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] 1 Introduction

This paper develops the fundamental aspects of the theory of martingale pricing of derivative securities in a setting where the cumulative gains processes are Itô processes, while the cumulative dividend processes of both the underliers and the derivative securities are general enough to cover all the ways in which dividends are modeled in practical applications. The most general cumulative dividend processes that arise in practice are Itô processes and finite-variation processes. Itô processes arise as cumulative dividend processes of continuously resettled futures contracts. Finite-variation processes include the cumulative dividend processes arising from continuous flows of dividends and random or deterministic lump-sum dividends paid at random or deterministic discrete dates. Continuous flows are used to model stocks and stock indexes, currencies, and commodities (where the dividend flow is called convenience yield). Lump sum dividends at discrete dates are used to model stocks, bonds and swaps (where the dividends are interest payments), and American options (which pay a random payoff at a random date). A key operation that needs to be covered in the theory is changing the unit 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 absolutely continuous 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 not seem to be needed in applications. The more complicated case is where the price of the new unit is an Itô process. Examples include real as opposed to nomial units, units of a foreign currency or a foreign money market account, units of a commodity, or units of a stock with reinvested dividends, as in Schroder [30, 1999]. Margrabe [26, 1978] may have 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 the risky assets, thereby reducing the problem to that of pricing a standard call option (Margrabe attributed the idea to Stephen Ross). Since then, the idea has been further developed and stressed by Geman, El Karoui, and Rochet

1 [15, 1995] and Schroder [30, 1999] and used in numerous applications. Thus the theory needs to cover cumulative dividend processes that may be either Itô processes or finite-variation processes, and it needs to prescribe how to change the unit of account of these processes when the price of the new unit is an Itô process. In fact, the theory developed here covers virtually all possible cumulative dividend processes, both for the underliers and for the derivative securities. They generally only need to need to be measurable and adapted. When their unit is being changed, they also need to satisfy a minimal integrability condition which allows them to be integrated with respect to the price of the new unit. This level of generality comes essentially for free, since it is simpler to develop a general theory than to develop a theory that is narrowly designed for Itô processes and finite-variation processes. Since finite-variation processes and Itô processes are covered as special cases, the pricing of American options and of continuously resettled futures contracts 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 trading strategy is transformed under a change of unit in a securities market model where 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 every cumulative dividend process is in fact the cumulative dividend process of a trading strategy in some very simple securities market model. Therefore, we use the same formula for changing the unit of account of cumulative dividend processes in general, including the cumulative dividend processes of the basic securities when these are non-zero. The general use of the formula is justified by four of its properties: it obeys unit-invariance for trading strategies, it satisfies a consistency property when the unit is changed twice in a row, it gives the correct results in well-known and 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 the units of a cumulative dividend process, there are two ways of changing the units of a trading strategy’s cumulative dividend process. One is to calculate

2 it in old units and then re-measure it in new units. The other is to re-measure the basic securities prices and dividends in new units and then calculate the trading strategy’s dividend process with respect to these re-measured prices and dividends. The unit-invariance result, which is stated in Theorem 1, says that these two are equivalent. The result extends a similar result by Huang [21, 1985] to general dividend processes and modifies a related statement by Duffie [9, 1991, Section 5]. The consistency property, which is stated in Theorem 2, says that changing from one unit to a second and then from the second unit to a third is the same as changing from the first unit to the third in one go. Special cases include cumulative dividend processes that are semi-martingales, and in particular Itô processes or processes with finite variation. Our formula for changing the unit of account is consistent with the literature in the case of finite-variation processes but not in the general case of semi-martingales. It is consistent with the current literature but not the earlier literature in thecaseofItô processes. Specifically, Duffie and Zame [13, 1989] and Duffie [9, 1991] change the unit of account by integrating the price of the old unit in terms of the new unit with respect to the cumulative dividend process. This is appropriate if the cumulative dividend process has finite variation, but it is not appropriate if it is an Itô process, as assumed by Duffie and Zame [13, 1989], or more generally if it is a semi-martingale, as assumed by Duffie [9, 1991]. We finally demonstrate how the theory of martingale valuation of derivatives extends to general dividend processes for both the basic and the derivative securities. The third and final main result, Theorem 3, states that the value of a dividend process equals the value of a claim to dividends to be accumulated from today on plus the value of a flow of interest on the cumulative dividends at each point in time. The rest of this introduction consists in an overview of how the literature has modeled dividends and an overview of the organization of the paper. We distinguish between dividends on the underlying securities and the dividends on the derivative security. In the literature on optimal consumption and investment choice, the derivative security corresponds to the cumulative consumption process.

3 underlying securities (or basic securities in the literature on optimal consumption and investment choice) and derivative security (or optimal consumption process in the Apart from Duffie [9, 1991, Section 5], the abstract literature on martingale pricing of derivatives has so far considered only underlying securities without dividends or with continuous dividend flows. Duffie [9, 1991, Section 5] allows for cumulative dividend processes that are semi-martingales. The applied literature on pricing of equity derivatives considers continuous proportional dividend flows and, in addition, various forms of discrete dividends. 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 are functions of stock price and time. The literature on optimal portfolio and consumption choice and dynamic equilibrium, like the literature on derivatives pricing, usually assumes that the basic securities pay no dividends or else pay continuous proportional dividend flows. When it relaxes this assumption, it requires the cumulative dividend processes on the basic securities to be semi-martingales, or at least to be right-continuous with left limits. Duffie and Zame [13, 1989], Dana and Jeablanc-Picqué [6, 1994], Dana, Jeanblanc-Picqué, 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, and Aase [1, 2002] assume that the cumulative dividend processes on the basic securities are Itô processes. Huang [21, 1985], Dybvig and Huang [14, 1988], Cox and Huang [4, 1991], Huang and Pagès [22, 1992], and Hindy [19, 1995] make assumptions which imply that the cumulative dividend processes on the basic securities have finite 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 into units of security zero, which is assumed to have finite variation. Cuoco [5, 1997] makes no explicit assumptions about the cumulative dividend processes on the basic securities but must implicitly be assuming that they

