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IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Dividend Discount Approach to Equity Derivatives Modelling
Oliver BrockhausMathFinance AG
oliver.brockhaus@mathfinance.com
Cass Business SchoolLondon, 3 December 2014
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 1 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Agenda
1 IntroductionResearchEquity dynamicsDividend discount modelsCalibration
2 Affine dividendsBos-VandermarkAffine stock processAffine dividend dynamics
3 Homogenous dividend dynamicsIndicator functionKorn-RogersPiecewise linearExponential
4 ConclusionSummaryReferences
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 2 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
ResearchEquity dynamicsDividend discount modelsCalibration
Research
There are various directions of research on the topic of dividends:
Empirical properties, forecasting:van Binsbergen et al. [vBBK10]
Dividends within equity dynamics:Bos-Vandermark [BV02] (closed form formula with discrete dividends)Overhaus et al. [OFK+02] and [OBB+07] (affine model)Vellekoop, Nieuwenhuis [VN06] (model survey)
Dividend discount models:Korn-Rogers [KR05] (dividend announcement)Bernhart, Mai [BM12] (general framework)
Dividend derivatives modelling:Tunaru [Tun14] (uncertain timing, cum dividend process)Buhler et al. [BDS10] (discrete dividends with stochastic yield)
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 4 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
ResearchEquity dynamicsDividend discount modelsCalibration
Equity dynamics
When modelling equity dynamics there are two popular ways of incorporatingdividends:
Continuous dividend payment stream, proportional to spot level S , namelycontinuous dividend yield q:
dSt
St= (rt − qt)dt + σtdWt
Discrete dividends with known future ex-dividend dates0 < t1 < t2 < . . . , namely
D iti = fi (Sti−)
with deterministic fi . In practice one often assumesdeterministic payments in the near future: fi = ciproportional dividends for the distant future: fi (Sti−) = ciSti−Positivity of S is preserved if fi is capped at Sti− or fi (x) ≤ x .
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 5 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
ResearchEquity dynamicsDividend discount modelsCalibration
Dividend discount models
Dividend discount models assume that the stock price St can be viewed as thepresent value of all future dividends, namely
St =∑ti>t
D it (1)
where D it denotes the value discounted to t of the i-th dividend going ex at
time ti . Also, for any given stock model (St , t ≥ 0) with discrete dividends onehas
Sti− − Sti = D iti (2)
In this presentation we address the following questions:
1 Given a stock model (St , t ≥ 0) with discrete dividends, are thereprocesses D i such that (1) holds?
2 Do these processes D i have ‘reasonable’ properties?
3 What is the dynamics of (St , t ≥ 0) if one starts with ‘reasonable’processes D i?
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 6 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
ResearchEquity dynamicsDividend discount modelsCalibration
Calibration
Let Ft and Pt denote equity forward and discount factor, respectively. One hasthe calibration condition
PtFt = S0 −∑ti≤t
D i0 =
∑ti>t
D i0 (3)
If a stock does not pay dividends then one may assume a single dividendpayment at infinity with value Dt = St at time t. Otherwise one can assume D i
0
to be known for all i up to some index k.With exponential growth thereafter, namely
D i0 = Dk
0 e−q∆(i−k) and ti = tk + ∆(i − k)
for all i ≥ k and some time increment ∆ one computes
q∆ = logS0 −
∑i<k D
i0
S0 −∑
i≤k Di0
= logPtk−1Ftk−1
PtkFtk
from (1) or (3) with t = 0. Growth parameter q can be interpreted as longterm dividend yield.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 7 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Bos-VandermarkAffine stock processAffine dividend dynamics
Bos-Vandermark
In a famous article Bos and Vandermark [BV02] present a variant of the BlackScholes formula, namely
CBV (K , t) = PtBlack(P−1t (S0 − X n
t ),K + P−1t X f
t , σ2t)
(4)
with deterministic functions
X nt =
∑ti≤t
(t − ti )+
tD i
0 X ft =
∑ti≤t
ti ∧ t
tD i
0
and volatility σ.
This formula is a good approximation for a European option price withdeterministic dividend drops and volatility σ in between. Additionally, theformula is continuous both as valuation date moves across a dividend andmaturity moves across a dividend. The model as defined in their paper dependson the maturity t of the European option.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 9 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Bos-VandermarkAffine stock processAffine dividend dynamics
Bos-Vandermark
We now fix t independently of the product, namely
X n,Tt =
∑ti≤t
(T − ti )+
TD i
0 X f ,Tt =
∑ti≤t
ti ∧ T
TD i
0
Inserting X n,Tt ,X f ,T
t rather than X nt ,X
ft into formula (4) implies a stock
process S given as
PtSt = (S0 − X n,Tt )Mt − X f ,T
t
where M is a geometric Brownian motion with mean 1 and volatility σ. Thisgives rise to affine dividends, namely
Sti− − Sti = P−1ti D i
0
((T − ti )
+
TMti +
ti ∧ T
T
)= P−1
ti D i0
((T − ti )
+
T
PtiSti− + X f ,Tti−
S0 − X n,Tti−
+ti ∧ T
T
)= αi + βiSti−
Note that αi ↘ 0 for ti ↘ 0 and βi ↘ 0 for ti ↗ T .
