Noname manuscript No. (will be inserted by the editor)

On the law of one price

Jean-Michel Courtault^{1}, Freddy Delbaen^{2} ^{⋆}, Yuri Kabanov^{3}, Christophe Stricker^{4}

1 L.I.B.R.E., Université de Franche-Comté, Avenue de l’Observatoire, F-25030 Besançon Cedex, France (e-mail: jean-michel.courtault@univ-fcomte.fr)

2 Department für Mathematik, Eidgenössische Technische Hochschule Zürich, ETH-Zentrum, CH-8092 Zürich, Schweiz (e-mail: delbaen@math.ethz.ch )

3 Université de Franche-Comté, 16 Route de Gray, F-25030 Besançon Cedex, France, and Central Economics and Mathematics Institute, Moscow, Russia (e-mail: kabanov@math.univ-fcomte.fr)

4 Université de Franche-Comté, 16 Route de Gray, F-25030 Besançon Cedex, France (e-mail: stricker@math.univ-fcomte.fr)

The date of receipt and acceptance will be inserted by the editor

Abstract We consider the standard discrete-time model of a frictionless financial market and show that the law of one price holds if and only if there exists a martingale density process with strictly positive initial value. In contrast to the classical no-arbitrage criteria, this density process may change its sign. We also give an application to the CAPM.

Key words Law of one price, Harrison–Pliska theorem, Dalang–Morton–Willinger theorem, market portfolio, CAPM

Mathematics Subject Classification (2000): 60G44

JEL Classification Numbers: G13, G11

⋆ This research was done during the stay of the author at Université de Franche-Comté.

1 Introduction

To explain our interest in the question discussed in this paper we recall several facts and concepts from the foundations of stochastic finance. The fundamental point of this theory is that there are basic securities with prices evolving as random processes; the randomness is inherited by the investors portfolios. In the simplest one-period model where we have t ∈ {0, T } (the terminal date T may be denoted by 1), the numéraire is a traded asset, the price increments of d basic securities are described simply by a random d-dimensional vector ξ. The portfolio strategy is just a (deterministic) vector h ∈ R^{d} and the portfolio increment is the scalar product hξ. One says that the market is arbitrage free (briefly: satisfies the NA property) if the inequality hξ ≥ 0 may hold only if hξ = 0. This property plays a key role for the pricing of derivative securities. In the case of finite Ω it is an easy exercise on the use of the separation theorem to check that the NA property is equivalent to the existence of a (scalar) random variable Z which is strictly positive (a.s.), EZ = 1, and the vector EZξ is zero. In the economic literature the probability space (Ω, F, P̃) with P̃ = ZP is called the “risk-neutral world”. Rather remarkably, this simple exercise happened to be the germ of an important development in mathematical finance. The first steps of this development are now classical. The Harrison–Pliska theorem gives an extension to the multi-period model (with finite Ω) claiming that NA holds iff there exists a strictly positive (a.s.) martingale Z = (Z_{t}) such that EZ_{t}ξ_{t} = 0 for all t. The Dalang–Morton–Willinger theorem, a result which is mathematically much more delicate, extends this assertion to an arbitrary Ω, adding also that one can always choose a bounded density process Z. At the moment, no-arbitrage criteria are obtained for numerous models (continuous-time models, models with transaction costs, models with constraints), see the handbook [2].

However, the NA property is not a single one isolated in the economic literature. A weaker property, called the law of one price (we shall use the abbreviation L1P), for the one-period model can be formulated as follows: the identity x + hξ = x′ + h′ξ where h, h′ ∈ R^{d} implies that x = x′, see, e.g. [7]. In other words, if a contingent claim has a price, namely, the replication price, this price is unique. Clearly, this property is always fulfilled if NA holds. Again, in this elementary case, it is an easy exercise on a finite-dimensional separation theorem to check that L1P holds iff there exists a random variable Z (not necessary positive) with EZ = 1 such that the vector EZξ is zero. Similarly, an extension to the multi-period model with a finite number of states of nature (i.e. to the setting of the Harrison–Pliska theorem) does not pose new mathematical difficulties. Our aim here is to analyze the law of one price for the multi-period model with a general probability space in the same spirit as it was done in the recent note [5]. It is worthy to mention that similarly to NA criteria, there are several strategies of proof. We opt that of the mentioned note.

