The fundamental theorem of asset pricing for self-financing portfolios
aa r X i v : . [ q -f i n . M F ] M a y THE FUNDAMENTAL THEOREM OF ASSET PRICING FORSELF-FINANCING PORTFOLIOS
ECKHARD PLATEN AND STEFAN TAPPE
Abstract.
Consider a financial market with nonnegative semimartingaleswhich does not need to have a numéraire. We are interested in the absenceof arbitrage in the sense that no self-financing portfolio gives rise to arbitrageopportunities, where we are allowed to add a savings account to the market.We will prove that in this sense the market is free of arbitrage if and only ifthere exists an equivalent local martingale deflator which is a multiplicativespecial semimartingale. In this case, the additional savings account relates tothe finite variation part of the multiplicative decomposition of the deflator. Byfocusing on self-financing portfolios, this result clarifies links between previousresults in the literature and makes the respective concepts more realistic. Introduction
There exists now a rich literature on no-arbitrage concepts and their relation-ships. Unfortunately, this literature involves often lengthy proofs. Some no-arbitrageconcepts remain also difficult to interpret from a practical perspective because thetradeability of the securities involved is not a focus. Within this paper we aim todemonstrate a way of deriving and relating to each other many of the existingresults and concepts by focusing on self-financing portfolios, which allows us toavoid lengthy proofs and brings more clarity into the links between the existingconcepts. In this way, we aim to contribute conceptually and mathematically to therich rather theoretical no-arbitrage literature by giving it a slightly more practicalorientation and providing short proofs.Before giving in this introduction a brief description of the main results of thepaper let us first list some of the important papers in the no-arbitrage literaturewhich relate to our results. These include the papers [18, 19, 8, 40, 36, 37, 23,28] and the textbook [14], which treat the fundamental theorem of asset pricing(FTAP) in discrete time. The papers [9, 11] and the textbook [12] establish theFTAP in continuous time and its connection between NFLVR and the existence ofa martingale measure, and the papers [10, 26, 39, 35, 30, 15, 16, 32, 7, 22] and thetextbook [25] treat further developments and related topics concerning the FTAP.The papers [5, 31, 41] present versions of the FTAP, which connect the notionsNA , NAA and NUPBR with the existence of a martingale deflator. Finally, thearticles [29, 6, 21, 17, 27] study related topics.As will become clear throughout the paper, the conceptual way we approach thequestions of no-arbitrage, including the fundamental NFLVR condition, allows usto rely on proofs that remain rather short, the theory of no-arbitrage concepts intopological vector lattices, which we have developed in [34], and well-known resultsfrom stochastic analysis. Date : 12 May, 2020.2010
Mathematics Subject Classification.
Key words and phrases. fundamental theorem of asset pricing, self-financing portfolio, no-arbitrage concept, equivalent local martingale deflator.We are grateful to Martin Schweizer and Josef Teichmann for valuable discussions.
ECKHARD PLATEN AND STEFAN TAPPE
Consider a financial market S = { S , . . . , S d } with nonnegative semimartingales,and fix a finite time horizon T > . For the moment, assume that S d = 1 . Thenthe market S can be interpreted as discounted price processes of risky assets withrespect to some savings account S d . The fundamental theorem of asset pricing(FTAP) tells us that the market is free of arbitrage opportunities if and only if thereexists an equivalent local martingale measure (ELMM). The notion of no-arbitrageused here is No Free Lunch with Vanishing Risk (NFLVR). More precisely, we havethe following result.1.1.
Theorem.
The following statements are equivalent: (i) I adm0 ( S ) satisfies NFLVR. (ii) There exists an ELMM Q ≈ P for S . Here I adm0 ( S ) is the convex cone of all admissible stochastic integrals starting inzero evaluated at the terminal time T . Since S consists of discounted assets, we mayregard the convex cone I adm0 ( S ) as the set of outcomes of wealth processes startingin zero. A first version of the FTAP can be found in [19]. The stated Theorem 1.1follows from [11]; we also refer to the earlier paper [9]. All these results can also befound in the textbook [12], and a related reference is [26].Another approach is to work exclusively under the physical probability measure P . This is done, for example, under the Benchmark Approach in [33]. Then theappropriate concept replacing an ELMM is that of an equivalent local martingaledeflator (ELMD), and the appropriate no-arbitrage concept is No Unbounded Profitwith Bounded Risk (NUPBR), see [29]. In this context, the following result has beenestablished.1.2.
Theorem.
The following statements are equivalent: (i) I +1 ( S ) satisfies NUPBR. (ii) There exists an ELMD Z for S with Z ∈ M loc . Here I +1 ( S ) is the convex set consisting of all nonnegative stochastic integralswith initial value one evaluated at the terminal time T . The proof of Theorem 1.2follows from [41]. We also refer to [5] and [31] for earlier versions of this result.In this paper, we consider a financial market S = { S , . . . , S d } with nonnegativesemimartingales which does not need to have a numéraire S d = 1 . In this respect,we would like to mention the recent papers [3, 4, 2], where it has been pointedout that different discounting of the same original market can lead to different no-arbitrage properties, and the paper [20], where a numéraire-independent modelingframework has been presented.Our goal of this paper is to provide characterizations of no-arbitrage conceptsfor self-financing portfolios. After this question is answered one could ask in asecond step which other wealth or value processes make sense in the given setting.For instance, it would be then natural to permit value processes that extend theabsence of arbitrage from the set of self-financing portfolios to other wealth andvalue processes. This makes good sense because as soon as some value processbecomes liquidly traded one can, in practice, form self-financing portfolios withthis asset and is in a situation that the first step has covered already.When we allow to add a savings account to the market, then our main resultsessentially state that the market is free of arbitrage opportunities if and only ifthere exists an ELMD which is a multiplicative special semimartingale. More pre-cisely, denote by P +sf , ( S ) the convex set of all nonnegative, self-financing portfolioswith initial value one evaluated at the terminal time T . This is equivalent to look-ing at all nonnegative, self-financing portfolios with strictly positive initial values.Furthermore, we call every predictable, strictly positive process B of locally finitevariation a savings account . A condensed version of our result reads as follows: HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 3
Theorem.
The following statements are equivalent: (i)
There exists a savings account B such that P +sf , ( S ∪{ B } ) satisfies NUPBR. (ii) There exists an ELMD Z for S which is a multiplicative special semimartin-gale. In this case, such a savings account B has to fit into the multiplicative decom-position Z = DB − of the deflator. For details, we refer to Theorem 7.5 and theprevious, more detailed Theorem 7.4 later on in the paper.We also provide such a characterization for the following set of admissible self-financing portfolios. Denoting by P admsf , ( S ) the convex cone of all admissible, self-financing portfolios starting in zero and evaluated at the terminal time T , a con-densed version of this result is as follows:1.4. Theorem.
The following statements are equivalent: (i)
There exists a savings account B such that B and B − are bounded, and P admsf , ( S ∪ { B } ) satisfies NFLVR. (ii) There exists an ELMD Z for S which is a multiplicative special semimartin-gale such that the local martingale part is a true martingale, and the finitevariation part and its inverse are bounded. (iii) There exist a savings account B such that B and B − are bounded, and anELMM Q ≈ P for the discounted market S B − . As in the previous result, such a savings account B has to fit into the multi-plicative decomposition Z = DB − of the deflator. Furthermore, the martingale D appearing in this decomposition is just the density process of the measure change Q ≈ P , which provides a connection to the classical FTAP by Delbaen and Schacher-mayer (see [9] and [11]). For details, we refer to Theorem 7.7 and the previous, moredetailed Theorem 7.6 later on in the paper.Our continuous time results can, in particular, be applied to the discrete timesetting. Denote by P sf , ( S ) the convex cone of all self-financing portfolios startingin zero and evaluated at the terminal time T , which is now a positive integer. Theno-arbitrage concept which we consider in this case is simply No Arbitrage (NA).Then, without much effort we derive the following result.1.5.
Theorem.
The following statements are equivalent: (i)
There exists a savings account B such that P sf , ( S ∪ { B } ) satisfies NA. (ii) There exists an EMD Z for S which is a multiplicative special semimartin-gale such that the local martingale part is a true martingale. (iii) There exist a savings account B and an EMM Q ≈ P for the discountedmarket S B − . Here EMD means equivalent martingale deflator , and EMM means equivalentmartingale measure . Note that this result provides a connection to the well-knownFTAP in discrete time; see for example [14]. For details, we refer to Theorem 8.3and the more detailed Theorem 8.2.Let us briefly outline some further applications of our main results, which wepresent in this paper. Consider a market of Black-Scholes type S = { S } with, forsimplicity, one asset S = E ( a · λ + σ · W ) , where a and σ > are suitable processes, λ denotes the Lebesgue measure and W is a standard Wiener process. Then there exist several ELMDs for S which aremultiplicative special semimartingales. Indeed, consider Z = DB − , where D = E ( − θ · W ) and B = exp( r · λ ) ECKHARD PLATEN AND STEFAN TAPPE with suitable processes θ and r . Then Z is an ELMD if and only if θ = a − rσ , or equivalently r = a − σθ. Since r is still free to choose, there exists for each r a respective ELMD. If theoriginal market consists of several risky assets, then we have still a similar resultwhere its extension with a savings account allows many choices for r leading to arespective ELMD. We refer to Section 9 for more details.Our main result can also be used in order to construct arbitrage free marketsby means of contingent claims H , . . . , H d . Indeed, choose a multiplicative specialsemimartingale Z = DB − with a local martingale D ∈ M loc and a savings account B , and define the market S = { S , . . . , S d } by the real-world pricing formula S it := Z − t E [ H i Z T | F t ] , t ∈ [0 , T ] for i = 1 , . . . , d . Then the market is free of arbitrage, and the same technique canbe used in order to extend arbitrage free markets. We refer to Section 10 for furtherdetails.Moreover, our result can also be used in order to construct reasonable portfolioswhich may not be self-financing; this can be crucial for risk management purposesand gives an answer to the question raised above: Which other wealth or valueprocesses make sense in the given setting? Consider an arbitrage free market S = { S , . . . , S d } , and let Z = DB − be an ELMD for S which is a multiplicative specialsemimartingale with a savings account B . Then for every self-financing strategy δ for the market S , the strategy ν = ( δ, η ) , where η = Z is simply the deflator,provides a locally real-world mean self-financing dynamic strategy for the extendedmarket S ∪ { B } . We refer to Section 11 for more details.Note that for all applications outlined above a desirable feature of the ELMD Z is its tradeability; that is, its inverse ¯ Z = Z − can be realized as a self-financingportfolio. This property is desirable because otherwise we obtain strategies whichare difficult to implement in practice. In this case, we expect that such an equivalentlocal martingale numéraire portfolio ¯ Z will be the growth optimal portfolio, whichprovides a link to the Benchmark Approach in [33]; see also Section 12 for an outlookabout subsequent research.The remainder of this paper is organized as follows. In Section 2 we review therequired results about no-arbitrage concepts, and how they are related. In Section3 we review the transformation result for self-financing portfolios and draw someconsequences for the no-arbitrage concepts under consideration. In Section 4 wepresent the required results about ELMDs and related concepts. In Section 5 wepresent some results with sufficient conditions for the absence of arbitrage. In Sec-tion 6 we review the well-known FTAPs in our present framework, and presentsome minor extensions. In Section 7 we present our main results and show some oftheir consequences. In Section 8 we deal with the particular situation of financialmodels in discrete time. In Section 9 we present an example of Black-Scholes type,in Section 10 we construct arbitrage free markets by means of contingent claims,and in Section 11 we show some consequences of our main result for the construc-tion of locally real-world mean self-financing dynamic trading strategies. Section12 concludes with an outlook about subsequent research topics. For convenience ofthe reader, in Appendix A we provide the required results about vector stochasticintegration, in Appendix B we provide further results from the theory of stochas-tic processes, and in Appendix C we provide the required results about stochasticprocesses in discrete time. HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 5 No-arbitrage concepts
In this section we review several no-arbitrage concepts. For more details we referto [34, Sec. 7].From now on, let (Ω , F , ( F t ) t ∈ R + , P ) be a stochastic basis satisfying the usualconditions, see [24, Def. I.1.3]. Furthermore, we assume that F = { Ω , ∅} . Thenevery F -measurable random variable is P -almost surely constant. Let L be thespace of all equivalence classes of adapted, càdlàg processes X : Ω × R + → R ,where two processes X and Y are identified if X and Y are indistinguishable, thatis if almost all paths of X and Y coincide; see [24, I.1.10]. Let ( K α ) α ≥ be a familyof subsets of L such that for each α ≥ and each X ∈ K α we have X = α .Throughout this section, we make the following assumptions:2.1. Assumption.
