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A NEW LOOK AT THE FUNDAMENTAL THEOREM OF ASSET PRICING

JIA-AN YAN

ABSTRACT. In this paper we consider a security market whose asset price process is a vector semimartingale. The market is said to be fairif there exists an equivalent martingale measure for the price process, deflated by a numeraire asset. It is shown that the fairness of a market is invariant under the change of numeraire. As a conse- quence, we show that the characterization of the fairness of a market is reduced to the case where the deflated price process is bounded.

In the latter case a theorem of Kreps (1981) has already solved the problem. By using a theorem of Delbaen and Schachermayer (1994) we obtain an intrinsic characterization of the fairness of a market, which is more intuitive than Kreps' theorem. It is shown that the arbitrage pricing of replicatable contingent claims is independent of the choice of numeraire and equivalent martingale measure. A sufficient condition for the fairness of a market, modeled by an Ita process, is given.

1. Introduction

In the early 70's Black and Scholes (1973) made a breakthrough in option pricing theory by deriving the celebrated Black-Scholes formula for pricing European options via a "hedge approach". This work was fur- ther elaborated and extended by Merton (1973). Cox and Ross (1976) is the overture of a modern theory of option pricing-the risk-neutral valuation or arbitrage pricing. A key step in this direction was made

Received February 24, 1998.

1991 Mathematics Subject Classification: 60G44, 60H05.

Key words and phrases: equivaIent martingale measure, fair market, allowable strategy, no-arbitrage, replicatable contingent claim, arbitrage pricing, Ita process.

Research supported by the Liu Bie Ju Center for Mathematical Sciences of the City University of Hong Kong and the National Natural Science Foundation of China.

This is an invited paper to the International Conference on Probability Theory and its Applications.

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in Harrison and Kreps (1979). They remarked that the hedge approach is not mathematically rigorous unless one excludes doubling-like strate- gies. Harrison and Kreps imposed some "admissibility" condition on the trading strategy and showed that the existence of an equivalent martingale measure for the deflated price processes implies the absence of arbitrage. Since then many attempts have been devoted to show the converse statement. Harrison and Pliska (1981) solved this problem in discrete-time and finite-state case. This result is referred to as thefunda- mental theorem of asset pricing. In the general state and discrete-time with finite and infinite horizon case, this problem has' been solved by Dalang-Morton-Willinger (1990) and Schachermayer (1994) respectively.

However, inthe continuous-time case and the discrete-time with infinite horizon case the absence of arbitrage is no longer a sufficient condition for the existence of an equivalent martingale measure. A "no-free-IunCh"

condition, slightly stronger than no-arbitrage condition, was introduced by Kreps (1981). Under a mild but irrelevant separability assumption Kreps proved thatifthe deflated price process is bounded then the mar- ket is fair if and only if the market has no free-lunch. See Schachermayer ,(1994)for a transparent proof ofthis result. Without knowing this result of Kreps, the problem was attacked by Stricker (1990), who discovered that a result of Yan (1980) (or more precisely, the method of its proof) is an appropriate tool for solving the problem. The result of Stricker was re-examinated and extended by Delbaen (1992), Kusuoka (1993), Lakner (1993), Delbaen and Schachermeyer (1994), Frittelli and Lakner (1994).

In this paper we consider a semimartingalemodel for a market. The market is said to befairifthere exists an equivalent martingale measure for the deflated price process. In section 2 we show that the fairness of a market is invariant under the change of numeraire and give a charac- terization of self-financing strategies. In section 3, by augmenting the original market with a new asset we show that the characterization of the fairness of a market can be reduced to the case, where the deflated price process is bounded. By using a theorem of Delbaen and Schacher- mayer (1994) we obtain an intrinsic characterization of the fairness of a market. Ifthe asset price process is continuous, a theorem of Del- baen (1992) implies a more elegant result. Insection 4 we show that a fair market has no arbitrage with allowable strategies and the arbitrage pricing of replicatable contingent claiIns is independent of the choice of

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numeraire and equivalent martingale measure. Finally, in section 5 we give a sufficient condition for the fairness of a market modeled by an Ita process.

2. The characterization of self-financing strategies and fair market

We fix a finite time-horizon [0,T] and consider a security market which consists of m +1 assets whose price processes (S;),i = 0,··· , m are assumed to be strictly positive semimartingales, defined on a filtered probability space(n,:F,:Ft,P) satisfying the usual conditions. Moreover, we assume that :Fois the trivial a-algebra. For notational convenience, we take asset 0 as the numeraire asset. We set ')'t~(Sf)-l and call ')'t the deflatorat timet. We set St = (Si,· .. ,Sf) andSt = (Si,··· ,Sf), where S: ='YtS:,1 :s; i :s; m. We call (St) the deflated price process of the assets. Note that the deflated price process of asset 0isthe constant 1.