At last, we introduce the notion of normalized excess expected return of a strategy and show that such a week condition as LPT1 implies already the existence of a market portfolio.

2 The law of one price

Let (Ω, F, P) be a probability space equipped with a finite discrete-time filtration (F_{t}), t = 0, ..., T, F_{T} = F, and let S = (S_{t}) be an adapted d-dimensional process. Let R_{T} := {ξ : ξ = H · S_{T} , H ∈ P} where P is the set of all predictable d-dimensional processes (i.e. H_{t} is F_{t−1}-measurable) and let

H · S_{T} := ∑_{t=1}^{T} H_{t}ΔS_{t}, ΔS_{t} := S_{t} − S_{t−1}.

We denote by L^{0}(F_{t}) the set of finite F_{t}-measurable random variables.

The linear subspace R_{T} is closed in L^{0}(F_{T}). This fact, fundamental in the sequel, was established in [5].

We say that the model satisfies the law of one price (L1P) at the date t = 0 if the equality ζ + H · S_{T} = ζ′ + H′ · S_{T} where H, H′ ∈ P and ζ, ζ′ ∈ L^{0}(F_{0}), implies that ζ = ζ′ (a.s).

It is easily seen that this condition, L1P, at t = 0, can be written as follows: R_{T} ∩ L^{0}(F_{0}) = {0}.

Theorem 1 The following conditions are equivalent:

(a) R_{T} ∩ L^{0}(F_{0}) = {0};

(b) there is a bounded martingale Z = (Z_{t})_{t≤T} with EZ_{T} = 1 and Z_{0} > 0 such that the process ZS is a martingale.

Proof. (a) ⇒ (b) Take an arbitrary non-null F_{0}-measurable set A. By the assumption I_{A} ∉ R_{T}. Choose a probability measure P̃ ∼ P with the bounded density ρ := dP̃/dP and such that all S_{t} are integrable with respect to P̃. Since R_{T} is closed in probability, the set R_{T}^{1} := R_{T} ∩L^{1}(P̃) is a closed linear space in L^{1}(P̃). Thus, there exists a bounded random variable Z̃_{T}^{A} such that ẼZ̃_{T}^{A}η = 0 for all η ∈ R_{T}^{1} but ẼZ̃_{T}^{A}I_{A} > 0. Putting Z_{T}^{A} := ρZ̃_{T}^{A} and normalizing, if necessarily, we can rephrase this as follows: there exists Z_{T}^{A} with |Z_{T}^{A}| ≤ 1 such that EZ_{T}^{A}η = 0 for all η ∈ R_{T}^{1} but EZ_{T}^{A}I_{A} > 0. The set R_{T}^{1} is stable under multiplication by the indicator functions of F_{0}-measurable sets. It follows that the above properties remain valid if we replace Z_{T}^{A} by I_{A}I_{{E(Z_{T}^{A}|F_{0})>0}}. Hence, we may assume without loss of generality that E(Z_{T}^{A}|F_{0}) ≥ 0.

The usual exhaustion arguments ensure that there is a bounded random variable Z_{T} with E(Z_{T} |F_{0}) > 0 such that EZ_{T} η = 0 for all η ∈ R_{T}^{1} . For the reader’s convenience we recall them. Let C be the family formed by all sets of the form {E(Z_{T}^{A}|F_{0}) > 0}, A ∈ F_{0}. Let a := sup_{Γ ∈C} P (Γ). The supremum here is attained: it is sufficient to consider the set A = ∪_{n}A_{n} and the bounded random variable Z_{T}^{A} := ∑_{n} 2^{−n}Z_{T}^{A_{n}} where A_{n} ∈ F_{0} are such that P (A_{n}) → a. It remains to notice that P (A) = 1 (otherwise we could increase the supremum with Z^{A^{c}}).

We conclude by putting Z_{t} := E(Z_{T} |F_{t}) and by observing that the martingale property of ZS holds because ξΔS_{t} ∈ R_{T}^{1} for every bounded F_{t−1}-measurable random variable ξ.

