stochastic proportional dividends

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Electronic copy available at: http://ssrn.com/abstract=1706758 Stochastic Proportional Dividends Working Paper Hans Buehler, Anissa Dhouibi, Dimitri Sluys JP Morgan Equity Derivatives Group Quantitative Research London [email protected] [email protected] [email protected] First draft: January 2010 - This draft: December 2010, Revision 1.013 Abstract Motivated by recently increased interest in trading derivatives on div- idends, we present a simple, yet efficient equity stock price model with discrete stochastic proportional dividends. The model has a closed form for European option pricing and can therefore be calibrated efficiently to vanilla options on the equity. It can also be simulated efficiently with Monte-Carlo and has fast analytics to aid the pricing of derivatives on dividends. While its efficiency makes the model very appealing, it has the twin drawbacks that dividends in this model can become negative, and that it does not price in any skew on either dividends or the stock price. We present the model and also discuss various extensions to stochastic interest rates, local volatility and jumps. Keywords: Options on Dividends, Stochastic Dividends, Dividend Yield 1 Introduction The recent years have seen an increased interest in trading more directly one of the most basic features of equity stock prices, its dividends. Dividends are comparable to coupons of a bond since they provide the equity holder with a stream of income. The main difference, of course, is that a company can decide 1

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Motivated by recently increased interest in trading derivatives on div-idends, we present a simple, yet ecient equity stock price model withdiscrete stochastic proportional dividends. The model has a closed formfor European option pricing and can therefore be calibrated eciently tovanilla options on the equity. It can also be simulated eciently withMonte-Carlo and has fast analytics to aid the pricing of derivatives ondividends. While its eciency makes the model very appealing, it has thetwin drawbacks that dividends in this model can become negative, andthat it does not price in any skew on either dividends or the stock price. We present the model and also discuss various extensions to stochasticinterest rates, local volatility and jumps.

TRANSCRIPT

Page 1: Stochastic Proportional Dividends

Electronic copy available at: http://ssrn.com/abstract=1706758

Stochastic Proportional Dividends

Working Paper

Hans Buehler, Anissa Dhouibi, Dimitri Sluys

JP Morgan Equity Derivatives GroupQuantitative Research

London

[email protected]

[email protected]

[email protected]

First draft: January 2010 - This draft: December 2010,Revision 1.013

Abstract

Motivated by recently increased interest in trading derivatives on div-idends, we present a simple, yet efficient equity stock price model withdiscrete stochastic proportional dividends. The model has a closed formfor European option pricing and can therefore be calibrated efficiently tovanilla options on the equity. It can also be simulated efficiently withMonte-Carlo and has fast analytics to aid the pricing of derivatives ondividends. While its efficiency makes the model very appealing, it has thetwin drawbacks that dividends in this model can become negative, andthat it does not price in any skew on either dividends or the stock price.

We present the model and also discuss various extensions to stochasticinterest rates, local volatility and jumps.

Keywords: Options on Dividends, Stochastic Dividends, Dividend Yield

1 Introduction

The recent years have seen an increased interest in trading more directly oneof the most basic features of equity stock prices, its dividends. Dividends arecomparable to coupons of a bond since they provide the equity holder with astream of income. The main difference, of course, is that a company can decide

1

Page 2: Stochastic Proportional Dividends

Electronic copy available at: http://ssrn.com/abstract=1706758

how much dividend it will pay, hence the dividend amounts are a function ofthe company’s performance.

In the derivatives worlds, dividends have traditionally been traded either di-rectly via forms of over-the-counter (OTC) equity swaps or implicitly by tradingcalendar spreads of forwards. The latter in particular allows to trade the real-ized dividends against the assumed, or “implied”, dividends. Starting in 2001,dividend swaps have become a popular way for dealers to hedge their dividendexposure [11]. A dividend swap pays directly the aggregated value of all (gross)dividends a company pays in exchange for the initially agreed strike price.

The next step in listed product development was to introduce dividend fu-tures on an exchange. The first such product, the “Dow Jones EURO STOXX50 Index Dividend Futures” was introduced by Eurex in June 2008 [1]. Figure 1shows the growth of the volume of dividend futures over the last two years.

Figure 1: Growth of dividend future volumes since beginning of 2009.

The culmination of this process so far was the introduction on May 25, 2010of options on the “S&P500 Annual Dividend Index (DIVD)” and the “S&P 500Dividend Index (DVS)” on the CBOE [12], and the introduction on the same dayof options on both “Euro STOXX50 Index Dividend Futures (FEXD)” and the“Euro STOXX50 Index Dividend Points (DVP)” on Eurex [2]. The exchangetraded contracts have the potential to improve price discovery for the volatilitymarket on dividends substantially.