We assume that K is a convex cone. Assumption.
We assume that aX + bY ∈ K aα + bβ . (2.1) for all a, b ∈ R + , α, β > with aα + bβ > and X ∈ K α , Y ∈ K β . Now, let T ∈ (0 , ∞ ) be a fixed terminal time. We define the family ( K α ) α ≥ ofsubsets of L = L (Ω , F T , P ) as K α := { X T : X ∈ K α } . (2.2)We define the convex cone C ⊂ L ∞ as C := ( K − L ) ∩ L ∞ . Moreover, we define the family ( B α ) α ≥ of subsets of L as B α := ( K α − L ) ∩ L , α ≥ , and we set B := B . By [34, Lemma 3.11] we have B α = α B for each α > , andhence for the upcoming no-arbitrage concepts it suffices to consider K rather thanthe family ( K α ) α> . The Minkowski functional p B : L → [0 , ∞ ] is given by p B ( ξ ) = inf { α > ξ ∈ B α } , ξ ∈ L . Definition.
We introduce the following concepts: (1) K satisfies NA if K ∩ L = { } , or equivalently C ∩ L ∞ + = { } . (2) K satisfies NFL if C ∗ ∩ L ∞ + = { } , where C ∗ denotes the closure withrespect to the weak- ∗ topology σ ( L ∞ , L ) . (3) K satisfies NFLBR if e C ∗ ∩ L ∞ + = { } , where e C ∗ denotes the sequentialclosure with respect to the weak- ∗ topology σ ( L ∞ , L ) . (4) K satisfies NFLVR if C ∩ L ∞ + = { } , where C is the denotes the closurewith respect to the norm topology on L ∞ . (5) K satisfies NUPBR if B is topologically bounded, or equivalently boundedin probability. (6) K satisfies NAA if B is sequentially bounded. (7) K satisfies NA if p B ( ξ ) > for all ξ ∈ L \ { } . As discussed in [34, Sec. 5], these concepts correspond to the well-known respec-tive concepts that are usually used in the finance literature.2.4.
Proposition. [34, Prop. 5.7]
We have the implications (i) ⇒ (ii) ⇒ (iii) ⇒ (iv), where: (i) K satisfies NFL. (ii) K satisfies NFLBR. (iii) K satisfies NFLVR. ECKHARD PLATEN AND STEFAN TAPPE (iv) K satisfies NA. Proposition. [34, Cor. 5.9]
Suppose that K − L is closed in L . Then thefollowing statements are equivalent: (i) K satisfies NFLVR. (ii) K satisfies NA. Now, we consider some particular examples for the family ( K α ) α ≥ . Let I = ∅ be an arbitrary nonempty index set, and let ( S i ) i ∈ I be a family of semimartingales.We assume that S i ≥ for each i ∈ I . We define the market S := { S i : i ∈ I } .For an R d -valued semimartingale X we denote by L ( X ) the set of all X -integrableprocesses in the sense of vector integration; see [39] or [24, Sec. III.6]. We also referto Appendix A for further details. For δ ∈ L ( X ) we denote by δ · X the stochasticintegral according to [39]. For a finite set F ⊂ I we define the multi-dimensionalsemimartingale S F := ( S i ) i ∈ F .2.6. Definition.
We call a process δ = ( δ i ) i ∈ I a strategy for S if there is a finiteset F ⊂ I such that δ i = 0 for all i ∈ I \ F and we have δ F ∈ L ( S F ) . Definition.
We denote by ∆( S ) the set of all strategies δ for S . Definition.
For a strategy δ ∈ ∆( S ) we set δ · S := δ F · S F , where F ⊂ I denotes the finite set from Definition 2.6. Definition.
For α ∈ R and strategy δ ∈ ∆( S ) we define the integral process I α,δ := α + δ · S . Definition.
For a strategy δ ∈ ∆( S ) we define the portfolio S δ := δ · S , wherewe use the short-hand notation δ · S := X i ∈ F δ i S i with F ⊂ I denoting the finite set from Definition 2.6. Definition.
A strategy δ ∈ ∆( S ) and the corresponding portfolio S δ are called self-financing for S if S δ = S δ + δ · S . Definition.
We denote by ∆ sf ( S ) the set of all self-financing strategies for S . The following auxiliary result is obvious.2.13.
Lemma.
For a strategy δ ∈ ∆( S ) the following statements are equivalent: (i) We have δ ∈ ∆ sf ( S ) . (ii) We have S δ = I α,δ , where α = S δ . Recall that a process X is called admissible if X ≥ − a for some constant a ∈ R + .2.14. Definition.
We introduce the following families: (1)
We define the family of all integral processes ( I α ( S )) α ≥ as I α ( S ) := { I α,δ : δ ∈ ∆( S ) } , α ≥ . (2) We define the family of all admissible integral processes ( I adm α ( S )) α ≥ as I adm α ( S ) := { X ∈ I α ( S ) : X is admissible } , α ≥ . (3) We define the family of all nonnegative integral processes ( I + α ( S )) α ≥ as I + α ( S ) := { X ∈ I α ( S ) : X ≥ } , α ≥ . HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 7 (4)
We denote by ( I α ( S )) α ≥ , ( I adm α ( S )) α ≥ and ( I adm α ( S )) α ≥ the respectivefamilies of random variables defined according to (2.2). Remark.
Consider the particular case where X i ≡ for some i ∈ I . Then themarket S can be interpreted as discounted price processes of risky assets with respectto some savings account, and the families ( I α ( S )) α ≥ , ( I adm α ( S )) α ≥ , ( I + α ( S )) α ≥ canbe regarded as wealth processes in this case. Definition.
We introduce the following families: (1)
We define the family of self-financing portfolios ( P sf ,α ( S )) α ≥ as P sf ,α ( S ) := { S δ : δ ∈ ∆ sf ( S ) and S δ = α } , α ≥ . (2) We define the family of admissible self-financing portfolios ( P admsf ,α ( S )) α ≥ as P admsf ,α ( S ) := { X ∈ P sf ,α ( S ) : X is admissible } , α ≥ . (3) We define the family of nonnegative self-financing portfolios ( P +sf ,α ( S )) α ≥ as P +sf ,α ( S ) := { X ∈ P sf ,α ( S ) : X ≥ } , α ≥ . (4) We denote by ( P sf ,α ( S )) α ≥ , ( P admsf ,α ( S )) α ≥ and ( P admsf ,α ( S )) α ≥ the respec-tive families of random variables defined according to (2.2). For each i ∈ I we denote by e i ∈ ∆( S ) the strategy with components e ji = ( , if j = i , , otherwise.2.17. Lemma. [34, Lemma 7.21]
For each i ∈ I we have e i ∈ ∆ sf ( S ) . Now, we have a series of results, which can be found in [34, Sec. 7].2.18.
Theorem.
Let ( K α ) α ≥ be one of the families ( I α ( S )) α ≥ , ( I adm α ( S )) α ≥ , ( I + α ( S )) α ≥ , ( P sf ,α ( S )) α ≥ , ( P admsf ,α ( S )) α ≥ , ( P +sf ,α ( S )) α ≥ . Then the following statements are equivalent: (i) K satisfies NUPBR. (ii) K satisfies NAA . (iii) K satisfies NA . (iv) We have T α> B α = { } . Proposition.
Let ( K α ) α ≥ be one of the families ( I α ( S )) α ≥ , ( I adm α ( S )) α ≥ , ( I + α ( S )) α ≥ . If K satisfies NA , then K satisfies NA. Proposition.
Let ( K α ) α ≥ be one of the families ( I α ( S )) α ≥ , ( I adm α ( S )) α ≥ . If K satisfies NFLVR, then K satisfies NA . Proposition.
Suppose we have S i > for some i ∈ I , and let ( K α ) α ≥ beone of the families ( P sf ,α ( S )) α ≥ , ( P admsf ,α ( S )) α ≥ , ( P +sf ,α ( S )) α ≥ . If K satisfies NA , then K satisfies NA. ECKHARD PLATEN AND STEFAN TAPPE
Proposition.
Suppose that S i > and S iT ∈ L ∞ for some i ∈ I , and let ( K α ) α ≥ be one of the families ( P sf ,α ( S )) α ≥ , ( P admsf ,α ( S )) α ≥ . If K satisfies NFLVR, then K satisfies NA . Market transformations
In this section, we review the well-known transformation result for self-financingportfolios and draw some consequences for the no-arbitrage concepts under con-sideration. As in Section 2, we consider a market S = { S i : i ∈ I } consisting ofnonnegative semimartingales S i ≥ for some index set I = ∅ . For a nonnegativesemimartingale Z ≥ we agree on the notation S Z := { S i Z : i ∈ I } . Furthermore, we introduce the notation ¯ S := S ∪ { } . Lemma.
Suppose that / ∈ S . Then there is a bijection between R × ∆( S ) and ∆ sf (¯ S ) , which is defined as follows: (1) For ¯ δ = (¯ δ S , ¯ δ ) ∈ ∆ sf (¯ S ) we assign ¯ δ ( x, δ ) := ( ¯ S ¯ δ , ¯ δ S ) ∈ R × ∆( S ) . (2) For ( x, δ ) ∈ R × ∆( S ) we assign ( x, δ ) ¯ δ = ( δ, x + δ · S − S δ )= ( δ, x + ( δ · S ) − − δ · S − ) ∈ ∆ sf (¯ S ) . Furthermore, for all ( x, δ ) ∈ R × ∆( S ) and the corresponding strategy ¯ δ ∈ ∆ sf (¯ S ) we have ¯ S ¯ δ = x + δ · S. Proof.
This is a consequence of [41, Lemma 5.1] and Lemma A.3. (cid:3)
Lemma. [41, Prop. 5.2]
Suppose that / ∈ S . Let δ ∈ ∆ sf (¯ S ) be a self-financingstrategy, and let Z be semimartingale with Z, Z − > . Then we also have δ ∈ ∆ sf (¯ S Z ) . Lemma.
Let Z ∈ S be such that Z, Z − > , and set S := S \ { Z } . Then foreach α ≥ we have I α ( S Z − ) = I α ( S Z − ) . Proof.
Noting that S Z − = S Z − ∪ { } , this is a consequence of Lemma A.3. (cid:3) Definition.
We call every predictable process B of locally finite variation with B = 1 and B, B − > a savings account (or a locally risk-free asset ). Lemma. If B is a savings account, then B − is also savings account withrepresentation B − = 1 − ( B − ) − · B + X s ≤• (cid:16) ( B s ) − − ( B s − ) − + ( B s − ) − ∆ B s (cid:17) . Proof.
This is a consequence of Itô’s formula; see [24, Thm. I.4.57]. (cid:3)
Lemma.
For each savings account B we have S B − B = S ∪ B . HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 9
Proof.
We have S B − B = ( S B − ∪ { } ) B = S ∪ { B } , completing the proof. (cid:3) Now, let T ∈ (0 , ∞ ) be a fixed terminal time. Recall that the definitions of theupcoming sets like P sf , ( S ∪ { B } ) depend on the time T .3.7. Proposition.
