[copenhagen business school, nielsen] dividends in the theory of derivative securities pricing

7/30/2019 [Copenhagen Business School, Nielsen] Dividends in the Theory of Derivative Securities Pricing

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Dividends in the Theory of Derivative

Securities Pricing1

Lars Tyge Nielsen2

First Draft: January 1995. This Version: February 2006

1The first version of the paper was entitled Dividends in the Theory of Deriv-

ative Securities Pricing and Hedging and was presented at ESSEC in 1995. The

initial research was carried out during a visit to the University of Tilburg in the

Fall of 1994. The author would like to thank Knut Aase and Darrel Duffie for

comments on an earlier version.2Copenhagen Business School, Department of Finance, DK-2000 Frederiksberg,

Denmark; e-mail: [email protected]


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1 Introduction

This paper develops the fundamental aspects of the theory of martingale pric-ing of derivative securities in a setting where the cumulative gains processesare Ito processes, while the cumulative dividend processes of both the under-liers and the derivative securities are general enough to cover all the ways inwhich dividends are modeled in practical applications.

The most general cumulative dividend processes that arise in practice are Itoprocesses and finite-variation processes. Ito processes arise as cumulative div-idend processes of continuously resettled futures contracts. Finite-variationprocesses include the cumulative dividend processes arising from continuous

flows of dividends and random or deterministic lump-sum dividends paid atrandom or deterministic discrete dates. Continuous flows are used to modelstocks and stock indexes, currencies, and commodities (where the dividendflow is called convenience yield). Lump sum dividends at discrete dates areused to model stocks, bonds and swaps (where the dividends are interestpayments), and American options (which pay a random payoff at a randomdate).

A key operation that needs to be covered in the theory is changing theunit of account. The simplest case consists in discounting at the rolled-over instantaneous interest rate, which is equivalent to changing to units of a

money market account. In this case, the price of the new unit is an absolutelycontinuous process. Some authors, including Karatzas and Shreve [25, 1998,Chapter 2], change to new units whose price processes have finite variation,but this case will not be considered in the present paper because it does notseem to be needed in applications.

The more complicated case is where the price of the new unit is an It o process.Examples include real as opposed to nomial units, units of a foreign currencyor a foreign money market account, units of a commodity, or units of a stockwith reinvested dividends, as in Schroder [30, 1999]. Margrabe [26, 1978] mayhave been the first to make substantive use of this type of change of unit.He derived his exchange option formula by changing to units of one of therisky assets, thereby reducing the problem to that of pricing a standard calloption (Margrabe attributed the idea to Stephen Ross). Since then, the ideahas been further developed and stressed by Geman, El Karoui, and Rochet

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[15, 1995] and Schroder [30, 1999] and used in numerous applications.

Thus the theory needs to cover cumulative dividend processes that may beeither Ito processes or finite-variation processes, and it needs to prescribehow to change the unit of account of these processes when the price of thenew unit is an Ito process.

In fact, the theory developed here covers virtually all possible cumulativedividend processes, both for the underliers and for the derivative securities.They generally only need to need to be measurable and adapted. Whentheir unit is being changed, they also need to satisfy a minimal integrabilitycondition which allows them to be integrated with respect to the price ofthe new unit. This level of generality comes essentially for free, since it issimpler to develop a general theory than to develop a theory that is narrowlydesigned for Ito processes and finite-variation processes.

Since finite-variation processes and Ito processes are covered as special cases,the pricing of American options and of continuously resettled futures con-tracts fits seamlessly into the theory.

Our prescription for how to change the unit of account is based on first prin-ciples. We first calculate how the cumulative dividend process of a tradingstrategy is transformed under a change of unit in a securities market modelwhere there are no dividends on the basic securities. This leads to a formula

which is reminiscent of integration by parts. It turns out that virtually everycumulative dividend process is in fact the cumulative dividend process of atrading strategy in some very simple securities market model. Therefore, weuse the same formula for changing the unit of account of cumulative dividendprocesses in general, including the cumulative dividend processes of the basicsecurities when these are non-zero.

The general use of the formula is justified by four of its properties: it obeysunit-invariance for trading strategies, it satisfies a consistency property whenthe unit is changed twice in a row, it gives the correct results in well-knownand uncontroversial special cases, and it fits perfectly into a generalization

of martingale valuation theory to general dividend processes.

Unit-invariance means the following. Given the formula for changing theunits of a cumulative dividend process, there are two ways of changing theunits of a trading strategys cumulative dividend process. One is to calculate

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underlying securities (or basic securities in the literature on optimal consump-

tion and investment choice) and derivative security (or optimal consumptionprocess in the

Apart from Duffie [9, 1991, Section 5], the abstract literature on martingalepricing of derivatives has so far considered only underlying securities withoutdividends or with continuous dividend flows. Duffie [9, 1991, Section 5] allowsfor cumulative dividend processes that are semi-martingales.

