VALUATION OF OVER-THE-COUNTER (OTC) DERIVATIVES WITH COLLATERALIZATION

VALUATION OF OVER-THE-COUNTER (OTC) DERIVATIVES WITH COLLATERALIZATION by LEON FELIPE GUERRERO RODRIGUEZ B.S. Universidad EAFIT, 997 B.S. Universiy of Cenral Florida, 20 A hesis submied in parial fulfilmen of he requiremens for he degree of Maser of Science in he Deparmen of Mahemaics in he College of Sciences a he Universiy of Cenral Florida Orlando, Florida Summer Term 203 Major Professor: Jiongmin Yong

203 Leon Felipe Guerrero Rodriguez ii

ABSTRACT Collaeralizaion in over-he-couner (OTC) derivaives markes has grown rapidly over he pas decade, and even faser in he pas few years, due o he impac of he recen financial crisis and he paricularly imporan aenion o he counerpary credi risk in derivaives conracs. The addiion of collaeralizaion o such conracs significanly reduces he counerpary credi risk and allows o offse liabiliies in case of defaul. We sudy he problem of valuaion of OTC derivaives wih payoff in a single currency and wih single underlying asse for he cases of zero, parial, and perfec collaeralizaion. We assume he derivaive is raded beween wo defaul-free counerparies and analyze he impac of collaeralizaion on he fair presen value of he derivaive. We esablish a uniform generalized derivaive pricing framework for he hree cases of collaeralizaion and show how differen approaches o pricing urn ou o be consisen. We hen generalize he resuls o include muli-asse and cross-currency argumens, where he underlying and he derivaive are in some domesic currency, bu he collaeral is posed in a foreign currency. We show ha he resuls for he single currency, muli-asse case are consisen wih hose obained for he single currency, single asse case. iii

To my family iv

ACKNOWLEDGMENTS I would like o hank my advisor, Dr. Jiongmin Yong, for his suppor, paience, and wonderful advice on my undergraduae and graduae sudies. I would like o hank Dr. Xin Li for his suppor and advice o help me ge hrough difficul imes in his life-changing journey. I would also like o hank Dr. Peer Bossaers from he California Insiue of Technology for his suppor, inspiraion, and moivaion o achieve higher goals and a higher level of performance. I also wan o hank Dr. Richie He from he Royal Bank of Canada for he inroducion o his imporan opic and for his suppor, encouragemen, and valuable pracical advice. The auhor also highly appreciaes and is graeful for he conribuions and helpful discussions wih Binbin Wang, Shi Jin, Tong Zhao, Henrike Koepke, and Elham Negahdary, who also paricipaed in he 6h Workshop on Mahemaical Modeling in Indusry for Graduae Sudens a he Insiue for Mahemaics and is Applicaions, Universiy of Minnesoa, and a he Deparmen of Mahemaics, Universiy of Calgary, 202. v

TABLE OF CONTENTS LIST OF FIGURES...................................... ix CHAPTER : INTRODUCTION.............................. Financial Markes and Producs...........................2 Financial Economics and Financial Engineering................. 3.3 Derivaives Markes and Securiy Design.................... 3.4 Collaeralizaion................................... 6.5 Securiy Pricing wih Collaeralizaion...................... 8.6 Mechanisms of Unsecured and Secured Trades wih Exernal Funding... 0.7 Brief Discussion on Some Relaed Works..................... 3.8 Our Conribuion.................................. 4 CHAPTER 2: FUNDAMENTAL MATHEMATICAL CONCEPTS.......... 7 2. General Probabiliy Theory............................ 7 2.2 Informaion and Sochasic Processes....................... 9 2.3 Wiener Processes and Brownian Moion..................... 2 2.4 Geomeric Brownian Moion and The Process for a Sock Price........ 24 vi

2.5 Iô Process, Iô s Formula, and he Feynman-Kac Theorem.......... 27 2.6 Some Addiional Conceps of Ineres in Securiy Pricing........... 30 2.6. On Ineres Raes.............................. 30 2.6.2 Risk-Neural Valuaion (Risk-neural World vs. Real World)..... 3 2.6.3 The Greeks: Dela of a Derivaive.................... 32 2.6.4 The Numeraire............................... 32 CHAPTER 3: SINGLE CURRENCY, SINGLE ASSET ANALYSIS........... 33 3. Pricing by Replicaion wihou Specified Collaeral Process.......... 33 3.. The Processes for he Underlying and he Derivaive......... 33 3..2 Self-financing Replicaing Porfolio and Black-Scholes Equaion wih Collaeral.................................. 34 3..3 Derivaive Pricing Framework wih Collaeralizaion......... 36 3..4 Derivaive Pricing Formulas for Perfec,Parial, and Zero Collaeralizaion................................... 39 3..5 The Concep of Liquidiy Value Adjusmen (LVA)........... 42 3.2 Pricing by Maringales for Coninuous and Perfec Collaeralizaion..... 44 CHAPTER 4: CROSS-CURRENCY, SINGLE ASSET ANALYSIS........... 48 vii

4. Pricing by Replicaion and LVA.......................... 48 4.2 Pricing by Maringales for Coninuous and Full Collaeralizaion...... 50 CHAPTER 5: SINGLE-CURRENCY, MULTI-ASSET ANALYSIS........... 54 5. The Processes for he Underlying Asses and he Derivaive......... 54 5.2 Self-financing Replicaing Porfolio and Black-Scholes Equaion wih Collaeral......................................... 54 5.3 Derivaive Pricing Framework wih Collaeralizaion............. 56 LIST OF REFERENCES................................... 58 viii

LIST OF FIGURES Figure.: Unsecured conrac wih exernal funding................ 0 Figure.2: Collaeralized (secured) conrac wih exernal funding......... ix