Let B be a savings account. Then the following statements areequivalent: (i) P sf , ( S ∪ { B } ) satisfies NA. (ii) I ( S B − ) satisfies NA.Proof. (i) ⇒ (ii): By Lemma 3.3 we may assume that B / ∈ S ; otherwise we consider S := S \ { B } rather than S . Let ξ ∈ ( I ( S B − ) − L ) ∩ L be arbitrary. Then there exists a strategy δ ∈ ∆( S B − ) such that (cid:0) δ · ( SB − ) (cid:1) T ≥ ξ. By Lemma 3.1 there is a self-financing strategy ¯ δ of the form ¯ δ = ( δ, η ) ∈ ∆ sf ( S B − ) for some predictable process η such that δ · ( SB − ) = ( δ, η ) · ( SB − ,
1) = δ · ( SB − ) + η. Therefore, we have δ · ( S B − ) + η = 0 ,δ T · ( S T B − T ) + η T ≥ ξ. By Lemmas 3.2 and 3.6 we have ¯ δ ∈ ∆ sf ( S B − B ) = ∆ sf ( S ∪ { B } ) . Furthermore, we have δ · S + η · B = 0 ,δ T · S T + η T · B T ≥ ξB T . In other words, we have ( S, B ) ¯ δ = 0 and ( S, B ) ¯ δT ≥ ξB T . Therefore, we deduce ξB T ∈ ( P sf , ( S ∪ { B } ) − L ) ∩ L . Since P sf , ( S ∪ { B } ) satisfies NA, it follows that ξB T = 0 , and hence ξ = 0 .Consequently, I ( S B − ) satisfies NA.(ii) ⇒ (i): By Lemma 3.3 we may assume that B / ∈ S ; otherwise we consider S := S \ { B } rather than S . Let ξ ∈ ( P sf , ( S ∪ { B } ) − L ) ∩ L be arbitrary. Then there exists a self-financing strategy ¯ δ = ( δ, η ) ∈ ∆ sf ( S ∪ { B } ) such that ( S, B ) ¯ δ = 0 and ( S, B ) ¯ δT ≥ ξ. Therefore, we have δ · S + η · B = 0 ,δ T · S T + η T · B T ≥ ξ, and hence δ · ( S B − ) + η = 0 ,δ T · ( S T B − T ) + η T ≥ ξB − T . By Lemmas 3.6 and 3.2 we have ¯ δ ∈ ∆ sf ( S B − ) . Therefore, we obtain (cid:0) δ · ( SB − ) (cid:1) T ≥ ξB − T . Since δ ∈ ∆( S ) and I ( S B − ) satisfies NA, it follows that ξB − T = 0 , and hence ξ = 0 . Consequently, P sf , ( S ∪ { B } ) satisfies NA. (cid:3) Proposition.
Let B be a savings account. Then the following statements areequivalent: (i) P +sf , ( S ∪ { B } ) satisfies NUPBR. (ii) I +1 ( S B − ) satisfies NUPBR.Proof. (i) ⇒ (ii): By Lemma 3.3 we may assume that B / ∈ S ; otherwise we consider S := S \ { B } rather than S . Let ξ ∈ \ α> B α be arbitrary, where B α := ( I + α ( S B − ) − L ) ∩ L for each α > .Furthermore, let α > be arbitrary. Then there exists a strategy δ α ∈ ∆( S B − ) such that α + δ α · ( SB − ) ≥ ,α + (cid:0) δ α · ( SB − ) (cid:1) T ≥ ξ. By Lemma 3.1 there is a self-financing strategy ¯ δ α of the form ¯ δ α = ( δ α , η α ) ∈ ∆ sf ( S B − ) for some predictable process η α such that α + δ α · ( SB − ) = ( δ α , η α ) · ( SB − ,
1) = δ α · ( SB − ) + η α . Therefore, we have δ α · ( S B − ) + η α = α,δ α · ( SB − ) + η α ≥ ,δ αT · ( S T B − T ) + η αT ≥ ξ. By Lemmas 3.2 and 3.6 we have ¯ δ α ∈ ∆ sf ( S B − B ) = ∆ sf ( S ∪ { B } ) . Furthermore, noting that B = 1 , we have δ α · S + η α · B = α,δ α · S + η α · B ≥ ,δ αT · S T + η αT · B T ≥ ξB T . In other words, we have ( S, B ) ¯ δ α = α, ( S, B ) ¯ δ α ≥ and ( S, B ) ¯ δ α T ≥ ξB T . HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 11
Since α > was arbitrary, we deduce ξB T ∈ \ α> ¯ B α , where we use the notation ¯ B α := ( P +sf ,α ( S ∪ { B } ) − L ) ∩ L for each α > .Since P +sf ( S ∪ { B } ) > satisfies NUPBR, by Theorem 2.18 it follows that ξB T = 0 ,and hence ξ = 0 . Consequently, I +1 ( S B − ) satisfies NUPBR.(ii) ⇒ (i): By Lemma 3.3 we may assume that B / ∈ S ; otherwise we consider S := S \ { B } rather than S . Let ξ ∈ \ α> B α be arbitrary, where we use the notation B α := ( P +sf ,α ( S ∪ { B } ) − L ) ∩ L for each α > .Let α > be arbitrary. Then there exists a self-financing strategy ¯ δ α = ( δ α , η α ) ∈ ∆ sf ( S ∪ { B } ) such that ( S, B ) ¯ δ α = α, ( S, B ) ¯ δ α ≥ and ( S, B ) ¯ δ α T ≥ ξ. Therefore, we have δ α · S + η α · B = α,δ α · S + η α · B ≥ ,δ αT · S T + η αT · B T ≥ ξ, and hence, since B = 1 , we obtain δ α · ( S B − ) + η α = α,δ α · ( SB − ) + η α ≥ ,δ αT · ( S T B − T ) + η αT ≥ ξB − T . By Lemmas 3.6 and 3.2 we have ¯ δ α ∈ ∆ sf ( S B − ) . Therefore, we have δ α ∈ ∆( S ) and α + δ α · ( SB − ) ≥ ,α + (cid:0) δ α · ( SB − ) (cid:1) T ≥ ξB − T . Since α > was arbitrary, we deduce that ξB T ∈ \ α> ¯ B α , where we use the notation ¯ B α := ( I + α ( S B − ) − L ) ∩ L for each α > .Since I +1 ( S B − ) satisfies NUPBR, by Theorem 2.18 it follows that ξB − T = 0 , andhence ξ = 0 . Consequently, P +sf , ( S ∪ { B } ) satisfies NUPBR. (cid:3) A savings account B is called bounded if there exists a finite constant K > such that B ≤ K on Ω × [0 , T ] up to an evanescent set.3.9. Proposition.
Let B be a savings account such that B is bounded. If P admsf , ( S ∪{ B } ) satisfies NFLVR, then I adm0 ( S B − ) satisfies NFLVR. Proof.
By Lemma 3.3 we may assume that
B / ∈ S ; otherwise we consider S := S \ { B } rather than S . Let ξ ∈ C ∩ L ∞ + be arbitrary, where C := ( I adm0 ( S B − ) − L ) ∩ L ∞ . Then there exists a sequence ( ξ j ) j ∈ N ⊂ C such that k ξ j − ξ k L ∞ → . Let j ∈ N bearbitrary. Then there exist a strategy δ j ∈ ∆( S B − ) and a constant a j ∈ R + suchthat δ j · ( SB − ) ≥ − a j , (cid:0) δ j · ( SB − ) (cid:1) T ≥ ξ j . By Lemma 3.1 there is a self-financing strategy ¯ δ j of the form ¯ δ j = ( δ j , η j ) ∈ ∆ sf ( S B − ) for some predictable process η j such that δ j · ( SB − ) = ( δ j , η j ) · ( SB − ,
1) = δ j · ( SB − ) + η j . Therefore, we have δ j · ( S B − ) + η j = 0 ,δ j · ( SB − ) + η j ≥ − a j ,δ jT · ( S T B − T ) + η jT ≥ ξ j . By Lemmas 3.2 and 3.6 we have ¯ δ j ∈ ∆ sf ( S B − ) = ∆ sf ( S ∪ { B } ) . Furthermore, we have δ j · S + η j · B = 0 ,δ j · S + η j · B ≥ − a j B,δ jT · S T + η jT · B T ≥ ξ j B T . In other words, we have ( S, B ) ¯ δ j = 0 , ( S, B ) ¯ δ j ≥ − a j B and ( S, B ) ¯ δ j T ≥ ξ j B T . Since B is bounded, the portfolio ( S, B ) ¯ δ j is admissible, and we have ξ j B T ∈ L ∞ .Therefore, we deduce that ξ j B T ∈ E , where E := ( P admsf , ( S ∪ { B } ) − L ) ∩ L ∞ . Since k ξ j − ξ k L ∞ → and B T ∈ L ∞ , we also have k ξ j B T − ξB T k L ∞ → . Therefore,we have ξB T ∈ E ∩ L ∞ + . Since P admsf , ( S ∪ { B } ) satisfies NFLVR, it follows that ξB T = 0 , and hence ξ = 0 . This proves that I adm0 ( S B − ) satisfies NFLVR. (cid:3) Proposition.
Let B be a savings account such that B − is bounded. If I adm0 ( S B − ) satisfies NFL, then P admsf , ( S ∪ { B } ) satisfies NFL.Proof. By Lemma 3.3 we may assume that
B / ∈ S ; otherwise we consider S := S \ { B } rather than S . Let ξ ∈ C ∗ ∩ L ∞ + be arbitrary, where C := ( P admsf , ( S ∪ { B } ) − L ) ∩ L ∞ . Then there exists a net ( ξ j ) j ∈ J ⊂ C for some index set J such that E [( ξ j − ξ ) ζ ] → for all ζ ∈ L . Let j ∈ J be arbitrary. Then there exist a self-financing strategy ¯ δ j = ( δ j , η j ) ∈ ∆ sf ( S ∪ { B } ) and a constant a j ∈ R + such that ( S, B ) ¯ δ j = 0 , ( S, B ) ¯ δ α ≥ − a j and ( S, B ) ¯ δ j T ≥ ξ j . HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 13
Therefore, we have δ j · S + η j · B = 0 ,δ j · S + η j · B ≥ − a j ,δ jT · S T + η jT · B T ≥ ξ j , and hence, we obtain δ j · ( S B − ) + η j = 0 ,δ j · ( SB − ) + η j ≥ − a j B − ,δ jT · ( S T B − T ) + η jT ≥ ξ j B − T . By Lemmas 3.6 and 3.2 we have ¯ δ j ∈ ∆ sf ( S B − ) . Therefore, we have δ j ∈ ∆( S ) and δ j · ( SB − ) ≥ − a j B − , (cid:0) δ j · ( SB − ) (cid:1) T ≥ ξ j B − T . Since B − is bounded, the integral process δ j · ( SB − ) is admissible, and we have ξ j B − T ∈ L ∞ . Therefore, we deduce that ξ j B T ∈ E , where E := ( I adm0 ( S B − ) − L ) ∩ L ∞ . Since E [( ξ j − ξ ) ζ ] → for all ζ ∈ L and B − T ∈ L ∞ , we also have E [( ξ j B − T − ξB − T ) ζ ] = E [( ξ j − ξ ) ζB − T ] → for all ζ ∈ L ,because ζB − T ∈ L . Therefore, we have ξB − T ∈ E ∗ ∩ L ∞ + . Since I adm0 ( S B − ) satisfies NFL, it follows that ξB − T = 0 , and hence ξ = 0 . This proves that P admsf , ( S ∪{ B } ) satisfies NFL. (cid:3) Equivalent local martingale deflators and related concepts
In this section we present results about local martingale deflators and relatedconcepts.4.1.
Definition.
Let X be a family of semimartingales, and let Z be a semimartin-gale such that Z, Z − > . (1) We call Z an equivalent martingale deflator (EMD) for X if XZ ∈ M for all X ∈ X . (2) We call Z an equivalent local martingale deflator (ELMD) for X if XZ ∈ M loc for all X ∈ X . (3) We call Z an equivalent σ -martingale deflator (E Σ MD) for X if XZ ∈ M σ for all X ∈ X . (4) We call Z a strict σ -martingale density (S Σ MD) for X if Z ∈ M loc and XZ ∈ M σ for all X ∈ X . Lemma.
