Disentangling Price, Risk and Model Risk: V&R measures

TL;DR

This paper introduces a novel framework for assessing intrinsic risk in financial positions by evaluating model and price uncertainties through a family of probability measures and derivative-based testing.

Contribution

It develops a new interpretation of quasiconvex duality in a Knightian setting and constructs Value&Risk measures based on derivative testing of pricing models.

Findings

New interpretation of quasiconvex duality in a Knightian context

Construction of Value&Risk measures using derivative testing

Framework for assessing additional capital needed for acceptability

Abstract

We propose a method to assess the intrinsic risk carried by a financial position $X$ when the agent faces uncertainty about the pricing rule assigning its present value. Our approach is inspired by a new interpretation of the quasiconvex duality in a Knightian setting, where a family of probability measures replaces the single reference probability and is then applied to value financial positions. Diametrically, our construction of Value\&Risk measures is based on the selection of a basket of claims to test the reliability of models. We compare a random payoff $X$ with a given class of derivatives written on $X$ , and use these derivatives to \textquotedblleft test\textquotedblright\ the pricing measures. We further introduce and study a general class of Value\&Risk measures $% R(p,X,\mathbb{P})$ that describes the additional capital that is required to make $X$ acceptable under a probability $\mathbb{P}$ and given the initial price $p$ paid to acquire $X$.

Equations

$$E_{\mathbb{Q}}[f(X)]=\int f_{X}d\mathbb{Q}=\int fd{Q}_X\text{,}$$

$$V(a,X;f)=\underset{\mathbb{Q}\in \mathcal{M}}{\sup }\left\{E_{\mathbb{Q}}[f(X)]\mid A(\mathbb{Q},X)\le a\right\}.$$

$${\Pi }_{\phi }(r,X,;\mathbb{Q}):=\underset{f\in K}{\inf }\left\{E_{\mathbb{Q}}\left[f(X)\right]\mid \phi (f,X)\ge r\right\}.$$

$$R_{\phi }(p,X;\mathbb{Q}):=\sup \{s\in \mathbb{R}\mid {\Pi }_{\phi }(s,X;\mathbb{Q})\le p\}.$$

$$(p,X,\mathbb{P})\in \mathbb{R}\times \mathcal{L}(\Omega ,\mathcal{F})\times \mathcal{P}(\Omega )$$

$$Q{\preccurlyeq }_{1}P\text{ if and only if }{F}_P(x)\le F_Q(x)\text{ for all }x\in \mathbb{R},$$

$${\mathbb{P}}^1{\preccurlyeq }_X{\mathbb{P}}^2\text{ if and only if }{P}_X^1{\preccurlyeq }_1{P}_X^2.$$

$$\mathbb{P}\circ (X-p{)}^{-1}=T_{-p}{P}_X.$$

$$R(p,X,\mathbb{P})$$

$$\begin{array}[]{ccccc}\text{P\&L Risk}&=&\text{Price of X}&+&\text{Risk of the Payoffs}\\ R(p,X,\mathbb{P})&=&p&+&\Phi(P_{X})\end{array}$$

$$R(p,X,\mathbb{Q})=p-E_{\mathbb{Q}}[X],$$

$$F_X(z)$$

$$F_Y(z)$$

$$R(p,X,\lambda {\mathbb{P}}^1+(1-\lambda ){\mathbb{P}}^2)\le R(p,X,{\mathbb{P}}^1)\vee R(p,X,{\mathbb{P}}^2).$$

$$R(p,\lambda X_1+(1-\lambda )X_2,\mathbb{P})\le R(p,X_1,\mathbb{P})\vee R(p,X_2,\mathbb{P}).$$

$$R(p,X,\mathbb{P}):=\Phi (T_{-p}{P}_X)=p+\Phi (P_X).$$

$$\Lambda V@R(P_X):=-\sup \left\{m\in \mathbb{R}\mid \mathbb{P}(X\le x)\le \Lambda (x),\forall x\le m\right\}.$$

$$R(p,X,\mathbb{P})$$

$$R(p,X,\mathbb{P})=p+{\Lambda }^{-p}V@R(P_X)$$

$$\text{(Sup-CA)}R(p,X+\alpha ,\mathbb{P})\ge R(p,X,\mathbb{P})-\alpha .$$

$$\text{(Sub-CA)}R(p,X+\alpha ,\mathbb{P})\le R(p,X,\mathbb{P})-\alpha .$$

$$R_{\mathbb{A}}(p,X,\mathbb{P})=\inf \{m\mid P_X\in {\mathcal{A}}_m^p\}.$$

$$R_{\mathbb{A}}(p,X,\mathbb{P})=\inf \{m\mid T_m{P}_X\in {\mathcal{A}}^p\},$$

$${\mathcal{M}}_X:=\left\{Q_X\in \mathcal{P}(\mathcal{B}(\mathbb{R}))\mid Q_X=\mathbb{Q}(X\le x)\text{, }\mathbb{Q}\in \mathcal{M}\right\}$$

$$\mathcal{M}=\{\mathbb{Q}\in \mathcal{P}(\Omega )\mid E_{\mathbb{Q}}[S]=S_0\text{ and }{E}_{\mathbb{Q}}[F^i]=F_0^i\text{, }i=1,...,N\}.$$

$$V_A(a,X;f):=\underset{\mathbb{Q}\in \mathcal{M}}{\sup }\left\{E_{\mathbb{Q}}\left[f(X)\right]\mid A(\mathbb{Q},X)\le a\right\}=\underset{\mathbb{Q}\in \mathcal{M}}{\sup }\left\{\int f(X)d\mathbb{Q}\mid A(\mathbb{Q},X)\le a\right\}.$$

$$V(a,X;f)=\underset{Q\in {\mathcal{M}}_X}{\sup }\left\{\int fdQ\mid \tilde{A}(Q)\le a\right\}.$$

$$v=V(a,X;f)$$

$$a=V^{-1}(v,X;f)=\inf \{s\in \mathbb{R}\mid V(s,X;f)\ge v\}$$

$$V^{-1}\left(\int fd{Q}_X,X;f\right)=\inf \left\{s\in \mathbb{R}\mid V(s,X;f)\ge \int fd{Q}_X\right\}$$

$${\alpha }_K(Q_X):=\underset{f\in K}{\sup }{V}^{-1}\left(\int fd{Q}_X,X;f\right)$$

Full text

Diametrically, our construction of Value&Risk measures is based on the selection of a basket of claims to test the reliability of models. We compare a random payoff $X$ with a given class of derivatives written on $X$, and use these derivatives to “test” the pricing measures.

We further introduce and study a general class of Value&Risk measures $R(p,X,\mathbb{P})$ that describes the additional capital that is required to make $X$ acceptable under a probability $\mathbb{P}$ and given the initial price $p$ paid to acquire $X$.

Keywords: Model Risk, Pricing Uncertainty, Test Functions, Value&Risk Measures, Law Invariant Risk Measures, Quasi-convex Duality.

1 Introduction

The art of finance is essentially related to the capacity of transferring the Risk: many notions (replicability, hedging trading strategies, superhedging, quantile hedging, partial hedging, indifference pricing, see for example [11]) are essentially based on some technique which aims at replacing the risk carried by one claim $X$ by the risk of some other object $Y$ that is considered sufficiently close to $X$ (whatever it means), provided that the risk of the auxiliary object $Y$ is easier to compute.

In this paper we take such approach in order to evaluate the intrinsic risk of a claim $X$ by comparing the value of $X$ with the value of a family of derivatives $f(X)$ on $X$, having a bounded level of risk. In this way, we will conclude that the intrinsic risk of $X$ corresponds to the maximal risk reduction we would obtain buying $X$ at price $p$ and selling a derivative $f(X)$, in the given class, with a price at most equal to $p$. This methodology is sketched below but will be analyzed in detail only in Section 4.2, as in the Introduction we will illustrate the main concepts only and defer to the subsequent sections the precise notations and mathematical details.

In the literature the approaches used are mainly based on the selection of a set of “calibrated” pricing model. In this setting, an important contribution is provided by Cont [7], where a quantitative framework to assess Model Uncertainty was introduced. The prices of a set of benchmark instruments written on the underlying was supposed to be known (allowing the possibility to belong to the bid-ask interval). Consequently arbitrage-free pricing models $\mathcal{Q}$ consistent with these benchmark prices lead to the natural definition of Coherent Measure of Model Risk as: $\mu_{\mathcal{Q}}(\cdot)=\sup_{Q\in\mathcal{Q}}E_{Q}[\cdot]-\inf_{Q\in\mathcal{Q}}E_{Q}[\cdot]$.

The absolute and relative measures of model risk, based on the specification of a set of alternative distributions around a reference one and on a worst- and best-case approach, are introduced in Barrieu and Scandolo [4].

Both the approaches in [7] and [4] are however very different from our analysis developed in Section 4.2.

We let $\mathcal{L}(\Omega,\mathcal{F})$ be the space of $\mathcal{F}$ measurable finite valued random variables with $\mathcal{F\subseteq B}(\Omega)$, the Borel sigma algebra of a Polish space $\Omega$. If $f:\mathbb{R}\rightarrow\mathbb{R}$ is a Borel function and $X\in\mathcal{L}(\Omega,\mathcal{F})$, the random variable $f(X)$ is interpreted as the terminal payoff of a contingent claim written on the underlying asset having terminal value $X$. Suppose that the price of this contingent claim is determined by the real function $f$ and by the distribution function $\mathbb{Q}(X\leq x)$ of $X$ with respect to a “pricing” probability measure $\mathbb{Q}$. As the choice of such pricing measure is clearly an important and problematic issue, in our approach we will contemplate a model risk function defined on a set of plausible models. The price under $\mathbb{Q}$ will be given by the formula:

[TABLE]

where $f_{X}:=f\circ X$ is the random variable $f(X)$ defined on $(\Omega,\mathcal{F})$ and $Q_{X}:=\mathbb{Q}\circ X^{-1}$ is the law of $X$ under $\mathbb{Q}$.

The reason of writing explicitly the above formula is that in the two approaches below we will simply exploit the “bilinear form” $\left\langle f,\mathbb{Q}\right\rangle_{X}:=E_{\mathbb{Q}}[f(X)]$, testing one variable via a set of the dual (testing) variables.

We stress the analogy of the two approaches that will be developed in Sections 4.1 and 4.2 and that are here briefly introduced.

Use Models to test Claims.

Consider an underlying $X\in\mathcal{L}(\Omega,\mathcal{F})$ and a claim $f:\mathbb{R}\rightarrow\mathbb{R}$.

In this approach, we “compare” the prices of the contingent claim $f(X)$ with respect to a given class of probability models $\mathbb{Q}\in\mathcal{M}$, and use these models to “test” $f(X)$. In other words we take the classical Knightian Uncertainty point of view and adopt a set of probability measures $\mathbb{Q}\in\mathcal{M}$ to asses possible prices of the claim. This idea is in agreement with the definition proposed by Cont [7]: the range of feasible prices varies from the minimal to the maximal one. Indeed an agent may incorporate her preferences, binding a maximal model risk she is willing to accept when choosing a pricing probability.

In our approach we further assume the existence of a model-risk function $A$ on $\mathcal{M}$ so that we may define the best (seller) price of the claim $f(X)$ relative to all possible choices of pricing measures under the constraint that the model risk is less than or equal to $a$, formally

[TABLE]

In this way $V(a,X;f)$ represents the maximum value of the contingent claim $f$ on the underlying $X$, for the level $a$ of model-risk in the choice of $Q$, i.e. is the best (seller) price of the claim $f(X)$ relative to all possible choices of pricing measures under the constraint that the model risk is less than or equal to $a$. By applying results from quasi-convex duality (see [13]) we then show under which conditions it is possible to recover the model risk function $A$ from the inverse function of $V.$

Use Claims to test Models.

Consider an underlying $X\in\mathcal{L}(\Omega,\mathcal{F})$ and a probability $\mathbb{Q}\in\mathcal{P}(\Omega)$.

In this novel approach, we “compare” $X$ with the derivatives on $X$, in a given class of derivatives $f\in K$, and use these derivatives to “test” $\mathbb{Q}$. Contrary to the above mentioned (Knightian Uncertainty) approach, here we select a class $K$ of derivatives to test the “reliability” of the model $\mathbb{Q}$.

An agent is willing to hold (or sell) the position $X$ but she is aware that she may face losses. In order to control these potential losses she will try to transfer/reduce the overall risk by buying (or selling) derivatives/insurances on $X$. We assume the existence of a risk reduction function, $\varphi(f,X)$, defined on the basket of claims $K$ (which in general is independent from a particular choice of the reference probability). Among those derivatives, which guarantee the same level of risk reduction $r$, the agent will choose the cheapest one with respect to the pricing rule she adopt, computing the minimal price

[TABLE]

The intrinsic risk for $X$, given that its present value is $p$ and $\mathbb{Q}$ is the pricing rule selected by the agent, is therefore provided by the left inverse of $\Pi,$ namely

[TABLE]

In fact if the optimization problems just mentioned can be solved then there will exists a derivative $f(X)$ such that the price $E_{\mathbb{Q}}[f(X)]$ is equal to $p$ and provides a risk reduction $R_{\varphi}(p,X;\mathbb{Q})$.

We analyze several properties of the map $R_{\varphi}$ (see Proposition 25 and 26) including the dependence of $R_{\varphi}$ from the set $K$ (Proposition 23).

In Section 5 we show how the choice of the class of test functions $K$ for $R_{\varphi}$ defined in (1), can be adapted to several different contexts. The key idea is that $K$ collects those derivatives which can be sold or acquired in order to cover unexpected/unbounded losses of the underlying $X$. In addition, we prove in Proposition 21 a quasi-convex duality result that allows us to recover the risk reduction function $\varphi$ from $R_{\varphi}$.

