FINITE HORIZON ARBITRAGE-FREE SECURITY MARKETS

1998,18(2) :203-211

FINITE HORIZON ARBITRAGE-FREE

SECURITY MARKETS

1

Zhang Shunm'ing ( *"pWi!lJl ) School of Economics and Management, Tsinghua University, Beijing 100084, China

IWang Yuyunl <1£«-i;I)

Institute of Systems Science, Academia Sinica, Beijing 100080, China

Abstract This paper studies the weakly and strictly arbitrage-free security markets. The authors extend the Farkas-Minkowski's Lemma and Stiemke's Lemma from two periods to finite periods and from finite-dimensional (Euclidean) space to locally convex topological space and separable Banach space, show weakly and strictly arbitrage-free security pricing theory, then obtain the conditional expectation form of weakly and strictly arbitrage-free security pricing formula.

Key words Farkas-Minkowski's Lemma, strictly arbitrage-free.

Stiemke's Lemma,

weakly arbitrage-free,

1 Introduction D.Duflie (1987, 1988, 1992) and M.Florenzano & P.Gourdel (1994) discussed stochastic (finite-period) economies, gave the no-arbitrage (strictly arbitrage-free) security pricing theory and proved the existence of general equilibrium with incomplete financial markets. In the model, they assumed the finite possible states of nature, and the finite-dimendsional commodity space. We extend their model to the general case of infinite possible states of nature, hence infinitedimensional space is assumed. We examine not only strictly arbitrage-free security markets, but also weakly arbitrage-free security markets. Section 2 presents the model-the security markets. Section 3 obtains the Extensions of Farkas-Minkowski's Lemma and Stiemke's Lemma in multiperiod model from finite-dimensional (Euclidean) space to locally convex topological space and separable Banach space. Sections 4 and 5 obtain the arbitrage-free security pricing theory by taking X = [1(3) and [1(3, P), respectively. Theorems 3 and 5 say weakly arbitrage-free security pricing theory, while Theorems 4 and 6 say strictly arbitrage-free security pricing theory. We obtain equivalent condition of arbitrage-free security price process. According to the result in Section 5, Section 6 shows the arbitrage-free security pricing formula. We obtain the conditional expectation form of (weakly and strictly) arbitrage-free security pricing formula, the market value of a trading strategy being the state-price discounted expected future dividends generated by the strategy at any time. 1 Received

may 29,1996, This paper is financial supported by a project of Financial Mathematics, Financial

Engineering and Financial Management, which is one of "Ninth Five-Year Plan" Major Projects of National Natural Science Foundation of China (Grant 79790130) and Xiao Lin-Shi Foundation of Chinese Economic Research in School of Economics and Management, 'I'singhua University.

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2 Security Dividends, Prices and Trading Strategies We use an event tree E to describe time, uncertainty and revelation of information over a

= {O, 1,···, T} denote the finite set of dircrete-time periods n be general set of possible states of nature. The revelation of information is described by a sequence of partitions of n, F = (:FO,:F1 , " ' , : FT ) , where the partition :Ft is finer than the partition :Ft for all t 2 1. At date t = 0 we assume that there exists no information so that :Fa = {0, n}. The information available at time t E T is assumed to be the same for all

finite horizon. More precisely, let T and let

- 1

agents in the economy (symmetric information). The set consisting of all nodes is called the event tree induced by F. The set of nodes which succeed a node

{e

{e

~

E ::: is called the subtree

{e

:::(0 = E::: Ie 2 0 with root ~ E ::: and :::+(0 = E::: Ie> 0 = E :::(0 Ie> o· The subset of nodes of :::(() at date T is denoted by :::r(O and the subset of nodes between

dates

teO

and

by :::r(()

T

When ~ is the initial node the notation is simplified to t(~)

+ 1}

is the set of immediate successors

of~.

