Market consistent valuations with financial imperfection
Abstract
In this paper, we study market consistent valuations in imperfect markets. In the first part of the paper, we observe that in an imperfect market one needs to distinguish two type of market consistencies, namely types I and II. We show that while market consistency of type I holds without very strong conditions, market consistency of type II (which in the literature is known as the usual definition of market consistency) is only well defined in perfect markets. This is important since the existing literature on market consistency considers perfect markets where the two market consistencies are equivalent. In the second part of the paper, by introducing a best estimator we find strong connections between hedging and market consistency of either type. We show under very general conditions, the type I and the type II market consistent evaluators are best estimators, and establish a twostep representation for the market consistent risk evaluators. In the third part of the paper, we present several families of market consistent evaluators in imperfect markets.
Keywords
Imperfect financial valuation Risk evaluation Hedging Market consistent valuationJEL Classification
G11 G13 C22 E441 Introduction
The main assumption of a market consistent valuation is that the fully hedged portfolios cannot improve the actuarial valuation; see, for example, Wüthrich et al. (2010), Wüthrich and Merz (2013), Pelsser and Stadje (2014), Happ et al. (2015) and Dhaene et al. (2017). But given that a perfect approach to hedging does not always exist, this assumption needs to be examined more carefully. Indeed, this assumption postulates that liquidly traded assets and payoffs replicable by them do not carry any risk as they can be converted to cash at any time. In this paper, we show that this assumption necessitates the hedging strategy to be perfect, i.e., the market pricing rule is cone linear over the cone of fully hedged portfolios. However, the possibility of a perfect hedging can be challenged in practice along many dimensions. For instance, nonzero ask–bid spreads, costly dynamic hedging or model risk are among the reasons that hedging strategies do not need to be perfect. We will discuss these particular reasons using a few examples in Sect. 2.3.
This paper considers a financial market where hedging is not necessarily perfect. In the first part, we argue that with an imperfect hedging strategy, we have to distinguish between two different (type I and II) market consistencies. Market consistency of type I only asserts that the valuation of a fully hedged position is the same as its market price, whereas market consistency of type II assumes further that hedging with hedgeable strategies cannot improve the valuation of the risky positions. The existing literature uses the type II market consistency as the usual definition. We characterize market consistent evaluators of either type and prove that a market consistent valuation of type II is well defined only in perfect markets.
In the second part of the paper, we observe that market consistent valuations are strongly related to hedging. Interestingly, this connection will help us characterize market consistent valuation by introducing a best estimator. We show that if some market principle conditions hold (e.g., compatibility^{1}), the type I and II market consistent evaluators are best estimators. In addition, we demonstrate how the best estimator characterization of market consistent evaluators can facilitate obtaining a twostep representation.
Finally, we introduce and discuss practical ways for constructing market consistent evaluators. First, we introduce a family of twostep market consistent evaluators. Second, inspired by superhedging pricing methods, we introduce the family of superevaluators.
The rest of the paper is organized as follows. Section 2 introduces the notation, provides some preliminary definitions and states the main problem. Section 3 discusses the concepts and studies the properties of market consistent evaluators. Section 4 develops a general framework for hedging in an imperfect market and shows its relation to the market consistencies of either type. In Sect. 5, we provide several examples of market consistent valuations. Section 6 concludes.
2 Preliminaries and analytical setup
For a time interval [0, T], we consider two rightcontinuous filtration \( \left\{ \mathcal {F}_{t}^{A}\right\} {}_{0\le t\le T}\) and \(\left\{ \mathcal {F}_{t}^{S}\right\} {}_{0\le t\le T}\), representing the flows of information for insurance and financial markets, respectively. We assume that \(\mathcal {F}_{0}^{A}=\mathcal {F}_{0}^{S}=\{\varnothing ,\varOmega \}\), and \(\mathcal {F}=\mathcal {F}_{T}^{A}\vee \mathcal {F}_{T}^{S}\) (the smallest sigmafield containing \(\mathcal {F}_{T}^{A}\) and \(\mathcal {F}_{T}^{S}\)). We also assume that \(\mathcal {F}_{T}^{S}\) contains all measure zero sets of \( \mathcal {F}\).