Let X be a family of nonnegative semimartingales. Then the followingstatements are equivalent: (i) Z is an ELMD for X . (ii) Z is an E Σ MD for X .Proof. This is an immediate consequence of Lemma A.6. (cid:3)
Definition.
Let X be a family of semimartingales, and let Q ≈ P be an equiv-alent probability measure on (Ω , F ∞− ) . (1) We call Q an equivalent martingale measure (EMM) for X if X is a Q -martingale for all X ∈ X . (2) We call Q an equivalent local martingale measure (ELMM) for X if X isa Q -local martingale for all X ∈ X . (3) We call Q an equivalent σ -martingale measure (E Σ MM) for X if X is a Q - σ -martingale for all X ∈ X . Lemma.
Let X be a family of admissible semimartingales, and let Q ≈ P bean equivalent probability measure on (Ω , F ∞− ) . Then the following statements areequivalent: (i) Q is an ELMM for X . (ii) Q is an E Σ MM for X .Proof. This is an immediate consequence of Lemma A.6. (cid:3)
Lemma.
Let Q ≈ P be an equivalent probability measure on (Ω , F ∞− ) , andlet D be the density process D of Q , relative to P . Then for every semimartingale X the following statements are equivalent: (i) XD is a P -local martingale. (ii) X is a Q -local martingale.Furthermore, the following statements are equivalent: (iii) XD is a P -martingale. (iv) X is a Q -martingale.Proof. (i) ⇒ (ii): This is an immediate consequence of [24, Prop. III.3.8.b].(ii) ⇒ (i): Let ( T n ) n ∈ N be a Q -localizing sequence for M . Since the probabilitymeasures P and Q are equivalent, it follows that P (lim n →∞ T n = ∞ ) = 1 . Therefore,the stated implication follows from [24, Prop. III.3.8.c].(iii) ⇔ (iv): This equivalence is a consequence of [24, Prop. III.3.8.a]. (cid:3) Lemma.
Let X be a family of semimartingales, let Q ≈ P be an equivalentprobability measure on (Ω , F ∞− ) , and let D be the density process of Q , relative to P . Then for every semimartingale Z with Z, Z − > the following statements areequivalent: (i) ZD is an ELMD for X . (ii) Z is a Q -ELMD for X .Furthermore, the following statements are equivalent: (iii) ZD is an EMD for X . (iv) Z is a Q -EMD for X .Proof. This is an immediate consequence of Lemma 4.5. (cid:3)
Lemma.
Let X be a family of semimartingales, let Q ≈ P be an equivalentprobability measure on (Ω , F ∞− ) , and let D be the density process of Q , relative to P . Then the following statements are equivalent: (i) D is an ELMD for X . (ii) Q is an ELMM for X .Furthermore, the following statements are equivalent: (iii) D is an EMD for X . (iv) Q is an EMM for X .Proof. This is an immediate consequence of Lemma 4.6. (cid:3)
Proposition.
The following statements are equivalent: (i)
There exists an ELMM Q ≈ P on (Ω , F ∞− ) for X . HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 15 (ii)
There exists an ELMD D for X such that D ∈ M with P ( D ∞ >
0) = 1 .Proof. (i) ⇒ (ii): Let D be the density process of Q , relative to P . By [24, Prop.III.3.5] we have D ∈ M with P ( D ∞ >
0) = 1 . Furthermore, by Lemma 4.7 theprocess D is an ELMD for X .(ii) ⇒ (i): Without loss of generality we may assume that D = 1 . According to [24,Prop. III.3.5] there exists an equivalent probability measure Q ≈ P on (Ω , F ∞− ) such that D is the density process of Q , relative to P . By Lemma 4.7 the measure Q is an ELMM for X . (cid:3) For the rest of this section, let S = { S i : i ∈ I } be a market with nonnegativesemimartingales, as considered in the previous sections. We introduce the unions I ( S ) := [ α ≥ I α ( S ) , I adm ( S ) := [ α ≥ I adm α ( S ) and I + ( S ) := [ α ≥ I + α ( S ) . Proposition.
For a semimartingale Z with Z, Z − > the following statementsare equivalent: (i) Z is a S Σ MD for S . (ii) Z is an ELMD for S , and we have Z ∈ M loc . (iii) Z is an ELMD for I adm ( S ) . (iv) Z is an E Σ MD for I ( S ) .Proof. (i) ⇔ (ii): This equivalence is a consequence of Lemma A.6.(ii) ⇒ (iv): Let α ∈ R + and δ ∈ ∆( S ) be arbitrary. By Proposition A.14 we have I α,δ Z = ( α + δ · S ) Z = αZ + ( δ · S ) Z ∈ M σ . (iv) ⇒ (iii): Using Lemma 4.2 this implication follows, because I adm ( S ) ⊂ I ( S ) .(iii) ⇒ (ii): We have S ⊂ I adm ( S ) . Therefore, the process Z is an ELMD for S .Furthermore, setting α := 1 and δ := 0 we obtain I α,δ = 1 , and hence Z = I α,δ Z ∈ M loc , completing the proof. (cid:3) Proposition.
For an equivalent probability measure Q ≈ P on (Ω , F ∞− ) thefollowing statements are equivalent: (i) Q is an ELMM for S . (ii) Q is an ELMM for I adm ( S ) . (iii) Q is an E Σ MM for I ( S ) .Proof. (ii) ⇒ (i): Since S ⊂ I adm ( S ) , this implication is obvious.(iii) ⇒ (ii): Since I adm ( S ) ⊂ I ( S ) , this implication follows with Lemma 4.4.(i) ⇒ (iii): Let α ∈ R + and δ ∈ ∆( S ) be arbitrary. Then by Lemma A.5 the process I α,δ = α + δ · S is a Q - σ -martingale. (cid:3) We introduce the unions P sf ( S ) := [ α ≥ P sf ,α ( S ) , P admsf ( S ) := [ α ≥ P admsf ,α ( S ) and P +sf ( S ) := [ α ≥ P +sf ,α ( S ) . Lemma.
We have S ⊂ P +sf ( S ) .Proof. This is an immediate consequence of Lemma 2.17. (cid:3)
Proposition.
The following statements are equivalent: (i) Z is an ELMD for S . (ii) Z is an ELMD for P admsf ( S ) . (iii) Z is an E Σ MD for P sf ( S ) .Proof. (i) ⇒ (iii): Let δ ∈ ∆ sf ( S ) be arbitrary. By Proposition A.15 we have S δ Z = ( δ · S ) Z ∈ M σ . (iii) ⇒ (ii): Using Lemma 4.2 this implication follows, because P admsf ( S ) ⊂ P sf ( S ) .(ii) ⇒ (i): By Lemma 4.11 we have S ⊂ P admsf ( S ) . Therefore, the process Z is anELMD for S . (cid:3) Sufficient conditions for the absence of arbitrage
In this section we present some results with sufficient conditions for the absenceof arbitrage. As in the previous sections, let S = { S i : i ∈ I } be a market withnonnegative semimartingales, and let T ∈ (0 , ∞ ) be a fixed time horizon. Fromnow on, deflators are only considered on the time interval [0 , T ] , and probabilitymeasures are only considered on (Ω , F T ) .5.1. Proposition.
Suppose that an ELMM Q ≈ P on (Ω , F T ) for S exists. Then I adm0 ( S ) satisfies NFL.Proof. By Proposition 4.10 the probability measure Q is also an ELMM for I adm ( S ) .Let Z be the density process of Q , relative to P . Furthermore, let ξ ∈ C ∗ ∩ L ∞ + bearbitrary, where C := ( I adm0 ( S ) − L ) ∩ L ∞ . Then there exists a net ( ξ j ) j ∈ J ⊂ C for some index set J converging to ξ withrespect to the weak- ∗ topology σ ( L ∞ , L ) . Let j ∈ J be arbitrary. Then thereexists a strategy δ j ∈ ∆( S ) such that δ j · S is admissible and ( δ j · S ) T ≥ ξ j . Since Q is an ELMM for I adm ( S ) , the process δ j · S is an admissible Q -local martin-gale, and hence by Lemma B.1 a Q -supermartingale. By Doob’s optional samplingtheorem for supermartingales (see Theorem B.2) we obtain E [ ξ j Z T ] = E Q [ ξ j ] ≤ E Q [( δ j · S ) T ] ≤ E Q [( δ j · S ) ] = 0 . Since the net ( ξ j ) j ∈ J converges to ξ with respect to the weak- ∗ topology σ ( L ∞ , L ) ,we obtain E [ ξ j Z T ] → E [ ξZ T ] , and hence E [ ξZ T ] ≤ . Since ξ ≥ and P ( Z T >
0) = 1 , this shows that ξ = 0 .Hence I adm0 ( S ) satisfies NFL. (cid:3) Proposition.
Suppose that an ELMD Z for S with Z ∈ M loc exists. Then I +0 ( S ) satisfies NFL.Proof. By Proposition 4.10 the process Z is also an ELMD for I + ( S ) . Let ξ ∈ C ∗ ∩ L ∞ + be arbitrary, where C := ( I +0 ( S ) − L ) ∩ L ∞ . Then there exists a net ( ξ j ) j ∈ J ⊂ C for some index set J converging to ξ withrespect to the weak- ∗ topology σ ( L ∞ , L ) . Let j ∈ J be arbitrary. Then thereexists a strategy δ j ∈ ∆( S ) such that δ j · S ≥ and ( δ j · S ) T ≥ ξ j . HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 17
Since Z is an ELMD for I + ( S ) , the process ( δ j · S ) Z is a nonnegative local mar-tingale, and hence by Lemma B.1 a supermartingale. By Doob’s optional samplingtheorem for supermartingales (see Theorem B.2) we obtain E [ ξ j Z T ] ≤ E [( δ j · S ) T Z T ] ≤ E [( δ j · S ) Z ] = 0 . Note that Z T ∈ L , because Z is a nonnegative local martingale, and hence byLemma B.1 a supermartingale. Since the net ( ξ j ) j ∈ J converges to ξ with respectto the weak- ∗ topology σ ( L ∞ , L ) , we obtain E [ ξ j Z T ] → E [ ξZ T ] , and hence E [ ξZ T ] ≤ . Since ξ ≥ and P ( Z T >
0) = 1 , this shows ξ = 0 . Hence I +0 ( S ) satisfies NFL. (cid:3) Proposition.
Suppose that an ELMD Z for S exists. Then P +sf , ( S ) satisfiesNUPBR.Proof. By Proposition 4.12 the process Z is also an ELMD for P +sf ( S ) . Let ξ ∈ \ α> B α be arbitrary, where B α := ( P +sf ,α ( S ) − L ) ∩ L for each α > .Let α > be arbitrary. Then there exists a self-financing strategy δ α ∈ ∆ sf ( S ) suchthat S δ α = α, S δ α ≥ and S δ α T ≥ ξ. Since Z is an ELMD for P +sf ( S ) , the process S δ α Z is a nonnegative local martingale,and hence by Lemma B.1 a supermartingale. By Doob’s optional stopping theoremfor supermartingales (see Theorem B.2) we obtain E [ ξZ T ] ≤ E [ S δ α T Z T ] ≤ E [ S δ α Z ] = αZ . Since α > was arbitrary, we deduce that E [ ξZ T ] = 0 . Since ξ ≥ and P ( Z T >
0) = 1 , this shows ξ = 0 . Therefore, by Theorem 2.18 the set P +sf , ( S ) satisfiesNUPBR. (cid:3) The fundamental theorems of asset pricing revisited
In this section we review the fundamental theorems of asset pricing in our presentframework, and present some minor extensions. Now, we consider a finite market S = { S , . . . , S d } with nonnegative semimartingales for some d ∈ N ; that is, theindex set I = { , . . . , d } is finite. As in the previous sections, we fix a terminal time T ∈ (0 , ∞ ) . We recall that the definitions of the upcoming sets like I +1 ( S ) dependon the time T , and that notions like deflators are considered on the time interval [0 , T ] .6.1. Theorem.
