Graduate School Master of Science in Finance Master Degree Project No. 2011:161 Supervisor: Alexander Herbertsson

Transcription

1 Valuaion of Ineres Rae Swaps in he Presence of Counerpary Credi Risk Robin Axelsson Graduae School Maser of Science in Finance Maser Degree Projec No. 0:6 Supervisor: Alexander Herbersson

2 Acknowledgemens I would like o hank my supervisor Alexander Herbersson for geing he privilege of working wih him. He is very knowledgable in he field of financial derivaives and credi risk modelling. His ideas have been very inspiraional and i has been a grea educaional experience o work wih him. To say he leas, I have learned a lo on his inense perilous journey from concepion o he final repor. 3

3 Absrac Insuring deb hrough credi defaul swaps (CDS) and collaeralized deb obligaions (CDO) has become increasingly more popular. Recen evens such as he financial crisis of 008 have shown ha he credi models for hese insurances have lacked severely in cerain aspecs. One commonly referred example of hese ramificaions ha have ensued is he AIG, he larges insurance company in he Unied Saes, ha were pu ino a serious liquidiy crisis back in 008 which promped a large bailou by he U.S. governmen. The AIG inciden made i eviden ha he insrumens being used didn properly address a par of he credi exposure ha is known as counerpary risk. Counerpary risk means he risk ha he counerpary (in his case he insurance company) fails o mee is conracual obligaions. Several models ha accoun for his ype of risk have been inroduced during he pas wo decades. The purpose of his hesis is o explore his idea of accouning for couner pary risk in financial derivaives and how i affecs he pricing adjusmen of ineres rae swaps. We esimae he value of IRS agreemens in he presence of counerpary risk by adding a credi value adjusmen ha is esimaed using an inensiy based approach. The inensiy is assumed o be piecewise consan and is calibraed agains observed marke CDS quoes using he boosrapping mehod. We find ha a 5 year IRS wih a low risk counerpary wih 95.4% survival probabiliy during his period yields a credi adjusmen of abou 40 basis poins whereas a 30 year IRS wih a high risk counerpary wih 3.5% survival probabiliy yields a credi value adjusmen of almos 000 basis poins. 5

4 INDEX INTRODUCTION AND OVERVIEW SWAPS AND THE SWAP MARKET INTRODUCTION TO CREDIT RISK....3 CREDIT DERIVATIVES AND USAGE OF SWAPS TO MANAGE CREDIT RISK....4 COUNTERPARTY CREDIT RISK AND CREDIT DERIVATIVES... 3 MODELING FRAMEWORK STOCHASTIC MODELING INTENSITY MODELING IRS VALUATION FRAMEWORK GENERAL VALUATION OF COUNTERPARTY RISK COUNTERPARTY RISK AND IRS VALUATION ENHANCING COUNTERPARTY RISK VALUATION RISK VALUATION UNDER STOCHASTIC DEPENDENCE USING BIVARIATE INTEREST RATE USING CIR++ BASED DEFAULT INTENSITY INTRODUCING CROSS CORRELATION NUMERICAL SIMULATION PROCEDURES RESULTS AND COMPUTATIONS CONCLUSION FINAL THOUGHTS... 5 APPENDIX REFERENCES

5 Inroducion and overview The presence of counerpary credi risk in he rades of financial insrumens has caugh he aenion since he afermah of he credi crisis of 008. The derivaives marke is huge reaching over US$ 600 rillion by he end of 00. They are popular because hey have opened up for a way o easily reallocae and more efficienly manage differen ypes of risk. The hree mos common risk ypes o hedge agains using derivaives are ineres rae risk, credi risk and currency risk. Before he credi crisis of 008 a lo of loans, morgages and corporae bonds wih poor credi qualiy were issued which creaed a demand for credi insurances which were issued hrough wha is called credi defaul swap agreemens (also known as CDS agreemens). A he end of 008 he AIG which is he larges insurance company in he U.S was on he brink of bankrupcy and he blame was pu on is vas porfolio of CDS agreemens. This made i eviden ha one big elemen of he credi risk was no accouned for in his siuaion; he counerpary defaul risk. The framework of managemen of counerpary credi risk exends beyond adding some exra premium on he exchange raes and he prices of financial insrumens. I also affecs he collaeralizaion and decision making process of a bank. I is oulined boh in he Basel II and he Basel III accords which are he regulaions for how a bank should conduc heir business in a safe and sound manner. In his hesis we inend o look a he valuaion of ineres rae swaps in he presence of counerpary credi risk. In order o accoun for counerpary credi risk we need o undersand credi risk and how we use CDS agreemens o coninuously quanify his risk in a given counerpary. The res of his paper is organized as follows; in Secion we ake a our on he swaps and derivaives marke, explore how hey are used o manage differen kinds of credi risk. This secion is finalized by discussing counerpary credi risk which is he focus of his paper and how i affecs he valuaion of financial derivaives. In Secion we esablish a modeling framework for valuaion of ineres rae swaps wih counerpary credi risk. The end of he secion presens a valuaion model of an ineres rae swap ha is adjused o accoun for counerpary credi risk, we es his model under differen risk scenarios and examine how hese scenarios affec he counerpary adjusmen.. Swaps and he Swap Marke A swap is an agreemen ha les wo eniies swap heir cash flows wih each oher. This is done wihou any iniial moneary ransacions which makes i more viable as an insrumen as no ransacion fees or limiaions due o bound capial have o be deal wih. Swaps can involve any kind of cash flows and he main idea is o le a floaing cash flow where here is a risk ha i can be eiher oo high or oo low be exchanged for a fixed cash flow or anoher floaing cash flow which has a differen risk profile. 7

