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c©Eberlein, Uni Freiburg, 1
Multiple Curve Lévy Forward Price ModelAllowing For Negative Interest Rates
Ernst Eberlein
Department of Mathematical Stochasticsand
Center for Data Analysis and Modeling (FDM)University of Freiburg
Joint work with Christoph Gerhart and Zorana Grbac
12th Financial Risks International ForumParis, France March 18–19, 2019
c©Eberlein, Uni Freiburg, 1
20−06−2005 01−12−2006 02−06−2008 01−12−2009 01−06−2011 03−12−2012
0
20
40
60
80
100
120
140
160
180
200
220
240b
pEURIBOR−EONIA OIS rate Spread
12m 9m 6m 3m 1m
c©Eberlein, Uni Freiburg, 2
Basic interest rates
B(t ,T ): price at time t ∈ [0,T ] of a default-free zero coupon bond
f (t ,T ): instantaneous forward rate (short rate r(t) = f (t ,t))
B(t ,T ) = exp(−∫ T
t f (t ,u) du)
L(t ,T ): default-free forward Libor rate for the interval T to T + δ
L(t ,T ) := 1δ
(B(t,T )
B(t,T+δ) − 1)
FB(t ,T ,U): forward price process for the two maturities T and U
FB(t ,T ,U) := B(t,T )B(t,U)
=⇒ 1 + δL(t ,T ) = B(t ,T )B(t ,T + δ)
= FB(t ,T,T + δ)
c©Eberlein, Uni Freiburg, 3
ECB, Frankfurt
c©Eberlein, Uni Freiburg, 4
15−09−2006 01−03−2009 01−09−2011 01−03−2014
0.60.7
0.80.9
1.0
Bonds
15−09−2008 01−03−2011 01−09−2013 01−03−2016
0.60.7
0.80.9
1.0
Bonds
15−09−2010 01−03−2013 01−09−2015 01−03−2018
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Bonds
17−09−2012 01−03−2015 01−09−2017 01−03−2020
0.70
0.75
0.80
0.85
0.90
0.95
1.00
Bonds
15−09−2014 01−03−2017 01−09−2019 01−03−2022
0.80
0.85
0.90
0.95
1.00
Bonds
15−09−2016 01−03−2019 01−09−2021 01−03−2024
0.80
0.85
0.90
0.95
1.00
Bonds
Evolution of Discount Curves 2006 – 2016
c©Eberlein, Uni Freiburg, 5
09−02−2017 01−08−2018 01−02−2020 01−08−2021 01−02−2023 01−08−2024 01−02−2026
0.90
0.92
0.94
0.96
0.98
1.00
1.02
Discount Curves, February 9, 2017
c©Eberlein, Uni Freiburg, 6
16−09−2013 01−03−2016 01−09−2018 01−03−2021
0.75
0.80
0.85
0.90
0.95
1.00
Bonds
15−09−2016 01−03−2019 01−09−2021 01−03−2024
0.75
0.80
0.85
0.90
0.95
1.00
Bonds
16−09−2013 01−03−2016 01−09−2018 01−03−2021
0.000
0.005
0.010
0.015
0.020
0.025
0.030
0.035
Instantaneous forward rates
15−09−2016 01−03−2019 01−09−2021 01−03−2024
−0.00
50.0
000.0
050.0
100.0
15
Instantaneous forward rates
Derived Instantaneous Forward Rates 2013 and 2016
c©Eberlein, Uni Freiburg, 7
Bootstrapping of Initial Curves
The quoted overnight indexed swap rate (OIS) for a discrete tenorstructure T = {T0, . . . ,Tn} with tenor δ can be expressed in the form
Son0 (T ) =Bd0(T0)− Bd0(Tn)∑n
k=1 δBd0(Tk )
Quotes of the swap rates are given for increasing maturities.
