A Proofs and technical results

Shadow prices for continuous processes1

Abstract

In a financial market with a continuous price process and proportional transaction costs we investigate the problem of utility maximization of terminal wealth. We give sufficient conditions for the existence of a shadow price process, i.e., a least favorable frictionless market leading to the same optimal strategy and utility as in the original market under transaction costs. The crucial ingredients are the continuity of the price process and the hypothesis of “no unbounded profit with bounded risk”. A counterexample reveals that these hypotheses cannot be relaxed.

MSC 2010 Subject Classification: 91G10, 93E20, 60G48

JEL Classification Codes: G11, C61

Key words: utility maximization, proportional transaction costs, convex duality, shadow prices, continuous price processes.

1 Introduction

In this paper, we analyze continuous -valued stock price processes under proportional transaction costs . We investigate the duality theory for portfolio optimization, sometimes also called the “martingale method”, under proportional transaction costs as initiated in the seminal paper [9] by Cvitanić and Karatzas.

We build on our previous paper [11], where the duality theory was analyzed in full generality, i.e., in the framework of càdlàg (right-continuous with left limits) processes . Our present purpose is to show that the theory simplifies considerably if we restrict ourselves to continuous processes . More importantly, we obtain sharper results than in the general càdlàg setting on the theme of the existence of a shadow price. This a price process such that frictionless trading for this price process leads to the same optimal strategy as trading in the original market under transaction costs. It is folklore going back to the work of Cvitanić and Karatzas [9] that, if the minimiser of a suitable dual problem is induced by a local martingale, rather than a supermartingale, there exists a shadow price process in the sense of Definition 2.12 below. Let us quote Cvitanić and Karatzas [9] on the hypothesis that the dual optimizer is induced by a local martingale:“This assumption is a big one!” To the best of our knowledge, previously to the present paper there have been no theorems providing sufficient conditions for this local martingale property to hold true. Our first main result (Theorem 3.2 below) states that, assuming that is continuous and satisfies the condition of “no unbounded profit with bounded risk” , we may conclude — assuming only natural regularity conditions — that the local martingale property holds true, and therefore there is a shadow price process in the sense of Definition 2.12 below. For this theorem to hold true, the assumption of is crucial. It is not possible to replace it by the assumption of the existence of a consistent price system for each level of transaction costs (abbreviated ), which at first glance might seem to be the natural hypothesis in the context of transaction costs. The example constructed in Proposition 4.1, which constitutes the second main result of this paper, yields a continuous process , satisfying for each , and such that there is no shadow price in the sense of Definition 2.12 below. In fact, satisfies the stickiness condition introduced by Guasoni [19].

The paper is organized as follows. In Section 2, we fix notations and formulate the problem. This section mainly consists of applying the general results obtained in [11] to the general case of càdlàg processes to the present case of continuous processes. Section 3 contains the main result, Theorem 3.2, which gives sufficient conditions for the existence of a shadow price process. In Section 4, we construct the above mentioned counterexample. The technicalities of this example are postponed to the Appendix.

2 Formulation of the Problem

We fix a time horizon and a continuous, -valued stock price process , based on and adapted to a filtered probability space , satisfying the usual conditions of right continuity and saturatedness. We also fix proportional transaction costs . The process models the bid and ask price of the stock respectively, which means that the agent has to pay a higher ask price to buy stock shares but only receives a lower bid price when selling them.

As in [11] we define trading strategies as -valued, optional, finite variation processes , modeling the holdings in units of bond and stock respectively, such that the following self-financing condition is satisfied:

(2.1)

for all . The integrals are defined as pathwise Riemann-Stieltjes integrals, and , denote the components of the Jordan-Hahn decomposition of . We recall that a process of finite variation can be decomposed into two non-decreasing processes and such that

There is a pleasant simplification as compared to the general setting of [11]. While in the case of a càdlàg process it does make a difference whether the jumps of are on the left or on the right side, this subtlety does not play any role for continuous processes . Indeed, if satisfies (2.1), then its left-continuous version as well as its right-continuous version also satisfy (2.1). Therefore, we are free to impose any of these properties. It turns out that the convenient choice is to impose that the process is right-continuous, and therefore càdlàg, which is formalized in Definition 2.1 below. Indeed, in this case is a semimartingale so that the Riemann-Stieltjes integrals in (2.1) may also be interpreted as Itô integrals and we are in the customary realm of stochastic analysis. But occasionally it will also be convenient to consider the left-continuous version , which has the advantage of being predictable. We shall indicate if we pass to the left-continuous version . Again by the continuity of , trading strategies can be assumed to be optional.

