Pricing a basket option when volatility is capped using affine jump-diffusion models

88 Pages·2013·1.88 MB·English

Pricing a basket option when volatility
is capped using affine jump-diffusion
models

D A N I E L K R E B S

Master of Science Thesis
Stockholm, Sweden 2013

Pricing a basket option when volatility is
capped using affine jump-diffusion
models

D A N I E L K R E B S

Degree Project in Mathematical Statistics (30 ECTS credits)
Degree Programme in Engineering Physics (270 credits)
Royal Institute of Technology year 2013
Supervisor at KTH was Camilla Johansson Landén
Examiner was Boualem Djehiche

TRITA-MAT-E 2013:27
ISRN-KTH/MAT/E--13/27--SE

Royal Institute of Technology
School of Engineering Sciences

KTH SCI
SE-100 44 Stockholm, Sweden

URL: www.kth.se/sci

Abstract
Thisthesisconsidersthepriceandcharacteristicsofanexoticoptioncalledthe
Volatility-Cap-Target-Level(VCTL)option. ThepayofffunctionisasimpleEuro-
peanoptionstylebuttheunderlyingvalueisadynamicportfoliowhichiscom-
prisedoftwocomponents:Ariskyassetandanon-riskyasset.Thenon-riskyasset
isabondandtheriskyassetcanbeafundoranindexrelatedtoanyassetcategory
suchasequities,commodities,realestate,etc.
Themainpurposeofusingadynamicportfolioistokeeptherealizedvolatil-
ityoftheportfolioundercontrolandpreferablybelowacertainmaximumlevel,
denotedastheVolatility-Cap-Target-Level(VCTL).Thisisattainedbyavariable
allocationbetweentheriskyassetandthenon-riskyassetduringthematurityofthe
VCTL-option.Theallocationisreviewedandifnecessaryadjustedevery15thday.
Adjustmentdependsentirelyupontherealizedhistoricalvolatilityoftheriskyasset.
Moreover, it is assumed that the risky asset is governed by a certain group
ofstochasticdifferentialequationscalledaffinejump-diffusionmodels. Allmod-
elswillbecalibratedusingout-of-themoneyEuropeancalloptionsbasedonthe
Deutsche-Aktien-Index(DAX).
ThenumericalimplementationoftheportfoliodiffusionsandtheuseofMonte
CarlomethodswillresultindifferentVCTL-optionprices. Thus,topriceanon-
standardproductandtocomplywithgoodriskmanagement,itisadvocatedthatthe
financialinstitutionuseseveralresearchmodelssuchastheSVSJ-andtheSepp-
modelinadditiontotheBlack-Scholesmodel.
Keywords: Exoticoption, basketoption, riskmanagement, greeks, affinejump-
diffusions, the Black-Scholes model, the Heston model, Bates model with log-
normal jumps, the Bates model with log-asymmetric double exponential jumps,
theStochastic-Volatility-Simultaneous-Jumps(SVSJ)-model,theSepp-model.

Acknowledgements
IwouldliketothankmysupervisoratRoyalInstituteofTechnology,Dr. CamillaLandénforexcellentguid-
anceintheworldofderivativesandsuperbfeedback. IwouldalsoliketoacknowledgeDr. ArturSepp,Vice
PresidentEquityDerivativesAnalytics,BankofAmericaMerrillLynch,forfruitfuldiscussionsaboutaffine
jump-diffusions and Professor Kenneth Holmström, Tomlab for insightful comments about calibration. I’m
alsogratefultoDubravkoSalcic,TomAndersson,NelaCekredziandRobertAxelssonatSEB,forhighlight-
ingthepracticalaspectsofstructuredderivatives. Finally,IwouldliketothankmymentorDr. Christianvon
Ledebur,andmyfamilyandfriendsfortheirvaluablesupportandencouragement.
DanielKrebs
([email protected])