Aside from the listed market, there also the market of OTC derivatives.Here, the potential for structuring ideas around dividends are wide: options ondividends are one way to expresss a view on dividends, but other products suchas Dividend Yield Swaps and Knock-Out Dividend Swaps can give a much moretailored exposure to the realized dividends of an index or a given equity.

In addition, there is also a natural interest in products which give exposureto the “yield gap” between the yield on bond investments and the yield derivedfrom dividends.

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Pricing with Dividends

In the light of these developments, it is surprising to find that there is actuallynot much literature on the modeling of dividends for derivatives pricing and riskmanagement, except on the question on whether these should be proportionalto the stock price, fixed in cash or a mixture of the two.1 Beyond this, the onlyreference to our knowledge on modeling the dividends of an equity is implicitlygiven by Gaspar’s theoretical work [7] on modeling the entire term-structure ofthe forward curve as a diffusion in an Hilbert-space. This “HJM-approach” isrelated to a version of our model where a dividend yield is modeled instead ofa stream of discrete dividends.

With this article, we aim to contribute to the modeling discussion on div-idends with a model where dividends are discrete and proportional; the (log-)proportionality factor itself is given by a function of an Ornstein-Uhlenbeckprocess, meaning that the resulting dividend yield is mean-reverting around along-term average yield level. While the drawback of using a normal process forthe proportionality factor is that it can lead to negative dividends, the upside isthat the model remains very tractable: pricing vanilla options on the equity forthe purpose of calibration, simulating the model with Monte-Carlo and evaluat-ing future forwards are all efficient operations. This gives the model the flavorof a robust “Black & Scholes”-style workhorse for pricing various variants ofstructures on dividends.

This article is structured as follows: first, we will introduce our setup anda set of interesting payoffs. We then present our main model and discuss itsproperties as well as a calibration procedure which we apply to market datafor STOXX50E. In the next section we show how an efficient Monte-Carloscheme can be implemented and how we can compute dividend future priceson the given Monte-Carlo paths. We also compute some sample option prices.A final section discusses extensions of the model to stochastic interest rates,credit and jump risk, and local volatility (with numerical methods). The sec-tion concludes with a comment on the connection to Gaspar’s HJM-frameworkfor the forward curve.

An appendix contains most of the actual calculations.

Acknowledgements

We are very grateful for the help of Christopher Jordinson, now at UBS, manyof whose ideas are incorporated in the model we present here. His input wascrucial to the development of the model.

1For a thorough review of this setting, cf. Bermudez et al. [3] and, in more detail,Buehler [6].

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2 Modeling Dividends

We aim to model an equity stock price process S = (St)t under an interestrate process r = (rt)t and with effective borrowing costs of µ = (µt)t. Both are

assumed to be deterministic. For notional convenience we write Rt := e∫ t0rs−µs ds

for the drift factor of the equity and Bt := e∫ t0rs ds for the cash account.2

To model the dividend stream of the equity, we assume that the stock isgoing to pay N random dividends ∆ = (∆i)i=1,...,N at dividend dates 0 < τ1 <· · · < τN .3 (We will assume that the dividend dates are fixed and known inadvance.) Modeling the dividends of the equity as stochastic means modelingeach paid dividend ∆i as a random variable which is “known” by the dividendtime τi.

4

Dividend Derivatives

A dividend swap on the equity between the dates T1 and T2 pays the accumu-lated dividends between these two dates against a fixed strike K, i.e.∑

i:T1≤τi≤T2

∆i −K .

Similarly, an option on realized dividends, or in short OOD, is a vanilla payoffon the same quantity, i.e. in the case of a call on realized dividends ∑

i:T1≤τi≤T2

∆i −K

+

where we used the notation x+ := max(0, x). Note that dividend futures usuallypay the realized dividends of the respective underlying index. Hence, an “optionon a dividend future” is effectively also an option on realized dividends for theperiod of the future.

More advanced OTC products on dividends are structures, where the payoffof the dividend is linked to the performance of the equity itself. One example isa knock-out dividend swap, i.e. a dividend swap which knocks out if the equity Strades below a given barrier. The payoff for this product is

1mint:T1<t≤T2 St<B

∑i:T1≤τi≤T2

∆i −K

. (1)

2We do not model the possibility of default here, but an extension to the case of a deter-ministic hazard rate model is well within our framework, c.f. the discussion in [6].

3For ease of exposure we assume that the payment dates are equal to the ex-div dates. Anextension to separate ex-div and payment dates is straight-forward.

4An exception here is Japan, where dividends are usually announced after the ex-div date.

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Another example is a dividend yield swap, which pays the sum of realized divi-dends over the monthly average spot of the equity:∑

i:T1≤τi≤T2∆i∑

k:T1≤tk≤T2

∑Stk

(2)

where (ti)i are monthly fixings (other variants scale the realized dividends bythe stock price at the end of the period, or divide each dividend by the stockprice of the previous trading day).