4 are semi-martingales, because he uses them as integrators. He changes to units of a money market account by integrating, which is unproblematic because the money market account has finite variation (in fact, it is absolutely continuous). Exceptions to the semi-martingale requirement include Duffie [8, 1986] and Schweizer [31, 1992]. They assume the basic cumulative dividend processes to be right-continuous with left limits, but they do not assume them to be semi-martingales. In Duffie [8, 1986], there is no need to change the unit of account. Schweizer [31, 1992] changes to units of the locally riskless asset by integrating with respect to the cumulative dividend processes. He observes that when the new numeraire is the locally riskless asset and the cumulative dividend processes are right-continuous with left limits, this integral is defined by integration by parts even if the cumulative dividend processes are not semi-martingales. The theoretical literature on martingale pricing of derivatives mostly considers 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 separately considers also the case of American contingent claims. Bensoussan [3, 1984] initially assumes that the derivative pays a flow of dividends and a random payout at maturity and then considers American options separately. Duffie [9, 1991, Section 5] assumes that the cumulative dividend process of the derivative security has finite variation. Dana and Jeanblanc-Picqué[6, 1994], Aase [1, 2002] and Dana, Jeanblanc-Picqué, and Koch [7, 2003] al- low the cumulative dividend process of the derivative security to be an Itô process. The most general setting available in this literature appears to be that of Karatzas and Shreve [25, 1998, Chapter 2], who assume that the cumulative dividend process is a semimartingale. In models of optimal consumption and portfolio selection or of dynamic equilibrium, the cumulative consumption process is analogous to the cumulative dividend process on a derivative security. Most such models assume that consumption is represented by a continuous flow. Those that go beyond consumption flows make assumptions which imply that the cumulative consumption process has finite variation. This is true, for example, of Huang [21, 1985], Dybvig and Huang [14, 1988], Hindy and Huang [20, 1993], Jin and Deng [23, 1997], and Karatzas [24, 1997].

5 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 unit does to cumulative dividend processes that are semi-martingales, including Itˆo processes and processes with ﬁnite variation, and what changing to units of the money market account does to a general dividend process. It also reviews an alternative way of changing units that has been proposed in the literature. Section 5 states and proves the consistency property: changing the unit twice in a row is the same as changing it in one go. Sections 6 and 7 generalize the standard concepts and results of derivatives pricing to general dividends and demonstrates a general formula for valuing a dividend process as a sum of a claim to the total accumulated nominal dividends and a claim to a stream of interest payment.

2 Securities and Trading Strategies

We consider a securities market where the uncertainty is represented by a complete probability space (Ω, F,P) with a filtration F = {Ft}t∈T and a K-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 D with D(0) = 0. Suppose a security has cumulative dividend process D and price process S. Define the cumulative gains process G of this security as the sum of the price process and the cumulative dividend process:

G = S + D

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

6 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 dimension N + 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 Itˆo process with respect to F and W . The process G¯ = S¯ + G¯ is the cumulative gains processes corresponding to (S,¯ D¯).

Write t t G¯(t)=G¯(0) + µds¯ + σdW¯ 0 0 whereµ ¯ is an N +1 dimensional vector process in L1 andσ ¯ is an (N +1)×K dimensional 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 integrable processes. 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¯)isthe process ∆¯ S¯. The set of trading strategies ∆¯ such that ∆¯¯ µ ∈L1 and ∆¯¯ σ ∈L2, will be denoted L(G¯). Generally, if X is an n-dimensional Itˆo process,

t t X(t)=X(0) + ads+ bdW 0 0 then L(X) is the set of adapted, measurable, (n × K)-dimensional processes γ such that γµ ∈L1 and γσ ∈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

t G(∆;¯ G¯)(t)=∆(0)¯ G¯(0) + ∆¯ dG¯ 0 for all t ∈T.

7 A trading strategy ∆in¯ L(G¯)isself-ﬁnancing with respect to (S,¯ D¯)if ∆¯ S¯ = G(∆;¯ G¯) or t ∆(¯ t)S¯(t)=∆(0)¯ S¯(0) + ∆¯ dG¯ 0

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¯)isthe process D(∆;¯ S,¯ D¯) deﬁned by ∆¯ S¯ + D(∆;¯ S,¯ D¯)=G(∆;¯ G¯) The process D(∆;¯ S,¯ D¯) is adapted and measurable and has initial value D(∆;¯ S,¯ D¯)(0) = 0.

3 Changing the Unit of Account

Let π be a one-dimensional Itô process: t t π(t)=π(0) + µπ ds + σπ dW 0 0 1 where µπ is a one-dimensional process in L and σπ is a K dimensional row vector process in L2. If D is a cumulative dividend process such that D ∈L(π), define a process Dπ by t Dπ(t)=π(t)D(t) − Ddπ 0 The process Dπ is adapted and measurable, and, hence, it is a cumulative dividend process. The purpose of assuming that D ∈L(π)istomakesure that the integral in this expression is well defined. The following proposition shows how the cumulative dividend process of a trading strategy is re-measured in a new unit of account in the case where there are no dividends on the basic securities. The original securities market model is (S,¯ 0), and the new one is (πS,¯ 0). The original dividend process D of the trading strategy is replaced by Dπ.

8 Proposition 1 Suppose D¯ =0.Let∆¯ ∈L(S¯),andsetD = D(∆;¯ S,¯ 0). Then ∆¯ ∈L(πS¯) if and only if D ∈L(π),inwhichcaseD(∆;¯ πS,¯ 0) = Dπ.

