1 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 two-step representation.
Finally, we introduce and discuss practical ways for constructing market consistent evaluators. First, we introduce a family of two-step market consistent evaluators. Second, inspired by super-hedging pricing methods, we introduce the family of super-evaluators.
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
We consider a probability triple
\((\varOmega ,\mathcal {F},\mathbb {P})\), where
\( \varOmega \) is a set of scenarios,
\(\mathcal {F}\) is a set of events and
\(\mathbb { P}\) is a probability measure. We denote the expectation by
E and for any other probability triple
\(\left( \varOmega ,\mathcal {F},\mathbb {Q}\right) \), the associated expectation is denoted by
\(E^{\mathbb {Q}}\). For
\(p\in [1,\infty ]\), let
\(L^{p}\) be the Banach space defined as:
$$\begin{aligned} L^{p}=\left\{ x:\varOmega \rightarrow \mathbb {R}\vert \mathcal {F}\text {-measurable random variables and }E\left( \left| x\right| ^{p}\right) <\infty \right\} . \end{aligned}$$
\(L^{p}\) is endowed with the norm
\(\left\| x\right\| _{L^{p}}=E\left( \left| x\right| ^{p}\right) \). Consider
\(L^{p}\) and
\(L^{q}\) where
\(1/p+1/q=1\). There is a duality relation between
\(L^{p}\) and
\(L^{q}\) defined as
$$\begin{aligned} (y,y^{\prime })\mapsto E(yy^{\prime }),\forall (y,y')\in L^{ p}\times L^{q}. \end{aligned}$$
The smallest topology induced by
\(L^{q}\) on
\(L^{p}\) is denoted by
\( \sigma (L^{p},L^{q})\). Similarly, one can introduce the topology
\( \sigma (L^{q},L^{p})\). For any sub-sigma-field
\(\mathcal {G}\subseteq \mathcal {F }\),
\(L^{p}(\mathcal {G})\) and
\(L^{q}(\mathcal {G})\) represent the same spaces as described above just for
\(\mathcal {G}\) measurable random variable. Note that
\(L^{p}=L^{p}(\mathcal {F})\) and
\(L^{q}=L^{q}(\mathcal {F})\). For any set of random variables
\(x_{1},\ldots ,x_{n}\), the smallest sigma-field generated by them is denoted by
\(\varsigma (x_{1},\ldots ,x_{n})\).
For a time interval [0, T], we consider two right-continuous 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 sigma-field containing \(\mathcal {F}_{T}^{A}\) and \(\mathcal {F}_{T}^{S}\)). We also assume that \(\mathcal {F}_{T}^{S}\) contains all measure zero sets of \( \mathcal {F}\).
To fix the terminology and the notation, it is useful to briefly review some concepts from convex analysis. We assume that all mappings
\( f:L^{p}\rightarrow \left( -\infty ,\infty \right] \) in this paper are
\( \sigma (L^{p},L^{q})\)-lower semi-continuous, i.e., for any number
\(a\in \mathbb {R}\), the set
\(\{x\in L^{p}|f(x)\le a\}\) is
\(\sigma (L^{p},L^{q})\)-closed in
\(L^{p}\). This assumption implies the existence of a dual representation (see Proposition 4.1, Ekeland and Témam
1999) for a lower semi-continuous convex mapping
\(f: L^p\rightarrow (-\infty ,+\infty ]\) as follows,
$$\begin{aligned} f(x)=\sup \limits _{z\in L^{q}}\left\{ E(zx)-f^{*}(z)\right\} , \end{aligned}$$
where
\(f^{*}:L^{q}\rightarrow \left( -\infty ,\infty \right] \) is the dual of
f defined as
$$\begin{aligned} f^{*}(z)=\sup \limits _{x\in L^{p}}\left\{ E(zx)-f(x)\right\} . \end{aligned}$$
It can be easily seen that for any positive homogeneous (i.e.,
\(f(\lambda x)=\lambda f(x),\forall \lambda \ge 0,x\in L^{p}\)) and convex function
f,
\( f^{*}\) is 0 on a closed convex set
\(\varDelta _{f}\), and infinity otherwise. Therefore, the dual representation of a positive homogeneous and convex function
f has the form
$$\begin{aligned} f(x)=\sup \limits _{z\in \varDelta _{f}}E(zx). \end{aligned}$$
Then, it is straightforward to see that
\(\varDelta _{f}=\left\{ z\in L^{q}|E(zx)\le f(x),\forall x\in L^{p}\right\} \).