:::r

and E". (+

Every node

~

= {e

E

:::(0 I tee) =

has a unique node (- as the

predecessor of (. A security is a claim to a dividend process, say d. .. with d(O denoting the dividend paid by the security at node F. Each security has a security-price process S, so that S(O is the price of the security, ex dividend, at node

~.

That is. at node

C

the security pays its dividend d( ~)

and is then available for trade at the price S((). This convention implies that d((o) plays no role in determining ex-dividend price d( ~o)

S(O

+ d(~).

= O.

The number of securities is some integer J

The cum-dividend security price at node ( is

2 1. Each security

dj is assigned with a price

process Sj, j E .:J = {1,···,.J}. In other words, Sj (0 is the price of dj (0 as the market value of dj at node f E :::. It will be convenient to treat Sj (0 as the market value of d j after the dividend dj

(()

has been "declared" and paid. The RJ -valued process d

dividend vector is thus assigned a process S

= (Sl,"', Sl) of price processes.

= (rh, .. ·, d 1 )

of

We assume thsat

= 0, ( E :::T' The pair (d, S) is a complete characterization of trading opportunities, or a market system. S(~)

A traging strategy 8

= (81,00.,8 1 )

is a function on

:::T-1

x.:J, a process in RJ. The

scalar 8j (0 represents the number of units of security dj held at node

~

when strategy

e is

followed. We adopt the convention that 8(0 represents the portfolio held after trading at node ~ has occured and dividends d(O E RJ are paid. That is, a spot market valus e(o . d(O accrues to strategy 8 at node F. Because trades occur at pre-dividend security value

S(~)+d(f),

our conventions are consistent. The pre-trade holdings of any strategy 8 at the root node is

=

8( ((;) 0 by a notational convention, since no agent will be endowed initially with securities. 8((-) . [S(O + d(O] is the number of units of account paid in gains at node (, e(o . S(() is the market value of the trading strategy 8 bought node ( The dividend process (it! generated hy a

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trading strategy 8 is defined by

8((0)·5((0), bO( ( )

=

{

(= (0,

8((-) . [5(0 + d(()] - 8(0·5(0, (E ST-l - (0, 8((-) . d((),

that is, 15°(0 = 8((-) . [5(0

+ d(O] -

(E ST,

8(0·5(0,

(E S.

3 Arbitrage-free Security Markets Let X be a locally convex topological space consisting of processes in R=.-eo, and 0 a subspace of R =.T- 1 «a with 150 E R x X. 0 is the space of trading strategies {8 E R =.T - 1 x:r I 0° E R x X}. Thus the marketed subspace M

= {15° E R x X 18 E 0} of dividend process generated

by trading strategies is a linear subspace of the space R x X. Let X· be the dual space composed ofthe continuous linear functionals on X, X+ the positive cone of X, the positive cone of X· (the set of all positive continuous linear functionals

x.+.

on X) and X+.+ the set of all strictly positive continuous linear functionals on X: x.++ ~ X.:j... o E R x X, then 0 = (0((0),<5) where 0((0) E Rand 8 E X. Similarly, A E R x X', then

= ().(~o»:) where A(~o) E R

and ~ EX'. The idea of arbitrage and the absence of arbitrage opportunities is a basic concept of finance. We usually apply strictly arbitrage-freeness in studying GEl, thus the following Theorem 2 is of basically importance. The principle mathematical tool used to prove the following Theorems 1 and 2 is the Separating Hyperplane Theorem S.A.Clark (1993, 1994). Definition 1 A security price process 5 is weakly arbitrage-free for the dividend process d if any trading strategy 8 E 0 has a positive market value at root node, 8((0) . 5((0) 2:: 0, whenever it hag a positive net dividend at every node after root node, 8° E X+, that is,

A

0°(0 2:: O,~ E S - ~o, specifically, 8((-)· [5(0 + d(e)] - 8(e) ·5(0 2:: O,e E ST-l - eo and 8(e-)· d(O 2:: O,~ EST. Theorem 1 Let X be a locally convex topological space, then the security price process 5 is weakly arbitrage-free for the dividend process d if and only if there exists a postive process ). E R++ x X.:j.. such that

8(eo)5(eo)

1

--0

= A(eo) (A,b ).

