Modelling Credit Value Adjustment Using Defaultable Options Approach

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Sidita Zhabjaku, The University of Western Ontario
Supervisor: Matt Davison, The University of Western Ontario
A thesis submitted in partial fulfillment of the requirements for the Master of Science degree in
Statistics and Actuarial Sciences
© Sidita Zhabjaku 2013
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MODELLING CREDIT VALUE ADJUSTMENT USING DEFAULTABLE
OPTIONS APPROACH
(Thesis format: Monograph)
by
Sidita Zhabjaku
Graduate Program in Statistics and Actuarial Science
A thesis submitted in partial fulfillment
of the requirements for the degree of
Master of Science
The School of Graduate and Postdoctoral Studies
The University of Western Ontario
London, Ontario, Canada
(cid:13)c Sidita Zhabjaku 2013
Abstract
This thesis calculates Credit Value Adjustment on defaultable options. The prices of default-
able European options are computed through analytical, quadrature approximation and Monte
Carlosimulationsundertheassumptionofaconstantrateofdefault. Subsequently,wepropose
toinverselyrelatethecompany’sinstantaneousrateofdefaulttoitsunderlyingstockprice,re-
sultinginanon-constantrateofdefault. Thisallowsforanewapproachtoestimatethedefault
of company different from previous work where default is calculated through historical data.
Therationalebehindthisideareliesonthefactthatpriceofthestockplungesbeforetheevent
of default. For a given set of option parameters, we show that it is possible to find an optimal
intensity, which produces the same prices of European options under a simpler framework.
However, this intensity fluctuates with changes in other parameters. Implementation details
andanalysisoftheresultsareprovided.
Keywords: DefaultableOptions,CreditValueAdjustment,PoissonProcess
ii
Acknowledgements
I would like to take this opportunity and say thank you to my supervisor Dr. Matt Davison for
hisguidance,encouragementandundividedsupportthroughoutmyMastersprogram. Without
hisusefulcomments,remarksandinfinitepatiencethisthesiswouldnothavebeencompleted.
I would also like to thank all faculty, staff and graduate students in the department of Ac-
tuarial & Statistical Science for always being helpful and kind. A special thanks goes to my
officematesformakingtheofficelikemysecondhome.
TomysisterBesmiraforalwaysencouragingmeandteachingmethatanythingispossible
andtoKyle,foralwaysbelievinginmeandgivingmehisunconditionalsupportthroughoutall
myyearsatWestern.
Last but not least, I would like to express my deepest appreciation to my loving parents,
MyfitandRudina,whoalwayssacrificedeverythingformyhappiness. Withouttheirlovenone
ofthiswouldhavebeenpossible.
iii
Tomyparents...
iv
Contents
Abstract ii
Acknowledgements iii
ListofFigures viii
ListofAppendices xii
1 IntroductionandMotivation 1
1.1 IntroductiontoCreditValueAdjustment . . . . . . . . . . . . . . . . . . . . . 1
1.2 BackgroundonCVAanditsChangefromBaselIItoIII . . . . . . . . . . . . . 3
1.3 ThesisataGlance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 BackgroundReview 9
2.1 CounterpartyCreditRisk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 CVAEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 CVACalculationsintheLiterature . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Black-ScholesEquation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 StructuralModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5.1 MertonModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5.2 BlackCoxModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.6 ReducedFormModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3 DetailedAnalysisofConstantDefaultRateCase 22
3.1 IntroductiontoConstantDefaultArrivals . . . . . . . . . . . . . . . . . . . . . 22
3.2 HomogeneousPoissonProcesses . . . . . . . . . . . . . . . . . . . . . . . . . 23
v
3.3 MonteCarloSimulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 AnalyticalApproximationforDefaultableOptionPrices . . . . . . . . . . . . 28
3.4.1 TaylorSeriesApproximationMethodforStandardNormal . . . . . . . 30
3.4.2 DiscretizationofTaylorSeriesApproximation,SecondTerm . . . . . . 33
3.4.3 DiscretizationofTaylorSeriesApproximation,ThirdTerm . . . . . . . 36
3.5 ProblemswithTaylorSeriesApproximationandErrors . . . . . . . . . . . . . 38
3.6 QuadratureSolutiontoDefaultableOptionwithConstantDefaultArrival . . . . 41
3.6.1 RiskFreeInterestRateofZero . . . . . . . . . . . . . . . . . . . . . . 41
3.6.2 NonZeroRiskFreeInterestRate . . . . . . . . . . . . . . . . . . . . . 42
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 PricingResultsforVariableDefaultRateCase 45
4.1 Non-ConstantDefaultRates . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.1.1 InhomogeneousPoissonProcess . . . . . . . . . . . . . . . . . . . . . 45
