Term- Structure Models Damir Filipovi´ c Outlines Part 1: Interest Rates and Related Contracts Part 2: Estimating the Term-Structure Part 3: Arbitrage Theory Part 4: Short Rate Models Part 5: Heath– Jarrow–Morton (HJM) Methodology Part 6: Forward Measures Part 7: Forwards and Futures Part 8: Consistent Term-Structure Parametrizations Part 9: Affine Processes Part 10: Market Models Term-Structure Models A Graduate Course Damir Filipovi´ c Version 5 November 2009
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Term-StructureModels
DamirFilipovic
Outlines
Part 1: InterestRates andRelatedContracts
Part 2:Estimating theTerm-Structure
Part 3:Arbitrage Theory
Part 4: ShortRate Models
Part 5: Heath–Jarrow–Morton(HJM)Methodology
Part 6: ForwardMeasures
Part 7: Forwardsand Futures
Part 8:ConsistentTerm-StructureParametrizations
Part 9: AffineProcesses
Part 10: MarketModels
Term-Structure ModelsA Graduate Course
Damir Filipovic
Version 5 November 2009
Term-StructureModels
DamirFilipovic
Outlines
Part 1: InterestRates andRelatedContracts
Part 2:Estimating theTerm-Structure
Part 3:Arbitrage Theory
Part 4: ShortRate Models
Part 5: Heath–Jarrow–Morton(HJM)Methodology
Part 6: ForwardMeasures
Part 7: Forwardsand Futures
Part 8:ConsistentTerm-StructureParametrizations
Part 9: AffineProcesses
Part 10: MarketModels
Course Book
Term-StructureModels
DamirFilipovic
Outlines
Part 1: InterestRates andRelatedContracts
Part 2:Estimating theTerm-Structure
Part 3:Arbitrage Theory
Part 4: ShortRate Models
Part 5: Heath–Jarrow–Morton(HJM)Methodology
Part 6: ForwardMeasures
Part 7: Forwardsand Futures
Part 8:ConsistentTerm-StructureParametrizations
Part 9: AffineProcesses
Part 10: MarketModels
Outline
Part 1: Interest Rates and Related ContractsPart 2: Estimating the Term-StructurePart 3: Arbitrage TheoryPart 4: Short Rate ModelsPart 5: Heath–Jarrow–Morton (HJM) MethodologyPart 6: Forward MeasuresPart 7: Forwards and FuturesPart 8: Consistent Term-Structure ParametrizationsPart 9: Affine ProcessesPart 10: Market Models
Term-StructureModels
DamirFilipovic
Outlines
Part 1: InterestRates andRelatedContracts
Part 2:Estimating theTerm-Structure
Part 3:Arbitrage Theory
Part 4: ShortRate Models
Part 5: Heath–Jarrow–Morton(HJM)Methodology
Part 6: ForwardMeasures
Part 7: Forwardsand Futures
Part 8:ConsistentTerm-StructureParametrizations
Part 9: AffineProcesses
Part 10: MarketModels
Outline of Part 1
1 Zero-Coupon Bonds
2 Interest Rates
3 Money-Market Account and Short Rates
4 Coupon Bonds, Swaps and YieldsFixed Coupon BondsFloating Rate NotesInterest Rate SwapsYield and Duration
5 Market Conventions
6 Caps and FloorsBlack’s Formula
7 SwaptionsBlack’s Formula
Term-StructureModels
DamirFilipovic
Outlines
Part 1: InterestRates andRelatedContracts
Part 2:Estimating theTerm-Structure
Part 3:Arbitrage Theory
Part 4: ShortRate Models
Part 5: Heath–Jarrow–Morton(HJM)Methodology
Part 6: ForwardMeasures
Part 7: Forwardsand Futures
Part 8:ConsistentTerm-StructureParametrizations
Part 9: AffineProcesses
Part 10: MarketModels
Outline
Part 1: Interest Rates and Related ContractsPart 2: Estimating the Term-StructurePart 3: Arbitrage TheoryPart 4: Short Rate ModelsPart 5: Heath–Jarrow–Morton (HJM) MethodologyPart 6: Forward MeasuresPart 7: Forwards and FuturesPart 8: Consistent Term-Structure ParametrizationsPart 9: Affine ProcessesPart 10: Market Models
Figure: forward swap rates from 25 Sep 09 (for illustration)
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Swap Example
Swaps were developed because different companies couldborrow at fixed or at floating rates in different markets.Example:
• company A is borrowing fixed at 5 12 %, but could borrow
floating at LIBOR plus 12 %;
• company B is borrowing floating at LIBOR plus 1%, butcould borrow fixed at 6 1
2 %.
By agreeing to swap streams of cash flows both companiescould be better off, and a mediating institution would alsomake money:
• company A pays LIBOR to the intermediary in exchangefor fixed at 5 3
16 % (receiver swap);
• company B pays the intermediary fixed at 5 516 % in
exchange for LIBOR (payer swap).
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Swap Example cont’d
The net payments are as follows:
• company A is now paying LIBOR plus 516 % instead of
LIBOR plus 12 %;
• company B is paying fixed at 6 516 % instead of 6 1
2 %;
• the intermediary receives fixed at 18 %.
Figure: A swap with mediating institution.
Everyone seems to be better off (why?)
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Interest Rate Swap Markets
• interest rate swap markets are over the counter
• but swap contracts exist in standardized form, e.g. by theISDA (International Swaps and Derivatives Association,Inc.).
• swap markets are extremely liquid
• maturities from 1 to 30 years are standard, swap ratequotes available up to 60 years
• gives market participants, such as life insurers, opportunityto create synthetically long-dated investments
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Outline
1 Zero-Coupon Bonds
2 Interest Rates
3 Money-Market Account and Short Rates
4 Coupon Bonds, Swaps and YieldsFixed Coupon BondsFloating Rate NotesInterest Rate SwapsYield and Duration
5 Market Conventions
6 Caps and FloorsBlack’s Formula
7 SwaptionsBlack’s Formula
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Zero-Coupon Yield
• zero-coupon yield is the continuously compounded spotrate R(t,T ):
P(t,T ) = e−R(t,T )(T−t).
• T 7→ R(t,T ) is called (zero-coupon) yield curve
Figure: Yield curve T 7→ R(t,T ).
• note: term “yield curve” is ambiguous
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Yield-to-Maturity
• consider fixed coupon bond (short hand: cn contains N)
p =n∑
i=1
P(0,Ti )ci
• bond’s “internal rate of interest”: (continuouslycompounded) yield-to-maturity y : unique solution to
p =n∑
i=1
cie−yTi .
• Schaefer [47]: yield-to-maturity is inadequate statistic forbond market:
• coupon payments occurring at the same point in time arediscounted by different discount factors, but
• coupon payments at different points in time from the samebond are discounted by the same rate.
In reality, one would wish to do exactly the opposite !
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Macaulay duration
• bond price change as function of y : Macaulay duration:
DMac =
∑ni=1 Ticie−yTi
p
= weighted average of the coupon dates T1, . . . ,Tn (“meantime to coupon payment”)
= first-order sensitivity of bond price w.r.t. changes in theyield-to-maturity:
dp
dy=
d
dy
(n∑
i=1
cie−yTi
)= −DMacp
(interest rate risk management!)
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Duration
• write yi = R(0,Ti )
• duration of the bond
D =
∑ni=1 Ticie−yiTi
p=
n∑i=1
ciP(0,Ti )
pTi
= first-order sensitivity of bond price w.r.t. parallel shifts ofyield curve:
d
ds
(n∑
i=1
cie−(yi +s)Ti
)|s=0 = −Dp.
→ duration is essentially for bonds (w.r.t. parallel shift of theyield curve) what delta is for stock options
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Convexity
• bond equivalent of gamma is convexity:
C =d2
ds2
(n∑
i=1
cie−(yi +s)Ti
)|s=0 =
n∑i=1
cie−yiTi (Ti )2
→ second-order approximation for bond price change ∆pw.r.t. parallel shift ∆y of yield curve:
∆p ≈ −Dp∆y +1
2C (∆y)2
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Outline
1 Zero-Coupon Bonds
2 Interest Rates
3 Money-Market Account and Short Rates
4 Coupon Bonds, Swaps and YieldsFixed Coupon BondsFloating Rate NotesInterest Rate SwapsYield and Duration
5 Market Conventions
6 Caps and FloorsBlack’s Formula
7 SwaptionsBlack’s Formula
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Day-Count Conventions
• convention: measure time in units of years
• market evaluates year fraction between t < T in differentways
• examples of day-count conventions δ(t,T ):• actual/365: year has 365 days
δ(t,T ) =actual number of days between t and T
365.
• actual/360: as above but year counts 360 days• 30/360: months count 30 and years 360 days. Let
t = d1/m1/y1 and T = d2/m2/y2
δ(t,T ) =min(d2, 30) + (30− d1)+
360+
(m2 −m1 − 1)+
12+y2−y1.
Example: t = 4 January 2000 and T = 4 July 2002:
δ(t,T ) =4 + (30− 4)
360+
7− 1− 1
12+ 2002− 2000 = 2.5.
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Coupon Bonds
Coupon bonds issued in the American (European) marketstypically have semiannual (annual) coupon payments.Debt securities issued by the US Treasury are divided into threeclasses:
• Bills: zero-coupon bonds with time to maturity less thanone year.
• Notes: coupon bonds (semiannual) with time to maturitybetween 2 and 10 years.
• Bonds: coupon bonds (semiannual) with time to maturitybetween 10 and 30 years.1
STRIPS (separate trading of registered interest and principal ofsecurities): synthetically created zero-coupon bonds, tradedsince August 1985
130-year Treasury bonds were not offered from 2002 to 2005.
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Accrued Interest, Clean Price andDirty Price
• recall coupon bond price formula
p(t) =∑Ti≥t
ciP(t,Ti )
→ systematic discontinuities of price trajectory at t = Ti
• accrued interest at t ∈ (Ti−1,Ti ] is defined by
AI (i ; t) = cit − Ti−1
Ti − Ti−1
• clean price (quoted) of coupon bond at t ∈ (Ti−1,Ti ] is
pclean(t) = p(t)− AI (i ; t)
→ dirty price (to pay) is
p(t) = pclean(t) + AI (i ; t)
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Yield-to-Maturity
see course book Section 2.5.4
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Outline
1 Zero-Coupon Bonds
2 Interest Rates
3 Money-Market Account and Short Rates
4 Coupon Bonds, Swaps and YieldsFixed Coupon BondsFloating Rate NotesInterest Rate SwapsYield and Duration
5 Market Conventions
6 Caps and FloorsBlack’s Formula
7 SwaptionsBlack’s Formula
Term-StructureModels
DamirFilipovic
Zero-CouponBonds
Interest Rates
Money-MarketAccount andShort Rates
CouponBonds, Swapsand Yields
Fixed CouponBonds
Floating RateNotes
Interest RateSwaps
Yield andDuration
MarketConventions
Caps andFloors
Black’s Formula
Swaptions
Black’s Formula
Caplets
• caplet with reset date T and settlement date T + δ: paysthe holder difference between simple market rateF (T ,T + δ) (e.g. LIBOR) and strike rate κ
ill-posed: first-order condition C>C d = C>p, anddimension of solution space =dim ker(C>C ) = dim ker(C ) ≥ N − n
• moreover: as many parameters as there are cash flowdates, and there is nothing to regularize the discount curvefound from the regression ⇒ discount factors of similarmaturity can be very different ⇒ ragged yield and forwardcurves
• alternative and better method: estimate a smooth yieldcurve parametrically from market rates . . .
Table: Overview of estimation procedures by several central banks.Bank for International Settlements (BIS) 1999 [5]. NS is forNelson–Siegel, S for Svensson, wp for weighted prices
• major problem in term-structure estimation: highdimensionality
• aim: find basis shapes of the yield curve (increments)
• principal component analysis (PCA): dimension reductiontechnique in multivariate analysis
• key mathematical principle: spectral decomposition forsymmetric n × n matrix Q: Q = ALA> where
• L = diag(λ1, . . . , λn) is the diagonal matrix of eigenvaluesof Q with λ1 ≥ λ2 ≥ · · · ≥ λn;
• A is an orthogonal matrix (that is, A−1 = A>) whosecolumns a1, . . . , an are the normalized eigenvectors of Q(Qai = λiai ), which form orthonormal basis of Rn.
• can show: E[Y ] = 0 andCov[Y ] = A>QA = A>ALA>A = L
⇒ principal components of X are uncorrelated, havevariances Var[Yi ] = λi
• can show: Var[a>1 X ] = maxVar[b>X ] | b>b = 1
: Y1
has maximal variance among all standardized linearcombinations of X . For i = 2, . . . , n: Yi has maximalvariance among all such linear combinations orthogonal tofirst i − 1 linear combinations
• application: X = high-dimensional stationary model for(daily changes of) the forward curve. If first k principalcomponents Y1, . . . ,Yk explain significant amount (e.g.99%) of variability in X then approximate
X ≈ µ+k∑
i=1
Yiai .
⇒ loadings a1, . . . , ak are main components of stochasticforward curve movements
• empirical mean µ and covariance matrix Q standardestimators for true parameters µ and Q, if observationsX (t) are either independent or at least seriallyuncorrelated (i.e. Cov[X (t),X (t + h)] = 0 for all h 6= 0)
• if this kind of stationarity of time series X (t) is in doubt,the standard practice is to differentiate and to consider theincrements ∆X (t) = X (t)− X (t − 1)
Table: Explained variance of the principal components
PC Explainedvariance (%)
1 92.172 6.933 0.614 0.245 0.03
6–8 0.01
• First 3 PCs explain more than 99% of variance of x :forward curves can be approximated by linear combinationof first three loadings, with small relative error
• these features are very typical (stylized facts), to beexpected in most PCA of forward curve (increments). Seealso Carmona and Tehranchi [14, Section 1.7]. PCA offorward curve goes back to Litterman andScheinkman [37].
• Infinite time horizon (w.l.o.g.): F = F∞ = ∨t≥0Ft
• Sometimes also : Ft = FWt (for hedging/completeness)
Term-StructureModels
DamirFilipovic
StochasticCalculus
StochasticDifferentialEquations
FinancialMarket
Self-FinancingPortfolios
Numeraires
Arbitrage andMartingaleMeasures
MartingaleMeasures
Market Price ofRisk
AdmissibleStrategies
The FirstFundamentalTheorem ofAsset Pricing
Hedging andPricing
CompleteMarkets
Arbitrage Pricing
Stochastic Processes
• Convention: “X = Y ” means “X = Y a.s.”(P[X = Y ] = 1)
• Borel σ-algebras: B[0, t], B(R+), or simply B, etc.
• Stochastic process X = X (ω, t) is called:• adapted if Ω 3 ω 7→ X (ω, t) is Ft-measurable ∀t ≥ 0,• progressively measurable if Ω× [0, t] 3 (ω, s) 7→ X (ω, s) isFt ⊗ B[0, t]-measurable ∀t ≥ 0.
• Progressive implies adapted
• Progressive ⇒∫ t
0 X (s) ds, X (t ∧ τ) (stopping time τ),etc. are adapted
• Notation: Prog = progressive σ-algebra, generated by allprogressive processes, on Ω× R+. Fact: progressive ⇔Prog-measurable (Proposition 1.41 in [42])
Term-StructureModels
DamirFilipovic
StochasticCalculus
StochasticDifferentialEquations
FinancialMarket
Self-FinancingPortfolios
Numeraires
Arbitrage andMartingaleMeasures
MartingaleMeasures
Market Price ofRisk
AdmissibleStrategies
The FirstFundamentalTheorem ofAsset Pricing
Hedging andPricing
CompleteMarkets
Arbitrage Pricing
Stochastic Integrands
• L2 := set of Rd -valued progressive processesh = (h1, . . . , hd) with
E[∫ ∞
0‖h(s)‖2 ds
]<∞
• L := set of Rd -valued progressive processesh = (h1, . . . , hd) with∫ t
0‖h(s)‖2 ds <∞ for all t > 0
• Obvious: L2 ⊂ L
Term-StructureModels
DamirFilipovic
StochasticCalculus
StochasticDifferentialEquations
FinancialMarket
Self-FinancingPortfolios
Numeraires
Arbitrage andMartingaleMeasures
MartingaleMeasures
Market Price ofRisk
AdmissibleStrategies
The FirstFundamentalTheorem ofAsset Pricing
Hedging andPricing
CompleteMarkets
Arbitrage Pricing
Stochastic Integral
Theorem (Stochastic Integral)
For every h ∈ L one can define the stochastic integral
(h •W )t =
∫ t
0h(s) dW (s) =
d∑j=1
∫ t
0hj(s) dWj(s).
with the following properties:
1 The process h •W is a continuous local martingale.
2 Linearity: (λg + h) •W = λ(g •W ) + h •W , for g , h ∈ Land λ ∈ R.
3 For any stopping time τ , the stopped integral equals∫ t∧τ
0h(s) dW (s) =
∫ t
01s≤τh(s) dW (s) for all t > 0.