Remark 1
As it is common in the literature, we have to specify whether we are working with loss/profit or deficit/surplus variables. However, in this paper, we do not need to do this. Indeed, our framework is flexible to cover both approaches; i.e., one can decide if the set of random variables represents losses (or deficit)—which is of interest to actuaries, or if it represents profits (surplus)—which is of interest to financial modelers. We will come back to this point later after introducing risk evaluators and pricing rules.
2.1 Risk evaluator
 (P1)
\(\varPi (\lambda x)=\lambda \varPi (x)\), for all \(\lambda \ge 0\) and \(x\in L^{p}\) (positive homogeneity);
 (P2)
\(\varPi (x+y)\le \varPi (x)+\varPi (y)\), for all \(x,y\in L^{p}\) (subadditivity);
 (P3)
\(\varPi (\lambda x+(1\lambda )y)\le \lambda \varPi (x)+(1\lambda )\varPi (y)\), for all \(x,y\in L^{p}\) and \(\lambda \in [0,1]\) (convexity).
Definition 1
 1.
\(\varPi \) is sublinear if it satisfies P1 and P2.
 2.
\(\varPi \) is convex if it satisfies P3.
Remark 2
Following Remark 1, a few remarks regarding the cash invariance and monotonicity, as they are commonly used in the literature, seem warranted. For those properties, we need to specify whether \(L^{p}\) represents loss variables (deficit) or profit (surplus). More specifically, if we suppose random variables model the losses (profits) then for all \(x,y\in L^{p}\), \(x\le y\) a.s. and \( \forall c\in \mathbb {R}\), cash invariance and monotonicity are \(\varPi (x+c)=\varPi (x)+c\) and \(\varPi (x)\le \varPi (y)\), (\(\varPi (x+c)=\varPi (x)c\) and \(\varPi (x)\ge \varPi (y)\)), respectively. But, our approach in this paper does not need to specify if we are working with loss/deficit or profit/surplus variables, since our theory is not dependent on cash invariance or monotonicity property.
2.2 Pricing rule
 (S1)
Positive homogeneity \(\lambda \mathcal {X}\subseteq \mathcal {X}\), for all \(\lambda \ge 0\);
 (S2)
Subadditivity \(\mathcal {X}+\mathcal {X}\subseteq \mathcal {X}\);
 (S3)
Convexity \(\lambda \mathcal {X}+(1\lambda )\mathcal {X}\subseteq \mathcal {X}\) for all \(\lambda \in (0,1)\).
A pricing rule \(\pi :\mathcal {X}\rightarrow \mathbb {R}\) is a mapping from \( \mathcal {X}\) to the set of real numbers \(\mathbb {R}\) which maps each random variable in \(\mathcal {X}\) to a real number representing its price, with an additional property \(\pi (0)=0\). Just realize that in principle the main difference between the definition of \(\pi \) and \(\varPi \) is the domain of these two mappings. If \(\pi \) satisfies properties P1, P2, or P3, \(\mathcal {X}\) has to satisfy properties S1, S2, or S3, respectively. Jouini and Kallal (1995a, b, 1999) argue that for a wide range of market imperfections the pricing rule is sublinear, i.e., \(\pi \) has P1 and P2. That is why in this paper we develop our theoretical framework for sublinear pricing rules.
Remark 3
Like in Remarks 1 and 2, to emphasize the generality of our framework, note that we do not need to specifically assume that if \(\pi \) is cash invariant or monotone. Just a further attention needed to be paid when considering cash invariance or monotonicity: if \(\pi \) is cash invariance we have to consider \(\mathbb {R} \subseteq \mathcal {X}\) and if it is nondecreasing (nonincreasing), \(\forall x\in \mathcal {X}\) and \(y\ge x\), (\(y\le x\)), \(y\in \mathcal {X}\).
Next, we introduce more rigorously a perfect market:
Definition 2
A pricing rule \(\pi \) on a cone \(\mathcal {X}\) is perfect if \(\pi (x+\lambda y)=\pi (x)+\lambda \pi (y),\forall x,y\in \mathcal {X} ,\lambda >0\). When \(\pi \) is perfect we say the market and the hedging strategy are perfect.