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 10 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Bos-VandermarkAffine stock processAffine dividend dynamics
Affine stock process
It is now standard to model dividends as affine functions of the spot at theex-date
D iti = αi + βiSti− (5)
with for example βi = 0 for ti < 1y and αi = 0 if ti ≥ 5y .
This is equivalent to the stock process being an affine function in the sense that
St = (Ft − Ct)Mt + Ct (6)
holds for some martingale M and a deterministic function C .
This approach dates back to [BFF+00] p7f. More recent references includeOverhaus et al. [OFK+02] p116ff as well as [OBB+07] p6ff.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 11 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Bos-VandermarkAffine stock processAffine dividend dynamics
Affine stock process
More formally, one has (compare Overhaus et al. [OBB+07] p7):
Proposition (affine stock model): Assume a stock process S with jumps dueto dividends only. The following two statements are equivalent:
1 The stock process S is positive and pays positive affine dividends (5)satisfying
αi = 0 for all i ≥ n for some nβi ∈ [0, 1) for all i
Ptiαi ≥ −βi∑
j≥i Ptjαj
∏jk=i (1− βk)−1 for all i
2 The stock process S is an affine function (6) of a positive continuousmartingale M with M0 = 1 where C satisfies
Ct = 0 for all t ≥ T for some T∆Ft ≤ ∆Ct ≤ 0 for all t
dCt = rCt for all t /∈ {t1, t2, . . . }
with notation ∆ft ≡ ft − ft− for a given cadlag function f .
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 12 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Bos-VandermarkAffine stock processAffine dividend dynamics
Affine stock process
(2)⇒ (1): Substituting (6) into the dividend drop Sti− − Sti yields
Sti− − Sti = (−∆Fti + ∆Cti )Mti− −∆Cti
= (−∆Fti + ∆Cti )Sti− − Cti−
Fti− − Cti−−∆Cti ≥ 0
and hence
αi =Cti−∆Fti − Fti−∆Cti
Fti− − Cti−βi = −∆(Fti − Cti )
Fti− − Cti−
One computes
αi + βiCti− = −∆Cti = Cti− −Pti+1
Pti
Cti+1−
and hence
PtiCti− =Ptiαi
1− βi+
Pti+1Cti+1−
1− βi= . . . =
∑j≥i
Ptjαj
j∏k=i
(1− βk)−1
The properties of C imply the required properties of αi and βi for all i as wellas positivity of S .
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 13 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Bos-VandermarkAffine stock processAffine dividend dynamics
Affine stock process
(1)⇒ (2): Both C and M can be constructed by induction on [ti−1, ti ) if givenon t ≥ ti such that (6) holds:
For t ≥ tn define Ct = 0 as well as Mt through St = FtMt .If C is defined for t ≥ ti set
−∆Cti = αi + βiCti− =αi + βiCti
1− βi(7)
This defines C and hence M through (6) on t ≥ ti−1, and one has
∆Sti = (∆Fti −∆Cti )Sti− − Cti−
Fti− − Cti−+ ∆Cti = . . . = −βiSti− − αi
where we used
−∆Fti = αi + βiFti− =αi + βiFti
1− βiNote that as seen above (7) yields via induction
PtiCti− =∑j≥i
Ptjαj
j∏k=i
(1− βk)−1
Hence C has the required properties. Continuity of M at ti can also beestablished.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 14 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Bos-VandermarkAffine stock processAffine dividend dynamics
Examples
Special cases include
Proportional dividends: Ct = 0
Deterministic dividends up to some final date T :
PtCt =∑
T≥ti>t
D i0
Without T one has the deterministic case St = Ft due to (3).
Blend model:
PtCt =∑ti>t
(1− τi )D i0 (8)
with some increasing sequence τ in [0, 1].
We have seen above that Bos-Vandermark is a model with affinedividends. Since Ct ≤ 0 the stock process can be negative.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 15 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Bos-VandermarkAffine stock processAffine dividend dynamics
Affine dividend dynamics
So far we only referred to expected discounted dividend D i0 as well as dividend
drop D iti . The next step is to define D i
t for t < ti .
