Table Of ContentCreditRisk
Modellingcreditriskaccuratelyiscentraltothepracticeofmathematicalfinance.
Themajorityofavailabletextsareaimedatanadvancedlevel,andaremoresuitable
forPhDstudentsandresearchers.ThisvolumeoftheMasteringMathematicalFinance
seriesaddressestheneedforacourseintendedformaster’sstudents,final-year
undergraduates,andpractitioners.Thebookfocusesonthetwomainstreammodelling
approachestocreditrisk,namelystructuralmodelsandreduced-formmodels,andon
pricingselectedcreditriskderivatives.Balancingrigoroustheorywithexamples,it
takesreadersthroughanaturaldevelopmentofmathematicalideasandfinancial
intuition.
marek capin´skiisProfessorofAppliedMathematicsatAGHUniversityof
ScienceandTechnology,Krako´w.Hisresearchinterestsincludemathematicalfinance,
corporatefinance,andhydrodynamics.Hehasbeenteachingforover35years,has
heldvisitingfellowshipsinPolandandtheUK,andhaspublishedoverfiftyresearch
papersandninebooks.
tomasz zastawniakisChairinMathematicalFinanceattheUniversityof
York.Hisresearchinterestsincludemathematicalfinance,stochasticanalysis,
stochasticoptimisationandconvexanalysis,andmathematicalphysics.Hehas
previouslytaughtatnumerousinstitutionsinPoland,theUSA,Canada,andtheUK,
andhaspublishedoverfiftyresearchpublicationsandeightbooks.
Mastering Mathematical Finance
Mastering Mathematical Finance is a series of short books that cover all
coretopicsandthemostcommonelectivesofferedinMaster’sprogrammes
inmathematicalorquantitativefinance.Thebooksarecloselycoordinated
and largely self-contained, and can be used efficiently in combination but
alsoindividually.
TheMMFbooksstartfinanciallyfromscratchandmathematicallyassume
only undergraduate calculus, linear algebra, and elementary probability
theory.Thenecessarymathematicsisdevelopedrigorously,withemphasis
onanaturaldevelopmentofmathematicalideasandfinancialintuition,and
thereadersquicklyseereal-lifefinancialapplications,bothformotivation
and as the ultimate end for the theory. All books are written for both
teachingandself-study,withworkedexamples,exercises,andsolutions.
[DMFM] DiscreteModelsofFinancialMarkets,
MarekCapin´ski,EkkehardKopp
[PF] ProbabilityforFinance,
EkkehardKopp,JanMalczak,TomaszZastawniak
[SCF] StochasticCalculusforFinance,
MarekCapin´ski,EkkehardKopp,JanuszTraple
[BSM] TheBlack–ScholesModel,
MarekCapin´ski,EkkehardKopp
[PTRM] PortfolioTheoryandRiskManagement,
MaciejJ.Capin´ski,EkkehardKopp
[NMFC] NumericalMethodsinFinancewithC++,
MaciejJ.Capin´ski,TomaszZastawniak
[SIR] StochasticInterestRates,
DaraghMcInerney,TomaszZastawniak
[CR] CreditRisk,
MarekCapin´ski,TomaszZastawniak
[SCAF] StochasticControlAppliedtoFinance,
SzymonPeszat,TomaszZastawniak
Series editors Marek Capin´ski, AGH University of Science and Technol-
ogy, Krako´w; Ekkehard Kopp, University of Hull; Tomasz Zastawniak,
UniversityofYork
Credit Risk
MAREK CAPIN´ SKI
AGHUniversityofScienceandTechnology,Krako´w
TOMASZ ZASTAWNIAK
UniversityofYork
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Informationonthistitle:www.cambridge.org/9781107002760
10.1017/9781139047432
©MarekCapin´skiandTomaszZastawniak2017
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Contents
Preface pagevii
1 Structuralmodels 1
1.1 Tradedassets 1
1.2 Mertonmodel 7
1.3 Non-tradeableassets 21
1.4 Barriermodel 24
2 Hazardfunctionmodelandnoarbitrage 39
2.1 Marketmodel 39
2.2 Defaulttimeandhazardfunction 41
2.3 Defaultindicatorprocess 48
2.4 Noarbitrageandrisk-neutralprobability 57
3 Defaultablebondpricingwithhazardfunction 71
3.1 Initialpricesandcalibration 73
3.2 Processofprices 76
3.3 Recoveryschemes 79
3.4 Martingaleproperties 81
3.5 Martingalepropertiesofintegrals 86
3.6 Bondpricedynamics 91
4 Securitypricingwithhazardfunction 94
4.1 Martingalerepresentationtheorem 94
4.2 Pricingdefaultablesecurities 96
4.3 CreditDefaultSwap 109
5 Hazardprocessmodel 114
5.1 Marketmodelandrisk-neutralprobability 114
5.2 Martingaleswithrespecttotheenlargedfiltration 116
5.3 Hazardprocess 121
5.4 Canonicalconstructionofdefaulttime 125
5.5 Conditionalexpectations 130
5.6 Martingaleproperties 136
6 Securitypricingwithhazardprocess 146
6.1 Bondpricedynamics 146
6.2 Tradingstrategies 150
v
vi Contents
6.3 Zero-recoverysecurities 157
6.4 Martingalerepresentationtheorem 162
6.5 Securitieswithpositiverecovery 169
A Appendix 176
A.1 Lebesgue–Stieltjesintegral 176
A.2 PassagetimeandminimumofWienerprocess 181
A.3 Stochasticcalculuswithmartingales 185
Selectbibliography 191
Index 193
Preface
Credit risk is concerned with the possibility of bankruptcy happening as
a result of unpaid debt obligations. This applies to companies partially fi-
nanced by debt. The key question is to find the value of debt or, equiv-
alently, the amount of interest charged by the debt holders to offset their
possible loss due to bankruptcy. Our goal is to explain the main issues in
thesimplestpossiblecases,tokeepthetheoryfreeoftechnicalcomplica-
tions.