Term-StructureModels
DamirFilipovic
StochasticCalculus
StochasticDifferentialEquations
FinancialMarket
Self-FinancingPortfolios
Numeraires
Arbitrage andMartingaleMeasures
MartingaleMeasures
Market Price ofRisk
AdmissibleStrategies
The FirstFundamentalTheorem ofAsset Pricing
Hedging andPricing
CompleteMarkets
Arbitrage Pricing
Stochastic Integral cont’d
Theorem (Stochastic Integral cont’d)
4 If h ∈ L2 then h •W is a martingale and the Ito isometryholds:
E
[(∫ ∞0
h(s) dW (s)
)2]
= E[∫ ∞
0‖h(s)‖2 ds
].
5 Dominated convergence: if (hn) ⊂ L is a sequence withlimn hn = 0 pointwise and such that |hn| ≤ k for somefinite constant k then limn sups≤t |(hn •W )s | = 0 inprobability for all t > 0.
Proof.See [45, Section 2, Chapter IV].
Term-StructureModels
DamirFilipovic
StochasticCalculus
StochasticDifferentialEquations
FinancialMarket
Self-FinancingPortfolios
Numeraires
Arbitrage andMartingaleMeasures
MartingaleMeasures
Market Price ofRisk
AdmissibleStrategies
The FirstFundamentalTheorem ofAsset Pricing
Hedging andPricing
CompleteMarkets
Arbitrage Pricing
Ito processes• Convention: stochastic integrands = row vectors,
Brownian motion = column vector• Ito process := drift + continuous local martingale
X (t) = X (0) +
∫ t
0a(s) ds +
∫ t
0ρ(s) dW (s)
where ρ ∈ L and a is progressive process with∫ t0 |a(s)| ds <∞ ∀t > 0
LemmaThe above decomposition of X is unique in the sense that
X (t) = X (0) +
∫ t
0a′(s) ds +
∫ t
0ρ′(s) ds
implies a′ = a and ρ′ = ρ dP⊗ dt-a.s.
Proof.This follows from Proposition (1.2) in [45, Chapter IV].
Term-StructureModels
DamirFilipovic
StochasticCalculus
StochasticDifferentialEquations
FinancialMarket
Self-FinancingPortfolios
Numeraires
Arbitrage andMartingaleMeasures
MartingaleMeasures
Market Price ofRisk
AdmissibleStrategies
The FirstFundamentalTheorem ofAsset Pricing
Hedging andPricing
CompleteMarkets
Arbitrage Pricing
Ito processes cont’d
• Notation:dX (t) = a(t) dt + ρ(t) dW (t)
or, shorter,dX = a dt + ρ dW
• L2(X ) := set of progressive h = (h1, . . . , hd) withE[∫∞
0 |h(s)a(s)|2 ds]<∞ and hρ ∈ L2
• L(X ) := set of progressive h = (h1, . . . , hd) with∫ t0 |h(s)a(s)| ds <∞ for all t > 0 and hρ ∈ L
• For h ∈ L(X ): define stochastic integral w.r.t. X as∫ t
0h(s) dX (s) =
∫ t
0h(s)a(s) ds +
∫ t
0h(s)ρ(s) dW (s)
Term-StructureModels
DamirFilipovic
StochasticCalculus
StochasticDifferentialEquations
FinancialMarket
Self-FinancingPortfolios
Numeraires
Arbitrage andMartingaleMeasures
MartingaleMeasures
Market Price ofRisk
AdmissibleStrategies
The FirstFundamentalTheorem ofAsset Pricing
Hedging andPricing
CompleteMarkets
Arbitrage Pricing
Quadratic Variation andCovariation
• let Y (t) = Y (0) +∫ t
0 b(s) ds +∫ t
0 σ(s) dW (s) be anotherIto process
• Define covariation process of X and Y as
〈X ,Y 〉t =
∫ t
0ρ(s)σ(s)>ds
• 〈X ,X 〉 called quadratic variation process of X
• Fact (Theorem (1.8) and Definition (1.20) in [45, ChapterIV]):
〈X ,Y 〉t = limm∑
i=0
(Xti+1 − Xti )(Yti+1 − Yti ) in probability,
for any sequence of partitions 0 = t0 < t1 < · · · < tm = twith maxi |ti+1 − ti | → 0
Term-StructureModels
DamirFilipovic
StochasticCalculus
StochasticDifferentialEquations
FinancialMarket
Self-FinancingPortfolios
Numeraires
Arbitrage andMartingaleMeasures
MartingaleMeasures
Market Price ofRisk
AdmissibleStrategies
The FirstFundamentalTheorem ofAsset Pricing
Hedging andPricing
CompleteMarkets
Arbitrage Pricing
Levy’s Characterization Theorem
• Fact ([49, Section 6.4] or Exercise (1.27) in [45, ChapterIV]): 〈Wi ,Wj〉t = δij t
Theorem (Levy’s Characterization Theorem)
An Rd -valued continuous local martingale X vanishing at t = 0is a Brownian motion if and only if 〈Xi ,Xj〉t = δij t for every1 ≤ i , j ≤ d.
Proof.See Theorem (3.6) in [45, Chapter IV].
Term-StructureModels
DamirFilipovic
StochasticCalculus
StochasticDifferentialEquations
FinancialMarket
Self-FinancingPortfolios
Numeraires
Arbitrage andMartingaleMeasures
MartingaleMeasures
Market Price ofRisk
AdmissibleStrategies
The FirstFundamentalTheorem ofAsset Pricing
Hedging andPricing
CompleteMarkets
Arbitrage Pricing
Multivariate Ito Processes
• Definition: X = (X1, . . . ,Xn)> n-dimensional Ito process ifevery Xi is an Ito process
• Definition: L2(X ) (L(X )) := set of progressive processesh = (h1, . . . , hn) such that hi is in L2(Xi ) (L(Xi )) for all i
• In this sense: L2 = L2(W ) and L = L(W )
• Stochastic integral of h ∈ L(X ) w.r.t. to X is definedcoordinate-wise as
(h • X )t =
∫ t
0h(s) dX (s) =
n∑i=1
∫ t
0hi (s) dXi (s)
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Ito’s Formula
Core formula of stochastic calculus:
Theorem (Ito’s Formula)
Let f ∈ C 2(Rn). Then f (X ) is an Ito process and
f (X (t)) = f (X (0)) +n∑
i=1
∫ t
0
∂f (X (s))
∂xidXi (s)
+1
2
n∑i ,j=1
∫ t
0
∂2f (X (s))
∂xi∂xjd〈Xi ,Xj〉s .
Proof.See Theorem (3.3) in [45, Chapter IV].
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Integration by Parts Formula
Corollary (for f (x , y) = xy): integration by parts formula
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Definitions
Ingredients:
• b : Ω× R+ × Rn → Rn Prog ⊗ B(Rn)-measurable
• σ : Ω× R+ × Rn → Rn×d Prog ⊗ B(Rn)-measurable
• ξ: some F0-measurable initial value
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Definitions
DefinitionA process X is solution of the stochastic differential equation
dX (t) = b(t,X (t)) dt + σ(t,X (t)) dW (t)
X (0) = ξ(1)
if X is an Ito process satisfying
X (t) = ξ +
∫ t
0b(s,X (s)) ds +
∫ t
0σ(s,X (s)) dW (s).
We say that X is unique if any other solution X ′ of (1) isindistinguishable from X , that is, X (t) = X ′(t) for all t ≥ 0 a.s.If b(ω, t, x) = b(t, x) and σ(ω, t, x) = σ(t, x): solution X of(1) is also called a (time-inhomogeneous) diffusion withdiffusion matrix a(t, x) = σ(t, x)σ(t, x)> and drift b(t, x).
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Existence and Uniqueness
TheoremSuppose b(t, x) and σ(t, x) satisfy the Lipschitz and lineargrowth conditions
for all t ≥ 0 and x , y ∈ Rn, where K is some finite constant.Then, for every time–space initial point (t0, x0) ∈ R+ × Rn,there exists a unique solution X = X (t0,x0) of the stochasticdifferential equation
Existence and UniquenessNote: existence and uniqueness hold sometimes withoutLipschitz condition on σ(t, x), see “Affine Processes” below
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Markov Property
TheoremSuppose b(t, x) and a(t, x) = σ(t, x)σ(t, x)> are continuous in(t, x), and assume that for every time–space initial point(t0, x0) ∈ R+ × Rn, there exists a unique solution X = X (t0,x0)
of the stochastic differential equation (2). Then X has theMarkov property. That is, for every bounded measurablefunction f on Rn, there exists a measurable function F onR+ × R+ × Rn such that
E[f (X (T )) | Ft ] = F (t,T ,X (t)), t ≤ T .
In words, the Ft-conditional distribution of X (T ) is a functionof t, T and X (t) only.
Proof.Follows from [35, Theorem 4.20].
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Remarks
• Continuity assumption on the diffusion matrix a(t, x)rather than on σ(t, x), since a(t, x) determines law of X .Ambiguity with σ(t, x): σDD>σ> = σσ> for anyorthogonal d × d-matrix D.
• For most practical purposes: assume σ(t, x) itself iscontinuous in (t, x)
• Time-inhomogeneous diffusion X in (2) can be regardedas R+ × Rn-valued homogeneous diffusion(X ′0, . . . ,X
′n)(t) = (t0 + t,X (t)). Calendar time at
inception (t = 0) is then X ′0(0) = t0. Accordingly, tmeasures relative time with respect to t0.
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Stochastic ExponentialDefine: stochastic exponential of an Ito process X by
Et(X ) = eX (t)− 12〈X ,X 〉t
LemmaLet X and Y be Ito processes.
1 U = E(X ) is a positive Ito process and the unique solutionof the stochastic differential equation
dU = U dX , U(0) = eX (0).
2 E(X ) is continuous local martingale if X is localmartingale.
3 E(0) = 1.4 E(X )E(Y ) = E(X + Y ) e 〈X ,Y 〉.5 E(X )−1 = E(−X ) e 〈X ,X 〉.
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Arbitrage Portfolios
• Definition: arbitrage portfolio := self-financing portfolio φwith value process satisfying
V (0) = 0 and V (T ) ≥ 0 and P[V (T ) > 0] > 0
for some T > 0.
• If no arbitrage portfolios exist for any T > 0 we say themodel is arbitrage-free.
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Example of Arbitrage
LemmaSuppose there exists a self-financing portfolio with valueprocess
dU = U k dt,
for some progressive process k. If the market is arbitrage-freethen necessarily
r = k, dP⊗ dt-a.s.
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Proof I
After discounting with S0 we obtain
U(t) =U(t)
S0(t)= U(0)e
∫ t0 (k(s)−r(s)) ds .
Then
ψ(t) = 1k(t)>r(t)
yields a self-financing strategy with discounted value process
V(t) =
∫ t
0ψ(s) dU(s)
=
∫ t
0
(1k(s)>r(s)(k(s)− r(s))U(s)
)ds ≥ 0.
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Proof II
Hence absence of arbitrage requires
0 = E[V(T )]
=
∫N
(1k(ω,t)>r(ω,t)(k(ω, t)− r(ω, t))U(ω, t)
)︸ ︷︷ ︸>0 on N
dP⊗ dt
where
N = (ω, t) | k(ω, t) > r(ω, t)
is a measurable subset of Ω× [0,T ]. But this can only hold ifN is a dP⊗ dt-nullset. Using the same arguments withchanged signs proves the lemma (→ exercise).
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Equivalent Martingale Measures
• When is a model arbitrage-free?
• For simplicity of notation: fix S0 as a numeraire
DefinitionAn equivalent (local) martingale measure (E(L)MM) Q ∼ Phas the property that the discounted price processes Si areQ-(local) martingales for all i .
• Need to understand how W transforms under equivalentchange of measure . . .
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Girsanov’s Theorem
Theorem (Girsanov’s Change of Measure Theorem)
Let γ ∈ L be such that the stochastic exponential E(γ •W ) isa uniformly integrable martingale with E∞(γ •W ) > 0. Then
dQdP
= E∞ (γ •W )
defines an equivalent probability measure Q ∼ P, and theprocess
W ∗(t) = W (t)−∫ t
0γ(s)>ds
is a Q-Brownian motion.
Proof.See Theorem (1.12) in [45, Chapter VIII].
Note: dQdP |Ft = Et (γ •W )
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Novikov’s Condition
Theorem (Novikov’s Condition)
IfE[e
12
∫∞0 ‖γ(s)‖2 ds
]<∞
then E(γ •W ) is a uniformly integrable martingale withE∞(γ •W ) > 0.
Proof.See Proposition (1.15) in [45, Chapter VIII] for uniformintegrability of E(γ •W ), and Proposition (1.26) in [45,Chapter IV] for finiteness of (γ •W )∞ which is equivalent toE∞(γ •W ) > 0.
Note: Novikov’s condition is only sufficient but not necessary,see exercise in “Affine Processes”.
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An Arbitrage Strategy
• Local martingales ⇒ be alert to pitfalls!
• Example: ∃φ ∈ L s.t. local martingaleM(t) =
∫ t0 φ(s) dW (s) satisfies M(1) = 1 (Dudley’s
Representation Theorem 12.1 in [49])
• Looks like discounted value process of a self-financingstrategy (in the Bachelier model [3] S = W )
⇒ arbitrage!
• Solution: M is unbounded from below (a true localmartingale). In reality, no lender would provide us with aninfinite credit line.
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Admissible Strategies
DefinitionA self-financing strategy φ is admissible if its discounted valueprocess V is a Q-martingale for some ELMM Q.
• Caution: admissibility is sensitive with respect to thechoice of numeraire (see Delbaen and Schachermayer [22])
Useful local martingale property result (generalizing“Stochastic Integral” Theorem):
LemmaThe discounted value process V of an admissible strategy is aQ-local martingale under every ELMM Q.
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Proof
Proof.By assumption, dV = φ dS is the stochastic integral withrespect to the continuous Q-local martingale S. The statementnow follows from Proposition (2.7) in [45, Chapter IV] andProposition (1.5) in [45, Chapter VIII].
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LemmaSuppose there exists an ELMM Q. Then the model isarbitrage-free, in the sense that there exists no admissiblearbitrage strategy.
Proof.Indeed, let V be the discounted value process of an admissiblestrategy, with V(0) = 0 and V(T ) ≥ 0. Since V is aQ-martingale for some ELMM Q, we have
0 ≤ EQ[V(T )] = V(0) = 0,
whence V(T ) = 0.
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As for converse statement:
• Absence of arbitrage among admissible strategies notsufficient for the existence of an ELMM
• Delbaen and Schachermayer [21]: “no free lunch withvanishing risk” (some form of asymptotic arbitrage) isequivalent to the existence of an ELMM
• Technical details far from trivial, beyond this course
• Custom in financial engineering: consider existence of anELMM as “essentially equivalent” to the absence ofarbitrage
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Contingent Claims
DefinitionA contingent claim due at T (or T -claim) is an FT -measurablerandom variable X .
• Examples: cap, floor, swaption
Main problems:
• How can one hedge against the financial risk involved intrading contingent claims?
• What is a fair price for a contingent claim X ?
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Attainable Claims
DefinitionA contingent claim X due at T is attainable if ∃ admissiblestrategy φ which replicates, or hedges, X . That is, its valueprocess V satisfies V (T ) = X
• Simple example: S1 = price process of T -bond.Contingent claim X ≡ 1 due at T : attainable bybuy-and-hold strategy φ0 ≡ 0, φ1 = 1[0,T ], with valueprocess V = S1.
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Representation TheoremWe know: stochastic integral w.r.t. W is a local martingale.The converse holds true if the filtration is not too large:
Theorem (Representation Theorem)
Assume that the filtration
(Ft) is generated by the Brownian motion W . (4)
Then every P-local martingale M has a continuous modificationand there exists ψ ∈ L such that
M(t) = M(0) +
∫ t
0ψ(s) dW (s).
Consequently, every equivalent probability measure Q ∼ P canbe represented in the form (15.3) for some γ ∈ L.
Proof.See Theorem (3.5) in [45, Chapter V]. The last statement
follows since M(t) = E[
dQdP | Ft
]is a positive martingale.
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Complete Markets
DefinitionThe market model is complete if, on any finite time horizonT > 0, every T -claim X with bounded discounted payoffX/S0(T ) is attainable.
• Note: completeness 6⇔ absence of arbitrage
• However . . .