Proposition 1 will follow from Theorem 1 below. In the expression for Dπ, the term π(t)D(t) is the dividends cumulated in the old units and expressed in the new unit. This ignores the fact that at each point in time, already accumulated dividends change value. The integral term corrects for that. The term D(s) dπ(s) represents the change during instant s in the value of the dividends D(s) that have been accumulated up to that time. The integral represents the cumulative value of these changes. If, for example, the process π tends to increase over time, it means that the value of the old unit of account increases. The term π(t)D(t) overstates the cumulative dividends in the new unit of account, because most of the dividends have been accumulated at times when they were worth less than π(t) per old unit. Proposition 1 tells us how to change the unit of account of a cumulative dividend process that arises as the cumulative dividend process of a trading strategy. But as we shall now see, every cumulative dividend process is in fact the cumulative dividend process of a trading strategy in some very simple securities market model. Let D be a cumulative dividend process. Consider a securities market model where there is a money market account, and assume for simplicity that the money market account has zero interest rate and value process M =1. Consider the trading strategy which consists in holding, at every point in time t, −D(t) units of the money market account. The value process is −D. Since the trading strategy only invests in the money market account, which has zero interest rate, its cumulative gains process is zero. Hence, the cumulative dividend process is D. This observation, taken together with Proposition 1, suggests that in general, for any cumulative dividend process D, the process Dπ can be interpreted as D re-measured in the new unit of account. Therefore, we will use the same procedure to change the unit of account on the basic securities in the case where the basic securities pay dividends. If (S,¯ D¯) is a securities market model such that D¯ ∈L(π), then we deﬁne

9 the transformed cumulative dividend process D¯ π in the new unit of account entry by entry by ⎛ ⎞ D¯ π ⎜ N ⎟ π ⎜ . ⎟ D¯ = ⎝ . ⎠ π D¯ 0 or t D¯ π(t)=π(t)D¯(t) − Ddπ¯ 0 The transformed securities price process is πS¯, and the transformed securities market model is (πS,¯ D¯ π). The corresponding cumulative gains process G¯π is given by ⎛ ⎞ G¯π ⎜ N ⎟ π π ⎜ . ⎟ G¯ = πS¯ + D¯ = ⎝ . ⎠ π G¯0 or t G¯π(t)=π(t)G¯(t) − Ddπ¯ 0

The next proposition exhibits the Itˆo diﬀerential of G¯π.

Proposition 2 Let π be an Itˆo process, and assume that D¯ ∈L(π).Then G¯π is an Itˆoprocesswith π dG¯ = πµ¯ + Sµ¯ π +¯σσπ dt + πσ¯ + Sσ¯ π dW

Proof: Since t G¯π(t)=π(t)G¯(t) − Ddπ¯ 0 G¯π is obviously an Itˆo process, and

π dG¯ = πdG¯ + Gdπ¯ +¯σσπ dt − Ddπ¯ ¯ ¯ = πd G + Sdπ+¯σσπ dt = πµ¯ +(G¯ − D¯)µπ +¯σσπ dt + πσ¯ +(G¯ − D¯)σπ dW = πµ¯ + Sµ¯ π +¯σσπ dt + πσ¯ + Sσ¯ π dW 2

10 We now need to show that even when there are dividends on the basic securities, the cumulative dividend process of a trading strategy will undergo the transformation D → Dπ In other words, if D is the cumulative dividend process of a trading strategy measured in the old prices, then Dπ is the cumulative dividend process of the same trading strategy measured in the new prices. This general unit- invariance result is the content of Theorem 1 below.

Theorem 1 Let π be an Itˆo process, assume that D¯ ∈L(π),andlet∆¯ ∈ L(G¯).Then∆¯ ∈L(G¯π) if and only if ∆¯ S¯ ∈L(π),andifandonlyif D(∆;¯ S,¯ D¯) ∈L(π).Ifso,then

D(∆;¯ πS,¯ D¯ π)=Dπ and G(∆;¯ G¯π)=Gπ In particular, if ∆¯ is self-ﬁnancing with respect to (S,¯ D¯),then∆¯ ∈L(G¯π) and ∆¯ is self-ﬁnancing with respect to (πS,¯ D¯ π).

Proof: Set S = ∆¯ S¯, D = D(∆;¯ S,¯ D¯), and G = G(∆;¯ G¯). Since G is a continuous process, it is in L(π); and since G = S + D, one of the processes S and D is in L(π) if and only if the other one is. Observe that ∆¯ πµ¯ + Sµ¯ π +¯σσπ = π∆¯¯ µ + Sµπ + ∆¯¯ σσπ and ∆[¯ πσ¯ + Sσ¯ π]=π∆¯¯ σ + Sσπ 1 1 1 Now π∆¯¯ µ ∈L because π is continuous and ∆¯¯ µ ∈L; ∆¯¯ σσπ ∈L because 2 2 2 ∆¯¯ σ and σπ are in L ;andπ∆¯¯ σ ∈L because π is continuous and ∆¯¯ σ ∈L. Hence, 1 ∆¯ πµ¯ + Sµ¯ π +¯σσπ ∈L 1 if and only if Sµπ ∈L,and

2 ∆[¯ πσ¯ + Sσ¯ π] ∈L

11 2 π if and only if Sσπ ∈L. It follows that ∆¯ ∈L(G¯ ) if and only if S ∈L(π). Assume that this is so. Then

