Affine LIBOR models with multiple curves: John Schoenmakers, WIAS Berlin joint with Z. Grbac, A. Papapantoleon & D. Skovmand 9th Summer School in Mathematical Finance Cape Town, 18-20 February 2016
18-20.02.2016
John Schoenmakers ()
MC ALM
18-20.02.2016
1 / 34
Outline 1 Introduction
Pre-crisis markets In-crisis markets LIBOR mechanics 2 Multi-curve LIBOR models
The new landscape 3 Affine LIBOR models with multiple curves
General framework The MC affine LIBOR model Calibration Old and new examples Connections to other frameworks 4 Summary and Outlook 1 / 42
Introduction
Pre-crisis markets
Market size According to the Bank for International Settlements: 6/06
6/07
6/08
6/09
6/10
6/11
Foreign exchange Interest rate Equity-linked Commodity Credit default swaps Unallocated
38 262 6 6 20 35
49 347 8 7 42 61
63 458 10 13 57 82
49 437 6 4 36 72
53 452 6 3 30 38
65 554 7 3 32 46
Total
370
516
684
604
583
707
Table: Notional amounts outstanding for OTC derivatives, billions of US$
2 / 42
Introduction
Pre-crisis markets
Interest rates I 2
Comparison of estimated interest rates (least squares Svensson)
3 2 0
1
interest rate in percent
4
5
Euroland Japan Switzerland USA
0
2
4
6
8
10
time to maturity
Figure: Term structure of EUR, USD, JPY, CHF interest rates; 17.2.2004 Termstructure, February 17, 2004
3 / 42
Introduction
Pre-crisis markets
Interest rates II
0
1-month rate in % 5 10
15
3
1970
1975
1980
1985
1990
1995
time
month rateof at one-month the German market, March 01, 1967rate, – March 31, 1997 Figure:One Evolution German interest 3.1967 – 3.1997
!"#$#%&'()!"#*+,!.- "#/ 01 23 45-6!879 +;:8 < >= <? =@"A>BDC"E ? >= <.? =F1A>A.C
film
4 / 42
Introduction
Pre-crisis markets
Evolution of interest rates
Figure: Evolution of interest rate term structure, 2003–2004 5 / 42
Introduction
Pre-crisis markets
LIBOR rates Tenor structure: T = {0 < T1 < T2 < · · · < TN },
I = {1, . . . , N}
B(t,T ): zero coupon bond L(t,T ): discretely compounded forward LIBOR t LIBOR FRA
T
T +δ
1
1 + δL(t, T ) B(t,T ) B(t,T +δ)
FRA: buy 1 T -bond, sell
B(t,T ) B(t,T +δ)
[T + δ]-bonds
“Master” equation
L(t,T ) =
1 δ
B(t,T ) −1 B(t,T + δ)
(1) 6 / 42
Introduction
In-crisis markets
What happened during the ‘credit crunch’ ? 250 200
Euribor - Eoniaswap spreads spread 1m spread 3m spread 6m spread 12m
bp
150 100 50 0
2006
2007
2008
2009
2010 7 / 42
Introduction
LIBOR mechanics
How is LIBOR really computed? LIBOR panels: 8-16 banks scale, reputation, expertise Question: “At what rate could you borrow funds, were you to do so by asking for and then accepting inter-bank offers in a reasonable market size?” Funds: unsecured interbank cash
8 / 42
Introduction
LIBOR mechanics
Euribor and Eonia mechanics Euribor Panel: 39 EU & 4 international banks scale, reputation, expertise
Question: “What rate do you believe one prime bank is quoting to another prime bank for interbank term deposits within the euro zone?” Eonia = Euro Over Night Index Average / OIS Panel: same Each panel bank submits the total volume of overnight unsecured lending transactions that day and the weighted average lending rate for these transactions Negligible counterparty credit and liquidity risk market proxy to a risk free rate.
