross currecy swaps. The floatig leg for a cross currecy swap would geerally have oe leg at zero spread to Libor (geerally the USD leg) ad the leg of the other currecy would be at a spread to its Libor. There are a umber of ways i which such a curve ca be costructed. The two maor methods possible with curretly quoted products would be: Maitai the OIS curve for discoutig both iterest rate ad cross currecy swaps. A atural cosequece of this approach is that Cross Currecy Libor rates bootstrapped usig this method would be differet to bootstrapped Swap Libor rates This would ot be the preferred method of discoutig the cross currecy swaps because the Libor iterest flows would ot et off betwee a swap ad cross currecy swap. The other method is to use the Swap Libor curves for a cross currecy swap as well. This will esure that idetical iterest flows are geerated for Iterest Rate ad Cross Currecy Swaps. Usig this Libor curve meas that oe has to reverse bootstrap the discout curve off the Libor (plus spread) (A) USD OIS (C) USD Libor I recet times, the relative cost of placig collateral i differet currecies may vary greatly. A good example of this was i the year subsequet to the fiacial crisis i September 008, the effective fudig rate i EUR was much less tha the fudig rate of USD due to the actios of the ECB compared to the FED ad also the demad for USD assets resultig from a flight to quality i times of crisis. I practice, may market participats do ot price i the effect of multi-currecy collateral placig, but there is a wide recogitio of the effects of the selectio of currecy o swap pricig. May participats are keepig a watchig brief o this issue ad will modify swap pricig i the future to accout for arbitrage opportuity arisig from the selectio of collateral currecy. (B) EUR OIS (D) EUR Libor (E) EUR Cross Currecy Figure : Cross Currecy Bootstrappig Sequece The term Cross Currecy Swap is used to refer to Cross Currecy Basis Swaps as they are also ofte called.

5 Figure shows the sequece for bootstrappig the cross currecy curve for a example USD-EUR cross currecy swap. The startig poit for the sequece is the USD ad EUR OIS curves, A ad B. Usig Iterest Rate swap rates, the USD & EUR Libor curves (C & D) are bootstrapped. The USD Leg of the Cross Currecy Swap would discout usig the USD OIS curve. It will be see that the USD leg will ot price to Par because the USD OIS curve would ot equal the USD Libor ; cosequetly, because of the bootstrappig process the other currecy leg will ot price to Par as well. This has a large impact o the maagemet of currecy deltas i the pricig of the cross currecy swap. It is expected that cross currecy swaps based o floatig OIS rates will begi to be quoted i the market ad will provide the pricig bechmarks for other cross currecy swaps with Libor teors. OIS based cross currecy swaps will also provide the bechmarks for correct calculatio of currecy deltas. How this will work (we assume) will be as follows, refer to Figure where a USD/EUR example is give: The USD discoutig curve will be the USD OIS curve (A). The USD iterest flows o the OIS cross currecy swap will be calculated with the same OIS curve (B). This will result i full Par pricig of USD legs. For the o USD leg, the iterest is geerated off a stadard OIS curve plus the cross currecy spread (C), the spread is quoted i the market. The discoutig curve (D) is reverse-bootstrapped off the iterest geeratio curve (C) whist esurig that the leg prices to Par. For other Libor cross currecy swaps, the same Libor curves used for iterest rate swaps should be used ad the discout curves (A &D) used. (A) USD OIS (B) USD OIS relevat FED fuds rate, this will be the case with the bulk of trasactios betwee large market players. However, i the case where trasactios are doe that do ot fall uder such CSA agreemets a differet approach to deal pricig is required. The pricig of such deals are based o the computatio of a credit value adustmet (CVA) which affects the deal pricig cosiderably. It is ot the itetio of this paper to deal with the pricig of such deals, but a future paper will be produced to itroduce the cocepts ad methodologies ivolved. Coclusio It was almost uthikable prior to 007 that such a fudametal reassessmet of iterest rate derivative pricig was goig to take place i