a term-structure model for dividends and interest … · a term-structure model for dividends and...
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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
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
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 1 / 29
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.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 2 / 29
Overview
1 Introduction
2 Dividend Futures and Bonds
3 Dividend Paying Stock
4 Empirical Analysis
5 Derivative Pricing
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 2 / 29
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 .
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 3 / 29
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).
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 4 / 29
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.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 5 / 29
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
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 6 / 29
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).
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 7 / 29
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.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 8 / 29
Overview
1 Introduction
2 Dividend Futures and Bonds
3 Dividend Paying Stock
4 Empirical Analysis
5 Derivative Pricing
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 8 / 29
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.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 9 / 29
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+ .
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 10 / 29
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).
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 11 / 29
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).
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 12 / 29
Overview
1 Introduction
2 Dividend Futures and Bonds
3 Dividend Paying Stock
4 Empirical Analysis
5 Derivative Pricing
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 12 / 29
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.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 13 / 29
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
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 14 / 29
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.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 15 / 29
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.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 16 / 29
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
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 17 / 29
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.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 18 / 29
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
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 19 / 29
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
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Sw
ap
ra
te (
%)
-1
0
1
2
3
4Maturity 2 yrs
Filtered
Observed
Time
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Sw
ap
ra
te (
%)
-1
0
1
2
3
4Maturity 3 yrs
Filtered
Observed
Time
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Sw
ap
ra
te (
%)
-1
0
1
2
3
4Maturity 5 yrs
Filtered
Observed
Time
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Sw
ap
ra
te (
%)
-1
0
1
2
3
4Maturity 7 yrs
Filtered
Observed
Time
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Sw
ap
ra
te (
%)
-1
0
1
2
3
4Maturity 10 yrs
Filtered
Observed
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 20 / 29
Filtered Dividend Futures Prices
Time
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Fu
ture
s p
rice
60
70
80
90
100
110
120
130
140Maturity 1-2 yrs
Filtered
Observed
Time
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Fu
ture
s p
rice
60
70
80
90
100
110
120
130
140Maturity 2-3 yrs
Filtered
Observed
Time
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Fu
ture
s p
rice
60
70
80
90
100
110
120
130
140Maturity 3-4 yrs
Filtered
Observed
Time
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Fu
ture
s p
rice
60
70
80
90
100
110
120
130
140Maturity 4-5 yrs
Filtered
Observed
Time
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Fu
ture
s p
rice
60
70
80
90
100
110
120
130
140Maturity 6-7 yrs
Filtered
Observed
Time
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Fu
ture
s p
rice
60
70
80
90
100
110
120
130
140Maturity 9-10 yrs
Filtered
Observed
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 21 / 29
Filtered Index Level
Time
Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16
Ind
ex le
ve
l
1500
2000
2500
3000
3500
4000
4500
5000
5500
6000
Filtered
Observed
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 22 / 29
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.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 23 / 29
Overview
1 Introduction
2 Dividend Futures and Bonds
3 Dividend Paying Stock
4 Empirical Analysis
5 Derivative Pricing
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 23 / 29
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).
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 24 / 29
Derivative Pricing with Polynomial Expansions
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)].
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 25 / 29
Derivative Pricing with Polynomial Expansions
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?
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 26 / 29
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.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 27 / 29
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.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 28 / 29
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
)+
.
Sander Willems (SFI@EPFL) A TSM for Dividends and Interest Rates May 24th 2017 29 / 29
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.
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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