The following statements are equivalent: (i) I +1 ( S ) satisfies NUPBR. (ii) I +1 ( S ) satisfies NAA . (iii) I +1 ( S ) satisfies NA . (iv) There exists an ELMD Z for S such that Z ∈ M loc . (v) There exists a S Σ MD Z for S .If the previous conditions are fulfilled, then I +0 ( S ) satisfies NFL, NFLBR, NFLVRand NA. Proof. (i) ⇔ (ii) ⇔ (iii): These equivalences follow from Theorem 2.18.(iv) ⇔ (v): This equivalence is a consequence of Proposition 4.9.(i) ⇔ (v): This equivalence follows from [41, Thm. 2.6].The additional statement is a consequence of Propositions 5.2 and 2.4. (cid:3) Theorem.
The following statements are equivalent: (i) I adm0 ( S ) satisfies NFL. (ii) I adm0 ( S ) satisfies NFLBR. (iii) I adm0 ( S ) satisfies NFLVR. (iv) There exists an ELMD Z for S such that Z ∈ M . (v) There exists an ELMM Q ≈ P on (Ω , F T ) for S .If the previous conditions are fulfilled, then I adm0 ( S ) satisfies NA, and I adm1 ( S ) satisfies NUPBR, NAA and NA .Proof. (i) ⇒ (ii) ⇒ (iii): These implications follow from Proposition 2.4.(iv) ⇔ (v): This equivalence is a consequence of Proposition 4.8.(iii) ⇔ (v): Noting Lemma 4.4, this is an immediate consequence of the MainTheorem from [11].(v) ⇒ (i): This follows from Proposition 5.1.The additional statements follow from Propositions 2.4, 2.20 and Theorem 2.18. (cid:3) Note that the FTAP in [9] and [11] characterizes the existence of an ELMM interms of the property NFLVR. The equivalence of NFL, NFLBR and NFLVR wasproven in [26]. Here we obtain a rather simple proof of this result.7.
The main results and their consequences
In this section we present our main results and show some of their consequences.We begin with multiplicative decompositions of semimartingales.7.1.
Definition.
A semimartingale Z with Z, Z − > is called a multiplicativespecial semimartingale if it admits a multiplicative decomposition Z = DC (7.1) with a local martingale D ∈ M loc and a predictable càdlàg process C with locallyfinite variation such that D, D − > and C, C − > . Theorem. [24, Thm. II.8.21]
For a semimartingale Z with Z, Z − > thefollowing statements are equivalent: (i) Z is a multiplicative special semimartingale. (ii) Z is a special semimartingale.If the previous conditions are fulfilled, then the processes D and C appearing in themultiplicative decomposition (7.1) are unique up to an evanescent set. In the situation of Theorem 7.2 we call D the local martingale part of the mul-tiplicative decomposition (7.1), and we call C the finite variation part of the mul-tiplicative decomposition (7.1).Now, consider a finite market S = { S , . . . , S d } with nonnegative semimartin-gales for some d ∈ N .7.3. Lemma.
Let Z = DB − be a multiplicative special semimartingale with a localmartingale part D ∈ M loc and a savings account B . Then the following statementsare equivalent: (i) Z is an ELMD for S . (ii) Z is an ELMD for S ∪ { B } .Proof. Since BZ = D ∈ M loc , the proof is immediate. (cid:3) HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 19
As in the previous sections, we fix a terminal time T ∈ (0 , ∞ ) . Once again, weemphasize that the definitions of the upcoming sets like P +sf , ( S ∪ { B } ) depend onthe time T , and that notions like deflators are considered on the time interval [0 , T ] .Now, we are ready to state our main results.7.4. Theorem.
Let B be a savings account. Then the following statements areequivalent: (i) P +sf , ( S ∪ { B } ) satisfies NUPBR. (ii) P +sf , ( S ∪ { B } ) satisfies NAA . (iii) P +sf , ( S ∪ { B } ) satisfies NA . (iv) There exists a local martingale D ∈ M loc with D, D − > such that Z = DB − is an ELMD for S . (v) There exists an ELMD D for S B − such that D ∈ M loc .If the previous conditions are fulfilled, then P +sf , ( S ∪ { B } ) satisfies NA.Proof. The equivalences (i) ⇔ (ii) ⇔ (iii) follow from Theorem 2.18.(i) ⇒ (v): By Proposition 3.8 the set I +1 ( S B − ) satisfies NUPBR. Hence, by The-orem 6.1 there exists an ELMD D for S B − such that D ∈ M loc .(v) ⇒ (iv): This implication is obvious.(iv) ⇒ (i): By Lemma 7.3 the process Z is also an ELMD for S ∪ { B } . Hence, byProposition 5.3 the set P +sf , ( S ∪ { B } ) satisfies NUPBR.The additional statement follows from Proposition 2.21. (cid:3) Theorem.
The following statements are equivalent: (i)
There exists a savings account B such that P +sf , ( S ∪{ B } ) satisfies NUPBR. (ii) There exists a savings account B such that P +sf , ( S ∪ { B } ) satisfies NAA . (iii) There exists a savings account B such that P +sf , ( S ∪ { B } ) satisfies NA . (iv) There exists an ELMD Z for S which is a multiplicative special semimartin-gale. (v) There exist a savings account B and an ELMD D for S B − such that D ∈ M loc .If the previous conditions are fulfilled, then we can choose an ELMD Z for S withmultiplicative decomposition Z = DB − , where B is a savings account as in (i)–(iii).Proof. This is an immediate consequence of Theorem 7.4. (cid:3)
Theorem.
Let B be a savings account such that B and B − are bounded. Thenthe following statements are equivalent: (i) P admsf , ( S ∪ { B } ) satisfies NFL. (ii) P admsf , ( S ∪ { B } ) satisfies NFLBR. (iii) P admsf , ( S ∪ { B } ) satisfies NFLVR. (iv) There exists a martingale D ∈ M with D, D − > such that Z = DB − isan ELMD for S . (v) There exists an ELMM Q ≈ P for S B − .If the previous conditions are fulfilled, then P admsf , ( S ∪ { B } ) satisfies NA, and P admsf , ( S ∪ { B } ) satisfies NA , NAA and NUBBR.Proof. (i) ⇒ (ii) ⇒ (iii): These implications follow from Proposition 2.4.(iii) ⇒ (iv): By Proposition 3.9 the set I adm0 ( S B − ) satisfies NFLVR. Hence, byTheorem 6.2 there exists an ELMD D for S B − such that D ∈ M . Therefore, theprocess Z = DB − is an ELMD for S . (iv) ⇒ (v): Note that D is an ELMD for S B − . Without loss of generality, wemay assume that E [ D T ] = 1 . Let Q ≈ P be the equivalent probability measure on (Ω , F T ) with density process D . By Lemma 4.7 the measure Q is an ELMM for S B − .(v) ⇒ (i): By Proposition 5.1 the set I adm0 ( S B − ) satisfies NFL. Thus, by Propo-sition 3.10 the set P admsf , ( S ∪ { B } ) satisfies NFL as well.The remaining statements follow from Proposition 2.4, Proposition 2.22 and The-orem 2.18. (cid:3) Theorem.
The following statements are equivalent: (i)
There exists a savings account B such that B and B − are bounded, and P admsf , ( S ∪ { B } ) satisfies NFL. (ii) There exists a savings account B such that B and B − are bounded, and P admsf , ( S ∪ { B } ) satisfies NFLBR. (iii) There exists a savings account B such that B and B − are bounded, and P admsf , ( S ∪ { B } ) satisfies NFLVR. (iv) There exists an ELMD Z for S which is a multiplicative special semimartin-gale such that the local martingale part is a martingale, and the finite vari-ation part and its inverse are bounded. (v) There exists a savings account B such that B and B − are bounded, andan ELMM Q ≈ P for S B − .Proof. This is an immediate consequence of Theorem 7.6. (cid:3)
In the previous results (Theorems 7.4–7.7) the savings account B could alreadybe contained in the market S . As the next result shows, an arbitrage free marketcan have at most one savings account.7.8. Proposition.
Suppose that P +sf , ( S ) satisfies NUPBR (or, equivalently, NAA or NA ), and let B, ˆ B ∈ S be two savings accounts. Then we have B = ˆ B up to anevanescent set.Proof. By Theorem 7.4 there exists a local martingale D ∈ M loc with D, D − > such that Z = DB − is an ELMD for S . In particular, setting A := B − ˆ B we have DA = Z ˆ B ∈ M loc . Applying Lemma B.3 gives us A = 1 up to an evanescent set,and hence B = ˆ B up to an evanescent set. (cid:3) In the situation of the previous results (Theorems 7.4–7.7) the savings account B , and hence the ELMD Z = DB − , do not need to be unique. However, as thefollowing result shows, for a given local martingale D ∈ M loc there is at most onesuitable savings account fitting into the multiplicative decomposition Z = DB − .7.9. Proposition.
Suppose that S i , S i − > for some i ∈ { , . . . , d } . Let D ∈ M loc be a local martingale with D, D − > . Furthermore, let B, ˆ B be two savings accountssuch that the multiplicative special semimartingales Z = DB − and ˆ Z = D ˆ B − areELMDs for the market S . Then we have B = ˆ B up to an evanescent set.Proof. We have S i DB − ∈ M loc and S i D ˆ B − ∈ M loc . Note that A := B ˆ B − isanother savings account, and that ( S i DB − ) A ∈ M loc . Applying Lemma B.3 gives us that A = 1 up to an evanescent set, and hence wehave B = ˆ B up to an evanescent set. (cid:3) Remark.
In this section we have considered a finite market S = { S , . . . , S d } .However, note that for an arbitrary market S = { S i : i ∈ I } with nonnegative HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 21 semimartingales and an arbitrary index set I the existence of an appropriate ELMD,which is a multiplicative special semimartingale, is sufficient for the absence ofarbitrage. More precisely, in such a more general market the following implicationsstill hold true: • (iv) ⇒ (i), (v) ⇒ (i) in Theorems 7.4 and 7.5. • (iv) ⇒ (i), (v) ⇒ (i) in Theorems 7.6 and 7.7. Financial models in discrete time
Using our previous results for continuous time models, we can also derive a resultfor discrete time models in a rather simple manner. This result is in accordance withthe well-known result concerning the absence of arbitrage in discrete time finance.In this section we assume that a discrete filtration ( F k ) k ∈ N with F = { Ω , ∅} isgiven, and we consider a finite market S = { S , . . . , S d } consisting of nonnegative,adapted processes. As shown in [24, page 14], this can be regarded as a particularcase of the continuous time setting, which we have considered so far. The terminaltime T is assumed to be an integer T ∈ N . Note that every R d -valued predictableprocess δ belongs to ∆( S ) , and that the stochastic integral δ · S = ( δ · S t ) t ∈ N isgiven by δ · S = 0 ,δ · S t = t X k =1 δ k · ( S k − S k − ) , t ∈ N . For our upcoming result, we require the following theorem.8.1.
Theorem. If I ( S ) satisfies NA, then I ( S ) − L is closed in L .Proof. This is a consequence of [28, Thm. 1]. (cid:3)
Theorem.
Let B be a savings account. Then the following statements areequivalent: (i) P sf , ( S ∪ { B } ) satisfies NA. (ii) There exists a martingale D ∈ M with D > such that Z = DB − is anEMD for S . (iii) There exists an EMM Q ≈ P for S B − .Proof. (i) ⇒ (ii): By Proposition 3.7 the set I ( S B − ) also satisfies NA. By The-orem 8.1 the set I ( S B − ) − L is closed in L . Hence, by Proposition 2.5 theset I ( S B − ) also satisfies NFLVR. By Proposition 2.20 it follows that I ( S B − ) satisfies NA , and hence NUPBR. Of course, the subset I +1 ( S B − ) also satisfiesNUPBR. Therefore, by Proposition 3.8 the set P +sf , ( S ∪ { B } ) satisfies NUPBR.Hence by Theorem 7.4 there exists a local martingale D ∈ M loc with D > suchthat Z = DB − is an ELMD for S . By virtue of Lemma C.3 we have D ∈ M and Z is an EMD for S .(ii) ⇒ (iii): Note that D is an EMD for S B − . Without loss of generality, we mayassume that E [ D T ] = 1 . Let Q ≈ P be the equivalent probability measure on (Ω , F T ) with density process D . By Lemma 4.7 the measure Q is an EMM for S B − .(iii) ⇒ (i): Let ξ ∈ I ( S B − ) ∩ L be arbitrary. Then there exists a strategy δ ∈ ∆( S B − ) such that ( δ · ( SB − )) T = ξ . Since Q is an EMM for S B − , theprocess δ · ( SB − ) is a d -martingale transform under Q . Therefore, by TheoremC.1 the process δ · ( SB − ) is a Q -local martingale. Since ( δ · ( SB − )) = 0 and ξ ∈ L , by Lemma C.2 we deduce that ξ = 0 . Hence I ( S B − ) satisfies NA, andby Proposition 3.7 it follows that P sf , ( S ∪ { B } ) satisfies NA. (cid:3) Theorem.
