Pricing under dynamic risk measures

Abstract In this paper, we study the discrete-time super-replication problem of contingent claims with respect to an acceptable terminal discounted cash flow. Based on the concept of Immediate Profit, i.e., a negative price which super-replicates the zero contingent claim, we establish a weak version of the fundamental theorem of asset pricing. Moreover, time consistency is discussed and we obtain a representation formula for the minimal super-hedging prices of bounded contingent claims.


Introduction
In mathematical nance, it is very classical to solve the problem of super-replicating a contingent claim under a no-arbitrage condition (NA). In particular, in frictionless markets, the so-called fundamental theorem of asset pricing (FTAP) characterising NA condition has been studied by numerous authors, see [1][2][3] in discrete time and [4,5] in continuous time. It states that NA condition holds if and only if there exist equivalent martingale measures (EMM). In complete markets, such a martingale measure Q ∼ P is unique and the (replicating) price of a derivative is uniquely computed as the expectation of the discounted payo under Q. However, in incomplete markets, there exists an in nite number of EMM and the (minimal) super-hedging price is di cult to compute in practice. Indeed, this is a supremum of the expected discounted payo over all probability measures (see [6] and [7, Theorem 2.1.11]).
A new pricing technique called No Good Deal (NGD) pricing has been proposed in [8,9]. A good deal is a trade with an unusually high pro t/loss or Sharpe ratio. Cherny [10] introduced the concept of good deal with respect to a risk measure as a trade with negative risk. Contrarily to the classical approach where super-replication holds almost surely, Cherny assumes that the agent seller accepts some non null risk for its portfolio not to super-hedge the payo . In the setting of coherent risk measures, Cherny [10] provides a version of the FTAP under absence of NGD.
Risk measures are more studied and known on the space L ∞ , i.e. the space of essentially bounded random variables. And the space L p , p ∈ [ , ∞) is a natural extension, see [11,12]. Actually, working on the restricted subspaces of L , such as L ∞ and L p , is mainly motivated by the robust representation of risk measures. However, the space L , equipped with the topology of convergence in probability, is more adapted for some classical nancial and actuarial problems such as hedging, pricing, portfolio choice, equilibrium and optimal reinsurance with respect to risk measures.
Delbaen in [13,14] extends the coherent risk measure to the space L by enlarging its range to R ∪ {+∞} as there is no real-valued coherent risk measure on L when the probability space (Ω, F, P) is atomless [14,Theorem 5.1]. A robust representation with respect to a set of probability measures is then given [14,Theorem 5.4]. As the space L contains non integrable random variables, Delbaen in [14] truncates the random variables from above, i.e. only considers possible future wealth up to some threshold. It is then possible to compute the risk measures as in L ∞ and then make n tend to in nity [14,De nition 5.3]). Therefore, the robust representation on L ∞ appears to be the key point to extend coherent risk measure to L , see ([10, De nition 2.2] and [15]), which allow to formulate a FTAP with respect to NGD and solve super-replication problems. In this approach, coherent risk measures remain characterised through families of probability measures which are not necessarily easy to handle in practice, see e.g. the explicit representation of this family for the Weighted VaR risk measure [16,17].
In this paper, we de ne risk measures on the space L with values in R = [−∞, +∞]. They are naturally de ned through the concept of acceptable set, i.e. a risk measure is seen as the minimal capital requirement added to the position for it to be acceptable. Under some natural assumptions satis ed by the acceptable set, we show that a risk measure is lower semi-continuous. This allows to compute ω-wise risk measure using similar new results on conditional essential supremum [18]. Inspired by [10], the aim of this paper is to reconsider the super-replication problem in discrete-time with respect to a risk measure without using a dual representation. The minimal super-hedging prices of a contingent claim are recursively de ned in the spirit of [18].
Based on the concept of immediate pro t, introduced in [18], we establish a weak version of FTAP to equivalently characterise the condition of absence of immediate pro t (AIP). Moreover, we show that for bounded non-negative contingent claims, the minimal super-hedging price may be computed through a conditional (dynamic) coherent risk measure derived from the underlying risk measure. At last, we discuss the time consistency, i.e. coherent evaluations of risk in time, since it is a very important concept developed in the literatures for dynamic risk measures, see [19,20].
The paper is organized as follows. Section 2 gives the de nition of risk measures and some important properties for these risk measures are showed. Section 3 introduces the model of super-replication with respect to acceptable sets. We simplify the problem of minimal super-hedging price involving the essential in mum into a classic minimization problem just with in mum. In Section 4, a weak version of fundamental theorem of asset pricing is proved. Section 5 gives a price representation for the bounded non-negative contingent claims.