6 When enering ino such a conrac i is se up so ha boh cash flows in he conrac has he same expeced ne presen value, i.e. he conrac is se up so ha i is fair o boh paries. This basically means ha he value of he conrac is zero when being enered ino bu may change value over ime depending on circumsances. From a risk neural perspecive i is hard o see any reason for anyone o ener ino a swap agreemen since i is a fair game and here is no comparaive advanage for eiher pary o ener ino such an agreemen in an arbirage free world. In he real world however, an insiuion or company may face limiaions ha can only be overcome by enering ino such agreemens. One pary may have legally bound conracs ha no longer mach his risk profile due o changed circumsances or he may face any oher ype of insiuional fricions. There may be ax differences beween he wo paries or informaion asymmeries may moivae a pary o ener ino a swap agreemen wih anoher. In oher words he incenives behind enering ino a swap agreemen canno be quanified by risk neural measuremens. Swaps are rarely raded direcly beween paries, unless he paries are financial insiuions. In he case of financial insiuions, rades are more direc and hey generally know exacly wih whom hey ener heir swap agreemens. Oherwise swaps are usually raded over he couner hrough financial inermediaries and i is generally no known wih whom one swaps one s cash flows. These inermediaries serve he funcion of aking he opposie side of each ransacion of he swaps and carry he responsibiliy of maching and covering for defauling counerparies in he swap agreemen. The spread inheren in he swap agreemen is inended o cover for he defaul risk involved in he counerparies managed by he financial inermediary (Saunders, Corne 006, Bodie, Kane, Marcus 008, Hull 006). The financial inermediary may have a whole porfolio of eniies ha are in a swap agreemen wih each oher and a his disposal he may have a se of ools for managing and miigaing he risk ha any of he paries would defaul on his or her liabiliies. The mos popular ype of swaps involves ineres rae swaps (IRS) where one pary exchanges a floaing rae loan for a fixed rae loan. The ne presen value of he fixed cash flows of an IRS is called he fixed leg and he expeced ne presen value of he floaing cash flows is called he floaing leg (Lando 004). If he fixed leg is paid and he floaing leg is received we call he agreemen a payer IRS whereas if he floaing leg is received and he fixed leg is paid we call i a receiver IRS or receiver swap. A ypical life ime of a swap ranges from o 5 years (Hull 006). Each cash flow, be i annual, semiannual or quarerly can be represened by a forward conrac ha maures In a complee and arbirage free marke here is a measure of valuaion ha is independen of invesor s risk aversion. An invesor s individual choice wih regards o his risk aversion is hen explained by Fischer s separaion principle (Copeland Weson 005). 8

7 a ha dae. Therefore during his lifeime he swap agreemen can be seen (and herefore be valued) as a sream or porfolio of forward rae agreemens (FRA) wih differen mauriies up o he end of he swap period (Brigo, Mercurio 008). The mos popular floaing ineres rae is he LIBOR rae which is he London Inerbank Offer Rae. I is esimaed from he average rae a which banks lend and borrow unsecured funds in he London wholesale money marke. I is widely acceped as a reference rae in he valuaion of financial insrumens such as ineres rae swaps, foreign currency opions and forward rae agreemens. The LIBOR rae is esimaed for, 3, 6 and monhs mauriies only (Bodie, Kane, Markus 008). So if we are dealing wih insrumens ha have a longer life span han ha, oher sources of reference would be needed. The second mos popular ype of swap is he credi defaul swap (CDS). In his case he swap acs as an insurance policy agains defaul risk. We illusrae he roles of he paries wihin a CDS swap agreemen in Figure.. Le us say ha one pary A has a borrower, or reference asse C who may defaul on paymens and herefore exposes he lending pary A o credi risk. The lending pary A may ener ino a swap agreemen wih a proecion seller B which generally is an invesmen bank or a financial insiuion. From his swap agreemen, pary A can exchange wih B a fixed sream of cash flows (premium leg) which is he insurance premium, for a compensaion (defaul leg) if he insured reference asse C would defaul on paymens (Lando 004). A paymens premium leg C defaul leg B Figure.: A CDS swap agreemen ha is esablished beween a proecion buyer A and a proecion seller B where B insures agains he credi risk inheren in a reference asse C (usually a borrower or a bond seller). Source: Lando 004 The proecion seller has in his siuaion a beer abiliy o diversify agains his risk (over perhaps 000 differen cliens) han he buyer of he insurance may have. So, in his siuaion i is easy o see he incenives for boh paries o ener ino such an agreemen. As credi defaul swaps have become increasingly more popular over he pas 0 years, hey have become a useful ool in he assessmen of he credi risk of a company. The premium rae of CDS conracs is denoed by heir CDS spreads which are noed publicly for 9

8 banks and larger companies. As shown in Figure., a noable even in he hisory of CDS rades is ha he spread of many of hem were considerably higher han normal abou half a year before he credi crisis of 008. The CDS spreads of four large banks (Barclays Bank, BNP Paribas, Deusche Bank and Royal Bank of Scoland) sared o move urbulenly afer July 007 and a peak around March 008 came when Bear Searns was on he brink of bankrupcy and was offered o be acquired by J P Morgan (Roddy March 008 Forune). The acquisiion by J P Morgan was finalized on May 30. We also see ha he sock marke crash a Ocober 008 gave rise o a spike, especially for he Royal Bank of Scoland. So he CDS marke gave clear indicaions ha somehing bad was abou o happen and ha he financial indusry were aware of i before i happened. Figure.: The CDS spreads of four large banks illusraing he urbulence prior o he crisis of 008. Source: Reuers/GFI (Through Alexander Herbersson s lecure noes) There are oher ypes of swap agreemens such as currency swaps where wo paries usually exchange fixed rae ineres paymens in one currency wih he same ineres rae paymens in anoher currency. We also have commodiy swaps which can be seen as a se of forward conracs on a cerain commodiy. They are mos commonly used by airline companies on crude oil o hedge agains sudden price increases. We also have equiy swaps where a variable cash flow from equiy is exchanged for cash flows from a deb wih a fixed or floaing ineres rae. The Bank for Inernaional Selemens repors saisical daa on swap rades semiannually and he char in Figure.3 shows he oal amoun ousanding for he hree mos popular swaps. 0