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Theorem (Second Fundamental Theorem of Asset Pricing)
Assume Ft = FWt and there exists an ELMM Q. Then the
following are equivalent:
1 The model is complete.
2 The ELMM Q is unique.
3 The n × d-volatility matrix σ = (σij) is dP⊗ dt-a.s.injective.
4 The market price of risk −γ is dP⊗ dt-a.s. unique.
Under any of these conditions, every T -claim X with
EQ
[|X |
S0(T )
]<∞ is attainable.
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Second Fundamental Theorem ofAsset Pricing
Note: Property 3 ⇒ number of risk factors d ≤ n number ofrisky assets: randomness generated by the d noise factors dWcan be fully absorbed by the n discounted price incrementsdS1, . . . , dSn
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Proof I
1⇒ 2: Let A ∈ FT . By definition there exists an admissiblestrategy φ with discounted value process V satisfyingV(t) = EQ[1A | Ft ] for some ELMM Q. This implies that|V| ≤ 1. In view of Lemma 15.6, V is thus a martingale underany ELMM. Now let Q′ be any ELMM. ThenQ′[A] = V(0) = Q[A], and hence Q = Q′.2⇒ 3: See Proposition 8.2.1 in [41].3⇒ 4⇒ 2: This follows from the linear market price of riskequation (2) and the last statement of the representationtheorem.3⇒ 1: Let X be a claim due at some T > 0 satisfying
EQ
[|X |
S0(T )
]<∞ for some ELMM Q (this holds in particular if
X/S0(T ) is bounded). We define the Q-martingale
Y (t) = EQ
[X
S0(T )| Ft
], t ≤ T .
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Proof II
By Bayes’ rule we obtain
Y (t)D(t) = D(t)EQ[Y (T ) | Ft ] = E[Y (T )D(T ) | Ft ],
with the density process D(t) = dQ/dP|Ft = Et(γ •W ).Hence YD is a P-martingale and by the representation theoremthere exists some ψ ∈ L such that
Y (t)D(t) = Y (0) +
∫ t
0ψ(s) dW (s).
Applying Ito’s formula yields
d
(1
D
)= − 1
Dγ dW +
1
D‖γ‖2 dt,
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Proof IIIand
dY = d
((YD)
1
D
)= YD d
(1
D
)+
1
Dd(YD) + d
⟨YD,
1
D
⟩=
(1
Dψ − Y γ
)dW −
(1
Dψ − Y γ
)γ>dt
=
(1
Dψ − Y γ
)dW ∗
where dW ∗ = dW − γ dt denotes the Girsanov transformedQ-Brownian motion. Note that we just have shown that themartingale representation property also holds for W ∗ under Q.Since σ is injective, there exists some d × n-matrix-valuedprogressive process σ−1 such that σ−1σ equals thed × d-identity matrix. If we define φ = (φ1, . . . , φn) via
φi =
((1Dψ − Y γ
)σ−1
)i
Si,
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Proof IV
it follows that
dY =
(1
Dψ − Y γ
)σ−1σ dW ∗ =
n∑i=1
φiSiσi dW ∗ =n∑
i=1
φi dSi .
Hence φ yields an admissible strategy with discounted valueprocess satisfying
V(t) = Y (t) = EQ
[X
S0(T )
]+
n∑i=1
∫ t
0φi (s) dSi (s),
and in particular V(T ) = Y (T ) = X/S0(T ). Notice that φ isadmissible since V is by construction a true Q-martingale. Thisalso proves the last statement of the theorem.
• sufficient conditions for existence and uniqueness available
• recall: Markov property of r
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SomeStandardModels
Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Term-Structure Equation
LemmaLet T > 0, and Φ continuous function on Z. AssumeF = F (t, r) in C 1,2([0,T ]×Z) is a solution to boundary valueproblem on [0,T ]×Z (term-structure equation for Φ):
∂tF (t, r) + b(t, r)∂r F (t, r) +1
2σ2(t, r)∂2
r F (t, r)− rF (t, r) = 0
F (T , r) = Φ(r).
Then M(t) = F (t, r(t))e−∫ t
0 r(u) du, t ≤ T , is a localmartingale.If in addition either:
1 EQ
[∫ T0
∣∣∣∂r F (t, r(t))e−∫ t
0 r(u) duσ(t, r(t))∣∣∣2 dt
]<∞, or
2 M is uniformly bounded,
then M is a true martingale, and
F (t, r(t)) = EQ
[e−
∫ Tt r(u) duΦ(r(T )) | Ft
], t ≤ T .
Term-StructureModels
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Generalities
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Inverting theForward Curve
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Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Term-Structure Equation: Proof
Proof.We can apply Ito’s formula to M and obtain
dM(t) =(∂tF (t, r(t)) + b(t, r(t))∂r F (t, r(t))
+1
2σ2(t, r)∂2
r F (t, r(t))− r(t)F (t, r(t)))e−
∫ t0 r(u) du dt
+ ∂r F (t, r(t))e−∫ t
0 r(u) duσ(t, r(t)) dW ∗(t)
= ∂r F (t, r(t))e−∫ t
0 r(u) duσ(t, r(t)) dW ∗(t).
Hence M is a local martingale.Either Condition 1 or 2 implies M is true martingale. Since
M(T ) = Φ(r(T ))e−∫ T
0 r(u) du
we obtain
F (t, r(t))e−∫ t
0 r(u) du = M(t) = EQ
[e−
∫ T0 r(u) duΦ(r(T )) | Ft
].
Multiplying both sides by e∫ t
0 r(u) du yields the claim.
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Vasicek Model
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Term-Structure Equation: PricingEfficiency
• term-structure equation for Φ: solution F (t, r(t) = priceof T -claim Φ(r(T ))
• for Φ ≡ 1: P(t,T ) = F (t, r(t); T )
• pricing algorithm computationally efficient?
• ok: solving PDEs in less than three space dimensionsnumerically feasible, but . . .
• . . . nuisance: have to solve a PDE for every T > 0, just toget bond prices
• problem: term-structure calibration
⇒ short-rate models admitting closed-form solutions for bondprices favorable
Term-StructureModels
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Generalities
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Examples
Inverting theForward Curve
Affine Term-Structures
SomeStandardModels
Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Outline
16 Generalities
17 Diffusion Short-Rate ModelsExamplesInverting the Forward Curve
18 Affine Term-Structures
19 Some Standard ModelsVasicek ModelCIR ModelDothan ModelHo–Lee ModelHull–White Model
Term-StructureModels
DamirFilipovic
Generalities
DiffusionShort-RateModels
Examples
Inverting theForward Curve
Affine Term-Structures
SomeStandardModels
Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Examples
1 Vasicek [52]: Z = R,
dr(t) = (b + βr(t)) dt + σ dW ∗(t),
2 Cox–Ingersoll–Ross (henceforth CIR) [19]: Z = R+, b ≥ 0,
dr(t) = (b + βr(t)) dt + σ√
r(t) dW ∗(t),
3 Dothan [23]: Z = R+,
dr(t) = βr(t) dt + σr(t) dW ∗(t),
4 Black–Derman–Toy [7]: Z = R+,
dr(t) = β(t)r(t) dt + σ(t)r(t) dW ∗(t),
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Inverting theForward Curve
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Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Examples continued
5 Black–Karasinski [8]: Z = R+, `(t) = log r(t),
d`(t) = (b(t) + β(t)`(t)) dt + σ(t) dW ∗(t),
6 Ho–Lee [29]: Z = R,
dr(t) = b(t) dt + σ dW ∗(t),
7 Hull–White extended Vasicek [30]: Z = R,
dr(t) = (b(t) + β(t)r(t)) dt + σ(t) dW ∗(t),
8 Hull–White extended CIR [30]: Z = R+, b(t) ≥ 0,
dr(t) = (b(t) + β(t)r(t)) dt + σ(t)√
r(t) dW ∗(t).
Term-StructureModels
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Generalities
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Examples
Inverting theForward Curve
Affine Term-Structures
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Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Outline
16 Generalities
17 Diffusion Short-Rate ModelsExamplesInverting the Forward Curve
18 Affine Term-Structures
19 Some Standard ModelsVasicek ModelCIR ModelDothan ModelHo–Lee ModelHull–White Model
Term-StructureModels
DamirFilipovic
Generalities
DiffusionShort-RateModels
Examples
Inverting theForward Curve
Affine Term-Structures
SomeStandardModels
Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Inverse Problem
• specification of short-rate model parameters fully specifiesinitial term-structure T 7→ P(0,T ) = F (0, r(0); T ) andhence forward curve
• conversely: invert term-structure equation to match giveninitial forward curve
• example Vasicek model: P(0,T ) = F (0, r(0); T , b, β, σ)parameterized curve family with three degrees of freedomb, β, σ (for given (r(0))
• often too restrictive: poor fit of the current data
⇒ time-inhomogeneous short-rate models, such as theHull–White extensions: time-dependent parameters =infinite degree of freedom ⇒ perfect fit of any given curve
• usually functions b(t) etc. are fully determined by theinitial term-structure
• explicit examples in the following . . .
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Hull–WhiteModel
Outline
16 Generalities
17 Diffusion Short-Rate ModelsExamplesInverting the Forward Curve
18 Affine Term-Structures
19 Some Standard ModelsVasicek ModelCIR ModelDothan ModelHo–Lee ModelHull–White Model
− major drawback: explosion of the money-market account:
EQ[B(∆t)] = EQ
[e∫ ∆t
0 r(s) ds]≈ EQ
[e
r(0)+r(∆t)2
∆t]
• fact: EQ
[e eY]
=∞ for Gaussian Y
⇒ EQ[B(∆t)] =∞ for arbitrarily small ∆t
• similarly: Eurodollar future price = ∞ (see later)
• idea of lognormal rates taken up in mid-90s by Sandmannand Sondermann [46] and others → market models withlognormal LIBOR or swap rates (studied below)
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Generalities
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Examples
Inverting theForward Curve
Affine Term-Structures
SomeStandardModels
Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Outline
16 Generalities
17 Diffusion Short-Rate ModelsExamplesInverting the Forward Curve
18 Affine Term-Structures
19 Some Standard ModelsVasicek ModelCIR ModelDothan ModelHo–Lee ModelHull–White Model
Term-StructureModels
DamirFilipovic
Generalities
DiffusionShort-RateModels
Examples
Inverting theForward Curve
Affine Term-Structures
SomeStandardModels
Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Ho–Lee: dr = b(t) dt + σ dW ∗
• ATS equations (7)–(8) become
∂tA(t,T ) =σ2
2B2(t,T )− b(t)B(t,T ), A(T ,T ) = 0,
∂tB(t,T ) = −1, B(T ,T ) = 0
• explicit solution
B(t,T ) = T − t,
A(t,T ) = −σ2
6(T − t)3 +
∫ T
tb(s)(T − s) ds
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SomeStandardModels
Vasicek Model
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Ho–Lee: dr = b(t) dt + σ dW ∗
⇒ forward curve
f (t,T ) = ∂T A(t,T ) + ∂T B(t,T )r(t)
= −σ2
2(T − t)2 +
∫ T
tb(s) ds + r(t)
⇒ b(s) = ∂s f0(s) + σ2s gives a perfect fit of observed initialforward curve f0(T )
• plugging back into ATS:
f (t,T ) = f0(T )− f0(t) + σ2t(T − t) + r(t)
• integrate:
P(t,T ) = e−∫ Tt f0(s) ds+f0(t)(T−t)−σ
2
2t(T−t)2−(T−t)r(t)
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Ho–Lee Model
Hull–WhiteModel
Ho–Lee: dr = b(t) dt + σ dW ∗
• interesting:
r(t) = r(0)+
∫ t
0b(s) ds+σW ∗(t) = f0(t)+
σ2t2
2+σW ∗(t)
⇒ r(t) fluctuates along the modified initial forward curve
⇒ forward vs. future rate: f0(t) = EQ[r(t)]− σ2t2
2
Term-StructureModels
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Generalities
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Examples
Inverting theForward Curve
Affine Term-Structures
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Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Outline
16 Generalities
17 Diffusion Short-Rate ModelsExamplesInverting the Forward Curve
18 Affine Term-Structures
19 Some Standard ModelsVasicek ModelCIR ModelDothan ModelHo–Lee ModelHull–White Model
Term-StructureModels
DamirFilipovic
Generalities
DiffusionShort-RateModels
Examples
Inverting theForward Curve
Affine Term-Structures
SomeStandardModels
Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Hull–White:dr = (b(t) + βr) dt + σ dW ∗
• ATS equation for B(t,T ) as in Vasicek model:
∂tB(t,T ) = −βB(t,T )− 1, B(T ,T ) = 0
• explicit solution
B(t,T ) =1
β
(eβ(T−t) − 1
).
• ATS equation for A(t,T ):
A(t,T ) = −σ2
2
∫ T
tB2(s,T ) ds +
∫ T
tb(s)B(s,T ) ds
Term-StructureModels
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Examples
Inverting theForward Curve
Affine Term-Structures
SomeStandardModels
Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
Hull–WhiteModel
Hull–White:dr = (b(t) + βr) dt + σ dW ∗
• notice ∂T B(s,T ) = −∂sB(s,T ):
f0(T ) = ∂T A(0,T ) + ∂T B(0,T )r(0)
=σ2
2
∫ T
0∂sB2(s,T ) ds +
∫ T
0b(s)∂T B(s,T ) ds
+ ∂T B(0,T )r(0)
= − σ2
2β2
(eβT − 1
)2
︸ ︷︷ ︸=:g(T )
+
∫ T
0b(s)eβ(T−s) ds + eβT r(0)︸ ︷︷ ︸
=:φ(T )
.
• φ satisfies
∂Tφ(T ) = βφ(T ) + b(T ), φ(0) = r(0).
Term-StructureModels
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Examples
Inverting theForward Curve
Affine Term-Structures
SomeStandardModels
Vasicek Model
CIR Model
Dothan Model
Ho–Lee Model
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Hull–White:dr = (b(t) + βr) dt + σ dW ∗
• since φ = f0 + g : conclude
b(T ) = ∂Tφ(T )− βφ(T )
= ∂T (f0(T ) + g(T ))− β(f0(T ) + g(T )).
• plugging in (. . . ):
f (t,T ) = f0(T )− eβ(T−t)f0(t)
− σ2
2β2
(eβ(T−t) − 1
)(eβ(T−t) − eβ(T+t)
)+ eβ(T−t)r(t).
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Part V
Heath–Jarrow–Morton (HJM) Methodology
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Overview
• Have seen above: short-rate models not always flexibleenough to calibrating them to the observed initialterm-structure
• In late 1980s: Heath, Jarrow and Morton (henceforthHJM) [27] proposed a new framework for modeling theentire forward curve directly
• This chapter: provides essentials of HJM framework
Term-StructureModels
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Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Outline
20 Forward Curve Movements
21 Absence of Arbitrage
22 Implied Short-Rate Dynamics
23 HJM ModelsProportional Volatility
24 Fubini’s Theorem
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Outline
20 Forward Curve Movements
21 Absence of Arbitrage
22 Implied Short-Rate Dynamics
23 HJM ModelsProportional Volatility
24 Fubini’s Theorem
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Recap: Usual Stochastic Setup
• filtered probability space (Ω,F , (Ft)t≥0,P)
• usual conditions:• completeness: F0 contains all of the null sets• right-continuity: Ft = ∩s>tFs for t ≥ 0
• infinite time horizon (w.l.o.g.): F = F∞ = ∨t≥0Ft
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Assumptions
• given R- and Rd -valued stochastic process α = α(ω, t,T )and σ = (σ1(ω, t,T ), . . . , σd(ω, t,T )) s.t.
(HJM.1) α and σ are Prog ⊗ B-measurable
(HJM.2)∫ T
0
∫ T0 |α(s, t)| ds dt <∞ for all T
(HJM.3) sups,t≤T ‖σ(s, t)‖ <∞ for all T (note: thisis a ω-wise boundedness assumption)
• given integrable initial forward curve T 7→ f (0,T )
• for every T : forward rate follows Ito dynamics (t ≤ T )
f (t,T ) = f (0,T ) +
∫ t
0α(s,T ) ds +
∫ t
0σ(s,T ) dW (s) (10)
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Fubini’sTheorem
Assumptions
⇒ very general setup: only substantive economic restrictionis continuous sample paths assumption (and the finitenumber of random drivers W1, . . . ,Wd)
• integrals in (10) well defined by (HJM.1)–(HJM.3)
• from Fubini corollary below: implied short-rate process
r(t) = f (t, t) = f (0, t) +
∫ t
0α(s, t) ds +
∫ t
0σ(s, t) dW (s)
has progressive modification and satisfies∫ t
0 |r(s)| ds <∞a.s. for all t
• hence money-market account B(t) = e∫ t
0 r(s)ds well defined
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Properties
LemmaFor every maturity T , the zero-coupon bond price
P(t,T ) = e−∫ Tt f (t,u) du follows Ito process (t ≤ T )
P(t,T ) = P(0,T ) +
∫ t
0P(s,T ) (r(s) + b(s,T )) ds
+
∫ t
0P(s,T )v(s,T ) dW (s),
where
v(s,T ) = −∫ T
sσ(s, u) du,
is the T -bond volatility and
b(s,T ) = −∫ T
sα(s, u) du +
1
2‖v(s,T )‖2.
