Transaction costs, Shortselling Constraints and Taxes : A Unified Approach ∗ Laurence Carassus
†
and Ely`es Jouini
‡
First Version : December 1996 This version : November 1997
Abstract In this paper, we propose a general model for discrete and finite setup. First, we prove an absence of arbitrage result. Then, using this key result, we derive all the well-known no arbitrage theorems on imperfect markets, extending them to the case of assets paying dividends. We also prove some new results on taxes.
Keywords : Arbitrage, market’s imperfections, stochastic process, martingale, Farkas’s Lemma.
1
Introduction
The valuation of contingent claims is prominent in the theory of modern finance. It has been initiated by the well known works of Black-Scholes (1973) and Merton (1971). Later the papers of Harisson-Kreps (1979), HarissonPliska (1981), Kreps (1981) and Dalang-Morton-Willinger (1990) formalize the theory of pricing by arbitrage in frictionless markets. Loosely speaking, an arbitrage opportunity is a way to produce nonnegative wealth with ∗
Abbreviated title : Arbitrage and imperfections CREST, Timbre J 320, 15 Boulevard Gabriel P´eri, 92245 Malakoff Cedex, FRANCE, CEREMADE, Universit´e de Paris 9 Dauphine and CERMSEM Universit´e de Paris 1 †
‡
CREST, CERMSEM Universit´e de Paris 1, Laboratoire d’Econom´etrie de l’Ecole Polytechnique (Paris) and Laboratoire d’Economie et de Gestion de l’Ecole Polytechnique de Tunisie
1
positive expected value out of nothing. The fundamental Theorem of asset pricing (FTAP) state as follows : there is no arbitrage opportunity if and only of there exists an equivalent probability measure which turns the price process (appropriately renormalized) into a martingale. This result holds for a frictionless framework. An important body of literature has been developed to take frictions into account. In the transaction costs framework, the fundamental idea was introduce by Bensaid-Lesne-Pag`es-Scheinkman (1992) : pricing by replication is not optimal. Instead of replicating strategies, they consider dominating ones and prove that they can be costless. Using this domination price concept, Jouini-Kallal (1995 a) provide an extension of the FTAP to the transaction costs case using a stronger version of the no-arbitrage concept : the no free lunch one. More precisely, they prove that the absence of free lunch is equivalent to the existence of at least an equivalent probability measure that transforms some processes between the bid and the ask price processes of traded securities into a martingale. In the shortselling constraints framework, Jouini-Kallal (1995 b) provided an extension of the FTAP. If shortselling is possible but costly, they show that a securities price system is arbitrage free if and only if there exist a numeraire and an equivalent probability measure for which the normalized (by the numeraire) price processes of traded securities that cannot be sold short are supermartingale and the normalized price processes of the securities that cannot be hold in nonpositive quantities are submartingale. Sch¨ urger (1996) proves an extension of the FTAP considering the no-arbitrage condition and he obtains the existence of τ -martingale measure. Finally, in the taxe’s framework, Ross (1987) extends the Arbitrage Pricing Theory (1976) to taxe’s set-up. He considers a local version of arbitrage, and under technical conditions he obtains an absence of arbitrage theorem. Nevertheless, he studies only a two period model, and the problem becomes much more complicate in a multiperiod framework. The tax payment value depends on the whole history of investment. In this paper, we focus on the martingale approach of the valuation by arbitrage and we study the previous different kind of frictions : transaction costs, no shortselling, shortselling costs and taxes. In section 2, we present a general model for discrete and finite financial markets. The key idea is to represent every asset by its cash-flow as in Dermody-Rockaffelar (1991, 1995), Cantor-Lippman (1985, 1995), Milne (1996), Adler-Gale (1997) and Carassus-Jouini (1995, 1997). We consider a large family of cash-flows, including the strategy’s randomness in the asset. For example, consider a stock described by the price process (St )0≤t≤T , where T is the time horizon and paying dividends (Dt )1≤t≤T −1 . If an investor buys at time zero a share 2
of stock and and sells it at time T , the associated cash-flow sequence Φ will be : Φ0 = −S0 , Φt = Dt for all t ∈ {1, ..., T − 1}, and ΦT = ST . Notice that we include the asset price in the cash flow sequence. Now considering all the possible buying and selling dates, we obtain a family of cash-flows, which represents the stock. In section 2, we prove an absence of arbitrage result for cash flows sequences, using an adapted version of Farkas’s Lemma. Then, in section 3, we apply this result to financial models. Successively, we study the case of complete and incomplete markets, shortselling constraints and transaction costs. We find again the literature’s classical results, extending them to the case of assets paying dividends. In addition of the unified approach, the proofs are particularly simple. Moreover, this approach allows us to treat other kind of imperfections, as taxes.
2
The general model
We work in a discrete and finite setting. We suppose that there is only a finite number of observations, indexed by the set T = {0, 1, ..., T } of a financial world. This system can only be in a finite number of states, denoted by the set Ω. The information is modeled by a filtration IF = {Ft , t ∈ T }. We suppose that F0 = ∅ and that FT contains the whole information. Every asset is represented through its cash-flow. We suppose that the model contains N assets. Each asset ι has a finite time horizon Tι , where Tι is less or equal to T . The asset ι will be described by a stochastic process, Φι . More precisely, Φιt (ω) is the cash-flow received (positively or negatively) from the investment ι at time t and in state ω. We suppose that each Φι is adapted to the filtration IF . The investor is allowed to subscribe in every nonnegative fraction of a cash-flow (shortselling constraints). A strategy is a sequence (λι )ι=0,...,N of nonnegative numbers. This sequence is chosen at time zero, and does not contain any uncertainty. First, the investor chooses if she/he want or not to subscribe to a cash flow sequence. Then, she/he chooses the number of subscription to the selected sequence. Therefore, in this model the whole uncertainty is taken into account by considering a large family of cash flows, sufficiently large to cover all possible strategies choice. The payoff at time PN λι Φιt (ω). t, in state ω associated to the strategy (λι )ι=0,...,N is equal to ι=0 We will say that a cash flow Φ is nonnegative (resp. positive, equal to zero) if for all (t, ω) ∈ T × Ω, Φ(t, ω) ≥ 0 (resp. Φ(t, ω) > 0, Φ(t, ω) = 0). An arbitrage opportunity will be a possibility to choose a strategy leading to a nonnegative payoff at each date and in each state of the world, and to
3
a positive one in at least one date and one state of the world. Let E be the expectation operator under the uniform probability π, the next theorem characterizes the absence of arbitrage opportunity. Theorem 2.1 If the assets Φι can not be sold short, the absence of arbitrage is equivalent to the existence of a positive process H with H0 = 1, such that for all asset ι, X E[Ht Φιt ] ≤ 0. t∈T
Notice that Theorem 2.1 works without assuming the existence of a numeraire. In order to prove Theorem 2.1, we need an adapted version of Farkas’s Lemma, which proof is carried out later. Let us define α IR+ := {x = (xi ) ∈ IRα /xi ≥ 0, for i = 1, . . . α}
and α := {x = (xi ) ∈ IRα /xi > 0, for i = 1, . . . α}. IR++
Theorem 2.2 If Z is a set of vectors in IRα then exactly one of the following two alternatives must occur. 1. There is a nonnegative linear combination of vectors of Z which belongs α and is not equal to zero. to IR+ α 2. There exists a vector of IR++ which makes a nonpositive scalar product with all elements of Z. Proof of Theorem 2.1 : Let Z be the set of IF -adapted processes Φι . Thus, the set Z is included in IRT ×Ω , and Z can be identified to a sub-vectorspace of IR(T +1)|Ω| . The absence of arbitrage opportunities implies that the first alternative of Theorem 2.2 can not hold. Hence, there exists a positive process H ∗ , such that for ι = 1, . . . N , X Ht∗ (ω)Φιt (ω) ≤ 0. t∈T ,ω∈Ω
Let Ht = E[Ht∗ /Ft ], H is a positive IF -adapted process. Recalling that Φι is IF -adapted, we find that, X
t∈T
E[Ht Φιt ] =
X
E[Ht∗ Φιt ] =
t∈T
≤ 0.