Definition 2.1.

Fix the level of transaction costs.

For an -valued process , we define the liquidation value at time by

(2.2)

The process is called admissible if

(2.3)

for all .

For , we denote by the set of admissible, -valued, optional, càdlàg, finite variation processes , starting with initial endowment and satisfying the self-financing condition (2.1).

As we deal with the right-continuous processes , we have the usual notational problem of a jump at time zero. This is done by distinguishing between the value above and . In accordance with (2.1), we must have

i.e.,

We can now define the (primal) utility maximization problem. Let be an increasing, strictly concave, and smooth function, satisfying the Inada conditions and , as well as the condition of “reasonable asymptotic elasticity” introduced in [27]

(2.4)

Denote by the convex subset in

(2.5)

which equals the set as defined in [11].

For given initial endowment , the agent wants to maximize expected utility at terminal time , i.e.,

(2.6)

In our search for a duality theory, we have to define the dual objects. The subsequent definition formalizes the concept of consistent price processes. It was the insight of Jouini and Kallal [22] that this is the natural notion which, in the case of transaction costs, corresponds to the concept of equivalent martingale measures in the frictionless case.

Definition 2.2.

Fix and the continuous process as above. A -consistent price system is a two dimensional strictly positive process with , that consists of a martingale and a local martingale under such that

(2.7)

for .

We denote by the set of -consistent price systems. By we denote the set of processes as above, which are only required to be non-negative (where we consider (2.7) to be satisfied if ).

We say that satisfies the condition of admitting a -consistent price system, if is non-empty.

We say that satisfies locally the condition , if there exists a strictly positive process and a sequence of -valued stopping times, increasing to infinity, such that each stopped process defines a consistent price system for the stopped process .

Remark 2.3.

The central question of this paper, namely the existence of a shadow price, turns out to be of a local nature. Hence the condition of satisfying locally will turn out to be the natural one compare Definition 2.5 below. This is analogous to the frictionless setting where , which is the local version of the condition of “no free lunch with vanishing risk” , turns out to be the natural assumption for utility maximization problems.

The definitions above have been chosen in such a way that the following result which is analogous to [6, Theorem 3.5] holds true. For an explicit proof in the present setting see [11, Lemma A.1].

Theorem 2.4.

Fix , transaction costs and the continuous process as above. Suppose that satisfies locally for all .

Then the convex set in is closed and bounded with respect to the topology of convergence in measure.

More precisely, has the following convex compactness property (compare [35, Proposition 2.4]): given a sequence in , there is a sequence of forward convex combinations such that converges a.s. to some

The main message is the closedness (resp. the convex compactness) property of the set of attainable claims over which we are going to optimize. It goes without saying that such a closedness property is of fundamental importance when we try to optimize over as in (2.6). In the frictionless case, such a closedness property is traditionally obtained under the assumption of “no free lunch with vanishing risk” (compare [14]). It was notably observed by Karatzas and Kardaras [25] (in the frictionless setting) that – as mentioned in Remark 2.3 – it is sufficient to impose this property only locally when we deal with trading strategies which at all times have a non-negative value. Compare also [8], [33], [18] and [34].

Similarly, in the present setting of Theorem 2.4 it turns out that it suffices to impose a local assumption, namely the local assumption of , for all , as has been observed by [2].

We now translate Definition 2.2 into the language of local and supermartingale deflators as introduced in [25] and [26] in the frictionless setting, and in [2] and [11] in the setting of transaction costs.

Definition 2.5.

Fix and the continuous process as above.

The set (resp. ) of -consistent local martingale deflators consists of the strictly positive (resp. non-negative) processes , normalized by , such that there exists a localizing sequence of stopping times so that is in (resp. ) for the stopped process .