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Fubini’sTheorem
ProofUsing the classical and stochastic Fubini Theorem below twice,we calculate
logP(t,T ) = −∫ T
tf (t, u) du
= −∫ T
tf (0, u) du −
∫ T
t
∫ t
0α(s, u) ds du −
∫ T
t
∫ t
0σ(s, u) dW (s) du
= −∫ T
tf (0, u) du −
∫ t
0
∫ T
tα(s, u) du ds −
∫ t
0
∫ T
tσ(s, u) du dW (s)
= −∫ T
0f (0, u) du −
∫ t
0
∫ T
sα(s, u) du ds −
∫ t
0
∫ T
sσ(s, u) du dW (s)
+
∫ t
0f (0, u) du +
∫ t
0
∫ t
sα(s, u) du ds +
∫ t
0
∫ t
sσ(s, u) du dW (s)
= −∫ T
0f (0, u) du +
∫ t
0
(b(s,T )− 1
2‖v(s,T )‖2
)ds +
∫ t
0v(s,T ) dW (s)
+
∫ t
0
(f (0, u) +
∫ u
0α(s, u) ds +
∫ u
0σ(s, u) dW (s)
)︸ ︷︷ ︸
=r(u)
du
= log P(0,T ) +
∫ t
0
(r(s) + b(s,T )− 1
2‖v(s,T )‖2
)ds +
∫ t
0v(s,T ) dW (s).
Ito’s formula now implies the assertion.
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Fubini’sTheorem
Discounted Price
Corollary
For every maturity T , the discounted bond price processsatisfies
P(t,T )
B(t)= P(0,T ) +
∫ t
0
P(s,T )
B(s)b(s,T ) ds
+
∫ t
0
P(s,T )
B(s)v(s,T ) dW (s).
Term-StructureModels
DamirFilipovic
Forward CurveMovements
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ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Outline
20 Forward Curve Movements
21 Absence of Arbitrage
22 Implied Short-Rate Dynamics
23 HJM ModelsProportional Volatility
24 Fubini’s Theorem
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Absence of Arbitrage
• investigate restrictions on the dynamics (10) under noarbitrage
• assume given dQ/dP = E∞(γ •W ) for some γ ∈ L• Girsanov transformed Q-Brownian motion:
dW ∗ = dW − γ>dt
• Q an ELMM for the bond market if P(t,T )B(t) is a Q-local
martingale for every T
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HJM Drift Condition
Theorem (HJM Drift Condition)
Q is an ELMM if and only if
b(t,T ) = −v(t,T ) γ(t)> for all T , dP⊗ dt-a.s.
In this case, the Q-dynamics of the forward rates f (t,T ) are ofthe form
f (t,T ) = f (0,T )+
∫ t
0
(σ(s,T )
∫ T
sσ(s, u)>du
)︸ ︷︷ ︸
HJM drift
ds+
∫ t
0σ(s,T ) dW ∗(s),
and the discounted T -bond price satisfies
P(t,T )
B(t)= P(0,T )Et(v(·,T ) •W ∗)
for t ≤ T .
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Fubini’sTheorem
Proof I
In view of “Discounted Price” corollary we find that
dP(t,T )
B(t)=
P(t,T )
B(t)
(b(t,T ) + v(t,T ) γ(t)>
)dt
+P(t,T )
B(t)v(t,T ) dW ∗(t).
Hence P(t,T )B(t) , t ≤ T , is a Q-local martingale if and only if
b(t,T ) = −v(t,T ) γ(t)> dP⊗ dt-a.s. Since v(t,T ) andb(t,T ) are both continuous in T , we deduce that Q is anELMM if and only if b(t,T ) = −v(t,T ) γ(t)> for all T ,dP⊗ dt-a.s.Differentiating both sides in T yields
−α(t,T ) + σ(t,T )
∫ T
tσ(t, u)>du = σ(t,T ) γ(t)>
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Fubini’sTheorem
Proof II
for all T , dP⊗ dt-a.s. Insert this in (10). The expression forP(t,T )/B(t) now follows from Stochastic Exponential Lemma.
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Market Price of Risk
• follows from Theorem above:
dP(t,T ) = P(t,T )(
r(t)− v(t,T ) γ(t)>)
dt
+ P(t,T )v(t,T ) dW (t)
⇒ −γ = market price of risk for the bond market
• striking feature of HJM framework: Q-law of f (t,T ) onlydepends on volatility σ(t,T ), not on P-drift α(t,T )
⇒ option pricing only depends on σ, similar to Black–Scholesstock price model
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Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
When Is P(t,T )B(t) Q-martingale?
Corollary
Suppose HJM drift condition holds. Then Q is EMM if
1 Novikov condition EQ
[e
12
∫ T0 ‖v(t,T )‖2 dt
]<∞ for all T ; or
2 forward rates are nonnegative: f (t,T ) ≥ 0 for all t ≤ T .
Proof.Novikov condition is sufficient forP(t,T )B(t) = P(0,T )Et(v(·,T ) •W ∗) to be a Q-martingale.
If f (t,T ) ≥ 0, then 0 ≤ P(t,T )B(t) ≤ 1 is a uniformly bounded
local martingale, and hence a true martingale.
Term-StructureModels
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HJM Models
ProportionalVolatility
Fubini’sTheorem
Outline
20 Forward Curve Movements
21 Absence of Arbitrage
22 Implied Short-Rate Dynamics
23 HJM ModelsProportional Volatility
24 Fubini’s Theorem
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Simple Example
• interplay between short-rate models and HJM framework?
Suppose f (0,T ), α(t,T ) and σ(t,T ) are differentiable in T
with∫ T
0 |∂uf (0, u)| du <∞ and such that (HJM.1)–(HJM.3)are satisfied for α(t,T ) and σ(t,T ) replaced by ∂Tα(t,T ) and∂Tσ(t,T ).Then the short-rate process is an Ito process of the form
r(t) = r(0) +
∫ t
0ζ(u) du +
∫ t
0σ(u, u) dW (u)
where
ζ(u) = α(u, u)+∂uf (0, u)+
∫ u
0∂uα(s, u) ds+
∫ u
0∂uσ(s, u) dW (s).
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
ProofRecall first that
r(t) = f (t, t) = f (0, t) +
∫ t
0α(s, t) ds +
∫ t
0σ(s, t) dW (s).
Applying the Fubini Theorem 25.1 below to the stochasticintegral gives∫ t
0σ(s, t) dW (s) =
∫ t
0σ(s, s) dW (s) +
∫ t
0(σ(s, t)− σ(s, s)) dW (s)
=
∫ t
0σ(s, s) dW (s) +
∫ t
0
∫ t
s∂uσ(s, u) du dW (s)
=
∫ t
0σ(s, s) dW (s) +
∫ t
0
∫ u
0∂uσ(s, u) dW (s) du.
Moreover, from the classical Fubini Theorem we deduce in asimilar way that∫ t
0α(s, t) ds =
∫ t
0α(s, s) ds +
∫ t
0
∫ u
0∂uα(s, u) ds du,
and finally
f (0, t) = r(0) +
∫ t
0∂uf (0, u) du.
Combining these formulas, we obtain the desired result.
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Outline
20 Forward Curve Movements
21 Absence of Arbitrage
22 Implied Short-Rate Dynamics
23 HJM ModelsProportional Volatility
24 Fubini’s Theorem
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
HJM Models
• HJM model: σ(ω, t,T ) = σ(t,T , f (ω, t,T )) forappropriate function σ
• simplest choice: deterministic σ(t,T ) not depending on ω
⇒ Gaussian distributed forward rates f (t,T ) ⇒ simple bondoption price formulas (see later)
• assume: σ(t,T , f ) uniformly bounded, jointly continuous,and Lipschitz continuous in f
• assume: continuous initial forward curve f (0,T )
• then ∃ unique jointly continuous solution f (t,T ) of
df (t,T ) =
(σ(t,T , f (t,T ))
∫ T
tσ(t, u, f (t, u)) du
)dt
+ σ(t,T , f (t,T )) dW (t)
• remarkable: boundedness condition on σ cannot besubstantially weakened as following example shows . . .
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Outline
20 Forward Curve Movements
21 Absence of Arbitrage
22 Implied Short-Rate Dynamics
23 HJM ModelsProportional Volatility
24 Fubini’s Theorem
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Proportional Volatility
• single Brownian motion (d = 1)
• σ(t,T , f (t,T )) = σf (t,T ) for some constant σ > 0:positive and Lipschitz continuous but not bounded
• solution of HJM equation must satisfy
f (t,T ) = f (0,T )eσ2∫ t
0
∫ Ts f (s,u) du dseσW (t)−σ
2
2t (11)
• claim: there is no finite-valued solution to this expression(following Avellaneda and Laurence [2, Section 13.6])
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Proportional Volatility
• assume for simplicity: f (0,T ) ≡ 1 and σ = 1
• differentiating both sides of (11) in T :
∂T f (t,T ) = f (t,T )
∫ t
0f (s,T ) ds =
1
2∂t
(∫ t
0f (s,T ) ds
)2
• integrating from t = 0 to t = 1 and interchanging order ofdifferentiation and integration:
∂T
∫ 1
0f (s,T ) ds =
1
2
(∫ 1
0f (s,T ) ds
)2
• solving this differential equation path-wise forX (T ) =
∫ 10 f (s,T ) ds, T ≥ 1, we obtain as unique
solution
X (T ) =X (1)
1− X (1)2 (T − 1)
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Proportional Volatility
• since X (1) > 0: X (T ) ↑ ∞ for T ↑ τ where τ = 1 + 2X (1)
is a finite random time
• conclude: f (ω, t, τ(ω)) must become +∞ for some t ≤ 1,for almost all ω.
• nonexistence of HJM models with proportional volatilityencouraged development of LIBOR market models (seelater)
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Outline
20 Forward Curve Movements
21 Absence of Arbitrage
22 Implied Short-Rate Dynamics
23 HJM ModelsProportional Volatility
24 Fubini’s Theorem
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Fubini’s Theorem
Theorem (Fubini’s theorem for Stochastic Integrals)
Consider the Rd -valued stochastic process φ = φ(ω, t, s) withtwo indices, 0 ≤ t, s ≤ T , satisfying the following properties:
1 φ is ProgT ⊗ B[0,T ]-measurable;
2 supt,s ‖φ(t, s)‖ <∞.2
Then λ(t) =∫ T
0 φ(t, s) ds ∈ L, and there exists aFT ⊗ B[0,T ]-measurable modification ψ(s) of∫ T
0 φ(t, s) dW (t) with∫ T
0 ψ2(s) ds <∞ a.s.
Moreover,∫ T
0 ψ(s) ds =∫ T
0 λ(t) dW (t), that is,∫ T
0
(∫ T
0φ(t, s) dW (t)
)ds =
∫ T
0
(∫ T
0φ(t, s) ds
)dW (t).
2Note that this is a ω-wise boundedness assumption.
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Fubini’s Theorem: Proof
see course book Section 6.5
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Fubini’s Theorem: Corollary
Corollary
Let φ be as in Theorem 25.1. Then the process∫ s
0φ(t, s) dW (t), s ∈ [0,T ],
has a progressive modification π(s) with∫ T
0 π2(s) ds <∞ a.s.
Proof.For φ(ω, t, s) = K (ω, t)f (s), with bounded progressive processK and bounded measurable function f , the process∫ s
0φ(t, s) dW (t) = f (s)
∫ s
0K (t) dW (t)
is clearly progressive and path-wise square integrable. Now usea similar monotone class and localization argument as in theproof of Theorem 25.1 (→ exercise).
Term-StructureModels
DamirFilipovic
Forward CurveMovements
Absence ofArbitrage
ImpliedShort-RateDynamics
HJM Models
ProportionalVolatility
Fubini’sTheorem
Monotone Class Theorem
Theorem (Monotone Class Theorem)
Suppose the set H consists of real-valued bounded functionsdefined on a set Ω with the following properties:
1 H is a vector space;
2 H contains the constant function 1Ω;
3 if fn ∈ H and fn ↑ f monotone, for some bounded functionf on Ω, then f ∈ H.
If H contains a collection M of real-valued functions, which isclosed under multiplication (that is, f , g ∈M impliesfg ∈M). Then H contains all real-valued bounded functionsthat are measurable with respect to the σ-algebra which isgenerated by M (that is, σf −1(A) | A ∈ B, f ∈M).
Proof.see e.g. Steele [49, Section 12.6].
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Part VI
Forward Measures
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Overview
• we replace risk-free numeraire by another traded asset,such as the T -bond
• change of numeraire technique proves most useful foroption pricing and provides the basis for the marketmodels studied below
• we derive explicit option price formulas for Gaussian HJMmodels
• this includes the Vasicek short-rate model and someextension of the Black–Scholes model with stochasticinterest rates
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Outline
25 T -Bond as Numeraire
26 Bond Option PricingExample: Vasicek Short-Rate Model
27 Black–Scholes Model with Gaussian Interest RatesExample: Black–Scholes–Vasicek Model
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Outline
25 T -Bond as Numeraire
26 Bond Option PricingExample: Vasicek Short-Rate Model
27 Black–Scholes Model with Gaussian Interest RatesExample: Black–Scholes–Vasicek Model
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
T -Forward Measure
• HJM setup from above: ∃ EMM Q for all T -bonds, W ∗
the respective Q-Brownian motion
• recall: T -bond volatility v(t,T ) = −∫ Tt σ(t, u) du
• fix T > 0: dQT
dQ = 1P(0,T )B(T ) defines probability measure
QT ∼ Q on FT (why?) and for t ≤ T :
dQT
dQ|Ft = EQ
[dQT
dQ| Ft
]=
P(t,T )
P(0,T )B(t)
• QT is called the T -forward measure
• from above: dQT
dQ |Ft = Et (v(·,T ) •W ∗)
⇒ Girsanov’s Theorem: W T (t) = W ∗(t)−∫ t
0 v(s,T )>ds,t ≤ T , is a QT -Brownian motion
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Fundamental Property
LemmaFor any S > 0, the T -bond discounted S-bond price process
P(t, S)
P(t,T )=
P(0,S)
P(0,T )Et(σS,T •W T
), t ≤ S ∧ T
is a QT -martingale, where we define
σS ,T (t) = −σT ,S(t) = v(t, S)− v(t,T ) =
∫ T
Sσ(t, u) du.
(12)Moreover, the T - and S-forward measures are related by
dQS
dQT|Ft =
P(t, S) P(0,T )
P(t,T ) P(0, S)= Et
(σS,T •W T
), t ≤ S ∧ T .
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
ProofLet u ≤ t ≤ S ∧ T . Bayes’ rule gives
EQT
[P(t,S)
P(t,T )| Fu
]=
EQ
[P(t,T )
P(0,T )B(t)P(t,S)P(t,T ) | Fu
]P(u,T )
P(0,T )B(u)
=
P(u,S)B(u)
P(u,T )B(u)
=P(u, S)
P(u,T ),
which proves that P(t,S)/P(t,T ) is a martingale. Thestochastic exponential representation follows from StochasticExponential Lemma and the representation of P(t,T )/B(t)(→ exercise). The second claim follows from the identity
dQS
dQT|Ft =
dQS
dQ|Ft
dQdQT
|Ft .
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Pricing with Stochastic InterestRates
⇒ collection of EMMs: each QT corresponds to differentnumeraire: T -bond
• Q is called the risk-neutral (or spot) measure
• simpler pricing formulas: let X be a T -claim s.t.
EQ
[|X |
B(T )
]<∞
• arbitrage price at t ≤ T : π(t) = B(t)EQ
[X
B(T ) | Ft
]• computation: have to know joint distribution of 1/B(T )
and X ⇒ double integral (rather hard work)
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Pricing with Stochastic InterestRates
• if 1/B(T ) and X were independent under Q conditionalon Ft : π(t) = P(t,T )EQ [X | Ft ]
• a much simpler formula, since:• only have to compute single integral EQ[X | Ft ];• P(t,T ) observable at t, and does not have to be
computed within the model
• but: independence of 1/B(T ) and X unrealisticassumption for interest rate sensitive claims X !
• good news: above simple formula holds—under QT . . .
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Forward Measure Pricing
Proposition
Under above assumptions: EQT [|X |] <∞ and
π(t) = P(t,T )EQT [X | Ft ] .
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Proof
Proof.Bayes’ rule yields
EQT [|X |] = EQ
[|X |
P(0,T )B(T )
]<∞ (by assumption).