2.1 Risk evaluator
A
risk evaluator \(\varPi \) is a mapping from
\(L^{p}\) to the set of real numbers
\(\mathbb {R}\) which maps each random variable in
\(L^{p}\) to a real number representing its risk with the additional property
\(\varPi (0)=0\). Each risk evaluator can have one or more of the following properties:
(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 \(x,y\in L^{p}\) (sub-additivity);
(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).
There are several families of risk evaluators introduced and studied in different areas: for instance, coherent risk measure, convex risk measures and expectation bounded risks. Coherent and convex risk measures are introduced by Artzner et al. (
1999) and Föllmer and Schied (
2002), respectively, while expectation bounded risks are first defined in Rockafellar et al. (
2006).
The following examples are risk evaluators when
\(L^{p}\) represents the loss variables (or deficit). A mean-variance risk evaluator on
\(L^{2}\), is defined as
$$\begin{aligned} MV_{\delta }(x)=\delta \sigma (x)+E(x), \end{aligned}$$
where
\(\sigma (x)\) is the standard deviation of
x and
\(\delta \) is a nonnegative number representing the level of risk aversion. One can replace
\(\sigma \) by semi
pth moment to have the following family of risk evaluators on
\(L^{p}\)$$\begin{aligned} MV_{\delta }^{p}(x)=\delta E\left( \max \left\{ x-E(x),0\right\} {}^{p}\right) +E(x). \end{aligned}$$
One popular risk evaluator on
\(L^{p},\) (even on the set of all random variables) is the value at risk defined as
$$\begin{aligned} \mathrm {\mathrm {VaR}}_{\alpha }(x)=\inf \left\{ a\in \mathbb {R}|\mathbb {P} [x>a]\le \alpha \right\} . \end{aligned}$$
Here
\(\alpha \in (0,1)\) is the risk aversion parameter. Therefore, one can introduce the following risk evaluator on
\(L^{p},\)$$\begin{aligned} \varPi _{\delta }^{\mathrm {VaR}_{\alpha }}(x)=\delta \mathrm {VaR} _{\alpha }(x-E(x))+E(x). \end{aligned}$$
In contrast, the conditional value at risk (CVaR), expressed as the sum over all
\({\mathrm {VaR}}\) above
\(1-\alpha \) percent
$$\begin{aligned} \mathrm {CVaR}_{\alpha }(x)=\frac{1}{1-\alpha }\int _{1-\alpha }^{1}\mathrm { \mathrm {VaR}}_{\beta }(x)\mathrm{d}\beta , \end{aligned}$$
(1)
is a coherent risk evaluator on
\(L^{p}\). Accordingly, one can introduce the following risk evaluator based on CVaR on
\(L^{p},\)$$\begin{aligned} \varPi _{\delta }^{\mathrm {CVaR}_{\alpha }}(x)=\delta \mathrm {CVaR} _{\alpha }(x-E(x))+E(x). \end{aligned}$$
In this paper, we use the following two categories of risk evaluators in our statements.
As discussed earlier, it is clear that every sub-linear evaluator can be represented as
$$\begin{aligned} \varPi (x)=\sup \limits _{z\in \varDelta _{\varPi }}E(zx),\quad \forall x\in L^{p}, \end{aligned}$$
(2)
for a closed convex set
\(\varDelta _{\varPi }\) of
\(L^{q}\).