Proof Let E = R x X and E+ = R+ x X+, then E+ is a (closed convex) positive cone of E with vertex at the origin. Let M = {0° EEl 8 E 0}, a linear subspace of E, then M is closed and convex cone of E with vertex at the origin. The security price process 5 is weakly arbitrage-free for the dividend process d if and only if M n E+ = {0° E E+ I if 8° E X+, then oO(~o) = 0 for 8 E 0} == Mo. Note that (0,0) E M o and M o is a convex cone. Let N E+ \Mo, then N {15° E E+ I if 8° E X +, then 15° (eo) > 0 for 8 E 0} is a (nonempty) convex cone. Both M and N are nonempty disjoint convex cones MnN = Mn(E+ \Mo ) = Mn(E+ \(MnE+» = Mn(E+ \M) = 0, and (M - N) ::f. E from (1, 0) ~ (M - N), then there exists a nonzero continuous linear functional 1 : E -+ R on E separating N from M, that is, I(n) 2:: 0 for all n E Nand I(m) ::; 0 for all mE M. Moreover, 1(1,0) > 0 (S.A.Clark, 1994).

=

=

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Thus I(n) 2: 0 for any n E E+, that is, 1 is a positive continuous linear functional -+ 1?- on E. Thus 1 is represented by some positive process ,X E 1?-++ x X'+ by 1(6) =

= 'x((0)6((0) + (~, b) for any 6 E 1?- x X. Since M is a linear space, then I(m) = 0 for all mE M, that is, 0 = ('x,69) = ,x ((o)bll ((0) + - -9. _ 9 _ 1 - -9 '. _ 1 --9 ('x,6 ) for all () E e, then -6 ((0) - 'x((o) (,x, 6 ), that IS, ()((o)' 5((0) - 'x((o) (,x, 6 ). ('x,6)

The converse is obvious. Definition 2 A security price process 5 is strictly arbitrage-free for the dividend process d provided there is no trading strategy () E e such that 69 2: 0 and 69 01 0, that is

{() Eel 69 2: 0 and 69

01 O} = 0.

It is impossible to generate positive non-zero dividends.

A security price process 5 is strictly arbitrage-free for -the dividend process d if (1) any trading strategy () E e has a positive non-zero market value at root node,

B(~o)

.

5((0) > 0, whenever it has a positive net dividend at every node after root node, J E X+ \ {O}, 2: 0,( E:=: - (0, and a positive non-zero net dividend at, at least, one node; (2) any trading strategy () E e has a zero market value at root node, ()((o) . 5((0) = 0, whenever it has a zero net dividend at every node after root node, '6 = 0 EX, that is, 69(0 = 0,( E:=: - (0. Specifically, a security price process 5 is strictly arbitrage-free for the dividend process d if (1) any trading strategy () E e has a positive non-zero market value at root node, ()((o) . 5((0) > 0, whenever it has a positive net dividend at every node after root node, ()(C)· [5(()+ d(()] - ()(() . 5(02: 0,( E :=:T-l - (0 and ()((-) . d(() 2: 0,( E :=:T, and a positive non-zero net dividend at, at least, one node; (2) any trading strategy () E e has a zero market value at root node, B((o) . S((o) = 0, whenever it has a zero net dividend at every node after root node, ()((-) . [5(() + d(()] - ()(() . 5(() = 0, ( E :=:T-I - (0 and ()((-) . d(() = 0, ( E :=:T. Theorem 2 Let X be a separable Banach space, then the security price process 5 is strictly arbitrage-free for the dividend process d if and only if there exists a strictly postive that is, 69(0

process ,x E 1?-++ x X+-+ such that

=

=

Proof Let E 1?-xX, E+ = 1?-+ xX+ and E++ = E+ \ {(O, O)}, then E+ E++U{(O, OJ} 11 is a (closed convex) positive cone of E with vertex at the origin. Let M = {6 EEl () E O}, a linear subspace of E, then M is closed and convex cone of E with vertex at the origin. The security price process 5 is strictly arbitrage-free for the dividend process d if and only if M and E+ intersect precisely at (0,0), that is, M n E+ = {(O,On. If the security price process 5 is strictly arbitrage-free for the dividend process d, that is, M n E+ = {(O, O)}, then