π dG(∆;¯ G¯ )=π∆¯ dG¯ + Sdπ+ ∆¯¯ σσπ dt and d (πG)=d πG(∆;¯ G¯) = πdG(∆;¯ G¯)+G(∆;¯ G¯) dπ + ∆¯¯ σσπ dt = π∆¯ dG¯ + Gdπ+ ∆¯¯ σσπ dt Hence, d (πG) − dG(∆;¯ G¯π)=[G − S] dπ = Ddπ This implies that t G(∆;¯ G¯π)(t)=π(t)G(t) − Ddπ= Gπ(t) 0 and D(∆;¯ πS,¯ D¯ π)(t)=G(∆;¯ G¯π)(t) − π(t)∆(¯ t)S¯(t) = Gπ(t) − π(t)S(t) = Dπ(t) 2

It is remarkable that π and D are all we need in order to calculate Dπ.Once we know π and D = D(∆;¯ S,¯ D¯), we do not need ∆or¯ S¯ or G¯ in order to calculate Dπ = D(∆;¯ πS,¯ D¯ π). Theorem 1 extends Proposition 4.2 of Huang [21, 1985] to general cumulative dividend processes (in the case where π is an Itô process). Huang’s proposition assumes that the cumulative dividend processes on both the basic securities and the trading strategy have finite variation. It involves changing the unit of account by integrating with respect to the cumulative dividend process. It will be shown in Proposition 4 below that our procedure for changing the unit coincides with Huang’s in the case of finite variation. Theorem 1 modifies Proposition 9 of Duffie [9, 1991, Section 5] (in the case where, as here, π is an Itô process). Duffie’s proposition changes the unit by

12 integrating with respect to the cumulative dividend processes. This is fine in the case of the cumulative dividend process of the trading strategy, which is assumed to have finite variation, but it is not suitable in the case of the cumulative dividend processes of the basic securities, which are assumed to be semi-martingales. Proposition 5 is a counterexample to Duffie’s proposition.

4 Examples and Counterexamples

In this section, we shall first calculate D1/M for a general cumulative dividend process D when M is the value process of a money market account. Next, we shall calculate the process Dπ for two types of cumulative dividend processes: Itô processes and processes of finite variation. The cumulative dividend process of a continuously resettled futures contract would be an example of the former, and random discrete dividends at random times are a special case of the latter. Finally, we shall show that if the cumulative dividend processes of the basic securities are Itô processes, and if they are re-measured in a new unit by simple integration, then the unit-invariance result of Theorem 1 generally does 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 and instantaneously riskless (has zero dispersion).Wedenoteitsvalueprocess by M: M = ¯bS¯. If M is the value process of a money market account, then it must have the form t M(t)=M(0)η[r, 0](t)=M(0) exp rds 0 for some r ∈L1 (the interest rate process) and some M(0) > 0. We use the general notation η[α, β] for the stochastic exponential, which is the process defined by t 1 t η[α, β](t)=exp α(s) − β(s)β(s) ds + β(s)dW (s) 0 2 0

13 Here, α and β are processes in L1 and L2, of dimension one and K, respec- tively. Let us now change the unit of account by π =1/M . Let D be a cumulative dividend process. Then Dr ∈L1 if and only if D ∈L(1/M )=L(M). If so, then t r D1/M (t)=D(t)/M (t)+ D ds 0 M Schweizer [31, 1992, Equation 2.2] is similar to this equation except that we assume D(0) = 0, while he assumes D to be right-continuous with left limits and 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 only be done if this limit exists. The term D(t)/M (t) in the formula is the nominal cumulative dividends expressed as a number of units of the money market account, ignoring the interest earned at each point in time on already accumulated dividends in the form of appreciation of the money market account. The integral represents the cumulative value of this interest, expressed in units of the money market account. Next, we consider cumulative dividend processes that are semi-martingales, including Itô processes and processes with finite variation.

Proposition 3 If D is a semi-martingale, then t Dπ(t)= πdD+[D, π](t) 0

Proof: By the formula for integration by parts for semi-martingales (Protter [28, 1990, Chapter 2, Corollary 6.2]), t t D(t)π(t)= Ddπ+ πdD+[D, π](t) 0 0 Hence, t t Dπ(t)=D(t)π(t) − Ddπ= πdD+[D, π](t) 0 0 2

14 Corollary 1 If D is a cumulative dividend process which is and Itˆoprocess with dD = fds+ gdW then D ∈L(π) and

t t π D (t)= πdD+ gσπ ds 0 0

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

t [D, π](t)= gσπ ds 0 2

Notice in particular from Corollary 1 that when D is an Itˆo process, Dπ in general is not equal to the integral of π with respect to dD:

t Dπ(t) = πdD 0

π However, if π and D have zero instantaneous covariance (gσπ = 0), then D is indeed equal to the integral of π with respect to dD:

t Dπ(t)= πdD 0

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

t D(t)= fds 0 In this case, t t Dπ(t)= πdD= πf ds 0 0 In other words, Dπ has continuous dividend ﬂow πf.Thisisexactlywhat we would expect.

15 Proposition 4 If D is right-continuous and has ﬁnite variation, then t Dπ(t)= πdD 0

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

t t 1/2 t 1/2 |d[D, Wk]|≤ d[D, D] d[Wk,Wk] 0 0 √ 0 =0× t =0

Hence, [D, Wk]=0foreachk,andso[D, W] = 0. Letting I be the stochastic integral process t I(t)= σπ dW 0 it follows from Protter [28, 1990, Chapter 2, Theorem 6.29] that t [D, I](t)= σπd[D, W]=0 0 Hence [D, π]=0. 2

Consider an alternative way of changing the unit of account, which has been used by Duffie and Zame [13, 1989] for cumulative dividend processes that are Itô 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 the process D(π) by simple integration: t D(π)(t)= πdD 0 Similarly, suppose (S,¯ D¯) is a securities market model such that D¯ is a semi- martingale, define the process D¯ (π) by t D¯ (π)(t)= πdD¯ 0

