a term-structure model for dividends and interest … · a term-structure model for dividends and...

A Term-Structure Model for Dividends and InterestRates

Sander WillemsJoint work with Damir Filipovic

School and Workshop on Dynamical Models in FinanceMay 24th 2017

Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 1 / 29

Overview

1 Introduction

2 Dividend Futures and Bonds

3 Dividend Paying Stock

4 Empirical Analysis

5 Derivative Pricing


A new market for dividend derivatives

How can we trade dividends?I Synthetic replication.I Dividend swaps (OTC) or dividend futures (on exchange).I Latest innovations: single names, options, dividend-rates hybrids, . . .

Asset pricing: term structure of equity risk premium.I Lettau and Wachter (2007), Binsbergen et al. (2012), Binsbergen et al.

(2013), Binsbergen and Koijen (2017).

Dividend derivative pricing.I Buehler et al. (2010), Tunaru (2014), Buehler (2015), Kragt et al.

(2016).

Derivative pricing with dividend paying stockI Deterministic dividends: Bos and Vandermark (2002), Bos et al.

(2003), Vellekoop and Nieuwenhuis (2006).I Proportional dividends: Merton (1973), Korn and Rogers (2005).

Interest rates: hybrid products, long maturity dividend claims.


Overview

1 Introduction






State Process

Filtered probability space (Ω,F ,Ft ,Q), with Q the risk-neutralpricing measure.

Multivariate state process Xt in E ⊆ Rd with linear drift:

dXt = κ(θ − Xt)dt + dMt ,

for κ ∈ Rd×d , θ ∈ Rd , and some d-dimensional martingale Mt .

First moment is linear in the state:

Et

[(1XT

)]= eA(T−t)

(1Xt

), A =

(0 0κθ −κ

), t ≤ T .


Dividend Futures

Instantaneous dividend rate:

Dt = p + q>Xt ,

for p ∈ R, q ∈ Rd such that p + q>x ≥ 0 for all x ∈ E .

Dividend futures price:

Dfut(t,T1,T2) = Et

[∫ T2

T1

Ds ds

]=(p q>

) ∫ T2

T1

eA(s−t) ds

(1Xt

).

If κ is invertible:

Dfut(t,T1,T2) =(T2 − T1)(p + q>θ

)−

q>κ−1(e−κ(T2−t) − e−κ(T1−t)

)(Xt − θ).


Dividend Seasonality

Time

2009 2010 2011 2012 2013 2014 2015 2016 2017

DV

P I

nd

ex p

oin

ts

0

10

20

30

40

50

60

70

Figure: Monthly dividend payments by Eurostoxx 50 constituents (in index points)from October 2009 until October 2016. Source: Eurostoxx 50 DVP index,Bloomberg.


Dividend Seasonality

Standard choice to model annual cycles:

δ(t) = ρ0 + ρ>Γ(t), Γ(t) =

sin(2πt)cos(2πt)

...sin(2πKt)cos(2πKt)

.

Remark, Γ(t) is the solution of a linear ODE:

dΓ(t) = blkdiag((

0 2π−2π 0

), . . . ,

(0 2πK

−2πK 0

))Γ(t)dt.

→ We can add Γ to the state vector!

For example:

dXt = κ(δ(t)− Xt)dt + dMt


Interest Rates

Risk-neutral discount factor:

ζt = ζ0e−

∫ t0 rs ds , t ≥ 0,

where rt denotes the short rate.

Directly specify dynamics for ζt :

ζt := e−γtYt , dYt = λ(φ+ ψ>Xt − Yt)dt,

for φ, λ, γ ∈ R and ψ ∈ Rd such that Yt > 0.

Cfr. Filipovic et al. (2017), Ackerer and Filipovic (2016).