The applied literature on pricing of equity derivatives considers continuousproportional dividend flows and, in addition, various forms of discrete div-idends. For example, Roll [29, 1977], Geske [16, 1979], and Whaley [32,1979] introduced non-random discrete dividends, and Wilmott, Dewynne,and Howison [33, 1993] explored a model with discrete dividends that arefunctions of stock price and time.

The literature on optimal portfolio and consumption choice and dynamicequilibrium, like the literature on derivatives pricing, usually assumes thatthe basic securities pay no dividends or else pay continuous proportionaldividend flows. When it relaxes this assumption, it requires the cumulativedividend processes on the basic securities to be semi-martingales, or at leastto be right-continuous with left limits.

Duffie and Zame [13, 1989], Dana and Jeablanc-Picque [6, 1994], Dana,

Jeanblanc-Picque, and Koch [7, 2003], Duffie [10, 1992] and [11, 1996], Chap-ter 6, Sections K and L, Duffie [12, 2001], Chapter 6, Sections L and M, andAase [1, 2002] assume that the cumulative dividend processes on the basicsecurities are Ito processes.

Huang [21, 1985], Dybvig and Huang [14, 1988], Cox and Huang [4, 1991],Huang and Pages [22, 1992], and Hindy [19, 1995] make assumptions whichimply that the cumulative dividend processes on the basic securities havefinite variation.

Back [2, 1991] assumes that the cumulative dividend processes on the basic

securities are semi-martingales, but he changes the unit of account only intounits of security zero, which is assumed to have finite variation.

Cuoco [5, 1997] makes no explicit assumptions about the cumulative dividendprocesses on the basic securities but must implicitly be assuming that they

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are semi-martingales, because he uses them as integrators. He changes to

units of a money market account by integrating, which is unproblematicbecause the money market account has finite variation (in fact, it is absolutelycontinuous).

Exceptions to the semi-martingale requirement include Duffie [8, 1986] andSchweizer [31, 1992]. They assume the basic cumulative dividend processesto be right-continuous with left limits, but they do not assume them to besemi-martingales. In Duffie [8, 1986], there is no need to change the unit ofaccount. Schweizer [31, 1992] changes to units of the locally riskless asset byintegrating with respect to the cumulative dividend processes. He observesthat when the new numeraire is the locally riskless asset and the cumulative

dividend processes are right-continuous with left limits, this integral is de-fined by integration by parts even if the cumulative dividend processes arenot semi-martingales.

The theoretical literature on martingale pricing of derivatives mostly consid-ers only derivatives that pay a random payout at one future point in time.This is true of Harrison and Kreps [17, 1979], Harrison and Pliska [18, 1981],Karatzas [24, 1997], and Nielsen [27, 1999], except that Karatzas separatelyconsiders also the case of American contingent claims. Bensoussan [3, 1984]initially assumes that the derivative pays a flow of dividends and a ran-dom payout at maturity and then considers American options separately.

Duffie [9, 1991, Section 5] assumes that the cumulative dividend process ofthe derivative security has finite variation. Dana and Jeanblanc-Picque [6,1994], Aase [1, 2002] and Dana, Jeanblanc-Picque, and Koch [7, 2003] al-low the cumulative dividend process of the derivative security to be an It oprocess. The most general setting available in this literature appears to bethat of Karatzas and Shreve [25, 1998, Chapter 2], who assume that thecumulative dividend process is a semimartingale.

In models of optimal consumption and portfolio selection or of dynamic equi-librium, the cumulative consumption process is analogous to the cumulativedividend process on a derivative security. Most such models assume that

consumption is represented by a continuous flow. Those that go beyondconsumption flows make assumptions which imply that the cumulative con-sumption process has finite variation. This is true, for example, of Huang[21, 1985], Dybvig and Huang [14, 1988], Hindy and Huang [20, 1993], Jinand Deng [23, 1997], and Karatzas [24, 1997].

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The paper is organized as follows. Section 2 sets up the model and the

basic notation, which is as close as possible to that of Nielsen [27, 1999].Section 3 lays out how to change the unit of account and proves the gener-alized unit-invariance result. Section 4 shows what changing to a new unitdoes to cumulative dividend processes that are semi-martingales, includingIto processes and processes with finite variation, and what changing to unitsof the money market account does to a general dividend process. It alsoreviews an alternative way of changing units that has been proposed in theliterature. Section 5 states and proves the consistency property: changingthe unit twice in a row is the same as changing it in one go. Sections 6 and 7generalize the standard concepts and results of derivatives pricing to generaldividends and demonstrates a general formula for valuing a dividend process

as a sum of a claim to the total accumulated nominal dividends and a claimto a stream of interest payment.