E++ n (M - E+) = 0 and (M - E++) = M - E+, thus, E++ n (M - E++) = 0. Since both M and E+ are non-empty convex cones in the separable Banach space E. Theorem 5 in S.A.Clark (1993) states that there exists a nonzero continuous linear functional 1 : E -+ 1?- on E strictly separating N from M, that is, I(n) > 0 for all n E E++ and I(m) :S 0 for all m EM.

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> 0 for all n E E++ implies that f is a strictly positive continuous linear functional R on E. Thus f is represented by some strictly positive process .\ E R++ X X+-+ by

f( n)

f :E

--->

f(b) = (.\, b) = .\(~o)b(~o) + (~,8) for any s E xX. The final part is the same as the corresponding part on Proof of Theorem 1. The converse is obvious. Remark The two Theorems are the Extensions of Farkas-Minkowski's Lemma and Stiemke's Lemma which imply that the present value of the securities prices at date 0 is the value of their returns over the state set.

4 Arbitrage-free Security Pricing Theory (1) Take X

= 11(3-~o), we have the arbitrage-free security pricing theory-weakly arbitrage-free

security pricing theory (Theorem 3) and strictly arbitrage-free security pricing theory (Theorem 4). e is the space of trading strategies {B E R=-T-l «r I blJ E R x 11(3 - ~o)}. Theorem 3 The following three claims are equivalent: (1) A security price process S is weakly arbitrage-free for the dividend process tl;

(2) There exists a postive process .\ E 1+(3) with .\(05(0 =

L

.\(~o)

> 0 in R such that

.\(1})[5Cf)) + d('f))]'

l1H+

(3) There exists a postive process .\ E 1+(3) with .\(~())

.\(05(~) = .\(0

L

> 0 in R such that

.\('r))d(1}),

11>~

Proof Which is in my dissertation, is similar to the following proof of Theorem 5. Theorem 4 The following three claims are equivalent: (1) A security price process 5 is strictly arbitrage-free for the dividend process d;

(2) There exists a strictly postive process .\ E int(l+ (3)) such that

(3) There exists a strictly postive process .\ E int(1+(3)) such that

Proof Which is in my dissertation, is similar to the following proof of Theorem 6. Remark

(1) The security price process 5 is (weakly or strictly) arbitrage-free for the

dividend process rl if and only if there exists a (positive or strictly positive) stochastic state price (present value) process; (2) The present value of the security prices at node ~ E 3 is the present value of their dividend and capital values over the set of immediate sucessors ~+; (3) The current value of each security at node ~ E 3 is the present value of its future

dividend stream over all succeeding nodes

e> (

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5 Arbitrage-free Security Pricing Theory (2) In order to handle rnultiperiod issue, however, we will treat uncertainty a bit more formally as a probability space (0, F, P), with F denoting the tribe of subsets of 0 that are events (and can therefore be assigned a probability), and with P a probability measure assigning to any event Bin F its probability P(B). The filtration F revealed through time, F t l Take X

<::;;

F t 2 whenever t 1

= (Fa, F 1 , " ::;

FT) represents how information is

' ,

t 2 , meaning that events are never "forgotten".

= [1(:=:-~0' P), we have the arbitrage-free security pricing theory-weakly arbitrage-

free security pricing theory (Theorem 5) and strictly arbitrage-free security pricing theory (Thel orem 6). e is the space of trading strategies {8 En=7- x.'! lOll En x [1(3 - ~o, P)}. Theorem 5 The following three claims are equivalent: (1) A security price process 5 is weakly arbitrage-free. for the dividend process il: (2) There exists a postive process A E S(~)A(OP(O

[+ (3, P) with

= [1(3 -

e,

= [+(3 -

~o,

Pl.