the ear future. The chages have bee so fudametal that may baks have bee usure of exactly how to price certai derivative products which i the past were cosidered stadard. The chages see i the market however have perversely ot highlighted ew issues iheret i the market, but have rather exposed basis risks which were always preset i the products, but which had bee previously bee igored or deemed to be isigificat. The field of iterest rate derivative pricig is chagig rapidly ad as the market becomes more comfortable with the pricig of the credit ad liquidity risks i these products, we fully expect that further ehacemets ad refiemets will be made to the way i which the products are priced. Edu-Risk Iteratioal Edu-Risk Iteratioal is a Irish based fiacial risk maagemet cosultacy ad traiig compay owed ad maaged by Justi Clarke. Justi has 0 years experiece i the bakig idustry, maily i the area of Risk Maagemet ad related fields. More iformatio may be obtaied at: (D) EUR ImpliedOIS Cross Currecy (C) EUR OIS + CCS Spread Figure : Proposed OIS-based discoutig methodology for Cross currecy swaps. Iterest Rate Derivative Pricig i the Absece of a Collateralised CSA The discussio preseted above deals with situatios where the trades beig priced are subect to a typical CSA with daily collateral calls ad the collateral accrues at the All material, icludig but ot limited to, discussios, iformatio, methodologies, algorithms, software ad examples i this documet has bee produced for iformatio purposes oly ad is ot iteded to replace idividual research. The material cotaied i this documet is provided for the purpose of illustratig the cocepts i the documet ad busiess decisios should ot be take which rely o the material herei. Edu-Risk Iteratioal makes o represetatios or warraties of ay kid, express or implied, about the accuracy, reliability, completeess or suitability of the iformatio cotaied i this documet or iformatio preseted. To the maximum extet permitted by law, Edu-Risk Iteratioal takes o resposibility whatsoever for you or ayoe else for ay loss or damage 5

6 suffered of ay type as a result of or i coectio with the use of this documet or ay of its cotet. This icludes, but is ot limited to, reuse of ay equatios, methodologies, iformatio or diagrams Appedix: Method to Bootstrap the Libor Assume that we have a geeralised discout curve {D} used to preset value the flows of iterest rate derivatives. I practice this curve would be derived off OIS rates. The discout curve would be able to produce discout factors d i at each t i. I the equatio above, oe sees that the preset value of the floatig flows implied off the Libor curve would equate to the value of the fixed leg P, mius the preset value of the pricipal at maturity o the float leg. If oe had to solve the equatio above for the a i s oe ca the calculate a d for the Libor basis curve {D } each time period t thus: d i ai The problem is to bootstrap a basis curve that ca be used to geerate the flows ad hece price ay series of products which trade at a basis to the OIS curve. A example of such a curve would be a -moth Libor curve. For the purposes of this discussio, we will assume that we wish to bootstrap a swap basis curve. Let us call this curve {D } ad the discout factors read off this curve d i To calculate {D } we use the followig methodology: Assume we have a series of swap prices {S} where we have equally spaced swap rates which are cosistet with the swap coupo dates. Assume the swap rate for the swap maturig at time t i is s i. A price P i for each fixed leg of each swap is obtaied from the followig equatio (usig the OIS discout curve {D} ad treatig the fixed leg of the swap as a bod): s 00 s s s s 00 s s s 00 s d P d P d P s 00 d P The property of a swap is that the price of the fixed leg must equal the price of the floatig leg. Usig the otatio a i, where a i is the accumulatio factor (geeralised relatioship of a to a discout factor d is: a =/d- ad also 00a i is the iterest o the floatig leg i period i) for each of the forward periods i the Libor iterest geeratio curve applicable to the swap, we have the followig equatio: a 00d P 00d a a 00d P 00d a a a 00d P 00d a a a a 00d P 00d (for easier computatio): or alteratively 00d 00d 00d 00d 00d 00d 00d 00d 00d a P 00d a P 00d a P 00d 00d a P 00d 6