It is clear that a perfect pricing rule is sublinear.
2.3 Examples of pricing rules
Using several examples, we show how all different types of markets and pricing rules exist. For simplicity, in all examples, we assume that there is a variable (loss or profit) \(h\in L^{1}\) for the insurance company. We assume that the insurance information is given by \(\mathcal {F} _{t}^{A}=\left\{ \varnothing ,\varOmega \right\} \) for \(0\le t<T\), and \( \mathcal {F}_{T}^{A}=\varsigma \left( h\right) \). Therefore, we only need to focus our attention on introducing the financial part.
Example 1
Example 2
Example 3
Example 4
Example 5
(A Financial Approach) This example is based on an approach that is popular in the financial literature and is strongly related to factor models. It is also referred to as a nonparametric approach due to the lack of explicit assumptions about the asset models. In this approach, we need a set of test assets \( x_{0},x_{1},\ldots ,x_{N}\) with associated prices \(p_{0},p_{1},\ldots ,p_{N}\), which are assumed to be the most liquid assets in the market. We assume all pricing rules are able to correctly price the test assets. This implies that for any stochastic discount factor \(z\in L^{q}\), we have \( E(zx_{i})=p_{i},\forall i=0,1,\ldots ,N\). Also note that if we need to impose a noarbitrage condition, then we have to assume \(z\ge 0\). In this approach, one can consider any closed cone \(\mathcal {X}\) that is a subcone of all portfolios \(\{ \sum \nolimits _{i=0}^{N}a_{i}x_{i}\left a_{i}\in \mathbb {R} ,i=0,1,\ldots ,N\right. \} \), containing the origin. If we denote the set of all stochastic discount factors by SDF, then any sublinear pricing rule \(\pi \) can be expressed as (2), where \(\varDelta _{\pi }\subseteq \mathrm {\text {SDF}}\). In the financial literature, the typical set of test assets consists of excess returns^{2} on a Fama–French portfolio (either the FF25, FF 50 or FF 100), plus the risk free asset. The associated prices are \(p_{0}=1\), for the risk free and \(p_{1}=\dots =p_{N}=0\) , for the other assets in the portfolio. One important implication of this approach is that for any sublinear pricing rule \(\pi \), it is linear on the set \(\mathcal {X}\). Here, one can introduce a simple filtration \(\mathcal {F} _{t}^{S}=\left\{ \varnothing ,\varOmega \right\} \), for \(t<T\) and \(\mathcal {F} _{T}^{S}=\varsigma \left( x_{0},x_{1},\ldots ,x_{N}\right) \). It is important to observe that the pricing rules are linear on the set of all hedgeable positions \(\mathcal {X}\). Although this appears to be a credible assumption for the test assets which are liquidly traded in the market, in reality the ask and bid prices are different even for the most liquid assets. The implication of the bid–ask price spread is that the prices cannot be linear for hedgeable assets. Therefore, markets are usually imperfect and pricing rules are sublinear on \(\mathcal {X}\).
Remark 4
Even though the above examples provide a convincing argument for the generality and the practical relevance of the theoretical framework we considered in this paper, there are other important approaches to market consistent valuation that do not fit our framework. For instance, using a utility indifference pricing, in general, does not induce a sublinear pricing rule.
3 Market consistent valuation
Let us now proceed with the definition of the market consistency.
Definition 3
Type II consistency states that hedging strategies cannot have an effect on the evaluation of the economic risks, i.e., it makes it neither better nor worse. Type I consistency does not have such an implication and only implies that for hedgeable positions, market and risk evaluators have similar valuation of risk. We will see that while the type II consistency holds only in perfect markets, the type I consistency can hold under very general conditions.
We have the following immediate result from the definition of market consistencies.
Proposition 1
Market consistency of type II implies market consistency of type I.
However, the opposite is not true as it is shown in the following example.
Example 6
Consider a risk evaluator \(\varPi \) with properties P3 which is not linear. Consider the pricing rule \(\pi =\varPi \). If \(\mathcal {X}=L^{p}\) then the market consistency of type I holds, while II does not!