Bond Prices

Time-t price of zero-coupon bond maturing at T :

P(t,T ) =1

ζtEt [ζT ]

=e−γ(T−t)

Yte>d+2 e

B(T−t)

1Xt

Yt

, B =

0 0 0κθ −κ 0λφ λψ> −λ

.

Implied short rate:

rt = γ + λ− λφ+ ψ>Xt

Yt.

If all eigenvalues of κ have positive real parts and λ > 0:

limT→∞

− log(P(t,T ))

T − t= γ.


Overview

1 Introduction






GARCH Diffusion

Specify martingale part Mt as follows

dXt = κ(θ − Xt)dt + diag(X1t , . . . ,Xdt)ΣdBt , (1)

with Bt a standard d-dimensional Brownian motion and Σ ∈ Rd×d

lower triangular with Σii > 0.

Used before for stochastic volatility (Nelson (1990), Barone-Adesiet al. (2005)), energy markets (Pilipovic (1997)), interest rates(Brennan and Schwartz (1979)), and Asian option pricing (Linetsky(2004)).

Attractive features:I Unique positive solution.I Flexible correlation structure.I Moments in closed-form.


GARCH Diffusion

Proposition

Consider the following system of SDEs:dXt = κ(θ − Xt)dt + diag(X1t , . . . ,Xdt)ΣdBt ,dYt = λ(φ+ ψ>Xt − Yt)dt

If (X0,Y0) ∈ Rd+1+ , and

(κθ)i , ψi ≥ 0, λ, φ ≥ 0, κij ≤ 0 for i 6= j ,

then the system has a unique strong solution in Rd+1+ .


Moment Formula

(Xt ,Yt) is a polynomial diffusions, cfr. Filipovic and Larsson (2015).

Fix basis for Poln(Rd × R), n ≥ 1:

Hn(x , y) = (1, h1(x , y), . . . , hNn(x , y))>,

with Nn =(n+d+1

n

)− 1.

There exists matrix Gn such that for any z ∈ Poln(Rd × R)

Et [z(XT ,YT )] = ~z>eGn(T−t)Hn(Xt ,Yt),

where ~z is the coordinate representation of z with respect to thechosen basis.

Efficient algorithms exist for ~z>eGn(T−t), e.g. Al-Mohy and Higham(2011).


Dividend Paying Stock

Absence of arbitrage:

St =1

ζtEt [ζTST ] +

1

ζtEt

[∫ T

tζsDs ds

], ∀T ≥ t.

If <(eig(G2)) < γ, then

1

ζtEt

[∫ ∞t

ζsDs ds

]=

1

Yt~v> (γI − G2)−1 H2(Xt ,Yt) <∞,

with ~v the coordinate vector of (x , y) 7→ y(p + q>x) wrt H2(x , y).

Stock price representation:

St =Ltζt

+1

ζtEt

[∫ ∞t

ζsDs ds

], t ≥ 0,

for some non-negative martingale Lt , cfr. Buehler (2010).


Overview

1 Introduction






Data

Eurostoxx 50 stock index

Eurostoxx 50 dividend futuresI Underlying: sum of declared ordinary gross cash dividends (or cash

equivalent) with ex-date during one calendar year, divided by indexdivisor on ex-date.

I Fixed maturity dates in Dec of each year, up to 10y.I Example: Today we can trade in maturities Dec(17+k), k = 0, . . . , 9.

Payoff of Dec(17+k) contract is sum of dividends in [Dec(17+(k − 1)),Dec(17+k)].

I We use 2nd, 3rd, 4th, 5th, 7th, and 10th contract.

Euribor interest rate swapsI Fixed leg pays annual, floating leg semi-annual.I Fixed time to maturities: 1,2,3,5,7, and 10 years.

Daily observations from October 2009 until October 2016,(1 + 6 + 6)× 1827 = 23, 751 observations in total.