The set (resp. ) of -consistent supermartingale deflators consists of the strictly positive (resp. non-negative) processes , normalized by , such that takes values in and such that, for every , the value process

(2.8)

is a supermartingale under

Contrary to [11], where we were forced to consider optional strong supermartingales, in the present setting of continuous we may remain in the usual realm of (càdlàg) supermartingales in the above definition - compare Proposition 3.3 and Proposition 3.4. We use the letter to denote supermartingales rather then the letter , which will be reserved to (local) martingales.

Obviously , for , amounts to requiring that holds true locally.

Using the notation as in (2.7), we may rewrite the value process as

(2.9)

Comparing (2.9) to the liquidation value in (2.2), we infer that as takes values in . The admissibility condition (2.3) therefore implies the non-negativity of . Looking at formula (2.9) one may interpret as a valuation of the stock position by some element in the bid-ask spread while plays the role of a deflator, well known from the frictionless theory.

The next result states the rather obvious fact that supermartingale deflators are a generalization of local martingale deflators. It will be proved in the Appendix.

Proposition 2.6.

Fix and a continuous process . Then

i.e., a -consistent local martingale deflator is a -consistent supermartingale deflator.

We now are in a position to define the set of dual variables which will turn out to be polar to the set of primal variables defined in (2.5).

Definition 2.7.

Fix and a continuous process as above. For we denote by the set of supermartingale deflators , starting at . More formally, consists of all non-negative supermartingales such that and

for all , and such that is a supermartingale for all . We denote by the set .

We denote by the set of random variables such that there is a supermartingale deflator , whose first coordinate has terminal value . We denote by the set .

The definition of supermartingale deflators is designed so that the following closedness property holds true. A subset in is called solid, if and implies that .

Proposition 2.8.

Fix and the continuous process .

Then the set is a convex, solid subset of bounded in norm by one, and closed with respect to convergence in measure. In fact, for a sequence in , there is a sequence of convex combinations such that converges a.s. to some

This proposition goes back to [27] and was explicitly stated and proved in the frictionless case in [25]. In the present transaction cost setting, it was proved in [11, Lemma A.1] in the framework of càdlàg processes.

Now we can state the polar relation between and . In fact, these sets satisfy verbatim the conditions isolated in ([27, Proposition 3.1]). Here is the precise statement. For an explicit proof in the present setting see [11, Lemma A.1].

Proposition 2.9.

Fix the continuous process and . Suppose that satisfies locally for all . We then have:

  1. The sets and are solid, convex subsets of which are closed with respect to convergence in measure.
    Denoting by the set of terminal values , where ranges in , the set equals the closed, solid hull of .

  2. For we have that iff we have for all .
    For we have that iff we have for all .

  3. The set is bounded in and contains the constant function

We can now conclude from the above Proposition 2.9 that the theorems of the duality theory of portfolio optimization, as obtained in [27, Theorem 3.1 and 3.2], carry over verbatim to the present setting as these theorems only need the validity of this proposition as input. We recall the essence of these theorems.

Theorem 2.10 (Duality Theorem).

In addition to the hypotheses of Proposition 2.9 suppose that there is a utility function satisfying (2.4) above. Define the primal and dual value function as

(2.10)
(2.11)

where

is the conjugate function of , and suppose that for some Then the following statements hold true.

  1. The functions and are finitely valued, for all , and mutually conjugate

    The functions and are continuously differentiable and strictly concave (resp. convex) and satisfy

  2. For all , the solutions in (2.10) and in (2.11) exist, are unique and take their values a.s. in . There are and such that

  3. If and are related by , or equivalently , then and are related by the first order conditions

    (2.12)

    and we have that

    (2.13)

    In particular, the process is a uniformly integrable -martingale.

After these preparations, which are variations of known results, we now turn to the central topic of this paper.

The Duality Theorem 2.10 asserts the existence of a strictly positive dual optimizer , which implies that there is an equivalent supermartingale deflator such that . We are interested in the question whether the supermartingale can be chosen to be a local martingale. We say “can be chosen” for the following reason: it follows from above that the first coordinate of is uniquely determined; but we made no assertion on the uniqueness of the second coordinate .