The following statements are equivalent: (i)
There exists a savings account B such that P sf , ( S ∪ { B } ) satisfies NA. (ii) There exists an EMD Z for S which is a multiplicative special semimartin-gale such that the local martingale part is a martingale. (iii) There exist a savings account B and an EMM Q ≈ P for S B − .Proof. This is an immediate consequence of Theorem 8.2. (cid:3) An example of Black-Scholes type
In this section, we present an example of Black-Scholes type, where we discuss,for simplicity, the case of a single risky asset but could instead have a set of riskyassets. Let λ be the Lebesgue measure on ( R + , B ( R + )) , and let W be a R -valuedstandard Wiener process. We assume that the original market S = { S } is given bythe single asset S = E ( a · λ + σ · W ) , where a ∈ L ( λ ) and σ ∈ L ( W ) with σ > . We are looking for a savings ac-count B such that the extended market S ∪ { B } = { S, B } is free of arbitrage.Hence, according to Theorem 7.4 we have to look for an ELMD Z for S which is amultiplicative special semimartingale. As a candidate, we consider a multiplicativespecial semimartingale of the form Z = DB − , where D = E ( − θ · W ) and B = exp( r · λ ) (9.1)with integrable processes θ ∈ L ( W ) and r ∈ L ( λ ) . By Yor’s formula (see [24,II.8.19]) we have SZ = E (cid:0) ( a − r − σθ ) · λ + ( σ − θ ) · W (cid:1) . Hence Z is an ELMD if and only if θ = a − rσ , (9.2)or equivalently r = a − σθ. (9.3)Since at this stage the choice for r is free, there exists for each possible r a respectiveELMD. Note that the latter identity (9.3) confirms Proposition 7.9 concerning theuniqueness of the savings account for a given local martingale. Summing up, thereare several ELMDs for S which are multiplicative special semimartingales. Theyare all of the form Z = DB − with D and B given by (9.1), and where these twoprocesses are linked by (9.2) and (9.3). By Theorem 7.4 the set P +sf , ( S ∪ { B } ) satisfies NUPBR. Furthermore, if the savings accounts B and B − are bounded,then by Theorem 7.6 the set P admsf , ( S ∪ { B } ) satisfies NFL if and only if D ∈ M .This is, in particular, the case in the classical Black-Scholes setting, where theprocesses a , σ and r are constant.10. Markets given by contingent claims
In this section we construct arbitrage free markets by means of contingent claims.We fix a savings account B , a local martingale D ∈ M loc with D, D − > anddefine the multiplicative special semimartingale Z := DB − . Let H , . . . , H d benonnegative F T -measurable contingent claims for some d ∈ N such that H i Z T ∈ L for all i = 1 , . . . , d .(10.1)We define the market S = { S , . . . , S d } by setting S it := Z − t E [ H i Z T | F t ] , t ∈ [0 , T ] (10.2) HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 23 for all i = 1 , . . . , d . Then Z is an ELMD for S , and by Theorem 7.4 the set P +sf , ( S ∪{ B } ) satisfies NUPBR.Now, suppose that the savings accounts B and B − are bounded. By Theorem7.6 the set P admsf , ( S ∪ { B } ) satisfies NFL if and only if D ∈ M . Suppose that D ∈ M . Then, also by Theorem 7.6, there exists an ELMM Q ≈ P for S B − , andits density process is given by D , where, without loss of generality, we may assumethat D = 1 . By (10.1) we have H i B − T ∈ L ( Q ) for all i = 1 , . . . , d .Hence, using [24, III.3.9] we obtain the representation S it = B t E Q [ H i B − T | F t ] , t ∈ [0 , T ] (10.3)for all i = 1 , . . . , d . Note that (10.3) is the well-known risk-neutral pricing formulafor contingent claims, whereas (10.2) is the real-world pricing formula, which alsoshows up in the Benchmark Approach in [33].The method above can also be applied in order to extend an arbitrage free mar-ket. Let S = { S , . . . , S d } be a market with nonnegative semimartingales for some d ∈ N . Suppose there exists a savings account B such that P +sf , ( S ∪ { B } ) satisfiesNUPBR. By Theorem 7.4 there exists an ELMD Z for S , which is a multiplicativespecial semimartingale of the form Z = DB − with a local martingale D ∈ M loc .Now, let e ∈ N with e > d be an arbitrary integer, and let H d +1 , . . . , H e be nonnega-tive F T -measurable contingent claims such that H i Z T ∈ L for all i = d + 1 , . . . , e .We extend the market S = { S , . . . , S d } to a market ˆ S = S ∪ { S d +1 , . . . , S e } = { S , . . . , S e } by setting S it := Z − t E [ H i Z T | F t ] , t ∈ [0 , T ] for all i = d + 1 , . . . , e . Then Z is also an ELMD for ˆ S , and by Theorem 7.4 the set P +sf , (ˆ S ∪ { B } ) satisfies NUPBR.11. Dynamic trading strategies
For risk management purposes trading strategies that may not be self-financingcan be crucial. In this section we show how Theorem 7.4 can be used in order toconstruct such strategies. Of course, when looking for trading strategies which arenot self-financing, not all strategies can be allowed. In order to construct reasonablestrategies that may not be self-financing, the concept of a locally real-world meanself-financing dynamic trading strategy (see [13]) turns out to be fruitful. In the risk-neutral context, this notion has been introduced in [38]. As in the previous sections,we consider a finite market S = { S , . . . , S d } with nonnegative semimartingalestogether with a savings account B .11.1. Definition. A dynamic trading strategy ν = ( δ, η ) consists of a self-financingstrategy δ ∈ ∆ sf ( S ) and a real-valued optional process η such that the portfolio V ν := S δ + B η , where we use the common notations S δ := δ · S and B η := η · B , satisfies V ν = S δ + δ · S + B η . In this case, we call δ the self-financing part of ν . Remark.
Note that V ν = V ν + δ · S + B η − B η . Definition.
Let ν = ( δ, η ) be a dynamic trading strategy. The profit and loss(P&L) process C ν is defined as C ν := V ν − δ · S − V ν . Note that C ν = 0 and C ν = B η − B η . Therefore, we have V ν = V ν + δ · S + C ν . Thus, the P&L process C ν monitors the cumulative inflow and outflow of extracapital. We will consider such dynamic trading strategies ν for which the portfolio V ν is locally in mean self-financing. More precisely, we introduce the followingconcept.11.4. Definition.
A dynamic trading strategy ν is called locally real-world meanself-financing if C ν ∈ M loc . The following result shows how to construct locally real-world mean self-financingdynamic trading strategies, provided that the market S ∪ { B } is free of arbitragein the sense of Theorem 7.4.11.5. Proposition.
Suppose there is an ELMD Z which is a multiplicative specialsemimartingale of the form Z = DB − with D ∈ M loc , and let δ ∈ ∆ sf ( S ) be aself-financing strategy. Then ν := ( δ, η ) , where η := Z , is a locally real-world meanself-financing dynamic trading strategy.Proof. We have B η = ZB = D ∈ M loc , and hence C ν = B η − B η ∈ M loc . (cid:3) Conclusion
In this paper we have considered a market S = { S , . . . , S d } with nonnegativesemimartingales which does not need to have a numéraire S d = 1 . Provided we areallowed to add a savings account B to the market, we have proven that the marketis free of arbitrage if and only if there exists an ELMD Z which is a multiplicativespecial semimartingale, and that in this case the savings account B has to fit intothe multiplicative decomposition Z = DB − of the deflator.There are some connected questions, which give rise to future research projects.Here is an outline:(1) Given a candidate B for the savings account, can we provide a systematicoverview of all ELMDs of the form Z = DB − with a local martingale D ∈ M loc ?(2) Given a candidate B for the savings account, can we find an ELMD of theform Z = DB − with a local martingale D ∈ M loc such that its inverse ¯ Z = Z − can be realized as a self-financing portfolio constructed in theextended market S ∪ { B } ? From a practical point of view, the tradeabilityof the deflator is of particular interest because otherwise we obtain strate-gies which are difficult to implement. In this case, we expect that such anequivalent local martingale numéraire portfolio ¯ Z will be the growth opti-mal portfolio, which provides a link to the Benchmark Approach in [33]. Appendix A. Vector stochastic integration
In this appendix we provide the required results about vector stochastic inte-gration. First, we briefly review stochastic integration with respect to a local mar-tingale. Let M ∈ M d loc be arbitrary. Let the R d × d -valued process C be given by C ij := [ M i , M j ] for all i, j = 1 , . . . , d , and consider a factorization C ij = c ij · F, HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 25 where c is an optional R d × d -valued process, and F ∈ V + . We denote by L ( M ) the set of all R d -valued predictable processes H such that p ( cH · H ) · F ∈ A +loc . Furthermore, for each H ∈ L ( M ) we denote by (M) H · M ∈ M loc the stochastic integral according to [39, Sec. 3.1]. For every simple process of theform H = H { } + m X k =1 H k ( τ k ,τ k +1 ] with R d -valued and F τ k -measurable random variables H k for k = 0 , . . . , m thestochastic integral is defined as (M) H · M := m X k =1 H k · ( M τ k +1 − M τ k ) , (A.1)and for a general H ∈ L ( M ) the stochastic integral is defined as a H -limit ofstochastic integrals of simple processes. Next, we briefly review stochastic integra-tion with respect to a process of locally finite variation. Let A ∈ V d be arbitrary.Consider a factorization A i = a i · F, where a is an optional R d -valued process, and F ∈ V + . We denote by L var ( A ) bethe set of all R d -valued predictable processes H such that | H · a | · F ∈ V + . For H ∈ L var ( A ) we define the Lebesgue Stieltjes integral (LS) H · A := ( H · a ) · F ∈ V , (A.2)see [39, Sec. 3.2]. Now, we review the stochastic integral with respect to a multi-dimensional semimartingale, as defined in [39, Sec. 3.3]. Let X ∈ S d be arbitrary.A.1. Definition.
A process H is called X -integrable if there exists a decomposition X = M + A with M ∈ M d loc and A ∈ V d such that H ∈ L ( M ) ∩ L var ( A ) . In thiscase, the vector stochastic integral is defined by H · X := (M) H · M + (LS) H · A. The space of X -integrable processes is denoted by L ( X ) . This definition is correct due to the following result.A.2.
Proposition. [39, Cor. 3.11]
Let X ∈ S d . Let X = M + A and X = M ′ + A ′ with M, M ′ ∈ M d loc and A, A ′ ∈ V d be two semimartingale decompositions suchthat H ∈ (cid:0) L ( M ) ∩ L var ( A ) (cid:1) ∩ (cid:0) L ( M ′ ) ∩ L var ( A ′ ) (cid:1) . Then we have (M) H · M + (LS) H · A = (M) H · M ′ + (LS) H · A ′ . For every H ∈ L ( X ) we have H · X ∈ S and ∆( H · X ) = H · ∆ X . Furthermore,note that every predictable, locally bounded process H belongs to L ( X ) .A.3. Lemma.