2 Securities and Trading Strategies

We consider a securities market where the uncertainty is represented by acomplete probability space (, F, P) with a filtration F = {Ft}tT and aK-dimensional process W, which is a Wiener process relative to F.

A cumulative dividend process D measures the cumulative value of distribu-tions, dividends, interest payments or other cash flows, positive or negative,of a security or trading strategy.

Formally, a cumulative dividend process is a measurable adapted process Dwith D(0) = 0.

Suppose a security has cumulative dividend process D and price process S.Define the cumulative gains processG of this security as the sum of the priceprocess and the cumulative dividend process:

G = S+ D

Assume that G is an Ito process. It follows that G will be continuous,adapted, and measurable. Since D is adapted and measurable, so is S. SinceD(0) = 0, G(0) = S(0).

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An (N + 1)-dimensional securities market model (based on F and W) will

be a pair (S,

D) of measurable and adapted processes

S and

D of dimensionN + 1, interpreted as a vector of price processes and a vector of cumulative

dividends processes, such that D(0) = 0 and such that G = S+ D is an Itoprocess with respect to F and W. The process G = S+ G is the cumulativegains processes corresponding to (S, D).

Write

G(t) = G(0) +t0

ds +t0

dW

where is an N+ 1 dimensional vector process in L1 and is an (N+ 1)Kdimensional matrix valued process in L2.Here, L1 is the set of adapted, measurable, and pathwise integrable processes,and L2 is the set of adapted, measurable, and pathwise square integrableprocesses.

A trading strategy is an adapted, measurable (N+1)-dimensional row-vector-valued process .

The value process of a trading strategy in securities model (S, D) is theprocess S.

The set of trading strategies such that L1 and L2, will bedenoted L(

G).

Generally, if X is an n-dimensional Ito process,

X(t) = X(0) +t0

a ds +t0

b dW

then L(X) is the set of adapted, measurable, (n K)-dimensional processes such that L1 and L2.If is a trading strategy in L(G), then the cumulative gains process of, measured relative to the securities market model (S, D), is the process

G(;

G) defined by

G(; G)(t) = (0)G(0) +t0

dG

for all t T.

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A trading strategy in L(G) is self-financing with respect to (S, D) ifS = G(; G)

or

(t)S(t) = (0)S(0) +t0

dG

Generally, if is a trading strategy in L(G) which may not be self-financing,then the cumulative dividend process of with respect to (S, D) is theprocess D(; S, D) defined by

S+ D(; S, D) = G(; G)

The process D(; S, D) is adapted and measurable and has initial valueD(; S, D)(0) = 0.

3 Changing the Unit of Account

Let be a one-dimensional Ito process:

(t) = (0) +t0

ds +t0

dW

where is a one-dimensional process in L1 and is a K dimensional rowvector process in L2.If D is a cumulative dividend process such that D L(), define a processD by

D(t) = (t)D(t) t0

D d

The process D is adapted and measurable, and, hence, it is a cumulativedividend process. The purpose of assuming that D L() is to make surethat the integral in this expression is well defined.

The following proposition shows how the cumulative dividend process of atrading strategy is re-measured in a new unit of account in the case wherethere are no dividends on the basic securities. The original securities marketmodel is (S, 0), and the new one is (S, 0). The original dividend process Dof the trading strategy is replaced by D.

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Proposition 1 Suppose D = 0. Let L(S), and set D = D(; S, 0).Then

L(

S) if and only if D L(), in which case D(

;

S, 0) = D

.

Proposition 1 will follow from Theorem 1 below.

In the expression for D, the term (t)D(t) is the dividends cumulated in theold units and expressed in the new unit. This ignores the fact that at eachpoint in time, already accumulated dividends change value. The integralterm corrects for that. The term D(s) d(s) represents the change duringinstant s in the value of the dividends D(s) that have been accumulated upto that time. The integral represents the cumulative value of these changes.If, for example, the process tends to increase over time, it means that

the value of the old unit of account increases. The term (t)D(t) overstatesthe cumulative dividends in the new unit of account, because most of thedividends have been accumulated at times when they were worth less than(t) per old unit.

Proposition 1 tells us how to change the unit of account of a cumulativedividend process that arises as the cumulative dividend process of a tradingstrategy. But as we shall now see, every cumulative dividend process is in factthe cumulative dividend process of a trading strategy in some very simplesecurities market model.

Let D be a cumulative dividend process. Consider a securities market modelwhere there is a money market account, and assume for simplicity that themoney market account has zero interest rate and value process M = 1.Consider the trading strategy which consists in holding, at every point intime t, D(t) units of the money market account. The value process isD. Since the trading strategy only invests in the money market account,which has zero interest rate, its cumulative gains process is zero. Hence, thecumulative dividend process is D.