the best available
9 / 42
Multi-curve LIBOR models
The new landscape
The new landscape Tenor structures: T x = {0 < T0x < T1x < · · · < TNx = TN }, x ∈ {1, 3, 6, 12}M
T0 T1 T2 T3 T4 T5 T6 T0 T0
T1x1
Tn−1TN
T2x1
TNx1x1
T1x2
TNx2x2
Discount curve: OIS zero coupon bonds T 7→ B(0,T ) = B OIS (0,T ) Forward measures Pxk – numeraire B(·,Tkx ) 10 / 42
Multi-curve LIBOR models
The new landscape
The new landscape – II Definition x The time-t OIS forward rate for [Tk−1 , Tkx ] is
Fkx (t)
1 = δx
x B(t,Tk−1 ) −1 B(t,Tkx )
(2)
Definition (Mercurio) The time-t FRA rate Lxk (t) is the fixed rate to be exchanged at Tk for the LIBOR x rate L(Tk−1 ,Tkx ) such that the swap has value zero: x Lxk (t) = Exk [L(Tk−1 ,Tkx )|Ft ]
(3)
FRA – OIS spread: Skx (t) := Lxk (t) − Fkx (t) 11 / 42
Multi-curve LIBOR models
The new landscape
The new landscape – III Requirements: Fkx (t) ≥ 0 and Fkx is a Pxk -martingale Lxk (t) ≥ 0 and Lxk is a Pxk -martingale Skx (t) ≥ 0 ⇐⇒ Lxk (t) ≥ Fkx (t) (market observations) Options: 1
Model Fkx and Lxk
2
Model Fkx and Skx
3
Model Lxk and Skx
Mercurio [2010]: “The first choice . . . is the most convenient in terms of model tractability and calibration. . . The problem is that there is no guarantee that the implied spreads will have a realistic behavior in the future, in particular preserving the positive sign.” 12 / 42
Affine LIBOR models with multiple curves
General framework
Constructing ordered martingales > 1 1
Take a random variable YTu > 1, FT -measurable and integrable, and set: Mtu = E[YTu |Ft ],
(4)
then (Mt )0≤t≤T is a martingale and Mt > 1 for all t ∈ [0, T ].
13 / 42
Affine LIBOR models with multiple curves
General framework
Constructing ordered martingales > 1 1
Take a random variable YTu > 1, FT -measurable and integrable, and set: Mtu = E[YTu |Ft ],
(4)
then (Mt )0≤t≤T is a martingale and Mt > 1 for all t ∈ [0, T ]. 2
Require that u 7→ YTu is increasing, then u 7→ Mtu
(5)
is a.s. increasing for all t ∈ [0, T ].
14 / 42
Affine LIBOR models with multiple curves
General framework
Constructing ordered martingales > 1 1
Take a random variable YTu > 1, FT -measurable and integrable, and set: Mtu = E[YTu |Ft ],
(4)
then (Mt )0≤t≤T is a martingale and Mt > 1 for all t ∈ [0, T ]. 2
Require that u 7→ YTu is increasing, then u 7→ Mtu
(5)
is a.s. increasing for all t ∈ [0, T ]. 3
Affine LIBOR model: (Keller-Ressel, P., Teichmann) YTu = ehu,XT i X positive affine process, u ∈ Rd>0 Mtu = E[ehu,XT i |Ft ] = exp(φT −t (u) + hψT −t (u), Xt i)
(6) 15 / 42
Affine LIBOR models with multiple curves
The MC affine LIBOR model
Ansatz Modeling OIS forward rates: (ukx )k≥0 ux
1+
δx Fkx (t)
x ) B(t, Tk−1 Mt k−1 = = ux B(t, Tkx ) Mt k
(7)
16 / 42
Affine LIBOR models with multiple curves
The MC affine LIBOR model
Ansatz Modeling OIS forward rates: (ukx )k≥0 ux
1+
δx Fkx (t)
x ) B(t, Tk−1 Mt k−1 = = ux B(t, Tkx ) Mt k
(7)
Modeling FRA rates: (vkx )k≥0 vx
1+
δx Lxk (t)
=
Mt k−1 ux
(8)
Mt k
17 / 42
Affine LIBOR models with multiple curves
The MC affine LIBOR model
Ansatz Modeling OIS forward rates: (ukx )k≥0 ux
1+
δx Fkx (t)
x ) B(t, Tk−1 Mt k−1 = = ux B(t, Tkx ) Mt k
(7)
Modeling FRA rates: (vkx )k≥0 vx
1+
δx Lxk (t)
=
Mt k−1 ux
(8)
Mt k x
The Pxk -martingale property is obvious (M uk is the density)
18 / 42
Affine LIBOR models with multiple curves
The MC affine LIBOR model
Ansatz Modeling OIS forward rates: (ukx )k≥0 ux
1+
δx Fkx (t)
x ) B(t, Tk−1 Mt k−1 = = ux B(t, Tkx ) Mt k
(7)
Modeling FRA rates: (vkx )k≥0 vx
1+
δx Lxk (t)
=
Mt k−1 ux
(8)
Mt k x
The Pxk -martingale property is obvious (M uk is the density) Orderings: x uk−1 ≥ ukx ⇒ Fkx ≥ 0 automatic: initial values x x vk−1 ≥ uk−1 ⇒ Lxk ≥ Fkx
(?) 19 / 42
Affine LIBOR models with multiple curves
The MC affine LIBOR model
Analytical tractability X : affine process under PN Ex,N ehw ,Xt i = exp φTN −t (w ) + hψTN −t (w ), xi
(9)
20 / 42
Affine LIBOR models with multiple curves
The MC affine LIBOR model
Analytical tractability X : affine process under PN Ex,N ehw ,Xt i = exp φTN −t (w ) + hψTN −t (w ), xi
(9)
Proposition The process X is a time-inhomogeneous affine process under the measure Pxk , for every x, k, with hw ,Xt i k,x k,x x (10) Ex,k e = exp φt (w ) + hψt (w ), xi , where x x φk,x t (w ) := φt ψTN −t (uk ) + w − φt ψTN −t (uk ) ψtk,x (w ) := ψt ψTN −t (ukx ) + w − ψt ψTN −t (ukx ) .