16 Proposition 5 Suppose (S,¯ D¯) is a securities market model such that D¯ is an Itˆoprocesswith dD¯ = fdt¯ +¯gdW Suppose ∆¯ ∈L(G¯) is a trading strategy such that D(∆;¯ S,¯ D¯) has ﬁnite variation. Then the following are equivalent:

1. For every t, D(∆;¯ S,¯ D¯ (π))(t)=D(∆;¯ S,¯ D¯)(π)(t) with probability one

2. ∆¯¯ gσπ =0almost everywhere (the trading strategy has zero instantaneous covariance with π)

Proof: It follows from Corollary 1 that t t (π) π D¯ (t)= πdD¯ = D¯ (t) − gσ¯ π ds 0 0 The transformed securities market model is (πS,¯ D¯ (π)), and the corresponding cumulative gains process G¯(π) is given by t (π) (π) π G¯ (t)=π(t)S¯(t)+D¯ (t)=D¯ (t) − gσ¯ π ds 0 In the transformed securities market model, the cumulative gains process of ∆is¯ t G(∆;¯ G¯(π))(t)=∆(0)¯ G¯(0) + ∆¯ dG¯(π) 0 t t π = ∆(0)¯ G¯(0) + ∆¯ dG¯ − ∆¯¯ gσπ ds 0 0 t π = G(∆;¯ G¯ )(t) − ∆¯¯ gσπ ds 0 and the cumulative dividend process is

D(∆;¯ S,¯ D¯ (π))(t)=G(∆;¯ G¯(π))(t) − ∆(¯ t)S¯(t) t π = G(∆;¯ G¯ )(t) − ∆¯¯ gσπ ds − ∆(¯ t)S¯(t) 0 t π = D(∆;¯ πS,¯ D¯ )(t) − ∆¯¯ gσπ ds 0

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

D(∆;¯ S,¯ D¯)(π) = D(∆;¯ S,¯ D¯)π whereas, as shown above,

t (π) π D(∆;¯ S,¯ D¯ )(t)=D(∆;¯ S,¯ D¯) (t) − ∆¯¯ gσπ ds 0 Hence, for all t, D(∆;¯ S,¯ D¯ (π))(t)=D(∆;¯ S,¯ D¯)(π)(t) with probability one if and only if for all t,

t ∆¯¯ gσπ ds =0 0

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

Proposition 5 contradicts Proposition 9 of Duffie [9, 1991, Section 5] and implies that it is not appropriate in general to change the unit of a cumulative dividend process by simple integration, even if the integral is well defined because the cumulative dividend process is a semi-martingale. In particular, this procedure is not appropriate if the cumulative dividend process is an Itô process, as recognized in Duffie [10, 1992], [11, 1996], and [12, 2001], and Aase [1, 2002] Duffie and Zame [13, 1989, page 1287] assume that the cumulative dividend processes of the basic securities are Itô process. They re-measure them in units of a consumption good by simple integration, thus defining the real cumulative dividend processes. Duffie [9, 1991, Section 4, page 1640] also re-measures a cumulative dividend process (which is assumed to be a semi- martingale) in units of a consumption good by simple integration. One should be careful with the interpretation of this procedure, because it does not satisfy unit invariance. Proposition 6 of Duffie [9, 1991, Section 4] similarly involves changing the unit of account by simple integration. Its Equation 16 contradicts Equation 2.3 of Aase [1, 2002]. The source of this discrepancy seems to be the step in the proof where Fubini’s theorem for conditional expectations is applied.

18 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 and Jeanblanc-Picqu´e [6, 1994] and Dana, Jeanblanc-Picqu´e, and Koch [7, 2003]. The last equation of this proof is essentially an application of Fubini’s theorem for conditional expectations to a stochastic integral.

5 Consistency

Suppose we change the unit of account of a cumulative dividend process D by π,calculatingDπ, and we then further change the unit of account by a process ν,calculating(Dπ)ν. The result should be the same as if we changed the unit of account by νπ in one step, calculating Dνπ. This consistency property of changes of the unit of account is veriﬁed in the following theorem.

Theorem 2 Let π and ν be one-dimensional Itˆo processes, and let D be a cumulative dividend process such that D ∈L(π) and Dπ ∈L(ν).Then D ∈L(νπ) and ν (Dπ) = Dνπ

Proof: First, observe that πD ∈L(ν). This follows from the fact that Dπ ∈L(ν)andDπ − πD belongs to L(ν) because the latter is a continuous process. Write dν = µν dt + σν dW 1 2 where µν ∈L and σν ∈L.Then d(νπ)=νdπ + πdν+ σν σπ dt = νµπ + πµν + σν σπ dt +[νσπ + πσν] dW

1 2 Since D ∈L(π)andπD ∈L(ν) by assumption, Dµπ ∈L, Dσπ ∈L, 1 2 Dπµν ∈L,andDπσν ∈L.Sinceν is continuous, it follows that Dνµπ ∈ 1 2 L , Dνσπ ∈L, and consequently, D ∈L(νπ).

19 Now,

t ν (Dπ) (t)=ν(t)Dπ(t) − Dπ dν 0 t t t s = ν(t)π(t)D(t) − ν(t) Ddπ− πD dν + Ddπ dν 0 0 0 0 and t Dνπ(t)=ν(t)π(t)D(t) − Dd(νπ) 0 t t t = ν(t)π(t)D(t) − Dν dπ − Dπ dν − Dσν σπ ds 0 0 0 The diﬀerence between these two processes is

t t ν (Dπ) (t) − Dνπ(t)=ν(t)π(t)D(t) − ν(t) Ddπ− πD dν 0 0 t s + Ddπ dν − ν(t)π(t)D(t) 0 0 t t t + Dν dπ + Dπ dν + Dσνσπ ds 0 0 0 t t s = −ν(t) Ddπ+ Ddπ dν 0 0 0 t t + Dν dπ + Dσνσπ ds 0 0 It is an Itˆo process with diﬀerential

t π ν νπ d ((D ) − D )=−νD dπ − Ddπ dν − Dσν σπ dt 0 s + Ddπ dν + Dν dπ + Dσν σπ ds 0 =0