To the best of our knowledge, the approach of using a fixed basket of claims to test the reliability of models was not yet developed in the mathematical finance literature and it represents the first main contribution of this paper (see Section 4.2). The second one is the analysis and axiomatization on the Value and Risk (V&R) measures that we now illustrate.

1.1 On Value and Risk Measures

In Section 3 we propose a systematic study which allows to answer to the controversy about whether one should consider the future value of a position or the change in values as the argument of a risk measure (see the following excerpt from [3]).

“Although several papers (including an earlier version of this one) define risk in terms of changes in values between two dates, we argue that because risk is related to the variability of the future value of a position, due to market changes or more generally to uncertain events, it is better to instead consider future values only. Notice indeed that there is no need for the initial costs of the components of the position to be determined from universally defined market prices (think of over-the-counter transactions). The principle of bygones are bygones leads to this future wealth approach.”, Section 2.1, Artzner et al. [3].

Differently from what is suggested in [3], it is a common practice to apply standard risk measures as the Value at Risk or the Expected Shortfall to Profit and Loss (P&L) distributions. Given the triple $(p,X,\mathbb{P})$ with $p$ being the observed present value of $X$ and $\mathbb{P}$ a reference probability, the P&L distribution is the induced distribution of the variation $X-p$ with respect to $\mathbb{P}$. Indeed the P&L approach has the benefit to incorporate the price component in the risk assessment. On the other hand it is not possible to distinguish which source contributes mostly to the risk exposure, either a potential mis-pricing of $X$ or the future realization of $X$. This is clarified in Example 5 where we consider two random payoffs $X$ and $Y$ whose initial values are respectively $x,y\in\mathbb{R}$ and show that, even if the payoff $X$ is “riskier” than $Y$ by any Risk Measure (which is monotone decreasing with respect to the first stochastic order), when considering the P&L distributions of $X-x$ and $Y-y$ the risk order may be reverted, if the price $y$ is too large.

To overcome this drawback in Section 3 we will thus consider the triple**

[TABLE]

as the argument of a Value&Risk Measure $R(p,X,\mathbb{P})$, where $p$ is the observed initial value of $X$ or is assigned by a pricing functional.

Informally, $R(p,X,\mathbb{P})$ should describe the additional capital that is required to make $X$ acceptable under $\mathbb{P\in}\mathcal{P}(\Omega)$ and given the initial price $p$ paid to acquire $X$. We propose an axiomatic approach to define such Value&Risk (V&R) measures by describing some desirable minimal properties that $R(p,X,\mathbb{P})$ should satisfy.

Indeed risk measures defined on Profit and Loss distributions can be recovered as a particular case in the family of Value&Risk measures by defining $R(p,X;\mathbb{P})=\rho_{\mathbb{P}}(X-p)$ with $\rho_{\mathbb{P}}$ being a risk measure defined on some vector space $\mathcal{X}\subseteq L^{0}(\Omega,\mathcal{F},\mathbb{P})$. This case also suggests which are the reasonable properties that a V&R measure should satisfy.

On the other hand the map $R_{\varphi}$ as defined in (1) is a Value&Risk measure (see Theorem 22), which exceeds the common use of Profit and Loss distributions.

We point out that $R(p,X,\mathbb{P})$ can be interpreted:

• •

As an index of feasibility of the measure $\mathbb{P}$, with $X$ acting the role of a fixed parameter; in this case $R(p,X,\mathbb{\cdot})$ should behave as a model risk measure over the laws $P_{X}\in\mathcal{P}(\mathbb{R})$ (see Section 2 for a review of such notion);

• •

As a measure of the risk we are facing buying $X$ at price $p$, with $\mathbb{P}$ acting as the agent model belief; in this case $R(p,\cdot,\mathbb{P})$ should behave as a risk measure on random variables $X\in\mathcal{L}(\Omega,\mathcal{F})$.

One relevant feature of considering such V&R Measures $R(p,X,\mathbb{P})$ is the possibility to disentangle the three most important sources of uncertainty: the price $p$ (which in general might not be unique, but rather belong to a bid-ask interval), the random payoff $X$ and the probability $\mathbb{P}$. This differs to the common practice of concentrating these three information in a unique object which is the Profit and Loss distribution.

As a consequence this reflects into the behavior of $R(p,X,\mathbb{P})$ with respect to the addition of a cash amount $\alpha\in\mathbb{R}$. Note that there are several reasonable properties regarding “cash invariance”, corresponding to the different ways one may add cash: $R(p+\alpha,X,\mathbb{P});$ $R(p,X+\alpha,\mathbb{P});$ $R(p,X,\mathbb{P}^{\alpha})$. In Section 3 we will explicitly characterize the V&R Measures satisfying three distinct cash invariance properties and show the relevance of taking into account the initial amount $p$ needed to buy $X$. We will therefore conclude (Proposition 10 and Remark 11), that the choice between “future value only” versus “P&L” is not arbitrary and it rests on the type of cash invariance one is willing to accept.

2 Risk Measures on $\mathcal{P}(\mathbb{R})$.

Notations.

Let $(\Omega,\mathcal{B}(\Omega))$ be a probability space with $\Omega$ Polish and $\mathcal{B}(\Omega)$ the Borel sigma algebra induced by the metric. Let $\mathcal{L}(\Omega,\mathcal{F})$ be the space of $\mathcal{F}$ measurable finite valued random variables, endowed with the pointwise partial order $\leq$ and $\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ its subspace of bounded random variables. We denote respectively by $\mathcal{P}(\Omega),\;\mathcal{P}(\mathbb{R})$ the set of all probability measures on $(\Omega,\mathcal{B}(\Omega))$, $(\mathbb{R},\mathcal{B}(\mathbb{R}))$.

Notice that for $\mathbb{P}\in\mathcal{P}(\Omega)$ and $X\in\mathcal{L}(\Omega,\mathcal{F})$ the expectation $E_{\mathbb{P}}[X]$ might not be even defined and for this reason we will make use of the convention $E_{\mathbb{P}}[X]=E_{\mathbb{P}}[X^{+}]-E_{\mathbb{P}}[X^{-}]$ with $\infty-\infty=-\infty$.

For any Borel function $f:\mathbb{R}\rightarrow\mathbb{R}$ and $X\in\mathcal{L}(\Omega,\mathcal{F})$, the random variable $f(X)$ is interpreted as the terminal payoff of a contingent claim written on the underlying asset having terminal value $X$.

If a probability $\mathbb{P}\in\mathcal{P}(\Omega)$ is fixed we define $L^{0}(\Omega,\mathcal{F},\mathbb{P})$ be the space of $\mathcal{F}$ measurable random variables that are $\mathbb{P}$ almost surely finite, endowed with the $\mathbb{P}$-almost sure partial order $\leq_{\mathbb{P}}$.

For any fixed $\mathbb{P}\in\mathcal{P}(\Omega)$ the random variable $X\in\mathcal{L}(\Omega,\mathcal{F})$ induces a probability measure $P_{X}\in\mathcal{P}(\mathbb{R})$ by $P_{X}=\mathbb{P}\circ X^{-1}$. We refer to [1] Chapter 15 for a detailed study of the convex sets $\mathcal{P}(\Omega)$ (resp. $\mathcal{P}(\mathbb{R})$). If $\mathbb{P}(X=x)=1$ for some $x\in\mathbb{R}$ then $P_{X}$ is the Dirac distribution, denoted by $\delta_{x},$ that concentrates the mass in the point $x\in\mathbb{R}$. Similarly we denote by $\delta_{\omega}\in\mathcal{P}(\Omega)$ the Dirac distribution on $\omega\in\Omega$.

Definition 1

We consider the following partial order for probability measures.

1. (i)

The first order stochastic dominance on $\mathcal{P}(\mathbb{R})$ is given by:

[TABLE]

where $F_{P}(x)=P(-\infty,x]$ and $F_{Q}(x)=Q(-\infty,x]$ are the distribution functions of $P,Q\in\mathcal{P}(\mathbb{R})$. 2. (ii)

*For any fixed $X\in\mathcal{L}(\Omega,\mathcal{F})$ we define the following partial order on *$\mathcal{P}(\Omega)$

[TABLE]

Notice that when $\mathbb{P}^{1}\preccurlyeq_{X}\mathbb{P}^{2}$ then $\mathbb{P}^{2}$ is a safer scenario than $\mathbb{P}^{1}$ for $X$. Observe also that, for $\mathbb{P}\in\mathcal{P}(\Omega)$ and any $X,Y\in\mathcal{L}(\Omega,\mathcal{F})$, $X\leq Y$ implies $X\leq_{\mathbb{P}}Y$ which implies $P_{X}\preccurlyeq_{1}P_{Y}$.

We shall always refer to $C^{0}(\mathbb{R})=\{f:\mathbb{R}\rightarrow\mathbb{R}\mid f\text{ continuous }\}$ and $C_{+}^{0}(\mathbb{R})=\{f\in C^{0}(\mathbb{R})\mid f\text{ increasing }\}$. Let $C_{b}(\Omega)$ be the space of bounded continuous function $f:\Omega\rightarrow\mathbb{R}$ and $ca(\Omega)$ the space of countably additive signed measures $\mu:\mathcal{B}(\Omega)\rightarrow\mathbb{R}$. We endow $ca(\Omega)$ with the weak*∗* topology $w^{\ast}=\sigma(ca(\Omega),C_{b}(\Omega))$. The dual pairing $\langle\cdot,\cdot\rangle:C_{b}(\Omega)\times ca(\Omega)\rightarrow\mathbb{R}$ is given by $\langle f,\mu\rangle=\int fd\mu$ and the function $\mu\mapsto\int fd\mu$ ($f\in C_{b}(\Omega)$) is $w^{\ast}$ continuous.

Risk Measures on $\mathcal{P}(\mathbb{R})$ for a fixed reference

probability.

We refer to [13] for a detailed analysis of risk measures defined on $\mathcal{P}(\mathbb{R}).$ Recall that, when $\mathbb{P}\in\mathcal{P}(\Omega)$ is fixed, a map $\rho_{\mathbb{P}}:L\rightarrow\overline{\mathbb{R}}:=\mathbb{R}\cup\left\{-\infty\right\}\cup\left\{\infty\right\}$, defined on given subset $L\subseteq L^{0}(\Omega,\mathcal{F},\mathbb{P}),$ is called law invariant if $X,Y\in L$ and $P_{X}=P_{Y}$ implies $\rho_{\mathbb{P}}(X)=\rho_{\mathbb{P}}(Y)$.

Therefore, when considering law invariant risk measures $\rho_{\mathbb{P}}:L^{0}(\Omega,\mathcal{F},\mathbb{P})\rightarrow\overline{\mathbb{R}}$ it is natural to shift the problem to the set $\mathcal{P}(\mathbb{R})$ by defining the new map $\Phi:\mathcal{P}(\mathbb{R})\rightarrow\overline{\mathbb{R}}$  as $\Phi(\mathbb{P}\circ X^{-1})=\rho_{\mathbb{P}}(X)$. This map $\Phi$ is well defined on the entire $\mathcal{P}(\mathbb{R})$, since there exists a bi-injective relation between $\mathcal{P}(\mathbb{R})$ and the quotient space $\frac{L^{0}}{\sim}$ (provided that $(\Omega,\mathcal{F},\mathbb{P})$ supports a random variable with uniform distribution), where the equivalence is given by $X\sim_{\mathcal{D}}Y$ $\Leftrightarrow P_{X}=P_{Y}$. However, $\mathcal{P}(\mathbb{R})$ is only a convex set and the usual operations on $\mathcal{P}(\mathbb{R})$ are not induced by those on $L^{0}$, namely $(P_{X}+P_{Y})(A)=P_{X}(A)+P_{Y}(A)\neq P_{X+Y}(A)$, $A\in\mathcal{B}_{\mathbb{R}}$. From [13] we recall the following

Definition 2

A Risk Measure on $\mathcal{P}(\mathbb{R})$ is a map $\Phi:\mathcal{P}(\mathbb{R})\rightarrow\mathbb{R}\cup\{+\infty\}$ such that:

(Mon)

$\Phi$* is $\preccurlyeq_{1}$-monotone decreasing: $P\preccurlyeq_{1}Q$ implies $\Phi(P)\geq\Phi(Q)$;*

(QCo)

$\Phi$* is quasi-convex: $\Phi(\lambda P+(1-\lambda)Q)\leq\Phi(P)\vee\Phi(Q)$, $\lambda\in[0,1].$*

Quasiconvexity can be equivalently reformulated in terms of sublevel sets: a map $\Phi$ is quasi-convex if for every $c\in\mathbb{R}$ the set $\mathcal{A}_{c}=\{P\in\mathcal{P}(\mathbb{R})\mid\Phi(P)\leq c\}$ is convex.

As suggested by [19], we define the translation operator $T_{p}$ on the set $\mathcal{P}(\mathbb{R})$ by: $T_{p}P(-\infty,x]:=P(-\infty,x-p]$, for every $p\in\mathbb{R}$. Equivalently, if $P_{X}$ is the probability distribution of a random variable $X$ we define the translation operator as $T_{p}P_{X}=P_{X+p}$, $p\in\mathbb{R}$. As a consequence we map the distribution $F_{X}(x)$ into $F_{X}(x-p)$. Notice that $P\preccurlyeq_{1}T_{p}P$ for any $p>0$. We will interpret $T_{-p}P_{X}=\mathbb{P}\circ(X-p)^{-1}$ as the Profit and Loss distribution of the random payoff $X$ whose initial value is $p$.

Definition 3

We consider the following additional property for a risk measure $\Phi:\mathcal{P}(\mathbb{R})\rightarrow\mathbb{R}\cup\{+\infty\}$:

(TrI)

$\Phi$* is translation invariant if $\Phi(T_{p}P)=\Phi(P)-p$ for any $p\in\mathbb{R}.$*

Notice that (TrI) corresponds exactly to the notion of cash additivity for risk measures defined on a space of random variables as introduced in [3].