This observation, taken together with Proposition 1, suggests that in general,for any cumulative dividend process D, the process D can be interpreted as

D re-measured in the new unit of account. Therefore, we will use the sameprocedure to change the unit of account on the basic securities in the casewhere the basic securities pay dividends.

If (S, D) is a securities market model such that D L(), then we define

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the transformed cumulative dividend process D in the new unit of account

entry by entry by

D =

DN...

D0

or

D(t) = (t)D(t) t0

D d

The transformed securities price process is S, and the transformed securitiesmarket model is (S, D). The corresponding cumulative gains process G

is given by

G = S+ D =

GN...

G0

or

G(t) = (t)G(t) t0

D d

The next proposition exhibits the Ito differential of G.

Proposition 2 Let be an Ito process, and assume that D L(). ThenG is an Ito process with

dG = + S +

dt +

+ S

dW

Proof: Since

G(t) = (t)G(t) t0

D d

G is obviously an Ito process, and

dG = dG + G d + dt D d= dG + S d + dt

= + (G D) +

dt +

+ (G D)

dW

= + S +

dt +

+ S

dW

2

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We now need to show that even when there are dividends on the basic secu-

rities, the cumulative dividend process of a trading strategy will undergo thetransformationD D

In other words, if D is the cumulative dividend process of a trading strategymeasured in the old prices, then D is the cumulative dividend process ofthe same trading strategy measured in the new prices. This general unit-invariance result is the content of Theorem 1 below.

Theorem 1 Let be an Ito process, assume that D L(), and let

L(G). Then

L(G) if and only if S

L(), and if and only if

D(; S, D) L(). If so, thenD(; S, D) = D

andG(; G) = G

In particular, if is self-financing with respect to (S, D), then L(G)and is self-financing with respect to (S, D).

Proof: Set S = S, D = D(; S, D), and G = G(; G). Since G is acontinuous process, it is in L(); and since G = S+ D, one of the processesS and D is in L() if and only if the other one is.Observe that

+ S +

= + S +

and[ + S] = + S

Now L1 because is continuous and L1; L1 because and are in L2; and L2 because is continuous and L2.Hence,

+ S +

L1

if and only ifS L1, and[ + S] L2

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if and only ifS L2. It follows that L(G) if and only if S L().Assume that this is so. Then

dG(; G) = dG + S d + dtand

d (G) = d

G(; G)

= dG(; G) + G(; G) d + dt= dG + G d + dt

Hence,

d (G) dG(; G

) = [G S] d = D dThis implies that

G(; G)(t) = (t)G(t) t0

D d = G(t)

and

D(; S, D)(t) = G(; G)(t) (t)(t)S(t)= G(t) (t)S(t)= D(t)

2

It is remarkable that and D are all we need in order to calculate D. Oncewe know and D = D(; S, D), we do not need or S or G in order tocalculate D = D(; S, D).Theorem 1 extends Proposition 4.2 of Huang [21, 1985] to general cumu-lative dividend processes (in the case where is an Ito process). Huangsproposition assumes that the cumulative dividend processes on both the basicsecurities and the trading strategy have finite variation. It involves changingthe unit of account by integrating with respect to the cumulative dividendprocess. It will be shown in Proposition 4 below that our procedure forchanging the unit coincides with Huangs in the case of finite variation.

Theorem 1 modifies Proposition 9 of Duffie [9, 1991, Section 5] (in the casewhere, as here, is an Ito process). Duffies proposition changes the unit by

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integrating with respect to the cumulative dividend processes. This is fine

in the case of the cumulative dividend process of the trading strategy, whichis assumed to have finite variation, but it is not suitable in the case of thecumulative dividend processes of the basic securities, which are assumed to besemi-martingales. Proposition 5 is a counterexample to Duffies proposition.

4 Examples and Counterexamples

In this section, we shall first calculate D1/M for a general cumulative dividendprocess D when M is the value process of a money market account.

Next, we shall calculate the process D for two types of cumulative dividendprocesses: Ito processes and processes of finite variation. The cumulativedividend process of a continuously resettled futures contract would be anexample of the former, and random discrete dividends at random times area special case of the latter.

Finally, we shall show that if the cumulative dividend processes of the basicsecurities are Ito processes, and if they are re-measured in a new unit bysimple integration, then the unit-invariance result of Theorem 1 generallydoes not hold.

A money market account for (S, D) is a self-financing trading strategy b(or a security that pays no dividends) whose value process is positive andinstantaneously riskless (has zero dispersion). We denote its value processby M: M = bS.