From the Bd0(Tk ) we derive for each pair of consecutive datesTk−1,Tk ∈ T the discretely compounded forward reference rates
Ld(0,Tk−1,Tk ) =1δ
(Bd0(Tk−1)Bd0(Tk )
− 1)
c©Eberlein, Uni Freiburg, 8
0
1
2
3
2005−0
9−07
2006−0
9−07
2007−0
9−07
2008−0
9−07
2009−0
9−07
2010−0
9−07
Maturity
FRA R
ates (
in bp
s)
colour
3m
6m
Basic
(a)
0
1
2
3
4
5
2007−1
2−27
2008−1
2−27
2009−1
2−27
2010−1
2−27
2011−1
2−27
2012−1
2−27
Maturity
FRA R
ates (
in bp
s)
colour
3m
6m
Basic
(b)
0
1
2
3
4
2009−1
1−26
2010−1
1−26
2011−1
1−26
2012−1
1−26
2013−1
1−26
2014−1
1−26
Maturity
FRA R
ates (
in bp
s)
colour
3m
6m
Basic
(c)
0
1
2
2012−0
3−15
2013−0
3−15
2014−0
3−15
2015−0
3−15
2016−0
3−15
2017−0
3−15
Maturity
FRA R
ates (
in bp
s)
colour
3m
6m
Basic
(d)
0.0
0.4
0.8
2014−1
0−02
2015−1
0−02
2016−1
0−02
2017−1
0−02
2018−1
0−02
2019−1
0−02
Maturity
FRA R
ates (
in bp
s)
colour
3m
6m
Basic
(e)
−0.25
0.00
0.25
2016−0
9−15
2017−0
9−15
2018−0
9−15
2019−0
9−15
2020−0
9−15
2021−0
9−15
Maturity
FRA R
ates (
in bp
s)
colour
3m
6m
Basic
(f)
Historical Evolution of the Tenor-Dependent FRA Curves
c©Eberlein, Uni Freiburg, 9
The Initial 6-month CurveStart with its value at the maturity of six months by using the quoteddeposit rate R6m0 (0.5)
B6m0 (0.5) =1
1 + 0.5 · R6m0 (0.5).
For mid and long term maturities from one year upwards proceed bybootstrapping. Use quoted swap rates based on a 6-month floating legaccording to
S0(T 6m, T ) =0.5∑n6m
k=1 Bd0(T
6mk )L
6m(0,T 6mk−1,T6mk )
δ∑n
l=1 Bd0(Tl)
where
L6m(0,T 6mk−1,T6mk ) =
10.5
(B6m0 (T
6mk−1)
B6m0 (T6mk )
− 1
).
Values for the maturities of one and three months are added by usingrates of forward rate agreements
c©Eberlein, Uni Freiburg, 10
Literature on Modelling MultipleCurves
Kijima, Tanaka and Wong (2009)
Kenyon (2010), Mercurio (2009, 2010)
Bianchetti (2010), Morini (2009)
Crépey, Grbac and Nguyen (2012)
Crépey, Douady (2013)
Filipović and Trolle (2013)
Henrard (2010, 2014)
Grbac and Runggaldier (2015)
Cuchiero, Fontana and Gnoatto (2016, 2017)
Grbac, Papapantoleon, Schoenmakers and Skovmand (2015)
Crépey, Macrina, Nguyen and Skovmand (2016)
Eberlein and Gerhart (2018)
Macrina and Mahomed (2018)
Afeus, Grasselli and Schlögl (2018)
c©Eberlein, Uni Freiburg, 11
The Driving Process
L = (L1, . . . , Ld) is a d-dimensional time-inhomogeneous Lévyprocess, i.e. L has independent increments and the law of Lt is given bythe characteristic function
E[exp(i〈u, Lt〉)] = exp∫ t
0θs(iu) ds with
θs(z) = 〈z, bs〉+12〈z, csz〉+
∫Rd
(e〈z,x〉 − 1− 〈z, x〉
)Fs(dx)
where bt ∈ Rd , ct is a symmetric nonnegative-definite d × d-matrixand Ft is a Lévy measure
c©Eberlein, Uni Freiburg, 12
Description in Terms of ModernStochastic Analysis
L = (Lt) is a special semimartingale with canonical representation
Lt =∫ t
0bs ds +
∫ t0
c1/2s dWs +∫ t
0
∫Rd
x(µL − ν)(ds, dx)
and characteristics
At =∫ t
0bs ds, Ct =
∫ t0
cs ds, ν(ds, dx) = Fs(dx) ds
W = (Wt) is a standard d-dimensional Brownian motion,
µL the random measure of jumps of L and ν is the compensator of µL
c©Eberlein, Uni Freiburg, 13
Simulation of a GH Lévy motionNIG Levy process with marginal densities
t
valu
es o
f NIG
(100
,0,1
,0)
Levy
pro
cess
0.0 0.5 1.0 1.5 2.0
104.
610
4.8
105.
010
5.2
105.
4
c©Eberlein, Uni Freiburg, 14
The Basic Discount Curve (1)(following Eb., Özkan (2005))
Tenor structure T : 0 ≤ T0 < T1 < · · · < Tn = T ∗ with Tk −Tk−1 = δ
Bdt (T ): bond price at time t maturing at T
Ld(t ,Tk−1,Tk ): forward reference rate for the interval Tk−1 to Tk
Ld(t ,Tk−1,Tk ) :=1δ
(Bdt (Tk−1)Bdt (Tk )
− 1)
F d(t ,Tk−1,Tk ): forward price process for the maturities Tk−1, Tk
F d(t ,Tk−1,Tk ) :=Bdt (Tk−1)Bdt (Tk )
⇒ F d(t ,Tk−1,Tk ) = 1 + δLd(t ,Tk−1,Tk )
c©Eberlein, Uni Freiburg, 15
The Basic Discount Curve (2)Assumptions
(DFP.1) The initial term structure of bond prices Bd0 (T ) for T ∈ [0,T ∗]is given. This defines the starting values of the forwardprocesses
F d(0,Tk−1,Tk ) =Bd0 (Tk−1)Bd0 (Tk )
(DFP.2) For any maturity Tk−1 ∈ T there is a bounded, continuous anddeterministic function
λd(·,Tk−1) : [0,T ∗]→ Rd+.