The continuous trading is modeled by a stochastic integral. In order to be able to define a trading strategy we need the notion of integration w.r.t. a vector-valued semimartingale (see Jacod (1980)). Such inte- gral is defined globally and not componentwise. A basic fact is that a vector-valued predictable process H is integrable w.r.t. a vector-valued semimartingale X if and only if the sequence (IflHI$nJH).X converges in the semimartingale topology. In this case the limit gives the inte- gral H.X. Consequently, if H = (Ho, ... , Hm) is integrable w.r.t. a semimartingale (Xo, ... ,xm) and HO is integrable w.r.t. XO, then we have

(2.1) H.(Xo, ... ,xm) = SO.Xo+(Hl, ... ,Hm).(Xl, ... ,Xm).

Atrading strategyis a Rrn+I-valued Ft-predictable process 4>={0°, O}, where

O(t) = (OI(t),··· ,orn(t)),

such that 4>isintegrable w.r.t semimartingale(SO, S) withS = (St,··· , srn). Oi(t) represents the numbers of units of asset i held at time t.

This notion of trading strategyisnot very realistic. However it is conve- nient for mathematical studies. The wealth l-'t(4)) at timet of a trading strategy 4> = {0°, O} is

(2.2) l-'t(4)) = OO(t)S/+O(t) .St,

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whereO(t) . St =E:lOi(t)SI· The deflated wealth at timet is lIte</»~=

"Vt(</>ht. A trading strategy {0°,O} is said to be self-financing, if

(2.3)

In this paper we always use notation J;HudXuor (H.X)t to stand for the integral of H w.r.t. X over the interval (0,tJ. In particular, we have (H.X)o = O.

It is easy to see that for any given Rm-valued predictable process 0 which is integrable w.r.t (St) and a real number x there exists a real- valued predictable process (O~)such that {0°,O} isa self-financing strat- egy with initial wealth x.

A process {0°,O}issaid to be elementary,ifthere exist a finite parti- tion of (0,TJ: 0= to ::;t1 < ... < tn = T and a sequence of Rm+l-valued random variables (6, ... ,~n),with each~i being Ft;_l-measurable, such that

n

Oi(t) =L e1Ictk_l,tk](t), t E [0,T], 0:Si :Sm.

k=l

Ifwe take stopping times ~s instead of deterministic times, the corre- sponding process is said to be simple. Iffurthermore (6,'" ,~n) are elementary random variables (i.e. taken only a finite number of values), the corresponding process issaid to be very simple.

DEFINITION 2.1. A security market is said to befairif there exists a probability measureQ equivalent to the "objective" probability measure P such that the deflated price processes (St) is a (vector-valued) Q- martingale. We call such a Q an equivalent martingale measurefor the market.

We denote by Mi the set of all equivalent martingale measures for the market, if asset j is taken as the numeraire asset.·

The following theorem shows that the definition of fair market does not depend on the choice of numeraire.

THEOREM 2.2. The fairness ofa market is invariant under the change of numeraire.

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o

Proof. Assume that MO f=0. For a P* E MO we define a probability measure Qby

(2.4) dQ = S~(~)-1Sj.

dP* S6 T T

We denote Q by hj(P*). We are going to show that hj is a bijection from MO onto Mj. Let 'Y~= (Si)-1 and put

S:='Y~S:, O~i~m.

Since (S?)-lSi =si is a P*-martingale, we must have (2.5) Mt :=E* [ dQdP*I:F,]t = 58S~(SO)-l Sit t , 0 <_ t <_T.

From the fact that

~. . l ' sg-.

MtS; =Mt(Sf)- S; = sgS;, 0~i ~ m we know that QE Mj. The theorem is proved.

A strategy is said to be admissible, if its wealth process is non- negative. A strategy is said to be tame, if its deflated wealth process is bounded from below by some real constant. The weakness of the notion of tame strategy is that it is not invariant under the change of numeraire. Moreover, all bounded elementary or simple strategies are not tame. We propose below to extend the notion of tame strategy to a notion of "allowable strategy" .