One can prove the following theorem by following discussions in Pelsser and Stadje (2014).
Theorem 1
 1.
Consistency of type I holds.
 2.
Consistency of type II holds.
 3.\(\varPi \) can be presented as follows,for a penalty function \(c:\left\{ z\in L^{1}\left E_{\mathcal {G} }(z)=1\right. \right\} \rightarrow \left[ 0,\infty \right] \).$$\begin{aligned} \varPi (x)=\sup _{\left\{ z\in L^{1}\left E_{\mathcal {G}}(z)=1\right. \right\} }\left\{ E(zx)c(z)\right\} , \end{aligned}$$
Even though one can easily introduce market consistent evaluators of type I, the same is not true for market consistent evaluators of type II in an imperfect market. Indeed, we will see in Theorem 3 that under general conditions, unless the market is perfect we cannot introduce a market consistent valuation of type II. For that, we need to introduce further propositions and theorem in the following.
Proposition 2
Now, let us begin with the following theorem, which looks very similar to the results in Pelsser and Stadje (2014).
Theorem 2
 1
\(\varPi (x)\le \pi (x)\) for all \(x\in \mathcal {X}\).
 2
\(\varPi (x+y)\le \pi (x)+\varPi (y)\) for all \(x\in \mathcal {X}\) and \(y\in L^{p}\).
 3\(\varPi \) accepts the following representationfor a penalty function \(c\ge 0\).$$\begin{aligned} \varPi (x)=\sup _{z\in \varDelta _{\bar{\pi }}}\left\{ E(zx)c(z)\right\} , \end{aligned}$$(10)
 4
\(\{\varPi ^{*}<\infty \}\subseteq \varDelta _{\bar{\pi }}=\varDelta _{\pi }+\mathcal {X} ^{\circ }\).
Proposition 3
Let \(x,y\in L^{p}\) be two random variables such that \( f_{1}\Box f_{2}(y)=f_{1}(x)+f_{2}(yx)\). If \(f_{1}\) is finite on a neighborhood around x, then \(f_{1}\Box f_{2}\) is proper, i.e., \(\forall x\in L^{p},f_{1}\Box f_{2}(x)>\infty \) and \(\exists x\in L^{p},f_{1}\Box f_{2}(x)<\infty \).
Proposition 4
Now, we can prove Theorem 2.
Proof of Theorem 2
(\(2\Leftrightarrow 4\)). It is clear that the statement 2 is equivalent to \(\varPi (y)\le \bar{\pi }(x)+\varPi (yx),\forall x,y\in L^{p}\). This on its own is equivalent to \(\varPi (y)\le \inf _{x\in L^{p}}\left\{ \bar{\pi }(x)+\varPi (yx)\right\} , \forall y\in L^{p}\). But the right hand side is the infconvolution between \(\bar{\pi }\) and \(\varPi \), so, \(\varPi \le \bar{\pi }\square \varPi \). Therefore, the last inequality is equivalent to \(\varPi ^{*}\ge (\bar{\pi }\square \varPi )^{*}=\bar{\pi }^{*}+\varPi ^{*}\). This is also equivalent to \(\{\varPi ^{*}<\infty \}\subseteq \varDelta _{\bar{\pi }}\). In particular, if \(\varPi \) is sublinear then \(\{\varPi ^{*}<\infty \}=\varDelta _{\varPi }\).
An immediate corollary is the following.
Corollary 1
Assume \(\varPi \) is convex and \(\pi \) is sublinear. If \(\varPi \) is market consistent of either type, then \(\varPi \) can be represented by (10).
Remark 5
As one can see, if we instead of consistency of type I and II accept conditions 1 and 2 in Theorem 2, respectively, which can be interpreted as “subconsistency,” then we can prove the equivalence of the “subconsistencies”. However, as it will be made clear, we cannot always go further than this. That is why we will study consistency of types I and II separately.
Theorem
 1.
