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MARTINGALE MEASURES IN THE MARKETWITH RESTRICTED INFORMATION

YANG JIANQI, YAN HAIFENG, AND LIU LIMIN

Received 7 May 2005; Accepted 31 January 2006

This paper considers the problem of the market with restricted information. By con-structing a restricted information market model, the explicit relation of arbitrage and theminimal martingale measure between two different information markets are discussed.Also a link among all equivalent martingale measures under restricted information mar-ket is given .

Copyright © 2006 Yang Jianqi et al. This is an open access article distributed under theCreative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The formula of Black and Scholes for the valuation of options has led to the great de-velopment of mathematical finance. Mathematical finance is attracting more and moreattention of researchers. Some useful work has been done, but the majority of discus-sions are based on perfect markets. A perfect market includes the following conditions:(1) many buyers; (2) many sellers; (3) individual trades do not affect the market; (4) theunits of goods sold by different sellers are the same; (5) there is perfect information, thatis, all buyers and sellers have complete information on the price being asked and offeredin other parts of the market; (6) there is perfect freedom of entry to and exit from the mar-ket. Real financial markets are imperfect markets. In fact there are some investors differentto general investors in the financial market. Because of their conditions, for example, theylive in the country, the investors cannot know all market information such as some in-vest policies, construction plans, and so on, which are known by general investors. Theymight only know price information of risky assets. These make the investor’s informationincomplete. It is well known that hedging market risk and capturing arbitrage opportu-nity are closed to market information. So it conforms to financial application to discussfinancial markets under different information. There are several recent papers dealingwith restricted information in finance. Schweizer [15] presents risk-minimizing hedging

Hindawi Publishing CorporationJournal of Applied Mathematics and Decision SciencesVolume 2006, Article ID 74864, Pages 1–7DOI 10.1155/JAMDS/2006/74864

http://dx.doi.org/10.1155/S1173912606748647

2 Martingale measures in the market with restricted information

strategies of contingent claims under restricted information, Pham [12] researches theproblem of mean-variance hedging for partially observed drift processes, and Frey andRunggaldier [5] focus on the computation of the optimal hedging strategies when assetprice processes are observed at discrete random times. The utility maximization problemwhen only stock prices are observed was studied by Lakner [9]. Initiated by Cox and Ross[1] and Harrison and Kreps [8], the “martingale method” of pricing derivative is one oftwo approaches to the pricing of derivative securities. This approach consists of writingthe value of the security as the expected value of the discounted payoff under a martingalemeasure. If the market is incomplete, then there are many equivalent martingale mea-sures. It may be reasonable to suppose that there should be a special martingale measurewhich determines the prices of contingent claims. As the candidates of such measures,several martingale measures are proposed: minimal martingale measure (Follmer andSchweizer [4]), variance-optimal martingale measure (Schweizer [17] or Delbaen andSchachermayer [3]), canonical martingale measure (Miyahara [10]), and so forth. Theexamples which are given by Schachermayer [14] are useful for the understanding andthe investigation of the relations among the measures above. The importance of minimalmartingale measure is described in Miyahara [10], and so forth. Recently, it is mentionedthat minimal martingale measure is related to the exponential utility function and to thefair prices of options (see Davis [2] and Frittelli [6]). Different from those above, thepaper focuses on the relation of market completeness, arbitrage, and minimal martin-gale measure between markets with different information, which is important to hedgingcontingent claims. To our knowledge, the relation has not been discussed. In the paper,the explicit relation of arbitrage and the minimal martingale measure between two dif-ferent information markets are discussed by constructing restricted information markets.Also the relation of equivalent martingale measures is given under restricted informationmarkets.

2. Market with restricted information

Assume that (Ω,F,�,P) is a probability space with filtration. The filtration �= (�t)0≤t≤Tsatisfies the following assumptions: (1) � is right continuous, that is, �t =

⋂s>t �s; (2) �0

contains all P-null sets in F. On (Ω,F,�,P), define a financial market as follows: assumethat S is a local bounded d-dimensional semimartingale. With S we denote the movementof d risky assets. Also assume that there is a riskless asset denoted by B. For simplicity weassume B ≡ 1 (i.e., S is the discounted asset price). Assume that market participants’s in-vestment behaviors are based on their valid market information. We denote by �t thevalid market information that general investors know up to t. Assume in the above mar-ket, besides general information investors, that there are another investors who know lessmarket information than general investors. We call them restricted information investorsor incomplete information investors. More explicitly, we assume that the restricted infor-mation investors only acquire market information denoted by minor σ-filtration � ratherthan �, where

� = (�t)

0≤t≤T , �St ⊂�t ⊂�t . (2.1)

Yang Jianqi et al. 3

�S = (�St )0≤t≤T denotes the σ-filtration generated by asset price processes S. The market

for restricted information investors, that is, the market with information set �, is calledrestricted information market.