(11) (12) 21 / 42
Affine LIBOR models with multiple curves
The MC affine LIBOR model
Caplet pricing Easy: express the payoff of a caplet as: x x δx (L(Tk−1 , Tkx ) − K )+ = (1 + δx L(Tk−1 , Tkx ) − 1 + δx K )+ x x + A +B ·X = e k k Tk−1 − K
where K := 1 + δx K . Apply Fourier methods x C0 (K , Tkx ) = δx B(0, Tkx ) Exk (L(Tk−1 , Tkx ) − K )+ Z ΛAxk +Bkx ·XTk−1 (R − iv ) KB(0, Tkx ) Kiv −R dv = 2π (R − iv )(R − 1 − iv ) R
(13)
(14)
where ΛAxk +Bkx ·XTk is the Pxk -mgf (known).
22 / 42
Affine LIBOR models with multiple curves
The MC affine LIBOR model
Swaption pricing In-crisis: no cancelations . . . x S+ 0 (K , Tpq )
=
B(0, Tpx )
q X
Exp
δx B(Tpx , Tix )
K
Tpx
x (Tpq )
−K
+
i=p+1
= B(0, Tpx ) Exp
q X
δx B(Tpx , Tix )Lxi (Tpx )
i=p+1
−
q X
+ δx B(Tpx , Tix )K
i=p+1
Efficient numerical approximation?
23 / 42
Affine LIBOR models with multiple curves
The MC affine LIBOR model
Swaption pricing II Main ingredients: Affine property under forward measures x S+ 0 (K , Tpq ) = B(0, TN ) EN
q X
M
x vj−1 Tpx
−
j=p+1
= B(0, TN ) EN
q X j=p+1
= B(0, TN )
− Kx
q X
q X
+ ujx Tpx
Kx M
j=p+1
M
x vj−1 Tpx
−
q X
ujx Tpx
Kx M 1{f (XT x )≥0} p
j=p+1
h i x vx M0 j−1 Ej−1 1{f (XT x )≥0} p
j=p+1 q X
h i B(0, Tjx ) Exj 1{f (XT x )≥0} p
j=p+1
Linear approximation of the exercise boundary (Singleton & Umantsev) 24 / 42
Affine LIBOR models with multiple curves
The MC affine LIBOR model
Swaption pricing III Swaption 2Y2Y
Swaption 2Y2Y
0.58
0.05 Error in bp
Implied Volatility
0.56
0.06 True Approximation
0.54 0.52 0.5
0.04 0.03 0.02 0.01
0.48 0.005
0.01
0.015 Strike
0.02
0 0.005
0.025
0.01
0.015 Strike
Swaption 2Y8Y
0.02
0.025
Swaption 2Y8Y 0.7
True Approximation
0.6 0.5
0.22 Error in bp
Implied Volatility
0.24
0.2 0.18
0.3 0.2 0.1
0.16 0.01
0.4
0.015
0.02
0.025 Strike
0.03
0.035
0 0.01
0.04
0.015
0.02
Swaption 5Y5Y 0.205
0.035
0.04
0.3 0.25
0.195 0.19 0.185
0.2 0.15 0.1
0.18 0.175 0.01
0.03
0.35 True Approximation Error in bp
Implied Volatility
0.2
0.025 Strike Swaption 5Y5Y
0.05 0.015
0.02
0.025 0.03 Strike
0.035
0.04
0.045
0 0.01
0.015
0.02
0.025 0.03 Strike
0.035
0.04
0.045
25 / 42
Affine LIBOR models with multiple curves
The MC affine LIBOR model
Basis swaption pricing −3
Basis Swaption 2Y2Y 50
2.6 True Approximation
48
5.5
42 40
4.5
2
Pct.