CHAPTER INTRODUCTION. Financial Markes and Producs Financial markes exis because hey enable an efficien allocaion of resources across ime and across differen saes of naure. Take, for example, he case of a young individual who jus enered he job marke wih a high salary. If here are financial markes available, he earned income could be invesed in financial insrumens, such as socks and bonds, o finance he cos of home ownership, more educaion, or reiremen. The salary may be high in he presen bu his may no always be he case. This implici uncerainy makes necessary o prepare for unforunae saes of naure by rying o move resources from he presen ime o unknown imes. If here were no financial markes available, all his individual could do is consume since here would no be mechanisms in place o ransfer money from one sae of naure, where income is readily available, o anoher sae of naure, where he individual may have more limied income or no income a all. Now, consider he siuaion of a farmer who produces oranges in Florida. Wih financial markes, he farmer could use a derivaive conrac o hedge orange prices. A forward conrac, a fuures conrac, or a weaher derivaive would be very appropriae o prepare for poenial losses caused by unexpeced and damaging seasonal effecs. If here are no financial markes available, he farmer is jus subjec o whaever happens a any paricular ime, wih no mechanisms in place o preven or offse losses. Financial markes, hen, have hree main imporan roles ha makes hem essenial o our sociey. Firs, hey aggregae informaion from muliple sources, organize i, and make i available o all paricipans and ineracing agens in he marke. Markes also aggregae liquidiy,

prevening fragmenaion in order o make supply and demand work ogeher. Finally, markes promoe efficiency and fairness o all paricipans, paricularly when here is ransparency in prices, eliminaing insider informaion pracices ha preven markes from being well-funcioning. In all cases, financial funcions and implici risks due o uncerainy define risk-sharing as one of he mos imporan funcions of financial markes. Financial producs are creaed (engineered) in financial markes o saisfy paricular needs. For example, our aforemenioned farmer may use derivaives o hedge risk. On he oher hand, he same producs could also be used for speculaion (which would be he case of collaeralized deb obligaions). Producs also allow o raise capial (venure capialiss) o fund risky projecs or organizaions and expec reurn back from hem. They can also be used o fund liabiliies, such as buying a house or paying for educaion (for example hrough he use of annuiies). Financial producs are ypically raded in markes so ha heir price ges discovered by looking up and processing informaion ha is readily available o all agens in a fair marke. In our work, we assume ha he marke has he srucure of a perfecly compeiive marke in which here are large numbers of buyers and sellers, sellers can easily ener ino or exi from he marke, and buyers and sellers are wellinformed. We also assume ha he marke is arbirage-free, a siuaion in which all relevan asses are priced in such an appropriae way ha i is no possible for any individual gains o oupace marke gains wihou aking on addiional risk. This is commonly known as he no-arbirage condiion. Almos every produc in he financial marke is priced using he no-arbirage condiion wih respec o some underlying primary asse like a sock, a bond, or a commodiy of some kind. The siuaion is also known as he no-free lunch argumen: every nonzero, nonnegaive payoff comes wih a cos. 2

.2 Financial Economics and Financial Engineering Financial economics is concerned wih seing ineres raes and pricing bonds, equiies, and oher primary financial asses by using fundamenal equilibrium argumens. Financial engineering relies on financial economics by usually assuming ha ineres raes and prices of equiies are given and uses ha informaion o price derivaives based on no-arbirage argumens wih respec o some underlying primary asse like a sock or a bond. Financial engineering is comprised by hree major areas: securiy pricing, porfolio selecion, and risk managemen. Securiy pricing deals wih pricing derivaives securiies, such as forwards, swaps, fuures, opions, collaeralized deb obligaions (CDOs), and collaeralized morgage obligaions (CMOs). All hese producs are developed by financial engineers and i becomes necessary o assign a price o hem. The work presened in his hesis falls precisely under his caegory. In porfolio selecion, he goal is o choose a porfolio composiion and a rading sraegy o maximize he expeced uiliy wih respec o consumpion and final wealh. The oher major area, paricularly afer he recen financial crisis, is risk managemen, which aims o undersand he risks inheren in a porfolio and deermine he probabiliy of large losses..3 Derivaives Markes and Securiy Design The use of derivaives in financial markes has become increasingly imporan over he las four decades, even hough hese producs have pracically been around in some form for hundreds of years. Many differen ypes of derivaives are raded regularly by financial insiuions and fund managers in he over-he-couner (OTC) markes, as well as on Primary asses refer o hose from which oher asses are consruced 3

many exchanges hroughou he world. Derivaives are frequenly added o bond issues, used for execuive and employee compensaion plans, and embedded in many imporan capial invesmen opporuniies. This is why i is crucial o undersand how derivaives work, how hey can be used, and how hey can be priced fairly. In general erms, a derivaive can be defined as a financial insrumen whose value depends on he price of oher, more basic, underlying variables, such as he prices of asses (commodiies) ha can be raded, or even he amoun of rain falling a a cerain region in some paricular season. In a general conex, a financial insrumen can simply be hough of as a conrac or agreemen beween wo paries. A sock opion, for example, is a ype of derivaive whose value is deermined by he price of a sock. In his case, we refer o he sock as he underlying asse of he derivaive. For he case of he amoun of rain or any oher uncerain condiion, a derivaive can be hough of as a be on he price of a commodiy dependen on a fuure oucome of he underlying (weaher, in his case), which can be used o provide some kind of insurance and hedge he paries involved agains poenially unfavorable oucomes. This conrac is called a weaher derivaive. Noe ha invesors could also use his kind of conrac simply o speculae on he price of he commodiy, bu in his case he conrac would no be insurance. Hence, he riskreducing naure of a derivaive does no depend on he derivaive conrac iself bu on how he conrac is used and who is using i. There are hree disinc perspecives on derivaives ha define how we hink abou hem and how we use hem for a specific purpose [3]. The end-user perspecive considers corporaions, invesmen managers, and invesors as end-users. These users ener ino derivaive conracs wih specific goals in mind, such as o manage risk, speculae (as a way of invesing), reduce financial ransacion coss, or circumven regulaory resricions, and are more concerned abou how a derivaive can help hem o achieve he goals. The 4