Moreover, again by Bayes’ rule,
π(t) = P(0,T )B(t)EQ
[X
P(0,T )B(T )| Ft
]= P(0,T )B(t)
P(t,T )
P(0,T )B(t)EQT [X | Ft ]
= P(t,T )EQT [X | Ft ] ,
as desired.
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Application: ExpectationHypothesis
Lemma (Expectation Hypothesis)
If σ(·,T ) ∈ L2, the expectation hypothesis holds under theT -forward measure:
f (t,T ) = EQT [r(T ) | Ft ] for t ≤ T .
Proof.Under QT we have
f (t,T ) = f (0,T ) +
∫ t
0σ(s,T ) dW T (s). (13)
Hence, if σ(·,T ) ∈ L2 then f (t,T ), t ≤ T , is aQT -martingale.
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
A Word of Warning
In view of equation (13) it is tempting to “specify” a forwardrate model by postulating the dynamics of f (·,T ) under QT
for each maturity T separately without reference to someunderlying Q. However, it is far from clear whether a commonrisk-neutral measure Q, tying all QT ’s, exists in this case. Onthe other hand, we note that this is exactly the approach in theLIBOR market model developed below. The importantdifference being that there one considers finitely manymaturities only.
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Application:Dybvig–Ingersoll–Ross Theorem
• Dybvig–Ingersoll–Ross [24]: long rates can never fall
• recall: zero-coupon yield R(t,T ) = 1T−t
∫ Tt f (t, s) ds
• define: asymptotic long rate R∞(t) = limT→∞ R(t,T )
Lemma (Dybvig–Ingersoll–Ross Theorem)
For all s < t the long rates satisfy R∞(s) ≤ R∞(t) if they exist.
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
ProofLet s < t be such that R∞(s) and R∞(t) exist. Then
p(u) = limT→∞ P(t,T )1T = e−R∞(u) exist for u ∈ s, t, and
it remains to prove that p(s) ≥ p(t).Under the t-forward measure Qt , we have
P(s,T )
P(s, t)= EQt [P(t,T ) | Fs ],
and thusp(s) = lim
T→∞EQt [P(t,T ) | Fs ]
1T .
Now let X ≥ 0 be any bounded random variable withEQt [X ] = 1. Using the Fs -conditional versions of Fatou’slemma, Holder’s inequality and dominated convergence, weobtain
EQt [X p(t)] = EQt
[lim inf
T→∞X P(t,T )
1T
]≤ EQt
[lim inf
T→∞EQt
[X P(t,T )
1T | Fs
]]≤ EQt
[lim inf
T→∞EQt
[X
TT−1 | Fs
]T−1T EQt [ P(t,T ) | Fs ]
1T
]= EQt [X p(s)] .
Since X was arbitrary with the stated properties, we concludethat p(t) ≤ p(s), and the lemma is proved.
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Outline
25 T -Bond as Numeraire
26 Bond Option PricingExample: Vasicek Short-Rate Model
27 Black–Scholes Model with Gaussian Interest RatesExample: Black–Scholes–Vasicek Model
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Bond Option Pricing: General
• consider European call option on S-bond, expiry dateT < S , strike price K
• ATM cap prices and Black volatilities for: t0 = 0 (today),T0 = 1/4 (first reset date), and Ti − Ti−1 ≡ 1/4,i = 1, . . . , 119 (maturity of the last cap is T119 = 30)
• contrast to market curve: Vasicek model cannot producehumped volatility curves
Figure: Vasicek ATM cap Black volatilities.
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Vasicek Model
Table: Vasicek ATM cap prices and Black volatilities
26 Bond Option PricingExample: Vasicek Short-Rate Model
27 Black–Scholes Model with Gaussian Interest RatesExample: Black–Scholes–Vasicek Model
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Generalized Black–Scholes
• Black–Scholes model [9]: stock S , money-market accountB following Q-dynamics
dB = Br dt, B(0) = 1,
dS = Sr dt + SΣ dW ∗, S(0) > 0,
• constant volatility Σ = (Σ1, . . . ,Σd) ∈ Rd
• new: r stochastic within above Gaussian HJM setup
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Generalized Black–Scholes
• consider European call option on S , maturity T , strikeprice K
• arbitrage price at t = 0 (for simplicity):
π = EQ
[1
B(T )(S(T )− K )+
]= EQ
[S(T )
B(T )1S(T )≥K
]− KEQ
[1
B(T )1S(T )≥K
]• recall T -forward measure QT
• similarly: choose S as numeraire and define EMMQ(S) ∼ Q on FT via
dQ(S)
dQ=
S(T )
S(0) B(T )= ET (Σ •W ∗)
• Girsanov: W (S)(t) = W ∗(t)− Σ>t, t ≤ T ,Q(S)-Brownian motion
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Generalized Black–Scholes
• change of measure (Bayes):π = S(0)Q(S)[S(T ) ≥ K ]− KP(0,T )QT [S(T ) ≥ K ]
• remains: compute probabilities
Q(S)[S(T ) ≥ K ] = Q(S)
[P(T ,T )
S(T )≤ 1
K
],
QT [S(T ) ≥ K ] = QT
[S(T )
P(T ,T )≥ K
]• observe: P(t,T )
S(t) is Q(S)-martingale and S(t)P(t,T ) is QT
martingale for t ≤ T
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Generalized Black–Scholes
• Ito’s formula (to practice stochastic calculus):
dS(t)
P(t)=
1
P(t)dS(t)− S(t)
P(t)2dP(t)
− 1
P(t)2d〈S ,P〉t +
S(t)
P(t)3d〈P,P〉t
= (· · · ) dt +S(t)
P(t)(Σ− v(t,T )) dW ∗(t)
(omitted parameter T in P(t,T ))
• recall: T -bond volatility v(t,T ) = −∫ Tt σ(t, u) du
• no need to compute drift term: volatility unaffected bychange of measure ⇒ QT -dynamics:
dS(t)
P(t,T )=
S(t)
P(t,T )(Σ− v(t,T )) dW T (t)
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Generalized Black–Scholes
⇒ stochastic exponential equation:
S(T )
P(T ,T )=
S(0)
P(0,T )ET(
(Σ− v(·,T )) •W T)
is lognormally distributed under QT
• along similar calculations (. . . ):
P(T ,T )
S(T )=
P(0,T )
S(0)ET(−(Σ− v(·,T )) •W (S)
)is lognormally distributed under Q(S)
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Generalized Black–Scholes OptionPrice Formula
Proposition
In the above generalized Black–Scholes model, the option priceis
π = S(0)Φ[d1]− KP(0,T )Φ[d2],
where Φ is the standard Gaussian cumulative distributionfunction and
d1,2 =log[
S(0)KP(0,T )
]± 1
2
∫ T0 ‖Σ− v(t,T )‖2 dt√∫ T
0 ‖Σ− v(t,T )‖2 dt. (14)
Note that v(t,T ) = 0 yields the classical Black–Scholes optionprice formula for constant short rate.
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Outline
25 T -Bond as Numeraire
26 Bond Option PricingExample: Vasicek Short-Rate Model
27 Black–Scholes Model with Gaussian Interest RatesExample: Black–Scholes–Vasicek Model
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Black–Scholes–Vasicek
• special case of generalized Black–Scholes formula: Vasicekshort-rate model
• dim W ∗ = d = 2
• Vasicek short-rate dynamics:
dr = (b + βr) dt + σ dW ∗,
where σ = (σ1, σ2)
• note: this corresponds to standard representationdr = (b + βr) dt + ‖σ‖ dW∗ for the one-dimensional
Q-Brownian motion W∗ =σ1 W ∗1 +σ2 W ∗2
‖σ‖
⇒ R2-valued T -bond volatility
v(t,T ) = −σ∫ T
teβ(T−s) ds =
σ
β
(1− eβ(T−t)
)
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Black–Scholes–Vasicek
• tedious but elementary computation yields∫ T
0‖Σ− v(t,T )‖2 dt
= ‖Σ‖2 T + 2Σσ>eβT − 1− βT
β2
+ ‖σ‖2 e2βT − 4eβT + 2βT + 3
2β3
for aggregate volatility in (14)
• relation between option price π and instantaneouscovariation d〈S , r〉/dt = Σσ> of S and r : π monotone
increasing in∫ T
0 ‖Σ− v(t,T )‖2 dt, which again isincreasing in Σσ> since eβT − 1− βT > 0
Term-StructureModels
DamirFilipovic
T -Bond asNumeraire
Bond OptionPricing
Example:VasicekShort-RateModel
Black–ScholesModel withGaussianInterest Rates
Example: Black–Scholes–VasicekModel
Black–Scholes–Vasicek
⇒ π increases with increasing covariation between S and r
• for negative covariation, π may be smaller than classicalBlack–Scholes option price with constant short rates(σ = 0)
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Part VII
Forwards and Futures
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Overview
• we discuss two common types of term contracts:
• forwards: mainly traded over the counter (OTC)
• futures: actively traded on many exchanges
• underlying in both cases: a T -claim Y, e.g. exchange rate,interest rate, commodity such as copper, any traded ornon-traded asset, an index, etc.
• special discussion: interest rate futures, futures rates andforward rates in the Gaussian HJM model
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Outline
28 Forward Contracts
29 Futures ContractsInterest Rate Futures
30 Forward vs. Futures in a Gaussian Setup
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Outline
28 Forward Contracts
29 Futures ContractsInterest Rate Futures
30 Forward vs. Futures in a Gaussian Setup
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Forward Contracts
• assume HJM setup, and let Y denote a T -claim
• forward contract on Y, contracted at t, with time ofdelivery T > t and forward price f (t; T ,Y) defined byfollowing payment scheme:
• at T : holder (long position) pays f (t; T ,Y) and receivesY from underwriter (short position)
• at t: forward price chosen such that present value offorward contract is zero:
EQ
[e−
∫ Tt
r(s) ds (Y − f (t; T ,Y)) | Ft
]= 0
• this is equivalent to
f (t; T ,Y) =1
P(t,T )EQ
[e−
∫ Tt r(s) dsY | Ft
]= EQT [Y | Ft ]
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Forward Contracts
• examples: the forward price at t of:
1 a dollar delivered at T is 12 an S-bond delivered at T ≤ S is P(t,S)
P(t,T )
3 any traded asset S delivered at T is S(t)P(t,T )
• note: forward price f (s; T ,Y) has to be distinguished from(spot) price at time s of the forward contract entered attime t ≤ s, which is
EQ
[e−
∫ Ts r(u) du (Y − f (t; T ,Y)) | Fs
]= P(s,T ) (f (s; T ,Y)− f (t; T ,Y))
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Outline
28 Forward Contracts
29 Futures ContractsInterest Rate Futures
30 Forward vs. Futures in a Gaussian Setup
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Futures Contracts
• futures contract on Y with time of delivery T defined as:• at every t ≤ T : there is a market quoted futures price
F (t; T ,Y), which makes the futures contract on Y, ifentered at t, equal to zero
• at T : holder (long position) pays F (T ; T ,Y) and receivesY from underwriter (short position)
• marking to market or resettlement: during any infinitesimaltime interval (t, t + ∆t]: holder receives (or pays, ifnegative) F (t; T ,Y)− F (t + ∆t; T ,Y)
⇒ continuous cash flow between the two parties of a futurescontract, they are required to keep certain amount ofmoney as safety margin
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Futures Contracts
• trading volumes in futures are huge
• one of reasons: often difficult to trade/hedge directly inunderlying object (e.g. an index including illiquidinstruments, or a commodity such as copper, gas orelectricity)
• holding short position in futures: no need to physicallydeliver the underlying object if you exit contract beforedelivery date
• selling short makes it possible to hedge against theunderlying
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Futures Price Process
• Suppose EQ[|Y|] <∞, then futures price process =Q-martingale:
F (t; T ,Y) = EQ [Y | Ft ]
• consequence: forward = futures price if interest ratesdeterministic
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Heuristic Argument
• w.l.o.g F (t) = F (t; T ,Y) is an Ito process
• cumulative discounted cash flow of futures contract in(t,T ]: V = limN VN with
VN =N∑
i=1
1
B(ti )(F (ti )− F (ti−1))
limit over sequence of partitions t = t0 < · · · < tN = Twith maxi |ti − ti−1| → 0 for N →∞
• can rewrite
VN =N∑
i=1
1
B(ti−1)(F (ti )− F (ti−1))
+N∑
i=1
(1
B(ti )− 1
B(ti−1)
)(F (ti )− F (ti−1))
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Heuristic Argument
• B continuous ⇒ 1/B ∈ L(F )
• elementary stochastic calculus: VN → V in probabilitywith
V =
∫ T
t
1
B(s)dF (s) +
∫ T
td
⟨1
B,F
⟩s
=
∫ T
t
1
B(s)dF (s)
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Heuristic Argument
• present value = zero:
EQ
[∫ T
t
1
B(s)dF (s) | Ft
]= 0
• consequence:
M(t) =
∫ t
0
1
B(s)dF (s) = EQ
[∫ T
0
1
B(s)dF (s) | Ft
]is Q-martingale, t ≤ T
• assume: EQ
[∫ T0 B(s)2 d〈M,M〉s
]= EQ [〈F ,F 〉T ] <∞
⇒ B ∈ L2(M) and
F (t) =
∫ t
0B(s) dM(s)
is Q-martingale for t ≤ T , as desired
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Outline
28 Forward Contracts
29 Futures ContractsInterest Rate Futures
30 Forward vs. Futures in a Gaussian Setup
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Eurodollar Futures
• Interest rate futures: can be divided into futures onshort-term instruments and futures on coupon bonds
• here: we only consider an example from the first group
• Eurodollars = deposits of US dollars in institutions outsideof the US
• LIBOR is the interbank rate of interest for Eurodollar loans
• Eurodollar futures contract is tied to LIBOR
• introduced by the International Money Market (IMM) ofthe Chicago Mercantile Exchange (CME) in 1981
• designed to protect its owner from fluctuations in the3-month (=1/4 year) LIBOR
• maturity (delivery) months are March, June, Septemberand December
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Formal Definition
• fix maturity T
• L(T ) = 3-month spot LIBOR for period [T ,T + 1/4]
• market quote of Eurodollar futures contract on L(T ) att ≤ T is
1− LF (t,T ) [100 per cent]
where LF (t,T ) = corresponding futures rate (comparewith bootstrapping example above)
• futures price, used for the marking to market, defined by
F (t; T , L(T )) = 1− 1
4LF (t,T ) [million dollars]
⇒ change of 1 basis point (0.01%) in futures rate LF (t,T )leads to cash flow of 106 × 10−4 × 1
4 = 25 [dollars]
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Formal Definition
• definition: LF (T ,T ) = L(T )
⇒ final price F (T ; T , L(T )) = 1− 14 L(T ) = Y: underlying Y
is a synthetic value, no physical delivery at maturity,settlement is made in cash
• since F (t; T , L(T )) = EQ [F (T ; T , L(T )) | Ft ] we obtainexplicit formula for the futures rate
LF (t,T ) = EQ [L(T ) | Ft ]
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
A Mathematicians Intuition
• underlying Y = P(T ,T + 1/4)
• futures price = EQ [P(T ,T + 1/4) | Ft ]
• exact: P(T ,T + 1/4) = 1− 14 L(T )P(T ,T + 1/4)
• approximate: P(T ,T + 1/4) ≈ 1− 14 L(T )
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Outline
28 Forward Contracts
29 Futures ContractsInterest Rate Futures
30 Forward vs. Futures in a Gaussian Setup
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Forward vs. Futures
• let S be price process of a traded asset with Q-dynamics
dS(t)
S(t)= r(t) dt + ρ(t) dW ∗(t)
• fix delivery date T
• forward price of S for delivery at T :
f (t; T ,S(T )) =S(t)
P(t,T )
• futures price of S for delivery at T
F (t; T ,S(T )) = EQ[S(T ) | Ft ]
• aim: establish relationship between the two prices underGaussian assumption
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Forward vs. Futures
Proposition
Suppose ρ(t) and T -bond volatility v(t,T ) are deterministic.Then
F (t; T , S(T )) = f (t; T ,S(T )) e∫ Tt (v(s,T )−ρ(s)) v(s,T )>ds
for t ≤ T .
Hence, if the instantaneous covariation of S(t) and P(t,T ) isnegative,
d〈S ,P(·,T )〉tdt
= S(t)P(t,T ) ρ(t) v(t,T )> ≤ 0,
then the futures price dominates the forward price.
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Proof
Proof.Write µ(s) = v(s,T )− ρ(s). It is clear that
f (t; T ,S(T )) =S(0)
P(0,T )Et(µ•W ∗) exp
(∫ t
0µ(s) v(s,T )>ds
).