If M is the value process of a money market account, then it must have theform

M(t) = M(0)[r, 0](t) = M(0) expt

0r ds

for some r L1 (the interest rate process) and some M(0) > 0.

We use the general notation [, ] for the stochastic exponential, which isthe process defined by

[, ](t) = expt

0

(s) 1

2(s)(s)

ds +

t0

(s)dW(s)

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Here, and are processes in L1 and L2, of dimension one and K, respec-tively.Let us now change the unit of account by = 1/M.

Let D be a cumulative dividend process. Then Dr L1 if and only ifD L(1/M) = L(M). If so, then

D1/M(t) = D(t)/M(t) +t0

Dr

Mds

Schweizer [31, 1992, Equation 2.2] is similar to this equation except that weassume D(0) = 0, while he assumes D to be right-continuous with left limitsand allows M to have finite variation rather than be absolutely continuous.

When M is not absolutely continuous, it is necessary to replace D by D(the limit from the left) in the integral on the right, which of course can onlybe done if this limit exists.

The term D(t)/M(t) in the formula is the nominal cumulative dividendsexpressed as a number of units of the money market account, ignoring theinterest earned at each point in time on already accumulated dividends in theform of appreciation of the money market account. The integral representsthe cumulative value of this interest, expressed in units of the money marketaccount.

Next, we consider cumulative dividend processes that are semi-martingales,including Ito processes and processes with finite variation.

Proposition 3 If D is a semi-martingale, then

D(t) =t0

dD + [D, ](t)

Proof: By the formula for integration by parts for semi-martingales (Protter[28, 1990, Chapter 2, Corollary 6.2]),

D(t)(t) =t0 D d +

t0 dD + [D, ](t)

Hence,

D(t) = D(t)(t) t0

D d =t0

dD + [D, ](t)

2

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Corollary 1 If D is a cumulative dividend process which is and Ito process

with dD = f ds + g dW

then D L() and

D(t) =t0

dD +t0

g ds

Proof: Since D is continuous, it is in L(). The formula follows fromProposition 3 and the fact that

[D, ](t) =t0 g

ds

2

Notice in particular from Corollary 1 that when D is an Ito process, D ingeneral is not equal to the integral of with respect to dD:

D(t) =t0

dD

However, if and D have zero instantaneous covariance (g = 0), then D

is indeed equal to the integral of with respect to dD:

D(t) =t0

dD

This includes the special case where has zero dispersion ( = 0). It alsoincludes the case where D simply involves a continuous dividend flow:

D(t) =t0

f ds

In this case,

D

(t) =t0 dD =

t0 f ds

In other words, D has continuous dividend flow f. This is exactly whatwe would expect.

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Proposition 4 If D is right-continuous and has finite variation, then

D(t) =

t

0 dD

Proof: Referring to Proposition 3, we just need to verify that the quadraticcovariation [D, ] is zero. The quadratic variation of the Wiener process Wis time: [W, W](t) = t, and the quadratic variation of D is zero. By theKunita-Watanabe inequality (Protter [28, 1990, Chapter 2, Theorem 6.25]),for each k = 1, . . . , K ,

t0

|d[D, Wk]| t

0d[D, D]

1/2 t0

d[Wk, Wk]1/2

= 0 t= 0

Hence, [D, Wk] = 0 for each k, and so [D, W] = 0. Letting I be the stochasticintegral process

I(t) =t0

dW

it follows from Protter [28, 1990, Chapter 2, Theorem 6.29] that

[D, I](t) =t0

d[D, W] = 0

Hence [D, ] = 0. 2

Consider an alternative way of changing the unit of account, which has beenused by Duffie and Zame [13, 1989] for cumulative dividend processes thatare Ito processes and by Duffie [9, 1991] for the more general case of semi-martingales.

If D is a cumulative dividend process which is a semi-martingale, define theprocess D() by simple integration:

D()(t) =

t

0 dD

Similarly, suppose (S, D) is a securities market model such that D is a semi-martingale, define the process D() by

D()(t) =t0

dD

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Proposition 5 Suppose (S, D) is a securities market model such that D is

an Ito process with dD = f dt + g dW

Suppose L(G) is a trading strategy such that D(; S, D) has finite vari-ation. Then the following are equivalent:

1. For every t,D(; S, D())(t) = D(; S, D)()(t)

with probability one

2. g = 0 almost everywhere (the trading strategy has zero instanta-

neous covariance with )

Proof: It follows from Corollary 1 that

D()(t) =t0

dD = D(t) t0

g ds

The transformed securities market model is (S, D()), and the correspondingcumulative gains process G() is given by

G()(t) = (t)S(t) + D()(t) = D(t) t0

g ds

In the transformed securities market model, the cumulative gains process of is

G(; G())(t) = (0)G(0) +t0

dG()