4
X 1 H ∗ (ω)Φιt (ω) |Ω| t∈T ,ω∈Ω t
The process H is positive and we can normalize H by the condition H0 = 1. Conversely, suppose that there exists a positive process H such that for P ι = 1, . . . N , t∈T E[Ht Φιt ] is nonpositive. Now, suppose that there exists an arbitrage opportunity. Let (λι ) be one of the nonnegative strategy leading to a nonnegative and non-zero payoff. If we consider the expected value of the product of this payoff with H, we find a positive number. It is equal P P ι to N t∈T E[Ht Φt ], but this quantity is nonpositive by assumption. ι=1 λι
In Theorem 2.1, we require that no investment can be sold. The world ”sold” is unappropriated in this context because an investor facing a cash flow sequence have only two choices : subscribing or not. If she/he subscribes to the sequence, she/he must keep it until the end of the sequence. In this context, it is equivalent to sell an asset or to subscribe at the opposite of the asset’s cash flow sequence. So if the shortselling assumption for some cash-flow sequence does no correspond to an economic reality, we can avoid it assuming that the model contains the opposite of the cash flow sequence. This is done in the following corollary. Corollary 2.1 If there is no shortselling constrains on the assets Φι , the absence of arbitrage is equivalent to the existence of a positive process H with H0 = 1, such that for all asset i, X
E[Ht Φtι ] = 0.
t∈T
Proof of Corollary 2.1: Apply Theorem 2.1 to the cash flow Φι and −Φι . Proof of Theorem 2.2: We introduce the following notations : Z = {z1 , ..., zp }, and vect+ Z = P { pι=1 λι zι /λι ∈ IR+ , ι ∈ {1, ..., p}}. Suppose that the property 1 is not satisfied, then α vect+ Z ∩ IR+ = {0}. α/ Denoting by ∆α the simplex of IRα , that is, ∆α = {x = (xi ) ∈ IR+ 1},we find that, vect+ Z ∩ ∆α = ∅.
Pn
i=1 xi
=
It is well known that the set vect+ Z is a closed, convex cone of IRα (see Farkas’s Lemma) and that the simplex is a convex, compact set of IRα .
5
Hence using Minkowski Theorem, there exist h ∈ IR∗α , and b1 , b2 ∈ IR such that, < h, z >≤ b1 < b2 ≤< h, s >, ∀z ∈ vect+ Z, ∀s ∈ ∆α .
Notice that if z belongs to vect+ Z, then for all positive λ, λz also belongs to vect+ Z, and therefore λ < h, z >≤ b1 . Thus, b1 can be chosen equal to 0 and we obtain for all z ∈ vect+ Z that < h, z >≤ 0. Now choosing successively α , and for s the vectors of the canonical basis of IRα , we find that h ∈ IR++ the property 2 is proven. It remains to prove that properties 1 and 2 are incompatible. To do that suppose that they occur simultaneously. For all ι from 1 to p, < h, zι >≤ 0. Taking a linear combination with nonnegative coefficient, we find that P P α and h ∈ IRα , we find < h, pι=1 λι zι >≤ 0. Recalling that pι=1 λι zι ∈ IR+ ++ Pp that < h, ι=1 λι zι >> 0. This contradiction ends the proof of theorem 2.2.