9 Figure.3: Ineres rae swaps is he mos raded ype of financial insrumen in he derivaives markes reaching almos US$ 400 rillion by December 00. The char also shows ha he rading in credi defaul swaps declined afer he financial crisis of 008. Source: Bank for inernaional selemens (bis.org) We see in Figure.3 ha he marke for ineres rae conracs and foreign exchange conracs (currency swaps) is expanding whereas he populariy for he credi defaul swaps has become a bi ained since he financial crisis of 008 and AIG s ensuing liquidiy crisis semming from is large porfolio of ousanding CDS agreemens and collaeralized deb obligaions (CDO). Mos swaps have hisorically been raded over he couner whereas some of hem are also raded on public fuures markes. In recen years however, regulaions in he Basel III accords have forced swaps and paricularly CDS swaps o be raded via so called cenral counerpary clearing houses or CCP s (bis.org 00). So experience from he AIG inciden shows ha when rading in he OTC marke, he credi qualiy of he counerpary is quie imporan.. Inroducion o Credi Risk Credi risk is defined as he risk ha an obligor fails o mee obligaions owards crediors. When his happens he obligor is said o defaul. A defaul happens for example when a company goes bankrup, or fails o pay a coupon on one of is issued bonds in ime or when a household fails o mee is amorizaion schedule. So credi risk and defaul risk are prey much equivalen. The credi worhiness or credi risk of an eniy (be i an individual, a company, or even a whole counry) is commonly assessed by a credi bureau or a raing agency which gives i a credi raing (Hull 006). The agencies Moody s and Sandards & Poor classify heir raings ino brackes saring from AAA/Aaa (Sandard & Poor s / Moody s) which is he highes raing proceeding wih AA/Aa, A/A, BBB/Baa, BB/Ba, B/B and CCC/Caa. Each bracke is associaed wih a probabiliy of defaul where a higher raing implies a lower probabiliy of defaul (S&P Whiepaper 009). From his

10 raing a risk premium is added o he ineres rae of a loan or a bond ha is issued by his eniy. The scale of credi raing is raher coarse and he raing bureaus increase he granulariy by dividing he lower brackes ino subcaegories (such as Aa, Aa, or A+, A, A, ). The challenge is o ge a good esimae of he probabiliy of defaul. Credi risk is no saic. I varies over ime and wha is ineresing o noe is ha for a bond wih a high credi raing he defaul probabiliy end o increase wih ime whereas he defaul probabiliy end o decrease over ime for a bond wih a poor credi raing (Bodie, Kane, Marcus 008). The reason for his is ha for poor raing bonds he firs couple of years may be criical whereas for high raing bonds here is he possibiliy ha he financial healh will decline over ime. There are several ways o esimae he credi worhiness. The credi bureaus commonly look a financial hisory and he balance of asses and liabiliies (Hull 006). The downside of his is ha hese raings are revised quie infrequenly. So, financial insiuions who deal wih credi derivaives ha require a more coninuous assessmen use more sophisicaed saisical mehods in heir assessmens. If we ignore he influences from exernal marke facors we have he following elemens o consider involving credi risk and he modeling hereof (Schönbucher 003): Arrival risk This is also known as he probabiliy of defaul wihin a given ime period. Timing risk There is an uncerainy of he precise ime of defaul, i.e. o know he iming of a defaul. In order o know he ime of defaul one also needs o know abou he arrival risk for all possible ime horizons. This ype of risk involves his ype of uncerainy and one can say ha iming risk is more deailed and specific han he arrival risk. Recovery risk A defaul of an obligor doesn generally imply ha he loss on he credior s behalf will be 00%. The amoun he credior can claim from a defaul or bankrupcy is called he recovery. The risk involved wih he severiy of a defaul given ha i had occurred is herefore called recovery risk. Defaul dependency risk This is he risk ha several obligors defaul ogeher. I is also known as defaul correlaion risk and ime has proven i o be one of he mos crucial elemens o consider when i comes o credi risk..3 Credi derivaives and usage of swaps o manage credi risk Like one can have insurance for one s car or one s house i is possible o insure a loan or a bond. The mos popular way of insuring loans and bonds is hrough wha is called credi defaul swaps (CDS). When one eners ino a

11 CDS agreemen one agrees o pay a sream of paymens o he insurer for lending money o someone or buying obligaions. Should he obligor defaul, he CDS agreemen will erminae and he insurer will compensae for (he insured par of) whaever losses ha will be incurred from he defaul. In derivaives erminology, as a buyer of a CDS i is said ha one is long he premium leg (i.e. one pays he premium for he conrac) and shor he defaul leg (i.e. one receives paymens from he conrac in he even of a defaul) (Lando 004). Anoher way o manage ineres rae risk is hrough ineres rae swaps where one can exchange a floaing ineres rae for a fixed rae. Managers of credi porfolios can also manage credi risk wihin a porfolio by buying collaeralized deb obligaions (CDO) which is a more complicaed ype of credi proecion conneced o a whole porfolio of bonds, loans or morgages (Kane 008). There are differen ypes of CDOs bu he mos common is he synheic CDO which is consruced synheically from a porfolio of underlying CDS agreemens. So he risk exposure in a synheic CDO is aken on credi defaul swaps raher han direcly on he bonds ha he CDS agreemens apply o. The obligaions in he credi porfolio can be divided ino ranches of asse classes and here are differen CDOs ha cover differen ranches in a credi porfolio. All hese insurances are classified as credi derivaives..4 Counerpary Credi Risk and Credi Derivaives In he ransacion of credi derivaives (and oher OTC raded derivaives) here is also a risk ha he issuer of he derivaives defauls on his par of he conrac and fails o honor he agreemens wihin ha conrac. This ype of risk is called counerpary credi risk (CCR) and recen evens such as he financial crisis have marked heir imporance when dealing wih credi derivaives. For years i has been a sandard pracice in he indusry o mark porfolios of credi derivaives o marke wihou aking his ype of risk ino accoun (Pykhin, Zhu 007). In he early days of he credi derivaives markes, only he financial insiuions wih he highes credi raing were dealing wih hem. They were and are offered over he couner, i.e. on he OTC marke. Insiuions wih lower credi worhiness were excluded enirely or had o mee addiional rading requiremens such as paying subsanial premiums or were bound o rigid collaeral erms. This was and is done by wha is called margin agreemens (Algorimics Whiepapers 0). A margin agreemen limis he poenial exposure by means of collaeral requiremens should he unsecured exposure exceed a pre specified hreshold. Whenever his hreshold is exceeded, he oher counerpary mus supply addiional collaeral ha is sufficien o cover his excess (Cesari, 009). This is very much like he margin calls in a fuures conrac. In he early 000 s he rade wih derivaives grew rapidly and he ousanding noional of derivaives ransacions reached over $500 rillion by early 008 and over $600 rillion by 3