We obtain for the affine model (6) from (2)
D iti =
(P−1ti D i
0 −∆Cti
)Mti + ∆Cti
For t ≤ ti we may replace D iti ,Pti ,Mti by D i
t ,Pt ,Mt , namely
PtDit =
(D i
0 − Pti ∆Cti
)Mt + Pti ∆Cti
Summation over all i with ti > t yields
PtSt =
PtFt −∑ti>t
Pti ∆Cti
Mt +∑ti>t
Pti ∆Cti
= Pt(Ft − Ct)Mt + PtCt
We summarize:
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 16 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Bos-VandermarkAffine stock processAffine dividend dynamics
Affine dividend dynamics
Proposition (affine dividends): Dividend dynamics
dD it = rtD
itdt + P−1
t
(D i
0 − Pti ∆Cti
)dMt
= rtDitdt + P−1
t D i0τidMt (9)
together with (1) implies the affine stock model (6).
Remarks:
1 Dividend volatility is homogenous in t in the sense that volatility does notdecrease as t approaches ti (assuming M has constant volatility).
2 Dividend dynamics is not homogenous in i since dividend volatility scaleswith τi . The effect is non-homogenous equity returns: near returnsexperience deterministic dividends while far returns have proportionaldividends.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 17 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Indicator functionKorn-RogersPiecewise linearExponential
Homogenous dividend dynamics
Definition: Homogenous dividend dynamics assumes an increasing function τwith values in [0, 1] and τ0 = 0 as well as
dD it = rtD
itdt + P−1
t D i0τti−tdMt (10)
The difference with the blend model (9) is that the impact dMt is scaled downas the ex-dividend date approaches.
The solution is given as
PtDit = D i
0
(1 +
∫ t
0
τti−sdMs
)In case of continuous τ integration by parts yields
PtDit = D i
0
(1 + τti−tMt − τtiM0 −
∫ t
0
Msdτti−s
)Note that M ≥ 0 and M0 = 1 guarantees positive dividends.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 19 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Indicator functionKorn-RogersPiecewise linearExponential
Homogenous dividend dynamics
Proposition (homogenous dividends): With homogenous dividend dynamicsthe stock process is given as sum of an affine function and an integral of themartingale M. More specifically:
St = Ft + P−1t
(T tt Mt − T t
0 M0 −∫ t
0
MsdT ts
)with T t
s for 0 ≤ s ≤ t defined as
T ts =
∑ti>t
D i0τti−s
Proof: One has
PtSt = Pt
∑ti>t
D it
=∑ti>t
D i0
(1 + τti−tMt − τtiM0 −
∫ t
0
Msdτti−s
)
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 20 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Indicator functionKorn-RogersPiecewise linearExponential
Examples: indicator function
Dividend announcement at a fixed time interval a before the ex-date isrepresented by the indicator function
τt = 1{t>a}
Assuming M0 = 1 one has
PtDit = D i
0
(1 +
∫ t
0
1{ti−s>a}dMs
)= D i
0Mt∧(ti−a)+
and
St =∑ti>t
D it = P−1
t
∑ti−a>t
D i0Mt +
∑ti>t≥ti−a
D i0M(ti−a)+
(11)
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 21 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Indicator functionKorn-RogersPiecewise linearExponential
Examples: Korn-Rogers
The situation discussed by Korn and Rogers [KR05] is obtained by furtherspecializing
ti = ih, a = (1− ε)h ≥ 0, Mt = e−µtXt/X0, Pt = e−rt
where X is an exponential Levy process with mean
IE [ Xt ] = X0eµt
as well asD i
0 = λX0Pti−aeµ(ti−a)
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 22 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Indicator functionKorn-RogersPiecewise linearExponential
Examples: Korn-Rogers
Formula (11) yields
St = λ∑
(i−1+ε)h>t
e(µ−r)((i−1+ε)h−t)Xt
+ λ∑
ih>t≥(i−1+ε)h
er(t−(i−1+ε)h)X(i−1+ε)h
The second sum is either empty if the next dividend after t has not beenannounced yet or has exactly one term.