Summary Statistics

Mean Median Std Min Max

Swap rates (%)1 yrs 0.66 0.42 0.63 −0.23 2.032 yrs 0.76 0.52 0.72 −0.25 2.463 yrs 0.91 0.67 0.80 −0.25 2.795 yrs 1.25 1.07 0.91 −0.18 3.227 yrs 1.57 1.44 0.96 −0.03 3.5010 yrs 1.94 1.87 0.97 0.24 3.77

Dividend futures1-2 yrs 109.42 110.30 6.63 87.10 125.302-3 yrs 104.93 106.80 8.88 81.20 119.903-4 yrs 101.57 103.10 9.95 74.10 120.604-5 yrs 99.60 100.80 10.50 70.80 122.906-7 yrs 96.85 97.40 11.85 69.90 126.509-10 yrs 95.69 95.80 14.19 63.00 132.50

Eurostoxx 50 index 2,877.62 2,890.35 363.15 1,995.01 3,828.78


Model Specification

We model Xt = (X It ,X

Dt ) and set d = 2× 2:

dX I1t = κI1(X I

2t − X I1t) dt + X I

1tΣI1 dB

Pt

dX I2t = κI2(θI − X I

2t)dt + X I2tΣ

I2 dB

Pt

dXD1t = κD1 (XD

2t − XD1t ) dt + XD

1tΣD1 dBP

t

dXD2t = κD2 (θD − XD

2t ) dt + XD2tΣD

2 dBPt

,

with BPt a 4-dimensional P-Brownian motion and

Σ =

ΣI

1

ΣI2

ΣD1

ΣD2

=

ΣI

11

0 ΣI22

ΣDI11 0 ΣD

11

0 ΣDI22 0 ΣD

22

, ψ =

1000

, q =

0010

.

Constant market price of risk vector Λ ∈ R4:

EPt

[dQdP

]= exp

(Λ>BP

t −1

2‖Λ‖2t

).

Restrictions on parameters such that: (Xt ,Yt) > 0,limT→∞ EP[H2(XT ,YT )] <∞, and St <∞, ∀t > 0.


Parameter Estimation

Quasi-maximum likelihood + Kalman filtering.

Transition equation: Zt = Φ0 + Φ1Zt−1 + εt , εt ∼ N (0, q(Zt−1)) .

Measurement equation: Mt = h(Zt) + νt , νt ∼ N (0, σ2MId).

I Dividend futures: linear.I Swap rates and stock price: non-linear.→ Unscented Kalman filter.

All observations are scaled by their sample mean.

Three estimation steps:I Estimate κD1 , κ

D2 , θD ,Σ

D11,Σ

D22 from dividend futures.

I Estimate κI1, κI2, θI ,Σ

I11,Σ

I22 from swap rates.

I Re-estimate all parameters using all instruments.


Parameter Estimates

εI1 εI2 εD1 εD2 ΣI11 ΣI

22 ΣD11

0.124∗ 0.005∗∗∗ 0.122∗∗∗ 0.000 0.017∗∗ 0.110∗∗∗ 0.120∗∗∗

(0.076) (0.001) (0.015) (0.002) (0.009) (0.004) (0.012)

ΣD22 Λ1 Λ2 Λ3 Λ4 ΣDI

1,1 ΣDI2,2

0.162∗∗∗ −0.413∗∗∗ −0.300∗∗∗ −0.426∗∗∗ 0.202∗∗∗ −0.024∗∗ −0.022∗∗

(0.011) (0.040) (0.030) (0.082) (0.042) (0.011) (0.010)

θD γ λ σM L × 10−4

0.778∗∗∗ 0.032∗∗∗ 0.132∗∗ 0.044∗∗∗ 3.852(0.049) (0.002) (0.079) (0.000)

Table: Quasi-maximum likelihood estimates with asymptotic standard deviationsin parenthesis.