= (0)G(0) +t0

dG t0

g ds

= G(; G)(t) t0

g ds

and the cumulative dividend process is

D(; S, D())(t) = G(; G())(t) (t)S(t)= G(; G)(t)

t0

g ds (t)S(t)

= D(; S, D)(t) t0

g ds

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Since D(; S, D) has finite variation, it follows from Proposition 4 that

D(; S, D)() = D(; S, D)

whereas, as shown above,

D(; S, D())(t) = D(; S, D)(t) t0

g ds

Hence, for all t,D(; S, D())(t) = D(; S, D)()(t)

with probability one if and only if for all t,

t

0g ds = 0

with probability one, which is true if and only if g = 0 almost everywhere.2

Proposition 5 contradicts Proposition 9 of Duffie [9, 1991, Section 5] andimplies that it is not appropriate in general to change the unit of a cumulativedividend process by simple integration, even if the integral is well definedbecause the cumulative dividend process is a semi-martingale. In particular,this procedure is not appropriate if the cumulative dividend process is an

Ito process, as recognized in Duffie [10, 1992], [11, 1996], and [12, 2001], andAase [1, 2002]

Duffie and Zame [13, 1989, page 1287] assume that the cumulative dividendprocesses of the basic securities are Ito process. They re-measure them inunits of a consumption good by simple integration, thus defining the realcumulative dividend processes. Duffie [9, 1991, Section 4, page 1640] alsore-measures a cumulative dividend process (which is assumed to be a semi-martingale) in units of a consumption good by simple integration. One shouldbe careful with the interpretation of this procedure, because it does not satisfyunit invariance.

Proposition 6 of Duffie [9, 1991, Section 4] similarly involves changing the unitof account by simple integration. Its Equation 16 contradicts Equation 2.3of Aase [1, 2002]. The source of this discrepancy seems to be the step inthe proof where Fubinis theorem for conditional expectations is applied.

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This theorem applies to time integrals of conditional expectations but not to

general stochastic integrals.A similar calculation appears in the proof Proposition 7.3.2 in Dana andJeanblanc-Picque [6, 1994] and Dana, Jeanblanc-Picque, and Koch [7, 2003].The last equation of this proof is essentially an application of Fubinis theo-rem for conditional expectations to a stochastic integral.

5 Consistency

Suppose we change the unit of account of a cumulative dividend process Dby , calculating D, and we then further change the unit of account by aprocess , calculating (D). The result should be the same as if we changedthe unit of account by in one step, calculating D .

This consistency property of changes of the unit of account is verified in thefollowing theorem.

Theorem 2 Let and be one-dimensional Ito processes, and let D bea cumulative dividend process such that D L() and D L(). ThenD L() and

(D

)

= D

Proof: First, observe that D L(). This follows from the fact thatD L() and D D belongs to L() because the latter is a continuousprocess.

Writed = dt + dW

where L1 and L2. Thend() = d + d+

dt

= + +

dt + [ + ] dW

Since D L() and D L() by assumption, D L1, D L2,D L1, and D L2. Since is continuous, it follows that D L1, D L2, and consequently, D L().

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Now,

(D) (t) = (t)D(t) t0

D d

= (t)(t)D(t) (t)t0

D d t0

Dd+t0

s0

D d

d

and

D(t) = (t)(t)D(t) t0

D d()

= (t)(t)D(t)

t

0

D d

t

0

Dd

t

0

D

ds

The difference between these two processes is

(D) (t) D(t) = (t)(t)D(t) (t)t0

D d t0

Dd

+t0

s0

D d

d (t)(t)D(t)

+t0

D d +t0

Dd+t0

D

ds

= (t) t

0D d +

t

0 s

0D d d

+

t

0D d +

t

0D

ds

It is an Ito process with differential

d ((D) D) = Dd t

0D d

d D dt

+s

0D d

d+ D d + D

ds

= 0

Since the processes (D

)

and D

have the same initial value, it follows that

(D) = D

2

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6 State Prices and Risk Adjustment

This section generalizes some standard concepts and results to general divi-dends: state price process, prices of risk, the martingale property, and risk-adjusted probabilities.

Consider a securities market model of the form (S, D).

Let be a positive one-dimensional Ito process. It must have the form

= (0)[r,]for some (0) > 0 and processes r L1 and L2. Given a positiveconstant M(0), set M = M(0)[r, 0]

Then

= (0)M(0)[0, ]

MSay that is a state price process for (S, D) if Dr L1 and the process

[0, ]G1/M

has zero drift.

This definition is consistent with the standard definition for the case where

D = 0: is a state price process for (S, 0) if and only if

[0, ]GM = [0, ] 1M

S =1

M(0)(0)S

has zero drift.