3
Applications to imperfect markets
As previously done, we denote by T = {0, 1, ..., T } the set of dates and by Ω the states of the world’s set. The information structure is given by the filtration IF = {Ft , t ∈ T }. We assume that F0 is trivial and FT contains the whole information. The market contains N risky assets. We denote by S i the real-valued and IF -adapted stochastic price process of asset i. The random variable Sti (ω) represents the market value of asset i at time t, in the state of the world ω. The market also contains an (N + 1)th asset indexed by 0 and denoted by R. This asset have positive returns between its buying date and its expiration date. Without loss of generality, we can choose R0 = 1. We assume that there is no short selling constrains on asset R. Remark 3.1 In particular, if there exists a lending and a borrowing rate which are equal, we can choose for R this common rate. More precisely, let rt+1 be the interest rate between time t and time t + 1. We suppose that rt+1 is Ft -measurable and r is a positive process. A particular case of asset R = 1 + rt+1 . The absence of short selling constraints on R is given by RRt+1 t implies that one can borrow and lend at the same rate r. The key idea of the paper is to represent an asset by the cash-flows that it may generate. We will use simple strategies, that is strategies where there is only one buying date and one selling date. If an investor want to follow a strategy involving several dates of trading, she/he will make linear 6
combinations with nonnegative coefficients of simple strategies. We will give later some examples. The set I, which inedexs the set of strategies, will contain the following informations. It points out the considered asset i, a date t and an event A ∈ Ft . The event A conditions buying and selling times τ1 and τ2 , which are stopping times occurring after t. More precisely, let I = {ι = (i, t, A, τ1 , τ2 )/i = 1, . . . , N, t ∈ T , A ∈ Ft , (τ1 , τ2 ) ∈ St,T × St,T }, where St,T is the class of stopping time τ, such that t ≤ τ ≤ T. A simple strategy (λι )ι=(i,t,A,τ1 ,τ2 )∈I is a sequence of non negative number λι . Each λι represents the number of asset i bought at time τ1 if the event A ∈ Ft occurs and sold at τ2 always if A occurs. We will see that if there is no shortselling constraints, it is not necessary to precise if the buying date takes place before the selling date or not. Taking into account the new investments ι instead of i allows us to transfer the randomness of the strategy in the classical sense in the investment. In the next section, we will precise the exact form of the investments (Φι )ι∈I depending from the imperfection under consideration. As in the previous section, the payoff associated to a strategy (λι )ι∈I is P equal to ι∈I λι Φι . We keep the same definition for an arbitrage opportunity, i.e. the choice of a strategy leading to a nonnegative payoff, which is positive in at least one date and one state of the world. First, we present a consequence of Theorem 2.1 to the particular case of a financial market containing the asset R. The representation of asset R is the following : let ι = (0, t, A, τ1 , τ2 ) ∈ I, Φιt (ω) = −Rt (ω)IA (ω)Iτ1 (t, ω) + Rt (ω)IA (ω)Iτ2 (t, ω). We have used the classical notations : IA (ω) = 1 if ω ∈ A, and zero else; Iτ1 (t, ω) = 1 if τ1 (ω) = t, and zero elsewhere. The cash-flow Φι corresponds to the following strategy : buy a unit of asset i at τ1 if the event A occurs and sell it at τ2 always if A occurs. The absence of shortselling constraints on R implies that −Φ(0,t,A,τ1 ,τ2 ) also belongs to the model. Note that −Φ(0,t,A,τ1 ,τ2 ) = Φ(0,t,A,τ2 ,τ1 ) and the absence of shortselling constrains appears then to be equivalent to the absence of ordering conditions between the buying and the selling date. Recall that the particular assumption on the returns of R implies that R is positive. Theorem 3.1 If the market contains an asset R with positive returns and if the assets Φι can not be sold short, the absence of arbitrage is equivalent 7
to the existence of an equivalent probability π 0 , such that, that for all asset P Φtι i, t∈T Eπ0 [ Rt ] is nonpositive. We deduce immediately the following corollary.
Corollary 3.1 If there are no shortselling constraints, the absence of arbitrage is equivalent to the existence of an equivalent probability π 0 , such that, P Φtι that for all asset i, t∈T Eπ0 [ Rt ] is equal to zero.
Proof of Theorem 3.1 : Applying Theorem 2.1 to Φ0,0,Ω,0,T (buy one unit of R at time 0 and sell it at time T ) and to Φ0,0,Ω,T,0 (sell one unit of asset R at time 0 and buy it at time T ), we find that 1 = E[H0 ] = E[HT RT ]. Let π 0 be the probability defined by π 0 (A) = E[HT RT IA ], for all A ∈ FT . The process H and R are positive and thus π 0 and π are equivalent. Now applying Theorem 2.1 to Φ0,t,A,t,T (buy of one unit of R at t if A ∈ Ft and sell it at T if A), and to its opposite, we find that E[(Rt Ht − RT HT )IA ] = 0. From the arbitrariness of A in Ft , we obtain that E[RT HT /Ft ] = Rt Ht . Now consider an investment ι ∈ I, X
E[Φιt Ht ] =
X
E[
Φιt E[RT HT /Ft ]] Rt
E[
Φιt RT HT ], because Rt
t∈T
t∈T
=
X
t∈T
=
X
t∈T
Eπ 0 [
Φιt Rt
is Ft -measurable
Φιt ]. Rt
Then applying Theorem 2.1 or Corollary 2.1 permit then to conclude. In the next, we will always suppose that there exists an asset R with positive returns and without shortselling constraints.
3.1
Complete and incomplete markets
As usually done in the FTAP, we will focus on the existence of an equivalent martingale measure and we do not work on it uniqueness. First, we associate to the following N risky assets some cash-flows indexed by I. Let ι = (i, t, A, τ1 , τ2 ) ∈ I, Φιt (ω) = −Sti (ω)IA (ω)Iτ1 (t, ω) + Sti (ω)IA (ω)Iτ2 (t, ω). 8
The cash-flow Φι corresponds to the following strategy : buy of one unit of asset i at time τ1 if the event A occurs and sell it at time τ2 always if A occurs. Notice that −Φ(i,t,A,τ1 ,τ2 ) = Φ(i,t,A,τ2 ,τ1 ) . There is no shortselling constraints, and hence it is not necessary to specify if the buying date occurs before the selling date or not. Notice that the implicit associated strategies are simple. We can include more complicated strategies, using linear combination with positive coefficients of the preceding cash-flows. For example, if an investor wants to buy a share of asset i at time t1 , sell a half at time t2 , and the other half at time 1 ,Ω,t1 ,t3 1 ,Ω,t1 ,t2 . and a half unit of Φi,t t3 , he will choose a half unit of Φi,t t t We obtain then the classical result : Theorem 3.2 The securities price model is arbitrage free if and only if there exists a martingale measure π 0 such that every renormalized price process is a martingale with respect to the filtration IF and the probability measure π 0 , that is for all t ∈ T , Si Sti = Eπ0 [ t+1 /Ft ]. Rt Rt+1 Proof of Theorem 3.2 : If we apply corollary 3.1 to Φi,T,A,t,t+1 (and Φi,t,A,t+1,t ) for all t ∈ T and for all A ∈ Ft , we find that, Eπ0 [(− Recalling that find that,
P
t∈T
is Ft -measurable and A is a arbitrarily chosen in Ft , we Si Sti = Eπ0 [ t+1 /Ft ]. Rt Rt+1
Conversely, we prove that if h i Φιt Rt
Si R
is a π 0 -martingale then for all ι ∈ I,
Eπ0 = 0, which allows us to conclude from Corollary 3.1. Let ι ∈ I,
t∈T
X
Sti Rt
Si Sti + t+1 )IA ] = 0. Rt Rt+1
Eπ 0
ι Φ t
Rt
=
X
t∈T
Eπ 0 "
"
−Sti IA Iτ1 (t, .) + Sti IA Iτ2 (t, .) Rt #
#
−Sτi1 IA Sτi2 IA + , = Eπ0 R τ1 R τ2 from the definition of the conditional expectation and from A ∈ Ft , 9
"
= Eπ0 −Eπ0
"
recalling that = Eπ0
"
#
#
"
"
#
Sτi2 Sτi1 /Ft IA + Eπ0 Eπ0 /Ft IA Rτ1 Rτ2 S R
#
is a martingale under π 0 ,
Si Si − t IA + t IA Rt Rt
#
= 0
In the context of Remark 3.1, Sti =
3.2
Rt Rt+1
=
1 1+rt+1
and we find that,
1 i Eπ0 [St+1 /Ft ]. 1 + rt+1
Shortselling Constraints
In this context, the investor is not allowed to sell assets that she/he does not own. We obtain the following Theorem : Theorem 3.3 The securities price model is arbitrage free if and only if there exists an equivalent probability measure π 0 such that each renormalized price process is a supermartingale with respect to the filtration IF and the probability measure π 0 , that is for all t ∈ T , Si Sti ≥ Eπ0 [ t+1 /Ft ]. Rt Rt+1 Proof of Theorem 3.3: We use the same kind of representation for the N risky assets as in the previous section, but now the shortselling impossibility imposes that the investor must buy before he sells. Let ι = (i, t, A, τ1 , τ2 ) ∈ I, where τ1 ≤ τ2 , Φιt (ω) = −Sti (ω)IA (ω)Iτ1 (t, ω) + Sti (ω)IA (ω)Iτ2 (t, ω). Applying Theorem 3.1 to Φi,t,A,t,t+1 (buy one unit of asset i at time t if the event A ∈ Ft occurs and sell i at t + 1 if A occurs), for all t ∈ T and all A ∈ Ft , we find that Eπ0
"
#
Si Si (− t + t+1 )IA ≤ 0. Rt Rt+1
10
Sti is Ft -measurable R t i St+1 Eπ0 Rt+1 /Ft .
