Arbitrage free SABR Term structure modeling - Baruch MFE

36 Pages·2015·0.37 MB·English

ArbitragefreeSABR
Termstructuremodeling
StochasticvolatilityHull-Whitemodel
Interest Rate Volatility
III. Working with SABR
AndrewLesniewski
BaruchCollegeandPosnaniaInc
FirstBaruchVolatilityWorkshop
NewYork
June16-18,2015
A.Lesniewski InterestRateVolatility
ArbitragefreeSABR
Termstructuremodeling
StochasticvolatilityHull-Whitemodel
Outline
1 ArbitragefreeSABR
2 Termstructuremodeling
3 StochasticvolatilityHull-Whitemodel
A.Lesniewski InterestRateVolatility
ArbitragefreeSABR
Termstructuremodeling
StochasticvolatilityHull-Whitemodel
Arbitrage free approach
ThearbitragefreeapproachtoSABR[5]replacestheexplicitasymptotic
expressionsdiscussedinPresentationIIwithanefficientnumericalsolutionof
themodel.
Theprobabilitydensityfunction:
p(t,x,y;T,F,Σ)dFdΣ
(1)
=Prob(F <F(T)<F+dF,Σ<σ(T)<Σ+dΣ|F(t)=x,σ(t)=y)
satisfiestheforwardKolmogorovequation:
∂ p= 1 ∂2 (cid:0)Σ2C(F)2p(cid:1)+ρα ∂2 (cid:0)Σ2C(F)p(cid:1)+ 1α2 ∂2 (cid:0)Σ2p(cid:1), (2)
∂T 2 ∂F2 ∂F∂Σ 2 ∂Σ2
withtheinitialcondition:
p(t,x,y;t,F,Σ)=δ(F−x)δ(Σ−y). (3)
A.Lesniewski InterestRateVolatility
ArbitragefreeSABR
Termstructuremodeling
StochasticvolatilityHull-Whitemodel
Arbitrage free approach
Wehavethefollowingprobabilityconservationlaws:
(cid:90) ∞ ∂2 (cid:0)Σ2C(F)2p(cid:1)dΣ= ∂ (cid:0)Σ2C(F)2p(cid:1)(cid:12)(cid:12)∞
0 ∂F∂Σ ∂F (cid:12)0
=0,
(4)
(cid:90) ∞ ∂2 (cid:0)Σ2p(cid:1)dσ= ∂ (cid:0)Σ2p(cid:1)(cid:12)(cid:12)∞
0 ∂Σ∂Σ ∂Σ (cid:12)0
=0,
Introducenowthemoments:
(cid:90) ∞
Q(k)(t,x,y;T,F)= Σkp(t,x,y;T,F,Σ)dΣ, (5)
0
fork =0,1,....Clearly,Q(0)(t,x,y;T,F)istheterminalprobabilityofF,given
thestate(x,y)attimet.
Inthefollowing,wewillsuppresstheexplicitdependenceon(t,x,y)ofQ(k).
A.Lesniewski InterestRateVolatility
ArbitragefreeSABR
Termstructuremodeling
StochasticvolatilityHull-Whitemodel
Effective forward equation
IntegratingtheforwardKolmogorovequationoverallΣ’sandusingthe
probabilityconservationlaws(4)yieldsthefollowingequation:
∂ Q(0)= 1 ∂2 (cid:0)C(F)2Q(2)(cid:1). (6)
∂T 2 ∂F2
ThetimeevolutionofthemarginalPDFQ(0)dependsthusonthesecond
momentQ(2).
Now,eachofthemomentsQ(k)satisfiesthebackwardKolmogorovequation:
∂ 1 ∂2 ∂2 1 ∂2
Q(k)+ y2C(x)2 Q(k)+ραy Q(k)+ α2y2 Q(k)=0,
∂t 2 ∂x2 ∂x∂y 2 ∂y2 (7)
Q(k)(T,x,y;T,F)=ykδ(F−x).
Ratherthanfindinganexplicitsolutionto(7),weseektoexpressQ(2)intermsof
Q(0),inordertoclosetheforwardequation(6).