~. ./ For any ~ E .:::. and J E .:1, we define 8J (e) S(OA(~)P(O = E[SA1~]

={

+ rl(O] -

1,

0,

=

otherwise

+ rlj)l~+] - E[ASj1~] = 0, that = E[(S + d)A1~+], ~ E 3 T - 1 .

S(OA(OP(~)

E[dA1:=.+(~)].

8(0 . S(~)} rlP(O for any 8 E

e = ~ and . j' = .i 1,

= E[SA1~] = E[rlA1:='+(OL

then for all ( E 3 t -

S(()A(()P(() = E[SA1d = E[(S

~T -1

E[AS1~]

= 0 and

then S(~)A(~)P(O = E[SA1~]

If, for any t E {I,·", T - I} and

e.

then, for ~ E .:::.

+ rl)l~+] -

is, E[A(S

(2) ====> (3) We prove it by inductive method. If ~ E 3 T E[dA1~+]

> 0 in n such that

~o, P), then X*

that is, 0 = E[A . 011] = J:=. A(~){8(~-) . [5(0

and each j E.:1, E[A(Sj

A(~O)

= [00(3 - ~o, P), X+ = [~(3 - ~o, P) Using Theorem 1, we have 0 = (A,OIl) = A(~o)OIl(~o) + (.\)11) for all

Proof (1) ====> (2) Take X and Xi-

> 0 in n such that

= E[SA1d = E[(S + rl)A1~+],

(3) There exists a postive process A E [+(3, P) with

8E

A(~O)

~ E 3 r , (t ::;

T

::;

=

T - 1).

1,

+ rl)A1(+]

= E[SA1(+] + E[dA1(+] = E[dA1:=.+«+)] + E[dA1(+] = E[dA(l(+

+ 1:=.+«+»)] = E[dA1:=.+(O]'

[+

(3) ====> (1) If there exists a strictly positive process A E (3, P) with A(~O) > 0 in T 1 such that S(OA(OP(O = E[SA1~] = E[rlA1:=.+(O],~ E 3 - then, for ~ E 3 T - 1 - ~o, {8(~-)·

[5(0

+ d(O] - 8(0' S(~)}A(OP(O

= 8(~-)E[(S + d)A1~] - 8(OE[SA1~] = 8(~-){E[SA1~] + E[dA1~]} - 8(OE[SA1~] = 8(~-){E[dA1:=.+(oJ

= 8(~-)E[dA1:=.(O] -

+ E[dA1~]} -

8(~)E[dA1:=.+(oJ

8(OE[dA1:=.+(()].

n

+ d(~)]

If e(~-) . [S(()

- e(o . S(O 2: 0, ~ E 2 T -

e(~-)E[dAl::::(O] 2: e(~)E[dAl::::+(O],~ E 2 O,~

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T

-

1

-

~o

2: 0, ~ E 2 T , then and e(~-)E[dAl~] = e(~-)· d(~)A(OP(O 2: 1

-

~o and e(~-) . d(O

E 2T. Thus

= e(~o) . S(~o)A(~o)P(~o) = e(~o)E[SAl~o] = e(~o)E[dAl::::+(~oJl = E{e((i )E[dAl::::(~,)]I~lE~t} e(~o)' S(~o)A(~o)

= E{E{e((2 )E[dAl::::(~2)]1~2E~;t-}1~lE~t}

2:

E{e(6)E[dAl::::+(~I)]lelHt}

2:

E{E{8(6)E[dA1::::+(~2)]16E~;t- }1~lE~t}

2: ...