marke-maker perspecive considers raders or inermediaries as marke-makers who are ineresed in buying or selling derivaives o cusomers for a profi. In order o make money, marke-makers charge a spread: hey buy a a low price from cusomers who wish o sell and sell a a higher price from cusomers who wish o buy. Marke-makers ypically hedge he risk of supply and demand and hus hey are mainly concerned abou he mahemaical deails of derivaives pricing (valuaion) and hedging. The work presened in his hesis falls under he marke-maker perspecive. Finally, he economic observer perspecive gahers informaion and analyzes he general use of derivaives, he aciviies of he marke-makers, he organizaion of markes, and he differen elemens ha hold everyhing ogeher. This work deals precisely wih one of he major conceps in financial engineering and derivaives: i is possible o consruc a given financial produc from oher producs and creae a given derivaive payoff in muliple ways. This is why he general problem of valuaion of derivaives is so imporan and also cenral in undersanding how marke-making works. The marke-maker sells a derivaive conrac o an end-user. Wih he proper pricing, his creaes an offseing posiion ha pays he marke-maker if i becomes necessary o pay he cusomer. This suggess ha i is possible o cusomize he conrac o make i more appropriae for paricular siuaions. The idea of cusomizaion is based on he idea ha a given conrac can be replicaed, which is why we will use he concep of a replicaing porfolio as an imporan ool for derivaive pricing in his hesis work. Finally, i is worh emphasizing he wo differen kinds of derivaive rading markes: exchange-raded markes and over-he-couner markes (OTC markes). A derivaives exchange is a marke where individuals rade sandardized conracs ha have been defined by he exchange. Some radiional derivaives exchanges are The Chicago Board of Trade (CBOT), The Chicago Mercanile Exchange (CME), and he Chicago Board Opions Ex- 5

change (CBOE). Many oher exchanges all over he world now rade fuures and opions, wih foreign currencies, socks, and sock indices as underlying asses. The OTC marke is an imporan alernaive o exchanges ha has become increasingly popular and larger han he exchange-raded marke. Essenially, rades are carried ou over he phone and are generally beween wo financial insiuions or beween a financial insiuion and one of is cliens (such as a fund manager or corporae reasurer) as par of a elephone- and compuer- linked nework of dealers. A major and characerisic advanage of he OTC marke is ha he erms of a conrac may differ from hose specified by an exchange. Marke paricipans have he freedom o negoiae any deal ha benefis he paries involved. A resuling disadvanage is ha here may be some credi risk in a rade (i.e., a small risk ha he conrac will no be honored). This key elemen became increasingly imporan, paricularly afer he recen financial crisis, bringing abou he idea and need o incorporae he figure of collaeralizaion o OTC derivaives, which complees he framework of his hesis work. We ofen refer o OTC derivaives wih collaeralizaion simply as collaeralized OTC derivaives..4 Collaeralizaion In lending agreemens, collaeral is a deposi of specific propery (an asse) as recourse o he lender o secure repaymen in case he borrower defauls on he iniial loan. Collaeralizaion provides lenders a sufficien reassurance agains defaul risk. If a borrower defauls on a loan, hen he borrower mus forfei he collaeral asse and he lender akes possession of he asse. For example, in ypical morgage loan ransacions, he real esae being acquired by means of a loan serves as collaeral. Businesses can also use collaeralizaion for deb offerings hrough he use of bonds. In his case, such bonds would 6

clearly specify he asse being used as collaeral for he repaymen of he bond offering in case of defaul. In banking, lending wih collaeralizaion ofen refers o secured lending, asse-based lending, or lending secured by an asse. This ype of lending usually presens unilaeral obligaions secured by propery as collaeral. Over he pas few years, he use of more complex collaeralizaion agreemens o secure rade ransacions has increased rapidly, paricularly in response o he recen financial crisis. In his ype of agreemen, also known as capial marke collaeralizaion, he obligaions are ofen bilaeral and secured by more liquid asses such as cash or securiies wih corresponding ineres raes. Cash is he mos popular choice for collaeral due o he ease of valuaion, ransfer, and hold. To undersand how bilaeral agreemens work, suppose wo paries ener ino a swap, which is one ype of OTC derivaive ha ransforms one kind of cash flow ino anoher. As ineres raes change over ime, one pary will have a mark-o-marke (MTM) profi on he deal, while he oher pary will have a loss. If he pary losing money were o be in defaul, he pary wih MTM profi would have o replace he deal a curren marke prices and he profi would be los. Hence, a posiive MTM value on he swap is a credi exposure on he oher pary. For his reason, banks ofen sae credi risk on a swap as he MTM value plus some addiional value o offse he poenial fuure credi exposure. This makes collaeralizaion a very imporan risk managemen ool o miigae counerpary credi risk in derivaives conracs. Here, mark o marke refers o a measure of he fair value of accouns ha can change over ime, such as asses or liabiliies, under he noion of marke fairness o all paricipans, as menioned previously in Secion.. Collaeral managemen is he erm used o describe he process of reducing counerpary credi exposures in derivaives conracs. I is normally used wih OTC derivaives, such as swaps and opions. When wo paries agree o ener ino an agreemen wih 7

collaeralizaion, hey negoiae and execue a collaeral suppor documen ha conains he specific erms and condiions for he collaeralizaion. The rades subjec o collaeral are regularly marked-o-marke and heir ne valuaion is par of he agreemen. The pary wih negaive MTM on he rade porfolio mus pos collaeral o he pary wih posiive MTM, which, in urn, mus pay he counerpary he margin a he collaeral rae. As prices move and new deals are added, he valuaion of he rade porfolio will change. Depending on wha is agreed, he valuaion is repeaed a frequen inervals, ypically daily, weekly or monhly. However, he collaeral seled in a daily basis is he mos common pracice. This makes, in many cases, he collaeral rae o be he overnigh index rae of he collaeral currency in accordance o he specific erms of he agreemen. The collaeral posiion is hen adjused o reflec he new valuaion of he porfolio. The process is repeaed and he posed collaeral changes wih he value of he rades. The process coninues unless one of he paries defauls. In agreemens wih collaeralizaion, he rades can be erminaed in case of defaul and he collaeral can be used as repaymen of he conrac. If he collaeral is sufficien, he MTM profi is proeced and he credi risk is miigaed. Noe ha his process requires careful analysis of he rades involved in order o deermine heir accurae value. Collaeral managemen can be hough of as a process ha exchanges credi risk for operaional risk..5 Securiy Pricing wih Collaeralizaion Collaeralizaion in OTC derivaives markes has grown rapidly over he pas decade, and even faser in he pas few years, due o he impac of he recen financial crisis and he paricularly imporan aenion o he counerpary credi risk. According o he Inernaional Swaps and Derivaives Associaion (ISDA) Margin Survey [9], abou 8