2.2 Pricing rule
We now turn our attention to pricing rules. First, we need to fix a set of assets that are fully hedged. Let
\(\mathcal {X}\) be a closed subset of
\( L^{p}\left( \mathcal {F}_{T}^{S}\right) \) which contains the origin. In the subsequent discussions, we will assume that
\(\mathcal {X}\) possesses one or several properties from the following list:
(S1)
Positive homogeneity \(\lambda \mathcal {X}\subseteq \mathcal {X}\), for all \(\lambda \ge 0\);
(S2)
Sub-additivity \(\mathcal {X}+\mathcal {X}\subseteq \mathcal {X}\);
(S3)
Convexity \(\lambda \mathcal {X}+(1-\lambda )\mathcal {X}\subseteq \mathcal {X}\) for all \(\lambda \in (0,1)\).
If
\(\mathcal {X}\) has properties S1 and S2, it is called a convex cone and if S3 it is simply called convex.
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 sub-linear, i.e.,
\(\pi \) has P1 and P2. That is why in this paper we develop our theoretical framework for sub-linear pricing rules.
Next, we introduce more rigorously a perfect market:
It is clear that a perfect pricing rule is sub-linear.
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.
The first four examples use a standard mathematical finance setup to model portfolios with Brownian motions (e.g., see Karoui and Quenez
1995 for more details). Let
\(\left( \overrightarrow{W}_{t}\right) _{0\le t\le T}=\left( \left( W_{1,t},\ldots ,W_{d,t}\right) \right) _{0\le t\le T}\) be a standard Brownian motion, where all components are independent. For a natural number
\(N\ge d\), let
\(\overrightarrow{\mu } =\left( \mu _{1},\ldots \mu _{N}\right) \) be a vector of real numbers (representing the drifts) when also
\(\mu _{i}>r,i=1,\ldots ,d\), and
\(r>0\) is the interest rate. Let
\(\overrightarrow{\sigma }\left( t\right) =\left( \sigma _{i,j}\left( t\right) \right) _{0\le t\le T}\) be an
N by
d matrix of previsible volatility processes. Let
\(\mathcal {F}_{t}^{S}=\varsigma ( \overrightarrow{W}_{s},0\le s\le t)\). Introduce the discounted value of the
N assets
\(x_{i,t},i=1,\ldots ,N\) by
$$\begin{aligned} \mathrm{d}x_{i,t}=x_{i,t}\left( \left( \mu _{i}-r\right) \mathrm{d}t+\sum _{j=1}^{d}\sigma _{i,j}\left( t\right) \mathrm{d}W_{j,t}\right) ,\quad 0\le t\le T,\quad 1\le i\le N. \end{aligned}$$
It is known that for any previsible process
\(\overrightarrow{\theta }=\left( \theta _{1,t},\ldots ,\theta _{d,t}\right) \) that solves
\(\mu _{i}-r=\sum _{j=1}^{d}\sigma _{i,j}\left( t\right) \theta _{j,t}\) and
\( \int _{0}^{T}\theta _{s}^{2}\mathrm{d}s<\infty \) a.s., there is an equivalent martingale measure associated with
\(\overrightarrow{\theta }\) whose Radon–Nikodym derivative is given by
$$\begin{aligned} \frac{\mathrm{d}\mathbb {Q}}{\mathrm{d}\mathbb {P}}=\mathrm {exp}\left( -\int _{0}^{T}\sum _{j=1}^{d}\theta _{j,t}\mathrm{d}W_{j,t}-\frac{1}{2} \int _{0}^{T}\sum _{j=1}^{d}\theta _{j,t}^{2}\mathrm{d}t\right) . \end{aligned}$$
Such process
\(\overrightarrow{\theta }=\left( \theta _{1,t},\ldots ,\theta _{d,t}\right) \) is known as the market price of risk. It is known that if
\( N=d\), and if
\(\left( \sigma _{i,j}\left( t\right) \right) \) is a full rank matrix for every
t, then the market is complete and otherwise, incomplete. Let us also introduce the following set:
$$ \begin{aligned} \mathcal {A}=\left\{ c+\int _{0}^{T}\sum _{i=1}^{N}h_{i,t}\mathrm{d}x_{i,t}\left| \begin{array}{c} c\in \mathbb {R},h_{t}\text { is a previsible}\, \& \\ \exists a\in \mathbb {R},\int _{0}^{t}\sum \limits _{i=1}^{N}h_{i,t}\mathrm{d}x_{i,t}\ge a\text { a.s.},\forall 0\le t\le T \end{array} \right. \right\} . \end{aligned}$$
It is clear that
\(\mathcal A\) is a convex cone.