Let X ∈ S d be arbitrary, let H be a R d -valued predictable process,and let K be a R -valued predictable process. Then the following statements areequivalent: (i) We have H ∈ L ( X ) . (ii) We have ( H, K ) ∈ L (( X, .If the previous conditions are fulfilled, then we have H · X = ( H, K ) · ( X, . (A.3) Proof.
We define the R d +1 -valued semimartingale ˆ X := ( X, . Let X = M + A be an arbitrary semimartingale decomposition of X . Then there is a uniquesemimartingale decomposition ˆ X = ˆ M + ˆ A such that M i = ˆ M i and A i = ˆ A i for i = 1 , . . . , d ; namely ˆ M = ( M, and ˆ A = ( A, . With analogous notation as abovewe obtain ˆ c = (cid:18) c
00 0 (cid:19) and ˆ a = (cid:18) a (cid:19) . This proves the equivalence (i) ⇔ (ii), and by (A.1) and (A.2) we obtain the identity(A.3). (cid:3) A.4.
Lemma.
Let X ∈ S be a semimartingale, and let H, K be two predictable R d -valued processes such that H · K ∈ L ( X ) . Then we have HK ∈ L ( X R d ) andthe identity ( H · K ) · X = ( HK ) · ( X R d ) , where the R d -valued process HK has the components ( HK ) i := H i K i for each i = 1 , . . . , d .Proof. By assumption there exists a decomposition X = M + A with M ∈ M loc and A ∈ V such that H · K ∈ L ( M ) ∩ L var ( A ) , and we have ( H · K ) · X = (M)( H · K ) · M + (LS)( H · K ) · A. Furthermore, there exist optional R -valued processes c and a , and a process F ∈ V + such that [ M, M ] = c · F and A = a · F. Note that X R d = M R d + A R d is a semimartingale decomposition of the R d -valued semimartingale X R d . Let C be the R d × d -valued process given by C ij =[( M R d ) i , ( M R d ) j ] for all i, j = 1 , . . . , d . Then we have C = c R d × d · F and A R d = a R d · F. Furthermore, we have (cid:0) c R d × d · ( HK ) (cid:1) · ( HK ) = c (cid:18) d X i =1 H i K i (cid:19) R d · ( HK )= c d X j =1 (cid:18) d X i =1 H i K i (cid:19) H j K j = c (cid:18) d X i =1 H i K i (cid:19) = c ( H · K ) , and hence HK ∈ L ( M R d ) . Moreover, we have ( H · K ) a = a d X i =1 H i K i = ( HK ) · a R d , and hence HK ∈ L var ( A R d ) . It remains to prove that the respective integralscoincide. Concerning the finite variation part, we have (LS)( H · K ) · A = ( H · K ) a · F = (cid:0) ( HK ) · a R d (cid:1) · F = (LS)( HK ) · ( A R d ) . HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 27
Regarding the local martingale part, we may assume that H and K are simpleprocesses of the form H = H { } + m X k =1 H k ( τ k ,τ k +1 ] ,K = K { } + m X k =1 K k ( τ k ,τ k +1 ] with R d -valued and F τ k -measurable random variables H k , K k for k = 0 , . . . , m .Then we have H · K = ( H · K ) { } + m X k =1 ( H k · K k ) ( τ k ,τ k +1 ] , and hence (M)( H · K ) · M = m X k =1 ( H k · K k )( M τ k +1 − M τ k ) = m X k =1 (cid:18) d X i =1 H ik K ik (cid:19) ( M τ k +1 − M τ k )= d X i =1 m X k =1 ( H ik K ik )( M τ k +1 − M τ k ) = (M)( HK ) · ( M R d ) . Summing up, we obtain ( H · K ) · X = (M)( H · K ) · M + (LS)( H · K ) · A = (M)( HK ) · ( M R d ) + (LS)( HK ) · ( A R d ) = ( HK ) · ( X R d ) , which concludes the proof. (cid:3) A.5.
Lemma. [39, Lemma 5.6]
For all X ∈ M dσ and H ∈ L ( X ) we have H · X ∈ M σ . Recall that a process X is called admissible if X ≥ − a for some a ∈ R + .A.6. Lemma. [1, Cor. 3.5]
For every admissible process X ∈ M σ we have X ∈ M loc . As the next results show, the stochastic integral ( H, X ) H · X is bilinear.A.7. Theorem. [39, Thm. 4.1]
Let X , X ∈ S d . Furthermore, let H ∈ L ( X ) ∩ L ( X ) and α , α ∈ R . Then we have H ∈ L ( α X + α X ) and H · ( α X + α X ) = α ( H · X ) + α ( H · X ) . A.8.
Theorem. [39, Thm. 4.3]
Let X ∈ S d . Furthermore, let H , H ∈ L ( X ) and α , α ∈ R . Then we have α H + α H ∈ L ( X ) and ( α H + α H ) · X = α ( H · X ) + α ( H · X ) . Concerning the associativity of the stochastic integral, we have two results.A.9.
Theorem. [39, Thm. 4.6]
Let X ∈ S d and H ∈ L ( X ) be arbitrary, and let K be a R -valued predictable process. Then the following statements are equivalent: (i) We have K ∈ L ( H · X ) . (ii) We have KH ∈ L ( X ) . If the previous conditions are fulfilled, then we have K · ( H · X ) = ( KH ) · X. A.10.
Theorem. [39, Thm. 4.7]
Let X ∈ S d be arbitrary, and let H be a R d -valued process such that H i ∈ L ( X i ) for each i = 1 , . . . , d . We define the R d -valuedprocess Y as Y i := H i · X i for each i = 1 , . . . , d . Furthermore, let K be a R d -valued predictable process. We define the R d -valued process J as J i := K i H i foreach i = 1 , . . . , d . Then the following statements are equivalent: (i) We have K ∈ L ( Y ) . (ii) We have J ∈ L ( X ) .If the previous conditions are fulfilled, then we have K · Y = J · X. A.11.
Corollary.
Let X ∈ S d and H ∈ L ( X ) be arbitrary. Let K be a R -valuedpredictable, locally bounded process. Then we have K ∈ L ( H · X ) , KH ∈ L ( X ) , H ∈ L ( K · X ) and the identities K · ( H · X ) = ( KH ) · X = H · ( K · X ) , where K · X denotes the R d -valued process with components ( K · X ) i := K · X i foreach i = 1 , . . . , d .Proof. Since K is predictable and locally bounded, we have K ∈ L ( H · X ) , and byTheorem A.9 we obtain KH ∈ L ( X ) and K · ( H · X ) = ( KH ) · X. Since K is predictable and locally bounded, we also have K ∈ L ( X i ) for each i = 1 , . . . , d . Since KH ∈ L ( X ) , by Theorem A.10 we obtain H ∈ L ( K · X ) and H · ( K · X ) = ( KH ) · X, completing the proof. (cid:3) Concerning the quadratic variation of the stochastic integral, we have the fol-lowing result.A.12.
Theorem. [39, Thm. 4.19]
Let X ∈ S d , Y ∈ S e and H ∈ L ( X ) , K ∈ L ( Y ) be arbitrary. Let F ∈ V + and an optional R d × e -valued processes ρ be such that [ X i , Y j ] = ρ ij · F for all i = 1 , . . . , d and j = 1 , . . . , e .Then we have d X i =1 e X j =1 H i ρ ij K j ∈ L ( F ) and the identity [ H · X, K · Y ] = (cid:18) d X i =1 e X j =1 H i ρ ij K j (cid:19) · F. A.13.
Corollary.
Let X ∈ S d , Y ∈ S and H ∈ L ( X ) be arbitrary. Then we have H ∈ L ([ X, Y ]) and the identity [ H · X, Y ] = H · [ X, Y ] , where [ X, Y ] ∈ V d denotes the R d -valued process with components [ X i , Y ] for each i = 1 , . . . , d . HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 29
Proof.
Using the notation from Theorem A.12, we have e = 1 and K = 1 . Let F ∈ V + and an optional R d -valued processes ρ be such that [ X i , Y ] = ρ i · F, i = 1 , . . . , d.
By Theorem A.12 we obtain H · ρ ∈ L ( F ) , which means that | H · ρ | · F ∈ V + , and hence H ∈ L var ([ X, Y ]) . Furthermore, by Theorem A.12 and (A.2) we have [ H · X, Y ] = ( H · ρ ) · F = H · [ X, Y ] , completing the proof. (cid:3) A.14.
Proposition.
Let X ∈ S d and Y ∈ M σ be such that X i Y ∈ M σ for each i = 1 , . . . , d . Then for every H ∈ L ( X ) we have ( H · X ) Y ∈ M σ .Proof. Let i ∈ { , . . . , d } be arbitrary. Using integration by parts (see [24, Def.I.4.45]) we have X i Y + X i − · Y + Y − · X i + [ X i , Y ] = X i Y ∈ M σ . Since Y ∈ M σ , we have X i − · Y ∈ M σ , and hence Y − · X i + [ X i , Y ] ∈ M σ . By Corollary A.13 we have H ∈ L ([ X, Y ]) and [ H · X, Y ] = H · [ X, Y ] . Since Y − is predictable and locally bounded, by Corollary A.11 we have Y − ∈ L ( H · X ) , Y − H ∈ L ( X ) , H ∈ L ( Y − · X ) and Y − · ( H · X ) = ( Y − H ) · X = H · ( Y − · X ) . Therefore, using integration by parts (see [24, Def. I.4.45]) again, we obtain ( H · X ) Y = ( H · X ) − · Y + Y − · ( H · X ) + [ H · X, Y ]= ( H · X ) − · Y + H · ( Y − · X ) + H · [ X, Y ]= ( H · X ) − · Y + H · ( Y − · X + [ X, Y ]) ∈ M σ , completing the proof. (cid:3) A.15.
Proposition.
Let X ∈ S d and Y ∈ S be such that X i Y ∈ M σ for each i = 1 , . . . , d . Then for every H ∈ L ( X ) with H · X = H · X + H · X (A.4) we have ( H · X ) Y ∈ M σ .Proof. By (A.4) we have H · X ∈ S . Using integration by parts (see [24, Def.I.4.45]) we have ( H · X ) Y = ( H · X ) Y + ( H · X ) − · Y + Y − · ( H · X ) + [ H · X, Y ] . (A.5)By (A.4) and Corollary A.13 we have H ∈ L ([ X, Y ]) and [ H · X, Y ] = [ H · X, Y ] = H · [ X, Y ] . Furthermore, we have ( H · X ) − = H · X − ∆( H · X ) = H · X − ∆( H · X )= H · X − H · ∆ X = H · X − . Hence, by Lemma A.4 and Theorem A.10 we have HX − ∈ L ( Y ) , H ∈ L ( X − · Y ) and ( H · X ) − · Y = ( H · X − ) · Y = ( HX − ) · ( Y ) = H · ( X − · Y ) , (A.6)where X − · Y denotes the R d -valued process with components ( X − · Y ) i = X i − · Y for each i = 1 , . . . , d . Furthermore, by Corollary A.11 we have H ∈ L ( Y − · X ) and Y − · ( H · X ) = Y − · ( H · X ) = H · ( Y − · X ) , (A.7)where Y − · X denotes the R d -valued process with components ( Y − · X ) i = Y − · X i for each i = 1 , . . . , d . Consequently, using (A.5), (A.6), (A.7) and integration byparts (see [24, Def. I.4.45]) again, we deduce that ( H · X ) Y = ( H · X ) Y + H · (cid:0) X − · Y + Y − · X + [ X, Y ] (cid:1) = ( H · X ) Y + H · ( XY ) ∈ M σ , where XY ∈ M dσ denotes the local martingale with components X i Y for each i = 1 , . . . , d . (cid:3) Appendix B. Further auxiliary results about stochastic processes
In this appendix we provide further results from the theory of stochastic pro-cesses.B.1.
Lemma.