The phenomenon that the dual optimizer may be induced by a supermartingale only, rather than by a local martingale, is well-known in the frictionless theory ([27, Example 5.1 and 5.1’]). This phenomenon is related to the singularity of the utility function at the left boundary of its domain, where we have . If one passes to utility functions which take finite values on the entire real line, e.g., , the present “supermartingale phenomenon” does not occur any more (compare [31]).

In the present context of portfolio optimization under transaction costs, the question of the local martingale property of the dual optimizer is of crucial relevance in view of the subsequent Shadow Price Theorem. It states that, if the dual optimizer is induced by a local martingale, there is a shadow price. This theorem essentially goes back to the work of Cvitanić and Karatzas [9]. While these authors did not explicitly crystallize the notion of a shadow price, subsequently Loewenstein [28] explicitly formulated the relation between a financial market under transaction costs and a corresponding frictionless market. Later this has been termed “shadow price process” (compare also [24, 3] as well as [23, 17, 16, 7, 21] for constructions in the Black–Scholes model).

We start by giving a precise meaning to this notion (see also [11, Definition 2.1.]).

Definition 2.11.

In the above setting a semimartingale is called a shadow price process for the optimization problem (2.6) if

  1. takes its values in the bid-ask spread .

  2. The optimizer to the corresponding frictionless utility maximization problem

    (2.14)

    exists and coincides with the solution for the optimization problem (2.6) under transaction costs. In (2.14) the set consists of all non-negative random variables, which are attainable by starting with initial endowment and then trading the stock price process in a frictionless admissible way, as defined in [27].

  3. The optimal trading strategy (in the sense of predictable, -integrable process for the frictionless market , as in [27]) is equal to the left-continuous version of the finite variation process of the unique optimizer of the optimization problem (2.6).

The essence of the above definition is that the value function of the optimization problem for the frictionless market is equal to the value function of the optimization problem for under transaction costs, i.e.,

(2.15)

although the set contains the set defined in (2.5).

The subsequent theorem was proved in the framework of general càdlàg processes in [11, Proposition 3.7].

Theorem 2.12 (Shadow Price Theorem).

Under the hypothesis of Theorem 2.10 fix and such that . Assume that the dual optimizer equals , where is a local -martingale.

Then the strictly positive semimartingale is a shadow price process in the sense of Definition 2.11 for the optimization problem (2.6).

Remark 2.13.

Let be a the shadow price process as above and define the optional sets in

The optimizer of the optimization problem (2.6) for under transaction costs satisfies

for all , i.e., the measures associated to the increasing process (respectively ) are supported by (respectively ). This crucial feature has been originally shown by Cvitanić and Karatzas [9] in an Itô process setting. In the present form, it is a special case of [11, Theorem 3.5].

3 The Main Theorem

In the Shadow Price Theorem 2.12, we simply assumed that the the dual optimizer is induced by a local martingale . In this section, we present our main theorem, which provides sufficient conditions for this local martingale property to hold true.

For the convenience of the reader, we recall the definition of the condition of “no unbounded profit with bounded risk”, which is the key condition in the main theorem.

Definition 3.1.

A semimartingale is said to satisfy the condition of “no unbounded profit with bounded risk”, if the set

is bounded in , where

denotes the stochastic integral with respect to .

Theorem 3.2.

Fix the level of transaction costs and assume that the assumptions of Theorem 2.10 plus the assumption of are satisfied. To resume: is a continuous, strictly positive semimartingale satisfying the condition , and is a utility function satisfying the condition (2.4) of reasonable asymptotic elasticity. We also suppose that the value function in (2.10) is finite, for some .

Then, for each , the dual optimizer in Theorem 2.10 is induced by a local martingale . Hence, by Theorem 2.12, the process is a shadow price.

Before proving the theorem, let us comment on its assumptions. The continuity assumption on cannot be dropped. A two-period counterexample was given in [3], and a more refined version in [10]. These constructions are ramifications of Example 6.1’ in [27].

The assumption of satisfying , which is the local version of the customary assumption , is quite natural in the present context. Nevertheless one might be tempted (as the present authors originally have been) to conjecture that this assumption could be replaced by a weaker assumption as used in Proposition 2.9, i.e., that for every there exists a -consistent price system, at least locally. This would make the above theorem applicable also to price processes which fail to be semimartingales, e.g., processes based on fractional Brownian motion. Unfortunately, this idea was wishful thinking and such hopes turned out to be futile. In Proposition 4.1 below, we give a counterexample showing the limitations of Theorem 3.2.