Recalling that that,
Sti Rt
≥
and A arbitrarily chosen in Ft , we have
Conversely, let ι = (i, t, A, τ1 , τ2 ) ∈ I, such that τ1 ≤ τ2 , then, X
Eπ0
t∈T
ι Φ t
Rt
=
X
t∈T
Eπ 0 "
"
−Sti IA Iτ1 (t, .) + Sti IA Iτ2 (t, .) Rt #
#
S i IA S i IA = Eπ0 − τ1 + τ2 Rτ1 R τ2 using the definition of the conditional expectation and recalling that A ∈ Ft , " " # # # # " " Sτi1 Sτi2 = Eπ0 −Eπ0 /Ft IA + Eπ0 Eπ0 /Ft IA Rτ1 Rτ2 using again the definition of the conditional expectation and t ≤ τ1 ≤ τ2 , " # # " " # # # " " Sτi2 Sτi1 /Ft IA + Eπ0 Eπ0 Eπ0 /Fτ1 /Ft IA = Eπ0 −Eπ0 Rτ1 R τ2 using the super martingale property of ≤ Eπ 0
"
−Eπ0
"
#
"
Sτi1 Sτi1 /Ft + Eπ0 /Ft Rτ1 Rτ1
S
,
#!R #
IA
≤ 0
In the context of Remark 3.1, we find that, Sti ≥
3.3
h i 1 i Eπ0 St+1 /Ft . 1 + rt+1
Shortselling costs
The investor can now sell assets before buying them, but this opportunity will be costly. We suppose that there exist two real-valued IF -adapted 0 stochastic process, S i and S i . If the investor want to buy one unit of asset i at time t1 , she/he will pay Sti1 , and if she/he sells it at t2 , she/he will receive Sti2 . On the other hand, if she/he sells short one unit of asset i at time 0 t1 , she/he will receive St1i , and when she/he will buy it at t2 , she/he will 0 S pay St2i . With this modelization the return of a long position Stt1 is possibly different of the return of a short position We obtain the following Theorem : 11
St0 2 St0 . 1
2
Theorem 3.4 The securities price model is arbitrage free if and only if there exists an equivalent probability measure π 0 such that each renormalized 0 price process S i (resp. S i ) is a supermartingale (resp. submartingale) with respect to the filtration IF and the probability measure π 0 , that is for all t∈T, 0 0 Si Si Si Si Eπ0 [ t+1 /Ft ] ≤ t and t ≤ Eπ0 [ t+1 /Ft ]. Rt+1 Rt Rt Rt+1 Proof of Theorem 3.4 To take into account the shortselling costs, we consider two families of cash flows. Let ι = (i, t, A, τ1 , τ2 ) ∈ I such that τ1 ≤ τ2 , Φιt (ω) = −Sti (ω)IA (ω)Iτ1 (t, ω) + Sti (ω)IA (ω)Iτ2 (t, ω). 0
0
0
Φtι (ω) = Sti (ω)IA (ω)Iτ1 (t, ω) − Sti (ω)IA (ω)Iτ2 (t, ω).
Notice that if the cash flow Φi,t1 ,A,t1 ,t2 belongs to our portfolio, then between t1 and t2 and in the event A, we owne the ith asset. Therefore, if we sell the asset i at t3 ∈]t1 , t2 [ and then buy it at t4 ∈]t3 , t2 [, both in the event A, we 0 will not sell it short, and we will not use the process S i . To take this case into account, just consider the cash flow Φi,t1 ,A,t1 ,t3 and Φi,t1 ,A,t4 ,t2 (there is 0 no transaction cost). Note that the choice of Φi,t1 ,A,t1 ,t2 and Φ i,t1 ,A,t3 ,t4 is also possible but since it is costly, the agents will note consider it. The proof of Theorem 3.4 is similar as the one of Theorem 3.3.