Since E(µ •W ∗) is a Q-martingale and ρ(s) and v(s,T ) aredeterministic, we obtain
F (t; T ,S(T )) = EQ[f (T ; T , S(T )) | Ft ]
= f (t; T ,S(T )) e∫ Tt µ(s) v(s,T )>ds ,
as desired.
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Forward vs. Futures Rates
Lemma (Convexity Adjustments)
Assume Gaussian HJM framework. Then relation betweeninstantaneous forward and futures rates:
f (t,T ) = EQ[r(T ) | Ft ]−∫ T
t
(σ(s,T )
∫ T
sσ(s, u)>du
)ds
and simple forward and futures rates:
F (t; T ,S) = EQ[F (T ,S) | Ft ]
− P(t,T )
(S − T )P(t, S)
(e∫ Tt
(∫ ST σ(s,v) dv
∫ Ss σ(s,u)> du
)ds − 1
)
Term-StructureModels
DamirFilipovic
ForwardContracts
FuturesContracts
Interest RateFutures
Forwardvs. Futures ina GaussianSetup
Forward vs. Futures Rates
Hence, if
σ(s, v)σ(s, u)> ≥ 0 for all s ≤ u ∧ v
then futures rates are always greater than the correspondingforward rates.
Proof.Exercise
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Part VIII
Consistent Term-Structure Parametrizations
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Overview
• practitioners and academics have vital interest inparameterized term-structure models
• in this chapter: take up a point left open at the end ofestimation chapter: exploit whether parameterized curvefamilies φ(·, z), used for estimating the forward curve, gowell with arbitrage-free interest rate models
• recall BIS document [5]: rich source of cross-sectionaldata (daily estimations of the parameter z) for theNelson–Siegel and Svensson families
• suggests that calibrating a diffusion process Z for theparameter z would lead to an accurate factor model forthe forward curve
• conditions for absence of arbitrage can be formulated interms of the drift and diffusion of Z and derivatives of φ
• these conditions turn out to be surprisingly restrictive insome cases
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Outline
31 Multi-factor Models
32 Consistency Condition
33 Affine Term-Structures
34 Polynomial Term-StructuresSpecial Case: m = 1General Case: m ≥ 1
35 Exponential–Polynomial FamiliesNelson–Siegel FamilySvensson Family
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Outline
31 Multi-factor Models
32 Consistency Condition
33 Affine Term-Structures
34 Polynomial Term-StructuresSpecial Case: m = 1General Case: m ≥ 1
35 Exponential–Polynomial FamiliesNelson–Siegel FamilySvensson Family
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Short Rate Models Revisited
• seen earlier: every time-homogeneous diffusion short-ratemodel r(t) induces forward rates of the form
− unrealistic implications: e.g. family of attainable forwardcurves φ(·, r) | r ∈ R is only one-dimensional
→ term-structure movements explained by single statevariable r(t): conflicts with above principal componentanalysis (2-3 factors needed for statistically accuratedescription of forward curve movements)
− maturity-specific risk: if d = 1 e.g. a bond option withmaturity five years could be perfectly hedged by themoney-market account and a bond of maturity 30 years
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Multi-factor Models
• to gain more flexibility: multiple factors m ≥ 1
• m-factor model := interest rate model of the form
f (t,T ) = φ(T − t,Z (t))
• deterministic φ
• m-dim state space process Z
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Assumptions
• state space Z ⊂ Rm closed with non-empty interior
• φ ∈ C 1,2(R+ ×Z)
• b : Z → Rm continuous
• ρ : Z → Rm×d measurable, s.t. diffusion matrixa(z) = ρ(z)ρ(z)> continuous in z ∈ Z
• W ∗: d-dim Brownian motion on (Ω,F , (Ft),Q)
• ∀z ∈ Z ∃ unique Z-valued solution Z = Z z of
dZ (t) = b(Z (t)) dt + ρ(Z (t)) dW ∗(t)
Z (0) = z
• NA: Q is risk-neutral measure for bond prices
P(t,T ) = exp
(−∫ T−t
0φ(x ,Z z(t)) dx
)for all z ∈ Z
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Remark
Time-inhomogeneous models are included by identifying onecomponent, say Z1, with calendar time. We therefore setdZ1 = dt, which is equivalent to b1 ≡ 1 and ρ1j ≡ 0 forj = 1, . . . , d . Calendar time at inception is now Z1(0) = z1,and t, T , etc. accordingly denote relative time with respect toz1. The NA assumption for all z ∈ Z now means, in particular,that absence of arbitrage holds relative to any initial calendartime z1.
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
First Consequences
• short rates given by r(t) = φ(0,Z (t))
⇒ NA holds if and only if
exp(−∫ T−t
0 φ(x ,Z z(t)) dx)
exp(∫ t
0 φ(0,Z z(s)) ds) , t ≤ T ,
is a Q-local martingale, for all z ∈ Z• aim: find consistency condition for a, b and φ such that
NA holds
• possible way: apply Ito’s formula and set drift = 0
• our way: first embed in HJM framework and use HJMdrift condition . . .
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Outline
31 Multi-factor Models
32 Consistency Condition
33 Affine Term-Structures
34 Polynomial Term-StructuresSpecial Case: m = 1General Case: m ≥ 1
35 Exponential–Polynomial FamiliesNelson–Siegel FamilySvensson Family
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Consistency Condition: Derivation
apply Ito’s formula to f (t,T ) = φ(T − t,Z (t)):
df (t,T ) =(− ∂xφ(T − t,Z (t)) +
m∑i=1
bi (Z (t))∂ziφ(T − t,Z (t))
+1
2
m∑i ,j=1
aij(Z (t))∂zi∂zjφ(T − t,Z (t)))
dt
+m∑
i=1
d∑j=1
∂ziφ(T − t,Z (t))ρij(Z (t)) dW ∗j (t).
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Consistency Condition: Derivation
⇒ induced forward rate model is of the HJM type with
• to hold a.s. for all t ≤ T and z = Z (0), now let t → 0 . . .
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Consistency Condition
Proposition (Consistency Condition)
NA holds if and only if
∂xΦ(x , z) = φ(0, z) +m∑
i=1
bi (z)∂zi Φ(x , z)
+1
2
m∑i ,j=1
aij(z)(∂zi∂zj Φ(x , z)− ∂zi Φ(x , z)∂zj Φ(x , z)
)(15)
for all (x , z) ∈ R+ ×Z.
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Terminology
DefinitionThe pair of characteristics a, b and the forward curveparametrization φ are consistent if NA, or equivalently theabove consistency condition (15), holds.
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Interpretations of (15)
• pricing: take φ(0, z), a, b as given and solve PDE (15)with initial condition Φ(0, z) = 0
• inverse problem: given parametric estimation method φgiven, find a and b such that (15) is satisfied for all (x , z)
• it turns out that the latter approach is quite restrictive onpossible choices of a and b . . .
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
First Consequence
Proposition
Suppose that the functions
∂zi Φ(·, z) and1
2
(∂zi∂zj Φ(·, z)− ∂zi Φ(·, z)∂zj Φ(·, z)
),
for 1 ≤ i ≤ j ≤ m, are linearly independent for all z in somedense subset D ⊂ Z. Then there exists one and only oneconsistent pair a, b.
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Proof
Proof.Set M = m + m(m + 1)/2, the number of unknown functionsbk and akl = alk . Let z ∈ D. Then there exists a sequence0 ≤ x1 < · · · < xM such that the M ×M-matrix with kth rowvector built by
∂zi Φ(xk , z) and1
2
(∂zi∂zj Φ(xk , z)− ∂zi Φ(xk , z)∂zj Φ(xk , z)
),
for 1 ≤ i ≤ j ≤ m, is invertible. Thus, b(z) and a(z) areuniquely determined by (15). This holds for each z ∈ D. Bycontinuity of b and a hence for all z ∈ Z.
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Practical Implications
• suppose parameterized curve family φ(·, z) | z ∈ Z usedfor daily forward curve estimation in terms of statevariable z .
• above proposition: any consistent Q-diffusion model Z forz is fully determined by φ
• moreover: diffusion matrix a(z) of Z not affected byequivalent measure transformation
⇒ statistical calibration only possible for drift of the model(or equivalently, for the market price of risk), sinceobservations of z are made under objective measureP ∼ Q and dQ/dP is left unspecified by our consistencyconsiderations
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Outline
31 Multi-factor Models
32 Consistency Condition
33 Affine Term-Structures
34 Polynomial Term-StructuresSpecial Case: m = 1General Case: m ≥ 1
35 Exponential–Polynomial FamiliesNelson–Siegel FamilySvensson Family
⇒ can invert and solve the linear equation (16) for a and b(as in proof of above proposition)
• left-hand side of (16) affine in z ⇒ a, b affine:
aij(z) = aij +m∑
k=1
αk;ijzk ,
bi (z) = bi +m∑
j=1
βijzj ,
for parameters aij , αk;ij , bi and βij
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Affine Term-Structure (ATS)
plugging back into (16) and matching terms ⇒ Riccatiequations
∂xG0(x) = g0(0) +m∑
i=1
biGi (x)− 1
2
m∑i ,j=1
aijGi (x)Gj(x) (17)
∂xGk(x) = gk(0) +m∑
i=1
βkiGi (x)− 1
2
m∑i ,j=1
αk;ijGi (x)Gj(x)
(18)
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Summary
Proposition
If a, b is consistent with above ATS, then a and b are affine,and Gi solve system of Riccati equations (17)–(18) with initialconditions Gi (0) = 0.Conversely, suppose a and b are affine, and let gi (0) be somegiven constants. If the functions Gi solve the system of Riccatiequations (17)–(18) with initial conditions Gi (0) = 0, then theabove ATS is consistent with a, b.
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Discussion
• this proposition extends earlier result ontime-homogeneous affine short-rate models with
A(t,T ) = G0(T − t) and B(t,T ) = G1(T − t)
• note 1: we did not have to assume linear independence ofB and B2: this assumption becomes necessary as soon asm ≥ 2 (→ exercise)
• note 2: we have freedom to choose constants gi (0) relatedto short rates by
• typical choice: g1(0) = 1 and all other gi (0) = 0 ⇒ Z1(t)is (non-Markovian) short-rate process
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Outline
31 Multi-factor Models
32 Consistency Condition
33 Affine Term-Structures
34 Polynomial Term-StructuresSpecial Case: m = 1General Case: m ≥ 1
35 Exponential–Polynomial FamiliesNelson–Siegel FamilySvensson Family
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Polynomial Term-Structure (PTS)
• polynomial term-structure (PTS) φ(x , z) =∑n|i|=0 gi(x) z i
• multi-index notation: i = (i1, . . . , im), |i| = i1 + · · ·+ imand z i = z i1
1 · · · z imm
• n = degree of the PTS: maximal k with gi 6= 0 for some|i| = k
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Maximal Degree Problem
• n = 1: ATS
• n = 2: quadratic term-structure (QTS), intensively studiedin the literature (e.g. Ahn et al. [1])
• question: do we gain something by looking at n = 3 andhigher-degree PTS models?
• surprising answer: no
• we show that there is no consistent PTS for n > 2 . . .
• for simplicity m = 1 only (for general case see course bookSection 9.4.2)
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Outline
31 Multi-factor Models
32 Consistency Condition
33 Affine Term-Structures
34 Polynomial Term-StructuresSpecial Case: m = 1General Case: m ≥ 1
35 Exponential–Polynomial FamiliesNelson–Siegel FamilySvensson Family
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Maximal Degree Problem I
• special case m = 1: PTS now reads
φ(x , z) =n∑
i=0
gi (x) z i
• define Gi (x) =∫ x
0 gi (u) du
Theorem (Maximal Degree Problem I)
Suppose that Gi and GiGj are linearly independent functions,for 1 ≤ i ≤ j ≤ n, and that ρ 6≡ 0.Then consistency implies n ∈ 1, 2. Moreover, b(z) and a(z)are polynomials in z with deg b(z) ≤ 1 in any case (QTS andATS), and deg a(z) = 0 if n = 2 (QTS) and deg a(z) ≤ 1 ifn = 1 (ATS).
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Proof
Equation (15) can be rewritten
n∑i=0
(gi (x)− gi (0)) z i =n∑
i=0
Gi (x)Bi (z)−n∑
i ,j=0
Gi (x)Gj(x)Aij(z)
where we define
Bi (z) = b(z)iz i−1 +1
2a(z)i(i − 1)z i−2,
Aij(z) =1
2a(z)ijz i−1z j−1.
By assumption we can solve above linear equation for B and A,and thus Bi (z) and Aij(z) are polynomials in z of order lessthan or equal n.
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Proof cont’d
In particular, this holds for
B1(z) = b(z) and 2A11(z) = a(z).
But then, since a 6≡ 0 by assumption, 2Ann(z) = a(z)n2z2n−2
cannot be a polynomial of order less than or equal n unless2n − 2 ≤ n, which implies n ≤ 2. The theorem is thus provedfor n = 1. For n = 2, we obtain deg a(z) = 0 and thusB2(z) = 2b(z)z + a(z). Hence also in this case deg b(z) ≤ 1,and the theorem is proved for m = 1.
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Maximal Degree Problem IIRelax linear independence hypothesis on Gi , GiGj :
Theorem (Maximal Degree Problem II)
Suppose that:
1 supZ =∞;
2 b and ρ satisfy a linear growth condition
|b(z)|+ |ρ(z)| ≤ C (1 + |z |), z ∈ Z,
for some finite constant C;
3 a(z) is asymptotically bounded away from zero:
lim infz→∞
a(z) > 0.
Then consistency implies n ∈ 1, 2.Note: linear growth condition 2 is standard assumption fornon-explosion of Z
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Proof
Again, we consider equation (15), which reads
n∑i=0
(gi (x)− gi (0)) z i = b(z)n∑
i=0
Gi (x)iz i−1
+1
2a(z)
n∑i ,j=0
Gi (x)i(i − 1)z i−2 −
(n∑
i=0
Gi (x)iz i−1
)2 .
(19)
We argue by contradiction and assume that n > 2, whichimplies 2n − 2 > n. Dividing (19) by z2n−2, for z 6= 0, yields. . .
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Proof cont’d
1
2a(z)
(∑ni=0 Gi (x)iz i−1
)2
z2n−2=
b(z)
z
∑ni=0 Gi (x)iz i−1
z2n−3
+a(z)
2z2
∑ni ,j=0 Gi (x)i(i − 1)z i−2
z2n−4−∑n
i=0 (gi (x)− gi (0)) z i
z2n−2.
By assumption 1 this holds for all z large enough. Theright-hand side converges to zero, for z →∞, by assumption 2.Taking the lim inf of the left-hand side yields by 3, that
1
2lim infz→∞
a(z)G 2n (x)n2 > 0,
a contradiction. Thus n ≤ 2.
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Outline
31 Multi-factor Models
32 Consistency Condition
33 Affine Term-Structures
34 Polynomial Term-StructuresSpecial Case: m = 1General Case: m ≥ 1
35 Exponential–Polynomial FamiliesNelson–Siegel FamilySvensson Family
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Maximal Degree Problem I
Theorem (Maximal Degree Problem I)
Suppose that Giµ and GiµGiν are linearly independent functions,1 ≤ µ ≤ ν ≤ N, and that ρ 6≡ 0.Then consistency implies n ∈ 1, 2. Moreover, b(z) and a(z)are polynomials in z with deg b(z) ≤ 1 in any case (QTS andATS), and deg a(z) = 0 if n = 2 (QTS) and deg a(z) ≤ 1 ifn = 1 (ATS).
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Maximal Degree Problem II
Theorem (Maximal Degree Problem II)
Suppose that:
1 Z is a cone;
2 b and ρ satisfy a linear growth condition
‖b(z)‖+ ‖ρ(z)‖ ≤ C (1 + ‖z‖), z ∈ Z,
for some finite constant C;3 a(z) becomes uniformly elliptic for ‖z‖ large enough:
〈a(z)v , v〉 ≥ k(z)‖v‖2, v ∈ Rm,
for some function k : Z → R+ with
lim infz∈Z,‖z‖→∞
k(z) > 0.
Then consistency implies n ∈ 1, 2.Note: linear growth condition 2 is standard assumption fornon-explosion of Z
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Outline
31 Multi-factor Models
32 Consistency Condition
33 Affine Term-Structures
34 Polynomial Term-StructuresSpecial Case: m = 1General Case: m ≥ 1
35 Exponential–Polynomial FamiliesNelson–Siegel FamilySvensson Family
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Outline
31 Multi-factor Models
32 Consistency Condition
33 Affine Term-Structures
34 Polynomial Term-StructuresSpecial Case: m = 1General Case: m ≥ 1
35 Exponential–Polynomial FamiliesNelson–Siegel FamilySvensson Family
Only a22(z) is left as positive candidate among the componentsof a(z). The remaining terms are
q2(x) = (b3(z) + z3z5)x + b2(z)− z3 −a22(z)
z5+ z2z5,
q3(x) = (b4(z) + z4z6)x − z4,
q4(x) = a22(z)1
z5,
while q1 = q5 = q6 = 0.