(b) ⇒ (a) This claim is obvious. Indeed, we should check that ξ = 0 whenever ξ+H ·S_{T} = 0. For the process M := Z(ξ+H ·S) we have M_{t−1} = E(M_{t}|F_{t−1}) for t ≥ 1. As M is zero at the terminal date T, it is zero identically. In particular, its initial value Z_{0}ξ = 0. Since Z_{0} > 0, this implies that ξ = 0. □

3 Ramifications and comments

1. It is quite natural to have a look at the situation where the portfolio strategies are subjects some trading constrains. For instance, shortselling may be prohibited. To be specific, suppose that we are given a closed convex cone K ⊆ R^{d} and at each date t the vector of holdings in risky assets H_{t} belongs to L^{0}(K, F_{t−1}), i.e. H is a predictable d-dimensional process taking values in K. Let us denote the set of such processes by P^{K} and the corresponding set of “results” R_{T}^{K}.

The law of one price (at time zero), abbreviated in this model as L1P^{K}, means that the equality ζ +H ·S_{T} = ζ′ +H′ ·S_{T} where H, h′ ∈ P^{K} and ζ, ζ′ ∈ L^{0}(F_{0}) may hold only if ζ = ζ′ (a.s).

This property can be expressed by the relation (R_{T}^{K} − R_{T}^{K}) ∩ L^{0}(F_{0}) = {0}. Notice that

L^{0}(K, F_{t}) − L^{0}(K, F_{t}) = L^{0}(K − K, F_{t})

(to check the non-trivial implication ⊇ it is sufficient to fix a basis {v_{i}} in K − K and consider an arbitrary representation v_{i} = v′_{i} − v′′_{i} with v′_{i}, v′′_{i} ∈ K).

Recall that K −K = R^{d} if and only if int K ≠ ∅. Consequently, if int K ≠ ∅, then R_{T} = R_{T}^{K} − R_{T}^{K} and we arrive at the following theorem which covers, in particular, the model where shortselling is not allowed.

Theorem 2 Assume that int K ≠ ∅. Then the following conditions are equivalent: (a) L1P^{K};

(b) there is a bounded martingale Z = (Z_{t})_{t≤T} with EZ_{T} = 1 and Z_{0} > 0 such that the process ZS is a martingale.

Without difficulties this result can be extended to the case where K_{t} are F_{t−1}- measurable (as usual, this needs a bit of set-valued analysis).

2. It is economically reasonable that the law of one price holds globally, i.e. if a contingent claim is replicable when the trading starts at the date τ, then the replication price is unique. Formally, we can isolate the following GL1P property:

the equality ζ +H ·^{τ}S_{T} = ζ′+H′·^{τ}S_{T} where τ is a stopping time, H, H′ ∈ P, and ζ, ζ′ ∈ L^{0}(F_{τ}) may hold only if ζ = ζ′ (a.s).

Here we use the abbreviation^{τ}S_{t} := I_{]τ,T]} · S.

The following assertion is a corollary of Theorem 1 (cf. with the formulation of Th. 4.5 in [6]).

Theorem 3 The following conditions are equivalent:

(a) GL1P^{K};

(b) for every stopping time τ there exists a bounded martingale Z^{(τ)} with EZ_{T}^{(τ)} = 1 and Z_{τ}^{(τ)} > 0 such that the process Z^{(τ)} ^{τ}S is a martingale.

Note that replacing in the formulation of GL1P property the words “where τ is a stopping time” by “where τ

Note that using in the formulation of GL1P property only deterministic times gives an equivalent definition.

3. As usual, one can find in the literature on linear inequalities related results corresponding to the one-step model and formulated as alternative theorems. As an example, we give a relevant one: either the equation Ay = b has a solution or there is a vector z ≠ 0 such that zb = 1 and zA = 0. In the case where Ω := {ω_{1}, ..., ω_{N}}, the N ×d-matrix A is formed by the elements a^{ij} := ΔS^{j}(ω_{i}) and b = (1, ..., 1)^{T}.

4. Unfortunately, the above results have no natural counterparts for continuoustime models. “Natural” here means “for the standard concept of admissibility”. The latter requires that the value process is bounded from below. To see this, consider the model where S is just a Wiener process, T = ∞. The process V = e^{S− \frac{1}{2} ⟨S⟩} is a value process (corresponding to the admissible strategy H = V and the initial value x = 1) with V_{∞} = 0. Thus, we have that 1 + H · S = 0 · S violating the law of one price. Of course, an appropriate modification of this model provides an example where T is finite.

4 The law of one price and CAPM

The law of one price implies an interesting and important consequence: the existence of a market portfolio.