As mentioned before, there is a recent interest in rates-dividend hybrid prod-ucts, for example those which allow to manage an exposure to the difference individend and bond yields. Figure 2 shows that the implied yield on equity isrecently higher than the yield which can be derived from investing in bonds.A good example of a product which allows to hedge this “yield gap” would be a

Figure 2: Dividend yield is high in both absolute and relative terms compared tobond yields.

“Leveraged Div Yield Swap Certificate” which basically pays coupons which area difference between interest rates and dividends: We simplify the product forthe sake of simplicity. The simple version works as follows: define the realizeddividend yield between two times T1 and T2 as

ryld(T1, T2) :=1

ST2

∑i:T1<ti≤T2

∆i

and let CMSt(1Y ) denote the 1Y libor rate observed at time t. Then, theproduct pays

max

(100% + 5

∑t=1y,2y,3

{CMSt(1y)− ryld(t, t+ 1y)

}, floor

)where the floor is, for example, 30%. In this form, the product provides apositive exposure to a drop in dividend yield vs. interest rates.

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While we will mainly discuss our model in a deterministic interest-rate envi-ronment, section 3.1 explains how it can be incorporate stochastic interest ratesin order to risk manage rates-dividend hybrids.

The Equity and its Dividends

To model an equity process with a stream of stochastic dividends, the challengeis to find a distribution for the dividends which still allows pricing and riskmanagement of options on the underlying efficiently such that the model can becalibrated easily to observed prices of equity vanilla options.

To approach this topic, we first assume that we face an idealized liquidmarket of dividend swaps, which allows us to forward-trade any dividend priorto its dividend date. In particular, we assume that there is at any time t amarket for the dividends with dividend dates past t. We denote the forward-price at time t for the ith dividend accordingly by ∆i

t. As a consequence of ourassumptions, its value is given under any risk-neutral measure as

∆it := Et

[∆i]. (3)

Note that the process (∆it)t∈[0,τi] is a (local) martingale.

With the setup of the previous section, we can deduct a basic shape of anyequity model which is consistent with a given dividend stream: basically, theassumptions that dividends can be traded means that the stock price cannotdrop below the properly discounted value of future dividends at any point.Following the same arguments as laid out in Buehler [6], standard no-arbitragereplication arguments imply that the stock price process has to have the form

St = Rt

{S0Xt +

∑i:τi>t

∆it

Rτi

}

with S0 = S0−∑i ∆i/Rτi for a (local) martingale X. Accordingly, the forward

at time zero of the process is given as

Ft = Rt

S0 −∑i:τi≤t

∆i0

Rτi

.

Black & Scholes

The standard extension to Black & Scholes’ model is the classic case where thedividend is proportional to the equity, i.e.5

∆i = Sτi−(1− e−Di) with Di := − ln

(1− ∆i

0

Fτi−

). (4)

5This follows since both Fτi−e−Di = Fτi and Fτi− − ∆i = Fτi have to hold.

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The corresponding stochastic model is then given by

St = FtXt ,

where X is a log-normal model with “pure equity” volatility term structure σ.Its SDE is given as

dStSt

= (rt − µt) dt+ σt dWt −∑i:ti≤t

(1− e−Di) δτi(dt) (5)

(we use δx(·) to denote the Dirac-measure in x). In this model, the dividendstream is obviously stochastic – it exhibits the same volatility as the underlyingequity.

Stochastic Proportional Dividends

The model we want to propose here is given as well in the form

dStSt

= (rt − µt) dt+ σt dWt −∑i

(1− e−di) δτi(dt) (6)

but the dividend ratios di are random: to model them, we use an Ornstein-Uhlenbeck process

dyt = −κyt dt+ ν dBt

and setdi := (Di + Eiyτi) + Ci (7)

where Di is the Black & Scholes value for the proportional dividend as definedin (4). The constant Ei allows to blend between normal and log-normal volatilityfor the dividend yield by using Ei ≡ 1 or Ei ≡ Di, respectively. Finally, theconstant Ci is determined by matching the forward such that E[St] = Ft forall t using an analytical procedure; see appendix A.2 for details.

Remark 2.1 Our model can easily be extended to incorporate a fixed cash-dividend part for each dividend. For example, we could assume that a fraction αiof each dividend ∆i is fixed in cash. Compared to (4), the stochastic dividendin such a model becomes

∆i = Sτi−(1− e−Di) + (1− αi)∆i0 .

for a suitable Di which ensures that the forward of the process is correct. Ap-pendix A.1 provides some more details.

To motivate our model choice, let us define the “forward yield” between T1

and T2, seen at a time t, as the sum of the expected dividends between the tworeference times, divided by the current spot level:

yt(T1, T2) :=

∑i:T1<τi≤T2

∆it

St.