DEFINITION 2.3. A strategy 1J = {OO,O} is said to be allowable, if there exists a positive constant c such that the wealth (vt(1J)) at any timet is bounded from below by -cL~oS;.

It is easy to see that all bounded elementary or simple strategies are allowable, and the notion of allowable strategy does not involve the numeraire.

DEFINITION 2.4. A market is said to have no arbitrage with allowable strategies if there exists no aliowable self-financing strategy with initial wealth zero and a non-negative terminal wealth VT such that P (VT >

0) > O.

A key point of arbitrage pricing of contingent claims is the following characterization of the self-financing strategy.

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THEOREM 2.5. A strategy<jJ= {eo, B} is self-financingifand onlyif its wealth process("\It) satisfies

(2.6)

o

wherevt = vt't. In particular, the deflated wealth process ofan allow- able self-financing strategy is a local Q-martingale and a Q-supermartingale for any Q E MO.

Proof Assume that <jJ = {BO, B} is a self-financing strategy. First of all, by Ita's formula, we have

(2.7)

- ° . . . . 0 . . . . 0 °

del,St) = d(rtSt, ,tSt) = 'td(ui, St)+(Ut-, St)d,t+d([S ,,It, [S, ']t).

Secondly, by (2.1) we have

(2.8) <p(t)d(l, St) =B(t)dSt.

Thirdly, by (2.3) we have

L\vt =()o(t)t::..S£ +B(t) . L\St, .which together with (2.2) implies

(2.9) vt- =eo(t)S£_+()(t) . St-·

Finally, applying Ita's formula to the product vt,twe get from (2.3) and

(2.7)-(2.9)

dvt = vt-(<jJ)d,t+,t_dvt+d[V;,lt

(()O(t)st +()(t) . St- )d,t+'"Yt-(BO(t),()(t))d(S~, St) +()O(t)d[SO"lt+()(t). d[S"lt)

()(t)dSt.

Similarly, we can prove the "if' part.

Now assume that <P = {eo, B} is an allowable self-financing strat- egy. By definition there exists a positive constant c such that vt(<jJ) 2':

-c2:;:'0Si, t E [0,T]. Put

()i=()i +c, 0~i ~rn, <PI ={()~,()1}.

Then by (2.6) we haved~(<P1)= ()1(t)dStand

m

~(<jJI) =~(<jJ)+cI:Sf 2': o.

i=O

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By a theorem of Ansel and Stricker (1994), CVt(<Pl» is a local Q-martingale and Q-supermartingale. Thus so is (Vi) because CL:::oSf) is a Q- martingale.

As a corollary we obtain:

THEOREM 2.6. A fairmarket has no arbitrage with allowable strate- gies.

Proof Let Q E MO. Let {B,eo} be an allowable self-financing strat- egy with initial wealth zero. By Theorem 2.5 the deflated wealth process of<P is a Q-supermartingale. Therefore, we must have EQ[VT] ::; O. So the market has no arbitrage with allowable strategies. 0

3. The fundamental theorem of asset pricing

The fairness of a security market is the basis of the so-called ''pricing by arbitrage". By thefundamental theorem of asset pricing we mean a characterization of the fairness of a market. Roughly speaking, such a characterization states that the market is fair ifand only if the market has no "free-lunch". In the literature several notions of "free-lunch"

have been introduced in different circv.mstances. A common feature of these notions is that they involve an appropriate topological closure of the set V - L'f, where V is the set of all achievable gains by a certain bounded elementary (or simple) strategy. Ifthe deflated price processis a bounded (vector-valued) semimartingale, several characterizations of the fairness are available.

Now we introduce a new asset, indexed by m+ 1, whose price process

IS:

(3.1)

m

S;n+l = I:S;.

i=O

We augment the market with this new asset. It is readily seen that the new market is fair if and only if the old one is fair. According to Theorem 2.2 in order to characterize the fairness of the new market one can choose asset m+1asthe numeraire asset. In doing so the deflated price process becomes bounded. This trick not only reduces the problem to the easy case but also leads to an intrinsic characterization of the fairness of a market in the sense that no numeraire asset is involved.

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In the following we consider the augmented market and choose the new asset as the numeraire asset. We denote by (Xt) the deflated price process of asset i {Le. xl = (S;n+l)-lS1) and setXt = (Xp,· .. ,Xr).

A theorem of Lakner (1993, Theorem 2.1) implies immediately the following characterization of the fairness of a market.

THEOREM 3.1. Let process (Xt ) be definedas above. Put (3.2) V ={(H,X)T : H is a very simple process}.