Corresponding instantaneous correlation matrix:

X I1 X I

2 XD1 XD

2

X I1 1.00

X I2 0.00 1.00

XD1 −0.19 0.00 1.00

XD2 0.00 −0.13 0.00 1.00


Error Analysis

Maturities

Swap rates All 1 y 2 y 3 y 5 y 7 y 10 y

RMSE (bps) 6.62 3.98 4.59 5.92 3.40 6.60 11.65MAE (bps) 4.85 3.24 3.55 4.94 2.67 5.36 9.32

Dividend futures All 1-2 y 2-3 y 3-4 y 4-5 y 6-7 y 9-10 y

RMSRE (%) 2.70 2.86 1.72 2.37 2.29 2.24 4.09MARE (%) 1.93 2.17 1.14 1.72 1.76 1.66 3.09

Eurostoxx 50

RMSRE (%) 5.38MARE (%) 4.57

Table: The first five days of the sample are dropped when computing the errorstatistics to give the Kalman filter time to learn the current value of Xt . Theremaining sample period consists of 1,822 daily observations between October8, 2009 and October 1, 2016.


Filtered State

Jan09 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Jan17

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

Yt

XI

1t

XI

2t

(a) Interest rate factors

Jan09 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Jan17

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

XD

1t

XD

2t

(b) Dividend factors


Filtered Swap Rates

Time

Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16

Sw

ap

ra

te (

%)

-1

0

1

2

3

4Maturity 1 yrs

Filtered

Observed

Time


Sw

ap

ra

te (

%)

-1

0

1

2

3

4Maturity 2 yrs

Filtered

Observed

Time


Sw

ap

ra

te (

%)

-1

0

1

2

3

4Maturity 3 yrs

Filtered

Observed

Time


Sw

ap

ra

te (

%)

-1

0

1

2

3

4Maturity 5 yrs

Filtered

Observed

Time


Sw

ap

ra

te (

%)

-1

0

1

2

3

4Maturity 7 yrs

Filtered

Observed

Time


Sw

ap

ra

te (

%)

-1

0

1

2

3

4Maturity 10 yrs

Filtered

Observed


Filtered Dividend Futures Prices

Time


Fu

ture

s p

rice

60

70

80

90

100

110

120

130

140Maturity 1-2 yrs

Filtered

Observed

Time


Fu

ture

s p

rice

60

70

80

90

100

110

120

130

140Maturity 2-3 yrs

Filtered

Observed

Time


Fu

ture

s p

rice

60

70

80

90

100

110

120

130

140Maturity 3-4 yrs

Filtered

Observed

Time


Fu

ture

s p

rice

60

70

80

90

100

110

120

130

140Maturity 4-5 yrs

Filtered

Observed

Time


Fu

ture

s p

rice

60

70

80

90

100

110

120

130

140Maturity 6-7 yrs

Filtered

Observed

Time


Fu

ture

s p

rice

60

70

80

90

100

110

120

130

140Maturity 9-10 yrs

Filtered

Observed


Filtered Index Level

Time


Ind

ex le

ve

l

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

Filtered

Observed


Risk Premium Analysis

Mean (%) Std (%) Sharpe β

Dividend spot2 yrs 0.23 3.02 0.08 0.713 yrs 0.17 2.82 0.06 0.854 yrs 0.12 2.73 0.04 0.965 yrs 0.07 2.71 0.03 1.066 yrs 0.04 2.73 0.01 1.137 yrs 0.02 2.77 0.01 1.1910 yrs -0.02 2.86 -0.01 1.30

Bonds2 yrs 0.02 0.13 0.15 0.023 yrs 0.03 0.22 0.14 0.044 yrs 0.04 0.32 0.13 0.065 yrs 0.06 0.44 0.13 0.096 yrs 0.07 0.56 0.12 0.127 yrs 0.08 0.68 0.12 0.1510 yrs 0.11 1.03 0.11 0.23

Index 0.05 2.09 0.02 1.00

Table: Monthly returns in excess of the 1-month risk-free rate.