Every admissible local martingale X ∈ M loc is a supermartingale.Proof. Let ( T n ) n ∈ N be a localizing sequence for X , and let ≤ s ≤ t be arbitrary.By Fatou’s lemma for conditional expectations of random variables being boundedfrom below we obtain E [ X t | F s ] = E h lim n →∞ X t ∧ T n | F s i ≤ lim inf n →∞ E [ X t ∧ T n | F s ]= lim inf n →∞ X s ∧ T n = X s , which concludes the proof. (cid:3) We will use the following version of Doob’s optional sampling theorem.B.2.
Theorem.
Let X be an admissible supermartingale. For two finite stoppingtimes S ≤ T we have X S , X T ∈ L and E [ X T ] ≤ E [ X S ] . Proof.
This is a consequence of [24, Thm. I.1.39]. (cid:3)
B.3.
Lemma.
Let M ∈ M loc be a local martingale with M, M − > , and let A be apredictable process of locally finite variation with A, A − > and A = 1 such that M A ∈ M loc . Then we have A = 1 up to an evanescent set.Proof. Using integration by parts (see [24, Def. I.4.45]) we have
M A = M A + M − · A + A − · M + [ M, A ] . By [24, Prop. I.4.49.c] we have [ M, A ] ∈ M loc , and hence M − · A ∈ M loc ∩ V . Sincethis process is also predictable, by [24, Cor. I.3.16] we deduce that M − · A = 0 .Since M − > and A = 1 , it follows that A = 1 up to an evanescent set. (cid:3) HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 31
Appendix C. Stochastic processes in discrete time
In this appendix we provide the required results about stochastic processes indiscrete time. Let (Ω , F , ( F n ) n ∈ N , P ) be a filtered probability space with discretetime filtration such that F = { Ω , ∅} . As shown in [24, page 14], the discrete timesetting can be regarded as a particular case of the continuous time setting, whichwe have considered so far. Let X be an R d -valued adapted process. Note that every R d -valued predictable process H belongs to L ( X ) , and that the stochastic integral H · X = ( H · X n ) n ∈ N is given by H · X = 0 ,H · X n = n X k =1 H k · ( X k − X k − ) , n ∈ N . Let X be a process of the form X = X + H · M for a predictable R d -valued process H and an adapted R d -valued process M . If M is a martingale, then X is called a d -martingale transform , and if M is a localmartingale, then X is called a d -local martingale transform . An adapted R -valuedprocess X is called a generalized martingale if for each n ∈ N we have P -almostsurely E [ X n | F n − ] = X n − , (C.1)where we use the generalized conditional expectation ; see [24, I.1.1]. Note that (C.1)automatically implies that for each n ∈ N we have P -almost surely E [ | X n | | F n − ] < ∞ . Note that an adapted R -valued process X is a generalized martingale if and only iffor all n, k ∈ N with k ≤ n we have P -almost surely E [ X n | F k ] = X k . (C.2)Furthermore, a generalized martingale X is a martingale if and only if X n ∈ L for all n ∈ N .C.1. Theorem.
For every adapted process R -valued process X the following state-ments are equivalent: (i) We have X ∈ M loc . (ii) X is a generalized martingale. (iii) X is a d -martingale transform for all d ∈ N . (iv) X is a d -martingale transform for some d ∈ N . (v) X is a d -local martingale transform for all d ∈ N . (vi) X is a d -local martingale transform for some d ∈ N . (vii) We have X ∈ M σ .Proof. This is a consequence of [23, Thm. 1] and [24, Thm. III.6.41]. (cid:3)
C.2.
Lemma.
Let X be a local martingale such that P -almost surely X = 0 and X n ≥ for some n ∈ N . Then we have P -almost surely X k = 0 for all k = 0 , . . . , n .Proof. Noting that F = { Ω , ∅} , by Theorem C.1 and identity (C.2) we have P -almost surely E [ X n ] = E [ X n | F ] = X = 0 . Since X n ≥ , we deduce that P -almost surely X n = 0 . Using identity (C.2) again,we obtain P -almost surely X k = E [ X n | F k ] = 0 for all k = 0 , . . . , n . (cid:3) C.3.
Lemma.
Every nonnegative local martingale X is a martingale.Proof. Let n ∈ N be arbitrary. By Theorem C.1 and identity (C.2) we have P -almostsurely E [ X n ] = E [ X n | F ] = X < ∞ , and hence X n ∈ L . Therefore, the process X is a martingale. (cid:3) References [1] J. P. Ansel and C. Stricker. Couverture des actifs contingents.
Ann. Inst. Henry Poincaré ,30(2):303–315, 1994.[2] D. Á. Bálint and M. Schweizer. Large financial markets, discounting, and no asymptoticarbitrage.
Theory Probab. Appl. , 2019. to appear.[3] D. Á. Bálint and M. Schweizer. Making no-arbitrage discounting-invariant: A new FTAPbeyond NFLVR and NUPBR. Swiss Finance Institute Paper No. 18–23, 2019.[4] D. Á. Bálint and M. Schweizer. Properly discounted asset prices are semimartingales. SwissFinance Institute Paper No. 19–53, 2019.[5] T. Choulli and C. Stricker. Deux applications de la décomposition de Galtchouck–Kunita–Watanabe. In J. Azéma, editor,
Séminaire de Probabilités XXX , volume 1626 of
LectureNotes in Mathematics , pages 12–23, Berlin, 1996. Springer.[6] M. M. Christensen and K. Larsen. No arbitrage and the growth optimal portfolio.
StochasticAnal. Appl. , 25(1):255–280, 2007.[7] C. Cuchiero and J. Teichmann. A convergence result for the Emery topology and a variant ofthe proof of the fundamental theorem of asset pricing.
Finance Stoch. , 19(4):743–761, 2015.[8] R. C. Dalang, A. Morton, and W. Willinger. Equivalent martingale measures and no-arbitragein stochastic securities market model.
Stoch. Stoch. Rep. , 29(2):185–201, 1990.[9] F. Delbaen and W. Schachermayer. A general version of the fundamental theorem of assetpricing.
Math. Ann. , 300(3):463–520, 1994.[10] F. Delbaen and W. Schachermayer. The no-arbitrage property under a change of numéraire.
Stoch. Stoch. Rep. , 53(3-4):213–226, 1995.[11] F. Delbaen and W. Schachermayer. The fundamental theorem of asset pricing for unboundedstochastic processes.
Math. Ann. , 312:215–250, 1998.[12] F. Delbaen and W. Schachermayer.
The Mathematics of Arbitrage . Springer Finance.Springer, Berlin, 2008.[13] K. Du and E. Platen. Benchmarked risk minimization.
Math. Finance , 26(3):617–637, 2016.[14] H. Föllmer and A. Schied.
Stochastic Finance . De Gruyter, 4th edition edition, 2016.[15] C. Fontana. No-arbitrage conditions and absolutely continuous changes of measure. InC. Hillairet, M. Jeanblanc, and Y. Jiao, editors,
Arbitrage, Credit and Informational Risks ,volume 97, pages 3–18. World Scientific, 2014.[16] C. Fontana. Weak and strong no-arbitrage conditions for continuous financial markets.
Int.J. Theor. Appl. Finance , 18(1):1550005, 2015.[17] C. Fontana and W. J. Runggaldier. Diffusion-based models for financial markets withoutmartingale measures. In F. Biagini, A. Richter, and H. Schlesinger, editors,
Risk Measuresand Attitudes , EAA Series, pages 45–81. Springer, 2013.[18] J. M. Harrison and D. M. Kreps. Martingales and arbitrage in multiperiod securities markets.
J. Econ. Theory , 20(3):381–408, 1979.[19] J. M. Harrison and S. R. Pliska. Martingales and stochastic integrals in the theory of contin-uous trading.
Stoch. Proc. Appl. , 11(3):215–260, 1981.[20] M. Herdegen. No-arbitrage in a numéraire-independent modeling framework.
Math. Finance ,27(2):568–603, 2017.[21] H. Hulley and M. Schweizer. M – On minimal market models and minimal martingalemeasures. In C. Chiarella and A. Novikov, editors, Contemporary Quantitative Finance.Essays in Honour of Eckhard Platen , pages 35–51. Springer, 2010.[22] P. Imkeller and N. Perkowski. The existence of dominating local martingale measures.
FinanceStoch. , 19(4):685–717, 2015.[23] J. Jacod and A. N. Shiryaev. Local martingales and the fundamental asset pricing theoremsin the discrete-time case.
Finance Stoch. , 2:259–273, 1998.[24] J. Jacod and A. N. Shiryaev.
Limit Theorems for Stochastic Processes . Number 288 in Grund-lagen der mathematischen Wissenschaften. Springer, Berlin, second edition, 2003.
HE FUNDAMENTAL THEOREM FOR SELF-FINANCING PORTFOLIOS 33 [25] R. A. Jarrow.
Continuous-time Asset Pricing Theory . Springer Finance. Springer, Cham,2018.[26] Y. M. Kabanov. On the FTAP of Kreps-Delbaen-Schachermayer. In
The Liptser Festschrift ,Statistics and Control of Random Processes, pages 191–203. Steklov Mathematical Institute,1997.[27] Y. M. Kabanov, C. Kardaras, and S. Song. No arbitrage of the first kind and local martingalenuméraires.
Finance Stoch. , 20(4):1097–1108, 2016.[28] Y. M. Kabanov and C. Stricker. A teacher’s note on no-arbitrage criteria. In J. Azéma, editor,
Séminaire de Probabilités XXXV , volume 1755 of
Lecture Notes in Mathematics , pages 149–152, Berlin, 2002. Springer.[29] I. Karatzas and C. Kardaras. The numéraire portfolio in semimartingale financial models.
Finance Stoch. , 11(4):447–493, 2007.[30] C. Kardaras. Finitely additive probabilities and the fundamental theorem of asset pricing.In C. Chiarella and A. Novikov, editors,
Essays in Honour of Eckhard Platen , volume 97 of
Contemporary Quantitative Finance , pages 19–34, Berlin, 2010. Springer.[31] C. Kardaras. Market viability via absence of arbitrage of the first kind.
Finance Stoch. ,16(4):651–667, 2012.[32] J. Mancin and W. J. Runggaldier. On the existence of martingale measures in jump diffusionmarket models. In C. Hillairet, M. Jeanblanc, and Y. Jiao, editors,
Arbitrage, Credit andInformational Risks , volume 5 of
Peking University Series in Mathematics , pages 29–51,Singapore, 2014. World Scientific Publishing.[33] E. Platen and D. Heath.
A Benchmark Approach to Quantitative Finance . Springer Finance.Springer, Berlin, 2010.[34] E. Platen and S. Tappe. No-arbitrage concepts in topological vector lattices. (https://arxiv.org/abs/2005.04923) , 2020.[35] P. Protter and K. Shimbo. No arbitrage and general semimartingales. In
Markov Processesand related Topics: A Festschrift for Thomas G. Kurtz , volume 4 of
Proceedings of Symposiain Applied Mathematics , pages 267–283. Institute of Mathematical Statistics, IMS collections,2008.[36] W. Schachermayer. A Hilbert space proof of the fundamental theorem of asset pricing in finitediscrete time.
Insurance Math. Econom. , 11:249–257, 1992.[37] W. Schachermayer. Martingale measures for discrete time processes with infinite horizon.
Math. Finance , 4(1):25–56, 1994.[38] M. Schweizer. Option hedging for semimartingales.
Stoch. Proc. Appl. , 37(2):339–363, 1991.[39] A. N. Shiryaev and A. S. Cherny. Vector stochastic integrals and the fundamental theoremof asset pricing.
Proc. Steklov Inst. Math. , 237:12–56, 2002.[40] C. Stricker. Arbitrage et lois de martingale.
Ann. Inst. H. Poincaré Probab. Statist. , 26:451–460, 1990.[41] K. Takaoka and M. Schweizer. A note on the condition of no unbounded profit with boundedrisk.
Finance Stoch. , 18(2):393–405, 2014.
University of Technology Sydney, School of Mathematical and Physical Sciences,Finance Discipline Group, PO Box 123, Broadway, NSW 2007, Australia
E-mail address : [email protected] Karlsruhe Institute of Technology, Institute of Stochastics, Postfach 6980,76049 Karlsruhe, Germany
E-mail address ::