\(\varPi \) is market consistent of type I
 2.The following three statements hold together
 (a)
\(\varPi (x)\le \pi (x),\forall x\in \mathcal {X}\);
 (b)
\(\varPi \) is sublinear on \(\mathcal {X}\);
 (c)
\(\pi (x)=\sup \nolimits _{\{\varPi ^{*}<\infty \}}E(zx),\forall x\in \mathcal {X}\).
 (a)
Proof
Let us first prove \((1\Rightarrow 2)\). If \(\varPi \) is market consistent of type I then clearly a) and b) hold.
Now, let us prove \((2\Rightarrow 1)\). Since a) holds then we only need to prove \(\pi \le \varPi \) on \(\mathcal {X}\). It is clear that by b), \(\bar{\varPi }\) is sublinear. Therefore, \(\bar{\varPi }(x)=\sup \nolimits _{\left\{ \bar{\varPi }^{*}<\infty \right\} }E(zx).\) But since \(\varPi \le \bar{\varPi }\), we get \(\bar{\varPi }^{*}\le \varPi ^{*}\), and consequently \(\left\{ \varPi ^{*}<\infty \right\} \subseteq \left\{ \bar{\varPi }^{*}<\infty \right\} \). Combining this and c) with the last relation we get \(\forall x\in \mathcal {X},\bar{\varPi }(x)=\sup \nolimits _{\left\{ \bar{\varPi }^{*}<\infty \right\} }E(zx)\ge \sup \nolimits _{\left\{ \varPi ^{*}<\infty \right\} }E(zx)=\pi (x)\). This completes the proof of the second implication. \(\square \)
Remark 6
This theorem has a very interesting implication: if \(\varPi \) is market consistent of type I, the set of all true stochastic discount factors is equal to \(\left\{ \varPi ^{*}<\infty \right\} \).
Now, we have the following theorem for consistency of type II.
Theorem 3
 1.
\(\varPi \) is market consistent of type II
 2.
\(\forall z\in \{\varPi ^{*}<\infty \}\text { and }x\in \mathcal {X},\pi (x)=E(zx)\).
Proof
Let us assume that 1 holds. For a fixed \(x\in \mathcal {X}\), introduce the following two mappings \(\varPi _{1}(y):=\pi (x)+\varPi (y)\), and \(\varPi _{2}(y):=\varPi (x+y)\). It is easy to see that \(\varPi _{1}^{*}(z)=\pi (x)+\varPi ^{*}(z)\) and \(\varPi _{2}^{*}(z)=E(xz)+\varPi ^{*}(z)\). Since \(\varPi _{1}=\varPi _{2}\), and therefore \(\varPi _{1}^{*}=\varPi _{2}^{*}\), we get that \(\forall z\in \left\{ \varPi ^{*}<\infty \right\} \) and \(x\in \mathcal {X},\pi (x)=E(zx)\). This completes the proof of \((1\Rightarrow 2)\).
Remark 7
One can see that for a fixed \(z\in \left\{ \varPi ^{*}<\infty \right\} \) we have that \(\pi (x)=E(zx)\). This has two important consequences. First, in order to have the market consistency of type II, it is necessary to accept that the market is perfect. The second implication is that all members of \( \{\varPi ^{*}<\infty \}\) are true stochastic discount factors as mentioned in Remark 6.
Finally, the following corollary is very useful:
Corollary 2
If \(\varPi \) is convex and \(\pi \) is sublinear, then \(\varPi \) is market consistent of type II if and only if it is market consistent of type I and the market is perfect.
This means that once a market consistent valuation of type II exists, it has to be also market consistent of type I.
4 Compatibility and market consistency
In this section, we present a general hedging framework for pricing financial positions that cannot be perfectly hedged in an incomplete market. The hedging strategy, introduced below, is based on the concept of a best estimate for actuarial evaluation of an insurance position. We will show that under reasonable conditions, market consistency of either type is enough to guarantee that the risk estimator is a best estimator. We will also see how the best estimator representation of a market consistent evaluator can help us to obtain a twostep representation of market consistent evaluators.