The following proposition exhibits the standard equation for the prices ofrisk and identifies r as the interest rate.

Proposition 6 Let (0) > 0, M(0) > 0, r L1, and L2, and assumethat Dr L1. Then = (0)[r,] is a state price process for (S, D) ifand only if rS = almost everywhere. If so, and ifb is a money market account with initial valueb(0)S(0) = M(0), then b has value process bS = M(0)[r, 0] and interest rater.

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Proof: The drift of [0, ]G1/M is[0, ]

M

rS

Hence, = (0)[r, ] is a state price process for (S, D) if and only if

rS =

almost everywhere. If so, and if b is a money market account with initialvalue b(0)S(0) = M(0), then

b

rbS = b = 0

andd(bS) = dG(b; G) = b dG = b dt = rbS dt

Given the initial condition b(0)S(0) = M(0), this differential equation hasthe unique solution bS = M(0)[r, 0]. 2

The next proposition says that the zero-drift condition that defines the stateprice process holds for the discounted cumulative gains processes not only ofthe basic securities but of any trading strategies (whether self-financing ornot).

Proposition 7 If is a state price process for (S, D), and if L(G1/M),then

[0, ]G

, G1/M

has zero drift.

Proof: By Theorem 1, D L(1/M). Observe that

d

[0, ]G

, G1/M

= [0, ] dG , G1/M + G , G1/M d[0, ] [0, ] dt

Since [0, ] has zero drift, the coefficients to dt in G

, G1/M

d[0, ]and in (G1/M) d[0, ] are both zero, and the drift of[0, ]G

, G1/M

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equals the coefficient to dt in

[0, ] dG , G1/M + (G1/M) d[0, ] [0, ] dt=

[0, ] dG1/M+ G1/Md[0, ] [0, ] dt

= d

[0, ]G1/M

Since [0, ]G1/M has zero drift, the coefficient to dt is zero, and, hence[0, ]G

, G1/M

has zero drift. 2

Assume that D L(M).

Let be a trading strategy in L(G1/M

). It was just shown above that

[0, ]G(; G1/M)has zero drift. Say that is admissible for (S, D) and if this process is amartingale.

If is a self-financing trading strategy, then

[0, ]G

; G1/M

= [0, ]S/M = 1(0)M(0)

S

Hence, is admissible if and only if S is a martingale. This is consistentwith the definition of admissibility in Harrison and Pliska [18, 1981], eventhough now there may be dividends on the basic securities.

Requiring self-financing trading strategies to be admissible rules out self-financing arbitrage strategies when there are no dividends on the basic se-curities. The same will be seen to be true of trading strategies that arenot necessarily self-financing, and even if there are dividends on the basicsecurities.

We formally generalize the concept of arbitrage to basic securities with gen-

eral dividends and to trading strategies that are not necessarily self-financing,as follows.

An arbitrage trading strategy is a trading strategy

L

G1/M

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such that (0)S(0)/M(0) = 0 and for some t,

G ; G1/M (t) 0with probability one, and

G

; G1/M

(t) > 0

with positive probability.

Recall that a trading strategy is self-financing relative to (S, D) if andonly if it self-financing relative to (S/M, D1/M), in which case its normalizedcumulative gains process is

G ; G1/M = S/M

Therefore, a self-financing arbitrage trading strategy is a self-financing trad-ing strategy L(G) such that (0)S(0) = 0 and for some t, (t)S(t) 0with probability one, and (t)S(t) > 0 with positive probability.

Thus, the definition is consistent with the usual definition in the case wherethere are no dividends.

If is an arbitrage trading strategy, then

E[0, ](t)G

; G1/M

(t)

> (0)S(0)/M(0)

= [0, ](0)G ; G1/M

(0)Hence, [0, ]G(; G1/M) cannot be a martingale, and cannot be ad-missible. So, there are no arbitrage trading strategies among the admissibletrading strategies.

Define the K-dimensional process W by

W(t) =t0

ds + W(t)

We can express the differential of G1/M in terms of dW:

dG1/M

=1

M( rS) dt +1

M dW

=1

M( rS ) dt + 1

M dW

=1

M dW

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If L(G1/M) is a trading strategy, then

dG(; G1/M) = dG1/M = 1M

dW

Define the risk-adjusted probability measure Q (on the horizon T) correspond-ing to as the measure on (, F) which has density [0, ](T) with respectto the original probability measure P.

According to Girsanovs Theorem, if [0, ] is a martingale on [0, T], thenW is a Wiener processes with respect to F and Q on [0, T].

In terms of the risk-adjusted probabilities, a trading strategy inL

(G1/M) isadmissible if and only if the discounted cumulative gains process G(; G1/M)is a martingale under Q.