3.4
Transaction costs
If we assume that there is a bid-ask spread, then we can model this situation by two real-valued IF -adapted stochastic process. We denote by S i the 0 buying price process of the asset i and by S i it selling price. We obtain then the following Theorem : Theorem 3.5 The securities price model is arbitrage free if and only if there 0 exists an equivalent probability measure π 0 , and a process S ∗i between S i and S i , such that the renormalized process S ∗i is a martingale with respect to the filtration IF and the probability measure π 0 , that is for all t ∈ T , S ∗i St∗i = Eπ0 [ t+1 /Ft ]. Rt Rt+1
Proof of Theorem 3.5 : We consider the following family of cash flows indexed by ι = (i, t, A, τ1 , τ2 ) ∈ I in order to represent the N risky assets. 0
Φtι (ω) = −Sti (ω)IA (ω)Iτ1 (t, ω) + Sti (ω)IA (ω)Iτ2 (t, ω). 12
Notice that −Φ(i,t,A,τ1 ,τ2 ) = Φ(i,t,A,τ2 ,τ1 ) . There is no shortselling constraints and it is unnecessary to tell if we have bought or sold first. Using Theorem 3.1 with Φi,t,A,τ1 ,τ2 , we find that, Eπ 0
"
0
#
#
"
Sτi2 Sτi1 /Ft ≤ Eπ0 /Ft . Rτ2 R τ1
(3.1)
Consider the two following processes, 0 0 S˜ti Sτi 0 = max Eπ [ /Ft ], τ ∈St,T Rt Rτ
S˜ti Si = min Eπ0 [ τ /Ft ]. Rt τ ∈St,T Rτ It is straightforward to see from the definition of S˜0 and S˜ that, 0 0 S˜ i ≥ S i and S˜i ≤ S i .
S
0i
Let τ ∗ be an optimal stopping time of ( Rkk )k∈[t,T ] (the stopping times τ ∗ exists since the set St,T is finite). By definition, "
#
0 0 0 Sτi∗ Sτi S˜ i = max Eπ0 /Ft = t . τ ∈St,T Rτ ∗ Rτ Rt
This leads to,
Eπ0
"
#
"
#
"
#
0 0 0 0 S˜ i Sτi S˜ti Sτi∗ /Ft−1 = Eπ0 /Ft−1 ≤ max Eπ0 /Ft−1 = t−1 , τ ∈St−1,T Rt Rτ ∗ Rτ Rt−1
˜0 i
where we have used that τ ∗ ∈ St,T ⊂ St−1,T . The process SR is then a supermartingale with respect to the filtration IF and the probability measure π 0 . ˜i Using the same arguments, SR is a sub-martingale with respect to IF and π0 . If we consider the strategy that consists in buying and selling at the 0 same stopping time τ ∈ St,T , we find that Sτi ≤ Sτi . This is quite intuitive : the selling price is less or equal than the buying price. Next, we will show 0 that S˜ i ≤ S˜i . Let t ∈ [0, T ], and τ1 , τ2 ∈ St,T , recalling Equation (3.1), we have, " 0 # " # Sτi2 Sτi1 Eπ0 /Ft ≤ Eπ0 /Ft . Rτ2 R τ1 13
Taking the maximum on τ2 ∈ St,T in the left-handside of the previous equation, and then the minimum on τ1 ∈ St,T in the right-handside, we find 0 immediately that S˜ti ≤ S˜ti . 0 Finally, we have showed that there exists a process S˜ i (resp. S˜i ) which is after renormalization a supermartingale (resp. submartingale), and such that, 0 0 S i ≤ S˜ i ≤ S˜i ≤ S i . To end the proof, it remains to show that there exists a martingale S ∗i with 0 respect to IF and π 0 such that S˜ i ≤ S ∗i ≤ S˜i . To do that, we use the following lemma, which proof is carried out later.
Lemma 3.1 Let (Ut )t∈T be a supermartingale with respect to the filtration IF and the probability measure π, and (Vt )t∈T be a submartingale with respect to the filtration IF and the probability measure π, such that Ut ≤ Vt for all t ∈ T . Then, there exists a martingale (Mt∗ )t∈T with respect to the filtration IF and the probability measure π such that Ut ≤ Mt∗ ≤ Vt for all t ∈ T . 0 Choosing U i = S˜ i /R and V i = S˜i /R, we find that there exists a martingale M ∗i between U i and V i and we choose S ∗i = M ∗i R. Conversely if we have such a martingale, notice that,
"
#
"
Sτi1 Sτ∗i1 St∗i = Eπ0 /Ft ≤ Eπ0 /Ft Rt Rτ1 Rτ1 Eπ 0
"
t∈T
Eπ 0
ι Φ t
Rt
=
X
t∈T
Eπ 0 "
#
and
#
"
Sτi1 Sτ∗i1 S ∗i /Ft ≤ Eπ0 /Ft = t . R τ1 Rτ1 Rt
Consequently, let ι ∈ I, X
0
#
"
0
−Sti IA Iτ1 (t, .) + Sti IA Iτ2 (t, .) Rt 0
#
#
S i IA S i IA = Eπ0 − τ1 + τ2 Rτ1 R τ2 using the definition of the conditional expectation and A ∈ Ft , # # # # " " " 0 " Sτi2 Sτi1 /Ft IA + Eπ0 Eπ0 /Ft IA = Eπ0 −Eπ0 Rτ1 Rτ2 ≤ Eπ0
"
S ∗i S ∗i − t IA + t IA Rt Rt
≤ 0 14
#
Proof of Lemma 3.1 : We will argue by induction. Using the supermartingale (resp. submartingale) property of the process U (resp. V ) and denoting by E the expected value under π, we find that, E [V1 /F0 ] ≥ V0 and E [U1 /F0 ] ≤ U0 . Let M1λ = λV1 + (1 − λ)U1 , with 0 ≤ λ ≤ 1. i
h
It is straightforward to see that E M1λ /F0 runs throughout the stochastic interval [E [U1 /F0 ] , E [V1 /F0 ]] . An element of a closed, bounded interval is a convex combination of it extremal points, then there exists λ1∗ such that h
λ∗
i
U0 ≤ E M1 1 /F0 ≤ V0 .
(3.2)
max the range of λ∗ such that Equation (3.2) is We denote by λmin 1 , λ1 1 satisfied. Then, λmin λmax U1 ≤ M1 1 ≤ M1 1 ≤ V1 .
At the second step, we find that, λmax
E [V2 /F1 ] ≥ V1 ≥ M1 1
λmin
and E [U2 /F1 ] ≤ U1 ≤ M1 1
.
Let λ be a F1 -measurable random variable taking value in [0, 1] , we consider the following process,
h
M2λ = λV2 + (1 − λ)U2 .
i
Notice that E M2λ /F1 runs throughout the stochastic interval [E [U2 /F1 ] , E [V2 /F1 ]] . Now the stochastic interval
λmax λmin M1 1 , M1 1
is a subset of [U1 , V1 ] , which
is itself included in [E [U2 /F1 ] , E [V2 /F1 ]] , so there exists a F1 measurable random variable λ∗2 (it is build at every node of date 1, only from the informations of date 1) such that λmin
U1 ≤ M1 1
h
λ∗
i
λmax
≤ E M2 2 /F1 ≤ M1 1
≤ V1 .