Turning to the proof of Theorem 3.2, we split its message into the two subsequent propositions which clarify where the assumption of is crucially needed. We now drop and from and , respectively, for the sake of simplicity.

Proposition 3.3.

Fix . Under the assumptions of Theorem 2.10 where we do not impose the assumption , suppose that the liquidation value process associated to the optimizer

(3.1)

is strictly positive, almost surely for each .

Then the assertion of Theorem 3.2 holds true, i.e., the dual optimizer is induced by a local martingale .

Proposition 3.4.

Under the assumptions of Theorem 3.2, i.e., is a continuous semimartingale satisfying the condition , the liquidation value process in (3.1) is strictly positive, i.e., almost surely.

Obviously Proposition 3.3 and 3.4 imply Theorem 3.2. We start with the proof of the second proposition.

Proof of Proposition 3.4.

As shown by Choulli and Stricker [8, Théorème 2.9] (compare also [25, 26, 33, 34]), the condition implies the existence of a strict martingale density for the continuous semimartingale , i.e., a -valued local martingale such that is a local martingale. Note that is a semimartingale as we assumed to be optional and càdlàg, which makes the application of Itô’s lemma legitimate. Applying Itô’s lemma to the semimartingale and recalling that has finite variation, we get from (3.1)

By (2.1), the increment in the first bracket is non-positive. The two terms and are the increments of a local martingale. Therefore the process is a local supermartingale under . As , it is, in fact, a supermartingale.

Since is strictly positive and the terminal value is strictly positive a.s. by Theorem 2.10, we have that the trajectories of are a.s. strictly positive, by the supermartingale property of . This implies that the process is a.s. strictly positive (compare [32, Theorem 1.7]). ∎

Proof of Proposition 3.3.

Fix and assume without loss of generality that . We have to show that there is a local martingale deflator with and , where is the dual optimizer in Theorem 2.10 for .

By Proposition 2.9 , we know that there is a sequence of local martingale deflators such that

By the optimality of , we must have equality above. Using Lemma A.1 below, we may assume, by passing to convex combinations, that the sequence converges to a supermartingale, denoted by , in the sense of (A.2).

By passing to a localizing sequence of stopping times, we may assume that all processes are uniformly integrable martingales, that is bounded from above and bounded away from zero, and that the process is bounded.

To show that the supermartingale is a local martingale, consider its Doob-Meyer decomposition

(3.2)
(3.3)

where the predictable processes and are non-decreasing. We have to show that and vanish. By stopping once more, we may assume that these two processes are bounded and that and are true martingales.

We start by showing that and are aligned in the following way

(3.4)

which is the differential notation for the integral inequality

(3.5)

which we require to hold true for every optional subset . Turning to the differential notation again, inequality (3.4) may be intuitively interpreted that takes values in the bid-ask spread The proof of the claim (3.5) is formalized in the subsequent Lemma 3.5 below.

The process is a uniformly integrable martingale by Theorem 2.10. By Itô’s lemma and using the fact that is of finite variation, we have

Hence we may write the process as the sum of three integrals

The first integral defines a non-increasing process by the self-financing condition (2.1) and the fact that takes values in . The second integral defines a local martingale.

As regards the third term we claim that

(3.6)

defines a non-decreasing process. As is a martingale, this will imply that the process (3.6) vanishes.

We deduce from (3.5) that

As we have assumed that the liquidation value process satisfies a.s. and the process is non-decreasing, the vanishing of the process in (3.6) implies that vanishes. By (3.4) the processes and vanish simultaneously.

Summing up, modulo the (still missing) proof of (3.5), we deduce from the fact that is a martingale that and vanish. Therefore and are local martingales. ∎

Lemma 3.5.

In the setting of Proposition 3.3, let be the bounded, predictable processes in (3.2) and (3.3), and let be a stopping time. For , define

(3.7)

Then

(3.8)

Before starting the proof, we remark that it is routine to deduce (3.5) from the lemma.

Proof.

The processes and