We require that
λd(t ,Tk−1) = (0, . . . , 0) for t > Tk−1
Tn T *0 T T1 2 3TT =
c©Eberlein, Uni Freiburg, 16
Backward Induction (1)We postulate
F d(t ,Tn−1,Tn)
= F d(0,Tn−1,Tn) exp(∫ t
0λd(s,Tn−1)dLTns +
∫ t0
bd(s,Tn−1,Tn)ds)
where LTn = LT∗
is given in the form
LT∗
t =
∫ t0
√cs dW T
∗s +
∫ t0
∫Rd
x(µL − νT∗)(ds, dx)
with PdT∗ -Brownian motion WT∗ and compensator νT
∗(dt , dx).
Choose the drift bd(·,Tn−1,Tn) s.t. F d(·,Tn−1,Tn) becomes aPdT∗ -martingale
bd(t ,Tn−1,Tn) = −12λd(t ,Tn−1)ctλd(t ,Tn−1)>
−∫Rd(eλ
d (t,Tn−1)x − 1− λd(t ,Tn−1)x)F T∗
t (dx)
c©Eberlein, Uni Freiburg, 17
Backward Induction (2)
Define the forward martingale measure associated with Tn−1
dPdTn−1dPdTn
:=F d(Tn−1,Tn,Tn)F d(0,Tn−1,Tn)
By Girsanov’s theorem
W Tn−1t := WT∗t −
∫ t0
√cs λd(s,Tn−1)>ds
is a PdTn−1 -standard Brownian motion and
νTn−1(dt , dx) := exp(λd(t ,Tn−1)x)νT∗(dt , dx) = F Tn−1t (dx)dt
defines the PdTn−1 -compensator of µL.
c©Eberlein, Uni Freiburg, 18
Backward Induction (3)
Proceeding backwards along the tenor structure T one gets for eachk ∈ {1, . . . , n}
F d(t ,Tk−1,Tk )
= F d(0,Tk−1,Tk ) exp(∫ t
0λd(s,Tk−1)dL
Tks +
∫ t0
bd(s,Tk−1,Tk )ds)
where
LTkt =∫ t
0
√cs dW
Tks +
∫ t0
∫Rd
x(µL − νTk )(ds, dx)
and the drift term is chosen s.t. F d(·,Tk−1,Tk ) is a PdTk -martingale.
c©Eberlein, Uni Freiburg, 19
Multiple Term Structure Curves
Consider m tenor structures nested in T
T m ⊂ · · · ⊂ T 1 ⊂ T
T i = {T i0, . . . ,T ini }
0 ≤ T i0 = T0 < T i1 < · · · < T ini = Tn = T∗
year fractions: δi = δi(T ik−1,Tik ) independent of k
Li(T ik−1,Tik ): T
ik−1-spot Libor/Euribor rate
DefineLi(t ,T ik−1,T
ik ) := EdT ik [L
i(T ik−1,Tik ) | Ft ]
as the forward reference rate corresponding to δi .
c©Eberlein, Uni Freiburg, 20
Possibilities to Model the Spreads
(1) Additive forward spreads
si(t ,T ik−1,Tik ) := L
i(t ,T ik−1,Tik )− Ld(t ,T ik−1,T ik )
or
(2) Multiplicative forward spreads
S i(t ,T ik−1,Tik ) :=
1 + δiLi(t ,T ik−1,Tik )
1 + δiLd(t ,T ik−1,Tik )
=1 + δiLi(t ,T ik−1,T
ik )
F d(t ,T ik−1,Tik )
In the forward price framework the natural choice are multiplicativespreads.