44
1.8
3
1.4 0.1
0.15 0.2 Strike in pct.
0.25
1.2 0.05
0.3
0.1
Basis Swaption 2Y8Y
Basis Point
Basis Point
90 80
0.1
0.15 0.2 Strike in pct.
0.25
0.13 0.12
0.1
0.11
0.095
0.1
0.09
0.09
0.1
0.15 0.2 Strike in pct.
0.25
0.08 0.05
0.3
0.1
Absolute Error
0.15 0.2 Strike in pct.
0.25
0.3
0.15 0.2 Strike in pct.
0.25
0.3
0.25
0.3
Relative Error
0.095 True Approximation
0.14
0.09
60
55
0.135 0.085 Pct.
Basis Point
65
0.1
Relative Error
0.11
0.085 0.05
0.3
70
Basis Point
2.5 0.05
0.3
0.105
Basis Swaption 5Y5Y
50 0.05
0.25
Pct.
True Approximation
100
70 0.05
0.15 0.2 Strike in pct. Absolute Error
120 110
4 3.5
1.6
38
Relative Error
x 10
5
2.2 Basis Point
Basis Point
46
36 0.05
−3
Absolute Error
x 10
2.4
0.08
0.13
0.075
0.1
0.15 0.2 Strike in pct.
0.25
0.3
0.05
0.1
0.15 0.2 Strike in pct.
0.25
0.3
0.125 0.05
0.1
0.15 0.2 Strike in pct.
26 / 42
Affine LIBOR models with multiple curves
Calibration
Calibration – model structure Dynamics of FRA rates:
x x 1+δx Lxk (t) = exp φTN −t (vk−1 )−φTN −t (ukx )+ ψTN −t (vk−1 ) − ψTN −t (ukx ), Xt
x x Cancelations if vi,k−1 = ui,k x Constraints: ukx ≥ uk+1 and vkx ≥ ukx
M maturities, 2M drivers X = (X 1,1 , X 2,1 ), . . . , (X 1,M , X 2,M ) Dynamics dXt1,i
=
−λ1,i (Xt1,i
− θ1,i )dt + 2η1,i
dXt2,i = −λ2,i (Xt2,i − θ2,i )dt + 2η2,i
q
Xt1,i dWt1,i + dZti ,
(15)
q
Xt2,i dWt2,i ,
(16) 27 / 42
Affine LIBOR models with multiple curves
Calibration
Calibration – model structure II Hence, the structure is x vM/δ x −1
x v(M−1)/δ x −1 x v(M−2)/δx −1 .. . x v1/δ x −1
= = = =
(0, 0) (0, 0) (0, 0) .. . v¯1/δx −1
... ... ... . .. ...
(0, 0) (0, 0) v¯(M−2)/δx −1 .. . v¯(M−2)/δx −1
(0, 0) v¯(M−1)/δx −1 v¯(M−1)/δx −1 .. . v¯(M−1)/δx −1
v¯M/δx −1 v¯M/δx −1 v¯M/δx −1 .. . v¯M/δx −1
where u¯i , v¯i ∈ R2>0 , and x uM/δ x
x u(M−1)/δ x x u(M−2)/δx .. . x u1/δ x
= = = =
(0, 0) (0, 0) (0, 0) .. . u¯1/δx
... ... ... . .. ...
(0, 0) (0, 0) u¯(M−2)/δx .. . v¯(M−2)/δx −1
(0, 0) u¯(M−1)/δx v¯(M−1)/δx −1 .. . v¯(M−1)/δx −1
u¯M/δx v¯M/δx −1 v¯M/δx −1 .. . v¯M/δx −1 .
