Financial Modeling: A Backward Stochastic Differential Equations Perspective

463 Pages·2013·4.65 MB·English

by  Stéphane Crépey (auth.)

Springer Finance
Textbooks
Stéphane Crépey
Financial
Modeling
A Backward Stochastic
Diff erential Equations Perspective
Springer Finance
Textbooks
EditorialBoard
MarcoAvellaneda
GiovanniBarone-Adesi
MarkBroadie
MarkH.A.Davis
EmanuelDerman
ClaudiaKlüppelberg
WalterSchachermayer
Springer Finance Textbooks
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ThissubseriesofSpringerFinanceconsistsofgraduatetextbooks.
Forfurthervolumes:
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Stéphane Crépey
Financial
Modeling
A Backward Stochastic
Differential Equations Perspective
Prof.StéphaneCrépey
Départementdemathématiques,
LaboratoireAnalyse&Probabilités
Universitéd’EvryVald’Essone
Evry,France
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Password:[978-3-642-37112-7]
ISBN978-3-642-37112-7 ISBN978-3-642-37113-4(eBook)
DOI10.1007/978-3-642-37113-4
SpringerHeidelbergNewYorkDordrechtLondon
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MathematicsSubjectClassification: 91G20,91G60,91G80
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To Irèneand Camille
Preface
Thisisabookonfinancialmodelingthatemphasizescomputationalaspects.Itgives
a unified perspective on derivative pricing and hedging across asset classes and is
addressedtoallthosewhoareinterestedinapplicationsofmathematicstofinance:
students,quantsandacademics.
The book features backward stochastic differential equations (BSDEs), which
areanattractivealternativetothemorefamiliarpartialdifferentialequations(PDEs)
for representing prices and Greeks of financial derivatives. First, BSDEs offer the
mostunifiedsetupforpresentingthefinancialderivativespricingandhedgingtheory
(asreflectedbytherelativecompactnessofthebook,givenitsratherwidescope).
Second,BSDEsareatechnicallyveryflexibleandpowerfulmathematicaltoolfor
elaborating the theory with all the required mathematical rigor and proofs. Third,
BSDEs are also useful for the numerical solution of high-dimensional nonlinear
pricingproblemssuchasthenonlinearCVAandfundingissueswhichhavebecome
importantsincethegreatcrisis[30,80,81].
Structure oftheBook
Part I provides a course in stochastic processes, beginning at a quite elementary
levelinordertogentlyintroducethereadertothemathematicaltoolsthatareneeded
subsequently.PartIIdealswiththederivationofthepricingequationsoffinancial
claimsandtheirexplicitsolutionsinafewcaseswheretheseareeasilyobtained,al-
thoughtypicallytheseequationshavetobesolvednumericallyasisdoneinPartIII.
PartIVprovidestwocomprehensiveapplicationsofthebook’sapproachthatillus-
trate the versatility of simulation/regression pricing schemes for high-dimensional
pricingproblems.PartVprovidesathoroughmathematicaltreatmentoftheBSDEs
andPDEsthatareoffundamentalimportanceforourapproach.Finally,PartVIis
anextendedappendixwithtechnicalproofs,exercisesandcorrectedproblemsets.
vii
viii Preface
Outline
Chapters1–3provideasurveyofusefulmaterialfromstochasticanalysis.InChap.4
werecallthebasicsoffinancialtheorywhicharenecessaryforunderstandinghow
therisk-neutralpricingequationofagenericcontingentclaimisderived.Thischap-
tergivesa unifiedviewon thetheoryofpricingandhedgingfinancialderivatives,
using BSDEs as a main tool. We then review, in Chap. 5, benchmark models on
referencederivativemarkets.Chapter6isaboutMonteCarlopricingmethodsand
Chaps.7and8dealwithdeterministicpricingschemes:treesinChap.7andfinite
differencesinChap.8.
Notethatthereisnohermeticfrontierbetweendeterministicandstochasticpric-
ingschemes.Inessence,allthesenumericalschemesarebasedontheideaofprop-
agating the solution, starting from a surface of the time-space domain on which it
isknown(typically:thematurityofaclaim),alongsuitable(random)“characteris-
tics”oftheproblem.Here“characteristics”referstoRiemann’smethodforsolving
hyperbolic first-order equations (see Chap. 4 of [191]). From the point of view of
controltheory,allthesenumericalschemescanbeviewedasvariantsofBellman’s
dynamicprogrammingprinciple[26].MonteCarlopricingschemesmaythusbere-
gardedasone-time-stepmultinomialtrees,convergingtoalimitingjumpdiffusion
whenthenumberofspacediscretizationpoints(treebranches)goestoinfinity.The
difference between a tree method in the usual sense and a Monte Carlo method is
thataMonteCarlocomputationmeshisstochasticallygeneratedandnonrecombin-
ing.
Pricesofliquidfinancialinstrumentsaregivenbythemarketandaredetermined
bysupply-and-demand.Liquidmarketpricesarethusactuallyusedbymodelsinthe