3 Market consistent valuation
Let us now proceed with the definition of the market consistency.
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.
However, the opposite is not true as it is shown in the following example.
One can prove the following theorem by following discussions in Pelsser and Stadje (
2014).
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.
Consider a sub-linear pricing rule
\(\pi \) on a cone
\(\mathcal {X}\). In that case, we extend the range of
\(\pi \) to
\(\left( -\infty ,\infty \right] \)$$\begin{aligned} \bar{\pi }(x)=\left\{ \begin{array}{ll} \pi (x) ,&{}\quad x\in \mathcal {X} \\ +\infty ,&{}\quad \text { otherwise.} \end{array} \right. \end{aligned}$$
This extension allows us to use the dual representation of sub-linear pricing rules as
$$\begin{aligned} \bar{\pi }(x)=\sup \limits _{z\in \varDelta _{\bar{\pi }}}E(zx),\forall x\in L^{p}. \end{aligned}$$
(9)
In order to obtain the dual representation for
\(\bar{\pi }\), we need to introduce the dual polar of a scalar cone of random payoffs. If
\(\mathcal {X}\) is a cone, the dual polar of the set
\(\mathcal {X}\) is given by
$$\begin{aligned} \mathcal {X}^{\circ }:=\left\{ z\in L^{q}|E(zx)\le 0\,\forall x\in \mathcal {X} \right\} . \end{aligned}$$
Note that
\(\mathcal {X}^{\circ }\) is a closed convex cone in
\(L^{q}\). We then have the following proposition in convex analysis.
Now, let us begin with the following theorem, which looks very similar to the results in Pelsser and Stadje (
2014).
Before we prove the theorem we need to introduce the inf-convolution and state some related propositions which prove to be very useful in the continuation. Let
\(f_{1}\) and
\(f_{2}\) be two convex functions defined from
\( L^{p}\) to
\(\left( -\infty ,\infty \right] \). Then, the inf-convolution of
\(f_{1}\) and
\(f_{2}\) is defined as
$$\begin{aligned} f_{1}\square f_{2}(y)=\inf \limits _{x\in L^{p}}\left\{ f_{1}(x)+f_{2}(y-x)\right\} . \end{aligned}$$
The following proposition, which is a standard result in the literature of convex analysis, presents the necessary and sufficient conditions under which solution to the hedging problem exists (see Rockafellar
1997 for instance).
Now, we can prove Theorem
2.
An immediate corollary is the following.
Now, we have the following theorem for consistency of type II.
Finally, the following corollary is very useful:
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 two-step 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 non-hedgeable 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.
More specifically, let us introduce the hedging strategy by considering a position
y in an incomplete market whose risk has to be evaluated consistent with the market. To achieve this, we find a variable, among all variables in the set
\(\mathcal {X}\), that mimics
y most closely. In other words, we want to project
y on the set
\(\mathcal {X}\) as its best estimation (associated with the best estimate). Suppose for a moment that we know the best estimation and denote it by
\(x\in \mathcal {X}\). Hence,
y can be decomposed into two parts: a best estimation
x an unhedged part
\(y-x\), which is associated with the risk margin. The cost of the best estimation part is given by
\(\pi (x)\), and the risk generated by unhedged part, which cannot be diversified by any member of
\(\mathcal {X}\), is measured by
\(\varPi (y-x)\). We call
\(\pi (x)\) the best estimate and
\(\varPi (y-x)\) the risk margin. The idea is to minimize the aggregate cost of the hedging given as
\(\pi (x)+\varPi (y-x)\). Therefore, one can state the problem as follows,
$$\begin{aligned} \varPi _{\pi }(y):=\inf \limits _{x\in \mathcal {X}}\left\{ \pi (x)+\varPi (y-x)\right\} . \end{aligned}$$
(11)
In this case, the market imperfections are reflected by the (nonlinear) pricing rule
\(\pi \) and the risk evaluator
\(\varPi \) which capture the market incompleteness, respectively. From an insurance point of view, the minimum hedging cost can be considered as a normal practice if we see the pair
\((x,y-x)\) as a capital restructuring that can mitigate the risk of the insurance company. It is clear that all insurance companies will restructure their capital to achieve the minimum risk, justifying the infimum in (
11). Hence, we introduce the following concept:
Now we move toward addressing if
\(\varPi _{\pi }\) is a well-defined evaluator. Let us first state the following result for
\(\varPi _{\pi }\) defined in (
11) (for a proof see Barrieu and Karoui
2005).