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It is easy to see that in the current model, each dividend has the propertythat log ∆i

t/St is an affine function of yt, which implies that the log-yieldlog yt(T1, T2) is also approximately proportional to yt,

yt(T1, T2) ∼ ayt + b (8)

for some constants a and b.Looking then at historical data, we indeed see that the log-forward yield

seems to mean-revert around some fixed mean. During “the great moderation”of the early 2000’s in particular, we have seen a very stable pattern of mean-reversion. Even though this pattern has unsurprisingly been severely disturbedwith the onset of the financial crisis and, in particular, Lehman’s default inOctober 2008, the structure has since vaguely recovered, albeit to a regime witha lower level of average implied yield. Figure 3 illustrates this point. In general,

Figure 3: The graph shows historical log-dividend yields for future periods. Forexample, the 9Y/10Y point refers to the floating maturity forward dividend swapbetween 9Y and 10Y, divided by the spot at the observation point.

the assumption of a mean-reverting dividend yield in a non-distressed marketis a very natural from an economic point of view, in particular if applied toindices.

Model Properties

The first important observation is that the model’s stock price is log-normal:its explicit form is

St = RtS0e∫ t0σs dWs− 1

2

∫ t0σ2s ds−

∑i:τi≤t{Ci+Di+Ei[y0e

−κτi+ν∫ τi0 e

−κ(τi−s) dBs]} ,

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with log-variance

Var(t) =

∫ t

0

σ2s ds+

∑i:τi≤t

Ei

∑j:τj≤τi

Ej

[ν2 1− e−2κτj

2κ− 2ρν

∫ τi

0

σse−κ(τi−s) ds

] .

(9)This means that if we are given the dividend parameters y0, κ, ν and ρ, thenwe can bootstrap a piece-wise constant forward volatility (σk)k to matching anobserved term structure of Black & Scholes (spot-) implied volatilities (Σk)kdefined for maturities T1 < T2 < · · · (see appendix A.4 for details). Since this isan analytic procedure, this can be performed on-the-fly in order to ensure thatthe model always reprices a selected term structure of observed market impliedvolatilities on the stock price.

The task remains to determine reasonable values for the dividend volatilityterm structure via the two parameters ν and κ, and the correlation ρ betweendividend yield and the spot price (the initial value y0 can be left at zero). Forliquid indices, we can calibrate these parameters to ATM prices of options ondividends. As an example, we have used internal JP Morgan price indications asof 14 October 2010 for options on dividend prices for STOXX50E to calibratethe model. The prices provided for Dec10-11 and Dec11-12 ATM options ondividends are quoted in terms of Black & Scholes implied volatilities6 as 7.8%and 16.1%, respectively (the dividend swaps traded at 114.0 and 109.2).

To calibrate our model to just two maturities of options on dividend prices,we have fixed our correlation7 at −95% and then run a simple minimizer whichyielded a dividend volatility of ν = 27% and a mean-reversion speed of κ = 1.4.The resulting ATM implied BS volatilities for the two maturities in the modelare then 7.9% and 16.1%, respectively. These have been calibrated using aMonte-Carlo simulation with 10,000 paths.

In figure 4 we show the calibrated model prices against market price indica-tions for options on dividends. The data show that the market price indicationsexhibit far more skew than the model is able to replicate. This is in line withthe model’s basic structure where the sum of dividends is roughly log-normal,hence we do not expect much skew. Figure 5 shows the entire implied volatilitysurface for options on dividends.

The Role of Dividend Correlation

Correlation plays a particular role in this model. Recall that the yield in themodel is approximatively an affine function of yt, cf. equation (8). Figure 6illustrates the impact of a change of correlation on the relation between spotand yield – as expected, a positive correlation implies a positive yield withraising spot while a negative correlation implies a drop in yield.

6I.e. the forward in the Black & Scholes formula is set to the dividend swap fair strike andthen the respective implied volatility is calculated from observed market prices.

7The very negative correlation is not only necessary to be able to provide a decent fit tothe observed market prices; it is also economically sensible as we will explain below.

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Figure 4: Prices for options on dividends; the model is calibrated against the ATMstrike of the market price indications. The data has been priced as of mid October 2010.

Between the two, the latter is a much more realistic assumption since anincrease in spot is not usually immediately followed by an increase in dividendlevels; in a sense, the negative correlation between spot and dividends “sta-bilizes” the level of dividends and gives them some “constantness”. Figure 7shows visually how realistic paths generated by a negative correlation regimelook – until the crisis in September 2008. After the crisis, however, the twowere more ‘correlated’: dividends were cut along with the fall in stock prices.More recently, the market has recovered and we see again a more anticorrelatedbehavior.

This means that the correlation in our model should generally be significantlynegative in order to produce the well-known effect that short term dividendsare much more certain than longer term dividends. This is also borne out ofthe market prices for options on dividends: the above calibration only workswell for very negative correlations. If the correlation is set at, say, -50%, thenthere is no combination of mean-reversion speed and dividend volatility whichfits the provided market.