4.1 Best estimator and hedging
In the following discussion, we demonstrate the strong relation between a market consistent valuation, of either type, and hedging strategies as used in the literature on pricing (e.g., see Jaschke and Küchler 2001; Staum 2004; Xu 2006; Assa and Balbás 2011; Balbás et al. 2009a, b, 2010; Arai and Fukasawa 2014). We assume that the value of a variable is equal to the sum of a best estimate and a risk margin. For that, we assume any nonhedgeable position (i.e., \( L^{p}{\setminus } \mathcal {X}\)) can be decomposed into two parts: one which is fully hedged (associated with the best estimate) and a part which left and produces some risk (associated with the risk margin). However, for reasons that will be discussed below, we will extend the concept of a best estimate in a new direction.
Definition 4
For a risk evaluator \(\varPi \) and a pricing rule \(\pi \), the best estimator \( \varPi _{\pi }\) is introduced by (11).
Remark 8
If \(\varPi \) and \(\pi \) are sublinear, \(\varPi _{\pi }(x)\) is the Good Deal upper bound introduced in Staum (2004). We obtain the Good Deal upper bound within a general competitive pricing and hedging framework.
Now we move toward addressing if \(\varPi _{\pi }\) is a welldefined evaluator. Let us first state the following result for \(\varPi _{\pi }\) defined in (11) (for a proof see Barrieu and Karoui 2005).
Proposition 5
 1.
\(\varPi _{\pi }\) and \(\text {Dom}\left( \varPi _{\pi }\right) \) are positive homogeneous if \(\varPi \) and \(\pi \) are.
 2.
\(\varPi _{\pi }\) and \(\text {Dom}\left( \varPi _{\pi }\right) \) are subadditive if \(\varPi \) and \(\pi \) are.
 3.
\(\varPi _{\pi }\) and \(\text {Dom}\left( \varPi _{\pi }\right) \) are convex if \(\varPi \) and \(\pi \) are.
 4.
\(\varPi _{\pi }\) and \(\text {Dom}\left( \varPi _{\pi }\right) \) are translationinvariant if \(\varPi \) and \(\pi \) are.
 5.
\(\varPi _{\pi }\) is monotone if \(\varPi \) and \(\pi \) are monotone.
First, note that Proposition 5 does not say if \(\text {Dom }\left( \varPi _{\pi }\right) \) is equal to \(L^{p}\). Second, the proposition also does not say under which conditions \(\varPi _{\pi }(0)=0\). Actually if these two conditions hold then \(\varPi _{\pi }\) is a risk evaluator. Interestingly, it turns out that \(\text {Dom}\left( \varPi _{\pi }\right) =L^{p}\) and \(\varPi _{\pi }(0)=0\) hold under very general conditions, which will be discussed shortly.
Definition 5
For a risk evaluator \(\varPi \) and a pricing rule \(\pi \), compatibility holds if \( \varPi _{\pi }\) is a risk evaluator, i.e., if \(\text {Dom}\left( \varPi _{\pi } \right) =L^{p}\) and \(\varPi _{\pi }(0)=0\). In the sequel, we denote compatibility by (C).
Theorem 4
 1.
\(\varPi _{\pi }(0)=0\).
 2.
\(\text {Dom}\left( \varPi _{\pi }\right) =L^{p}\).
 3.
\(\varPi (x)+\pi (x)\ge 0,\forall x\in \mathcal {X}.\)
 4.
(C) holds.
Proof
Now we prove \((1\Leftrightarrow 2)\). Given that \(\varPi \) is finite on \(L^{p}\), in particular around a neighborhood of 0, and given that \(\varPi (0)+\pi (0)=0\), by Proposition 3 \(\varPi _{\pi }(0)=0\) is equivalent to \(\text {Dom}\left( \varPi _{\pi }\right) =L^{p}\).
The implication \((1\text { or }2\Leftrightarrow 4)\) is easy to prove; indeed, form the definition it is clear that since 1 is equivalent to 2, then both are equivalent to compatibility.
Corollary 3
One important question is to establish the conditions under which a market consistent risk evaluator is also a best estimator. For that, we first state the following obvious proposition without proof.