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.
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.
Combining Proposition
6 with Theorem
2, we get the following theorem.
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
\(\varPi \).
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.
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.
This corollary has a wide range of applications, since it shows how in a perfect market one can construct market consistent valuations.
4.3 Two-step evaluation and hedging
Pelsser and Stadje (
2014) establish that if
\(\pi (x)=E(zx)\), for a unique stochastic discount factor
z, then—under appropriate conditions—all market consistent risk evaluators can be represented within a two-step procedure
$$\begin{aligned} \varPi (x)=\pi \left( \varPi _{\mathcal {G}}(x)\right) =E\left( z\varPi _{\mathcal {G} }(x)\right) , \end{aligned}$$
for a particular mapping
\(\varPi _{\mathcal {G}}:L^{\infty }\rightarrow L^{\infty }(\mathcal {G})\), where at least
\(\varPi _{\mathcal {G}}(x)=x,\forall x\in L^{\infty }(\mathcal {G})\).
In order to have a two-step representation for market consistent risk evaluators in our setting, we generalize the concept of a two-step risk evaluator. First, for any
\(x\in \mathcal {X}\), let us introduce the following equivalent class
\(EQ_{\pi }(x)\)$$\begin{aligned} EQ_{\pi }(x)=\{y\in \mathcal {X}|\pi (y)=\pi (x)\}. \end{aligned}$$
Moreover, define the class of all equivalent classes as follows,
$$\begin{aligned} EQ_{\pi }=\{B\subseteq L^{p}|\exists x\in \mathcal {X};B\subseteq EQ_{\pi }(x)\}. \end{aligned}$$
We have the following definition for a two-step evaluator.
First, using the hedging approach, we show market consistent evaluators can be represented, under particular conditions, as a two-step evaluation. Let
\( \varPi \) be a risk evaluator and
\(\pi \) a pricing rule. For any
\(y\in L^{p}\) let
$$\begin{aligned} S_{\pi ,\varPi }(y)=\{x\in \mathcal {X}|x\text { is a solution to the hedging problem}\}. \end{aligned}$$
Then, we have the following theorem whose proof is straightforward and, hence, omitted.
Next, we combine this theorem with Theorem
5 to obtain the following 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 two-step estimators
Even though Theorem
8 assures that, under mild conditions, any market consistent evaluator can be represented as a two-step evaluator, it is not very helpful for constructing two-step evaluators in practice. For this reason, we opt to take a different path. Let us consider a mapping
\(\varPi _{\mathcal {X}}:L^{p}\rightarrow \{B:B\subseteq \mathcal {X}\}\), where
\(\varPi _{ \mathcal {X}}(x)=\{x\},\forall x\in \mathcal {X}\). It is clear that for any pricing rule
\(\pi \),
$$\begin{aligned} \varPi (y)=\min \limits _{x\in \varPi _{\mathcal {X}}(y)}\pi (x) \end{aligned}$$
is type I market consistent. Furthermore, if
\(\pi \) is cone linear on
\( \mathcal {X}\) and
\(\varPi \) is convex, then it is also type II market consistent. Therefore, the problem can be reduced to choosing an appropriate
\(\varPi _{\mathcal {X}}\). We propose the following strategy.
The first one is motivated by Pelsser and Stadje (
2014), as explained in Sect.