ν Since the processes (Dπ) and Dνπ have the same initial value, it follows that

ν (Dπ) = Dνπ

20 6 State Prices and Risk Adjustment

This section generalizes some standard concepts and results to general dividends: state price process, prices of risk, the martingale property, and risk- adjusted probabilities. Consider a securities market model of the form (S,¯ D¯). Let Π be a positive one-dimensional Itô process. It must have the form Π=Π(0)η[−r, −λ] for some Π(0) > 0 and processes r ∈L1 and λ ∈L2. Given a positive constant M(0), set M = M(0)η[r, 0] Then η[0, −λ] Π=Π(0)M(0) M Say that Π is a state price process for (S,¯ D¯)ifDr¯ ∈L1 and the process η[0, −λ]G¯1/M has zero drift. This definition is consistent with the standard definition for the case where D¯ = 0: Π is a state price process for (S,¯ 0) if and only if 1 1 η[0, −λ]G¯M = η[0, −λ] S¯ = ΠS¯ M M(0)Π(0) has zero drift. The following proposition exhibits the standard equation for the prices of risk λ and identifies r as the interest rate.

Proposition 6 Let Π(0) > 0, M(0) > 0, r ∈L1,andλ ∈L2, and assume that Dr¯ ∈L1.ThenΠ=Π(0)η[−r, −λ] is a state price process for (S,¯ D¯) if and only if µ¯ − rS¯ =¯σλ almost everywhere. If so, and if ¯b is a money market account with initial value ¯b(0)S¯(0) = M(0),then¯b has value process ¯bS¯ = M(0)η[r, 0] and interest rate r.

21 Proof: The drift of η[0, −λ]G¯1/M is

η[0, −λ] µ¯ − rS¯ − σλ¯ M Hence, Π = Π(0)η[−r, −λ] is a state price process for (S,¯ D¯) if and only if

µ¯ − rS¯ =¯σλ almost everywhere. If so, and if ¯b is a money market account with initial value ¯b(0)S¯(0) = M(0), then

¯bµ¯ − r¯bS¯ = ¯bσλ¯ =0 and d(¯bS¯)=dG(¯b; G¯)=¯bdG¯ = ¯bµdt¯ = r¯bSdt¯ Given the initial condition ¯b(0)S¯(0) = M(0), this diﬀerential equation has the unique solution ¯bS¯ = M(0)η[r, 0]. 2

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

Proposition 7 If Π is a state price process for (S,¯ D¯),andif∆¯ ∈L(G¯1/M ), then η[0, −λ]G ∆¯ , G¯1/M has zero drift.

Proof: By Theorem 1, D ∈L(1/M ). Observe that d η[0, −λ]G ∆¯ , G¯1/M = η[0, −λ] dG ∆¯ , G¯1/M + G ∆¯ , G¯1/M dη[0, −λ] − η[0, −λ]∆¯¯ σdt η , −λ dt G ¯ , G¯1/M dη , −λ Since [0 ] has zero drift, the coeﬃcients to in ∆ [0 ] andin(∆¯ G¯1/M ) dη[0, −λ] are both zero, and the drift of η[0, −λ]G ∆¯ , G¯1/M

22 equals the coeﬃcient to dt in η[0, −λ] dG ∆¯ , G¯1/M +(∆¯ G¯1/M ) dη[0, −λ] − η[0, −λ]∆¯¯ σdt = ∆¯ η[0, −λ] dG¯1/M + G¯1/M dη[0, −λ] − η[0, −λ]¯σdt = ∆¯ d η[0, −λ]G¯1/M

1/M Since η[0,−λ]G¯ has zero drift, the coeﬃcient to dt is zero, and, hence η[0, −λ]G ∆¯ , G¯1/M has zero drift. 2

Assume that D¯ ∈L(M). Let ∆¯ be a trading strategy in L(G¯1/M ). It was just shown above that

η[0, −λ]G(∆;¯ G¯1/M ) has zero drift. Say that ∆is¯ admissible for (S,¯ D¯) and Π if this process is a martingale. If ∆¯ is a self-ﬁnancing trading strategy, then 1 η[0, −λ]G ∆;¯ G¯1/M = η[0, −λ]∆¯ S/M¯ = Π∆¯ S¯ Π(0)M(0)

Hence, ∆¯ is admissible if and only if Π∆¯ S¯ is a martingale. This is consistent with the definition of admissibility in Harrison and Pliska [18, 1981], even though 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 securities. The same will be seen to be true of trading strategies that are not necessarily self-financing, and even if there are dividends on the basic securities. We formally generalize the concept of arbitrage to basic securities with general dividends and to trading strategies that are not necessarily self-financing, as follows. An arbitrage trading strategy is a trading strategy ∆¯ ∈L G¯1/M

23 such that ∆(0)¯ S¯(0)/M (0) = 0 and for some t, G ∆;¯ G¯1/M (t) ≥ 0 with probability one, and G ∆;¯ G¯1/M (t) > 0 with positive probability. Recall that a trading strategy ∆¯ is self-financing relative to (S,¯ D¯)ifand only if it self-financing relative to (S/M,¯ D¯ 1/M ), in which case its normalized cumulative gains process is G ∆;¯ G¯1/M = ∆¯ S/M¯ Therefore, a self-financing arbitrage trading strategy is a self-financing trading strategy ∆¯ ∈L(G¯) such that ∆(0)¯ S¯(0) = 0 and for some t, ∆(¯ t)S¯(t) ≥ 0 with probability one, and ∆(¯ t)S¯(t) > 0 with positive probability. Thus, the definition is consistent with the usual definition in the case where there are no dividends. If ∆¯ is an arbitrage trading strategy, then E η[0, −λ](t)G ∆;¯ G¯1/M (t) > ∆(0)¯ S¯(0)/M (0) = η[0, −λ](0)G ∆;¯ G¯1/M (0)