4.2 ConstantandNon-ConstantDefaultRates . . . . . . . . . . . . . . . . . . . . 47
4.2.1 Method1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2.2 Method2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2.3 Method1Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.2.4 Method2Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.3 ModelValidation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Non-ConstantDefaultAgainstConstantResultsandComparison . . . . . . . . 61
4.5 CreditValueAdjustment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
5 Conclusion 79
5.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
5.2 FinancialModellingImplications . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 FutureWork . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
References 83
A BaselCommitteeonBankingSupervisionCountries 86
vi
B DetailedAnalyticCalculationsforDefaultableOptions 87
C MonteCarloSimulationswithErrors 92
CurriculumVitae 100
vii
List of Figures
3.1 DefaultablecallandputoptionpricesagainstBlack-Scholesoptionpriceswith
r = 2%,S = K = $35,R = 70%,σ = 18%,T = 5,witha99%confidencelevel. 27
3.2 Defaultable call and put option prices with r = 0%, S = K = $35, σ = 18%,
R = 70%, T = 5, for error bars of the simulations refer to Appendix C, Figure
C.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3 Analyticalapproximationofequation(3.30)withdifferentmaturitydates,T=1,
5,10,20,σ = 18%,r = 0%,R = 70%andS = K = $35. . . . . . . . . . . . . 34
3.4 Analytical approximation of equation (3.36) using second term Taylor expan-
sion. Parameters: S = K = $35,σ = 18%,r = 0%,T = 5,R = 70%,n = 1000. 36
0
3.5 Analytical approximation with the second and third term of the Taylor Series,
comparing it to Black-Scholes. Parameters: S = K = $35, σ = 18%, r = 0%,
0
T = 5,R = 70%,n = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6 ErrorsofthefirstthreetermsoftheTaylorexpansionwithλfrom0.01to10. . 40
3.7 Comparisonbetweenaquadraturesolutionofequations(3.51)addedwith(3.52)
and the analytical solution that we derived in Section 3.4, with λ from 0.01 to
10,σ = 18%,r = 0%,T = 5,R = 70%,S = K = $35. . . . . . . . . . . . . . . 42
3.8 Comparisonbetweenquadratureandsimulatedresults,withλfrom0.01to10,
r = 2%,σ = 18%,R = 70%,S = K = $35andT = 5. . . . . . . . . . . . . . . 43
4.1 Method1comparisonbetweenconstantandnon-constantdefaultratesofacall
andputoption,throughMonteCarlosimulations,withC from5to50,r = 2%,
σ = 18%,S = K = $35,R = 70%andT = 5. Forerrorbarsoftheoptionsand
defaulttimesrefertoAppendixC,FigureC.2,andFigureC.3. . . . . . . . . . 53
viii
4.2 Method 1 relative difference between constant and non-constant default rates
of a call and put option with the Black-Scholes pricing of such options, with
parameters C from 5 to 50, r = 2%, σ = 18%, S = K = $35, R = 70% and
T = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Method2resultsonthecomparisonbetweenconstantandnon-constantdefault
rates of a call and put option, through Monte Carlo simulations, withC from 5
to50,r = 2%,σ = 18%,S = K = $35,R = 70%andT = 5. . . . . . . . . . . 55
4.4 Method 2 results on the comparison of relative difference between constant
and non-constant default rates of a call and put option with the Black-Scholes
pricing of a call and put, with parameters C from 5 to 50, r = 2%, σ = 18%,
S = K = $35,R = 70%andT = 5. . . . . . . . . . . . . . . . . . . . . . . . . 56
4.5 Comparison of formula (4.24) with parameters C from 5 to 50, r = 2%, σ =
18%,S = K = $35andT = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.6 Survival probability with the constant and non-constant default rate with pa-
rametersC from5to50,r = 2%,σ = 18%,S = K = $35andT = 5. Forerror
barsrefertoAppendixC,FigureC.4. . . . . . . . . . . . . . . . . . . . . . . . 58
4.7 Survival probability with the constant and non-constant default rate with pa-
rameters C from 5 to 50, r = 2%, σ = 35%, S = K = $35 and T = 5, with
errorbarsata99%confidencelevel. . . . . . . . . . . . . . . . . . . . . . . . 59
4.8 Put-Call Parity check holds for the defaultable options with C from 5 to 50,
r = 2%,σ = 35%,S = K = $35,R = 70%andT = 5. . . . . . . . . . . . . . . 60
4.9 Comparison of default and non default option pricing with a range of different
strike prices denoted by K, where C = 15, r = 2%, σ = 18%, S = $35,
R = 70%andT = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.10 Comparison of constant default and non-constant default rate option pricing
with a range of different C values, where R = 100%, r = 2%, σ = 18%,
S = K = $35andT = 5. FortheerrorbarsrefertoAppendixC,FigureC.5. . . 63
4.11 Comparison of default and non default option pricing with a range of different
Cvalues,whereR = 0%,r = 2%,σ = 18%,S = K = $35andT = 5. . . . . . . 64
ix