“reverse-engineering” mode that consists in calibrating a model to market prices.
This calibration process is the topic of Chap. 9. Once calibrated to the market, a
model can be used for Greeking and/or for pricing more exotic claims (Greeking
meanscomputingrisksensitivitiesinordertoset-uparelatedhedge).
Analogiesanddifferencesbetweensimulationanddeterministicpricingschemes
aremostclearlyvisibleinthecontextofpricingbysimulationclaimswithearlyex-
ercise features (American and/or cancelable claims). Early exercisable claims can
be priced by hybrid “nonlinear Monte Carlo” pricing schemes in which dynamic
programmingequations,similartothoseusedindeterministicschemes,areimple-
mentedonstochasticallygeneratedmeshes.Suchhybridschemesarethetopicsof
Chaps.10and11,indiffusionandinpurejumpsetups,respectively.Again,thisis
presentlybecomingquitetopicalforthepurposeofCVAcomputations.
Chapters 12–14 develop, within a rigorous mathematical framework, the con-
nection between backward stochastic differential equations and partial differential
equations. This is done in a jump-diffusion setting with regime switching, which
coversallthemodelsconsideredinthebook.
FinallyChap.15gathersthemostdemandingproofsofPartV,Chap.16isde-
votedtoexercisesforPartIandChap.17providessolvedproblemsetsforPartsII
andIII.
Preface ix
Fig.1 Gettingstartedwiththebook:roadmapof“afirstandpartialreading”fordifferentaudi-
ences.Green:Students.Blue:Quants.Red:Academics
Roadmap
Giventhedualnatureoftheproposedaudience(scholarsandquants),wehavepro-
videdmorebackgroundmaterialonstochasticprocesses,pricingequationsandnu-
mericalmethodsthanisneededforourmainpurposes.Yetwehavenotavoidedthe
sometimesdifficultmathematicaltechniquethatisneededfordeepunderstanding.
So, for the convenience of readers, we signal sections that contain advanced ma-
terial with an asterisk (*) or even a double asterisk (**) for the still more difficult
portions.
Our ambition is, of course, that any reader should ultimately benefit from all
partsofthebook.Weexpectthatanaveragereaderwillneedtwoorthreeattempts
atreadingatdifferentlevelsforachievingthisobjective.Toprovideadditionalguid-
ance,weproposethefollowingroadmapofwhata“firstandpartial”readingofthe
bookcouldbeforthree“stylized”readers(seeFig.1forapictorialrepresentation):
a student (in “green” on the figure), a quant (“blue” audience) and an academic
(“red”;the“blueandred”boxinthechartmayrepresentavaluablefirstreadingfor
bothquantsandacademics):
• for a graduate student (“green”), we recommend a first reading of the book at
a classical quantitative and numerical finance textbook level, as follows in this
order:
– startbyChaps.1–3(exceptforthestarredsectionsofChap.3),alongwiththe
accompanying(generallyclassical)exercisesofChap.16,1
1Solutionsoftheexercisesareavailableforcourseinstructors.
x Preface
– thenjumptoChaps.5to9,dothecorrectedproblemsofChap.17andrunthe
accompanyingMatlabscripts(http://extras.springer.com);
• foraquant(“blue”):
– startwiththestarredsectionsofChap.3,followedbyChap.4,
– thenjumptoChaps.10and11;
• foranacademicsoraPhDstudent(“red”):
– startwiththestarredsectionsofChap.3,followedbyChap.4,
– thenjumptoChaps.12to14,alongwiththerelatedproofsinChap.15.
TheRoleofBSDEs
Althoughthisbookisn’texclusivelydedicatedtoBSDEs,itfeaturestheminvarious
contextsasacommonthreadforguidingreadersthroughtheoreticalandcomputa-
tional aspects of financial modeling. For readers who are especially interested in
BSDEs,werecommend:
• Section3.5foramathematicalintroductionofBSDEsataheuristiclevel,
• Chapter4fortheirgeneralconnectionwithhedging,
• Section6.10andChap.10fornumericalaspectsand
• PartVandChap.15fortherelatedmathematics.
InSect.11.5wealsogiveaprimerofCVAcomputationsusingsimulation/regression
techniquesthataremotivatedbyBSDEnumericalschemes,eventhoughnoBSDEs
appear explicitly. More on this will be found in [30], for which the present book
shouldbeausefulcompanion.
BibliographicGuidelines
Toconcludethispreface,hereareafewgeneralreferences:
• on random processes and stochastic analysis, often with connections to finance
(Chaps.1–3):[149,159,167,174,180,205,228];
• onmartingalemodelinginfinance(Chap.4):[93,114,159,191,208,245];
• onmarketmodels(Chap.5):[43,44,58,131,146,208,230,241];
• onMonteCarlomethods(Chap.6):[133,176,226];
• ondeterministicpricingschemes(Chaps.7and8):[1,12,27,104,172,174,207,
248,254,256];
• onmodelcalibration(Chap.9):[71,116,213];
• onsimulation/regressionpricingschemes(Chaps.10and11):[133,136];
• onBSDEsandPDEs,especiallyinconnectionwithfinance(Chaps.12–14):[96,
113,114,122,164,197,224].