3 Value and Risk measures: $V\&R$

We consider a simple setting, in which the risk of a financial portfolio is evaluated over its (empirical) profit and loss (P&L) distribution in a one-period investment horizon. (i.e. we restrict the problem to two dates $t_{0}$ and $t_{1}$). For simplicity we can think of $t_{0}=0$ and $t_{1}=1$, but a (sufficiently long) market history is supposed to be known before time $t_{0}=0$. The risk manager can observe the present market values of a basket of financial tradable assets at $t_{0}$ and hence she will be able to compute the time $t_{0}$ price of any portfolio strategy. Tradable assets are described by a $d$-dimensional vector of initial prices $S^{0}\in\mathbb{R}^{d}$ and a $d$-dimensional random vector of payoffs $S:(\Omega,\mathcal{F})\rightarrow\mathbb{R}^{d}$. We are implicitly assuming the interest rate is zero or that the asset prices are already discounted. Given a random variable $X:(\Omega,\mathcal{F})\rightarrow\mathbb{R}$ any choice of the (historical) probability $\mathbb{P}\in\mathcal{P}(\Omega)$ and any price $p$ of $X$ will determine the Profit and Loss (P&L) distribution of $(X-p)$, namely

[TABLE]

The price $p$ could represent the observed initial price of $X$ or could be assigned via a pricing functional. In either cases, the P&L distribution will be given by: $T_{-p}P_{X}$. In addition, if a risk measure $\rho_{\mathbb{P}}$ is also assigned, then it will induce a risk measure on P&L distribution $\Phi:\mathcal{P}(\mathbb{R})\rightarrow\mathbb{R}\cup\left\{\infty\right\}$ by: $\rho_{\mathbb{P}}(X-p)=\Phi\left(T_{-p}P_{X}\right)$.

The drawback of such P&L approach is that usually the price component cannot be distinguished from the distribution component and this becomes a critical point as far as we are facing Uncertainty on the reference probability $\mathbb{P}$.

We will thus consider the triple $(p,X,\mathbb{P})\in\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})\times\mathcal{P}(\Omega)$ as the argument of our Value&Risk functional $R(p,X,\mathbb{P})$, where the initial value of $X$ is assigned by $p$.

Remark 4

Let $\mathbb{P}\in\mathcal{P}(\Omega)$. To better clarify the role of the sign of the variable $p$ we consider the following simple example: if $C=(S_{T}-k)^{+}$ is the payoff of a Call Option written on an underlying asset $S$, then the initial value of $C$ is positive and given by $c$. In the case we buy $C$ we will consider the triple $(c,C,\mathbb{P})$ as the argument of $R$. On the other hand if we are selling $C$ we will consider $(-c,-C,\mathbb{P})$. Thus in general for $R(p,X,\mathbb{P})$ the variable $p$ represents the value of $X$ at time [math]. In particular a positive value $p>0$ represents the price we paid to hold $X$ and a negative values $p<0$ corresponds to the amount $|p|$ we received selling $X$.

Illustrative observations.

Risk measurement is in general not only a binary answer to the question ‘is a portfolio acceptable?’. Any risk procedure allows us to quantify the level of risk exposure so that an extra capital requirement can be assessed to cover future unexpected losses. In order to develop the intuition leading to the following definition and properties of the V&R measures we present a common simple situation.

Consider a price/portfolio couple $(p,X)$ given by the selling of a call option by an agent whose personal belief is $\mathbb{P}$ (i.e $(p,X)=(-c,-C)$ with $c>0$). Obviously we expect that any rational agent will willingly sell $C$ if the statistical information guarantee that the risk $\Phi(\mathbb{P}_{-C})$ is low enough to be recovered by the amount $c$, i.e. if $R(-c,-C,\mathbb{P}):=-c+\Phi(\mathbb{P}_{-C})$ is non positive. Similarly an agent who is paying $c$ to acquire $C$ will be happy to be informed that no additional capital is required i.e. $R(c,C,\mathbb{P})=c+\Phi(\mathbb{P}_{C})$ is non-positive.

Informally we claim that the quantity $R(p,X,\mathbb{P})$ gives the eventual extra capital requirement the agent has to save if the level of risk $\Phi(P_{X})$ is too high.

Thus this extra capital requirement can be written in terms of acceptance set as follows:

[TABLE]

where $\mathcal{A}^{0}\subseteq\mathcal{P}(\mathbb{R})$ is the set of $P\&L$ distributions that are acceptable for the regulator. But this is only a particular case of a more general formulation that allow to conceive several reasonable cash additivity properties for $R(p,X,\mathbb{P}).$

The aforementioned situation can be summarized by a decomposition of the type

[TABLE]

Recall that usually regulators/risk managers focus their attention only on the component which estimates the risk of the Profit and Loss distribution. The interpretation is the following: the risk of the Profit and Loss distribution $T_{-p}P_{X}$ is strongly related to the price that was paid to hold $X$. When $R$ is defined as in (2) the total capital requirement will be given by the market price $p$ that was paid to acquire $X$ plus the risk of the payoff $\Phi(P_{X})$. Notice that usually if $p$ is positive (resp. negative) then $\Phi(P_{X})$ is expected to be negative (resp. positive) as suggested by the example of the Call Option described in Remark 4. The simplest example of such $V\&R$ measure is:

[TABLE]

which express in fact that the intrinsic risk of acquiring $X$ at a given price $p$ is exactly the discrepancy between $p$ and $E_{\mathbb{Q}}[X]$, assuming that $\mathbb{Q}$ is the pricing rule. However, this case and the decomposition in (2) may hold only in special cases of the general family of Value& Risk measures.

In this paper we generalize the usual form $p+\inf\{m\mid T_{m}P_{X}\in\mathcal{A}^{0}\}$. To explain this generalization we consider the following two steps.

First we consider a situation in which acceptance of a position $X$ has an explicit dependence on its price $p$. In such a case $R(p,X,\mathbb{P})=\inf\{m\mid T_{m}P_{X}\in\mathcal{A}^{p}\}$, and we recover the classical framework if $\mathcal{A}^{p}=\{T_{-p}P\mid P\in\mathcal{A}^{0}\}$.

Second we push the problem to the utmost general and interesting situation where $R(p,X,\mathbb{P})=\inf\{m\mid P_{X}\in\mathcal{A}_{m}^{p}\}$. We would like to stress that in the definition $R_{\mathbb{A}}(p,X,\mathbb{P})=\inf\{m\mid T_{m}P_{X}\in\mathcal{A}^{p}\}$ the position $T_{m}P_{X}$ has a different initial value with respect to $P_{X}$ which is naively speaking $p+m$. Notice that the set $\mathcal{A}_{m}^{p}$ is explicitly splitting the two components $p$ and $m$ corresponding respectively to the initial value of the position and the capital requirement to cover expected losses. Potentially these two components might be expressed in two different currencies and for this reason the quantity $T_{m}P$ might loose its meaning.

Example 5

We now consider two portfolios $X,Y$ whose initial values are respectively $x,y$ and suppose that the distribution of $X$ dominates the one of $Y,$ $P_{X}\preccurlyeq_{1}P_{Y}$ (which informally means $X$ is “riskier” than $Y$) and therefore for any Risk Measure $\Phi:\mathcal{P}\rightarrow\overline{\mathbb{R}}$ (which is monotone decreasing with respect to the first stochastic order) we have $\Phi(P_{X})\geq\Phi(P_{Y})$. It is also plausible that the initial price $y$ of $Y$ is not smaller than the one of $X$. However, if $y$ is “too large” compared to $x$ it is possible that the corresponding P&L distribution $T_{-y}P_{Y}$ is shifted too much to the left, the two distributions $T_{-y}P_{Y}$ and $T_{-x}P_{X}$ intersect each other and the risk order is reverted: $\Phi(T_{-x}P_{X})<\Phi(T_{-y}P_{Y})$.

For instance suppose that the distributions of $X$ and $Y$ are given by

[TABLE]

and take $\Phi=V@R_{\lambda}$ with $\lambda=0.01$. Then $V@R_{\lambda}(P_{X})=0.5-0.01=0,49>0.4=0.5-0.1=V@R_{\lambda}(P_{Y})$. But if $y>x+0.09$ then one easily checks that $V@R_{\lambda}(T_{-x}P_{X})<V@R_{\lambda}(T_{-y}P_{Y})$.

If we focus only on payoffs, an agent is induced to prefer $Y$ respect to $X$ since $P_{X}\preccurlyeq_{1}P_{Y}$. But obviously in order to hold position $Y$ the agent will have to pay an initial price which influences the risk profile: the first stochastic dominance makes sense as far as we compare positions having the same initial price.

We now provide the formal definition of Value&Risk measures and their properties.

Definition 6

A Value&Risk measure is any map $R:\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})\times\mathcal{P}(\Omega)\rightarrow\overline{\mathbb{R}}$ having the following four properties:

(1Mon)

for any fixed $(X,\mathbb{P})\in\mathcal{L}(\Omega,\mathcal{F})\times\mathcal{P}(\Omega)$ and $p\leq q$ we have $R(p,X,\mathbb{P})\leq R(q,X,\mathbb{P})$;

(2Mon)

for any fixed $(p,\mathbb{P})\in\mathbb{R}\times\mathcal{P}(\Omega)$ and $X\leq Y$ we have $R(p,Y,\mathbb{P})\leq R(p,X,\mathbb{P})$;

(3Mon)

for any fixed $(p,X)\in\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})$ and $\mathbb{P}^{1}\preccurlyeq_{X}\mathbb{P}^{2}$ we have $R(p,X,\mathbb{P}^{2})\leq R(p,X,\mathbb{P}^{1})$;

(QCo)

Quasiconvex on $\mathcal{P}(\Omega)$: for any $p\in\mathbb{R}$, $X\in\mathcal{L}(\Omega,\mathcal{F})$, $\mathbb{P}^{1},\mathbb{P}^{2}\in\mathcal{P}(\Omega)$ and $\lambda\in(0,1)$ we have

[TABLE]

(1Mon) is simply justified by observing that the higher is the price paid for $X,$ the higher is the risk. (2Mon) is the classical monotonicity property for risk measures on random variables. (3Mon) and (Qco) are the characteristic properties of risk measures on distributions (see Definition 2). Proposition 9 will characterize these different types of monotonicity in terms of acceptance sets.

The following condition is the appropriate extension, to this context, of the law invariant property of risk measures:

(CLI)

Cross-Law Invariant: for any fixed $(X,\mathbb{P}^{1}),(Y,\mathbb{P}^{2})\in\mathcal{L}(\Omega,\mathcal{F})\times\mathcal{P}(\Omega)$ such that $P_{X}^{1}=P_{Y}^{2}$ then $R(p,X,\mathbb{P}^{1})=R(p,Y,\mathbb{P}^{2})$ for all $p\in\mathbb{R}$.

An additional feature (which in general fails in examples like $R(p,X;\mathbb{P})=p+V@R_{\lambda}(P_{X})$) is the quasiconvexity of the $R$ with respect to the $X$ variable. This corresponds to the usual principle of diversification as introduced in [6].

(QCoX)

Quasiconvex on $\mathcal{L}(\Omega,\mathcal{F})$: for any $p\in\mathbb{R}$, $\mathbb{P}\in\mathcal{P}(\Omega)$, $X_{1},X_{2}\in\mathcal{L}(\Omega,\mathcal{F})$ and $\lambda\in(0,1)$ we have

[TABLE]

V&R measures and addition of cash.

In Definition 6 we do not require a priori any Cash Invariance property of $R$. We now introduce the three axioms (Aff), (CA) and (DI) that describe different level of invariancy with respect to additional cash and needs to be studied separately. We will give a characterization of these properties in Propositions 10.

Definition 7

Consider the following properties, with respect to addition of a cash amount $\alpha\in\mathbb{R}$, that a Value&Risk measure $R$ may satisfy:

(Aff)

Price Affinity: $R(p+\alpha,X,\mathbb{P})=R(p,X,\mathbb{P})+\alpha$;

(CA)

Cash Additivity: $R(p,X+\alpha,\mathbb{P})=R(p,X,\mathbb{P})-\alpha$;

(DI)

Deviation Invariancy: $R(p+\alpha,X+\alpha,\mathbb{P})=R(p,X,\mathbb{P})$;

(DCA)

Distribution Cash Additivity: $R(p,X,\mathbb{P}^{1})-\alpha=R(p,Y,\mathbb{P}^{2})$ if $T_{\alpha}P_{X}^{1}=P_{Y}^{2}$.

Finally we will also need the following property:

(Nor)

Normalization: $R(0,0,\mathbb{P})=0$ for all $\mathbb{P}\in\mathcal{P}(\Omega)$;

Remark 8

Easy computations show that for a V&R measure:

(Aff) and (CA) imply (DI); (Aff) and (DI) imply (CA); (CA) and (DI) imply (Aff). 2. 2.

(DCA) iff (CA) and (CLI). 3. 3.

(Nor) and (DI) imply that $R(p,p,\mathbb{P})=0$ for any choice of $\mathbb{P}\in\mathcal{P}(\Omega)$.

Examples.

• (1)

First we consider the case in which $\Phi:\mathcal{P}(\mathbb{R})\rightarrow\mathbb{R}\cup\left\{\infty\right\}$ is a Risk Measure on distribution, as in Definition 2, that also satisfies (TrI), as in the case of the V@R or the Entropic Risk Measure. Define $R(p,X,\mathbb{P}):=\Phi(T_{-p}P_{X})$. By the property (TrI), coherently with equation (2), we deduce

[TABLE]

Here the map $R(p,X,\mathbb{P})$ satisfies all the properties given in Definitions 6 and 7 and property (CLI) but not (Nor), unless $\Phi(\delta_{0})=0$ ($\delta_{0}\in\mathcal{P}(\mathbb{R})$ being the Dirac distribution on $0\in\mathbb{R}$).