Then the (original) marketis fairif and only if

(3.3) V - L'fnL':= {O},

whereV - L'fis the closure ofV - L'?; in theu(Loo,V(P))-topology.

REMARK. Condition (3.3) can be interpreted as "nerfree-Iunch" in a certain sense. In fact, ifcondition (3.3) is violated, then there is an

fo E L'?; \ {O} and a net (</>O,)a:EJ of very simple self-financing strategies with initial wealth 0 such that at the terminal time the agent "throws away" the amount of moneyha:S:;+! with ha: E L'?; the random variable (ST+1)-lVT(</>a:) - hOt. becomes close to fa: w.r.t u(Loo,L1(P))-topology.

On the other hand, according to Schachermayer (1994) the Kreps' "ner free-lunch" condition can be stated as follows:

(3.4) where

(Vo - Lt)nLoonLe: = {O},

(3.5) VC = {(H,X)T : H is an elementary process}.

So the economic meaning of Lakner's no-free-Iunch condition (3.3) is more convincing than the Kreps' one. We refer the reader to Kusuoka (1993)for another "nerfree-Iunch" condition which is similar to condition (3.3). In view of the economic meaning of nerfree-Iunch, an equivalent martingale measure is also called a risk-neutral probability measure.

As pointed out in Delbaen and Schachermayer (1994) the drawback of a variant of Kreps' theorem is twofold. First it is stated in terms of nets or topological closure, a highly non intuitive concept. Second it involves the use of very risky positions. The main theorem of Delbaen and Schachermayer (1994) remedies this drawback. By using this theer rem we obtafu the following intrinsic characterization of the fairness of a market.

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THEOREM 3.2. The market is fairifand onlyifthere is no sequence (4)n) ofallowable self-financing strategies with initial wealth 0 such that

( 1 m i

VT 4>n) 2:: -~2::i=OST' a..s., for all n 2:: 1 and such that VT(4)n) , a.s., tends to a non-negative random variable~ satisfyingP(~>0) > o.

Proof. Consider the market augmented with asset m +1 and choose asset m +1 as the numeraire asset. Let 4> = {4>0,... ,q;m} be an "ad- missible" integrand for the vector semimartingale X = (Xo, ... ,xm), in the sense of Delbaen and Schachermayer (1994) that there is a posi- tive constant c such that (4).X)r ~ -c. We can introduce a predictable process 4>m+1 such that4>together with4>m+1 constitutes a self-financing strategy with initial wealth 0 for the augmented market. By Theorem 2.5 we have

(3.6)

On the other hand, we have

lit(4>,4>m+1) = lit( +4>,,:+1Sr+l, O:s;t :s; T.

Thus, if we put

4>'; = 4>~+4>":+1, O:S;i :s; m,

then we have lIt(4)') = ltt(4),4>m+1). Consequently, by (3.6) 4>' is an al- lowable strategy for the original market. It is easy to see that 4>' is self-financing and its initial wealth is o. Conversely, for any allowable strategy4> for the original market, {4>,O} is a self-financing strategy for the augmented market and we have

(S;n+1)-ll1t(4» =(Sr+ 1)-lvt(4>,0) =

it

q;(t)dX

t.

Thus, by the vector versions of Theorem 1.1 and Corollary 3.7 of Delbaen and Schachermayer (1994) we can conclude the theorem. 0

REMARK. According to Delbaen and Schachermayer (1994) the con- dition in Theorem 3.2 is called the condition ofno free lunch with van- ishing risk.

Ifthe asset price process is continuous, a theorem of Delbaen (1992) (Theorem 5.1) gives us a more elegant characterization of the fairness of a market.

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THEOREM 3.3. Assume that the asset price process is continuous.

Then the market is fair if and only if the following condition is satisfied:

If(cPn) is a sequence of very simple self-financing strategies such that VO(cPn) = 0, !V(cPn)I :::; sm+l,\:In ~ 1 and VT(cPnt -+

°

in probability, then VT(cPn)+ -+ 0in probability.

4. Arbitrage pricing of contingent claims in a fair market In this section we will study the problem of the pricing of European contingent claims in a fair market. By a (European) contingent claim we mean a non-negative FT-measurable random variable. Let ~ be a contingent claim. One raises naturally a question: what is a "fair" price process ofe Assume that 'YT~is P*-integrable for some P* E MO. We put

(4.1)

If we consider (lit) as the price process of an asset, then the market augmented with this asset is still fair, because the deflated price process of this asset is a P*-martingale. So it seems that (Vi) can be considered as a candidate for a "fair" price process off However this definition of

"fair" price depends on the choice of equivalent martingale measure. We will show that for replicatable contingent claims (see Definition 4.1) this definition is reasonable.