Hence, η[0, −λ]G(∆;¯ G¯1/M ) cannot be a martingale, and ∆¯ cannot be admissible. So, there are no arbitrage trading strategies among the admissible trading strategies. Deﬁne the K-dimensional process W λ by t W λ(t)= λ ds + W (t) 0 We can express the diﬀerential of G¯1/M in terms of dW λ: 1 1 dG¯1/M = (¯µ − rS¯) dt + σdW¯ M M 1 1 = (¯µ − rS¯ − σλ¯ ) dt + σdW¯ λ M M 1 = σdW¯ λ M 24 If ∆¯ ∈L(G¯1/M ) is a trading strategy, then 1 dG(∆;¯ G¯1/M )=∆¯ dG¯1/M = ∆¯¯ σdWλ M

Deﬁne the risk-adjusted probability measure Q (on the horizon T ) corresponding to λ as the measure on (Ω, F) which has density η[0, −λ](T ) with respect to the original probability measure P . According to Girsanov’s Theorem, if η[0, −λ] is a martingale on [0,T], then W λ is a Wiener processes with respect to F and Q on [0,T]. In terms of the risk-adjusted probabilities, a trading strategy ∆in¯ L(G¯1/M )is admissible if and only if the discounted cumulative gains process G(∆;¯ G¯1/M ) is a martingale under Q.

7 Replication and Valuation

A trading strategy ∆issaidto¯ replicate a contingent claim Y (a random variable) at time T with respect to (S,¯ D¯) if it is self-ﬁnancing with respect to (S,¯ D¯)and∆(¯ T )S¯(T )=Y . If ∆¯ is a self-ﬁnancing trading strategy which replicates a contingent claim Y at time T with respect to (S,¯ D¯), and which is admissible for (S,¯ D¯)and Πon[0,T], then for t ≤ T , because of the martingale property, 1 ∆(¯ t)S¯(t)= E [Π(T )Y |Ft] Π(t)

If Π is any positive Itˆo process and Y is any claim such that Π(T )Y is integrable, deﬁne the value process or, for emphasis, its martingale value process of Y with respect to Π as the process V (Y ;Π;T )givenby 1 V (Y ;Π;T )(t)= E [Π(T )Y |Ft] Π(t) for 0 ≤ t ≤ T .

25 If Π is a state price process for (S,¯ D¯)andifY happens to be replicated at time T with respect to (S,¯ D¯) by a self-ﬁnancing strategy ∆,¯ then ∆is¯ admissible for (S,¯ D¯) and Π if and only if ∆¯ S¯ = V (Y ;Π;T )

A trading strategy ∆¯ will be said to replicate a cumulative dividend process D 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 that D¯ ∈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 Π,thenD ∈L(M), and for every t ∈ [0,T], G ∆;¯ G¯1/M (t)=V (D1/M (T ); η[0, −λ]; T )(t)

∆(¯ t)S¯(t)/M (t)=V (D1/M (T ); η[0, −λ]; T )(t) − D1/M (t) and ∆(¯ t)S¯(t)=V (D1/M (T )M(T ); Π; T )(t) − D1/M (t)M(t) with probability one.

Proof: Since ∆¯ is admissible, ∆¯ ∈L(G¯1/M ). By Theorem 1, Dr ∈L1,and ∆¯ replicates D1/M with respect to S/M¯ and D¯ 1/M . Furthermore, η[0, −λ]G ∆;¯ G¯1/M = η[0, −λ]∆¯ S/M¯ + η[0, −λ]D1/M is a martingale. Hence, η[0, −λ](T )G ∆;¯ G¯1/M (T )=η[0, −λ](T )D1/M (T ) is integrable, and for 0 ≤ t ≤ T , 1/M 1/M η[0, −λ](t)G ∆;¯ G¯ (t)=E η[0, −λ](T )D (T ) |Ft This implies that 1/M 1 1/M G ∆;¯ G¯ (t)= E η[0, −λ](T )D (T ) |Ft η[0, −λ](t) = V (D1/M (T ); η[0, −λ]; T )(t)

26 Hence, ∆(¯ t)S¯(t)/M (t)=G ∆;¯ G¯1/M (t) − D1/M (t) = V (D1/M (T ); η[0, −λ]; T )(t) − D1/M (t) and ∆(¯ t)S¯(t)=V (D1/M (T ); η[0, −λ]; T )(t)M(t) − D1/M (t)M(t) = V (D1/M (T )M(T ); Π; 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 η[0, −λ](T )D1/M (T ) is integrable. Define the value process or, for emphasis, the martingale value process V [D;Π;T ]ofD with respect to Π, on [0,T], by V [D;Π;T ]=V (D1/M (T )M(T ); Π; T ) − D1/M M = V (D1/M (T ); η[0, −λ]; T )M − D1/M M It may be interpreted as the present value of the dividends yet to be paid up to 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 value of a claim to the cumulative dividends from today on plus the value of a flow of interest on the cumulative dividends at each point in time.