• (2)

In Appendix B we describe the risk measure $\Lambda V@R$, introduced in [13], which depends on a Probability/Loss function $\Lambda:\mathbb{R}\rightarrow[0,1]$ and is defined as follows:

[TABLE]

Define $R:\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})\times\mathcal{P}(\Omega)\rightarrow\overline{\mathbb{R}}$ by

[TABLE]

By a simple change of variables and by defining the one parameter family $\Lambda^{-p}$ as $\Lambda^{-p}(x)=\Lambda(x-p)$ we get

[TABLE]

Here the map $R$ satisfies (1-2-3Mon), (QCo), (DI), (CLI) but not (CA) nor (Aff). Even though (CA) fails, we may deduce from equation (32) and (3) that $R(p,p,\mathbb{P})=0$ independently from the choice of $\mathbb{P}$. We have $\Lambda(\cdot+\alpha)\geq\Lambda(\cdot)$ for $\alpha>0$, which implies $\Lambda^{-p}V@R(P_{X+\alpha})=\Lambda^{-p+\alpha}V@R(P_{X})-\alpha\geq\Lambda^{-p}V@R(P_{X})-\alpha$ and therefore

[TABLE]

Similarly if $\alpha<0$

[TABLE]

Acceptance sets and Value&Risk measures.

We now consider a general family $\mathbb{A}=\{\mathcal{A}_{m}^{p}\}_{p,m\in\mathbb{R}}$, $\mathcal{A}_{m}^{p}\subseteq\mathcal{P}(\mathbb{R})$ for every $p,m\in\mathbb{R}$, and study the properties of the map

[TABLE]

As already mentioned the set $\mathcal{A}_{m}^{p}$ is intentionally splitting the two components $p$ and $m$ corresponding respectively to the initial value of the position and the capital requirement to cover expected losses.

We begin with the analysis of three different types of monotonicity and quasiconvexity.

Proposition 9

Consider a family $\{\mathcal{A}_{m}^{p}\}_{p,m\in\mathbb{R}}$ contained in $\mathcal{P}(\mathbb{R})$ and $R_{\mathbb{A}}:\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})\times\mathcal{P}(\Omega)\rightarrow\overline{\mathbb{R}}$ as defined in (4).

1. (m1)

If for every $m\in\mathbb{R}$, $\mathcal{A}_{m}^{p}\subseteq\mathcal{A}_{m}^{q}$ for $q\leq p$ then $R_{\mathbb{A}}$ is (1Mon). 2. (m2)

If for every $(m,p,\mathbb{P})\in\mathbb{R}\times\mathbb{R}\times\mathcal{P}(\Omega)$, $X\leq Y$ and $P_{X}\in\mathcal{A}_{m}^{p}$ imply $P_{Y}\in\mathcal{A}_{m}^{p}$, then $R_{\mathbb{A}}$ is (2Mon). 3. (m3)

If for every $(m,p,X)\in\mathbb{R}\times\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})$, $\mathbb{P}_{1}\preccurlyeq_{X}\mathbb{P}_{2}$ and $P_{X}^{1}\in\mathcal{A}_{m}^{p}$ imply $P_{X}^{2}\in\mathcal{A}_{m}^{p}$, then $R_{\mathbb{A}}$ is (3Mon). 4. (c)

Suppose that: (i) for all $p\in\mathbb{R}$ and all $\alpha\leq\beta,$ $\mathcal{A}_{\alpha}^{p}\subseteq\mathcal{A}_{\beta}^{p}$; (ii) $\mathcal{A}_{m}^{p}$ is convex for all $p,m\in\mathbb{R}$; then $R_{\mathbb{A}}$ is (QCo).

Viceversa take $R:\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})\times\mathcal{P}\rightarrow\overline{R}$ and define $\mathbb{A}=\{\mathcal{A}_{m}^{p}\}_{p,m}$ by: $\mathcal{A}_{m}^{p}=\{Q_{X}\in\mathcal{P}(\mathbb{R})\mid R(p,X,\mathbb{Q})\leq m\}$. Then:

1. (M1)

If $R$ is (1Mon) then $\mathcal{A}_{m}^{p}\subseteq\mathcal{A}_{m}^{q}$ for $q\leq p$ and $m\in\mathbb{R}$. 2. (M2)

If $R$ is (2Mon) then for every $(m,p,\mathbb{P})\in\mathbb{R}\times\mathbb{R}\times\mathcal{P}(\Omega)$, $X\leq Y$ and $P_{X}\in\mathcal{A}_{m}^{p}$ imply $P_{Y}\in\mathcal{A}_{m}^{p}$. 3. (M3)

If $R$ is (3-Mon) then for every $(m,p,X)\in\mathbb{R}\times\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})$, $\mathbb{P}^{1}\preccurlyeq_{X}\mathbb{P}^{2}$ and $P_{X}^{1}\in\mathcal{A}_{m}^{p}$ imply $P_{X}^{2}\in\mathcal{A}_{m}^{p}$. 4. (C)

If $R$ is (QCo) then $\mathcal{A}_{m}^{p}$ is convex for every $p,m\in\mathbb{R}$.

Proof. (m1) Let $q\leq p$, by assumption $\{m\in\mathbb{R}\mid P\in\mathcal{A}_{m}^{p}\}\subseteq\{m\in\mathbb{R}\mid P\in\mathcal{A}_{m}^{q}\}$ . Hence $R_{\mathbb{A}}(p,X,\mathbb{P})=\inf\{m\in\mathbb{R}\mid T_{m}P_{X}\in\mathcal{A}^{p}\}\geq\inf\{m\in\mathbb{R}\mid T_{m}P_{X}\in\mathcal{A}^{q}\}=R_{\mathbb{A}}(q,X,\mathbb{P})$. Similarly for (m2) and (m3).

(c) We fix $(p,X)\in\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})$ and consider the map $R_{\mathbb{A}}(p,X,\cdot)$. We want to show that the sublevels of this map are convex. Let $B_{a}=\{\mathbb{Q}\in\mathcal{P}(\Omega)\mid R_{\mathbb{A}}(p,X,\mathbb{Q})\leq a\}$ and $\mathbb{P}^{1},\mathbb{P}^{2}\in B_{a}$. Assume w.l.o.g. that $a\geq M:=R_{\mathbb{A}}(p,X,\mathbb{P}^{1})\geq R_{\mathbb{A}}(p,X,\mathbb{P}^{2})$. Fix any $\varepsilon>0.$ Then there exist $M^{i}\leq M+\varepsilon$ $(i=1,2)$ such that $P_{X}^{i}\in\mathcal{A}_{M^{i}}^{p}$. Since $\mathcal{A}_{M^{i}}^{p}\subseteq\mathcal{A}_{M+\varepsilon}^{p}$ and $\mathcal{A}_{M+\varepsilon}^{p}$ is convex, we deduce that $\lambda P_{X}^{1}+(1-\lambda)P_{X}^{2}\in\mathcal{A}_{M+\varepsilon}^{p}$ for any $\lambda\in(0,1).$ Then $R_{\mathbb{A}}(p,X,\lambda\mathbb{P}^{1}+(1-\lambda)\mathbb{P}^{2})=\inf\{m\mid\lambda P_{X}^{1}+(1-\lambda)P_{X}^{2}\in\mathcal{A}_{m}^{p}\}\leq M+\varepsilon$. As this holds for any $\varepsilon>0$ we obtain $R_{\mathbb{A}}(p,X,\lambda\mathbb{P}^{1}+(1-\lambda)\mathbb{P}^{2})\leq M\leq a$.

Items (M1-2-3) and (C) are straightforward consequences of the definitions.

We are interested in possible declinations of the family $\mathbb{A}=\{\mathcal{A}_{m}^{p}\}_{p,m\in\mathbb{R}}$ which leads to different types of behavior with respect to cash addition. The following Proposition fully characterizes those Cross-Law-Invariant maps $R$ that satisfy either (CA) or (Aff) or (DI).

Proposition 10

Consider a family $\{\mathcal{A}_{m}^{p}\}_{p,m\in\mathbb{R}}$ contained in $\mathcal{P}(\mathbb{R})$ and $R_{\mathbb{A}}:\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})\times\mathcal{P}(\Omega)\rightarrow\overline{\mathbb{R}}$ as defined in (4).

1. (CA)

If $\mathcal{A}_{m}^{p}=T_{-m}\mathcal{A}^{p}=\{T_{-m}P\mid P\in\mathcal{A}^{p}\}$ , for a given family $\{\mathcal{A}^{p}\}_{p\in\mathbb{R}}\subseteq\mathcal{P}(\mathbb{R})$, then

[TABLE]

*and $R_{\mathbb{A}}$ is (DCA) and hence (CA) and (CLI). *

Viceversa take $R:\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})\times\mathcal{P}(\Omega)\rightarrow\overline{\mathbb{R}}$ satisfying (DCA). Define $\mathbb{A}=\{T_{-m}\mathcal{A}^{p}\}_{p,m\in\mathbb{R}}$ where $\mathcal{A}^{p}=\{Q_{X}\in\mathcal{P}(\mathbb{R})\mid X\in\mathcal{L}(\Omega,\mathcal{F})$ and $R(p,X,\mathbb{Q})\leq 0\}$. Then $R_{\mathbb{A}}=R$. 2. (Aff)

*If $\mathcal{A}_{m}^{p}$ satisfies for every $\alpha$, $\mathcal{A}_{m}^{p+\alpha}=\mathcal{A}_{m-\alpha}^{p}$ then there exists $\beta:\mathcal{P}(\mathbb{R})\rightarrow\overline{\mathbb{R}}$ such that $R_{\mathbb{A}}(p,X,\mathbb{P})=p+\beta(P_{X})$ and $R_{\mathbb{A}}$ is (Aff) and (CLI). *

Viceversa take $R:\mathbb{R}\times\mathcal{P}\rightarrow\overline{R}$ satisfying (Aff) and (CLI) and define $\mathbb{A}=\{\mathcal{A}_{m}^{p}\}_{p,m}$ where $\mathcal{A}_{m}^{p}=\{Q_{X}\in\mathcal{P}(\mathbb{R})\mid X\in\mathcal{L}(\Omega,\mathcal{F})$ and $R(p,X,\mathbb{Q})\leq m\}$. Then $\mathcal{A}_{m}^{p+\alpha}=\mathcal{A}_{m-\alpha}^{p}$ for all $\alpha$ and $R_{\mathbb{A}}=R$. 3. (DI)

*If $\mathcal{A}_{m}^{p}=\{Q\in\mathcal{P}(\mathbb{R})\mid T_{-p}Q\in\mathcal{A}_{m}^{0}\}$ then there exists $\beta:\mathcal{P}(\mathbb{R})\rightarrow\overline{\mathbb{R}}$ such that $R_{\mathbb{A}}(p,X,\mathbb{P})=\beta(T_{-p}P_{X})$ and $R_{\mathbb{A}}$ is (DI) and (CLI). *

Viceversa take $R:\mathbb{R}\times\mathcal{P}\rightarrow\overline{R}$ satisfying (DI) and (CLI) and define $\mathbb{A}=\{\mathcal{A}_{m}^{p}\}_{p,m}$ where $\mathcal{A}_{m}^{0}=\{Q_{X}\in\mathcal{P}(\mathbb{R})\mid R(0,X,\mathbb{Q})\leq m\}$ and $\mathcal{A}_{m}^{p}=\{Q\in\mathcal{P}(\mathbb{R})\mid T_{-p}Q\in\mathcal{A}_{m}^{0}\}$. Then $R_{\mathbb{A}}=R$.

Proof. (CA): the first implication is straightforward. For the viceversa notice that $R_{\mathbb{A}}(p,X,\mathbb{P})=\inf\{m\mid P_{X}\in T_{-m}\mathcal{A}^{p}\}=\inf\{m\mid T_{m}P_{X}\in\mathcal{A}^{p}\}=\inf\{m\mid R(p,X+m,\mathbb{P})\leq 0\}=\inf\{m\mid R(p,X,\mathbb{P})\leq m\}=R(p,X,\mathbb{P})$.

(Aff): we show the existence of $\beta$ by observing $R_{\mathbb{A}}(p,X,\mathbb{P})=\inf\{m\mid P_{X}\in\mathcal{A}^{p}_{m}\}=p+\inf\{m-p\mid P_{X}\in\mathcal{A}^{p}_{m}\}=p+\inf\{m\mid P_{X}\in\mathcal{A}^{p}_{m+p}\}=p+\inf\{m\mid P_{X}\in\mathcal{A}^{0}_{m}\}$ and therefore $\beta(P_{X}):=\inf\{m\mid P_{X}\in\mathcal{A}^{0}_{m}\}$. The viceversa is similar to the case (CA).

(DI): both implications follows as in the previous cases.

Remark 11

A particular case of (DCA) is when the set $\mathcal{A}^{p}$ in $\mathcal{A}_{m}^{p}:=T_{-m}\mathcal{A}^{p}$ is independent from $p$ (in which case we may set $\mathcal{A}_{m}^{p}=T_{-m}\mathcal{A}^{0}$ ). Then $R_{\mathbb{A}}(p,X,\mathbb{P})=$ $R_{\mathbb{A}}(0,X,\mathbb{P})$, for any $p\in\mathbb{R}$, and this corresponds, as mentioned in the Introduction, to the intuition proposed in the original paper by Delbaen et al. [3] that bygones are bygones.

4 Model Risk and Intrinsic Risk

In Section 4.1 (resp. 4.2) we develop the two approaches sketched in the Introduction. In Section 4.2 we will introduce intrinsic risk maps which constitute particular $V$&$R$ measures.