The first approach covered in this book relates bankruptcy to the com-
pany’s performance. The resulting model is referred to as structural. His-
torically,thiswasthefirstsystematicapproachtocreditrisk,developedby
Mertonin1974.
Intheothermethodstudiedhere,calledthereduced-formapproach,de-
fault occurs as a result of external forces, not related directly to the func-
tioning of the company, and comes as a surprise at a random time. This
may be the time when some information is revealed about technological
advances, court decisions, or political events, which could wipe out the
company.
Wedevelopanapproachsimilartotheclassicalfinancialtheories,where
certainunderlyingsecuritiesarespecifiedandself-financingstrategiesare
constructedfromthese.Hereitistherisk-freeanddefaultablezero-coupon
bondsthatwillserveasunderlyingassets.Thismakesitpossibletopricea
rangeofdefaultablesecurities,includingCreditDefaultSwaps,bymeans
ofreplication.Then,afteraddingaBlack–Scholesstockasabuildingblock
of the market, the range of securities broadens, and this allows us to con-
clude with an example where the structural and reduced-form approaches
cometogether.
Readers of the book need to be familiar with material covered in some
earlier volumes in this series, namely [SCF] and [BSM], that is, with the
foundationsofstochasticcalculusandtheBlack–Scholesmodel.Exposure
tomeasuretheorybasedprobability,coveredin[PF],wouldalsobehelpful.
The text is interspersed with various examples and exercises. The nu-
mericalworkunderpinningmanyoftheexamples,andsolutionstotheex-
ercises,areavailableatwww.cambridge.org/creditrisk.
vii
1
Structural models
1.1 Tradedassets
1.2 Mertonmodel
1.3 Non-tradeableassets
1.4 Barriermodel
Consider a company launched at time 0, when some assets are purchased
forV(0).Fundingcomesfromtwosources.Shareholderscontribute E(0),
referred to as equity. The remaining amount D(0) = V(0)− E(0), called
debt,iseitherborrowedfromabankorraisedbysellingbondsissuedby
thecompany.
Weconsiderthiscompanyoveratimeintervalfrom0toT,duringwhich
theassetsareputtoworkinordertogeneratesomefunds,whicharethen
splitbetweenthetwogroupsofinvestorsattimeT.Thedebtisfirstrepaid
withinteresttothedebtholders,whohavepriorityovertheequityholders.
Anyremainingamountgoestotheequityholders.
The simplest way to raise money to make these payments is to sell the
assetsofthecompany.Webeginouranalysiswiththiscase,bymakingthe
necessaryassumptionthattheassetsaretradeable.
1.1 Tradedassets
We assume that there is a liquid market for the assets, and V(t) for t ∈
[0,T]representstheirmarketvalue.Wealsoassumethattheassetsgenerate
no additional cash flows. A practical example of such assets would be a
portfolio of traded stocks that pay no dividends, the company being an
investmentfund.
1
2 Structuralmodels
Payoffs
SupposethatthecompanyhastoclearthedebtattimeT,andthatthereare
no intermediate cash flows to the debt holders. The interest rate applying
to the loan will be quoted by the bank or implied by the bond price. We
denotethisloanratebyk withcontinuouscompounding,andbyK with
D D
annualcompounding.TheamountdueattimeT is
F = D(0)ekDT = D(0)(1+K )T.
D
(Throughoutthisvolumewetakeoneyearastheunitoftime.)Oneofthe
goalshereistofindtheloanratethatreflectstheriskforthedebtholders.
At time T we sell the assets and close down the business, at least hy-
pothetically, to analyse the company’s financial position at that time. It is
possible that the amount obtained by selling the assets is insufficient to
settle the debt, that is, V(T) < F. In this respect, we make an important
assumptionconcerningthelegalstatusofthecompany:ithaslimitedlia-
bility. This means that losses cannot exceed the initial equity value E(0).
If V(T) < F, the equity holders do not have to cover the loss from their
personalfunds.Thecompanyisdeclaredbankrupt,andtheequityholders
walkawayhavinglosttheirinitialinvestment.IfV(T)≥ F,theloancanbe
paidbackwithinterest,andtheequityholderskeepthebalance.Thefinal
value of equity is therefore a random amount equal to the payoff of a call
option,
E(T)=max{V(T)−F,0},
withthevalueoftheassetsastheunderlyingsecurityandthedebtrepay-
mentamountF asthestrikeprice.
Remark1.1
Acallisanoptiontobuytheunderlyingassetforaprescribedprice,which
soundsparadoxicalhere.However,itisconsistentwiththegeneralpractice
thatloansaresecuredonsomeassets.Theborrower’sownershiprightsin
theassetsarerestricteduntiltheloanisrepaid.Thefullrights(forinstance,
toselltheasset)areinasenseboughtbackwhentheloanissettled.
IfV(T) ≥ F,thedebtwillbepaidinfull,butinthecaseofbankruptcy,
whichwillbedeclaredifV(T)< F,thedebtholdersaregoingtotakeover
the assets and sell them for V(T). This is straightforward if the assets are
tradeable.TheamountreceivedattimeT willbe
D(T)=min{F,V(T)}= F−max{F−V(T),0}.