Negative Dividends

While most of the properties of our model are appealing, the model’s structurealso reveals its main drawback: the fact that the dividends themselves canbecome negative. This happens if di becomes negative. The probability of thishappens at a dividend date τi is

P

[Y ν

√1− e−2κτi

2κ≤ Di + Ci

Ei− y0e

−κτi

]

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Figure 5: The “implied volatility” surface for options on dividends in our model.Strikes are provided as offset to 100% ATM.

where Y is standard normal. For the calibrated model above, the annual prob-ability of having negative dividends is around 1% in the model as is shown

in figure 8. The same figure also shows that the size of the negative dividendsin relation to the total forward is (predictably) small.

However, we think that the advantage of analytical tractability of the modelfar outweighs the downside of having potentially negative dividends.

2.1 Using the Model

For most derivatives on dividends, we will need to revert to numerical methodsto compute the value of the payoff and risk manage it. We will here discussbriefly how to implement an efficient Monte-Carlo and will show how the modelprices some options on dividends.

In order to simulate the model (6), we assume that we have some “observa-tion dates” 0 < t1 < · · · < tn of interest at which points we wish to evaluateour payoff; for ease of exposure we assume w.l.g. that these dates include alldividend dates in the respective period.

A convenient feature of our essentially normal model is that we do not needto use Euler to simulate our SDE because the model is in fact normal betweentwo of the observation dates. Assuming constant equity vol of σk between tk−1

and tk = dtk + tk−1, the variance of the increment dy is Σyk = ν2 1−e−2κdtk

2κ , thevariance for the equity increment is ΣSk := σ2

kdtk and the covariance between

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Figure 6: Impact of correlation on the relation between spot and yield; the graphsshow expected dividends between Dec 11 and Dec 12 divided by the spot at observationtime. Dividend volatility was 50%, correlation -80% and mean-reversion speed 1.

Figure 7: Visual comparison of a negative correlation case above visually with historicdata.

the two is ρνσk1−e−κdtk

κ , which means that the correlation is

ρk := ρ1− e−κdtk

√1− e−2κdtk

√12κdtk

.

Hence, if we chose a iid sequence of standard normal random variables (Yk, Yk)k,we can simulate on big step between tk−1 and tk using

ytk = ytk−1e−κdtk +

√Σyk

(ρkYk +

√1− ρ2

kYk

)logStk = logStk−1

+ σk√dtkYk −

1

2σ2kdtk + log

FkFtk−1

− (Ekytk + Ck) .

(We used a slightly lax notation here: the Dk, Ek and Ck are either the relevantcoefficients from a dividend payable at tk or zero.)

Another strength of the model is that it is possible to efficiently calculate

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Figure 8: Illustration of the risk of having negative dividends for the calibratedmodel: the graph shows the annual probability of having negative dividends (aroundone percent) and the ratio of the expected negative part over the full expected sumover the same period, which hovers around 0.5%.

the forward price of the equity as a function of spot and the driving Ornstein-Uhlenbeck process u at a given future time efficiently since it has the form

Ft(T ) := Et[ST ] = StRTRt

eAt(T )yt+Bt(T )

for some deterministic functions A and B (see appendix A.3 below). This isimportant because it allows us8 computing future expected dividend values ona given Monte-Carlo path efficiently using

∆`t = Ft(τ`−)− Ft(τ`) = Ft(t ∨ τ`−1)

Rτ`Rt∨τ`−1

− Ft(τ`)

(x ∨ y := max(x, y)). Hence, the model permits efficient access to pricing divi-dend swaps analytically within a Monte-Carlo simulation. This means we canuse the model to price and risk manage not only options on realized dividends,but also on more involved trades which depend in value on futures on dividends.

As an example, figure 9 shows a sample path of both stock price and a div-idend future generated by the model.

To illustrate the pricing of derivatives with the model, we used it to pricetwo types of products with the parameters estimated before for STOXX50E.The first product is the aforementioned Knock-Out Equity Swap (1). Using thesame calibration as above, and a barrier of 90% of initial spot,9 we get

8We use the fact that by definition Ft(T ) = RT /Rt(St −∑i:t≤τi≤T Rt/Rτi∆t(i)).

9In practise, this model should not be used to price barriers due to its lack of skew for theequity. Below is a brief comment on how to incorporate skew.

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Figure 9: A Monte-Carlo path of spot and expected dividends divided by spot wherethe dividends are computed from the observation time to a fixed maturity (Dec 13).

Dec10-11 Dec11-12 Dec12-13 Dec13-1435.1 46.9 45.9 47.6

Another interesting product is the Dividend Yield Swap (2). It pays the sumof realized dividends over the monthly spot price average.

Our model gives us:

Dec10-11 Dec11-12 Dec12-13 Dec13-144.13% 4.09% 3.97% 3.97%

3 Extensions and Related Models

We briefly want to give some overview over possible extensions to the currentmodel.