70% of he rade volumes for OTC derivaives were collaeralized a he end of 2009, as opposed o barely 30% in 2003. Coverage has also gone up o 78% and 84% for all he OTC and fixed income derivaives, respecively [3], and now more han 80% of he collaeral posed is cash (abou 50% in USD). Agreemens in rading among dealers o collaeralize muual exposures (and hence reduce credi risk) are based on he Credi Suppor Annex (CSA) o he ISDA maser agreemen, which gives a deailed specificaion of all he erms of he ransacions. The CSA is essenially a legal documen regulaing credi suppor for derivaive ransacions. Collaeralized rades are ofen referred o as CSA rades. Collaeralizaion significanly reduces he counerpary credi risk (i.e., he pary wih negaive presen value of he derivaive). As collaeral is used o offse liabiliies in case of defaul, i could be hough of as an essenially risk-free invesmen, so he ineresed rae on he posed collaeral is usually se o be a proxy of a risk-free rae. Purchased asses are ofen posed as collaeral agains he funds used o buy hem, such as in he repurchase agreemen marke (simply known as he repo marke 2 ) for shares used in dela hedging. In his case, he goal is o reduce (hedge) he risk associaed wih price movemens in he underlying asse by offseing long and shor posiions []. Since his hesis work focuses mainly on he area of securiy pricing wih collaeralizaion, we do no consider credi risk facors in he valuaion of derivaives. We only consider he difference in pricing beween non-collaeralized and collaeralized derivaives, derived from he cos associaed wih he collaeralizaion. Moreover, since daily porfolio reconciliaion has rapidly become he marke sandard, we also assume ha he collaeral accoun is adjused coninuously, which urns ou o be a good approximaion. I is finally worh noing ha due o he naure of he derivaives we consider in he OTC marke, he specific 2 repo is he name given o a form of shor-erm borrowing for dealers in governmen securiies. The dealer sells he securiies o invesors on an overnigh basis and buys hem back he following day. This pracice is a repo for he he pary selling he securiy, which is also agreeing o repurchase i a a laer ime. 9

erms of collaeralizaion may vary from case o case. We only seek o cover a number of general scenarios for varying dynamics (processes) of he underlying and collaeral bu he resuls can be exended o differen cases. Oher exensions also allow for parial or perfec collaeralizaion, and he formulas presened in his work consider boh feaures in he general seing under he scenarios covered..6 Mechanisms of Unsecured and Secured Trades wih Exernal Funding To beer undersand he general effecs of collaeralizaion and is increasing imporance in he OTC derivaives marke, le us now briefly illusrae he mechanisms of unsecured and secured rades (conracs) wih exernal funding, following he descripion presened in [3]. The firs siuaion is depiced in Figure.. Figure.: Unsecured conrac wih exernal funding Consider wo paries, L (for lender) and B (for borrower), ha ener ino a derivaive conrac. L has a posiive presen value (PV) in he conrac wih B (which is assumed o have high credi qualiy). By a posiive PV we mean a receip of cash by L a a fuure ime from he conrac. From he perspecive of L, he siuaion is equivalen o providing a loan o he counerpary B wih he principal value equal o is PV. Since L has o wai for he paymen 0

from B unil he mauriy of he conrac, L has o finance is loan o B and hence he pricing of he conrac should reflec he funding cos o L. If L has and mainains LIBOR 3 credi qualiy, he funding cos is given by he LIBOR of is funding currency since i makes he PV of funding zero. This is he main reason why LIBOR is widely used as a proxy of he discouning rae in he OTC derivaive pricing. The above siuaion changes considerably when collaeral is added o he conrac, As menioned previously, we now assume ha he rade has been made wih a CSA, requiring cash collaeral wih zero minimum ransfer amoun. The siuaion wih collaeralizaion is depiced in Figure.2. Figure.2: Collaeralized (secured) conrac wih exernal funding In his case, L does no require exernal funding since B is posing an amoun of cash (collaeral) equal o he PV of he conrac. However, L has o pay he counerpary B he margin a he collaeral rae r c () (reurn rae of he collaeral) applied o he posed collaeral amoun (or ousanding collaeral). This makes he funding cos of he conrac equal o he collaeral rae. According o [9], he mos popular collaeral in he curren financial marke is he cash of he developed counries and he ypical choice of collaeral 3 London Iner-Bank Offered Rae

rae is he overnigh rae (ONR) of he corresponding currency. As menioned in [9], he impac of collaeralizaion o he valuaion of OTC derivaives is paricular higher when he borrowing rae of he derivaive desk (here denoed in general by L) is significanly higher han he collaeral rae designaed in he corresponding CSA. The convenional LIBOR-OIS 4 spread is usually regarded as an indicaor of such a gap. The models for discouning projeced cash flows of he derivaive wih he collaeral rae (ofen referred o as collaeral rae discouning) implies several main assumpions [, 2, 9]:. Full collaeralizaion: he posed collaeral amoun equals he PV (or MTM) of he derivaive; 2. Symmeric (or bilaeral) collaeralizaion: each counerpary poss collaeral when he MTM of he derivaive is negaive from is own perspecive and receives he same collaeral rae; 3. Coninuous adjusmen: he collaeral is adjused immediaely he MTM changes; 4. Domesic collaeralizaion: he collaeral and he derivaive payoff are in he same currency; 5. Cash-equivalen collaeral: he posed collaeral mus be essenially risk-free and have he highes qualiy. 6. No counerpary credi risk: he counerparies are boh assumed o be defaul-free 4 Overnigh Index Swap 2