4.1 Use Models to Test Claims

Here we adopt the Knightian uncertainty point of view. We consider a set $\mathcal{M}\subseteq\mathcal{P}(\Omega)$ of probability measures $\mathbb{Q}$ on $(\Omega,\mathcal{F})$, each representing a possible pricing rule, and for a given $X\in\mathcal{L}(\Omega,\mathcal{F})$ the corresponding set

[TABLE]

of associated probability distribution $Q_{X}$ on $(\mathbb{R},\mathcal{B}(\mathbb{R}))$. For example, $\mathcal{M}$ could be a set of calibrated martingale measures, i.e. those induced by a fixed set of benchmark contingent claims $F^{i}$ each having initial cost $F_{0}^{i}$

[TABLE]

In this approach we assume that we have a criterion to asses the correctness of our selection, by assuming the existence of a model-risk function $A:\mathcal{P}(\Omega)\times\mathcal{L}(\Omega,\mathcal{F})\rightarrow\overline{\mathbb{R}}$ which asses the risk (or level of ambiguity) in the choice of a probability $\mathbb{Q}$, whenever we are modelling a random payoff $X$. A small value of $A(\mathbb{Q},X)$ means that we are quite confident in our choice. We proceed in four steps.

• (i)

We “test” the claim $f(X)$ over the set $\mathcal{M}$ under the constraint $A(\mathbb{Q},X)\leq a$ and obtain a Value function.

Definition 12

Let $X\in\mathcal{L}(\Omega,\mathcal{F}),$ $\mathcal{M}\subseteq\mathcal{P}(\Omega)$, $a\in\mathbb{R}$ and $A:\mathcal{P}(\Omega)\times\mathcal{L}(\Omega,\mathcal{F})\rightarrow\overline{\mathbb{R}}$. Define the map $V_{A}:\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})\times C_{b}\rightarrow\overline{\mathbb{R}}$

[TABLE]

Here $a$ and $X$ are given and we test the price of the claim $f(X)$ over the set $\mathcal{M}$ (compare with Definition 16). We will omit the dependence of $V_{A}$ from $A$, whenever no confusion may arise.

Remark 13

(compare with Remark 17 and equation (29)). In many cases there might exist $\widetilde{A}:\mathcal{M}_{X}\rightarrow\mathbb{R}\cup\left\{\infty\right\}$ such that $A(\mathbb{Q},X)=\widetilde{A}(\mathbb{Q}\circ X^{-1}).$ If this is the case we reduce the problem to

[TABLE]

The value

[TABLE]

is the maximum value of the contingent claim $f$ on the underlying $X,$ for the level $a$ of model-risk in the choice of $Q$, i.e. is the best (seller) price of the claim $f(X)$ relative to all possible choices of pricing measures under the constraint that the model risk is less than or equal to $a$.

• (ii)

The Intrinsic Model Risk.

By defining the generalized inverse of $V(\cdot,X;f)$ we obtain:

[TABLE]

which represents the minimum model risk one has to accept relative to a claim $f$ on $X,$ having price larger than $v.$ In other words, $a$ is the smallest model risk the decision maker is forced to accept in order to find a pricing model that attributes to $f(X)$ the price $v$.

• (iii)

The Indirect Model-Risk function.

If a pricing model $\mathbb{Q\in}\mathcal{P}(\Omega)$ is determined, the quantity

[TABLE]

is the risk associated to the choice of the distribution $Q_{X}$, induced by $X$, for pricing the particular claim $f$. Let $K\subseteq C_{b}$ and let

[TABLE]

be the maximum (w.r.to $f\in K)$ model risk associated to $\mathbb{Q}$, given the underlying $X.$ We then see that starting from the a priori given model risk function $\widetilde{A}$ we end up with another map $\alpha_{K}:\mathcal{M}_{X}\rightarrow\overline{\mathbb{R}}$ induced by $\widetilde{A}$ and $K$, which can be interpreted as the “Indirect Model Risk” function.

• (iv)

Duality.

The natural problem now is to find conditions on the set $K$ for which $\alpha_{K}=\widetilde{A}.$ The solution is given by the following result, a reformulation of Proposition 31 in Appendix B.

Proposition 14

Let $\widetilde{A}:\mathcal{P}(\mathbb{R})\rightarrow\overline{\mathbb{R}}$ be quasi-convex, $\preccurlyeq_{1}$-monotone decreasing and $\sigma(\mathcal{P}(\mathbb{R}),C_{b}(\mathbb{R}))$-lsc and let $X\in\mathcal{L}(\Omega,\mathcal{F}).$ Then

[TABLE]

This also shows that whenever $K\subseteq C_{b}^{-}$ then the indirect model risk function is less conservative than $\widetilde{A},$ i.e. $\alpha_{K}\leq\widetilde{A}$.

4.2 Use Claims to Test Models

In this section we explain our approach that constitutes one of the main contributions of this paper. It can be considered as the dual formulation of the situation described in Section 4.1 and the presentation will intentionally follow the analogous four steps of the previous section. Given a position $X\in\mathcal{L}(\Omega,\mathcal{F})$ and a probability $\mathbb{Q}\in\mathcal{P}$, we look at all possible prices $E_{\mathbb{Q}}[f(X)],$ for $f$ belonging to a subset $K$ of $C^{0}:=C^{0}(\mathbb{R})$ the space of continuous functions on $\mathbb{R}$. The idea is to use the claims in $K$, or in the set

[TABLE]

to test the pricing rule $\mathbb{Q}$.

In this approach we assume the existence of a map $\varphi:K\times\mathcal{L}(\Omega,\mathcal{F})\rightarrow\overline{\mathbb{R}}$, where $\varphi(f,X)$ assigns the risk reduction the agent will benefit by introducing a derivative $f(X)$ to cover the losses of $X$. Such function $\varphi$ has the analogue role of the map $A$ introduced in Section 4.1, but a different interpretation (see the examples below). Indeed $\varphi(f,X)$ will determine all claims $f\in K$ having at most the same level of risk reduction and use these claims to test $\mathbb{Q}$. The nomenclature ‘risk reduction’ relates to the fact that we are looking to the effects that additional derivatives have on the overall risk.

Example 15

Some examples can be easily built up considering a classical risk measure $\rho:\mathcal{L}(\Omega,\mathcal{F})\rightarrow\overline{\mathbb{R}}$, namely:

[TABLE]

In this examples the choice of a specific risk measure could be strongly related to the knowledge of a reference probability if $\rho=\rho_{\mathbb{P}}$. If we do not want to rely on $\mathbb{P}$ nor on $X$ natural choices for $\varphi$ are:

[TABLE]

which assign the risk reduction led by selling/buying $f$ in the worst case scenario, whatever underlying we are considering and independently from the reference probability $\mathbb{P}$. The risk reductions defined in (6) (7) and (8) satisfy the following condition:

[TABLE]

Since $K$ could be very small, (9) may be weaker than requiring that $\varphi(f,X)$ is independent from $X$. We will explain better this fact in the examples provided in Section 5.

Similarly to the previous section, we proceed in four steps.

• (i)

We test $Q_{X}$ over the set $K$ under the constraint $\varphi(f,X)\geq r$ and obtain a Price function.

Definition 16

Let $X\in\mathcal{L}(\Omega,\mathcal{F})$, $K\subseteq C^{0},$ $r\in\mathbb{R}$ and $\varphi:\mathcal{L}(\Omega,\mathcal{F})\rightarrow\overline{\mathbb{R}}$ be a risk reduction. Define the map $\Pi_{\varphi}:\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})\times\mathcal{P}(\Omega)\rightarrow\overline{\mathbb{R}}$

[TABLE]

Here $r$ and $X$ are given and we test $Q_{X}$ over the set $K$ (compare with Definition 12).

Remark 17

(Compare with Remark 13) In some cases, for example when $\varphi(f,X)=-\rho(f(X))$, the function $\varphi$ can be written as $\varphi(f,X)=\widetilde{\varphi}(f(X))$ with $\widetilde{\varphi}:K_{X}\rightarrow\overline{\mathbb{R}}$. In this case

[TABLE]

The price

[TABLE]

corresponds to the cheapest $\mathbb{Q}-$price of any derivative (on $X$) in the class $K$ which guarantees a reduction (at least equal to $r$) of the level of risk. When $p=\Pi_{\varphi}(r,X;\mathbb{Q})$ then there exists some $f(X)\in K_{X}$ having $\mathbb{Q}$-price almost equal to $p$ and risk reduction at least equal to $r$. Notice that from the buyer point of view, an underlying $X$ bought at price $p_{2}$ is riskier than the same $X$ bought at the smaller price $p_{1}<p_{2}$.

• (ii)

The Intrinsic Risk:

Definition 18

The map $R_{\varphi}:\mathbb{R}\times\mathcal{L}(\Omega,\mathcal{F})\times\mathcal{P}(\Omega)\rightarrow\overline{\mathbb{R}}$ is the generalized left inverse of $\Pi_{\varphi}(\cdot,X;\mathbb{Q})$:

[TABLE]

with $R_{\varphi}(p,X;\mathbb{Q})=-\infty$ for $p<\inf_{f\in K}E_{\mathbb{Q}}[f(X)]$ and $R_{\varphi}(p,X;\mathbb{Q})=+\infty$ for $p>\sup_{f\in K}E_{\mathbb{Q}}[f(X)]$.

We will omit the dependence of $\Pi_{\varphi},R_{\varphi}$ from $\varphi$, whenever no confusion may arise.

Suppose that we buy $X$ at a price $p$ and that $\mathbb{Q}$ is the correct pricing rule. We set: $r:=\sup\{s\in\mathbb{R}\mid\Pi(s,X;\mathbb{Q})\leq p\}$. Then $r$ is the maximal risk reduction $s$ for which $\Pi(s,X;\mathbb{Q})\leq p$, i.e. the maximal risk reduction that allows to find a claim $f(X)$ having $\mathbb{Q}$-price not larger than $p$ and risk reduction at least equal to $r$. Therefore $r=R(p,X;\mathbb{Q})$ is the intrinsic risk of acquiring $X$ at price $p$, assuming that $Q$ is the correct pricing rule, and* it corresponds, for particular functions $\varphi(f,X)$,* to the maximal risk reduction we would obtain buying $X$ and selling a derivative $f(X)$ with a price at most equal to $p$ (see examples in sections 5.1 and 5.3).

Remark 19

The previous interpretation can be more precisely explained as follows. The maximal risk reduction an agent may obtain by selling derivatives (on $X)$ with $\mathbb{Q}$-price smaller than $p$ is given by the function

[TABLE]

with $\widetilde{\varphi}$ given in Remark 17. Then Proposition 28 will show that $R(\cdot,X;\mathbb{Q})$ is the right-continuous version of $H(\cdot,X;\mathbb{Q})$.

Example 20** (A simple case.)**

If we assume that $K=C_{+}^{0}=\{f\in C^{0}\mid f\text{ increasing }\}$ and $\varphi(f(X))=\inf_{x\in\mathbb{R}}(f(x)-x)$ then $\Pi(r,X,\mathbb{Q})=E_{\mathbb{Q}}[X]+r,$ with the abuse of notation that if $E_{\mathbb{Q}}[X]=\pm\infty$ then $\Pi(r,X,\mathbb{Q})=\pm\infty$. Thus

[TABLE]

and, as already mentioned, in this case the intrinsic risk $R(p,X,\mathbb{Q})$ of acquiring $X$ at price $p$ is exactly the discrepancy between $p$ and $E_{\mathbb{Q}}[X]$, assuming that $\mathbb{Q}$ is the pricing rule.

• (iii)

The Indirect Risk Reduction

If a contingent claim $f(X)$ is determined, then the quantity

[TABLE]

is the risk reduction we face buying $X$ at the price $E_{\mathbb{Q}}\left[f(X)\right]$. If $\mathcal{M}\subseteq ca(\Omega)$ then

[TABLE]

represents the smallest (with respect to all $\mathbb{Q}\in\mathcal{M}$) risk reduction associated to the claim $f(X)$.

• (iv)

Duality.

Such indirect risk reduction function $\Psi_{\mathcal{M}}$ should then be compared with the one we started from. The following proposition, an immediate consequence of Theorem 30 in Appendix, shows under which conditions we might recover $\varphi$ from $\Psi_{\mathcal{M}}$.

Proposition 21

Assume that $K_{X}\subset\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ is a convex cone, $\sigma(\mathcal{L}^{\infty},ca)$-closed. If $\varphi:K_{X}\rightarrow\overline{\mathbb{R}}$ is monotone increasing, quasiconcave and $\sigma(\mathcal{L}^{\infty},ca)$-upper semicontinuous. Then

[TABLE]

where $\mathcal{K}_{1}^{\circ}=\{\mu\in ca\mid\mu(\Omega)=1\text{ and }\int f(X)d\mu\geq 0\;\forall f\in K\text{ s.t. }f(X)\geq 0\}$. (Compare with Proposition 14).

The following theorem, a consequence of Proposition 25 in Section 6, shows that the maps $R$ defined in (12) are V&R measures.

Theorem 22

Suppose that $K\subseteq C_{+}^{0}(\mathbb{R})$ and that $\varphi:K\times\mathcal{L}(\Omega,\mathcal{F})\rightarrow\overline{\mathbb{R}}$ satisfies (9). Then $R_{\varphi}$ defined by (12) is a Value&Risk measure that satisfies (CLI).

If in addition $K\subseteq\left\{f\in C_{+}^{0}(\mathbb{R})\mid f\text{ concave}\right\}$ then $R_{\varphi}$ defined by (12) satisfies also (QCoX).

It is worth mentioning that in virtue of Proposition 25 (a1) and (a2) the properties (1Mon) and (QCo) for the map $R$ will hold independently from the properties of $K$ or $\varphi$. The following Proposition (which proof is postponed in Appendix A) considers a fixed couple $(X,\mathbb{Q})$ and studies three properties of $\Pi$ and $R$ with respect to monotonicity, convex combinations and Minkowski sum of the sets $K$ of testing claims.