12 he end of ha year. When i grows o such immensely huge amouns, i has urned ou ha counerpary credi risk plays a crucial role. By ha ime i was negleced because he financial insiuions could ac as clearing houses and limi he exposure by neing posiive and negaive posiions and offse hem wih respec o a defauling counerpary. While neing when used properly is a useful ool for miigaing risk, i complicaes quaniaive measuremens significanly. Accouning for counerpary credi risk in OTC ransacions is done by correcing by applying credi value adjusmens (CVA). While CVA pracices for adjusing wih respec o counerpary credi risk are specified wihin he Basel II accords (Basel Commiee on Banking Supervision July 005) and in he IAS39 accouning sandards, many insiuions negleced his par before 008 and herefore vasly underesimaed he CCR since he exposure was wih oo big o fail counerparies. In fac, he American banks were sill merely implemening he Basel I accords prior o he 008 sock marke crash. Evens such as he bankrupcy of Lehman Brohers, he bailou of Bear Searns and he AIG crisis have changed he aiude owards counerpary credi risk dramaically among invesors according o a recen survey (Algorihmics Whiepapers 0). I has been repeaedly called for an indusry overhaul and sricer regulaions have been pushed for in he Basel III accords since he swaps were blamed for being a chief conribuor o he collapse of Lehman Brohers and he AIG. In he meanime an inensive research is done in his field and new models addressing his ype of risk are being under heavy developmen. As has been discussed in e.g. Canabarro, Duffie (003) and Cesari (009) here are he following definiions involved wih counerpary credi risk: Curren Exposure (CE) This is he curren value of counerpary credi exposure Poenial Fuure Exposure (PFE) This is a saisical measure of fuure exposure generaed from a sochasic simulaion. E.g. a 95% PFE wih value 00 means ha he fuure exposure of he forecased horizon will no exceed 00 wih 95% confidence. Expeced Exposure (EE) This is he expeced value of he exposure up o he end of he forecased period. I is in paricular he expeced posiive exposure (EPE) ha is looked a in he assessmen of counerpary credi risk. These are he ools o assess fuure exposure. There are several differen models o esimae hese exposures ha are calibraed agains measures such as rades agreemens, legal eniies, opinions, collaeral holdings, limis ec, and hey are commonly esimaed hrough Mone Carlo simulaions. The shape of he models is also differen depending on he ime horizons. I is for example generally required o add jump diffusion processes for shorer ime horizons (Das, Sundaram 999) whereas jump diffusion becomes less imporan for longer ime horizons. There are mainly hree ways o miigae 4

13 his ype of exposure, some of which have already been menioned in his paper: Collaeralizaion A margin accoun is se up wih he counerpary and is generally managed in a fashion ha is similar o margin accouns for fuures posiions in he public exchange marke. The counerpary receives margin calls should he value of he posiions go below a cerain hreshold and generally he overdraw should exceed a cerain amoun o miigae he number of ransacions o his accoun. So, if he overdraw limi is se o say , an overdraw by will no generae a margin call bu an overdraw of will. The period ha defines he frequency a which collaeral is moniored and being called for is called he call period which is ypically one day. The ime inerval necessary o close ou he posiion wih he counerpary and re hedge is resuling marke risk should he counerpary defaul is called he cure period. This is he period o cure he wound ha is caused from he defaul of he counerpary. The oal ime inerval from he las exchange unil he defauling counerpary is closed ou is called margin period of risk which is he sum of he call period and he cure period. A collecion of rades whose values should be added in order o deermine he collaeral o be posed or received wih his collecion is called a margin node. Neing The exposure can be grealy reduced by wha is called neing agreemens. This agreemen is a legally binding conrac beween wo counerparies o aggregae he ransacions beween hem in he even of defaul. This means ha negaive exposure in one conrac will be offse by a posiive exposure in anoher which can grealy reduce he overall counerpary exposure. A collecion of rades wihin a posiion ha can be need is called a neing node. Credi Value Adjusmen (CVA) The premium or credi value is adjused o cover he risks ha are involved wih he posiion. The adjusmen can be eiher posiive or negaive depending on which pary of he conrac holds he highes risk. The CVA is generally calculaed from PFE and/or EE esimaions. The focus of his paper is on his adjusmen. Pykhin, Zhu (007) idenifies hree main componens in he calculaion of counerpary exposure: Scenario Generaion Fuure marke scenarios are simulaed for a given fixed se of daes using evoluion models of he risk facors Insrumen Valuaion For each simulaion dae and each realizaion of he underlying marke risk facors, he holder s insrumens are valued for each rade in he counerpary porfolio using he simulaed scenarios. I should be noed ha pah dependen insrumens such as American, Asian or Bermudan insrumens require a differen 5

14 approach han pah independen insrumens such as European insrumens. Porfolio Aggregaion For each simulaion dae and for each realizaion of he underlying marke risk facors, counerpary level exposure is obained by aggregaing he porfolio according o he holders neing agreemens There is also anoher elemen of risk involved wih counerpary exposure called righ way risk and wrong way risk. This ype of risk is involved wih he credi qualiy of he counerpary. I is wrong way if he exposure owards he counerpary ends o increase when he credi qualiy of he counerpary worsens and i is called righ way if he exposure ends o decrease wih he credi qualiy of he counerpary. A ypical wrong way risk scenario is when a bank eners ino a swap conrac wih an oil producer where he bank receives a fixed rae whereas i pays he oil producer he floaing crude oil price. Decreasing oil prices in his scenario will worsen he credi qualiy of he oil producer and increase he value of he swap o he bank. So he bank will be faced wih a wrong way risk scenario as he swap goes he wrong way for he oil producer. A righ way risk scenario will be faced by he bank if i insead akes he floaing rae and pays he fixed rae o he oil producer. This will be beneficial o he oil producer (alhough his credi qualiy will worsen) as i goes he righ way whereas i will be less beneficial for he bank. The emphasis wih his ype of risk is ha in eiher way, he bank will be faced wih a risk exposure wihin is swap posiion. There is no way for he bank o benefi from he increasing value of he swap if he oil producer defauls on his paymens. Modeling framework Credi risk modeling is usually done by sochasic models ha in one form or anoher use sochasic processes ha capure he flucuaing naure of facors such as sock prices, ineres raes ec. ha influence he naure of he credi risk wihin a given eniy. We discuss his furher in Secion.. The mos common way of assessing credi risk is by using so called inensiy models which we describe in Secion.. The resuls in inensiy models ofen ranslae ino a credi spread ha can be pu direcly on op of a given ineres rae in he valuaion of financial insrumens such as credi risk insurance policies of a given deb. Ineres rae swaps can be modeled in differen ways depending on wha facors ha are o be aken ino accoun. As we will see in Secion.3, he idea is o idenify he sream of paymens wihin an ineres rae agreemen as a se or porfolio of more rudimenary forward rae agreemens and value hem according o wha is a fair rae of he enire swap. Secion.4 discusses he how credi defaul risk is valued in general and Secion.5 describes how an ineres rae swap is valued in he presence of counerpary credi risk wih he framework of Secion.4 in mind. By assuming ha he counerpary 6