In his hesis work, as in [9], a collaeralized derivaive ha saisfies all he above assumpions is referred o as a perfecly collaeralized derivaive, whereas a fully collaeralized derivaive refers o perfec collaeralizaion wih collaeral currency differen from he payoff currency (assumpion 4 is relaxed). Similarly, a parially collaeralized derivaive refers o perfec collaeralizaion wih boh assumpions and 4 relaxed. In he developmen of he model for single currency, we will ofen relax he erminology a lile bi more and will simply consider parial collaeralizaion as ha occurring when only assumpion is relaxed..7 Brief Discussion on Some Relaed Works The heoreical foundaion of valuaion of OTC derivaives for single currency and single asse, under he aforemenioned assumpions, has been laid ou in he works of Pierbarg [], Fujii [2], and Casagna [6]. Each of hese auhors inroduce differen approaches in a general seing o develop valuaion models under specific consideraions for he underlying and collaeral processes. Pierbarg [] uses replicaing porfolio and self-financing argumens o derive he general derivaive pricing formula, which afer some manipulaion allows o obain specific expressions for he cases of zero and perfec collaeralizaion. Pierbarg does no sae any paricular process for he collaeral accoun bu his analysis includes a comprehensive replicaing porfolio wih underlying sock and a cash amoun spli among differen deailed accouns. Pierbarg s work only focuses on single currency and does no include parial collaeralizaion argumens. Fujii [2] inroduces a sochasic process for he collaeral accoun wih an appropriae 3

self-financing sraegy and a reinvesmen argumen ha makes he process dependen on he dynamics of he derivaive. In order o solve an implici dependence problem of he collaeral and he value of he derivaive in his argumen, Fujii considers a simple rading sraegy for he collaeral and he number of posiions of he derivaive ha allows him o obain he formula for a perfecly collaeralized derivaive in single currency. Fujii also suggess a differen valuaion approach for single currency and presens an equivalen formula for he cross-currency scenario, bu omis imporan deails in his analysis. Casagna [6] inroduces he concep of Liquidiy Value Adjusmen (LVA) and uses i o make a disincion beween a collaeralized derivaive and one wihou collaeral. Casagna follows he seps of Pierbarg by using he concep of a self-financing replicaing porfolio wih a modified version of he underlying asse and collaeral processes. His work is mainly focused on he developmen of a general derivaive valuaion model wih an exension o parial collaeralizaion. Even hough Casagna achieves he desired resuls, his work seems o have some inconsisencies in he derivaion ha are no clearly explained, paricularly regarding an apparen conflic in his proposed collaeral process and he parial collaeralizaion consrain..8 Our Conribuion We firs sudy he problem of valuaion of OTC derivaives wih single underlying asse and derivaive payoff in single currency for he cases of zero, parial, and perfec collaeralizaion. We assume he derivaive is raded beween wo defaul-free counerparies and analyze he impac of collaeralizaion on he fair presen value of he derivaive. We follow he differen ideas presened in [], [2], and [6], presen a complee mahemaical derivaion of he resuls, and seek o esablish a uniform derivaive pricing framework for 4

he hree cases of collaeralizaion. To achieve he goal, we presen differen approaches o pricing under specified condiions and show ha he resuls are uniformly consisen despie he differences in he sraegies. In paricular, we show how he general resuls presened in [] can be exended o allow for parial collaeralizaion and ha hese resuls are consisen wih hose presened in [2] and [6]. As described in [9], we firs show how a porfolio including an underlying asse for he derivaive and cash posiions wih various funding sources and corresponding shor raes is consruced o replicae he value of he derivaive. For he mahemaical derivaion, we use an approach based on a generalized Black-Scholes-Meron framework wih collaeral under CSA o obain a fundamenal PDE based on a self-financing condiion. Applying he Feynman-Kac formula o he PDE yields he desired pricing soluion for perfec collaeralizaion. Finally, we use he parial collaeralizaion argumen o exend his resul and obain a generalized pricing formula ha covers all saes of collaeralizaion. We also inroduce a his poin he concep of LVA and show how he value of a collaeralized derivaive is, in fac, equal o he value of he derivaive wihou collaeralizaion plus some adjusmen value, known as LVA. In he second par of he single currency, single asse analysis, and based on [2], we derive a derivaive pricing formula for perfec collaeralizaion by using Q-maringale argumens. We define some sochasic process and show ha such a process is a Q- maringale. Then, we show ha he price process of a collaeralized derivaive can be expressed as a funcion of a cerain maringale process, which afer some manipulaion leads o he perfecly collaeralized derivaive pricing formula. The second major par of our work focuses on esablishing a generalized pricing framework o include cross-currency and muli-asse argumens. Up o his poin, boh he 5

derivaive and he posed collaeral were in he same currency. Now, in he he crosscurrency, single asse analysis, he underlying asse and he derivaive are in some domesic currency, bu he collaeral is posed in a foreign currency. We refer o his as full collaeralizaion which is, in fac, he case of perfec collaeralizaion wih collaeral posed in differen currency. Finally, we include some general analysis for he case of single currency for a derivaive ha has muliple underlying asses, and show ha he resuls are consisen wih hose obained for he single asse. 6

CHAPTER 2 FUNDAMENTAL MATHEMATICAL CONCEPTS In order o develop a unified valuaion framework, i is essenial o presen he fundamenal mahemaical conceps, definiions, and heorems ha will be used hroughou his hesis work. No only his adds consisency o he work, given he diversiy in noaion and presenaion of he conceps used as reference, bu also allows o beer undersand and frame he derived resuls under a common se of suppor ools wih specified assumpions, consideraions, and even limiaions. The general purpose is for his hesis work o be as self-conained as possible. 2. General Probabiliy Theory Definiion 2.. (σ-algebra). Le Ω be a nonempy se. In paricular, le Ω = {ω, ω 2,..., ω n } be he se of all possible oucomes (basic evens) of a random experimen. We call Ω he sample space of he experimen. Le F 2 Ω be a collecion of subses of Ω. We say ha F is a σ-algebra provided ha:. Ω F, 2. if A is a se such ha A F, hen A c F, and 3. if A, A 2,... is a sequence of ses such ha A n F for all n, hen A n F n= This concep is essenial for our purposes because if we have a σ-algebra of ses, hen all he operaions we migh wan o do o he ses will give us oher ses in he σ-algebra. 7