Proposition 23

For fixed $X\in\mathcal{L}(\Omega,\mathcal{F})$ and $Q\in\mathcal{P}(\Omega)$, consider $K\subseteq C^{0}$ and denote

[TABLE]

Let $K^{1},K^{1}\subseteq C^{0}$.

• 1

If $K^{1}\subseteq K^{2}$ then $\Pi_{K^{1}}\geq\Pi_{K^{2}}$ and $R_{K^{1}}\leq R_{K^{2}}$.

• 2

If $\varphi$ is quasiconcave as a function of $f$ then for any fixed $\lambda\in[0,1]$

[TABLE]

where

[TABLE]

• 3

If, for each $r\in\mathbb{R}$, $\varphi(f^{1},X)\geq r$ and $\varphi(f^{2},X)\geq r$ implies $\varphi(f^{1}+f^{2},X)\geq r$ then:

[TABLE]

It is clear that for a larger set of testing claims the price $\Pi$ will decrease and the risk reduction $R$ will increase. From Item 2, we deduce that by taking convex combinations of two sets $K^{1}$ and $K^{2}$ the risk reduction $R_{K^{\lambda}}$ is always larger than the minimum $R_{K^{1}}\wedge R_{K^{2}}$ of the two single risk reductions, so that the operation of taking convex combination is encouraged.

5 Examples of V&R measures from Def. 18

5.1 Control of unbounded losses of the underlying by releasing

options.

In this first example we consider the case in which the underlying $X\in\mathcal{L}(\Omega,\mathcal{F})$ produces a potentially unbounded loss, in particular $\inf_{\omega\in\Omega}X(\omega)=-\infty$. We here consider a fairly general class of approximating test functions described by a family $\{f_{\alpha}\}_{\alpha\in\mathbb{R}}$ such that

$\{f_{\alpha}\}_{\alpha\in\mathbb{R}}\subset C^{0}_{+}$ and $f_{0}(x)=x$ for every $x\in\mathbb{R}$; 2. 2.

$f_{\alpha}(x)<f_{\beta}(x)$ for every $\alpha<\beta$ and any $x\in\mathbb{R}$; 3. 3.

for every $x\in\mathbb{R}$ we have $\lim_{\alpha\to 0^{\pm}}f_{\alpha}(x)=x$; 4. 4.

if we set $c(\alpha)=\sup_{x\in\mathbb{R}}\{f_{\alpha}(x)-x\}$ then $\lim_{\alpha\to 0^{\pm}}c(\alpha)=0$.

As in (6), we choose

[TABLE]

which represents the risk reduction we would benefit by selling $f_{\alpha}(X)$ jointly to the acquisition of $X$ (independently from the payoff of $X$). Equivalently we may interpret $\varphi(f_{\alpha},X)$ as the maximal benefit we would realize buying $f_{\alpha}$ and selling the underlying. Notice that it can be easily checked from the properties of the family $\{f_{\alpha}\}$ that $c(\cdot)$ is (strictly) increasing in $\alpha$.

The parameter $\alpha$, which indexes the family, represents the degree of approximation: the higher $\alpha$ is the higher payoff the derivative $f_{\alpha}(X)$ will grant. The identity, i.e. when $\alpha=0$, clearly corresponds to the case in which the risk completely annihilates by buying and selling $X$. On the other hand for $\alpha>0$ (resp. $\alpha<0$) we are considering testing functions which approximate from above (resp. from below) the identity: therefore the strategy $X-f_{\alpha}(X)<0$ (resp. $X-f_{\alpha}(X)>0$) will bring losses (resp. gains) which are controlled by $c(\alpha)$.

If we write explicitly the function $\Pi$ defined in (10) we obtain the following formulation

[TABLE]

with $c^{-1}$ being the left inverse of $c$.

From Proposition 22 we know that $R$ defined by (12) is a Value& Risk map (i.e. $R$ satisfies (1-2-3Mon), (QCo) and (CLI)). Moreover, from $\varphi(f_{\alpha},X)=c(\alpha),$ we obtain

[TABLE]

which interpretation is the following: the intrinsic Risk of acquiring $X$ at price $p$ if we assume $\mathbb{Q}$ as the correct pricing rule , corresponds to the maximal risk reduction we would obtain buying $X$ and selling a derivative $f_{\alpha}(X)$ with a price at most equal to $p$. Indeed implementing the strategy which buys $X$ and sells $f_{\alpha}(X)$, has initial zero cost and guarantees that potential losses are at most given by $\varphi(f_{\alpha},X)=c(\alpha)$.

Properties 24

Given the above definitions we have the following additional properties.

1. P1

If for some $\alpha\in\mathbb{R}$ we have $p=E_{\mathbb{Q}}\left[f_{\alpha}(X)\right]$ then $R(p,X;\mathbb{Q})=\varphi(f_{\alpha},X)=c(\alpha)$.

The proof of this property follows directly from the representation of $R$ given in (15) and the properties of the family $\{f_{\alpha}\}$. 2. P2

$R(a,a;\mathbb{Q})=0$* for $a\in\mathbb{R}$ and for every $\mathbb{Q}\in\mathcal{P}(\Omega)$.*

We notice that $f_{s}(a)\leq a$ if and only if $s\leq 0$ so that $R(a,a;\mathbb{Q})=\sup\{c(s)\mid f_{s}(a)\leq a\}\leq 0$. Moreover for $s=0$ we have $c(s)=0$ and hence the thesis. 3. P3

If $p\lesseqqgtr E_{\mathbb{Q}}[X]$ then $R(p,X;\mathbb{Q})\lesseqqgtr 0$.

To show this last property we recall that for $\alpha>0$ (resp. $\alpha<0$ or $\alpha=0$) we have $f_{\alpha}(x)>x$ (resp. $f_{\alpha}(x)<x$ or $f_{\alpha}(x)=x$) for any $x\in\mathbb{R}$ so that $E_{\mathbb{Q}}[f_{\alpha}(X)]>E_{\mathbb{Q}}[X]$ (resp. $E_{\mathbb{Q}}[f_{\alpha}(X)]<E_{\mathbb{Q}}[X]$ or $E_{\mathbb{Q}}[f_{\alpha}(X)]=E_{\mathbb{Q}}[X]$). Then

[TABLE]

and similarly for $p>E_{\mathbb{Q}}[X]$ or $p<E_{\mathbb{Q}}[X]$.

5.2 Control of potential losses of the underlying using call options.

Consider the set of call options $K=\{f(x)=(x-k)^{+}\mid k\in\mathbb{R}\}$ for any possible strike and assume that the agent is confident about the fact that the underlying $X$ is bounded from below (i.e. $\inf_{\omega\in\Omega}X(\omega)>-\infty$). For simplicity let the codomain of $X$ be some convex subset of $\mathbb{R}$ (i.e. there are no gaps in the codomain) of the type $(a,+\infty)$ or $[a,+\infty)$ (in particular $\sup_{\omega\in\Omega}X(\omega)=+\infty$). Buying a call option written on $X$ for $k<0$ and selling $X$, guarantees both a positive performance and no expected losses but the agent will have to pay a high price. If $k>0$ (for a lower price) the agent will face controlled losses (for $X\geq 0$) and gains up to a value $|a|$ for $X<0$. We consider

[TABLE]

which represents the (pessimistic) payoff we will face in the worst case scenario acquiring the derivative and selling the underlying.

We have immediately that

[TABLE]

To prove the previous equality we only notice that for $k_{n}\uparrow-r$ as $n\rightarrow\infty$, we have $\int(X-k_{n})^{+}d\mathbb{Q}\downarrow\int(X+r)^{+}d\mathbb{Q}$.

Properties From Proposition 25 we know that $R$ defined by (12) is (1-2-3Mon), (QCo) and (CLI) and hence is a Value and Risk map. Moreover

P1

$R(E_{\mathbb{Q}}[(X-k)^{+}],X;\mathbb{Q})=\varphi((x-k)^{+},X)=-k$.

This property follows immediately from the definition. It states that if we pay for $X$ the same price as a call options with $k<0$ (resp. $k>0$) then we are facing an intrinsic risk given by $-k>0$ (resp. $-k<0$). This intrinsic risk corresponds to the difference between the payoff of $(X-k)^{+}$ and $X$ in the worst case scenario.

P2

Cash Additivity: $R(p,X+a;\mathbb{Q})=R(p,X;\mathbb{Q})-a$.

To show this property we simply observe that

[TABLE]

and therefore $R(p,X+a;\mathbb{Q})=\sup\{s\mid\Pi(s+a,X;\mathbb{Q})\leq p\}=\sup\{s-a\mid\Pi(s,X;\mathbb{Q})\leq p\}$, which gives the thesis.

P3

Let $X\geq 0$. If $p=E_{\mathbb{Q}}[X]$ (resp. $p\gtrless E_{\mathbb{Q}}[X]$) then $R(p,X;\mathbb{Q})=0$ (resp. $R(p,X;\mathbb{Q})\gtrless 0$ ). In particular $R(a,a;\mathbb{Q})=0$ for $a\in[0,+\infty)$ and for every $\mathbb{Q}\in\mathcal{P}(\Omega)$.

The property follows from $\sup\left\{s\in\mathbb{R}\mid E_{\mathbb{Q}}[(X+s)^{+}]\leq E_{\mathbb{Q}}[X]\right\}=0$. Indeed for any $s>0$ $E_{\mathbb{Q}}[(X+s)^{+}]=E_{\mathbb{Q}}[X]+s>E_{\mathbb{Q}}[X]$ and since $X\geq 0$ we have $E_{\mathbb{Q}}[(X)^{+}]=E_{\mathbb{Q}}[X]$.

P4

Let $E_{\mathbb{Q}}[X]<0$ then $R(E_{\mathbb{Q}}[X],X;\mathbb{Q})=-\infty$. In particular $R(a,a;\mathbb{Q})=-\infty$ for $a\in(-\infty,0)$ and for every $\mathbb{Q}\in\mathcal{P}(\Omega)$.

The proof is straightforward. Clearly the set of call options is not feasible for testing a position $X$ which has a strong negative component from the $\mathbb{Q}$-perspective.

5.3 Asymmetric Tail Control with concave derivatives

In this example we consider the case in which the underlying $X\in\mathcal{L}(\Omega,\mathcal{F})$ produces a potentially unbounded loss together with potentially unbounded gains, in particular $\sup_{\omega\in\Omega}X^{+}(\omega)=\sup_{\omega\in\Omega}X^{-}(\omega)=+\infty$. We here consider a fairly general class of concave and increasing test functions described by a family $\{f_{\alpha}\}_{\alpha\in\mathbb{R}}$ such that

$\{f_{\alpha}\}_{\alpha\in\mathbb{R}}\subset C^{0}_{+}\cap C^{1}$, $f_{\alpha}$ concave for every $\alpha$ and $f_{0}(0)=0,f^{\prime}_{0}(0)=1$; 2. 2.

$f_{\alpha}(x)<f_{\beta}(x)$ for every $\alpha<\beta$ and any $x\in\mathbb{R}$;

The higher $\alpha$ is the higher payoff the derivative $f_{\alpha}(X)$ will grant. Moreover for $\alpha<0$ the derivative $f_{\alpha}(X)$ will be strictly dominated by $X$ in any possible state of nature.

The agent sells $f_{\alpha}(X)$ and keeps $X$: for $\alpha>0$ she will be guaranteed a positive payoff in any case (with a reduced performance on the positive tail). For $\alpha<0$ the agent may suffer a controlled loss on the set $\{\omega\in\Omega\mid a_{1}<X(\omega)<a_{2}\}$ with $a_{1},a_{2}\in\mathbb{R}\cup\{\infty\}$ being the intersections of $f_{\alpha}(x)$ with the identity function $id(x)=x$. The residual risk in the worst case scenario left over after selling $f_{\alpha}(X)$ jointly to the acquisition of $X$ (independently from the payoff of $X$) is given by

[TABLE]

Easy computations show that $c(\alpha)=\bar{x}-f_{\alpha}(\bar{x})$ with $\bar{x}$ solution of $f_{\alpha}^{\prime}(x)=1$.

Notice that from the concavity of $\{f_{\alpha}\}_{\alpha\in\mathbb{R}}$ and Theorem 22 we know that $R$ defined by (12) is a Value& Risk map (i.e. $R$ is (1-2-3Mon), (QCo) and (CLI)) satisfying in addition the property (QCoX).

To obtain more explicit results we specify $f_{\alpha}(x)=g(\alpha)-e^{-x}$ with $g(0)=1$ and $g$ strictly increasing. In this case $c(\alpha)=g(\alpha)-1$ and we can write explicitly the function $\Pi$ defined in (10)

[TABLE]

Moreover we can observe that

[TABLE]

which interpretation is the following: the intrinsic Risk of acquiring $X$ at price $p$ is the discrepancy between the price $p$ and the value of the derivative in [math] under the probability measure $\mathbb{Q}$.

5.4 Insured testing functions

Given a position $X\in\mathcal{L}(\Omega,\mathcal{F})$ we face the problem that the risk connected to $X$ might be unbounded but the agent is not willing to sell $X$. For this reason she aims at buying an insurance on $X$ in order to control the risk. The set of insurances on $X$ is denoted by $\mathcal{I}\subseteq C^{0}$ and we assume that the price of each insurance $i\in\mathcal{I}$ is exogenously determined by a (possibly non-liner) functional $c(\cdot)$. The only requirements on $c$ are positivity ($c(i)\geq 0$ for every $i\geq 0$) and monotonicity. Let $\mathbb{Q}\in\mathcal{P}$ be a pricing rule, then the agent will try to minimize the cost under $\mathbb{Q}$ under the risk constraint for the aggregated position. In this case

[TABLE]

For this reason the agent will face the minimization problem

[TABLE]

Here $\Pi(r,X;\mathbb{Q})$ represents the price under $\mathbb{Q}$ of the cheapest insured position which guarantees that the risk diminishes at least by $r$. If for some $r$ this problem is solved by $i^{\ast}$ and if $p$ is the observed price for $X$, then we have an equilibrium between observed prices and pricing beliefs under $\mathbb{Q}$ i.e.