Proposition 6
If for pricing rule \(\pi \) and risk evaluator \(\varPi \) we have \( \varPi (x+y)\le \pi (x)+\varPi (y),\forall x\in \mathcal {X},y\in L^{p}\), then (C) holds and \(\varPi _{\pi }=\varPi \).
Combining Proposition 6 with Theorem 2, we get the following theorem.
Theorem 5
If \(\pi \) is sublinear, \(\varPi \) is a convex market consistent risk evaluator of either type then \(\varPi _{\pi }=\varPi \).
Theorem 5 has two important implications: under the theorem’s conditions, first \(\varPi \) is a best estimator, second, \(\varPi _{\pi }\) is market consistent of the same type as \(\varPi \).
Remark 9
If \(\varPi \) is a coherent risk measure (i.e., monotone, cash invariant and sublinear) and if \(\pi \) is sublinear then condition 3 in Theorem 4 is equivalent to the No Good Deal assumption introduced and studied in Assa and Balbás (2011).
In general, it is not always true that \(\varPi _{\pi }\) is market consistent. Here, we illustrate with two examples that we cannot easily relax the assumptions in the previous theorem.
Example 7
Consider, \(z_{1},z_{2}\in L^{q}\), \(\pi (x)=E(z_{1}x)\) and \( \varPi (x)=\max \{E(z_{1}x),E(z_{2}x)\}\), and the cone \(\mathcal {X}=\{x\in L^{p}E(z_{1}x)\ge E(z_{2}x)\}\). First, observe that \(\varPi \) is market consistent of either type. One can easily see that \(\mathcal {X} ^{\circ }=\{z_{2}z_{1}\}\), and therefore, according to Theorem 4, \(\varPi _{\pi }(x)=E(z_{2}x)\). Again on the cone \(\{x\in L^{p}E(z_{1}x)>E(z_{2}x)\}\) the market consistency for \(\varPi _{\pi }\), of either type, does not hold.
Example 8
Consider two different members \(z_{1}\) and \(z_{2}\) in \(L^{q}\). Let \(\pi (x)=\max \{E(z_{1}x),E(z_{2}x)\}\) and \(\varPi (x)=E(z_{2}x)\) and let \(\mathcal { X}=L^{p}\). Then, according to Theorem 4 we have \( \varPi _{\pi }(x)=E(z_{2}x)\). This simply implies that on the cone \(\{x\in L^{p}E(z_{1}x)>E(z_{2}x)\}\), market consistency for \(\varPi _{\pi }\), of either type, does not hold.
4.2 Best estimator in a perfect markets
Without knowing anything about the consistency of \(\varPi \), it is difficult to prove whether the best estimator is market consistent. However, in a perfect market, we can answer this question by showing that \(\varPi _{\pi }\) is always market consistent of type I.
Theorem 6
Assume that \(\pi \) is superlinear (i.e., P1 and \( \pi (x+y)\ge \pi (x)+\pi (y),\forall x,y\in \mathcal {X}\)) and that \(\mathcal {X}\) is a vector space. If (C) holds, then \(\varPi _{\pi }\) is market consistent of type I.
Proof
On the other hand, if we let \(x=y\), then we get \(\pi (y)\ge \inf \nolimits _{x\in \mathcal {X}}\varPi (yx)+\pi (x)=\varPi _{\pi }(y).\) \(\square \)
Corollary 4
Assume that the market is perfect. If (C) holds, then \(\varPi _{\pi }\) is market consistent of type I.
This corollary has a wide range of applications, since it shows how in a perfect market one can construct market consistent valuations.
4.3 Twostep evaluation and hedging
Definition 6
Theorem 7
Let \(\pi :\mathcal {X}\rightarrow \mathbb {R}\) be a cash invariant pricing rule (i.e., \(\pi (x+c)=\pi (x)+c,\forall , x\in \mathcal X, c\in \mathbb R \)) and \(\varPi :L^{p}\rightarrow \mathbb {R}\) be a risk evaluator. If for any \( y\in L^{p}\), \(S_{\pi ,\varPi }(y)\not =\varnothing \) then \(\varPi _{\pi }\) can be represented as a twostep evaluator in (14) where \(\varLambda (y)=\{x+\varPi (yx)x\in S_{\pi ,\varPi }(y)\}\).