Remark 2.1. Obviously, the assumption for � is reasonable. On the one hand, there aresome investors who cannot achieve all the market information � actually and only knowthe restricted information �t ⊂�t. On the other hand, if the investor did not know thepast price information, he would not throw his money. So the assumption �S

t ⊂�t is alsoreasonable.

3. Main results and proofs

It is well known that market information set is an element of finance markets. Financemarkets vary with the market information set �. Obviously, the problem that market in-formation has what influence on the market completeness and arbitrage is worth study-ing. In this section we will discuss it.

For H ∈ {�,�}, we recall from Grorud and Pontier [7] and Pham [12] some defini-tions and notations.

Definition 3.1. A probability measure Q is called H-equivalent martingale (local mar-tingale) measure for S, if S is an (H ,Q) martingale (local martingale), and dQ/dP ∈L2(P,HT).

Define

M2e (P,H)=

{

Q ∼ P,dQ

dP∈ L2(P,HT

), S is an (H ,Q) martingale

}

,

M2loc(P,H)=

{

Q ∼ P,dQ

dP∈ L2(P,HT

), S is an (H ,Q) local martingale

} (3.1)

to be the equivalent martingale and the local martingale measure sets for S, respectively.

Definition 3.2. Let R ∈M2loc(P,�). If for every H ∈ L2(Ω,�T ,P), there exists an �0-

random variable a and a portfolio �-predictable ϑ,ϑ∈ L2(Ω× [0,T],dR× d[S,S]), suchthat

(C1)

: H= a+d∑

i=1

∫ T

0ϑidSi, (3.2)

then the market is complete for general information investors.

Also, we can define market completeness for the restricted information market.

Definition 3.3. Let R∈M2e (P,�). If for every H∈ L2(Ω,�T ,P), there exists a �0-random

variable a and a portfolio �-predictable ϑ,ϑ∈ L2(Ω× [0,T],dR×d[S,S]), such that

(C2)

: H= a+d∑

i=1

∫ T

0ϑidSi, (3.3)

then the market is complete for restricted information investors.


Remark 3.4. By the capital asset pricing basic theorem, we know that (1) if H0 is triv-ial and M2

loc(P,H) is singleton, the associated market is complete; (2) if M2loc(P,H) is

nonempty, there is no arbitrage in the associated market; the converse result is false. Butthat M2

loc(P,H) is nonempty is equivalent to a weaker property: the “no free lunch withvanish risk” (NFLVR in short).

The theorem below gives the relation of arbitrage between the markets with differentinformation.

Theorem 3.5. If M2e (P,�) �= ∅, then M2

e (P,�) �= ∅.

Proof. Since M2e (P,�) �= ∅, let Q ∈M2

e (P,�), then we know by definition dQ/dP ∈L2(P,�T), S is (�,Q) martingale. Let ZT = dQ/dP, Zt = E(ZT/�t), then Z = (Zt)0≤t≤Tis a strictly positive square integrable martingale on (P,�).

Let ζt = E[ZT/�t], obviously ζt ∈�t and

E[ζt�s

]

= E[E[Zt/�t

]

�s

]

= E[Zt

�s

]

= E[E[Zt/�s

]

�s

]

= E[Zs

�s

]

= ζs. (3.4)

So ζ = (ζt)0≤t≤T is (P,�) martingale. Thus we can define a probability measure Q as fol-lows:

dQ

dP= ζT . (3.5)

Next, we prove Q ∈M2e (P,�).

By definition we only prove three points as follows.(1) Q ∼ P. Since Q ∼ P, from the definition of Q, Q ∼ P holds obviously.(2) dQ/dP ∈ L2(P,�). In fact, only note that ζT = E[ZT/�T], ζT ∈�T ,

then we have

E(dQ

dP

)2

= E[

E(ZT

�T

)]2

≤ E[

E(Z2T

�T

)]

= EZ2T <∞. (3.6)

dQ/dP ∈ L2(P,�) is proved.(3) S is a (Q,�) martingale. By Protter [13], if Q ∼ P, Nt = E[dQ/dP/Ht], then S is

(Q,H) martingale if and only if SN is (P,H) martingale. Thus

E[Stζt�s

]

= E[StE(Zt/�t

)

�s

]

= E[E(StZt/�t

)

�s

]

= E[StZt

�s

]

= E[E(StZt/�s

)

�s

]

= E[SsZs

�s

]

= SsE[Zs

�s

]

= Ssζs.