Definiion 2..2 (Borel σ-algebra). If Ω = R, he Borel σ-algebra on Ω, denoed by B = B(R) σ(a), is he smalles σ-algebra on Ω conaining he se A of all open subses (inervals (a,b)), or equivalenly all closed subses, in R. Definiion 2..3 (Measurable Space). If F is a σ-algebra of subses of Ω, hen (Ω, F ) is called a measurable space and any se A F is called an even on Ω. Definiion 2..4 (Probabiliy Measure). Le Ω be a nonempy se, and le F be a σ-algebra of subses of Ω. A probabiliy measure P is a funcion ha assigns a number in [0, ] o every se A F. We call his number he probabiliy of A and wrie P(A). We require:. P(Ω) =, and 2. Whenever A, A 2,... is a sequence of disjoin ses in F, hen P A n = P(A n ). n= n= Definiion 2..5 (Probabiliy Space). The riple (Ω, F, P) is called a probabiliy space. Definiion 2..6 (Random Variable). Le (Ω, F, P) be a probabiliy space. A random variable is a real-valued funcion X defined on Ω (i.e., X : Ω R) wih he propery ha for every Borel subse B of R, he subse of Ω given by {X B} = {ω Ω X(ω) B} is in he σ-algebra F. A random variable X is essenially a numerical quaniy whose value is deermined by he random experimen of choosing any ω Ω. The index [0, ) of he random variables X X() admis a convenien inerpreaion as ime. In order o simplify noaion, we may use X or X() indisincively o denoe ha he variable X is a funcion of he parameer. This convenion applies o any oher variable ha is dependen on any oher parameer we may use hroughou his work. 8

2.2 Informaion and Sochasic Processes Definiion 2.2. (Firaion). A filraion {F n } is a sequence of σ-algebras F 0, F,..., F n wih he propery ha F 0 F F n F. The propery says ha each σ-algebra in he nondecreasing sequence conains all he ses conained by he previous σ-algebra. For our purposes, we use F o denoe he informaion available a ime. Then, he se {F } 0 is called an informaion filraion. So, E [ F ] denoes an expeced value ha is condiional on he informaion available up o ime. We usually wrie E [ F ] as E [ ]. Definiion 2.2.2 (Sochasic Process). A sequence of random variables X, X 2,..., X n is called a sochasic process. For our purposes, a sochasic process is a mahemaical model for he occurrence of a random phenomenon a each momen (ime ) afer a given iniial ime. The random naure is capured by he measurable space (Ω, F ) on which probabiliy measures can be placed. Thus, a sochasic process is a collecion of random variables X = {X 0 < } on (Ω, F ) which ake values in (R n, B(R n )). For a fixed sample poin ω Ω, he funcion X (ω), 0, is he sample pah (rajecory) of he process X associaed wih ω. The emporal feaure of a sochasic process suggess a flow of ime in which, a every momen 0, we can alk abou a pas, presen, and fuure and can ask how much an observer of he process knows abou i a presen, as compared o how much he knew a some poin in he pas or will know a some poin in he fuure. This allows o keep rack of informaion and provides he mahemaical model for a random experimen whose oucome can be observed coninuously in ime. For example, we hink of X as he price of some asse a 9

ime and F as he informaion obained by waching all he prices in he marke up o ime [6]. Definiion 2.2.3 (Adaped Sochasic Process). Le Ω be a nonempy sample space equipped wih a filraion {F }, 0 T. Le X be a collecion of random variables indexed by [0, T]. We say X is an adaped sochasic process, or ha X is adaped o he filraion {F } if, for each ime, he random variable X is F -measurable. Asse prices, porfolio processes (i.e., posiions), and wealh processes (i.e., values of porfolio processes) will all be adaped o a filraion ha we regard as a model of he flow of public informaion. Inuiively, his definiion says ha he informaion available a ime > 0 is sufficien o evaluae he sochasic process X a ha ime. If we know he informaion in F, hen we know he value of X. Consequenly, if we are a some ime 0, hen for some oher ime > 0 he value of he process X 0 is known bu he value of he process X is unknown. For our purposes, i is worh noing ha he no-arbirage heory of derivaive securiy pricing is based on coningency plans. In order o price a derivaive securiy, we deermine he iniial wealh we would need o se up a hedge of a shor posiion in he derivaive securiy. The hedge mus specify wha posiion we will ake in he underlying securiy a each fuure ime coningen on how he uncerainy beween he presen ime and ha fuure ime is resolved [6]. We mus also menion ha, in pracice, we do no observe sock prices following coninuous-variable, coninuous-ime processes. Sock prices are resriced o discree values (e.g., muliples of a cen) and changes can be observed only when he exchange is open. Neverheless, he coninuous-variable, coninuous-ime process proves o be a useful model for many purposes [2]. Definiion 2.2.4 (P-Maringale). Le (Ω, F, P) be a probabiliy space, le T be a fixed posiive number, and le F, 0 T, be a filraion of sub-σ-algebras of F. An adaped sochasic 20

process X is a maringale wih respec o he informaion filraion F and probabiliy measure P if E P [ X ] < (inegrabiliy condiion) and E P [X F s ] = X s for all 0 s T. Inuiively, a maringale is a sochasic process ha, on average, has no endency o rise or fall. Maringales and measures are criical elemens o a risk-neural valuaion framework. As we will see, a maringale is also defined as a zero-drif sochasic process. A measure is he uni in which we value securiy prices [2]. 2.3 Wiener Processes and Brownian Moion Here we follow [2] o discuss paricular ypes of sochasic processes and heir imporance in he realm of derivaive pricing. Definiion 2.3. (Markov Process and The Markov Propery). A Markov process is a paricular ype of sochasic process where only he PV of a variable is relevan for predicing he fuure. Tha is, he pas hisory of he variable and he way ha he presen has emerged from he pas are irrelevan. Sock prices are usually assumed o follow a Markov process. Since predicions for he fuure are uncerain, hey mus be expressed in erms of probabiliy disribuions. The Markov propery implies ha he probabiliy disribuion of he price a any paricular fuure ime does no depend on he paricular pah followed by he price in he pas. The Markov propery of sock prices is consisen wih he weak form of marke efficiency. This saes ha he presen price of a sock carries all he informaion conained in a record of pas prices. If his were no rue, hen financial analyss could make above-average 2