3.1 Stochastic Interest Rates

An interesting aspect of dividend modeling has to be the relation ship betweendividends and interest rates. From an economic point of view, both rates prod-ucts and dividend-paying equity play a similar role in providing investors witha stream of income. Consequently, we can assume that the market’s dividendyield is related to the level of interest an investor can earn by buying just acoupon-bearing bond.

While we will not discuss the economic situation much further here, we wouldlike to point out that the current model can easily be adapted to the standardHull & White interest model [9], and that it retains its analytical tractabilitysince the distribution of the stock price remains log-normal.

To start with, let us assume that we can observe at time zero a market ofzero coupon bonds (Pt)t for each maturity t. We are going to model the interest

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short rate as an Ornstein-Uhlenbeck process by setting:

drt = (θt − qrt) dt+ ξt dWrt .

Mean-reversion speed q, the interest volatility ξ and the correlation structurebetween the equity and the short rate are free parameters have to be calibratedor estimated. The level of mean-reversion, θ, is implicitly given by fitting themodel to the observed zero coupon bond term structure by stipulating

E[e−

∫ t0rs ds

]!= Pt

(in practical applications, one would never actually calculate θ but use it onlyin integrated form; see [3]). Given our Hull & White model, we can now definethe equity process via, once again,

dStSt

= (rt − µt) dt+ σt dWt −∑i

(1− e−di) δτi(dt)

with di given again as in (7) with adjusted C’s. Since r and therefore anyintegral over it are normal, it follows that the stock price process in this jointHybrid model is still log-normal. Consequently, the analytic tractability of themodel is preserved and it is a very convenient candidate if we want to pricejoint rates-dividend derivatives. We even maintain the ability to run an efficientMonte-Carlo by using the methods discussed in Brockhaus et.al. [5], page 36.

Remark 3.1 Both our original and our rates/dividends hybrid model allow theintroduction of several Ornstein-Uhlenbeck factors to drive the respective curves;see also Hull’s description for the 2-factor rates model in [8].

3.2 Jump Risk, Crash Risk, Credit Risk

Handling plain credit risk within a deterministic intensity model is straightforward following the recipe in [6]. If one is interested in adding less drasticjumps to the equity process which also impact the dividend, then the classisapproach by Merton [10]. One point of consideration is the behavior of theyield if the equity exhibits a sudden drop. We would assume that a suddendrop in the spot price will also lead to a drop in yield. This can be seen verywell in figure 3 on page 8 which shows that the stock market drop in October2008 lead to a substantial drop in dividend yield as well.

To model such a relationship, let us consider series of normal crash or jumpscenarios (γj)j=1,...,M for the stock price process, each with mean mj and volatil-ity sj , and driving Poisson processes (N j)j with crash intensities (λj)j . We alsoassume we have corresponding shock scenarios (gj)j for the dividend yield.

The diffusion for the stock price is then

dStSt

= (rt − µt) dt+ σt dWt −∑i

(1− e−di) δτi(dt)−

M∑j=1

γjdNjt + λj dt

.

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with λj := λj(e−mj+ 1

2 s2j − 1). In order to incorporate synchronous jumps

into the dividend yield as well, we additionally rewrite our driving Ornstein-Uhlenbeck process as

dyt = −κyt dt+ ν dBt −M∑j=1

gjdNjt .

This model has the desired properties of synchronous crashes in equity and yieldswhile it maintains enough flexibility to be adapted to “small” jumps which donot affect the yield itself (they will affect the dividends being proportional tothe spot price level). Note that one drawback of incorporating negative jumpsinto u is that the proportional dividend factors di are more likely to becomenegative.

In terms of analytical tractability this model is once again very convenient:conditional on the number of jumps for each Poisson-process this model has alog-normal distribution which means

that it can be used efficiently using Fourier-based pricing for European op-tions.

3.3 Fitting the Equity Smile

Since the proposed model’s stock is essentially log-normal, it does not have anyskew when pricing vanilla options on the equity. This obvious drawback canbe addressed at least in the case for deterministic interest rates by calibratingnumerically a local volatility function on top of the equity.

The corresponding SDE for the equity is

dStSt

= (rt − µt) dt+ σt(St) dWt −∑i

(1− e−Di) δτi(dt) .

The local volatility function itself can then be calibrated using forward-PDEmethods such as the one described in [3] for the calibration of a local volatilityfunction on top of a stochastic rates equity pricing model.