Lemma
Li(·,T ik−1,T ik ) is a PdT ik -martingale
⇔ S i(·,T ik−1,T ik ) is a PdT ik−1-martingale
c©Eberlein, Uni Freiburg, 21
Model Variant (a)
Choose bounded, continuous, deterministic volatilities γ i(·,T ik−1)and postulate for each pair T ik−1,T
ik ∈ T i
S i(t ,T ik−1,Tik )
= S i(0,T ik−1,Tik ) exp
(∫ t0γ i(s,T ik−1)dL
T ik−1s +
∫ t0
bi(s,T ik−1)ds)
where bi(·,T ik−1) is chosen s.t. S i(·,T ik−1,T ik ) is a PdT ik−1-martingale
→ maximum of tractabilitybasic forward reference rate as well as the δi - forward ratescan become negative (the initial rates can be negative)multiplicative spread not necessarily ≥ 1
c©Eberlein, Uni Freiburg, 22
Model Variant (a) Continued
We get the following explicit form for the δi -forward rate
1 + δiLi(t ,T ik−1,Tik ) = S
i(t ,T ik−1,Tik )F
d(t ,T ik−1,Tik )
=(
1 + δiLi(0,T ik−1,Tik ))
exp
(∫ t0
[∑j∈Jik
λd(s,Tj−1) + γ i(s,T ik−1)]dLT
ik
s
+
∫ t0
[bi(s,T ik−1) + 〈γ i(s,T ik−1),w(s,T ik−1,T ik )〉
+∑j∈Jik
[〈λd(s,Tj−1),w(s,Tj ,T ik )〉+ bd(s,Tj−1,Tj)
]]ds
).
c©Eberlein, Uni Freiburg, 23
Model Variant (b)
Choose again volatilities γ i(·,T ik−1) as before and postulate for eachpair T ik−1,T
ik ∈ T i
S i(t ,T ik−1,Tik )− 1
S i(0,T ik−1,Tik )− 1
= exp(∫ t
0γ i(s,T ik−1)dL
T ik−1s +
∫ t0
bi(s,T ik−1)ds
)where
bi(t ,T ik−1) = −
12γ i(t ,T ik−1)ctγ
i(t ,T ik−1)>
−∫Rd(eγ
i (t,T ik−1)x − 1− γ i(t ,T ik−1)x)FT ik−1t (dx).
→ reasonable tractabilitymultiplicative spread is > 1
c©Eberlein, Uni Freiburg, 24
Calibration
Pricing formula for caps with tenor length δl : T + δl = Tk
Cpl(t ,T , δl ,K )
= δlBdt (Tk )EdTk[ (
Ll(T ,Tk )− K)+| Ft]
= Bdt (Tk )EdTk[ (
1 + δlLl(T ,T ,Tk )− (1 + δlK ))+| Ft]
= Bdt (Tk )EdTk[(
F d(T ,T ,Tk )S l(T ,T ,Tk )− K̃ l)+| Ft]
= Bdt (Tk )(Zkt )−1EdT∗
[Z kT(
F d(T ,T ,Tk )S l(T ,T ,Tk )− K̃ l)+| Ft]
c©Eberlein, Uni Freiburg, 25
Calibration Continued
Assumption: volatility functions are decomposable
λd(t ,T ) = λd1(T )λ(t)
andγ l(t ,T ) = γ l1(T )λ(t)
Define XT :=∫ T
0λ(s)dLT
∗s
⇒ Cpl(0,T , δl ,K ) = Bd0 (Tk ) EdT∗ [fk,lK (XT )] for a function f
k,lK
Applying the Fourier-based approach one gets
Cpl(0,T , δl ,K ) =Bd0 (Tk )π
∫ ∞0
Re(ϕXT (u − iR) f̂
k,lK (iR − u)
)du
c©Eberlein, Uni Freiburg, 26
Calibration Results
Maturity
2.5
3.0
3.5
4.0
4.5
5.0
Strike
Rate
−0.2
0.0
0.2
0.4
Market and M
odel Volatility
0
20
40
60
80
(a) Model (a)
Maturity
2.5
3.0
3.5
4.0
4.5
5.0
Strike
Rate
−0.2
0.0
0.2
0.4M
arket and Model V
olatility
0
20
40
60
80
(b) Model (b)
Figure: Calibrated Volatility Surfaces on September 15, 2016
c©Eberlein, Uni Freiburg, 27
References
Ametrano, F. M. and Bianchetti, M.: Everything You Always Wanted toKnow About Multiple Interest Rate Curve Bootstrapping But Were Afraidto Ask. Available at SSRN 2219548, 2013
Eberlein, E., Gerhart, C.: A Multiple-Curve Lévy Forward Rate Model ina Two-Price Economy. Quantitative Finance, 18(4): 537-561, 2018
Eberlein, E., Gerhart, C., and Grbac, Z.: Multiple-Curve Lévy ForwardPrice Model Allowing for Negative Interest Rates. (to appear inMathematical Finance), 2018
Eberlein, E. and Özkan, F.: The Lévy Libor Model. Finance andStochastics, 9(3):327–348, 2005
Eberlein, E., Glau, K., and Papapantoleon, A.: Analysis of FourierTransform Valuation Formulas and Applications. Applied MathematicalFinance, 17(3):211–240, 2010