reurns by inerpreing chars of he pas hisory of sock prices, and here is no sufficien evidence ha his is acually possible. Finally, i is worh noing ha i is compeiion in he markeplace wha ends o ensure ha weak-form marke efficiency holds. Definiion 2.3.2 (Wiener Process). Le W be a variable ha follows a Markov sochasic process. Le W 0 be is curren value and le φ(m, v) represen he change in value during some ime uni v, where φ denoes a probabiliy disribuion ha is normally disribued wih mean change per uni ime, m, called drif rae, and variance per uni ime, v, called variance rae. A Markov sochasic process wih φ(m, v) = φ(0, ) is called a Wiener process. A drif rae of zero means ha he expeced value of W a any fuure ime is equal o is curren value. The variance rae of means ha he variance of he change in W in a ime inerval of lengh T equals T. A variable W follows a Wiener process if i has he wo following properies:. ΔW = ɛ Δ (change ΔW during a small period of ime Δ, where ɛ φ(0, )) 2. For any wo differen shor inervals of ime, Δ, he values of ΔW are independen By Propery, ΔW N(0, Δ). By Propery 2, he variable W follows a Markov process. We use dw o consider he change in he value of he variable W during a relaively long period of ime T. Hence, dw, wih he above properies for ΔW in he limi as ΔW 0, is a Wiener process. Definiion 2.3.3 (Generalized Wiener Process). A generalized Wiener process for a variable X(a, b) (i.e., wih parameers a,b), can be defined in erms of dw as: dx = a d + b dw (2.) 22

The a d erm implies ha W has an expeced drif rae of a. The b dw erm implies ha W has a variance rae of b 2 and i can be regarded as adding noise or variabiliy o he pah followed by W. The amoun of variabiliy is b imes a Wiener process. Thus, by he same properies of Definiion 2.3.2, i follows ha dw N(a, b 2 ) in any ime ineval [0, ]. Definiion 2.3.4 (Maringale). A maringale is a zero-drif sochasic process. A variable Θ follows a Maringale if is process has he form: dθ = σ dw (2.2) where dw is a Wiener process and σ is a parameer ha may also be sochasic. As implied in Definiion 2.2.4, a maringale has he propery ha is expeced value a any fuure ime T is equal o is value oday. Tha is: E[Θ T ] = Θ 0 (2.3) Definiion 2.3.5 (Brownian Moion). A coninuous, adaped sochasic process B(μ, σ) = {B, F 0 < }, defined on some probabiliy space (Ω, F, P), is said o be a Brownian moion wih parameers μ, σ (drif rae μ and variance rae σ) if i saisfies he following properies:. For 0 s <, he incremen B B s is independen of F s and is normally disribued wih mean zero and variance s. 2. For fixed imes 0 = 0 < < < n, he incremens (B B 0 ),..., (B n B n ) are muually independen wih mean zero (i.e., E[B i+ B i ] = E[dB ] = 0 and variance i+ i. 3. For any incremen τ > 0, (B +τ B ) N(μτ, σ 2 τ). 23

Definiion 2.3.6 (Sandard Brownian Moion). When μ = 0 and σ =, he process B(0, ) is called a sandard Brownian moion, denoed by W SB. We always assume ha his process begins a zero, so ha W SB = 0. Clearly, we also have W SB 0 N(0, ). If X B(μ, σ) wih X 0 = x, hen we can wrie X = x + μ + σw SB. Noe ha E[X ] = x + μ + σe[w SB ] = x + μ and Var(X ) = σ 2 Var ( ) W SB = σ 2. Hence, X N(x + μ, σ 2 ). The following sandard resuls are admied wihou proof. See [6]. Theorem 2.3.7. Sandard Brownian moion is a Markov process. Theorem 2.3.8. Sandard Brownian moion is a maringale. 2.4 Geomeric Brownian Moion and The Process for a Sock Price Definiion 2.4. (Geomeric Brownian Moion). A coninuous, adaped sochasic process {S, F 0 < } is a geomeric Brownian moion wih parameers μ, σ, and we wrie S GBM(μ, σ), if, for all 0, S = S 0 e (μ 2 σ2 ) + σw SB (2.4) and i saisfies he following properies:. If S > 0, hen S +s > 0 for any s > 0. 2. The disribuion of S +s S only depends on s and no on S. I can be shown ha geomeric Brownian moion in differenial form is given by he following sochasic differenial equaion: ds = μ S d + σ S dw SB (2.5) 24

The above properies sugges ha geomeric Brownian moion is a reasonable model for sock prices. This is in fac he asse-pricing model used in he Black-Scholes-Meron opion pricing formula. Definiion 2.4.2 (The Process for a Sock Price). Le W SB (0 T) be a sandard Brownian moion defined on a probabiliy space (Ω, F, P), wih filraion F (0 T) and parameers μ μ() and σ σ() adaped o he filraion. The sock price model is defined o be: ds = μ S d + σ S dw SB (2.6) Le us analyze briefly he above definiion. Le S be he sock price a ime. Le μ be he expeced rae of reurn of he sock (per uni of ime) a ime, expressed in decimal form. Define he volailiy of he sock price, σ, o be some measure of how uncerain we are abou he fuure sock price movemen. If he volailiy of he sock price is always zero, we can express he rae of change of he sock in a shor inerval of ime Δ as: ΔS Δ = μ S ΔS = μ S Δ (2.7) A reasonable assumpion for he volailiy σ is ha he variabiliy (uncerainy) of he percenage reurn of he sock in a shor period of ime Δ is he same regardless of he sock price. This suggess ha he sandard deviaion of he change in a shor period of ime Δ should be proporional o he sock price S. Hence, in he limi, he process for a sock price can be wrien as ds = ( ) μ S d + (σ S ) dw SB. By Definiion 2.4., his is a geomeric Brownian moion in differenial form which, in pracice, is he mos widely used model of sock price behavior. Noe ha he sock price model is compleely general and subjec only o he condiion ha he pahs of he process are coninuous [6]. 25