DEFINITION 4.1. Let P* E MO. A European contingent claim ~

is said to be P*-'replicatable (or attainable) if 'YT~ is P*-integrable and there exists an admissible self-financing strategy<Psuch that its terminal wealth is equal to ~ and its deflated wealth process is a P*-martingale (i.e. E*hT~J ='YoVO(cP)). Such a strategy is called a P*-hedging strategy for ~.

The following theorem shows that the "fair" price process of a repli- eatable contingent claim is uniquely determined.

THEOREM 4.2. Let P*,pl E MO and ~ be P*- and P/-replicatable.

Let (lit) (resp. (Ut)) be the wealth process of a P*- (resp. P/-)hedging strategy for~. Then (lit) and (Ut) are the same. Moreover, lit is given by (4.1) and we have

(4.2) lit= essinfQEMo'YtlEQ['YT~IFtJ.

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- - -

Proof· Put 11;,=/'t1l;" Ut =/'tUt· Then (11;,) is a P*-martingale and a P'-supermartingale and(fit) is a P'-martingale and a P*-supermartingale.

Note that UT =VT =~ and we have

E*[VTI Ft] =~ ~ E'[VTI Ft] =E'[fiTIJ=t] = fit.

Thus we have 11;, ~ Ut, a.s. Similarly, we have Ut ~ 11;" a.s. Hence V =U. The last assertion of the theoremisobvious. D REMARK. According to Theorem 4.2, for a P*-replicatable contingent claim ~it is natural to define its "fair" price at time t by (4.1). We call this method of pricing the arbitrage pricing (or pricing by arbitrage, or risk-neutral valuation).

The following theorem shows that the arbitrage pricing of replieatable contingent claims is independent of the choice of numeraire.

THEOREM 4.3. Let P* E MO and ~ be a P*-replicatable contingent claim and c/J be a fair hedging strategyfor~. Then for any 0 ::; j ::; m

~ is an hj(P*)-replicatable contingent claim, and its "fair" price process remains the same.

Proof We keep the notationsin the proof of Theorem 2.2. We have by (4.1)

EQb~~] =E*[MT/'~~] = ~E*bT~] ='Y~VO.

This implies that a P*-hedging strategy for~is also a Q-hedging strategy

for~. So~ is a Q-replicatable contingent claim. Moreover, by the Bayes rule we have

(I;)-lEQb~~1Ftl = (1;)-1Mt-1E*[MT'Y~~1Ft]

= It1E*['T~1Ft].

This proves that the "fair" price process of ~ is invariant under the change of numeraire.

Let P* E MO. The market is said to be P*-complete, if every con- tingent claim~ with'YT~being PO-integrable is P*-replicatable. Ifthere exists a unique martingale measure for the market then by a general result of Jacod-Yor (1977), the market is complete. In a complete mar- ket, the fair price process of a replicatable contingent claim~is given by

(4.1). D

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5. The Ito process model

Wefix a finite time-horizonT. Let B = (Bl,··· ,Bd) be a Brownian motion on a complete probability space(O,:F,P). We denote by(:Ft) the natural filtration of(Bt ) and by£ the set of all measurable (.1t)-adapted processes.

We consider a financial market which consists of m+1assets. The price process (SD of each asset i is assumed to be a strictly positive Ito process. Since its logarithm is also an Ito process, we can represent (SD as

(5.1) dS; = S;[ui(t)dBt+pi(t)dt], 8fJ=Pi' O:::;i:::;m.

We call p = (pO, ... ,pm) the vector of expected rate of return and u the volatility matrix.

We specify asset 0 as the numeraire asset and set rt := (82)-1. By Ito's formula we have

(5.2) drt = -rt[uO(t)dBt+(po(t) - juO(t)12)dt],

(5.3) dB;= B; [ai(t)dBt+bi(t)dt] , 1:::;i :::;m, where

ai(t) = ui(t) - uO(t); bi(t) = pi(t) - pO(t)+luO{i)12- ui(t) . uO(t).

In particular, Ifasset 0 is a bank account with interest rate process (r(t)), then

ai(t) =ui(t), bi(t) =pi(t) - r(t).