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Proof cont’dIf 2z5 6= z6 then also a22(z) = 0. If 2z5 = z6 then the conditionq3 + q4 = q2 = 0 leads to
a22(z) = z4z5,
b2(z) = z3 + z4 − 25z2,
b3(z) = −z5z3,
b4(z) = −2z5z4.
We derived the above results under the assumption (21). Butthe set of z where (21) holds is dense Z. By continuity of a(z)and b(z) in z , the above results thus extend for all z ∈ Z. Inparticular, all Zi ’s but Z2 are deterministic; Z1, Z5 and Z6 areeven constant.
Term-StructureModels
DamirFilipovic
Multi-factorModels
ConsistencyCondition
Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Proof cont’dThus, since
a(z) = 0 if 2z5 6= z6,
we only have a non-trivial process Z if
Z6(t) ≡ 2Z5(t) ≡ 2Z5(0). (22)
In that case we have, writing for short zi = Zi (0),
Z1(t) ≡ z1,
Z3(t) = z3e−z5t ,
Z4(t) = z4e−2z5t
(23)
and
dZ2(t) =(z3e−z5t + z4e−2z5t − z5Z2(t)
)dt+
d∑j=1
ρ2j(t) dW ∗j (t),
(24)where ρ2j(t) (not necessarily deterministic) are such that
d∑j=1
ρ22j(t) = a22(Z (t)) = z4z5e−2z5t .
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DamirFilipovic
Multi-factorModels
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Affine Term-Structures
PolynomialTerm-Structures
Special Case:m = 1
General Case:m ≥ 1
Exponential–PolynomialFamilies
Nelson–SiegelFamily
Svensson Family
Proof cont’dBy Levy’s characterization theorem we have that
W∗(t) =d∑
j=1
∫ t
0
ρ2j(s)√
z4z5e−z5sdW ∗
j (s)
is a real-valued standard Brownian motion. Hence thecorresponding short-rate process
r(t) = φS(0,Z (t)) = z1 + Z2(t)
satisfies
dr(t) =(z1z5 + z3e−z5t + z4e−2z5t − z5r(t)
)dt+√
z4z5e−z5t dW∗(t).
Hence the proposition is proved.
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
DefinitionWe call X affine if the Ft-conditional characteristic function ofX (T ) is exponential affine in X (t): there exist C- andCd -valued functions φ(t, u) and ψ(t, u), respectively, withjointly continuous t-derivatives such that X = X x satisfies
E[eu>X (T ) | Ft
]= eφ(T−t, u)+ψ(T−t, u)>X (t) (25)
for all u ∈ iRd , t ≤ T and x ∈ X .
• Re (φ(T − t, u) + ψ(T − t, u)>X (t)) ≤ 0
• φ, ψ uniquely determined by (25) with φ(0, u) = 0,ψ(0, u) = u
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
TheoremConversely, suppose the diffusion matrix a(x) and drift b(x) areaffine and suppose there exists a solution (φ, ψ) of the Riccatiequations (26) such that Re (φ(t, u) + ψ(t, u)>x) ≤ 0 for allt ≥ 0, u ∈ iRd and x ∈ X . Then X is affine with conditionalcharacteristic function (25).
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
Letting t → 0, by continuity of the parameters, we thus obtain
∂Tφ(T , u) + ∂Tψ(T , u)>x
= ψ(T , u)>b(x) +1
2ψ(T , u)>a(x)ψ(T , u)
for all x ∈ X , T ≥ 0, u ∈ iRd . Since ψ(0, u) = u, this impliesthat a and b are affine of the stated form. Plugging this backinto the above equation and separating first-order terms in xyields the Riccati equations (26).
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
Conversely, suppose a and b are affine. Let (φ, ψ) be a solutionof the Riccati equations (26) such that φ(t, u) + ψ(t, u)>x hasa nonpositive real part for all t ≥ 0 , u ∈ iRd and x ∈ X . ThenM, defined as above, is a uniformly bounded (substantial!)local martingale, and hence a martingale, withM(T ) = eu>X (T ). Therefore E[M(T ) | Ft ] = M(t), for allt ≤ T , which is (25), and the theorem is proved.
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
2 The domain DK = (t, u) ∈ R+ × Kd | t < t+(u) isopen in R+ × Kd and maximal in the sense that eithert+(u) =∞ or limt↑t+(u) ‖f (t, u)‖ =∞ for all u ∈ Kd .
3 The t-section DK (t) = u ∈ Kd | (t, u) ∈ DK is open inKd , and non-expanding in t:
Kd = DK (0) ⊇ DK (t1) ⊇ DK (t2) 0 ≤ t1 ≤ t2.
In fact, we have f (s,DK (t2)) ⊆ DK (t1) for all s ≤ t2 − t1.
4 If R is analytic on Kd then f is an analytic function onDK .
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
Suppose that either side of (25) is well defined for some t ≤ Tand u ∈ Rd . Then (25) holds, implying that both sides are welldefined in particular, for u replaced by u + iv for any v ∈ Rd .
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
Moreover, let DK (K = R or C) denote the maximal domainfor the system of Riccati equations. If either of the aboveconditions holds then DR(S) is a convex open neighborhood of0 in Rd , and S(DR(S)) ⊂ DC(S), for all S ≤ τ . Further:
E[e−
∫ Tt r(s) ds eu>X (T ) | Ft
]= eΦ(T−t,u)+Ψ(T−t,u)>X (t) (31)
for all u ∈ S(DR(S)), t ≤ T ≤ t + S and x ∈ Rm+ × Rn.
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
The theorem now follows from Theorem 38.3 once we setΦ(t, u) = φ′(t, u,−1) and Ψ(t, u) = ψ′1,...,d(t, u,−1).Indeed, it is clear by inspection thatDK (S) = u ∈ Kd | (u,−1) ∈ D′K (S) where D′K denotes themaximal domain for the system of Riccati equations (32).
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
For any maturity T ≤ τ , the T -bond price at t ≤ T is given as
P(t,T ) = e−A(T−t)−B(T−t)>X (t)
where we define A(t) = −Φ(t, 0), B(t) = −Ψ(t, 0).Moreover, for t ≤ T ≤ S ≤ τ , the Ft-conditional characteristicfunction of X (T ) under the S-forward measure QS is given by
EQS
[eu>X (T ) | Ft
]=
e−A(S−T )+Φ(T−t,u−B(S−T ))+Ψ(T−t,u−B(S−T ))>X (t)
P(t, S)(33)
for all u ∈ S(DR(T ) + B(S − T )), which contains iRd .
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
The bond price formula follows from the theorem with u = 0.Now let t ≤ T ≤ S ≤ τ . In view of the flow propertyΨ(T ,−B(S − T )) = −B(S), we know that−B(S − T ) ∈ DR(T ), and thus S(DR(T ) + B(S − T ))contains iRd . Moreover, for u ∈ S(DR(T ) + B(S − T )), weobtain from (31) by nested conditional expectation
TheoremSuppose either condition 1 or 2 of Theorem 39.1 is met forsome τ ≥ T , and let DR denote the maximal domain for thesystem of Riccati equations (30). Assume that f satisfies
f (x) =
∫Rq
e (v+iLλ)>x f (λ) dλ, dx-a.s.
for some v ∈ DR(T ) and d × q-matrix L, and some integrablefunction f : Rq → C, for a positive integer q ≤ d. Then theprice π(t) is well defined and given by the formula
π(t) =
∫Rq
eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t) f (λ) dλ. (34)
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
TheoremSuppose either condition 1 or 2 of Theorem 39.1 is met forsome τ ≥ T , and let DR denote the maximal domain for thesystem of Riccati equations (30). Assume that f is of the form
f (x) = e v>x h(L>x)
for some v ∈ DR(T ) and d × q-matrix L, and some integrablefunction h : Rq → R, for a positive integer q ≤ d. Define thebounded function
f (λ) =1
(2π)q
∫Rq
e−iλ>y h(y) dy , λ ∈ Rq.
1 If f (λ) is an integrable function in λ ∈ Rq then theassumptions of Theorem 39.3 are met.
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
2 If v = Lw, for some w ∈ Rq, andeΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t) is an integrable functionin λ ∈ Rq then the Ft-conditional distribution of theRq-valued random variable Y = L>X (T ) under theT -forward measure QT admits the continuous densityfunction
q(t,T , y) =1
(2π)q
∫Rq
e−(w+iλ)>y
× eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t)
P(t,T )dλ.
In either case, the integral in (34) is well defined and the priceformula (34) holds.
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
We denote by q(t,T , dy) the Ft-conditional distribution ofY = L>X (T ) under the T -forward measure QT . From (33) weinfer the characteristic function of the bounded (why?)
measure ew>y q(t,T , dy):∫Rq
e (w+iλ)>y q(t,T , dy) = E[e (w+iλ)>L>X (T ) | Ft
]=
eΦ(T−t,v+iLλ)+Ψ(T−t,v+iLλ)>X (t)
P(t,T ), λ ∈ Rq.
By assumption, this is an integrable function in λ on Rq. TheFourier inversion formula thus applies and the injectivity of thecharacteristic function (see e.g. [53, Section 16.6]) yields thatq(t,T , dy) admits the continuous density function as stated.
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
→ suggests to explore other types of diffusion processes thatadmit closed-form prices for some well specified basis ofpayoff functions (e.g. [15, 10]): open area of research
• examples of non-trivial Fourier decompositions? Yes! . . .
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
Proof.Let w < 0. Then the function h(y) = e−wy (K − e y )+ isintegrable on R. An easy calculation shows that its Fouriertransform
h(λ) =
∫R
e−(w+iλ)y (K − e y )+ dy =K−(w−1+iλ)
(w + iλ)(w − 1 + iλ)
is also integrable on R. Hence the Fourier inversion formulaapplies, and we conclude that the claimed identity holds forw < 0. The other cases follow by similar arguments (→exercise).
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
There exists some w− < 0 and w+ > 1 such that−B(S − T )w ∈ DR(T ) for all w ∈ (w−,w+), and the lineintegral
Π(w , t) =
∫R
eΦ(T−t,−(w+iλ)B(S−T ))
× eΨ(T−t,−(w+iλ)B(S−T ))>X (t) f (w , λ) dλ
is well defined for all w ∈ (w−,w+) \ 0, 1 and t ≤ T .Moreover, the time t prices of the European call and putoption on the S-bond with expiry date T and strike price K aregiven by any of the following identities: . . .
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
where I = (A(S − T ) + log K ,∞), and q(t,S , dy) andq(t,T , dy) denote the Ft-conditional distributions of thereal-valued random variable Y = −B(S − T )>X (T ) under theS- and T -forward measure, respectively.
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
• The pricing of European call and put bond options in thepresent d-dimensional affine factor model boils down tothe computation of a line integral Π(w , t), which is asimple numerical task!
• Moreover, in case the distributions q(t,S , dy) andq(t,T , dy) are explicitly known, the pricing is reduced tothe computation of the respective probabilities of theexercise events I and R \ I
• In the following two subsections, we illustrate thisapproach for the Vasicek and CIR short-rate models.
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
Noncentral χ2-Distribution cont’dNoncentral χ2 = generalization of distribution of the sum of thesquares of independent normal distributed random variables:
• fix δ ∈ N, reals ν1, . . . , νδ ∈ R, and define ζ =∑δ
i=1 ν2i
• let N1, . . . ,Nδ be independent standard normal distributedrvs
• define Z =∑δ
i=1(Ni + νi )2
• direct integration shows: characteristic function of Zequals
E[euZ ] =e
ζu1−2u
(1− 2u)δ2
, u ∈ C−
• above lemma ⇒ Z noncentral χ2-distributed with δdegrees of freedom and noncentrality parameter ζ
• good to know: noncentral χ2-distribution hard coded inmost statistical software packages
⇒ explicit bond option price formulas for the CIR model!
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
LemmaThere exists some invertible d × d-matrix Λ withΛ(Rm
+ × Rn) = Rm+ × Rn such that Λα(Λ−1y)Λ> is
block-diagonal of the form
Λα(Λ−1y)Λ> =
(diag(y1, . . . , yq, 0, . . . , 0) 0
0 p +∑
i∈I yiπi
)for some integer 0 ≤ q ≤ m and symmetric positivesemi-definite n × n matrices p, π1, . . . , πm. Moreover, Λb andΛBΛ−1 meet the admissibility conditions in lieu of b and B.
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
Since ΛaΛ> = a, the first assertion is proved (modulopermutation of first m indices).The admissibility conditions for Λb and ΛBΛ−1 can easily bechecked as well.
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
TheoremLet a, αi , b, βi be admissible parameters. Then there exists ameasurable function ρ : Rm
+ × Rn → Rd×d withρ(x)ρ(x)> = a +
∑i∈I xiαi , and such that, for any
x ∈ Rm+ × Rn, there exists a unique Rm
+ × Rn-valued solutionX = X x of the above affine SDE.Moreover, the law of X is uniquely determined by a, αi , b, βi ,and does not depend on the particular choice of ρ.
Term-StructureModels
DamirFilipovic
Definition andCharacteriza-tion of AffineProcesses
48 Monte Carlo Simulation of the LIBOR Market Model
49 Volatility Structure and CalibrationPrincipal Component AnalysisCalibration to Market Quotes
50 Continuous-Tenor Case
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Problem Formulation
• Consider a payer swaption with nominal 1, strike rate K ,maturity Tµ, underlying tenor Tµ, Tµ+1, . . . ,Tν (Tµ is thefirst reset date and Tν the maturity of the underlyingswap), for some µ < ν ≤ M
• Recall: payoff at maturity Tµ is
Π = δ
ν∑m=µ+1
P(Tµ,Tm)(L(Tµ,Tm−1)− K )
+
• By above pricing lemmas: swaption price at t = 0
π = P(0,Tµ)EQTµ [Π] = EQ∗
[Π
B∗(Tµ)
]⇒ Need joint distribution (under QTµ or Q∗) of
L(Tµ,Tµ), L(Tµ,Tµ+1), . . . , L(Tµ,Tν−1)
• No analytic formula in LIBOR market model . . .
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Outline
43 Heuristic Derivation From HJM
44 LIBOR Market ModelLIBOR Dynamics Under Different Measures
48 Monte Carlo Simulation of the LIBOR Market Model
49 Volatility Structure and CalibrationPrincipal Component AnalysisCalibration to Market Quotes
50 Continuous-Tenor Case
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Definition
• Corresponding forward swap rate at t ≤ Tµ is
Rswap(t) =P(t,Tµ)− P(t,Tν)
δ∑ν
k=µ+1 P(t,Tk)=
1− P(t,Tν)P(t,Tµ)
δ∑ν
k=µ+1P(t,Tk )P(t,Tµ)
.
⇒ Rswap(t) available in our LIBOR market model
• Define positive QTµ-martingale D(t) =∑ν
k=µ+1P(t,Tk )P(t,Tµ) ,
t ∈ [0,Tµ]
• Induces forward swap measure Qswap ∼ QTµ on FTµ by
dQswap
dQTµ=
D(Tµ)
D(0).
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Change of Numeraire
LemmaThe forward swap rate process Rswap(t), t ∈ [0,Tµ], is apositive Qswap-martingale.Moreover, there exists some d-dimensional Qswap-Brownianmotion W swap and an Rd -valued progressive swap volatilityprocess ρswap such that
dRswap(t) = Rswap(t)ρswap(t) dW swap(t), t ∈ [0,Tµ].
• Note: ρswap(t) explicitly available in principle
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Proof
Let 0 ≤ m ≤ M and 0 ≤ s ≤ t ≤ Tm ∧ Tµ. Then
EQswap
[P(t,Tm)
P(t,Tµ)D(t)| Fs
]=
1
D(s)EQTµ
[P(t,Tm)
P(t,Tµ)D(t)D(t) | Fs
]=
1
D(s)
P(s,Tm)
P(s,Tµ).
On the other hand, from above formula for Rswap(t):
Rswap(t) =1
δD(t)− P(t,Tν)
δP(t,Tµ)D(t).
Hence Rswap(t) is a positive Qswap-martingale.The representation of Rswap(t) in terms of W swap and ρswap
follows from above lemma for P(t,Tk)/P(t,Tm) andGirsanov’s theorem.