Contents
1 Introduction 1
2 Thedynamicportfolio 3
2.1 Theself-financingportfolio. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 TheVCTL-option. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Mathematicsbehindtherelativeweights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.4 Theannualizedrealizedhistoricalvolatilityandthediscretizedportfolio . . . . . . . . . . . . 6
2.5 Avolatilitycaptargetlevelandthecorrespondingallocationtable . . . . . . . . . . . . . . . 7
3 UsingFouriertransformforoptionpricingwheretheriskyassetsatisfiesdifferentaffinejump-
diffusions 9
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 ThegeneralandtransformedpricedynamicfortheriskyassetS . . . . . . . . . . . . . . . . 9
3.2.1 TheoptionpriceformulausingtheinverseandforwardFouriertransform . . . . . . . 10
3.2.2 Thecharacteristicformula: ExplicitexpressionscorrespondingtotheBlack-Scholes
model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2.3 Thecharacteristicformula:ExplicitexpressionscorrespondingtotheHestonmodel . 12
3.2.4 Thecharacteristicformula:ExplicitexpressionscorrespondingtotheBatesmodel . . 12
3.2.5 Thecharacteristicformula:ExplicitexpressionscorrespondingtotheSVSJmodel . . 13
3.2.6 Thecharacteristicformula:ExplicitexpressionscorrespondingtotheSEPPmodel . . 14
4 Modelcalibration 15
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Amulti-variatefunctionandestimationprocedure. . . . . . . . . . . . . . . . . . . . . . . . 15
4.2.1 Severalerrormeasuresandthechoiceoferrorfunctional. . . . . . . . . . . . . . . . 16
4.2.2 Themarketdata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
4.3 Boundariesandestimatedparametervalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Simulationofthetransformedportfolioaffinejump-diffusions 19
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Simple Euler discretization of the transformed portfolio dynamic: The risky asset evolves
accordingtotheBlack-Scholesmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset
evolvesaccordingtotheHestonmodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.4 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset
evolvesaccordingtotheBatesLNmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.5 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset
evolvesaccordingtotheBatesLDEmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
5.6 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset
evolvesaccordingtotheSVSJmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.7 Full truncation Euler discretization of the transformed portfolio dynamics: The risky asset
evolvesaccordingtotheSeppmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6 Thedistributionalfeaturesoftheportfolio 27
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2 ThemeanoftheportfolioX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
vi
6.2.1 Themeanversusvolatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2.2 ThemeanversusVCTL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
6.3 ThevarianceoftheportfolioX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.3.1 Thestandarddeviationoflog(X)versusmaturityfordifferentvolatilities . . . . . . . 30
6.4 Thestandarddeviationoflog(X)versusVCTL . . . . . . . . . . . . . . . . . . . . . . . . . 32
6.5 MeasuringskewnessandkurtosisoftheVolatilityCapPortfolio . . . . . . . . . . . . . . . . 32
6.6 ThekurtosisandskewnessoftheVolatilityCapPortfolio . . . . . . . . . . . . . . . . . . . . 33
7 TheVCTL-optionprice 37
7.1 TheVCTL-optionpriceversusVCTLundertheBS-model . . . . . . . . . . . . . . . . . . . 37
7.2 The VCTL-option price versus VCTL under the Heston-, BatesLN-,BatesLDE-,SVSJ- and
Sepp-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.3 TheVCTL-optionpriceversusvolatilitywhenVCTL=15% . . . . . . . . . . . . . . . . . . 41
7.4 TheVCTL-optionpriceversusvolatilitywhenVCTL=10% . . . . . . . . . . . . . . . . . . 44
7.5 Concludingremarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
8 ThesensitivemeasuresoftheVCTL-option 49
8.1 TheGreeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
8.2 Greeks for a European call option when the risky asset S evolves according to the Black-
Scholesmodel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8.2.1 Delta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
8.2.2 Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
8.2.3 Vega . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
8.3 GreeksfortheVCTL-option,whentheriskyassetSevolvesaccordingtotheBlack-Scholes
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
8.4 GreeksfortheVCTL-option,whentheriskyassetSevolvesaccordingtotheHeston-model . 53
8.5 Greeks for the VCTL-option, when the risky asset S evolves according to the BatesLN-,
BatesLDE-,SVSJ-andSepp-model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
8.6 DeltaforVCTL-optionversusvolatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
8.7 VegaforVCTL-optionversusvolatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
9 Impliedvolatilityandskeweffects 55
9.1 Impliedvolatilityversusthevolatilityofvolatilityε . . . . . . . . . . . . . . . . . . . . . . . 56
9.2 Impliedvolatilityversusthecorrelationρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
10 Conclusions 59
A Appendix 61
A.1 Allocationtableandrelativeweights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
A.2 Standarddeviationoflog(S)versusmaturity. . . . . . . . . . . . . . . . . . . . . . . . . . . 63
A.3 Thedrift(riskyassetS)versusvolatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
A.4 Thestandardvariationoflog(X)versusVCTL,maturity1year . . . . . . . . . . . . . . . . . 64
A.5 Thestandardvariationoflog(X)versusVOLATILITY.Maturity1year.VCTL=15% . . . . . 64
A.5.1 Kurtosisandskewnessforlog(S) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
A.6 Thestandardvariationoflog(X)versusVCTL,maturity1year . . . . . . . . . . . . . . . . . 65
A.7 Momentsforlog(X)versusVCTL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.8 GreeksfortheVCTL-option,whentheriskyassetSevolveseitheraccordingtotheBatesLN-
modelandBatesLDE-model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
A.9 GreeksfortheVCTL-option,whentheriskyassetSevolvesaccordingtotheSVSJ-model . . 69
A.10 GreeksfortheVCTL-option,whentheriskyassetSevolvesaccordingtotheSepp-model . . 69
A.11 GreeksforasimpleEuropeancalloptionunderaffinejump-diffusions . . . . . . . . . . . . . 70
A.12 Momentsfordifferentε. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
A.13 Impliedvolatility,3-dimensional,differentrho:s . . . . . . . . . . . . . . . . . . . . . . . . . 72
Bibliography 75