(3.7)

So Sζ is a martingale, and S is a (Q,�) martingale. �


Remark 3.6. (1) Generally speaking, because �-stopping times are not always �-stoppingtimes, M2

loc(P,�) �= ∅ does not imply M2loc(P,�) �= ∅.

(2) Under the condition that S is a bounded semimartingale, because M2e (P,H) =

M2loc(P,H), the nonarbitrage market for general information investor is also nonarbi-

trage for restricted information investor. That is a result in accordance with the marketfact that the investors with more information can easy gain more arbitrage opportunitiesthan those with less information. The converse result is false certainly.

For further discussion, we recall from Schweizer [16] the definition of minimal mar-tingale measure.

Let price process S be a P semimartingale with canonical decomposition St = S0 +Mt +At, M is a local martingale and A is a predictable finite variation process.

Definition 3.7. Suppose that S satisfies the structure condition (SC): S= S0 +M + λd〈M,M〉. Moreover, Z = ε(−∫ λdM) is a P-martingale, then call P defined by ZT = dP/dPminimal signal martingale measure for S, H-minimal martingale measure if in additionP ∈M2

e (P,H).

Remark 3.8. If H-minimal martingale measure for S exists, then it is unique.For discussing the relation of minimal martingale measure between markets with dif-

ferent information, we introduce a property which is an equivalent definition of minimalmartingale measure in essence. (see Pham [11] for detail)

Lemma 3.9. An equivalent martingale P is a minimal martingale measure for S if and onlyif any square integrable martingale under P and orthogonal to M remains a martingaleunder P.

Theorem 3.10. If every � martingale is also an � martingale, P� is an �-minimal mar-tingale measure, Z is P�’s density process with respect to P, then P� defined by (3.5) is alsoa �-minimal martingale measure.

Proof. Let S = S0 +M +A, K be a square integrable (P,�) martingale orthogonal to M.Thus K is also a square integrable (P,�) martingale. Using Lemma 3.9, we only provethat K is a (P�,�) martingale.

By the property of conditional expectation and the definition of martingale, for allt > s,

E[Ktζt�s

]

= E[KtE

(Zt/�t

)

�s

]

= E[E(KtZt/�t

)

�s

]

= E[KtZt

�s

]

= E[E(KtZt/�s

)

�s

]

= E[KsZs

�s

]

= KsE[Zs

�s

]

= Ksζs.

(3.8)

So Kζ is P-martingale, By Protter [13, Lemma, page 109], K is (P�,�) martingale. Theo-rem 3.10 follows immediately from Lemma 3.9.

Obviously, from the definitions of completeness, we know that under the condition�T = �T , if restricted information market is complete, then general information market


is also complete. Generally speaking, there is not close relation of completeness betweenmarkets with different information. But there is really a link among all the equivalentmartingale in M2

loc(P,�). �Theorem 3.11. In case of a complete market for the restricted information investor (i.e.,verifying (C2)) such that there exists Q ∈M2

loc(P,�) for which the discounted prices S are(Q,�S)-martingales, then every R∈M2

loc(P,�) is equal to f ·Q, where f ∈ L1(�S0,Q).

Proof. For all R∈M2loc(P,�), let dR/dQ=ZT , Zt=EQ[ZT/�S

t ], so we only prove ZT ∈L1(�S

0,Q). Let Q ∈M2loc(P,�), but S is an R-local martingale, thus SiZ is a Q-local mar-

tingale, By Ito formula, [Si,Z] = SiZ − ∫ Si−dZ −∫Z−dsi. Note that SiZ is a Q-local mar-

tingale, we have get [Si,Z]= 0, thus Z is orthogonal to price processes S. Since the marketis complete for the restricted information investor, there exist a �0-random variable a anda �0-predictable portfolio ϕ∈ L2(Ω× [0,T],dR⊗ [S]) such that ZT = a+

∑di=i∫ T

0 ϕidSi.Because Si is an (�S,Q)-martingale, ϕ · S is an (�S,Q)-martingale. Thus Zt − Z0 =

∑di=i∫ t

0 ϕidSi is strongly orthogonal to the stable space generalized by prices, then it is

orthogonal to itself; therefore Zt −Z0 = 0 and ZT is an �S0-measurable random variable.