(3.3)
the stochastic interval of F1 measurable As before, we denote λ2min , λmax 2 ∗ random variable λ2 such that Equation (3.3) is satisfied. 15
We construct by induction pairs of random variables respectively F2 , ..., max ), ..., (λmin , λmax ) such that, for all p ∈ FT −1 measurables, (λhmin 3 , λ3 T T i min , λmax , {0, ..., T − 1} and λ ∈ λp+1 p+1 λmin
Up ≤ Mp p
i
h
λ ≤ E Mp+1 /Fp ≤ Mpλ
max
≤ Vp .
max , Now, we choose a FT −1 -measurable random variable λ∗T ∈ λmin T , λT and we denote by MT∗ = λT VT + (1 − λT )UT ,
we have then that λmin
λmax
T −1 T −1 ≤ E [MT∗ /FT −1 ] ≤ MT −1 UT −1 ≤ MT −1 ≤ VT −1 .
If we denote MT∗ −1 = E [MT∗ /FT −1 ] .
h
i
max Then, there exists an FT −2 -measurable random variable λT −1 ∈ λmin T −1 , λT −1 , such that MT∗ −1 = λT −1 VT −1 + (1 − λT −1 )UT −1 .
Iterating the same reasoning, we construct step after step a martingale M ∗ with respect to IF and π 0 , such that U ≤ M ∗ ≤ V.
3.5
The dividends case
Suppose that every asset i pays dividends. We suppose that every asset pays dividends (possibly equal to zero) from the date following its buying date up to its selling date. Conversely, an investor buying this asset will pay dividends from the date following its buying date up to its buying date. The value Dti (ω) is the dividend paid by asset i in state ω and at time t. We suppose that the process Di is IF -adapted. 3.5.1
Complete and incomplete markets
In the complete and incomplete markets case, our main theorem states as follows : Theorem 3.6 The securities price model is arbitrage free if and only if there exists an equivalent probability measure π 0 such that for all i, that is
16
for all t ∈ T ,
Sti Rt
= Eπ0 [
i i +Dt+1 St+1 /Ft ]. Rt+1
Equivalently, for all t ∈ T , #
"
t+1 t i X St+1 Dui Sti X Dui + = Eπ0 + /Ft , Rt u=0 Ru Rt+1 u=0 Ru Si
P
i
Du t and the process Rtt + u=0 Ru is a martingale with respect to the filtration IF and the probability measure π 0 .
Proof : First, we define the associated family of cash-flows. Let ι = (i, t, A, τ1 , τ2 ) ∈ I, Φιt (ω) = −Sti (ω)IA (ω)Iτ1 (t, ω) + Sti (ω)IA (ω)Iτ2 (t, ω)
+ Dti (ω)Iτ1 <τ2 (ω)I]τ1 ,τ2 ] (t, ω)IA (ω) − Dti (ω)Iτ2 <τ1 (ω)I]τ1 ,τ2 ] (t, ω)IA (ω).
With the notations, Iτ1 <τ2 (ω) = 1 if τ1 (ω) < τ2 (ω) and zero else, and I]τ1 ,τ2 ] (t, ω) = 1 if t ∈]τ1 (ω), τ2 (ω)]. This means that it is possible to buy one unit of asset i if the event A occurs, at the random time τ1 , and to sell it back at the random time τ2 always if the event A occurs. If the buying date τ1 (ω) predecease the buying date τ2 (ω), the investor receive dividends between τ1 (ω) and τ2 (ω), else she/he must pay dividends between τ2 (ω) and τ1 (ω). Notice that −Φ(i,t,A,τ1 ,τ2 ) = Φ(i,t,A,τ2 ,τ1 ) , because there is no short-selling constraints. Now apply Corollary 3.1 to Φi,t,A,t,t+1 (buy at t one unit of i if the event t A ∈ Ft occurs and sell it at t + 1 always if A occurs), we get that, "
#
i i + Dt+1 St+1 Sti /Ft . = Eπ0 Rt Rt+1
Recalling that
Pt
i Du u=0 Ru
is Ft -measurable, #
"
t t+1 i X St+1 Sti X Dui Dui 0 + = Eπ + /Ft . Rt u=0 Ru Rt+1 u=0 Ru
Notice that using the law of the iterated expectations, S0i
= Eπ 0
"
#
T X STi Dti . + RT Rt t=1
17
Conversely, let ι ∈ I, we have that, X
E
π0
t∈T
ι Φ t
Rt
X
= −
Eπ 0 [
−Sti IA Iτ1 (t, .) + Sti IA Iτ2 (t, .) + Dti Iτ1 <τ2 I]τ1 ,τ2 ] (t, .) IA Rt
t∈T Dti Iτ2 <τ1 I]τ2 ,τ1 ] (t, .)
= Eπ0
"
Rt Si Si − τ 1 + τ2 Rτ1 Rτ2
IA ] !
If we remark that,
τ 2 X Dui u=0
Ru
−
!
τ1 X Dui
u=0
Ru
IA + Eπ0
τ2 X
u=τ1
τ
2
1 X Dui Dui Iτ1 <τ2 − Iτ <τ = R R 2 1 u=τ +1 u +1 u 2
Iτ1 <τ2 −
τ 1 X Dui u=0
Ru
−
τ2 X Dui
u=0
Ru
!
Iτ2 <τ1 =
τ2 X Dui
u=0
Ru
−
τ1 X Dui
u=0
Ru
.
Then, we get that, X
Eπ0
t∈T
ι Φ t
Rt
= Eπ 0
"
−
τ1 X Sτi1 Dui + Rτ1 u=0 Ru
!
τ2 X Si Dui + τ2 + Rτ2 u=0 Ru
!
IA
#
using the definition of the expected value and noting that A ∈ Ft , " # " #! # " τ1 τ2 X X Sτi2 Sτi1 Dui Dui + /Ft + Eπ0 + /Ft IA = Eπ 0 −Eπ0 Rτ1 u=0 Ru Rτ2 u=0 Ru i Pt Sti Du u=0 Ru , Rt + ! # t Dui Sti X IA + Rt u=0 Ru
using the martingale property of = Eπ0
"
−
= 0
3.5.2
t Sti X Dui + Rt u=0 Ru
!