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Swap Measure Swaption Pricing
• Recall: swaption payoff at maturity can be written as
δD(Tµ) (Rswap(Tµ)− K )+
• Hence the price equals
π = δP(0,Tµ)EQTµ
[D(Tµ) (Rswap(Tµ)− K )+]
= δP(0,Tµ)D(0)EQswap
[(Rswap(Tµ)− K )+]
= δ
ν∑k=µ+1
P(0,Tk)EQswap
[(Rswap(Tµ)− K )+]
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Lognormal Hypothesis
• Hypothesis (H): ρswap(t) is deterministic
• Consequence: log Rswap(Tµ) Gaussian distributed under
Qswap with mean = log Rswap(0)− 12
∫ Tµ0 ‖ρswap(t)‖2 dt
and variance =∫ Tµ
0 ‖ρswap(t)‖2 dt
⇒ Swaption price would be
π = δ
ν∑k=µ+1
P(0,Tk) (Rswap(0)Φ(d1)− K Φ(d2))
with
d1,2 =log(
Rswap(0)K
)± 1
2
∫ Tµ0 ‖ρswap(t)‖2 dt(∫ Tµ
0 ‖ρswap(t)‖2 dt) 1
2
• This is Black’s formula with volatilityσ2 = 1
Tµ
∫ Tµ0 ‖ρswap(t)‖2 dt
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Lognormal Hypothesis Valid?
• Fact (without proof): ρswap cannot be deterministic in ourlognormal LIBOR setup
• Alternative for swaption pricing: model forward swap ratesdirectly and postulate that they are lognormal under theforward swap measures (the so-called swap market model)
• Carried out by Jamshidian [32], and computationallyimproved by Pelsser [43]
• But (without proof): then forward LIBOR rate volatilitycannot be deterministic
⇒ Either one gets Black’s formula for caps or for swaptions,but not simultaneously for both
⇒ In lognormal forward LIBOR model swaption prices haveto be approximated
• via Monte Carlo methods• via analytic approximation . . .
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Outline
43 Heuristic Derivation From HJM
44 LIBOR Market ModelLIBOR Dynamics Under Different Measures
• Advantage of simulating Hm:• keeps L(t,Tm) = exp(Hm(t)) positive• Hm has Gaussian increments ⇒ improves convergence of
Euler scheme
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
General pricing problem
• General pricing problem: Tn-claim with payofff (Hn(Tn), . . . ,HM−1(Tn))
• Price at t = 0: π = EQ∗[
f (Hn(Tn),...,HM−1(Tn))B∗(Tn)
]
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Euler Scheme
• Fix time grid ti = i∆t, i = 0, . . . ,N, with ∆t = Tn/N forN large enough
• Z (1), . . . ,Z (N): sequence of independent standard normalrandom vectors in Rd
• Euler approximation for Hm: 1 ≤ i ≤ N
Hm(ti ) = Hm(ti−1) +αm(ti−1) ∆t + λ(ti−1,Tm) Z (i)√
∆t(40)
• Monte Carlo principle:
1 simulate via Euler scheme K independent copies
Π(1), . . . ,Π(K) of Π = f (Hn(Tn),...,HM−1(Tn))B∗(Tn)
2 estimate π via averaging: Π = 1K
∑Kj=1 Π(j)
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Efficiency
Three considerations are important for the efficiency of thissimulation estimator: bias, variance, and computing time
Bias: introduced via Euler approximation (40): EQ∗ [Π]differs from target value π = EQ∗ [Π]. Bias isreduced by increasing number of timediscretization steps N. (In our example: bias isalready negligible for ∆t = 1/12, can assumethat EQ∗ [Π] ≈ π.)
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Efficiency cont’d
Variance: central limit theorem: as number of replicationsK increases, the simulation estimation errorΠ− π approximately normal distributed withmean zero and approximate standard deviation of
sπ =
√∑Kj=1(Π(j) − Π)
K (K − 1).
sπ is called standard error of the Monte Carlosimulation: Π± sπ is an asymptotically (asK →∞) valid 68% confidence interval for thetrue value π.
Computing time: obvious trade-off between bias and variancefor a given computing capacity, which has to becarefully balanced in general.
Thorough treatment of Monte Carlo: e.g. Glasserman [26]
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Outline
43 Heuristic Derivation From HJM
44 LIBOR Market ModelLIBOR Dynamics Under Different Measures
48 Monte Carlo Simulation of the LIBOR Market Model
49 Volatility Structure and CalibrationPrincipal Component AnalysisCalibration to Market Quotes
50 Continuous-Tenor Case
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Sketch of PCA
• Assume: λ(t,Tm) = λ(Tm − t), for all m
• Given: N observations x(1), . . . , x(N) of random vectorsX (i) = (X1(i), . . . ,XM(i))> where (1 ≤ m ≤ M)
Xm(i) = log L(iδ, (i +m−1)δ)− log L((i−1)δ, (i +m−1)δ)
(sliding LIBOR curve increments)
• Euler approximation (40) & neglect drift term (see nextpage) ⇒
Xm(i) ≈ λ(mδ) Z (i)√δ (41)
⇒ X (i) approximately i.i.d. with zero mean
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Sketch of PCA: Drift Term
• Recall Q∗-drift term of log L(iδ, (i + m − 1)δ):
αm((i − 1)δ) = −1
2‖λ(mδ)‖2
+ λ(mδ)m∑
k=1
δL((i − 1)δ, (i + k − 1)δ)
1 + δL((i − 1)δ, (i + k − 1)δ)λ(kδ)>
• For P: to be corrected by
market price of risk× δ ‖λ(mδ)‖
⇒ Drift term of order
δ ‖λ(mδ)‖ ×max‖λ(mδ)‖, market price of risk
⇒ neglected
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Sketch of PCA cont’d• empirical mean µ = 1
N
∑Nt=1 x(t)
• empirical covariance matrix
Qij = Cov[xi , xj ] =1
N
N∑t=1
(xi (t)− µi )(xj(t)− µj)
• PCA decomposition:
x(i) = µ+M∑
j=1
aj yj(i) ≈M∑
j=1
aj yj(i)
with• Q = ALA>, loadings A = (a1 | · · · | an)• empirical principal components y = A>(x − µ): yj
uncorrelated, nonincreasing order Var[y1] ≥ Var[y2] ≥ · · ·• Compare with (41) ⇒ estimate (1 ≤ m ≤ M)
λj(mδ) =
√Var[yj ]
δajm
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
PCA Stylized Facts
• First 2–3 principal components yj enough to explain mostof the variance of x
• First three loadings aj (i.e. volatility curves s 7→ λj(s)) aretypically of the form as seen in Section “PCA of theForward Curve”: flat, upward (or downward) sloping, andhump-shaped
Figure: First three forward curve loadings
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Outline
43 Heuristic Derivation From HJM
44 LIBOR Market ModelLIBOR Dynamics Under Different Measures
⇒ v1, . . . , v19 as functions on β ( = degree of freedom)
• Sanity check: MC algorithm reproduces 3.5% market capprices independently the correlation specification . . .
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Price a Swaption
• Price the at-the-money 4× 6-swaption: maturity in 4years, underlying swap 6 years long
• Tenor: first reset date T8 = 4, and annual(!) couponpayments at T10 = 5, . . . ,T20 = 10
• Swaption price depends on β and on correlationspecifications I (d = 1, `m = 1) or II (d = M − 1,`m = e>m).
• Compute by MC and analytic approximation formula
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Results
Figure: The swaption price as function of β. The straight horizontalline indicates the real market quote of 248 bp. The upper curves arefor the correlation specification I, the lower curves are for specificationII. The solid lines show the Monte Carlo simulation based prices withstandard errors indicated by the dotted lines. The dashed lines showthe respective prices based on the analytic approximation formula.
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Discussion
• Actual market quote for this swaption was 248 bp• For correlation specification I: obtain an estimate β ≈ 0.07
• More systematic tests for quality of analyticalapproximation in [12, Chapter 8]
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Outlook: Volatility Smile
• Snag: bootstrapped caplet implied volatilities depend onstrike rate in general (volatility smile/skew)
⇒ Lognormal LIBOR market model can only matchterm-structure of caplet volatility for one strike rate at atime (no matter how many driving Brownian motions d)
• Situation similar to Black–Scholes stock market model:incapable of fitting market option prices across all strikes(note: Heston stochastic volatility model can producevolatility skews)
• Possible extensions:• λ(t,Tm) = λ(ω, t,Tm) progressive process (analogous to
Heston model)• replace driving Brownian motion by Levy process
• Much research over last decade to achieve a good fittingof market option data, see e.g. Brigo and Mercurio [12,Part IV] for a detailed overview
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Outline
43 Heuristic Derivation From HJM
44 LIBOR Market ModelLIBOR Dynamics Under Different Measures
• For T ∈ [TM−1,TM ] (B∗(T ) is FT -measurable andpositive): define T -forward measure QT ∼ Q∗ on FT by
dQT
dQ∗=
1
B∗(T )P(0,T )
• Note: dQT
dQTM= dQT
dQ∗dQ∗
dQTM= B∗(TM)P(0,TM)
B∗(T )P(0,T )
• Representation Theorem: ∃ unique σT ,TM∈ L such that
dQT
dQTM|Ft = EQTM
[B∗(TM)P(0,TM)
B∗(T )P(0,T )| Ft
]= Et
(σT ,TM
•W TM
), t ∈ [0,T ]
• Girsanov: W T (t) = W TM (t)−∫ t
0 σT ,TM(s)> ds is a
QT -Brownian motion, t ∈ [0,T ]
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
3rd Step: Backward Induction
• Fix T ∈ [TM−2,TM−1]
• Can now define forward LIBOR L(t,T ):
dL(t,T ) = L(t,T )λ(t,T ) dW T+δ(t),
L(0,T ) =1
δ
(P(0,T )
P(0,T + δ)− 1
)• This defines bounded progressiveσT ,T+δ(t) = δL(t,T )
δL(t,T )+1λ(t,T )
• T -Forward measure defined by
dQT
dQT+δ= ET
(σT ,T+δ •W T+δ
)
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
3rd Step cont’d
Note: σT ,TM= σT ,T+δ + σT+δ,TM
satisfies (t ∈ [0,T ])
dQT
dQTM|Ft =
dQT
dQT+δ|Ft
dQT+δ
dQTM|Ft
= Et(σT ,T+δ •W T+δ
)Et(σT+δ,TM
•W TM
)= Et
(σT ,TM
•W TM
)• Proceeding by backward induction ⇒ forward measure
QT , QT -Brownian motion W T for all T ∈ [0,TM ]
⇒ forward LIBOR L(t,T ) for all T ∈ [0,TM−1]
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Zero-Coupon Bonds
• Finally: obtain zero-coupon bond prices for all maturities
• Fix 0 ≤ T ≤ S ≤ TM
• Note: σT ,S = σT ,TM− σS,TM
satisfies (t ∈ [0,T ])
dQS
dQT|Ft =
dQS
dQTM|Ft
dQTM
dQT|Ft = Et
(−σT ,S •W T
)• By part “Forward Measures”: consistently define forward
price process
P(t, S)
P(t,T )=
P(0, S)
P(0,T )
dQS
dQT|Ft =
P(0,S)
P(0,T )Et(−σT ,S •W T
)
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
Zero-Coupon Bonds cont’d
• In particular for t = T :
P(T ,S) =P(0,S)
P(0,T )ET(−σT ,S •W T
)• Exercise: P(t,T )
B∗(t) is Q∗-martingale
• Note: P(T ,S) may be greater than 1, unless S − T = mδfor some m ∈ N
⇒ Even though all δ-period forward LIBOR L(t,T ) arenonnegative, there may be negative interest rates for otherthan δ periods.
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
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Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References II
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Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References III
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Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References IV
D. Brigo and F. Mercurio.Interest rate models—theory and practice.Springer Finance. Springer-Verlag, Berlin, second edition,2006.With smile, inflation and credit.
R. H. Brown and S. M. Schaefer.Why do long term forward rates (almost always) slopedownwards?London Business School working paper, November 1994.
R. A. Carmona and M. R. Tehranchi.Interest rate models: an infinite dimensional stochasticanalysis perspective.Springer Finance. Springer-Verlag, Berlin, 2006.
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References V
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P. Cheridito, D. Filipovic, and R. L. Kimmel.A note on the Dai–Singleton canonical representation ofaffine term structure models.Forthcoming in Mathematical Finance, 2009.
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Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References VI
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F. Delbaen and W. Schachermayer.A general version of the fundamental theorem of assetpricing.Math. Ann., 300(3):463–520, 1994.
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References VII
F. Delbaen and W. Schachermayer.The no-arbitrage property under a change of numeraire.Stochastics Stochastics Rep., 53(3-4):213–226, 1995.
M. Dothan.On the term structure of interest rates.J. of Financial Economics, 6:59–69, 1978.
P. Dybvig, J. Ingersoll, and S. Ross.Long forward and zero coupon rates can never fall.J. Business, 69:1–25, 1996.
D. Filipovic.Consistency problems for Heath-Jarrow-Morton interestrate models, volume 1760 of Lecture Notes inMathematics.Springer-Verlag, Berlin, 2001.
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References VIII
P. Glasserman.Monte Carlo methods in financial engineering, volume 53of Applications of Mathematics (New York).Springer-Verlag, New York, 2004.Stochastic Modelling and Applied Probability.
D. Heath, R. Jarrow, and A. Morton.Bond pricing and the term structure of interest rates: Anew methodology for contingent claims valuation.Econometrica, 60:77–105, 1992.
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Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References IX
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R. Jamshidian.Libor and swap market models and measures.Finance and Stochastics, 1(4):290–330, 1997.
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References X
N. L. Johnson, S. Kotz, and N. Balakrishnan.Continuous univariate distributions. Vol. 2.Wiley Series in Probability and Mathematical Statistics:Applied Probability and Statistics. John Wiley & Sons Inc.,New York, second edition, 1995.A Wiley-Interscience Publication.
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Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References XI
D. Lamberton and B. Lapeyre.Introduction to stochastic calculus applied to finance.Chapman & Hall, London, 1996.Translated from the 1991 French original by NicolasRabeau and Francois Mantion.
R. Litterman and J. K. Scheinkman.Common factors affecting bond returns.J. of Fixed Income, 1:54–61, 1991.
S. Lorimier.Interest Rate Term Structure Estimation Based on theOptimal Degree of Smoothness of the Forward Rate Curve.PhD thesis, University of Antwerp, 1995.
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References XII
K. Miltersen, K. Sandmann, and D. Sondermann.Closed form solutions for term structure derivatives withlognormal interest rates.Journal of Finance, 52:409–430, 1997.
A. J. Morton.Arbitrage and martingales.Technical Report 821, School of Operations Research andIndustrial Engineering, Cornell University, 1988.
M. Musiela and M. Rutkowski.Martingale methods in financial modelling, volume 36 ofStochastic Modelling and Applied Probability.Springer-Verlag, Berlin, second edition, 2005.
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References XIII
L. T. Nielsen.Pricing and Hedging of Derivative Securities.Oxford University Press Inc., New York, 1999.
A. Pelsser.Efficient methods for valuing interest rate derivatives.Springer Finance. Springer-Verlag London Ltd., London,2000.
R. Rebonato.Interest-Rate Option Models: Understanding, Analysingand Using Models for Exotic Interest-Rate Options.Wiley Series in Financial Engineering. John Wiley & SonLtd, Berlin, second edition, 1998.
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References XIV
D. Revuz and M. Yor.Continuous martingales and Brownian motion, volume 293of Grundlehren der Mathematischen Wissenschaften[Fundamental Principles of Mathematical Sciences].Springer-Verlag, Berlin, third edition, 1999.
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Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References XV
E. Sharef.Quantitative evaluation of consistent forward rateprocesses. An empirical study.Senior’s thesis, Princeton University, 2003.
J. M. Steele.Stochastic calculus and financial applications, volume 45 ofApplications of Mathematics (New York).Springer-Verlag, New York, 2001.
J. Steeley.Estimating the Gilt-edged term structure: basis splines andconfidence intervals.Jounal of Business Finance and Accounting, 18:513–530,1991.
Term-StructureModels
DamirFilipovic
HeuristicDerivationFrom HJM
LIBOR MarketModel
LIBORDynamics UnderDifferentMeasures
Implied BondMarket
ImpliedMoney-MarketAccount
SwaptionPricing
Forward SwapMeasure
AnalyticApproximations
Monte CarloSimulation ofthe LIBORMarket Model
VolatilityStructure andCalibration
PrincipalComponentAnalysis
Calibration toMarket Quotes
Continuous-TenorCase
References XVI
E. M. Stein and G. Weiss.Introduction to Fourier analysis on Euclidean spaces.Princeton University Press, Princeton, N.J., 1971.Princeton Mathematical Series, No. 32.
O. Vasicek.An equilibrium characterization of the term structure.J. of Financial Economics, 5:177–188, 1977.
D. Williams.Probability with martingales.Cambridge Mathematical Textbooks. Cambridge UniversityPress, Cambridge, 1991.