[TABLE]

with a risk residual equal to $r$.

Insuring using put options.

Consider the special case with $p$ representing the initial value of an unbounded from below risky position $X\in\mathcal{L}(\Omega,\mathcal{F})$ (i.e. $\inf_{\omega\in\Omega}X(\omega)=-\infty$). We insure $X$ using put options, so that

[TABLE]

and it is easy to show that $K\subset C_{+}^{0}$. Here $c:\mathbb{R}\rightarrow[0,+\infty)$ is assumed to be strictly increasing and $k-c(k)$ is strictly increasing with $\lim_{k\rightarrow+\infty}(k-c(k))>0$. Moreover we are interested at the case where $\inf_{\omega}X(\omega)=-\infty$ and choose consequently $\varphi(f,X)=\inf_{\omega}f(X(\omega))=k-c(k)$ which is independent from $X$ and represents the worst case payoff of the insured position $f(X)$. We compute

[TABLE]

where $b$ is the inverse of the function $y-c(y)$. Let $R$ be the map defined by (12). Simple inspections together with Proposition 25 show that $R$ is (1-2-3Mon), (QCo) and (CLI) and hence is a Value and Risk map. Indeed if $p=E_{\mathbb{Q}}[X]$ we have

[TABLE]

The proof follows from the definition. The interpretation is that the risk we face buying $X$ at price $E_{\mathbb{Q}}[X]$ corresponds to the index $s$ of the insuring derivative for which the exogenous price $c(s)$ corresponds to the price $E_{\mathbb{Q}}\left[(b(s)-X)^{+}\right]$ computed using $\mathbb{Q}$.

6 Appendix A

Let $\mathbb{Q}^{1},\mathbb{Q}^{2}\in\mathcal{P}(\Omega)$, $X\in\mathcal{L}(\Omega,\mathcal{F})$ and let $C_{+}^{0}:=\{f\in C^{0}\mid f\text{ increasing }\}.$ Recall the notation $\mathbb{Q}^{1}\preccurlyeq_{X}\mathbb{Q}^{2}$ given in Definition 1 and the fact that $\mathbb{Q}^{1}\preccurlyeq_{1}\mathbb{Q}^{2}$ implies $\int fdQ^{1}\leq\int fdQ^{2}$ for any $f\in C_{+}^{0}$.

The following Proposition collects the properties of $\Pi$ and $R$ and can be used to check that Theorem 22 holds true.

Proposition 25

[Monotonicity and convexity] Consider $\Pi$, $R$ defined respectively by (10) and (12). Then we have the following properties.

• (a1)

$\Pi(\cdot,X;\mathbb{Q})$* is monotone increasing and $R(\cdot,X;\mathbb{Q})$ is (1Mon).*

• (a2)

$\Pi(r,X;\cdot)$* is concave and $R(p,X;\cdot)$ is quasiconvex;*

• (b)

If $K\subseteq C_{+}^{0}(\mathbb{R})$ then:

• (b1)

$\mathbb{Q}^{1}\preccurlyeq_{X}\mathbb{Q}^{2}$* implies $\Pi(r,X;\mathbb{Q}^{1})\leq\Pi(r,X;\mathbb{Q}^{2})$ and $R$ is (3Mon);*

• (b2)

$\mathbb{Q}^{1}\preccurlyeq_{X}\mathbb{Q}^{2}$* and $p_{1}\geq p_{2}$ implies $R(p_{1},X;\mathbb{Q}^{1})\geq R(p_{2},X;\mathbb{Q}^{2})$;*

• (c)

Suppose that $\varphi$ satisfies

[TABLE]

Then $X\leq Y$ implies $\Pi(r,X;\mathbb{Q})\geq\Pi(r,Y;\mathbb{Q})$ and $R$ is (2Mon).

• (d)

If $K\subseteq\{f\in C_{+}^{0}(\mathbb{R})\mid f\text{ concave }\}$ and $\varphi$ satisfies (9) then $\Pi(r,\cdot;\mathbb{Q})$ is concave and $R(p,\cdot,\mathbb{Q})$ is quasiconvex.

• (e)

(CLI) Suppose that

[TABLE]

For any $(X,\mathbb{Q}^{1}),(Y,\mathbb{Q}^{2})\in\mathcal{L}(\Omega.\mathcal{F})\times\mathcal{P}(\Omega)$ such that $Q_{X}^{1}=Q_{Y}^{2}$ then $\Pi(r,X;\mathbb{Q}^{1})=\Pi(r,Y;\mathbb{Q}^{2})$ and $R(p,X;\mathbb{Q}^{1})=R(p,Y;\mathbb{Q}^{2})$, so that $R$ satisfies (CLI).

Notice that if $\varphi$ satisfies (9) then (18) and (19) hold true.

Proof. (a1) follows from the definition.

(a2) The concavity of $\Pi(r,X;\cdot)$ follows from its definition and the properties of the $\inf$. Take $\mathbb{Q}^{1},\mathbb{Q}^{2}$ and $\lambda\in(0,1)$ and let $R(p,X;\mathbb{Q}^{1}):=t_{1}\leq t_{2}:=R(p,X;\mathbb{Q}^{2})$. In this proof we omit the dependence on $X$. We need to prove that $R(p,\lambda\mathbb{Q}^{1}+(1-\lambda)\mathbb{Q}^{2})\leq t_{2}$. Note that $t_{i}=\sup\{s\mid\Pi(s,\mathbb{Q}^{i})\leq p\}$. Then: $\Pi(s,\mathbb{Q}^{i})>p$ for all $s>t_{i}$. Therefore, $\Pi(s,\mathbb{Q}^{1})>p$ and $\Pi(s,\mathbb{Q}^{2})>p$ for all $s>t_{2}$ implies: $\lambda\Pi(s,\mathbb{Q}^{1})+(1-\lambda)\Pi(s,\mathbb{Q}^{2})>p$ for all $s>t_{2}$. As a consequence:

[TABLE]

From the concavity of $\Pi(s,\cdot),$ $\lambda\Pi(s,\mathbb{Q}^{1})+(1-\lambda)\Pi(s,\mathbb{Q}^{2})\leq\Pi(s,\lambda\mathbb{Q}^{1}+(1-\lambda)\mathbb{Q}^{2})$, we obtain:

[TABLE]

Hence:

[TABLE]

(b1) Recall $\mathbb{Q}^{1}\preccurlyeq_{X}\mathbb{Q}^{2}$ implies $\int fdQ_{X}^{1}\leq\int fdQ_{X}^{2}$ for any $f\in C_{+}^{0}$. Clearly this shows that $\mathbb{Q}^{1}\preccurlyeq_{X}\mathbb{Q}^{2}$ implies $\Pi(s,X;\mathbb{Q}^{1})\leq\Pi(s,X;\mathbb{Q}^{2})$. As a consequence $\{s\mid\Pi(s,X;\mathbb{Q}^{1})\leq p\}\supseteq\{s\mid\Pi(s,X;\mathbb{Q}^{2})\leq p\}$ and $R(p,X;\mathbb{Q}^{1})\geq R(p,X;\mathbb{Q}^{2})$.

(b2) Follows from (a1) and (b1).

(c) Let $X\leq Y$. Then (18) implies $\{f\in K\mid\varphi(f,Y)\geq r\}\subseteq\{f\in K\mid\varphi(f,X)\geq r\}$ so that

[TABLE]

Moreover from this property we have $\{s\mid\Pi(s,Y;\mathbb{Q})\leq p\}\subseteq\{s\mid\Pi(s,X;\mathbb{Q})\leq p\}$ so that $R(p,Y;\mathbb{Q})\leq R(p,X;\mathbb{Q})$.

(d) The concavity of $\Pi(r,\cdot;\mathbb{Q})$ follows from $K\subseteq\{f\in C_{+}^{0}(\mathbb{R})\mid f\text{ concave }\}$, the properties of the $\inf$ and (9). Take $X_{1},X_{2}$ and $\lambda\in(0,1)$ and let $R(p,X;\mathbb{Q}):=t_{1}\leq t_{2}:=R(p,X;\mathbb{Q})$. In this proof we omit the dependence on $\mathbb{Q}$. We need to prove that $R(p,\lambda X_{1}+(1-\lambda)X_{2})\leq t_{2}$. As before $\Pi(s,X_{i})>p$ for all $s>t_{i}$. Therefore, $\Pi(s,X_{1})>p$ and $\Pi(s,X_{2})>p$ for all $s>t_{2}$. For any $\lambda\in(0,1)$ we have $\lambda\Pi(s,X_{1})+(1-\lambda)\Pi(s,X_{2})>p$ for all $s>t_{2}$. This implies:

[TABLE]

From the concavity of $\Pi(s,\cdot),$ we obtain:

[TABLE]

Hence:

[TABLE]

(e) Follows directly from the definitions and (19).

Proof of Proposition 23. The first property is straightforward. To prove (13) observe that

[TABLE]

The previous inequality is motivated by the quasi-concavity of $\varphi,$ namely

[TABLE]

Now we show (14).

[TABLE]

where the first inequality follows from (13).

The first inequality in Item 3 follows exactly with the same argument used in Item 2. The second inequality of Item 3 then is a consequence of:

[TABLE]

The proof of the following Proposition is omitted since is a straightforward consequence of the definitions.

Proposition 26** (Behavior with respect to cash)**

Consider $\Pi$, $R$ defined respectively by (10) and (12). Let $r,p,\alpha\in\mathbb{R}$ then we have the following properties.

1. (a)

$R(p,p;\mathbb{Q})=0$* if $\Pi(r,p;\mathbb{Q})=p+r$.* 2. (b)

$R(p+\alpha,X;\mathbb{Q})=\alpha+R(p,X;\mathbb{Q})$* if $\Pi(r+\alpha,X;\mathbb{Q})=\alpha+\Pi(r,X;\mathbb{Q})$.* 3. (c)

$R(p,X+\alpha;\mathbb{Q})=R(p,X;\mathbb{Q})-\alpha$* if $\Pi(r+\alpha,X;\mathbb{Q})=\Pi(r,X+\alpha;\mathbb{Q})$.* 4. (d)

$R(p+\alpha,X+\alpha;\mathbb{Q})=R(p,X;\mathbb{Q})$* if $\Pi(r,X+\alpha;\mathbb{Q})=\Pi(r,X;\mathbb{Q})+\alpha$.*

In the following Proposition we provide some explicit forms for $\Pi$, when we are able to find a representative class of one parameter functions.

Proposition 27

For any fixed $X\in\mathcal{L}(\Omega,\mathcal{F})$ consider $\varphi:K\times\mathcal{L}(\Omega,\mathcal{F})\rightarrow\mathbb{R}\cup{+\infty}$ and let $I_{\varphi}=\inf_{f\in K}\varphi(f,X)$. Assume that there exist $f^{0}\in K$ and a one parameter class of transformations $\{T_{\alpha}\}_{\alpha\geq I_{\varphi}}$ such that

• •

$T_{\alpha}:K\rightarrow K$* with *$T_{0}f^{0}=f^{0}$and $\varphi(T_{\alpha}f^{0},X)=\alpha$,

• •

$T_{\alpha}f^{0}<T_{\beta}f^{0}$* for $\alpha<\beta$,*

• •

for any $g\in K$ such that $\varphi(g,X)=\alpha$ we have $T_{\alpha}f^{0}\leq g$ for $\alpha\leq 0$ (resp. $T_{\alpha}f^{0}\geq f^{0}$ $\alpha\geq 0$).

Then

• (a)

$\Pi(r,X;\mathbb{Q})=\inf_{f\in K}\left\{E_{\mathbb{Q}}\left[f(X)\right]\mid\varphi(f,X)=r\right\}$.

• (b)

$\Pi(r,X;\mathbb{Q})=\inf_{\alpha\geq I_{\varphi}}\left\{E_{\mathbb{Q}}\left[T_{\alpha}\circ f^{0}(X)\right]\mid\varphi(T_{\alpha}\circ f^{0},X)\geq r\right\}$;

• (c)

$\Pi(r,X;\mathbb{Q})=\;E_{\mathbb{Q}}\left[T_{r}f^{0}(X)\right]$;

• (d)

$R(E_{\mathbb{Q}}[T_{\alpha}f^{0}(X)],X;\mathbb{Q})=\varphi(T_{\alpha}f^{0}(X))=\alpha.$**

Proof. (1) Obviously $\Pi(r,X;\mathbb{Q})\leq\inf_{f\in K}\left\{E_{\mathbb{Q}}\left[f(X)\right]\mid\varphi(f,X)=r\right\}$. By contradiction assume $<$. Then there exists $\overline{f}\in K$ such that $\varphi(\overline{f},X)>r$ and

[TABLE]

Take $\alpha=\varphi(\overline{f},X)$: in this way we have $\varphi(T_{\alpha}\circ f_{0},X)=\varphi(\overline{f},X)$ which implies $T_{\alpha}f^{0}\leq\overline{f}$. Moreover if $\varepsilon=\alpha-r>0$ we have $T_{\alpha-\varepsilon}f^{0}\leq T_{\alpha}f^{0}\leq\overline{f}$ with $\varphi(T_{\alpha-\varepsilon}f^{0},X)=r$ and thus a contradiction since $E_{\mathbb{Q}}[T_{\alpha-\varepsilon}\circ f_{0},X]\leq E_{\mathbb{Q}}[\overline{f}(X)]<\inf_{f\in K}\left\{E_{\mathbb{Q}}\left[f(X)\right]\mid\varphi(f,X)=r\right\}$.

(b) and (c) From the previous step

[TABLE]

By assumption we also have $T_{r}f^{0}\leq g$ for any $g\in K$ such that $\varphi(g,X)=r$ and hence $E_{\mathbb{Q}}\left[T_{r}f^{0}(X)\right]\leq\Pi(r,X;\mathbb{Q})$. Thus $T_{r}f^{0}$ corresponds to the minimizer.