2: E{ E{ E{' .. {E{ e(~T_l)E[dAl::::+(~T_,)]I~T_I E~;':_) I~T_2HL) ... } 1~2E~;t- }leI Eet } = E{E{E{··· {E{e(G)E[dAl~T]I~THLI}1~T_IE~i-)" .}I~oE~;t- }leIEet} If any trading strategy

2: O.

e E f) has a positive net dividend at every node after root node,

then it

yields a positive market value at root node. Theorem 6 The following three claims are equivalent:

(1) A security price process S is strictly arbitrage-free for the dividend process d; (2) There exists a strictly postive process A E int(l+, (2, P» such that S(OA(OP(O = E[SAl~] = E[(S

+ d)Al~+],

(3) There exists a strictly postive process A E int(l+,(2, P)) such that

= E[SAl~] = E[dAl::::+(o], ~o, P), then X* = lOO(2 -

S(~)A(OP(O

X;

Proof Take X

= l+, (2 -

~o,

= ll(2 -

~o, P), X+

= l~(2 -

~o, P) and

Pl. We can use Theorem 2. The next part is. similar to the corresponding

part of Proof of Theorem 5. Note that A E int( l+, (2, P» is a strictly positive process; and the last sentence is revised as following: If any trading strategy E e has a positive net dividend

e

at every node after root node, and a positive non-zero net dividend at, at least, one node, then it yields a positive non-zero market value at root node. If any trading strategy E has a zero

e e

net dividend at every node after root node, then it yields a zero market value at root node.

6 Arbitrage-free Security -Pr'icing Formula

= 0,1, ... ,T, let dt = (d(~tl; ~t E 2tl, St = (S(~tl; ~t E 2 t),et = (e(~t);~t E 2tl,or = WI(~tl;~t E 2 t), then a.s.e,» are adapted processes, and for t = 0,1"", T, let or = (08(etl; et E 2d be defined by of = et- 1 • (St + dt) - et . St and 08 is an This section folllows Section 5. For t

adapted process, then we can rewrite Theorems 5 and 6 as follows. Theorem 5' The following three claims are equivalent:

(1) A security price process S is weakly arbitrage-free for the dividend process d;

(2) There exists a postive process A E l+, (2, P) with A(eo) > 0 in R such that E[StAt]

= E[(St+l + dt+l».t+l], t = 0,1"", T -

(3) There exists a postive process A E l+'(2, P) with A(eo)

E[StAt]

=E

[

t

r=t+l

drAr] , t

1;

> 0 in R such that

= 0,1"", T

- 1.

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Theorem 6' The following three claims are equivalent: (1) A security price process S is strictly arbitrage-free for the dividend process d;

(2) There exists a strictly postive process A E int(l+,(3, P)) such that

(3) There exists a strictly postive process A E int(l+' (3, P)) such that E[StAt]

=E

[

t

r=t+l

drAr] , t

= 0,1, ... ,T -

1.

= O,I, .. ·,T-l in Theorem 5' (2) and (3), there exists a postive process A E 1+,(3, P) with A(~o) > o ,in 'R such that s,x, = E[(St+l + dt+1)At+l 1Ft], t = 0,1, ... ,T - 1, that is, StAt - E[St+lAt+l 1Ft] = E[dt+lAt+l 1Ft], t = Take conditional expectation E[· 1Ft ], for t

0,1, ... ,T - 1. Summing from t to T - 1 and taking conditional expectation E[· 1Ft ], then we

=

have StAt = L:;=t+l E[drA r 1Ft] E[L:;=t+l drAT 1Ft], t 5' (3). On tlie other hand, for t = 0,1,· . " T - 1,

L:;=t+l 8~ Ar

= L:;=t+Il8r- 1 . (Sr + dr) -

= 0,1, .. " T -

1, this is Theorem

8r . S; ]A r

= L:;=t+l[8r- l· (SrAr + drAr)

- 8r· SrAr] = L:;=t+1 {8r- l· (E[L:;':r+l dtlAtl IFr] + drAr) - 8r· E[L:;'=r+1 dtlAtllFr]} = L:;=t+d 8r-1 . E[L:;'=r dtl At/IFr] - 8r . ElL:;'=r+1 dtl Atl I F r]}

= 8t . ElL:;'=t+l dtl At/IFt+l] -

8t+1 . ElL:;'=t+2 dtl At/IFt+l]

T

+ ...