3.4 Stochastic Yield Dividends

One of the main features of our “stochastic proportional dividend” model is thatit keeps the dividend payments to exactly the payments found in the marketdata. An alternative is to model a yield on top of the prevalent forward curve:such a model is given by

dStSt

= (rt − µt − yt − Ct) dt+ σt dWt −∑i

(1− e−Di) δτi(dt)

where the Di are the proportional dividends from the Black & Scholes formu-lation (5), and where the continuous function C is chosen such that the model

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fits the forward, i.e. E[St] = Ft.10 While being numerically easier to handle

than the proportional dividend model above, it has the disadvantage that theyield will often be negative over periods between the input market dividenddates. As such, the model might be more useful if the equity forward itself isgiven as a yield – in fact, this model is related to Gaspar’s HJM-framework [7]for forward curves which we mentioned in the introduction; it is basically thenormal 1F-version in her framework.

We will not discuss the details of this approach further, but the interestedreader will find that most calculations are either similar or simpler compared toour proportional dividend model.

4 Conclusions

We have discussed a stochastic dividend model with very tractable analytics forcalibration towards the vanilla market and for the pricing and risk managementof dividend derivatives. To the best of our knowledge, this is the first attempt tomodel the dividend stream of an equity in a consistent manner for the purposeof derivatives pricing. The model’s simplicity qualifies it as a standard tool ina derivatives library, while the fact that its dividend are not always positiveand that it does not capture the implied volatility skew in either equity ordividends means that there is clearly a further need for development in thedividend modeling area.

A.1 Cash Dividends

In order to extend the model to support explicit cash dividends, we follow theapproach in [6]: first, we define for each dividend ∆i are ratio αi which denoteshow much of today’s expected dividend value ∆i

0 will certainly be paid (anextension to support simple credit risk is trivial). The model is then written asfollows: first define an adjusted spot

S0 := S0 −∑i

P (0, τi)αi ∆i0

and a ‘proportional forward’

Ft := Rt

S0 −∑i:τi≤t

(1− αi) ∆i0

Rτi

.

Accordingly, we define new proportionality factors

Di := − ln

(1− (1− αi) ∆i

0

Fτi−

).

10I.e. we use the relation e∫ t0Cs ds = E

[e∫ t0σs dWs−

12

∫ t0σ

2s ds−

∫ t0ys ds

]to find C.

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The proportional dividend for a given Ei will then be modelled as

di :=(Di + Eiyτi

)+ Ci

which gives us the diffusion

dSt

St= (rt − µt) dt+ σt dWt −

∑i

(1e− e−di) δτi(dt) .

The stock price is then given as

St := St +∑i:τi>t

P (t, τi)αi ∆i0

and will, with appropriately calculated Ci, reprice the forward. The calculationsfor C are the equivalent to the one for the model discussed in the text.

A.2 Matching the Forward

Fix some `. We want to compute

(∗) = Et[e∫ τ`0 σt dWt− 1

2

∫ τ`0 σ2

t dt−∑j:j≤` dj

].

As a first step, we change into the “equity measure” Q under which

dyt = (θt − κyt) dt+ ν dBt .

with θt := σtρν. Under this measure, we have

(∗) = EQ[e−

∑j:j≤` dj

].

Since we have deκtyt = eκtθt dt+ eκtν dBt, we get

yt = yue−κ(t−u) +

∫ t

u

e−κ(t−s)θs ds+ ν

∫ t

u

e−κ(t−s) dBs .

That means that for any p,

EQ [ e−pyt ∣∣Fu ] = e−pyuK(t,u)−pΘ(t,u)+p2Γ(t,u) . (10)

with Θ(t, u) :=

∫ tue−κ(t−s)θsds

K(t, u) := e−κ(t−u)

Γ(t, u) := 12ν

2 1− e−2κ(t−u)

2κ .

We also abbreviate

Θj := Θ(τj , τj−1) , Kj := K(τj , τj−1) and Γj := Γ(τj , τj−1) .

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Iteration

The aim is to make sure with out choice of (Cj)j=1,...,N that

1!=

EQ[e−

∑j:j≤` dj

]e−

∑j:j≤`Dj

= EQ[e−

∑j:j≤`(Ejyτj+Cj)

].

Consider the formula

c`(p) := logEQ[e−

∑j:j<`(Ejyτj+Cj)−(E`+p)yτ`

]which obviously yields with C` := c`(0) the desired correction terms. We willderive these functions iteratively: we start with ` = 1. Using (10) yields readily

c1(p) = −(E1 + p)K1y0 − (E1 + p)Θ1 + (E1 + p)2Γ1 .

The next step is ` > 1 to determine c`(p) assuming we know cj(q) and there-fore Cj for j < `.

ec`(p) = EQ[e−

∑j:j<`(Ejyτj+Cj)−(E`+p)yτ`

]= EQ

[e−

∑j:j<`(Ejyτj+Cj) EQ

τ`−1

[e−(E`+p)yτ`

] ]= EQ

[e−

∑j:j<`(Ejyτj+Cj)e−(E`+p)K`yτ`−1

−(E`+p)Θ`+(E`+p)2Γ`]

= EQ[e−

∑j:j<`−1(Ejyτj+Cj)−(E`−1+(E`+p)K`)yτ`−1

]e−(E`+p)Θ`+(E`+p)

2Γ`−C`−1

= e−(E`+p)Θ`+(E`+p)2Γ`+c`−1((E`+p)K`)−C`−1

which means that

c`(p) = −(E` + p)Θ` + (E` + p)2Γ` + c`−1 ((E` + p)K`)− C`−1 (11)

is well-defined.