Note that formula (11) also covers the case of longer announcement periods,namely a > h or ε < 0. In that case the second sum consisting of all dividendsannounced at t may have more than one term.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 23 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Indicator functionKorn-RogersPiecewise linearExponential
Examples: piecewise linear
An alternative to the indicator function is the piecewise linear function
τt =t
a∧ 1
where a is the time distance beyond which a given dividend has maximalvolatility. One obtains for t ≥ a
PtSt =
Pa+tFa+t +∑
a+t≥ti>t
D i0ti − t
a
Mt + a−1∑
a+t≥ti>t
D i0
∫ t
(ti−a)+
Msds
For t < a one has to add the term∑a≥ti>t
D i0
(1− ti
a
)representing the announced part of near dividends.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 24 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Indicator functionKorn-RogersPiecewise linearExponential
Examples: exponential
With
τt = 1− e−λt
one obtains
T ts = Pt
(Ft − eλsCt
)dT t
s = λPtCteλsds
where Ct is defined through
PtCt =∑ti>t
D i0e−λti
Thus
St = Ct +(Ft − eλtCt
)Mt + λCt
∫ t
0
eλsMsds
The corresponding non-homogenous affine dividend model (6) is
St = Ct + (Ft − Ct)Mt
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 25 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
Indicator functionKorn-RogersPiecewise linearExponential
Examples: exponential
λ = 0.5, σ = 20%, r = 5%, annual dividends d = 10% paid before maturity
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 26 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
SummaryReferences
Conclusion
Affine stock models (including Bos-Vandermark) can be viewed as dividenddiscount models where all dividends are driven by the same martingalefactor. The volatility level of each dividend does not change through time.
Homogenous dividend models behave like bonds in the sense that theirvolatility level decreases through time, reaching zero at the ex-date.
The stock return distribution of homogenous dividend models is morereasonable as one does not shift from affine for near returns toproportional for far returns.
Modelling homogenous dividend models is more involved since the stockprocess is a time average of a postive martingale.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 28 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
SummaryReferences
Bibliography I
H. Buhler, A. Dhouibi, and D. Sluys.
Stochastic proportional dividends.http://ssrn.com/abstract=1706758, 2010.
O. Brockhaus, M. Farkas, A. Ferraris, D. Long, and M. Overhaus.
Equity Derivatives and Market Risk Models.Risk Books, February 2000.
G. Bernhart and J. Mai.
Consistent modeling of discrete cash dividends.Technical report, Xaia Investment, 2012.
M. Bos and S. Vandermark.
Finessing fixed dividends.Risk, pages 157–158, 2002.
R. Korn and L. C. G. Rogers.
Stocks paying discrete dividends modeling and option pricing.Journal of Derivatives, 13(2):44–48, 2005.
M. Overhaus, A. Bermudez, H. Buhler, A. Ferraris, C. Jordinson, and A. Lamnouar.
Equity Hybrid Derivatives.John Wiley and Sons, 2007.
M. Overhaus, A. Ferraris, T. Knudsen, R. Milward, L. Nguyen-Ngoc, and G. Schindlmayr.
Equity Derivatives: Theory and Applications.John Wiley and Sons, February 2002.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 29 / 32
IntroductionAffine dividends
Homogenous dividend dynamicsConclusion
SummaryReferences
Bibliography II
R. Tunaru.
Dividend derivatives.ssrn, 2014.
J. van Binsbergen, M. Brandt, and R Koijen.
On the timing and pricing of dividends.Technical report, National Bureau of Economic Research, 2010.
M.H. Vellekoop and M.H. Nieuwenhuis.
Efficient pricing of derivatives on assets with discrete dividends.Applied Mathematical Finance, 13(3):265–284, 2006.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 30 / 32
Appendix: stochastic dividend yield
Buhler [BDS10] suggests an Ornstein-Uhlenbeck process y driving dividendyield. The resulting dynamics can be written as
St = FtNtiMt ti ≤ t < ti+1
where N is an adapted process of the form
Nt =S0
Fte−
∑ti≤t di =
S0
Fte−
∑ti≤t Ci+Di+Ei yt = e
−∑
ti≤t Ci+Ei yt
with IE [ MtiNti ] = 1, PtiFti = e−DiPti−1Fti−1 as well as Ei ∈ {1,Di}.
Remark: It is not required that N or NM is a martingale and indeed
IEti−1 [ Nti ] = Nti−1e−Ci IEti−1
[e−Ei yti
]6= Nti−1
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 31 / 32
Appendix: stochastic dividend yield
The dividend drop D iti satisfies
PtiDiti = −Pti ∆Sti =
(Pti−1Fti−1Nti−1 − PtiFtiNti
)Mti
A consistent dividend dynamics for t ≤ ti can be defined as
PtDit =
(Pti−1Fti−1Nti−1∧t − PtiFtiNti∧t
)Mt
For t ≤ ti−1 this simplifies to
PtDit = D i
0NtMt
One observes
PtSt = Pt
∑ti>t
D it = Pti−1Fti−1Nti−1Mt ti−1 ≤ t < ti
as required. Buhler [BDS10] also discusses affine dividends
D iti = αi + βiSti− = αi +
(1− Nti
Nti−1
)Sti−
Analysis of a version of this model with homogenous dividend dynamics is leftfor future research.
oliver.brockhaus@mathfinance.com Dividend Discount Approach to Equity Derivatives Modelling 32 / 32
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