Next, we combine this theorem with Theorem 5 to obtain the following representation.
Theorem 8
If \(\varPi \) is convex, \(\pi \) is sublinear and cash invariant (i.e., \(\pi (x+c)=\pi (x)+c,\forall , x\in \mathcal X, c\in \mathbb R \)) and \(\varPi \) is market consistent of either type and if for any \(y\in L^{p}\), \( S_{\pi ,\varPi }(y)\not =\varnothing \) then \(\varPi \) has a twostep representation.
5 Market consistent risk evaluators
So far, we have studied the conditions under which a risk estimator is market consistent. In the following sections we will introduce some families of market consistent evaluators and using discussions in Example 5 we show how one can construct them. Note that based on Corollary 2, in the following examples once we construct a convex market consistent risk evaluator \(\varPi \), where its pricing rule is linear on \(\mathcal {X}\), \(\varPi \) is automatically market consistent of type II. This is the main reason why we are mainly concerned with constructing market consistent valuations of type I.
5.1 A family of twostep estimators

\(\varPi (x)=\pi \left( \delta E_{\mathcal {G}}(\max \{xE_{\mathcal {G} }(x),0\}^{p})+E_{\mathcal {G}}(x)\right) ,\)

\(\varPi (x)=\pi \left( \delta \mathrm {VaR}_{\alpha }^{\mathcal {G}}(xE_{ \mathcal {G}}(x))+E_{\mathcal {G}}(x)\right) ,\)

\(\varPi (x)=\pi \left( \delta \mathrm {CVaR}_{\alpha }^{\mathcal {G}}(xE_{ \mathcal {G}}(x))+E_{\mathcal {G}}(x)\right) .\)
5.2 Superevaluators
It is clear that for every type I market consistent evaluator \(\varPi \), we have \(\varPi \le \bar{\pi }\). Indeed, if we assume for a moment that \(+\infty \) belongs to the range of a risk evaluator, we can say that \(\bar{\pi }\) is the largest type I market consistent risk evaluator. But the question is whether we can find the smallest type I market consistent evaluator.
Note that from a mathematical point of view, the super estimator is a best estimator when \(\varPi (y)=\chi _{\left\{ y\ge 0\right\} }= {\left\{ \begin{array}{ll} 0, &{}\quad y\ge 0 \\ +\infty , &{}\quad \text {otherwise} \end{array}\right. } \). We also have the following proposition.
Proposition 7
 1.
\(\tilde{\varPi }\) is positive homogeneous if \(\pi \) is.
 2.
\(\tilde{\varPi }\) is subadditive if \(\pi \) is.
 3.
\(\tilde{\varPi }\) is convex if \(\pi \) is.
 4.
\(\tilde{\varPi }\) is translationinvariant if \(\pi \) is.
 5.
\(\tilde{\varPi }\) is nondecreasing.
6 Conclusion
To the best of our knowledge, this is the first paper that considers market consistency in imperfect markets. We presented several examples that justify the necessity of studying market consistent valuation in imperfect markets, including both complete and incomplete markets. In the first part of the paper, we distinguished between market consistency of two types, namely, types I and II. The type I consistency bears the very meaning of “consistency” by assuming that the market and risk evaluator are equal on hedgeable positions, whereas type II consistency further ensures that hedging strategies cannot improve the valuation of risky positions. While market consistency of type II implies the type I consistency, the opposite only can happen in perfect markets. Indeed, we demonstrated that the market consistency of type II only exists if the hedging strategy is perfect. This means that once a market consistent valuation of type II exists, it has to be also market consistent of type I. In the existing literature with perfect markets, the two definitions are equivalent. In the second part of the paper, motivated by the literature on pricing and hedging in incomplete markets, we introduced a best estimator and a risk margin. We showed that if the compatibility holds (e.g., Good Deals are ruled out), then a market consistent valuation is equal to its best estimator. We also used this to demonstrate how market consistent valuations can be represented in a twostep manner. Finally, we showed how to construct market consistent valuations as twostep estimators and superevaluators.
Footnotes
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