+
Shortselling costs 0
There exists two IF -adapted stochastic process, S i and S i as in section 3.2. Using the same line of argument as before, we find that,
18
1 X Dui Dui Iτ1 <τ2 − Iτ2 <τ1 IA R R u u u=τ +1 +1
τ
τ2 X
u=τ1
#
Theorem 3.7 The securities price model is arbitrage free if and only if there exists an equivalent probability measure π 0 such that for all i, that is for all t ∈ T , Eπ 0
3.6
"
#
"
0
#
0
i + Di i i St+1 St+1 + Dt+1 Si Si t+1 /Ft ≤ t and t ≤ Eπ0 /Ft . Rt+1 Rt Rt Rt+1
Proportional transaction costs
As in section 3.3, we model the transaction costs with two IF -adapted 0 stochastic process, S i (buying price) and S i (selling price). In the transaction costs framework, the theorem states as follows, Theorem 3.8 The securities price model is arbitrage free if and only if there exists an equivalent probability measure π 0 and a process S ∗i , which i P 0 S ∗i u lies between S i and S i , and such that the process ( Rtt + tu=0 D Ru )t∈T is a martingale with respect to the filtration IF and the probability measure π 0 . Proof : Let ι = (i, t, A, τ1 , τ2 ) ∈ I, 0
Φtι (ω) = −Sti (ω)IA (ω)Iτ1 (t, ω) + Sti (ω)IA (ω)Iτ2 (t, ω)
+ Dti (ω)Iτ1 <τ2 (ω)I]τ1 ,τ2 ] (t, ω)IA (ω) − Dti (ω)Iτ2 <τ1 (ω)I]τ1 ,τ2 ] (t, ω)IA (ω).
In this case, shortselling is allowed and the cash-flows −Φ(i,t,A,τ1 ,τ2 ) = Φ(i,t,A,τ2 ,τ1 ) is feasible. Applying Theorem 3.1 to Φ(i,t,A,τ1 ,τ2 ) , we find that,
0
τ2 τ1 X X Si Si Dui Dui Iτ1 <τ2 /Ft ≤ Eπ0 τ2 + Iτ <τ /Ft . Eπ0 τ2 + Rτ2 u=τ +1 Ru Rτ2 u=τ +1 Ru 2 1 1
2
If we recall that,
τ
τ2 X
u=τ1
τ
τ
1 2 1 X X X Dui Dui Dui Dui Iτ1 <τ2 − Iτ2 <τ1 = − . R R R R +1 u u=τ +1 u u=0 u u=0 u 2
Then, this leads to, Eπ0
"
0
#
"
#
τ2 τ1 X X Sτi1 Sτi2 Dui Dui + /Ft ≤ Eπ0 + /Ft . Rτ2 u=0 Ru Rτ1 u=0 Ru
19
(3.4)
Let us define S˜ and S˜0 by, "
#
"
#
0 0 t τ S˜ti X Dui Sτi X Dui 0 + = max Eπ + /Ft , τ ∈St,T Rt u=0 Ru Rτ u=0 Ru
We get that,
t τ X Sτi S˜ti X Dui Dui + = min Eπ0 + /Ft . τ ∈St,T Rt u=0 Ru Rτ u=0 Ru
0 0 t t t t S˜ti X Dui Dui Dui Dui Sti X S˜i X Si X + ≥ + and t + ≤ t + . Rt u=0 Ru Rt u=0 Ru Rt u=0 Ru Rt u=0 Ru
S
0i
Let τ ∗ be the stopping time of the process ( Rkk + stopping time τ ∗ exists because St,T is finite).Then,
Pk
i Du u=0 Ru )k∈[t,T ]
(this
#
"
0 0 0 τ∗ τ t X S˜ i X Sτi X Sτi∗ Dui Dui Dui + = max Eπ0 + /Ft = t + . τ ∈St,T Rτ ∗ u=0 Ru Rτ u=0 Ru Rt u=0 Ru
This shows that, Eπ0
"
#
#
"
0 0 0 t t−1 τ∗ X X S˜ i S˜ti X Sτi∗ Dui Dui Dui + + /Ft−1 = Eπ0 /Ft−1 ≤ t−1 + , Rt u=0 Ru Rτ ∗ u=0 Ru Rt−1 u=0 Ru 0 S˜ i
P
i
u where we have notice that τ ∗ ∈ St,T ⊂ St−1,T . The process Rtt + tu=0 D Ru is a super-martingale with respect to the filtration IF and the probability i P S˜i u measure π 0 . Using the same arguments, Rtt + tu=0 D Ru is then a supermartingale with respect to the filtration IF and the probability measure π0. Taking the maximum on τ2 ∈ St,T in the left-handside of Equation (3.4) and then the minimum on τ1 ∈ St,T in the right-handside, it follows that, 0 t t S˜ti X S˜i X Dui Dui + ≤ t + . Rt u=0 Ru Rt u=0 Ru 0 S˜ i
P
The process Rtt + tu=0 (resp. submartingale), and,
i Du Ru
(resp.
S˜ti Rt
+
Pt
i Du u=0 Ru )
is a supermartingale
0 0 t t t t Dui Dui Dui Dui Sti X S˜ti X S˜i X Si X + ≤ + ≤ t + ≤ t + . Rt u=0 Ru Rt u=0 Ru Rt u=0 Ru Rt u=0 Ru
20
Using Lemma 3.1, we find that there exits a martingale Z ∗i with respect to the filtration IF and the probability measure π 0 , such that Zti∗ lies between 0
St i Rt
+
Pt
i Du u=0 Ru
and
Sti Rt
+
Pt
i Du u=0 Ru .