Definiion 2.4.3 (The Change in Wealh of an Invesor). Consider an invesor who begins wih non-random iniial wealh X 0 and holds Δ Δ() shares of sock a each ime (Δ can be random bu mus be adaped). Suppose he sock is modelled by a geomeric Brownian moion (in differenial form) given by Definiion 2.4.. Suppose also ha he invesor finances his invesing by borrowing or lending a ineres rae r. If X denoes he wealh of he invesor a each ime, hen dx = r X d + (μ r) Δ S d + Δ S σ dw SB (2.8) The hree erms in his equaion can be undersood as follows: an average underlying rae of reurn r on he porfolio, which is he erm r X d, a risk premium μ r for invesing in he sock, which is he erm (μ r) Δ S, and a volailiy erm proporional o he size of he sock invesmen, which is refleced in he erm Δ S σ dw SB. Remark: The process for a sock price developed in Definiion 2.4.2 involves wo parameers: μ and σ. The parameer μ is he (annualized) expeced reurn earned by an invesor in a shor period of ime. Since mos invesors require higher expeced reurns o induce hem o ake higher risks, i follows ha he value of μ should depend on he par of he risk ha canno be diversified away by he invesor. Moreover, i should also depend on he level of ineres raes in he economy. The higher he level of ineres raes, he higher he expeced reurn required on any given sock. The value of a derivaive wih a sock as underlying asse is, in general, independen of μ. In conras, he sock price volailiy σ plays a major role in he valuaion of many derivaives. The sock price volailiy is equal o he sandard deviaion of he coninuously compounded reurn provided by he sock in year. Typical values for σ for a sock are in he range of 0.5 (5%) o 0.60 (60%) [2]. 26

2.5 Iô Process, Iô s Formula, and he Feynman-Kac Theorem In his secion, we cover sochasic processes ha are more general han Brownian moion and define he inegrals, and heir properies, used o model he value of a porfolio ha resuls from rading asses in coninuous ime. Definiion 2.5. (Generalized Iô Process). A generalized Iô process is a generalized Wiener process (see Definiion 2.3.3) in which he parameers a and b are funcions of boh he value of he underlying variable X and ime. Hence, i can be wrien as: dx = a(x, ) d + b(x, ) dw (2.9) Definiion 2.5.2 (Iô Process). Le W SB ( 0) be a Brownian moion wih filraion F, ( 0). An Iô process is a sochasic process of he form X = X 0 + 0 Θ u u + 0 Δ u dw SB u (2.0) where X 0 is non-random and Θ and Δ are adaped sochasic processes. In differenial noaion, dx = Θ d + Δ dw SB (2.) Lemma 2.5.3 (Quadraic Variaion of he Iô Process). The quadraic variaion of he Iô process, in differenial form, is (dx ) (dx ) = Δ 2 d, given he differenial muliplicaion able (dw SB ) (dw SB ) = d, (d) (dw SB ) = (dw SB ) (d) = 0, and (d) (d) = 0. Theorem 2.5.4 (Iô-Doeblin Formula for an Iô Process). Le X, 0, be an Iô process, as saed in differenial form in Definiion 2.5.2, wih a drif rae of Θ and a variance rae of Δ 2. 27

Le F F(, X ) be a funcion wih defined and coninuous parial derivaives F, F X, and 2 F X 2. Then, for every T 0, df = F F d + dx + 2 F X 2 X (dx 2 )(dx ) = F F d + dx + 2 F X 2 2 X Δ2 d ( F = + ) ( ) 2 F F 2 Δ2 d + dx 2 (2.2) X X ( ) F = L(F) d + dx (2.3) X where L(F) F + 2 F 2 Δ2 (2.4) 2 X is called he sandard pricing operaor. Using dx = Θ d + Δ dw SB, we also have ( F df = + Θ F + ) ( 2 F X 2 Δ2 d + 2 X ( ) = L Θ F (F) d + Δ X Δ F X ) dw SB (2.5) dw SB (2.6) which implies ha F also follows an Iô process wih drif rae L Θ (F) F + Θ F X variance rae ( ) F 2. Δ X + 2 Δ2 2 F X 2 and The following imporan heorem relaes sochasic differenial equaions and parial differenial equaions. Before saing he heorem, we firs define an imporan concep regarding ineres raes and esablish an imporan convenion for very useful noaion. 28

Definiion 2.5.5 (Compounding and Discouning Processes). Suppose we have an adaped ineres rae process R R(). We define he following processes:. Compounding process: Φ R() a,b b e a R(u) du (2.7) 2. Discouning process: Φ R() a,b e b a R(u) du (2.8) Theorem 2.5.6 (Feynman-Kac Formula). Le X, 0, be an Iô process driven by he SDE dx = Θ(, X ) d + Δ(, X ) dw SB and le f (, x) f : [0, T] R R be he soluion of: wih iniial condiion X 0 = x. Le T > 0 be a fixed parameer f + Θ f + 2 f X 2 Δ2 X 2 = q(, x) f g(, x) (2.9) f (T, x) = h(x) (2.20) Then, f (, x) = E Q = E Q [ T T e s Φ q(x ),s ] q(x u ) du g(s, X s ) ds + e T q(x u ) du h(x T ) g(s, X s ) ds + Φ q(x ),T h(x T ) (2.2) (2.22) Noe ha Equaion (2.9) corresponds o he Black-Scholes-Meron differenial equaion wih an addiional erm g(, x). If X represens he process for a sock price, hen h(x T ) = h(x(t)) denoes he payoff a ime T of a derivaive securiy whose underlying asse is he geomeric Brownian moion (sock price model) saed in Definiion 2.4.2. 29

Because he sock price is Markov and he payoff is a funcion of he sock price alone, here is a funcion f (, x) such ha F() = f (, X()), where F() f (, x) may represen he value of a derivaive wih underlying sock a each ime. 2.6 Some Addiional Conceps of Ineres in Securiy Pricing 2.6. On Ineres Raes Two imporan raes for derivaive raders are Treasury raes and LIBOR raes. Treasury raes are hose paid by a governmen in is own currency. LIBOR raes are shor-erm lending raes offered by banks in he inerbank marke. Derivaive raders usually assume ha he LIBOR rae is a risk-free rae [2]. The risk-free rae affecs direcly he price of a derivaive. As ineres raes in he economy increase, he expeced reurn required by invesors from he sock ends o increase. In addiion, he PV of any fuure cash flow received by he holder of he derivaive decreases. In general, he discouning rae ha should be used for he expeced cash flow a a fuure ime T mus equal a leas an invesor s required reurn on he invesmen. We assume he risk-free rae r is he nominal rae and no he real (effecive) rae. We also assume r > 0. Oherwise, an invesmen a he risk-free rae would provide no advanages over cash. As menioned previously, i is also imporan o assume ha here are some marke paricipans, such as large invesmen banks, for which he following saemens are rue:. There are no arbirage opporuniies. 2. Borrowing and lending are possible a he risk-free ineres rae. 30