15 credi risk is sochasically independen from he underlying ineres rae of an IRS agreemen we can resor o a simplified framework ha calculaes he defaul probabiliies and he expeced cash flows from he IRS agreemen separaely using he basic Black Scholes model for swapions. Secion.6 inroduces a few conceps o furher enhance he valuaion of he counerpary risk adjusmen. We discuss an approach o accouning for when he ineres rae and he defaul inensiy follow a sochasic process and are correlaed. We consider he case for when he ineres rae follows a bivariae G++ process and when he defaul inensiy follows a CIR++ process. Calibraion procedures for hese models are also discussed and his Secion is ended wih a discussion abou numerical mehods for simulaion which will see are required when here is a correlaion beween he ineres rae and he defaul inensiy.. Sochasic modeling Credi risk is associaed wih a probabiliy of defaul and we wan o esimae his probabiliy in such a way ha we can find a premium o add on op of e.g. a lending rae. This is why so called inensiy models have become popular. Using sochasic processes in he inensiy models works very well bu hey add o he complexiy of he calculaions. The complexiy of a credi risk model increases dramaically wih he number of facors o be accouned for. The simpler models even have closed form expressions ha one can use direcly o calculae he credi risk. The more complex models don have such soluions and herefore require simulaions. Sochasic models such as he CIR model have become popular o use because hey have properies ha have been recognized in observaions of e.g. ineres rae movemens. The problem however is ha hey are difficul o fi o real world daa and are no as flexible as one would wan o wish for. Credi risk modeling in is radiional form generally assumes ha here is a sock value represening he obligor ha follows some form of sochasic process. The even of defaul is hen defined as he poin when his sock value dips below a cerain hreshold, usually represening he amoun of deb held by he obligor. An imporan and ground breaking credi risk model is he so called Meron model (Lando 004) where i simply looks a deb as a European pu opion and he sock as a European call opion. I is easy o use simply because i has closed form soluions which are found hrough he Black Scholes formula. The downside of his is ha he Meron framework is simplified and doesn accoun for all of he elemens involved wih credi risk, paricularly defaul dependency risk. A he hear of his framework is he Wiener process which is a process where all incremens are sochasically independen, i.e. hey are no correlaed. A way o make he Meron and he Black Scholes model accoun for defaul dependency risk is o base i upon a process where he incremens have some degree of correlaion. This makes calculaions more 7

16 complicaed and in mos cases here won be any closed form soluions. Also, some processes ha in such calculaions would replace he Wiener process in he Black Scholes model don have finie variance (i.e. Var(X ) = ) or second order momen (E[ X ] = ) which would complicae hings even furher. Such processes wih infinie variance are called lepokuric because heir incremens have a fa ailed disribuion which is a consequence of he infinie variance. The fa ail naure of hese disribuions makes hem appealing o use o capure unforeseeable evens such as sock marke crashes or oher observed phenomena ha are no mahemaically well behaved.. Inensiy Modeling In his Secion we sudy he assessmen of credi risk using inensiy models. We begin by explaining sopping imes which is he foundaion of inensiy models. We hen show how an inensiy ranslaes ino a probabiliy and how i can be applied e.g. in he valuaion of a risky bond. Inensiies are commonly assumed o be piecewise consan and are usually calibraed agains CDS quoes ha are observed on he marke. The mehod of calibraing a piecewise consan inensiy agains marke CDS spreads is called boosrapping which is explained furher a he end of his secion. The ime beween he presen and a given even of defaul is defined in sochasic calculus as he sopping ime (Klebaner 005, Shreve 008). Formally: A non negaive random variable given a filraion F is called a sopping ime if for each he even { } F. The erm filraion is a heoreical concep in sochasic processes which in layman s erms means informaion, i.e. i is condiional on ha a given series of evens has happened up unil ime. This means ha given he informaion in F i can be decided wheher { } has occurred or no. So if he filraion is generaed by a sochasic process {X }, hen by observing i up o ime, from he generaed values X 0, X,, X we can decide wheher he even { } has or has no occurred. 8

17 Figure.: The sochasic variable is defined as he firs ime a given sochasic process V crosses he hreshold D and is called a sopping ime. A sopping ime can be used o denoe e.g. a defaul ime which hen is he ime poin when he value of a company goes below he value of is deb. As can be seen in Figure., he sopping ime is he ime a which he sochasic process V his he hreshold D. If we look a a capial srucure of a company, he sochasic process represens he marke value of he company and he hreshold is he amoun of deb ousanding. In his framework he sochasic process is above he hreshold and he defaul occurs a he poin when he process inersecs his hreshold. A popular way of modeling credi risk is hrough wha is called he inensiy based approach (Schönbucher 003, Lando 004, Brigo, Mercurio 006). The inensiy is defined as a probabiliy of defaul wihin a given (infiniesimal) ime period, i.e. wih respec o a given filraion F and a defaul ime >. Given a sochasic process X and an mapping λ (X ) of ha process where λ ( X ) [ 0, ), he defaul probabiliy wihin an infiniesimal ime period d is 0 [ ) l( ) lim P, + F = X d, on τ > (.) and he sopping ime in he inensiy modeling framework is defined as 0 { λ( X ) 0 s ds E} = inf (.) for an exponenially disribued random variable E ha is independen of X. The mapping λ (X ) is also known as he inensiy for he random variable τ. I should be noed ha his inensiy is a condiional parameer, i.e. i is a measure of defaul probabiliy a ime condiional on no earlier defaul. So λ() is he probabiliy of defaul beween ime and + condiional on no 9