Allows for sequential calibration 28 / 42
Affine LIBOR models with multiple curves
Calibration
Calibration – caplet data Maturity = 2
1.2
1.2
1.1 1 0.9 0
0.05 Strike Maturity = 4
0
0.05 Strike Maturity = 5
0.62 0.6 0.58 0.05 Strike Maturity = 7
0.55 0.5
0
0.05 Strike Maturity = 8
0.45
0.4
0
0.05 Strike
0.1
0
0.05 Strike Maturity = 6
0.1
0
0.05 Strike Maturity = 9
0.1
0.55 0.5 0.45 0.4
0.1
0.5 Implied Volatility
0.5
0.8
0.6
0.6
0.45
0.1
0.82
0.78
0.1
Implied Volatility
Implied Volatility
Implied Volatility
0.9
0.65
0.64
0
Implied Volatility
1
0.8
0.1
0.66
0.35
1.1
0.45 Implied Volatility
0.8
Maturity = 3 0.84 Implied Volatility
1.3 Implied Volatility
Implied Volatility
Maturity = 1 1.3
0.45 0.4 0.35
0
0.05 Strike
0.1
Model Market
0.4 0.35 0.3 0.25
0
0.05 Strike
0.1
29 / 42
Affine LIBOR models with multiple curves
Calibration
Calibration – caplet data (multiple tenors) Maturity = 1
Maturity = 2
0.7 0.6
0
0.05 Strike
0.8 0.7 0.6 0.5
0.1
Implied Volatility
0.8
0.5
0
Maturity = 4
0.52
0.05 Strike
0.5
0.45
0.4
0.1
0
0.4
0.1
0.1
0.46 0.44 0.42 0.4 0.38
0.1
0
0.05 Strike
0.1
Maturity = 9 0.45 Implied Volatility
Implied Volatility
Implied Volatility
0.05 Strike
0.5
0.45
0.05 Strike
0.48
Maturity = 8
0.5
0.05 Strike
0
0.5
Maturity = 7
0
0.62
Maturity = 6
Implied Volatility
Implied Volatility
Implied Volatility
0.54
0
0.64
0.6
0.1
0.55
0.56
0.35
0.05 Strike
0.66
Maturity = 5
0.58
0.5
Maturity = 3 0.68
0.9 Implied Volatility
Implied Volatility
0.9
0.45 0.4 0.35
0
0.05 Strike
0.1
Model Market
0.4 0.35 0.3 0.25
0
0.05 Strike
0.1
30 / 42
Affine LIBOR models with multiple curves
Old and new examples
Affine processes I Model setup: 1
X = (Xt )0≤t≤TN a time-homogeneous Markov process in D = Rd>0
2
X is affine, if the moment generating function satisfies: Ex exphu, Xt i = exp φt (u) + hψt (u), xi
3
(17)
where x ∈ D, while φt (u) and ψt (u) are defined on [0, T ] × IT , where n o IT := u ∈ Rd : Ex ehu,XT i < ∞, for all x ∈ D , (18) the ’domain of exponential moments’.
4
The process X is a regular affine process in the spirit of DFS, and a semimartingale. 31 / 42
Affine LIBOR models with multiple curves
Old and new examples
Affine processes II
Lemma (Riccati equations) The functions φ and ψ satisfy the (generalized) Riccati ODEs ∂ φt (u) = F (ψt (u)), ∂t ∂ ψt (u) = R(ψt (u)), ∂t
φ0 (u) = 0,
(19a)
ψ0 (u) = u,
(19b)
for all suitable 0 ≤ t + s ≤ T and u ∈ IT .
32 / 42
Affine LIBOR models with multiple curves
Old and new examples
Affine processes III Lemma (Riccati equations II) [DFS] showed that F (u) :=
∂ φt (u) ∂t t=0+
and
R(u) :=
∂ ψt (u) ∂t t=0+
(20)
exist for all u ∈ IT and are continuous in u. Moreover, F and R satisfy L´evy–Khintchine-type equations: Z F (u) = hb, ui + ehξ,ui − 1i m(dξ) (21) D
and Ri (u) = hβi , ui +
Dα
i
2
E Z u, u +
ehξ,ui − 1 − hu, hi (ξ)i µi (dξ),
(22)
D
where (b, m, αi , βi , µi )1≤i≤d are admissible parameters.