7 Replication and Valuation

A trading strategy is said to replicate a contingent claim Y (a randomvariable) at time T with respect to (S, D) if it is self-financing with respectto (S, D) and (T)S(T) = Y.

If is a self-financing trading strategy which replicates a contingent claimY at time T with respect to (S, D), and which is admissible for (S, D) and on [0, T], then for t T, because of the martingale property,

(t)S(t) =1

(t)E[(T)Y | Ft]

If is any positive Ito process and Y is any claim such that (T)Y isintegrable, define the value process or, for emphasis, its martingale valueprocess of Y with respect to as the process V(Y; ; T) given by

V(Y; ; T)(t) =1

(t)E[(T)Y | Ft]

for 0 t T.

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If is a state price process for (S, D) and if Y happens to be replicated

at time T with respect to (S,

D) by a self-financing strategy

, then

isadmissible for (S, D) and if and only if

S = V(Y; ; T)

A trading strategy will be said to replicate a cumulative dividend processD up to time T with respect to (S, D) if L(G), D(; S, D) = D and(T)S(T) = 0.

Proposition 8 Let(S, D) be a securities market model such thatD L(M),and let D be a cumulative dividend process. If D is replicated with respect to

(S, D) up to time T by a trading strategy L(G) which is admissible for(S, D) and , then D L(M), and for every t [0, T],

G

; G1/M

(t) = V(D1/M(T); [0, ]; T)(t)

(t)S(t)/M(t) = V(D1/M(T); [0, ]; T)(t) D1/M(t)and

(t)S(t) = V(D1/M(T)M(T);; T)(t) D1/M(t)M(t)with probability one.

Proof: Since is admissible, L(G1/M). By Theorem 1, Dr L1, and replicates D1/M with respect to S/M and D1/M. Furthermore,

[0, ]G

; G1/M

= [0, ]S/M + [0, ]D1/M

is a martingale. Hence,

[0, ](T)G

; G1/M

(T) = [0, ](T)D1/M(T)is integrable, and for 0 t T,

[0, ](t)G

; G1/M

(t) = E

[0, ](T)D1/M(T) | Ft

This implies that

G

; G1/M

(t) =1

[0, ](t) E[0, ](T)D1/M(T) | Ft

= V(D1/M(T); [0, ]; T)(t)

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Hence,

(t)S(t)/M(t) = G ; G1/M (t) D1/M(t)= V(D1/M(T); [0, ]; T)(t) D1/M(t)

and

(t)S(t) = V(D1/M(T); [0, ]; T)(t)M(t) D1/M(t)M(t)= V(D1/M(T)M(T);; T)(t) D1/M(t)M(t)

with probability one. 2

We may want to use the formulas from Proposition 8 to value a cumulative

dividend process even if it is not replicated by an admissible trading strategy.Let D be a cumulative dividend process such that Dr L1 and such that[0, ](T)D1/M(T)

is integrable. Define the value process or, for emphasis, the martingale valueprocess V[D; ; T] of D with respect to , on [0, T], by

V[D; ; T] = V(D1/M(T)M(T);; T) D1/MM= V(D1/M(T); [0, ]; T)M D1/MM

It may be interpreted as the present value of the dividends yet to be paid upto time T.

We refer to this valuation procedure as the martingale method.

The final theorem says that the value of a dividend process equals the valueof a claim to the cumulative dividends from today on plus the value of a flowof interest on the cumulative dividends at each point in time.

Theorem 3 LetD be a cumulative dividend process in L(M) and let D bethe cumulative dividend process with continuous dividend flow Dr:

D(t) =t0

Drds

Assume that [0, ] is a martingale. If two of the variables (T)D(T),[0, ](T)D1/M(T), and [0, ](T)D1/M(T) are integrable, then so is thethird, and

V[D; ; T](t) = V(D(T);; T)(t) D(t) + V[D; ; T](t)

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Proof: The statement about integrability follows directly from the equation

(0)M(0)[0, ](T)D1/M(T)= (T)D(T) + (0)M(0)[0, ](T)

T0

Dr

Mds

= (T)D(T) + (0)M(0)[0, ](T)D1/M(T)If the relevant variables are indeed integrable, then

V[D; ; T](t)/M(t)

= EQD1/M(T) | Ft

D1/M(t)

= EQ[D(T)/M(T) | Ft] D(t)/M(t) + EQ T

t D

r

M ds Ft

= V(D(T);; T)(t)/M(t) D(t)/M(t) + V[D; ; T](t)/M(t)which implies that

V[D; ; T](t) = V(D(T);; T)(t) D(t) + V[D; ; T](t)2

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[copenhagen business school, nielsen] dividends in the theory of derivative securities pricing

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