�

Remark 3.12. By martingale pricing method, the price of a contingent claim is its ex-pected payoff under a special equivalent martingale measure. Because there are manyequivalent martingale measures in an incomplete market, there are many nonarbitrageprice for a contingent claim. Under an explicit market, we can deduce the link amongmany arbitrage-less prices using Theorem 3.11.

4. Conclusion

The paper discusses an imperfect market with restricted information. Based on con-structing restricted markets and martingale theory, we strictly prove the result that nonar-bitrage market for general information investors is also nonarbitrage for restricted infor-mation investors. Also the explicit relation of minimal martingale measure between twodifferent information markets is given. Finally, a link among all equivalent martingalemeasures in the class of restricted local martingale measures, which is important in pric-ing contingent claims, is derived.

Acknowledgments

This research is supported by Henan Province Science and Technology Commission SoftScience Research Project (0313062400) and Hunan University of Science and EngineeringScience Research Project. We thank the three referees for their remarks on the first draftof this paper.

References

[1] J. C. Cox and S. Ross, The valuation of options for alternative stochastic processes, Journal of Fi-nancial Economics 3 (1976), no. 1-2, 145–166.

[2] M. H. A. Davis, Option pricing in incomplete markets, Mathematics of Derivative Securities(Cambridge, 1995) (M. A. H. Dempster and S. R. Pliska, eds.), Publ. Newton Inst., vol. 15,Cambridge University Press, Cambridge, 1997, pp. 216–226.


[3] F. Delbaen and W. Schachermayer, The variance-optimal martingale measure for continuous pro-cesses, Bernoulli 2 (1996), no. 1, 81–105.

[4] H. Follmer and M. Schweizer, Hedging of contingent claims under incomplete information, Ap-plied Stochastic Analysis (London, 1989) (M. H. A. Davis and R. J. Elliot, eds.), StochasticsMonogr., vol. 5, Gordon and Breach, New York, 1991, pp. 389–414.

[5] R. Frey and W. J. Runggaldier, Risk-minimizing hedging strategies under restricted information: thecase of stochastic volatility models observable only at discrete random times, Mathematical Methodsof Operations Research 50 (1999), no. 2, 339–350.

[6] M. Frittelli, The minimal entropy martingale measure and the valuation problem in incompletemarkets, Mathematical Finance 10 (2000), no. 1, 39–52.

[7] A. Grorud and M. Pontier, Asymmetrical information and incomplete markets, InternationalJournal of Theoretical and Applied Finance 4 (2001), no. 2, 285–302.

[8] J. M. Harrison and D. M. Kreps, Martingales and arbitrage in multiperiod securities markets,Journal of Economic Theory 20 (1979), no. 3, 381–408.

[9] P. Lakner, Optimal trading strategy for an investor: the case of partial information, Stochastic Pro-cesses and Their Applications 76 (1998), no. 1, 77–97.

[10] Y. Miyahara, Canonical martingale measures of incomplete assets markets, Probability Theoryand Mathematical Statistics: Proceedings of the 7th Japan-Russia Symposium (Tokyo, 1995) (S.Watanabe, M. Fukushima, Yu. V. Prohorov, and A. N. Shiryaev, eds.), World Scientific, NewJersey, 1996, pp. 343–352.

[11] H. Pham, On quadratic hedging in continuous time, Mathematical Methods of Operations Re-search 51 (2000), no. 2, 315–339.

[12] , Mean-variance hedging for partially observed drift processes, International Journal ofTheoretical and Applied Finance 4 (2001), no. 2, 263–284.

[13] P. Protter, Stochastic Integration and Differential Equations, Applications of Mathematics (NewYork), vol. 21, Springer, Berlin, 1990.

[14] W. Schachermayer, A counterexample to several problems in the theory of asset pricing, Mathemat-ical Finance 3 (1993), 217–229.

[15] M. Schweizer, Risk-minimizing hedging strategies under restricted information, Mathematical Fi-nance 4 (1994), no. 4, 327–342.

[16] , On the minimal martingale measure and the Follmer-Schweizer decomposition, Stochas-tic Analysis and Applications 13 (1995), no. 5, 573–599.

[17] , Variance-optimal hedging in discrete time, Mathematics of Operations Research 20(1995), no. 1, 1–32.

Yang Jianqi: Department of Mathematics, Hunan University of Science and Engineering, Yongzhou,Hunan 425006, ChinaE-mail address: [email protected]

Yan Haifeng: School of Finance and Banking, Nanjing University of Finance and Economics,Nanjing, Jiangsu 210046, ChinaE-mail address: [email protected]

Liu Limin: Department of Mathematics, Henan Normal University, Xinxiang, Henan 453008, ChinaE-mail address: [email protected]

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