Overview

1 Introduction






Derivative Pricing with Polynomial Expansions

Denote Zt = (Xt ,Yt) and Z = (Zt1 , . . . ,Ztn) for some finite timepartition t1 < · · · < tn.

Time-t price of a (path dependent) derivative:

πt = Et [F (Z)],

for some discounted payoff function F on E = (Rd × R)n.

Denote by g(dz) the (unknown) conditional distribution of Z.

Let w(dz) be an auxiliary distribution such that g(dz) w(dz) and

g(dz) = `(z)w(dz).

Define Hilbert space L2w (E) with norm and scalar product

‖f ‖2w =

∫Ef (z)2w(dz), 〈f , h〉w =

∫Ef (z)h(z)w(dz).



Assumptions:

1 Pol(E) ⊂ L2w , 2 ` ∈ L2

w , 3 F ∈ L2w , 4 g w .

Let H = H0(z) = 1,H1(z),H2(z), . . . be an orthonormal set ofpolynomials spanning the closure Pol(E) in L2

w

Let F be the orthogonal projection of F onto Pol(E) in L2w .

Elementary functional analysis now gives:

πt = E[F (Z)] = 〈F , `〉w =∑k≥0

Fk`k ,

Fk = 〈F ,Hk〉w = 〈F ,Hk〉w =

∫EF (z)Hk(z)w(dz),

`k = 〈`,Hk〉H = Et [Hk(Z)].



Truncating the series for π, we get:

π(K)t =

K∑k=0

Fk lk

= πt + (πt − πt)︸︷︷︸projection bias

+ (π(K)t − πt)︸︷︷︸

truncation error

.

(πt − πt) = 0 if Pol(E) = L2w .

(π(K)t − πt)→ 0 as K →∞.

Crucial question: how to choose the auxiliary distribution?


The Auxiliary Distribution

We choose the multivariate log-normal distribution:

Definition

A random vector (X1, . . . ,Xk) ∈ Rk+, k ≥ 1, is said to have a multivariate

log-normal distribution LN (µ,Λ) if

(log(X1), . . . , log(Xk)) ∼ N (µ,Λ),

for some µ ∈ Rk and some positive semi-definite Λ ∈ Rk×k .

Very easy to simulate from a log-normal.

Finite moments of any order:

E[Xα11 · · ·X

αkk ] = exp

α>µ+

1

2α>Λα

<∞, ∀α ∈ Nk .

→ Assumption 1 is always satisfied

Moment indeterminate → projection bias.


The Auxiliary Distribution

Theorem

Let m = (d + 1)n. Suppose that the random vector Z = (Zt1 , . . . ,Ztn)admits a continuous density with support on Rm

+. Let w be the LN (µ,Λ)distribution with µ ∈ Rm and pos. def. Λ ∈ Rm×m. Define the matrixM ∈ Rn×n as

M =1

σ2T−1 − (In ⊗ 1>d+1)Λ−1(In ⊗ 1d+1),

where σ2 = max (ΣΣ>)ij | i , j = 1, . . . , d, and T = (ti ∧ tj)1≤i ,j≤n. If M ispositive semi-definite, then assumption 2 is satisfied.

Lemma

Suppose that Σ has strictly positive diagonal elements, λψ is differentfrom the zero vector, and (X0,Y0) ∈ Rd+1

+ . Then for any t > 0, therandom vector Zt = (Xt ,Yt) has an infinitely differentiable density.


Examples of (discounted) derivative payoffs

Swaption:

ζT0

ζt

(πswapT0

)+=

e−γT0

Yt

(w>swap H1(XT0 ,YT0)

)+.

Dividend option: Denote It =∫ t

0 Ds ds,

ζT1

ζt

(∫ T1

T0

Ds ds − K

)+

=e−γ(T1−t)

YtYT1 (IT1 − IT0 − K )+ ,

Stock option:

ζTζt

(ST − K )+ =e−γ(T−t)

Yt

(~v> (γI − G2)−1 H2(XT ,YT )− YTK

)+.