4.3. Let
\(\mathcal {X}=L^{p}(\mathcal {G})\), for a sub-sigma-algebra
\(\mathcal {G}\subseteq \mathcal {F}\), and
\(\pi :L^{p}( \mathcal {G})\rightarrow \mathbb {R}\) be any pricing rule. Let us also introduce
\(\varPi _{\mathcal {X}}=\) \(E_{\mathcal {G}}\),
\(\mathrm {VaR}_{\alpha }^{ \mathcal {G}}\) or
\(\mathrm {CVaR}_{\alpha }^{\mathcal {G}}\), where they are expectation, VaR and CVaR conditioned on
\(\mathcal {G}\), respectively (for more details on this see Pelsser and Stadje
2014).The following examples are market consistent evaluators:
-
\(\varPi (x)=\pi \left( \delta E_{\mathcal {G}}(\max \{x-E_{\mathcal {G} }(x),0\}^{p})+E_{\mathcal {G}}(x)\right) ,\)
-
\(\varPi (x)=\pi \left( \delta \mathrm {VaR}_{\alpha }^{\mathcal {G}}(x-E_{ \mathcal {G}}(x))+E_{\mathcal {G}}(x)\right) ,\)
-
\(\varPi (x)=\pi \left( \delta \mathrm {CVaR}_{\alpha }^{\mathcal {G}}(x-E_{ \mathcal {G}}(x))+E_{\mathcal {G}}(x)\right) .\)
However, a larger family of two-step evaluators can be constructed by using loss functions. Let
\(L:L^{p}\rightarrow \left[ 0,\infty \right] \) be a function so that
\(L(x)=0\) if and only if
\(x=0\). Let
\(\varPi _{L}(y)=\mathrm { argmin}_{x\in \mathcal {X}}L(y-x)\). For instance, one can consider
\( L(y)=\Vert y\Vert _{L^{2}}\). Therefore, the following market consistent evaluator can be constructed
$$\begin{aligned} \varPi ^{L}(y)=\min \limits _{\left\{ x\in L^{p}\left| L(y-x)=\min \limits _{x^{\prime }\in \mathcal {X}}L(y-x^{\prime })\right. \right\} }\pi (x). \end{aligned}$$
Note that, in many cases the minima
x in
\(\left\{ x\in L^{p}\left| L(y-x)=\min \nolimits _{x^{\prime }\in \mathcal {X}}L(y-x^{\prime })\right. \right\} \) is unique; for instance, if we take
\(L(y)=\Vert y\Vert _{L^{2}}\) and
\(\mathcal {X}\) is a subspace in
\(L^{2}\). In order to develop more practical examples of market consistent risk evaluators, let us combine this idea with the approach we developed in Example
5. Indeed, let us consider
\( N+1\) test assets
\(x_{0},x_{1},\ldots ,x_{N}\), and associated prices
\( p_{0},p_{1},\ldots .,p_{N}\), and consider
\(\mathcal {X}=\left\{ \sum \nolimits _{i=0}^{N}a_{i}x_{i}\left| a_{i}\in \mathbb {R} ,i=0,1,\ldots ,N\right. \right\} \).
First, let us take
\(L(u)=\Vert u\Vert _{L^{2}}\). Then, for any
\(y\in L^{2}\), we have to solve the problem
$$\begin{aligned} \min \limits _{(a_{0},a_{1},\ldots ,a_{N})\in \mathbb {R}^{N+1}}\sigma \left( y-\sum \limits _{i=0}^{N}a_{i}x_{i}\right) , \end{aligned}$$
where
\(\sigma (.)\), denoted the standard deviation. The solution to this minimization problem is the OLS estimator of
y on
\(x_{0},x_{1},\ldots ,x_{N}\), which we denote by
\(\hat{a}\). Recall that since the pricing rule is linear over
\(\mathcal {X}\) for any
\(z\in \text {SDF}\), the market consistent evaluator is
$$\begin{aligned} \varPi ^{\mathrm {OLS}}(y)=E(z(\hat{a}\cdot X))=\hat{a}\cdot p=p^{\prime }\hat{a} , \end{aligned}$$
where
\(X=(x_{0},x_{1},\ldots ,x_{N})\) and
\(p=(p_{0},p_{1},\ldots ,p_{N})\).