A.3 Future Forwards

A very similar calculation as the above allows us to compute the future forwardsof the model,

Ft(T ) := Et[ST ] .

We sketch the idea here: let n : τn ≤ T < τn+1 and k : τk−1 ≤ t < τk. First ofall,

Ft(T ) = StRTRt

EQt

[e−

∑j:k≤j≤n(Ejyτj+Cj+Dj)

].

The last term can be handled with the same method as above: define for ` : ` ≤ n

c`(t; p) := EQt

[e−

∑j:k≤j<` Ejyτj−(E`+p)yτ`

].

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The first term we need to know is

ck(t; p) := −(Ek + p)K(t, τk)yt − (Ek + p)Θ(t, τk) + (Ek + p)2Γ(t, τk) .

and all further terms have the same structure as (11), i.e.

c`(t; p) = −(E` + p)Θ` + (E` + p)2Γ` + c`−1 (t; (E` + p)K`) .

The forward is then given as

Ft(T ) = StRTRt

ecn(t;0)−∑j:k≤j≤n Cj+Dj ≡ St

RTRt

eAt(T )yt+Bt(T ) (12)

for some A and B (note that cn(t; 0) is a function of yt).

A.4 Fitting to Equity Vanillas

Assume we are given a term structure of reference maturities 0 < T1 < T2 < · · ·for which we are given Black & Scholes-implied volatilities (Σk)k on the stockprice. We want to bootstrap a piecewise linear equity forward volatility termstructure σ ≡ (σk)k which is constant over the intervals [Tk−1, Tk].

To do so, we iteratively solve the following quadratic equation in σk:

Var(Tk)−Var(Tk−1) = σ2k(Tk − Tk−1)

−2σk · ρν∑

i:Tk−1<τi≤Tk

e−κτiEiOik

+∑

i:Tk−1<τi≤Tk

Ei

{Ei − ρνe−κτi

∑r:r<i

σrKr

}

with

Kr :=eκTr − eκTr−1

κ, Oik :=

eκτi − eκTk−1

κ, Ei :=

∑j:j≤i

Ejν2 1− e−2κτj

2κ.

References

[1] B.Baldwin:The World of Equity Derivatives - The Essential Toolbox for Investores,Eurex, September 2008,http://www.eurexchange.com/download/documents/publications/TheWorldofEquityDerivatives.pdf

[2] B.Baldwin:“Dividend Derivatives: Introduction of Options on EURO STOXX50 IndexDividend Futures”, eurex circular 082/10, May 2010,hhttp://www.eurexchange.com/download/documents/circulars/cf0822010e.pdf

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[3] A.Bermudez, H.Buehler, A.Ferraris, C.Jordinson, A.Lamnouar,M.Overhaus:Equity Hybrid Derivatives, Wiley, 2006

[4] F.Black, P.Scholes:“The Pricing of Options and Corporate Liabilities”, Journal of PoliticalEconomy, 81, pp. 637-59, 1973

[5] O.Brockhaus, A.Ferraris, C.Gallus, D.Long, R.Martin, M.Overhaus:Modelling And Hedging Equity Derivatives, Risk Books, 1999

[6] H.Buehler:”Volatility and Dividends - Volatility Modelling with Cash Dividends andSimple Credit Risk”, WP, June 7, 2008http://ssrn.com/abstract=1141877

[7] R.Gaspar“Finite Dimensional Markovian Realizations for Forward Price Term Struc-ture Models”, Stochastic Finance, 2006, Part II, 265-320

[8] J.Hull: ”Interest Rate Derivatives: Models of the Short Rate”, Options,Futures, and Other Derivatives (6th ed), Upper Saddle River, N.J: PrenticeHall. pp. 657658, 2006

[9] J.Hull, A.White:”One factor interest rate models and the valuation of interest rate derivativesecurities”, Journal of Financial and Quantitative Analysis, Vol 28, No 2,(June 1993) pp 235254

[10] R.Merton: “Option Pricing When Underlying Stock Returns are Discon-tinuous”, Journal of Financial Economics 3 (1976) pp. 125-144.

[11] D.Wood:“Uncertain Dividends”, Risk Magazine, Oct 2007

[12] D.Wood:“Options on DIVD and DVS Indexes”, CBOE Website, May 2010,http://www.cboe.com/micro/dvs/introduction.aspx andhttp://www.cboe.com/micro/dvs/DIVDFAQ.pdf

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