multiple curve lévy forward price model allowing for ......jan 03, 2020 · multiple curve levy...

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


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


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)


ECB, Frankfurt


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


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


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


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)


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


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


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)


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


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


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


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 )


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 =


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)


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.


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.


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 .


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


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


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

).


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


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]


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


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


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

multiple curve lévy forward price model allowing for ......jan 03, 2020 · multiple curve levy...

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