18 earlier defaul. In his framework i can be shown ha he survival probabiliy (he complemen of he defaul probabiliy) up o ime is P ( 0 s ) X ds. (.3) [ > ] = E exp λ( ) The sochasic process X s can be chosen arbirarily and i can be designed o accoun for any ype of risk involved wih credi risk. Of course, he more facors ha are involved, he more difficul i will be o make a sensible esimaion. There is a lo more o say abou his bu ha would reach beyond he scope of his paper. The beauy of his formula can be illusraed by assuming ha he inensiy is deerminisic and consan wih respec o ime: Le P( 0, ) be he presen value (a ime 0) of a risk free bond wih a consan coninuously compounded ineres rae r ha pays one uni of a given numeraire a a fuure ime. Assuming ha he ineres rae is consan a risky bond P ( 0, ) wih defaul probabiliy P [ < ] can hen be valued as P ( 0, ) = e (r+λ) since P ( 0, ) = E P( 0, ) { > } = E[ P( 0, ) ] E { > } ( r+ λ ) = P( 0, ) P[ > ] = exp( r) E[ exp ( λ) ] = e. (.4) In his case he inensiy λ is a kind of a risk premium over he risk free rae ha is esimaed from credi risk assessmens and P[ > ] is he survival probabiliy of he risky bond, i.e. he probabiliy ha i will no defaul prior o. One very common implemenaion of λ is leing i follow a piecewise consan funcion as i is easier o calculae, compuaionally less inensive han sochasic funcions and easier o properly fi wih real world daa han for example CIR models. The usage of piecewise consan defaul inensiies is very common in he financial indusry when calibraing a defaul probabiliy disribuion from marke quoes of CDS spreads. The easies way o arihmeically define a piecewise consan funcion is o formulae i using he indicaor funcion. So we have N λ( ) = λj { T j < T j} j= N ( ) = λ j { T j 0} > { T j> 0} j= (.5) An indicaor funcion X is a funcion ha has he value if X (e.g. x < 5 or A B) is saisfied and 0 oherwise. A noable resul is ha he expeced value of an indicaor funcion is he probabiliy for X o happen, i.e. E[ X ] = P[X]. 0

19 for a se of N ime inervals wih N inensiy values for each ime period. Le m = m() be he larges inerval where [0, T m ], i.e. m() = max { [0, T j ]} for j 0,. The siuaion is illusraed in Figure.. ( ) λ ( ) λ λ 3 λ λ J+ T T T 3 Figure. The inensiy funcion λ() as a piecewise consan funcion where T m encloses he larges inerval [0, T ] ha does no conain. From his we have he survival probabiliy funcion m ( ) P[ > ] = exp λj( T j T j ) λj+ ( T J) (.6) j= which is a resul from he inegraion in Equaion (.6). The use of a piecewise consan inensiy has he obvious drawback of being disconinuous. Using insead a piecewise linear inensiy has proven o someimes yield srange resuls when exrapolaing up o 0 years (in some cases also wih negaive probabiliies) (Brigo, Pallavicini 008). Wha we can observe from real world daa is he CDS spreads on CDS conracs for differen mauriies. If we assume ha a) he accrued premium erm is ignored and b) a a defaul τ n n in he period,, he loss is paid a ime 4 4 n = /4, i.e. a he end quarer insead of immediaely a τ, we have he following formula for calculaing a CDS spread given he probabiliy above using piecewise consan inensiy m T m T m + ( ) RT = 4T ( ) ( φ ) D( ) F ( ) F ( ) n= 4T 4 n= n n n ( n ) ( ) ( ) D F n (.7) where φ is he recovery given defaul which is a percenage of he loss ha can be recovered from he defaul should i happen, D( n ) is a discoun facor and

20 F τ ( n ) is he cumulaive disribuion funcion ha represens he defaul probabiliy which is P[ > ], i.e. F τ ( n ) = P[ ]. The cash flows ha are exchanged in a CDS swap agreemen is illusraed in Figure.3. A ( ) RT N quarerly up o T 4 N, credi loss from C if < T B C N = nominal insured = credi loss in % = defaul ime for C Figure.3: Cash flows beween paries in a CDS agreemen. Here, he proecion buyer A pays B a quarerly fee given ha C has no defauled. If C defauls which happens when τ < T, B pays A for he credi loss incurred by C which is N. Source: Alexander Herbersson s Lecure noes We can esimae he inensiy parameers { λ } N j j= by recursively finding inensiy values ha yield a CDS spread R(T) ha maches he real world daa, saring a he firs ime inerval, moving on o he nex and so on. This way of esimaion is called boosrapping. Boosrapping is a mehod of calibraing he inensiy owards observed marke quoes of CDS spreads. The inensiy is assumed o be deerminisic and piecewise consan for differen periods of ime, see Figure.. If we have observed CDS spreads for credi defaul swaps wih J differen mauriies, he inensiy is hen given by λ ( ) λ if 0 < T λ if T < T = λ if T < T λj if T J < J J J for some erm srucure T = { T,..., TJ } (.8) and a given se of consan values { λ,..., λ J }. Saring wih he lowes mauriy we find a value λ for he inensiy such ha RT ( ) = RM ( T ) as of Equaion (.7) for an observed CDS spread R ( ) M T of a CDS agreemen wih mauriy T. In he nex sep we find a value λ given he prior value λ ha yields he same value as he observed marke CDS spread for a CDS conrac wih mauriy T, i.e. saisfies RT ( ) = RT ( λ) = RM ( T ). Then we proceed in a recursive manner o λ esimae he res of he inensiy values from he J observed CDS spreads