To show this we make an extra assumption that all S_{t} ∈ L^{2} and define, in the Hilbert space L^{2}, the closed linear subspace R_{T}^{2} = R_{T} ∩ L^{2} (recall that R_{T} is closed in L^{0}). It contains all terminal values of portfolios with bounded strategies and starting from zero. The random variable x + H · S_{T} is the terminal value (expressed in the units of the numéraire assumed to be a traded security) of a portfolio with the initial endowment x. Hence ξ := H · S_{T} is the surplus of this strategy with respect to holding x in the numéraire; it does not depend of the initial endowment, but for x = 1 it is usually referred in the economic literature as the excess return of the portfolio. We define as the (normalized) excess expected return of the strategy H the ratio r_{H} := \frac{Eξ}{σ(ξ)} where σ is the standard deviation; r_{H} is called the Sharpe ratio. By convention, \frac{0}{0} = 0. Since r_{λH} = r_{H} for λ > 0, this normalization allows us to compare “quality” of the portfolio composition independently of its “size”. We say that a strategy G with η := G · S_{T} ∈ R_{T}^{2} defines a market portfolio if for every strategy with the terminal value ξ := H · S_{T} in R_{T}^{2} we have the equality

r_{H} = β_{GH}r_{G}  (1)

where β_{GH} is the correlation coefficient ρ(ξ, η). Notice that (1) looks like the usual CAPM relation, see, e.g., [4].

For the case where P is a martingale measure the identity is trivial. We exclude this situation in the formulation below.

Proposition 4 Suppose that P is not a martingale measure and L1P holds. Then the market portfolio does exist.

Proof. The requirement is met by the strategy G corresponding to the projection η of the unit onto the subspace R_{T}^{2} . Indeed, by definition, E(1 − η)ξ = 0 for each ξ ∈ R_{T}^{2} . In particular, E(1 − η)η = 0 and, therefore, Eη = Eη^{2} ≥ 0. In fact, the inequality here is strict: otherwise η is zero and Eξ = 0 for each ξ ∈ R_{T}^{2} implying that P is a martingale measure. The case Eη^{2} = 1 is also impossible: since Eη^{2} + E(1 − η)^{2} = 1 we would have that η = 1 violating L1P. Thus, 0 < Eη < 1. Now we write the relation E(1 − η)ξ = 0 in a different way:

Eξ = Eξη = Cov (ξ, η) + EξEη

where Cov (ξ, η) is the covariance of ξ and η.

We deduce from here taking into account the equality Eη = Eη^{2} that

Eξ = \frac{Cov (ξ, η)}{1 − Eη} = \frac{Cov (ξ, η)Eη}{Eη − (Eη)^{2}} = \frac{Cov (ξ, η)Eη}{σ^{2}(η)} .

Hence,

\frac{Eξ}{σ(ξ)} = \frac{Cov (ξ, η)}{σ(ξ)σ(η)} \frac{Eη}{σ(η)}

and we obtain the required relation (1). □ Remark. In fact, the above proof is not needed: simple geometric considerations replace the above arguments. Indeed, r_{H}, r_{G}, β_{GH} are the cosines of angles between ξ and 1, η and 1, ξ and η. This observation shows that (1) means that the projection of the vector 1 onto ξ can be obtained in two steps: first we project 1 onto the plane of the vectors ξ and η and then project the result onto ξ. A picture makes this obvious.

References

1. Dalang R.C., Morton A., Willinger W.: Equivalent martingale measures and noarbitrage in stochastic securities market model. Stochastics and Stochastic Reports, 29, 185–201 (1990)

2. Handbooks in Mathematical Finance: Topics in Option Pricing, Interest Rates and Risk Management. Eds. E. Jouini, J. Cvitanić, M. Musiela. Cambridge: Cambridge University Press, 2001

3. Harrison M., Pliska S.: Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications, 11, 215–260 (1981)

4. Huang C.-F., Litzenberger R.H.: Foundations of Financial Economics. New York: North-Holland 1988

5. Kabanov, Yu.M., Stricker, Ch.: A teachers’ note on no-arbitrage criteria. Séminaire de Probabilités XXXV, Lect. Notes in Math., 1755, 149-152, Berlin Heidelberg New York: Springer 2001

6. Kabanov, Yu.M., Stricker, Ch.: Remarks on the true no-arbitrage property. Preprint (2003)

7. Pliska S.: Introduction to Mathematical Finance: Discrete Time Models. Cambridge: Blackwell 1997