Dividend-Rates hybrid:

ζTζt

(∫ TT−1 Ds ds

ST− L1y

T

)+

.


Thank you for your attention!

References I

Ackerer, D. and D. Filipovic (2016). Linear credit risk models. Swiss Finance InstituteResearch Paper (16-34).

Al-Mohy, A. H. and N. J. Higham (2011). Computing the action of the matrixexponential, with an application to exponential integrators. SIAM Journal onScientific Computing 33(2), 488–511.

Barone-Adesi, G., H. Rasmussen, and C. Ravanelli (2005). An option pricing formula forthe GARCH diffusion model. Computational Statistics & Data Analysis 49(2),287–310.

Binsbergen, J. H. v., M. W. Brandt, and R. S. Koijen (2012). On the timing and pricingof dividends. American Economic Review 102(4), 1596–1618.

Binsbergen, J. H. v., W. Hueskes, R. S. Koijen, and E. B. Vrugt (2013). Equity yields.Journal of Financial Economics 110(3), 503–519.

Binsbergen, J. H. v. and R. S. Koijen (2017). The term structure of returns: Facts andtheory. Journal of Financial Economics, Forthcoming.

Bos, M., A. Shepeleva, and A. Gairat (2003). Dealing with discrete dividends. Risk,109–112.

Bos, M. and S. Vandermark (2002). Finessing fixed dividends. Risk, 157–158.

References II

Brennan, M. J. and E. S. Schwartz (1979). A continuous time approach to the pricingof bonds. Journal of Banking & Finance 3(2), 133–155.

Buehler, H. (2010). Volatility and dividends-volatility modelling with cash dividends andsimple credit risk. Working Paper .

Buehler, H. (2015). Volatility and dividends II-Consistent cash dividends. WorkingPaper .

Buehler, H., A. S. Dhouibi, and D. Sluys (2010). Stochastic proportional dividends.Working Paper .

Filipovic, D. and M. Larsson (2015). Polynomial preserving diffusions and applications infinance. Finance and Stochastics, 14–54.

Filipovic, D., M. Larsson, and A. B. Trolle (2017). Linear-rational term structuremodels. Journal of Finance 72, 655–704.

Korn, R. and L. G. Rogers (2005). Stocks paying discrete dividends: modeling andoption pricing. Journal of Derivatives 13(2), 44–48.

Kragt, J., F. De Jong, and J. Driessen (2016). The dividend term structure. WorkingPaper .

Lettau, M. and J. A. Wachter (2007). Why is long-horizon equity less risky? Aduration-based explanation of the value premium. Journal of Finance 62(1), 55–92.

References III

Linetsky, V. (2004). Spectral expansions for Asian (average price) options. OperationsResearch 52(6), 856–867.

Merton, R. C. (1973). Theory of rational option pricing. Bell Journal ofEconomics 4(1), 141–183.

Nelson, D. B. (1990). ARCH models as diffusion approximations. Journal ofEconometrics 45(1), 7–38.

Pilipovic, D. (1997). Energy risk: Valuing and managing energy derivatives.

Tunaru, R. (2014). Dividend derivatives. Working Paper .

Vellekoop, M. H. and J. W. Nieuwenhuis (2006). Efficient pricing of derivatives onassets with discrete dividends. Applied Mathematical Finance 13(3), 265–284.

Open interest Daily volume

Time to maturity Mean Median Mean Median

1-2 yrs 178,143 177,906 4,692 3,6152-3 yrs 132,100 129,800 3,892 2,9973-4 yrs 87,059 85,116 2,461 1,8714-5 yrs 57,530 54,968 1,514 1,0256-7 yrs 23,169 22,211 421 1339-10 yrs 3,354 1,507 85 0

Table: Open Interest and Daily Volume of Dividend Futures

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