One raises naturally a question: what conditions we should impose on coefficients a and b of the Ito process (Bt ) such that the market is fair? The following theorem gives a partial answer to this question.

THEOREM 5.1. Ifthe marketisfair, the linear equation (5.4) a(t)~(t) = b(t), dt x dP-a.e., a.s., on [0,T] x 0

has a solution ~ E (£2)d, where £2 stands for the set of all adapted process c/J with JoTc/J2(u)du<00. Conversely, if

(5.5) E [exp{~lTlai(tWdt}] <00, 1:::;i:::;m,

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and equation (5.4) hasa solution 'ljJE (£2)d satisfying (5.6)

then the probability measure Q with Radon-Nikodym derivative ~ =

£(-'ljJ.B)r isan equivalent martingale measure.

Proof. Let Q E MO. Put

dQ Mt =E(dP1.1t)·

Then (Mt )is a P-martingale. By the martingale representation theorem for Brownian motion there exists </> E (,C2)d such that dMt = </>(t)dBt.

Set'ljJ(t) = -q;(t)/Mt. Then M =£(-'ljJ.B) and by Girsanov's theorem B; = Bt +J;'ljJ(s)ds is a Brownian motion under Q. Moreover, by a Theorem of Fujisaki, M., G. Kallianpur and H. Kunita (1972), (Bn has also the martingale representation property w.r.t. (Ft) under Q. Thus there exists someu* E (£2)mxd such that

iSt=u*(t)dB; =u*(t)(dBt+'ljJ(t)dt).

According to the uniqueness of the representation of Ito process(St) and the invariance of the stochastic integral under a change of probability, from (5.3) we know that u*(t) = Bta(t), dt x dP-a.e., a.s., and con- sequently, a(t)'ljJ(t) = b(t), dt x dP-a.e., a.s. So ('l/J(t)) is a solution of equation (5.4).

Now assume that a satisfies (5.5), 'ljJ is a solution of(5.4) and verifies (5.6). By the Novikov theorem £(-'ljJ.B) is a P-martingale. So we can define a pro~bility measure Q such that ~ = £(-'ljJ.B)T. In order to prove that (St) is a Q-martingale, it suffices to prove that the product

£(-'ljJ.B)Bis a P-martingale. By (5.3) we have

S; =S~exp{it [ai(s)dBs+bi(s)ds] - ~i t lai(sWds}.

Thus from (5.4) we know that

£(-'ljJ.B)tS; =S~exp{it (ai(s) - 'ljJ(s))dBs - ~i t lai(s) - 'ljJ(S)/2ds}.

Once again by the Novikov Theorem£(_'ljJ.B)Si is a P-martingale. 0

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REMARK. Without condition (5.5) Q is an equivalent local martin- gale measure but not necessarily a martingale measure. We refer the reader to Ansel and Stricker (1993) for an investigation on this subject.

The following theorem provides a sufficient condition for the existence of a unique equivalent martingale measures.

THEOREM 5.2. Assume that m ~ d, a satisfies (5.5) and aT(t)a(t) are non-degenerated for a.e. t, where aT(t) stands for the transpose of aCt). Put 'lj;(t) = (aT (t)a(t»-laT(t)b(t). If'lj; satisfies (5.4) and (5.6), then there exists a unique equivalent martingale measure P* for the market. Moreover, we have

E[~IFt] =exp { -it'lj;(s)dBs - ~it1'lj;(s)12ds}, 0::;t::;T.

Proof By Theorem 5.1 there exists an equivalent martingale measure.

To prove the uniqueness, let Q be an equivalent martingale measure.

There exists a (J E (£2)d such that ~ =£(-(J.B)T. By Theorem 5.1, we have a(t)()(t) = bet). Consequently, applying (aT(t)a(t»-laT(t) to the both sides of this equation we get (J(t) = '!/J(t). The uniqueness is thus

proved. 0

REMARK. Ifm = d,then'!/J satisfies(5.4)automatically. In this case if (5.5)-(5.6) are satisfied, then there exists a unique equivalent martingale measure if and onlyifaCt,w) is non-singular, for (t,w) E [0,T] xn, a.e., a.s. See Karatzas (1997).

ACKNOWLEDGEMENT. This work was done by the author during his visit to the Liu Bie Ju Center for Mathematical Sciences of the City University of Hong Kong in the autumn of 1997. The author wishes to thank Professor Roderick Wong for the kind invitation.

References

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Institute of Applied Mathematics Academia Sinica

Beijing, China

E-mail: jayan@amath7.amt.ac.cn

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