T '

+8 T - 2 . ElL:tl=T-I dtIAtIIFT_I] - 8T- 1 . ElL:tl=T de At/IFT-I]

+ 8T- I . E[dTAT 1FT],

Take conditional expectation El·IFt], then, for t = 0,1"", T - 1, ElL:;=t+18~ArIFt] = 8t . ElL:;: I ElL:;=t+1 a,»; 1Ft] = e.s,»; thus ElL:;:1 8~ x, 1.1'0] = 80S0AO, that is, 80So =

:fa-

8~ Ar

I .:Fo], therefore the security price process S is weakly arbitrage-free for the dividend process

d. Thus we have the weakly arbitrage-free security pricing formula.

Theorem 7 The following four claims are equivalent: (1) A security price process S is weakly arbitrage-free for the dividend process d;

(2) There exists a postive process A E 1+,(3, P) with

A(~O)

> 0 in

'R such that

(3) There exists a postive process A E 1+,(3,P) with

A(~o)

> 0 in

'R such that

s». = E

[

t

r=t+l

c,»; 1Ft] , t = 0,1"

.. ,T - 1

(4) There exists a postive process A E I+,(3, P) with A(~O) > 0 in 'R such that

e.s,», = E [

t

r:t+1

8~Ar I .:Ft]

t

= 0,1, ... ,T -

1

Similarly, we have the strictly arbitrage-free security pricing formula.

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Theorem 8 The following four claims are equivalent: (1) A security price process S is strictly arbitrage-free for the dividend process d; (2) There exists a strictly postive process A E int(I+'(S, P)) such that St

= At1 E[(St+l + dt+1)AH l 1Ft ], t = 0,1"

.. ,T - 1;

(3) There exists a strictly ,Postive process A E int(I+,(S, P)) such that

s, = ; E [ t

t a,»; 1Ft] ,

t = 0,1,···· ,T - 1;

r:t+1

(4) There exists a strictly postive process A E int(I+,(S, P)) such that

Remark Theorems 7 and 8 mean the conditional expectation forms of weakly and strictly arbitrage-free security pricing formula. The market value of a trading strategy is, at any time, the state-price discounted expected future dividends generated by the strategy. References

2 3 4 5 6

Clark S A. The Valuation Problem in Arbitrage Pricing Theory. Journal of Mathematical Economics,1993, 22: 463-478 Clark S A. Vector Space Methods in Additive Theory. Journal of Mathematical Economics, 1994 Duffie D. Stochastic Equilibria with Incomplete Financial Markets. Journal of Economic Theory, 1987,41: 405-416 Duffie D. Security Markets: Stochastic Models. Stanford University, California, Academic Press, 1988 Duffie D. Dynamic Asset Pricing Theory. Princeton University, California, Academic Pl'ess.1992 Florenzano M, Gourde! P. T-period Economies with Incomplete.Markets. Economics Letter, 1994,44:

91-97 7 Geanakoplos J. An Introduction to General Equilibrium with Incomplete Asset Markets. Jownal of Mathematical Economics, 1990,19: 1-38 8 Magill M, QuinaiiM. Infinite Horizon Incomplete Markets. Econometrica, 1994,62: 853-880 9 Magill bIl, Shafer W. Incomplet.e Market.s. In: Hildenbrand W, Sonnenschein H ed. Handbook of Mat.hemetical Economics, VolUlIln 4. 1991 10 Werner J. Structure of Financial Markets and Real Indeterminacy of Equilibria. JOW'llal of Methematical Economics, 1990,19: 217-232 11 Zhang'S M,Extension of Stiemke's Lemma and Equilibrium for Economy with infinite-dimensional commodity space and Incomplete Financial Markets. JourrrelofMethematical Economics,1995