33 / 42
Affine LIBOR models with multiple curves
Old and new examples
Affine processes IV Admissible parameters for R2>0 -valued affine processes: + b= , +
∗ β1 = , +
a = 0,
α1 =
+ ∗ , ∗ 0
+ β2 = ∗ 0 ∗ α2 = ∗ +
m, µ1 , µ2 are L´evy measures on R2>0 µ1 , µ2 can have infinite variation ’Classical’ examples: CIR, CIR with jumps, OU processes, . . . 34 / 42
Affine LIBOR models with multiple curves
Old and new examples
New examples I 2D CIR with dependent jumps q = − θ1 )dt + 2η1 Xt1 dWt1 + α1 dZt , q 2 2 dXt = −λ2 (Xt − θ2 )dt + 2η2 Xt2 dWt2 + α2 dZt . dXt1
−λ1 (Xt1
Functional characteristics F (u1 , u2 ) = λ1 θ1 u1 + λ2 θ2 u2 + ` 1 µ
Ri (u1 , u2 ) = −λi ui +
2ηi2 ui2 ,
u1 α1 + u2 α2 , − u1 α1 − u2 α2
i = 1, 2.
Explicit solution of the Riccati equations when λ1 = λ2 35 / 42
Affine LIBOR models with multiple curves
Old and new examples
New examples II Stochastic volatility/intensity on R2>0 p Xt dWt1 + dZt , p dVt = −λ2 (Vt − θ2 )dt + 2η2 Vt dWt2 , dXt = −λ1 (Xt − θ1 )dt + 2η1
where the intensity of Z is an affine function of V : Λt = `0 + `1 · Vt . Functional characteristics u1 , F (u1 , u2 ) = λ1 θ1 u1 + λ2 θ2 u2 + `0 1 µ − u1 R1 (u1 , u2 ) = −λ1 u1 + 2η12 u12 , R2 (u1 , u2 ) = −λ2 u2 + 2η22 u22 + `1
1 µ
u1 , − u1
Explicit solution of the Riccati equations when 2η12 = λ1 µ and λ2 = λ1 /2 36 / 42
Affine LIBOR models with multiple curves
Connections to other frameworks
Connection to LMMs Assume that X is an affine diffusion: OIS dynamics d √ dFkx 1 + δx Fkx X l x = ψT −t uk−1 − ψTl −t (ukx ) σlT X x,k;l dW x,k x x Fk δ x Fk l=1
FRA dynamics d √ 1 + δx Lxk X l dLxk = ψT −t (vkx ) − ψTl −t (ukx ) σlT X x,k;l dW x,k x x Lk δx Lk l=1
where x
W
x,k
=W
N
−
N d Z X X l=k+1 i=1
· x ψTi −t (ul−1 ) − ψTi −t (ulx )
q
Xti σi dt.
0
Built-in displacement Structure of volatility: completely determined by X 37 / 42
Affine LIBOR models with multiple curves
Connections to other frameworks
Connection to HJM & CVA
Extend the model to a continuous tenor: the instantaneous forward rate looks like f (t, T ) = p(t, T ) + q(t, T ) · Xt ,
(23)
while the short rate looks like rt = pt + qt · Xt ,
(24)
where p and q are known in terms of φ and ψ. Consistent CVA computations (work in progress)
38 / 42
Summary and Outlook
Summary and Outlook We have presented . . . a class of tractable LIBOR models for the multiple curve framework explicit examples (multi-dimensional, correlated) numerical methods for swaption pricing calibration to caplets
39 / 42
Summary and Outlook
Summary and Outlook We have presented . . . a class of tractable LIBOR models for the multiple curve framework explicit examples (multi-dimensional, correlated) numerical methods for swaption pricing calibration to caplets Open questions – future work: CVA computation, dependence strucures,. . .
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Summary and Outlook
Summary and Outlook We have presented . . . a class of tractable LIBOR models for the multiple curve framework explicit examples (multi-dimensional, correlated) numerical methods for swaption pricing calibration to caplets Open questions – future work: CVA computation, dependence strucures,. . . Z. Grbac, A. Papapantoleon, J. Schoenmakers, D. Skovmand Affine LIBOR models with multiple curves: theory, examples and calibration (SIAM J. on Financial Math. 2015)
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Summary and Outlook
Summary and Outlook We have presented . . . a class of tractable LIBOR models for the multiple curve framework explicit examples (multi-dimensional, correlated) numerical methods for swaption pricing calibration to caplets Open questions – future work: CVA computation, dependence strucures,. . . Z. Grbac, A. Papapantoleon, J. Schoenmakers, D. Skovmand Affine LIBOR models with multiple curves: theory, examples and calibration (SIAM J. on Financial Math. 2015)
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