Theorem 3 Let D be a cumulative dividend process in L(M) and let Dˆ be the cumulative dividend process with continuous dividend ﬂow Dr: t Dˆ(t)= Dr ds 0 Assume that η[0, −λ] is a martingale. If two of the variables Π(T )D(T ), η[0, −λ](T )D1/M (T ),andη[0, −λ](T )Dˆ 1/M (T ) are integrable, then so is the third, and V [D;Π;T ](t)=V (D(T ); Π; T )(t) − D(t)+V [Dˆ;Π;T ](t)

27 Proof: The statement about integrability follows directly from the equation

Π(0)M(0)η[0, −λ](T )D1/M (T ) T r =Π(T )D(T )+Π(0)M(0)η[0, −λ](T ) D ds 0 M =Π(T )D(T )+Π(0)M(0)η[0, −λ](T )Dˆ 1/M (T )

If the relevant variables are indeed integrable, then

V [D;Π;T](t)/M (t) 1/M 1/M = EQ D (T ) |Ft − D (t) T r = EQ[D(T )/M (T ) |Ft] − D(t)/M (t)+EQ D ds Ft t M = V (D(T ); Π; T )(t)/M (t) − D(t)/M (t)+V [Dˆ;Π;T ](t)/M (t) which implies that

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

References

[1] K. K. Aase. Equilibrium pricing in the presence of cumulative dividends following a diﬀusion. Mathematical Finance, 12(3):173–198, 2002.

[2] K. Back. Asset pricing for general processes. Journal of Mathematical Economics, 20:371–395, 1991.

[3] A. Bensoussan. On the theory of option pricing. Acta Applicandae Mathematicae, 2:139–158, 1984.

[4]J.C.CoxandC.Huang.Avariational problem arising in ﬁnancial economics. Journal of Mathematical Economics, 20:465–487, 1991.

[5] D. Cuoco. Optimal consumption and equilibrium prices with portfolio constraints and stochastic income. Journal of Economic Theory, 72:33– 73, 1997.

28 [6] R.-A. Dana and M. Jeanblanc-Picqu´e. March´es Financiers en Temps Continu: Valorisation et Equilibre´ . Recherche en Gestion. Economica, Paris, 1994.

[7] R.-A. Dana, M. Jeanblanc-Picqu´e, and H. F. Koch. Financial Markets in Continuous Time. Springer Finance. Springer-Verlag, 2003.

[8] D. Duﬃe. Stochastic equilibria: Existence, spanning number, and the ‘no expected ﬁnancial gain from trade’ hypothesis. Econometrica, 54:1161–1183, 1986.

[9] D. Duﬃe. The theory of value in security markets. In W. Hildenbrand and H. Sonnenschein, editors, Handbook of Mathematical Economics, volume 4, chapter 31, pages 1615–1682. North-Holland, 1991.

[10] D. Duﬃe. Dynamic Asset Pricing Theory. Princeton University Press, ﬁrst edition, 1992.

[11] D. Duﬃe. Dynamic Asset Pricing Theory. Princeton University Press, second edition, 1996.

[12] D. Duﬃe. Dynamic Asset Pricing Theory. Princeton University Press, third edition, 2001.

[13] D. Duﬃe and W. Zame. The consumption-based capital asset pricing model. Econometrica, 57:1279–1297, 1989.

[14] P. H. Dybvig and C. Huang. Nonnegative wealth, absence of arbitrage, and feasible consumption plans. Review of Financial Studies, 1:377–401, 1988.

[15] H. Geman, N. El Karoui, and J. Rochet. Changes of num´eraire, changes of probability measure and option pricing. Journal of Applied Probabil- ity, 32:443–458, 1995.

[16] R. Geske. A note on an analytical formula for unprotected American call options on stocks with known dividends. Journal of Financial Eco- nomics, 7:375–380, 1979.

[17] J. M. Harrison and D. Kreps. Martingales and arbitrage in multiperiod securities markets. Journal of Economic Theory, 20:381–408, 1979.

29 [18] J. M. Harrison and S. Pliska. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and Their Appli- cations, 11:215–260, 1981.

[19] A. Hindy. Viable prices in ﬁnancial markets with solvency constraints. Journal of Mathematical Economics, 24:105–135, 1995.

[20] A. Hindy and C. Huang. Optimal consumption and portfolio rules with durability and local substitution. Econometrica, 61:85–121, 1993.

[21] C. Huang. Information structures and viable price systems. Journal of Mathematical Economics, 14:215–240, 1985.

[22] C. Huang and H. Pages. Optimal consumption and portfolio policies with an inﬁnite horizon: Existence and convergence. The Annals of Applied Probability, 2:36–64, 1992.

[23] X. Jin and S. Deng. Existence and uniqueness of optimal consumption and portfolio rules in a continuous-time ﬁnance model with habit for- mation and without short sales. Journal of Mathematical Economics, 28:187–205, 1997.

[24] I. Karatzas. Lectures on the Mathematics of Finance, volume 8 of CRM Monograph Series. American Mathematical Society, 1997.

[25] I. Karatzas and S. E. Shreve. Methods of Mathematical Finance,volume 39 of Applications of Mathematics: Stochastic Modeling and Applied Probability. Springer, New York, 1970.

[26] W. Margrabe. The value of an option to exchange one asset for another. Journal of Finance, 33:177–186, 1978.

[27] L. T. Nielsen. Pricing and Hedging of Derivative Securities.Oxford University Press, 1999.

[28] P. Protter. Stochastic Integration and Diﬀerential Equations: A New Approach. Applications of Mathematics. Springer-Verlag, 1990.

[29] R. Roll. An analytical formula for unprotected American call options on stocks with known dividends. Journal of Financial Economics, 9:251– 258, 1977.

30 [30] M. Schroder. Changes in numeraire for pricing futures, forwards, and options. Review of Financial Studies, 12:1143–1163, 1999.

[31] M. Schweizer. Martingale densities for general asset prices. Journal of Mathematical Economics, 21:363–378, 1992.

[32] R. E. Whaley. A note on an analytical formula for unprotected Ameri- can call options on stocks with known dividends. Journal of Financial Economics, 7:375–380, 1979.

[33] P. Wilmott, J. N. Dewynne, and S. Howison. Option Pricing: Mathe- matical Models and Computation. Oxford Financial Press, 1993.