(4) Follows directly from the definition of $R(E_{\mathbb{Q}}[T_{\alpha}f^{0}(X)],X;\mathbb{Q})$.

6.1 Duality for testing functions cones

This section is devoted to the proof of Theorem 30, which is instrumental to Proposition 21. In this section we will often omit in the notations the dependence from $\mathbb{R}$ and write $ca$ for $ca(\mathbb{R})$, similarly for the other symbols.

Let $\mathcal{K}\subseteq\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ and $\varphi:\mathcal{K}\rightarrow\overline{\mathbb{R}}$. Let $\Pi_{\varphi}:\mathbb{R}\times ca\rightarrow\overline{\mathbb{R}}$ be defined by:

[TABLE]

and let $R_{\varphi}:\mathbb{R}\times ca\rightarrow\overline{\mathbb{R}}$ be the right inverse of the increasing function $\Pi(\cdot,\mu)$

[TABLE]

Let $H:\mathbb{R}\times ca\rightarrow\overline{\mathbb{R}}$ be defined by

[TABLE]

Notice that the three functions $\Pi_{\varphi}(\cdot,\mu),$ $R_{\varphi}(\cdot,\mu)$ and $H_{\varphi}(\cdot,\mu)$ are monotone increasing. In the proofs we will omit the label $\varphi$ in $\Pi_{\varphi},$ $R_{\varphi}$ and $H_{\varphi}.$

Proposition 28

Let $H_{\varphi}^{+}(\cdot,\mu)$ be the right continuous version of $H_{\varphi}(\cdot,\mu)$. Then:

[TABLE]

Proof. Since $R(\cdot,\mu)$ is the right inverse of the increasing function $\Pi(\cdot,\mu),$ $R(\cdot,\mu)$ is right continuous. To prove that $H^{+}(p,\mu)\leq R(p,\mu)$ it is sufficient to show that for all $t>p$ we have:

[TABLE]

Indeed, if (24) is true

[TABLE]

as both $H^{+}$ and $R$ are right continuous in the first argument.

Writing explicitly the inequality (24)

[TABLE]

and letting $\xi\in\mathcal{K}$ satisfying $E_{\mu}[\xi]\leq t$, we see that it is sufficient to show the existence of $r\in\mathbb{R}$ such that $\Pi(r,\mu)\leq t$ and $r\geq\varphi(\xi)$. If $\varphi(\xi)=\infty$ then $\Pi(r,\mu)\leq t$ for any $r$ and therefore $R(t,\mu)=H(t,\mu)=\infty$.

Suppose now that $\infty>\varphi(\xi)>-\infty$ and define $r:=\varphi(\xi).$ As $E_{\mu}[\xi]\leq t$ we have:

[TABLE]

Then $r\in\mathbb{R}$ satisfies the required conditions.

To obtain $H^{+}(p,\mu):=\inf_{t>p}H(t,\mu)\geq R(p,\mu)$ it is sufficient to prove that, for all $t>p,$ $H(t,\mu)\geq R(p,\mu)$, that is :

[TABLE]

Fix any $t>p$ and consider any $r\in\mathbb{R}$ such that $\Pi(r,\mu)\leq p$. By the definition of $\Pi$, for all $\varepsilon>0$ there exists $\xi_{\varepsilon}\in\mathcal{K}$ such that $\varphi(\xi_{\varepsilon})\geq r$ and $E_{\mu}[\xi_{\varepsilon}]\leq p+\varepsilon.$ Take $\varepsilon$ such that $0<\varepsilon<t-p$. Then $E_{\mu}[\xi_{\varepsilon}]\leq t$ and $\varphi(\xi_{\varepsilon})\geq r$ and (25) follows.

From now on we suppose that $\mathcal{K}\subseteq\mathcal{L}^{\infty}(\Omega,\mathcal{F})$ is a convex cone. Consider $\mathcal{K}^{+}=\mathcal{K}\cap\mathcal{L}_{+}^{\infty}$ and define

[TABLE]

Proposition 29

Let $\mathcal{M}\subseteq\mathcal{K}^{\circ}$ and $\Psi_{\varphi}:\mathcal{K}\rightarrow\overline{\mathbb{R}}$ be defined by:

[TABLE]

If $\Psi_{\varphi}$ is continuous from above ($Y_{n}\downarrow Y$ implies $\Psi_{\varphi}(Y_{n})\downarrow\Psi_{\varphi}(Y)$), then:

[TABLE]

Proof. Let $Y\in\mathcal{K}$ and $\left\{Y_{n}\right\}\subseteq\mathcal{K}$. If $Y_{n}\geq Y$ and $Q\in\mathcal{M}$ then $E_{Q}[Y_{n}]\geq E_{Q}[Y].$ Since $H(\cdot,Q)$ is increasing, for every $Q\in\mathcal{M}$ we obtain

[TABLE]

Therefore:

[TABLE]

The last equality in the Proposition follows from (23).

In the following theorem we provide the representation of $\varphi$ in terms of the dual functions $H_{\varphi}$ and $R_{\varphi}$ defined in (22) and (21).

Theorem 30

Suppose that $\mathcal{K\subseteq L}^{\infty}(\Omega,\mathcal{F})$ is a convex cone $\sigma(\mathcal{L}^{\infty},ca)$-closed and that $\varphi:\mathcal{K}\rightarrow\overline{\mathbb{R}}$ is monotone increasing, quasiconcave and $\sigma(\mathcal{L}^{\infty},ca)$-upper semicontinuous (using the relative topology on $\mathcal{K}$). Then for all $Y\in\mathcal{K}$

[TABLE]

Proof. Fix $Y\in\mathcal{K}$. As $Y\in\left\{\xi\in\mathcal{K}\mid E_{\mu}[\xi]\leq E_{\mu}[Y]\right\}$, by the definition of $H(E_{\mu}[Y],\mu)$ we deduce that, for all $\mu\in ca$

[TABLE]

hence

[TABLE]

We prove the opposite inequality. Let $\varepsilon>0$ and define the set

[TABLE]

As $\varphi$ is quasi-concave and $\sigma(\mathcal{L}^{\infty},ca)$-upper semicontinuous (on $\mathcal{K}$), $C_{\varepsilon}$ is convex and $\sigma(\mathcal{L}^{\infty},ca)$-closed. Suppose $\varphi(Y)>-\infty$ (if $\varphi(Y)=-\infty,$ we may take $C_{M}:=\left\{\xi\in L^{\infty}\mid\varphi(\xi)\geq-M\right\}$ and the following argument would hold as well). Since $Y\notin C_{\varepsilon}$, the Hahn Banach theorem implies the existence of a continuous linear functional that strongly separates $Y$ and $C_{\varepsilon},$ that is there exist $\mu_{\varepsilon}\in ca$ such that

[TABLE]

Hence

[TABLE]

and from (27)

[TABLE]

Therefore, $\varphi(Y)=\inf_{\mu\in ca}H(E_{\mu}[Y],\mu)$.

To show that the $inf$ can be taken over $\mathcal{K}^{\circ}$, it is sufficient to prove that $\mu_{\varepsilon}\in\mathcal{K}^{\circ}$. Let $\xi\in C_{\varepsilon}.$ Given that $\varphi$ is monotone increasing and that $\mathcal{K}$ is a convex cone, $\xi+nW\in C_{\varepsilon}$ for every $n\in\mathbb{N}$ and $W\in\mathcal{K}^{+}$. From (28), we have:

[TABLE]

As this holds for any $W\in\mathcal{K}^{+}$ we deduce that $\mu_{\varepsilon}\in\mathcal{K}^{\circ}$. Therefore, $\inf_{\mu\in\mathcal{K}^{\circ}}H(E_{\mu}[Y],\mu)=\varphi(Y)$. By definition of $H(p,\mu)$,

[TABLE]

Hence we deduce $\varphi(Y)=\inf_{Q\in\mathcal{K}_{1}^{\circ}}H(E_{Q}[Y],Q).$

The remaining equalities follows from Propositions (28) and (29), since $\varphi(Y)=\inf_{Q\in\mathcal{K}_{1}^{\circ}}H(E_{Q}[Y],Q)$ is continuous from above. Indeed, if $Y_{n}\downarrow Y$ then the monotonicity of $\varphi$ implies $\varphi(Y_{n})\geq\varphi(Y)$ and $\varphi(Y_{n})\downarrow$. Moreover the Monotone Convergence Theorem implies for every $\mu=\mu^{+}-\mu^{-}\in ca$ that $E_{\mu^{\pm}}[Y_{n}]\rightarrow E_{\mu^{\pm}}[Y]$ and therefore $Y_{n}\rightarrow Y$ in the $\sigma(\mathcal{L}^{\infty},ca)$. As $\mathcal{K}$ is closed and $\varphi$ is usc in the $\sigma(\mathcal{L}^{\infty},ca)$ topology, we can conclude $\limsup_{n}\varphi(Y_{n})=\lim\varphi(Y_{n})\leq\varphi(Y)$, and therefore $\lim\varphi(Y_{n})=\varphi(Y)$.

7 Appendix B: Dual representation

We recall, from [13], the dual representation of risk measures defined on $\mathcal{P}(\mathbb{R}).$ Consider the set

[TABLE]

Proposition 31** (Prop. 5.6 [13])**

Any $\sigma(\mathcal{P}(\mathbb{R}),C_{b}(\mathbb{R}))$-lsc Risk Measure $\Phi:\mathcal{P}(\mathbb{R})\rightarrow\mathbb{R}\cup\left\{\infty\right\}$ can be represented as

[TABLE]

where $V:\mathbb{R}\times C_{b}(\mathbb{R})\rightarrow\overline{\mathbb{R}}$ is given by:

[TABLE]

and

[TABLE]

We also mention that the $\sigma(\mathcal{P}(\mathbb{R}),C_{b}(\mathbb{R}))$ lower semicontinuity property can be characterized with an appropriate and simple continuity from above condition with respect to the first order stochastic dominance (see [13] Proposition 2.5).

Example 32** (The certainty equivalent)**

Fix any continuous, bounded from below and strictly decreasing function $f:\mathbb{R}\rightarrow\mathbb{R}$. Then the map $\Phi_{f}:\mathcal{P}(\mathbb{R})\rightarrow\mathbb{R}\cup\{+\infty\}$ defined by:

[TABLE]

is a Risk Measure on $\mathcal{P}(\mathbb{R})$. It is also easy to check that $\Phi_{f}$ is $\sigma(\mathcal{P}(\mathbb{R}),C_{b}(\mathbb{R}))-$lsc. In [13] it is shown that $\Phi_{f}$ can not be convex on $\mathcal{P}(\mathbb{R})$. By selecting the function $f(x)=e^{-x}$ we obtain $\Phi_{f}(P)=\ln\left(\int\exp\left(-x)dF_{P}(x)\right)\right)$, which is in addition (TrI). Its associated risk measure $\rho_{\mathbb{P}}:L^{0}(\Omega,\mathcal{F},\mathbb{P})\rightarrow\mathbb{R}\cup\{+\infty\}$ defined on random variables, $\rho_{\mathbb{P}}(X)=\Phi_{f}(P_{X})=\ln E_{\mathbb{P}}[e^{-X}],$ is the Entropic Convex Risk Measure.

On the $\Lambda V@R$.

All the details of the present section can be found in [13]. We consider a family of risk measures called $\Lambda V@R$ which depend on a Probability/Loss function $\Lambda$. This family provides one example of a $V\&R$ measure that exhibits the peculiar cash invariance property (31).

Fix the right continuous increasing function $\Lambda:\mathbb{R}\rightarrow[0,1]$ and define the family $\left\{F_{m}\right\}_{m\in\mathbb{R}}$ of functions $F_{m}:\mathbb{R}\rightarrow[0,1]$ by

[TABLE]

It is easy to show that if $\sup_{x\in\mathbb{R}}\Lambda(x)<1$ then the associated map $\Phi:\mathcal{P}(\mathbb{R})\rightarrow\mathbb{R}\cup\{+\infty\}$ defined by

[TABLE]

with

[TABLE]

is (Mon), (Qco) and $\sigma(\mathcal{P}(\mathbb{R}),C_{b})-$l.s.c.. This map was named $\Lambda V@R$ since

[TABLE]

If $\Lambda(x)=\lambda$ for every $x\in\mathbb{R}$ then

[TABLE]

coincides with the classical Value at Risk $V@R_{\lambda}$, and in particular if $\lambda=0$ we recover the worst case risk measure

[TABLE]

Both these risk measures are (TrI), whereas for a general $\Lambda$ we get the following property

[TABLE]

where $\Lambda^{\alpha}(x)=\Lambda(x+\alpha)$. Clearly by the definition $\Lambda V@R(\delta_{0})=0$ for every $\Lambda$ so that

[TABLE]

We also mention that the $\Lambda V@R$ is elicitable (depending on the selection of $\Lambda$) and is statistically consistent. We refer to [5] for details on this topic.

Regarding the dual representation of the $\Lambda V@R,$ the functions $V$ in (29) and $V^{-1}$ in (30) can be easily computed (see [13]):

[TABLE]

where $H_{f}^{l}$ is the left inverse of the function: $a\rightarrow\int_{-\infty}^{a}(1-\Lambda)df$.

As two particular cases, from (33) and (34), we get for the $V@R_{\lambda}$ (where $\Lambda(x)=\lambda)$ : $V^{-1}\left(v,f\right)=-f^{l}\left(\frac{v-\lambda f(-\infty)}{1-\lambda}\right)$; for the Worst Case risk measure $\rho_{w}$ (where $\Lambda(x)=0)$ we obtain $V^{-1}\left(v,f\right)=-f^{l}(v)$, where $f^{l}$ is the left inverse of $f$.