If we denote the time series observations of each test asset
\(x_{i}\) and the position
y by
\(x_{i}=\left( (x_{i,t})_{t=0}^{T}\right) ^{\prime }\) and
\( y=\left( (y_{t})_{t=0}^{T}\right) ^{\prime }\), then we know that the OLS estimator is obtained as
\(\hat{a}=(X^{\prime }X)^{-1}y^{\prime }X\), and therefore, the market consistent evaluation is given by
$$\begin{aligned} \varPi ^{\mathrm {OLS}}(y)=p^{\prime }(X^{\prime }X)^{-1}y^{\prime }X. \end{aligned}$$
Note that if we use the Fama–French portfolios, then we also know that
\( p=(1,0,0,\ldots ,0)\).
Now let us consider another loss function to replace OLS with the quantile regression of
y on
\(x_{0},x_{1},\ldots ,x_{N}\). For that we have to assume
$$\begin{aligned} L(u)=\rho _{1-\alpha }\left( u\right) =u\left[ (1-\alpha )\mathbb {I} _{\left\{ u>0\right\} }-\alpha \mathbb {I}_{\left\{ u\le 0\right\} }\right] , \end{aligned}$$
and
\(\mathbb {I}\{\cdot \}\) denotes the indicator function. For a given tolerance level
\(\alpha \), we have to solve the following problem
$$\begin{aligned} \min \limits _{(a_{0},a_{1},\ldots ,a_{N})\in \mathbb {R}^{N+1}}\frac{1}{T} \sum \limits _{t=1}^{T}\rho _{\alpha }\left( y_{t}-\sum a_{i}x_{i,t}\right) . \end{aligned}$$
(15)
It is interesting that one can also solve the quantile regression via solving the following problem
$$\begin{aligned} \mathrm {VaR}_{\alpha }\left( y-\sum _{i}a_{i}x_{i}\right) =0. \end{aligned}$$
In this case, a hedging strategy removes all the risk as measured by
\( \mathrm {VaR}_{\alpha }\). Therefore, the market consistent valuation is again
$$\begin{aligned} \varPi ^{\mathrm {Q}}(y)=p^{\prime }\hat{a}, \end{aligned}$$
where
\(\mathrm {VaR}_{\alpha }\left( y-\sum _{i}\hat{a}_{i}x_{i}\right) =0.\)
5.2 Super-evaluators
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.
In practice, pricing rules are non-decreasing since they have to be consistent with no-arbitrage condition. Thus, let us assume that
\(\pi \) is non-decreasing. Then, motivated by the super-hedging strategy (e.g., see Karoui and Quenez
1995) for pricing, define the super estimator
\(\tilde{\varPi }\) as follows,
$$\begin{aligned} \tilde{\varPi }(x)=\inf \limits _{\left\{ y\in \mathcal {X},y\ge x\right\} }\pi (x). \end{aligned}$$
(16)
If we consider
\(\mathcal {X=}L^{\infty }\), it is clear that
\(\pi \) is non-decreasing and as a result
\(\pi (-\Vert y\Vert _{\infty })\le \tilde{\varPi }(y)\le \pi (\Vert y\Vert _{\infty })\). Therefore,
\(\tilde{\varPi }\) is well defined. On the other hand, by construction
\(\tilde{\varPi }(x)=\pi (x),x\in \mathcal {X}\), so
\(\tilde{\varPi }\) is type I market consistent.
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.
Consider again the approach we developed in Example
5. More specifically, we have to solve the following problem:
$$\begin{aligned} \begin{array}{l} \inf p^{\prime }a \\ \text {s.t. }y\le \sum _{i}a_{i}x_{i} \end{array} \end{aligned}$$
(17)
In order to find the value of this linear programming problem, we solve the dual problem:
$$\begin{aligned} \begin{array}{l} \max E(\lambda y) \\ E(x_{i}\lambda )=p_{i},\quad i=0,\ldots ,N \\ \lambda \ge 0, \end{array} , \end{aligned}$$
(18)
where
\(\lambda \) is a Lagrangian multiplier. If we assume the risk free asset is equal to one, i.e.,
\(x_{0}=1\), then this corresponds to the so-called super-hedging price and
\(\lambda \)s are the members of the set of all SDF’s.
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 two-step manner. Finally, we showed how to construct market consistent valuations as two-step estimators and super-evaluators.