A.Lesniewski InterestRateVolatility
ArbitragefreeSABR
Termstructuremodeling
StochasticvolatilityHull-Whitemodel
Effective forward equation
AdetailedanalysisusingasymptoticanalysisofthethebackwardKolmogorov
equationforQ(0)andQ(2)showthat:
Q(2)(T,F)=y2(1+2ρζ+ζ2)eραyΓ(T−t)Q(0)(T,F)(cid:0)1+O(ε3)(cid:1)
=y2I(ζ)2eραyΓ(T−t)Q(0)(T,F)(cid:0)1+O(ε3)(cid:1),
where
α (cid:90) F du
ζ= ,
y x C(u)
(cid:113)
I(ζ)= 1+2ρζ+ζ2,
C(F)−C(x)
Γ= .
F−x
ThemarginalPDFQ(0)(T,F)satisfiesthustheeffectiveforwardequation:
∂ Q(0)= 1 ∂2 (cid:0)y2I(ζ)2eραyΓ(T−t)C(F)2Q(0)(cid:1). (8)
∂T 2 ∂F2
TheapproximationaboveisaccuratethroughO(ε2),whichisthesameaccuracy
astheoriginalSABRanalysis.
A.Lesniewski InterestRateVolatility
ArbitragefreeSABR
Termstructuremodeling
StochasticvolatilityHull-Whitemodel
Option prices
Topriceanoptionwethusproceedinthefollowingsteps.
Wesolvenumericallytheeffectiveforwardequation:
∂ Q(0)= 1 ∂2 (cid:0)y2I(ζ)2eραyΓ(T−t)C(F)2Q(0)(cid:1), (9)
∂T 2 ∂F2
withtheinitialcondition:
Q(0)(0,F)=δ(F−F ), atT =0. (10)
0
Weassumethat0<F <Fmax,whereFmaxisasuitablychosenmaximumvalue
oftheforward(say10%).
Weassumeabsorbing(Dirichlet)boundaryconditionssothatF(t)isa
martingale:
Q(0)=0, atF =0,
Q(0)=0, atF =Fmax.
A.Lesniewski InterestRateVolatility
ArbitragefreeSABR
Termstructuremodeling
StochasticvolatilityHull-Whitemodel
Numerical solution
Thereducedproblemisonedimensional.
(i) ItssolutionisimplementedusingthemomentpreservingCrank-Nicolson
scheme.
(ii) Itsruntimeisvirtuallyinstantaneous.
Furthermore,themethod
(i) guaranteesthatprobabilityisexactlypreserved,andthatF(t)isa
martingale:
(cid:90) ∞
p(T,F)dF =1,
0 (11)
(cid:90) ∞
Fp(T,F)dF =F ;
0
0
(ii) themaximumprincipleforparabolicequationsguaranteesthat
p(t,F)≥0, forallF. (12)
A.Lesniewski InterestRateVolatility
ArbitragefreeSABR
Termstructuremodeling
StochasticvolatilityHull-Whitemodel
Numerical solution
Optionpricesaregivenbytheintegrals:
(cid:90) ∞
Pcall=N(0) (F−K)Q(0)(T,F)dF,
K
(13)
(cid:90) K
Pput=N(0) (K −F)Q(0)(T,F)dF,
0
whicharecalculatednumerically.
ThePDFisindependentofthestrikeandcanbeusedforpricingoptionsofall
strikes.
Thenumericalsolutionisanarbitragefreemodel.
A.Lesniewski InterestRateVolatility
ArbitragefreeSABR
Termstructuremodeling
StochasticvolatilityHull-Whitemodel
Boundary layer
ArbitragefreeapproachyieldsnearlythesamevaluesastheexplicitSABR
formulasσn(T,K,F0,σ0,α,β,ρ),exceptforlowstrikesandforwards.
Usingasymptoticmethodstosolvetheeffectiveforwardequationleadstothe
sameexplicitformulasforσnasintheoriginalanalysis,unlesstheforwardor
strikeisnearzero.
Explicitformulasforσndonotholdinaboundarylayeraroundzero.
Boundarylayeroccurswhereasignificantfractionofthepathsgetabsorbedat0
beforeoptionexpiration.
A.Lesniewski InterestRateVolatility