St∗i
:=
"
Denoting by S ∗i the process
Zt∗i
−
t X Dui
u=0
Ru
#
Rt ,
we have proved the implication of the required result. Conversely, let ι = (i, t, A, τ1 , τ2 ) ∈ I, define first the process Z ∗i by, Zt∗i :=
t St∗i X Dui + . Rt R u=0 u
The process Z ∗i is a martingale with respect to the filtration IF and the probability measure π 0 , and we have that, Zti∗
h
= Eπ0
Eπ0
"
Zτi∗1 /Ft
i
≤ Eπ0
"
#
0
τ1 X Sτi1 Dui + /Ft Rτ1 u=0 Ru
#
and
τ1 h i X Sτi1 Dui + /Ft ≤ Eπ0 Zτi∗1 /Ft = Zti∗ . Rτ1 u=0 Ru
As previously done, for all ι ∈ I, X
t∈T
Eπ 0
ι Φ t
Rt
=
X
t∈T
= Eπ0
Eπ 0 "
"
0
S i IA S i IA − τ1 + τ2 R τ1 Rτ2
"
= Eπ0 −Eπ0 ≤ Eπ 0
0
−Sti IA Iτ1 (t, .) + Sti IA Iτ2 (t, .) Rt
"
"
#
#
#
"
"
#
0
#
Sτi1 Sτi2 /Ft IA + Eπ0 Eπ0 /Ft IA Rτ1 Rτ2
S ∗i S ∗i − t IA + t IA Rt Rt
≤ 0
21
#
#
4
Taxes
We focus now on the case of an investor paying taxes on capital gains and receiving tax credits for capital losses. We suppose that those tax payments and tax credits occurs immediately, that is when the investor actually realises those gains and losses by trading shares of stock and not at the end of the year. In order to justify our assumptions, we recall some rules of the French tax code. We refer to Lamorlette-Lamorlette (1995). First, we justify the assumption of credit for capital losses. For an individual, this assumption is not true for a loss on real estate and personal estate. The capital loss can neither be credited on the capital gain from same kind of assets nor from other assets. It must be treat as an unproductive asset. However, for transferable securities, the capital losses can be deducted against capital gains on the same kind of assets, in the same year or in the next five years. This gains could also be deducted against capital gains of the MONEP and the MATIF. For the industrial and commercial gains, the capital losses can also be deducted against capital gains. Of course in those cases, this is not exactly a credit in case of loss, but this justify the assumption of compensation in case of loss. Next, we present the different methods used to compute the capital gain. The first one consists in making the difference between the selling price and the buying price. Those two prices are face values. This is the most used method. Another one consists in making the difference between the selling price and the discounted buying price. This discounting takes into account the erosion by inflation or by use. For example, this method is used in the case of real estate and personal estate. The first method is a particular case of the second one with an erosion coefficient equal to one. In the next, we will use this formalization. More precisely, the erosion coefficient is modeled by an adapted and positive process U . If the investor buys one unit of asset i at time t1 , and if sells it at time t2 , the capital gain or loss will be equal to U Sti2 − Utt2 Sti1 . The tax or the credit consists in k percent of the capital gain or 1 loss. Here the strong assumption is to use the same coefficient k for the tax and the credit. Recall that the tax code imposes for some classes of investors, managing rules like First.In.First.Out or Last.In.Last.Out. We will not use this rules. To justify our position, we observe that the market contains many close substitutes assets so that investors can buy another security with similar risk and tax characteristic without violating the F.I.F.O or L.I.F.O rule (see Cadenillas-Pliska (1996), Constantinides (1983) and Dybvig-Koo (1995)). 22
Our main theorem states as follows : Theorem 4.1 If the market is arbitrage free then there exists an equivalent probability measure π 0 such that S i , discounted by a process C, is a martingale with respect to the filtration IF and the probability measure π 0 . This means that for all i, that is for all t, t0 ∈ T such that t0 ≥ t, Sti Ct = Eπ0 [Sti0 Ct0 /Ft ], with Ct :=
k UT 1 − Eπ 0 [ /Ft ]. Rt Ut RT i
This means that, S i is a martingale via an actualization by the non risky asset corrected by the expected ratio, taxed at k, of the erosion by the non risky asset. In the case of an erosion process U equal to R, we find that, Ct = − k),
1 Rt (1
Corollary 4.1 If U = R and if the market is arbitrage free then there exists an equivalent martingale measure π 0 such that S i /R is a martingale with respect to the filtration IF and the probability measure π 0 . Proof of Theorem 4.1 : We first represent the N risky assets through their cash-flows. Recall that we assume that an investor pays taxes on capital gains and receives credits losses. Let ι = (i, t, A, τ1 , τ2 ) ∈ I, Φιt (ω) = −Sti (ω)IA (ω)Iτ1 (t, ω) + Sti (ω)IA (ω)Iτ2 (t, ω) Ut i i (ω)Sτ1 (ω) IA (ω)Iτ2 (t, ω)Iτ1 <τ2 − k St (ω) − Uτ1 Ut i i + k St (ω) − (ω)Sτ2 (ω) IA (ω)Iτ2 (t, ω)Iτ2 <τ1 . Uτ2 Since short-selling is allowed, then −Φ(i,t,A,τ1 ,τ2 ) = Φ(i,t,A,τ2 ,τ1 ) is an available cash-flow. If we apply corollary 3.1 to Φi,t1 ,A,t1 ,t2 ,A (and Φi,t1 ,A,t2 ,t1 ), for all t1 , t2 ∈ T , with t1 ≤ t2 , and for all A ∈ Ft1 , we find that,
Sti2 − k(Sti2 − Si Eπ0 (− t1 + Rt1 Rt2 23
Ut2 i Ut1 St1 )
)IA = 0.
As
Sti 1 Rt1
is Ft1 -measurable, we find that, "
#
Sti2 Sti1 Ut k = Eπ 0 − (Sti2 − 2 Sti1 )/Ft1 . Rt1 R t2 Rt2 Ut 1
(4.5)
If we apply this equation between t and T , then between t and t0 , and finally between t0 and T , with t ≤ t0 ≤ T , we find that, "
#
(4.6)
Sti0 k Sti Ut0 i = Eπ0 − (Sti0 − S )/Ft , Rt Rt0 R t0 Ut t
"
#
(4.7)
UT i Sti0 STi k = Eπ0 S 0 )/Ft0 . − (STi − Rt0 RT RT Ut0 t
(4.8)
Sti k STi UT i = Eπ 0 S )/Ft , − (S i − Rt RT RT T Ut t
#
"
Equations (4.7) and (4.8) lead to, #
"
Sti STi UT i k k Ut0 i = Eπ0 − (S i − S )/Ft . S 0) − (S i0 − Rt RT RT T Ut0 t R t0 t Ut t Comparing with Equation (4.6), we get that, Eπ0
k k UT i UT i Ut 0 i ( St0 − St )/Ft = Eπ0 (Sti0 − S )/Ft . RT Ut0 Ut Rt0 Ut t
Considering Equation (4.7) again, we obtain that, #
"
Sti0 k UT i UT i Sti = Eπ0 − ( S t0 − S )/Ft . Rt Rt0 RT Ut0 Ut t As
Sti Ut
is Ft -measurable, we find that, Sti (
k UT UT 1 1 k − Eπ0 − /Ft ) = Eπ0 Sti0 ( )/Ft . Rt Ut RT R t0 Ut 0 R T
Now, if we denote by
k 1 UT − Eπ 0 Ct = /Ft , Rt Ut RT then we obtain that,
h
i
Sti Ct = Eπ0 Sti0 Ct0 /Ft .
24
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26