21 given he prior esimaions, i.e. in each ieraive sep we find a λ j such ha RT ( j λ, λ,..., λj ) = RM( T j) unil j = J. I should be noed ha a CDS conrac M( j) λ,..., j+ λj since i sops a mauriy T j. R T does no depend on { }.3 IRS Valuaion framework A forward ineres rae agreemen (FRA) is legally binding agreemen where he ineres rae of a given deb is fixed so ha i is consan unil mauriy. A he ime of mauriy one receives an ineres ha is accrued beween a fuure sar dae (Brigo calls i dae of expiry) and he dae of mauriy. So an FRA is effecively a swap agreemen where a floaing rae ineres paymen is swapped for a fixed rae paymen which is seled a he ime of mauriy. The siuaion is illusraed in Figure.4. Figure.4: Pary A eners an agreemen wih pary B o pay a fixed ineres rae paymen in exchange for a floaing paymen a mauriy T. This ype of agreemen is called forward rae agreemen or FRA in shor. The difference beween an ineres rae swap and a forward ineres rae agreemen is ha whereas a swap involves a sream of paymens unil mauriy here is only one paymen in a forward rae agreemen which occurs a mauriy. If we le T be he sar dae, S be he dae of mauriy, where T < S, τ(t, S) be a meric ha measures he ime beween T and S in number of years (his meric is also known as year fracion) and le N be he nominal value of he conrac, hen he value of he conrac is ( ) (, ) (, ) Nτ T S K L T S where L(T, S) is he spo rae reseing a ime T and mauring a ime S, and K is an agreed upon fixed rae. The value of he conrac is husly he difference beween he wo raes accrued over a given ime period τ(t, S). When enered ino a some ime < T, he fixed rae K is usually se so ha he expeced value of he conrac is zero and he value is discouned by a facor P(, T). The funcion P(, T) represens he value of a bond a ime < T ha pays one uni of currency a ime T. The floaing rae L(, T) is he simply compounded spo ineres rae a ime for he mauriy T (i.e. his rae may have a erm srucure) and is defined by he formula (, ) L T A ( ) ( ) ( T) P( T) Fixed ineres i Floaing ineres paymen a mauriy T ( ) ( T) P( T) B PTT, P T, P T, = = (.9),,,, 3

22 since P(T, T) =. This is he same formula as when calculaing he reurn from a sock over ime and convering i o he effecive annual reurn by dividing i by a year fracion erm. The so called LIBOR raes are ypically compounded his way which is why Brigo habiually denoes his rae by L (Brigo, Pallavicini 008). For he forward conrac o be rendered fair, is value should be se o zero. Thus if we se he rae K equal o he forward rae F ha makes he forward rae agreemen zero, we have for a forward rae agreemen FRA(T, S) beween T and S ha ( ) ( T S) Nτ ( T S) F L( T S) FRA, =,, = 0 = Nτ ( T, S) F τ PTS (, ) ( T, S) P( T, S) = N τ ( T, S) F +. PTS (, ) (.0) The agreemen is se up so ha i is zero, boh a he ime of enry and a mauriy. This is because he no arbirage argumen implies ha he payoff of he FRA is zero iff he curren value is zero. If we consider he agreemen in Equaion (.0) bu a a fuure ime poin where < T, hen we can define he discouned FRA(,T,S) prevailing a as ( T S) P( S) N ( T S) F P( T) P( S) FRA,, =,,, +,. (. ) FRA(, T, S ) = P (, S ) FRA( T, S ). From he arbirage argumen (Brigo, Mercurio 008) we can break up he discoun facor P(, S) such ha which in (. ) yields P(, S) = P(, T) P( T, S) ( T S) P( S) N ( T S) F P( T) P( S) FRA,, =,,, +,. (.) By seing Equaion (.) o zero jus like Equaion (.9), we have he simply compounded forward ineres rae F(, T, S) prevailing a ime wih sar dae T, and mauriy S defined as (, ) ( ) P T F( T,, S) =. (.3) ( TS, ) PS, 4

23 Mahemaically, an ineres rae swap can herefore be seen as a porfolio of forward rae agreemens where he mauriy of one FRA is he sar dae of anoher, i.e. we can formulae a payer IRS as β ( T T α β N K) = P( Ti) N ( Ti Ti) K F( Ti Ti) i= α + ( ) PFS,,,,,,,, (.4) where he effecive period of he IRS is beween T α and T β. Hence, he fair rae a ime ha ses he value of his swap agreemen o zero is ab, ( ) ( ;, ) S = S T T = a b i= a + (, a) P( T, b) P T b ( T, T) P( T, ) i i i. (.5) This formula doesn say anyhing a all abou he value of he conrac wihin he duraion of his agreemen, i merely saes ha he value is zero a he ime of enry and a he ime of mauriy. Since we canno be sure ha boh paries will honor heir commimens o such a swap agreemen we wan o know he poenial credi loss ha would be incurred for one of he paries a he defaul of he oher. The coninuous valuaion of an esablished swap agreemen can be done by looking a a swap as a swap opion or swapion. A swapion is an opion giving he buyer of i he righ bu no he obligaion o ener ino a swap agreemen a a fuure ime. We usually agree ha i is European, i.e. ha he opion can only be exercised a he given dae of mauriy. Since here is no incenive for a swapion holder o exercise i if he ne presen value of he swap is negaive, he payoff funcion is posiive and we can rea i as a call opion where we can value i using Black Scholes formula. A payer swapion is hen valued as ( ( ) ) ( ) β ( ) i ( i) PS KS,,, σ = NBl KS,, σ T, PT, (.6) αβ, αβ, αβ, αβ, αβ, α i= α + where τ i = τ(t i-, T i ) and Bl represens Black Scholes formula 5

24 ( ( a )) ( (, aβ, ( ), aβ, a )) ( ) ( σaβ, Ta ) ( ( ) ) = ( ) Φ ( ) Bl K, S, σ T, ω S ω ωd K, S, σ T aβ, aβ, α αβ, aβ, aβ, (, aβ( ), σaβ a ) d KS T,, (, aβ( ), σaβ a ) d KS T,, KωΦ ωd KS σ T Saβ, ln + = K σ Saβ, ln = K σ aβ, ( ) ( σaβ, Ta ) aβ, T T a a (.7) where Φ is a sandard Gaussian cumulaive disribuion, K he srike price, and σ α,β he volailiy a he fair price Sαβ, ( ) as given in Equaion (.5). The exra coefficien ω is mainly used o mark he sign of he srike par of he payer swapion where ω is + for a payer swapion and for a receiver swapion. For a more horough reamen, see Brigo, Mercurio

25 .4 General valuaion of counerpary risk The general procedure for valuing a cash flow in he presence of a counerpary defaul risk involves adding a premium erm ha represens his ype of risk. We le he filraion F denoe all observable marke quaniies bu he defaul even up o ime. Moreover we le H = σ({ u} : u ) be he righ coninuous filraion generaed by he defaul even. From his we se G := F H (so F is a sub filraion of G, i.e. F G ) and E [.: ] E. [ ] = G. If we le P D (, T) (which we also abbreviae o P D ()) be a discouned payoff funcion of a generic defaulable claim and CF(, T) be he cash flows beween ime and T from a coningen claim wihou counerpary risk, hen he ne presen value of hese cash flows a he defaul ime is defined as NPV() = E [CF()] for he filraion G τ a τ, and D P ( ) = { } CF (, T ) T + + ( ( )) + { T} CF (, ) + D(, ) φ( NPV ( ) ) NPV ( ) (.8) where D(u, v) is a sochasic discoun facor a ime u wih mauriy v, A is an indicaor funcion for he even A and φ is he recovery given defaul. The funcion x + is he maximum of x and 0, i.e. x + = max(0, x) and is commonly used o describe he payoff of a pu or a call opion. In his seing we assume ha he counerpary defaul risk is unilaeral, i.e. we assume ha only one counerpary may defaul whereas he oher counerpary is assumed o be risk free. The raionale of his is ha if he counerpary doesn defaul, hen he cash flows will go as usual unil mauriy T. Should a defaul happen a a ime prior o mauriy, i.e. < T, hen he cash flows will sop a ha poin of ime and if he ne presen value of he conrac is posiive, only a cerain percenage of wha is lef of he counerpary will be recovered. If he ne presen value is negaive, he enire value has o be paid o he counerpary if i defauls. In his framework i can be shown ha he expeced payoff of a defaulable claim is given by he following formula (Brigo, Mercurio 008): D + E P ( ) = E [ P ( ) ] LGD E { < T} D(, ) ( NPV ( ) ) )))))) ))))))( Posiive counerpary risk adjusmen (.9) where L GD is loss given defaul which is he same as ( φ) which is assumed o be deerminisic. This is a general model and i is no specified in he model how he defaul risk probabiliy is consruced. In his hesis we consruc he 7

26 counerpary defaul risk probabiliy from defaul inensiy models. The focus in his hesis is on his counerpary risk adjusmen erm..5 Counerpary Risk and IRS Valuaion Le he value of a risk free ineres rae swap agreemen be IRS(). In he general framework of Equaion (.9) we can hen express he value of a defaulable swap agreemen IRS D () as ( ) ( ) L ( ) D IRS = IRS DP. (.0) where he adjusmen erm DP() is defined as (Brigo, Masei 005) GD + DP( ) = E { < T} D(, ) ( NPV ( ) ) Tb = PS st, (, K, S( st ;, b), sst, ) dsq( s) Ta b b (.) where PS is he value of he payer swap as a swapion and K is he forward swap rae rendering he conrac fair a, i.e. K = S(; T a, T b ) as in Equaion (.5). Since we are going o express he values in erms of swap raes, he DP() adjusmen will be in erms of a spread and no a ne presen value in he regular sense as in Equaion (.9). If we assume ha he inensiy and he cash flows are independen, he calculaions become sraighforward. We can simplify furher wihou noable loss of accuracy by assuming ha defauls only occur a he poins of paymen T i. In his seing we can eiher assume ha he defaul occurs before he las paymen (he payoff is hen said o be posponed) or afer he las defaul (Brigo denoes his as anicipaed defaul). Under hese assumpions we have for he posponed payoff (P) and he anicipaed defaul (A) respecively ha b ( ) P DP ( ) = Q { ( T, T] } PS KS ;, ( ), σ i= a+ b i i ib, ib, ib, ( ) = ( Q ( > T ) Q ( > T) ) PS KS ;, ( ), σ i= a+ i i ib, ib, ib, (.) where PS i,b (;K,S i,b (),σ i,b ) is defined as in (.6) and b ( ) A DP ( ) = Q { ( T, T] } PS KS ;, ( ), σ i= a+ b i i i, b i, b i, b ( ) = ( Q ( > T ) Q ( > T) ) PS KS ;, ( ), σ. i= a+ i i i, b i, b i, b (.3) 8

27 Here Q is he condiional risk neural probabiliy measure for he defaul probabiliies of he counerpary, ha is Q (.)=Q (. F ). These probabiliies are compued from he boosrapped piecewise consan inensiy funcion and he payer swap is calculaed using Black Scholes formula for a European opion wih a payer swap payoff. For more deails on he derivaion of (.) and (.3) see Brigo Masei Enhancing counerpary risk valuaion The mehods presened previously for valuing counerpary risk are simplified and raher crude. In his secion we will discuss mehods for enhancing he valuaion of his adjusmen. One imporan enhancemen is o accoun for sochasic dependence beween he ineres rae and defaul inensiy. Accouning for his dependence ends o lead o complex calculaions and a limied se of opions if we seek o yield mahemaically feasible expressions. We conduc a furher discussion abou his in Subsecion.6.. In [Brigo Pallavicini 006] a valuaion of he counerpary risk in he presence of correlaion is assessed. There i is suggesed o le he ineres rae follow a G++ process and he defaul inensiy follow a CIR++ process. We consider hese cases in Subsecion.6. and.6.3 and discuss how hey can ge correlaed in Subsecion.6.4. Correlaed processes do no generally lead o closed form soluions so we need o conduc numerical simulaions o be able o assess he counerpary risk adjusmen under hese seings. In Subsecion.6.5 a he end of his secion we herefore inroduce some numerical mehods such as he Euler Maruyama mehod and he Milsein mehod for numerical simulaion..6. Risk valuaion under sochasic dependence We can enhance he counerpary risk adjusmen erm furher in he IRS valuaion by assuming ha here is some degree of sochasic dependence beween ineres raes and he counerpary risk. This can be done by assuming ha he sochasic process behind he ineres rae and he sochasic process behind he counerpary risk inensiy have some degree of correlaion. In [Lando 004] and [Brigo, Mercurio 006] a modeling of a join behaviour beween ineres raes and defaul inensiies are suggesed by incorporaing correlaed Brownian moions as drivers behind hese processes. Two approaches are suggesed o inroduce his correlaion. The firs approach is o ie he correlaion wih he noise erm. If we assume ha he ineres rae r and he defaul inensiy λ follow a mean reversing process hen he join behaviour can be modeled as 9