Notations:
L (R, F) is the metric space of all R-valued random variables which are F-measurable; L p (R, F, P), p ∈ [ , ∞) (resp. p = ∞), is the normed space of all R-valued random variables which are Fmeasurable and admit a moment of order p under the probability P (resp. bounded). Without any confusions, we omit the notation P and just denote L p (R, F); In the following, we consider a complete discrete-time stochastic basis (Ω, F := (F t ) t= ,··· ,T , P) where F t represents the available information of the market at time t; E P and E Q are the expectations of any integrable random variable with respect to the probability measure P and Q. In general, we denote E P as E without of any confusions. All equalities and inequalities of random variables are understood up to a negligible set.
The dynamic risk measure X → (ρ t (X)) t= ,··· ,T we consider is de ned on L . It is constructed from its acceptance sets de ned as follows: De nition 2.1. A dynamic acceptable set is a family (A t ) t= ,··· ,T of subsets of L (R, F T ) satisfying the following conditions: (4) k t X ∈ A t for any X ∈ A t and k t ∈ L (R+, F t ).
Any element of A t is said acceptable at time t. For any X ∈ L (R, F T ), we denote by De nition 2.2. Let (A t ) t= ,··· ,T be a dynamic acceptance set. The risk measure associated to (A t ) t= ,··· ,T is, at time t, the mapping ρ t : up to a negligible set.
Observe that ρ t (X) is the the minimal capital requirement we add to the position X for it to be acceptable at time t. The e ective domain of ρ t is denoted as In this paper, we just consider the positions whose risk measures are not in nite at any time t. In other words, we assume that ρ t (X) < +∞ for any X ∈ L (R, F T ).

Lemma 2.3.
For any X ∈ L (R, F T ), there exists a sequence Yn ∈ A X t such that ρ t (X) = limn→∞ ↓ Yn a.s.
Proof. We rst observe that the set To see it, we use conditions 1) and 4) of De nition 2.1. We then deduce that Therefore, there exists a sequence Yn ∈ A X t such that ρ t (X) = limn→∞ ↓ Yn a.s., see [7, Section 5.3.1.] The following proposition is straightforward due to the de nition. The proofs are showed in the Appendix C.

Minimal super-hedging prices
In the discrete-time model, let (S t ) ≤t≤T be the discounted price process of asset where S t ∈ L (R+, F t ). And (ρ t ) ≤t≤T is dynamic risk measure de ned in De nition 2.5. A contingent claim at time T is denoted by a real-valued F T -measurable random variable h T . The question is to nd a self-nancing strategy process (θ t ) ≤t≤T to super-replicate the contingent claim h T . Here we use the concept of super-replication in the sense of acceptable set, that is the resulting risk is negative, instead of super-hedging almost surely as the most literatures did. In fact, super-replication almost surely usually can not be realized in a real market.
First let us start with the one step model, that is to super-replicate the contingent claim h T at time T − . And the acceptable set A T− is assumed to be closed in this section. An notion of super-hedging with respect to the acceptable set is given as follows. In this paper, we just consider the contingent claims which can be super-hedged in the sense of the following de nition.

De nition 3.1. Contingent claim h T is said to be super-hedged at time T
The set P T− (h T ) consists of all super-hedging prices at time T − , that is Since we assume that the contingent claims of consideration can be super-hedged, that is to say, we may suppose that P T− (h T ) ≠ ∅. According to (2.2) and the cash invariance property of ρ T− , Then the set P T− (h T ) can be equivalently written as then the set of super-hedging prices can be expressed as Actually, we may construct a jointly measurable version of the random function g(ω, x) such that g( And we can prove that g(ω, x) is convex and lower semi-continuous in x for almost all ω under the assumption that the acceptable set A T− is closed.
Proof. Trivially G T− is closed and convex since A T− is supposed to be closed and a convex cone. And [21,Proposition 2.7]. Then, by a contradiction argument and using a measurable selection argument, we may show that G T− is convex as G T− .

s. convex and lower semi-continuous.
Proof. De ne the following random function is a.s. lower-semi continuous. Consider a sequence x n ∈ R which converges to x ∈ R. Let us denote β n := g(x n ). We have (x n , β n ) ∈ G T− from the above discussion. In the case where infn β n = −∞, g(ω, x) − > βn for n large enough (up to a subsequence) hence ( . This contradicts the de nition of g. Moreover, the inequality g(x) ≤ lim infn β n is trivial when the right hand side is +∞. Otherwise, β ∞ := lim infn β n < ∞ and (x , β ∞ ) ∈ G T− as G T− is closed. It follows by de nition of g that g(x ) ≤ lim infn g(x n ), i.e. g is lower-semi continuous.

Corollary 3.4. We have g(X)
The inequality being trivial on the complementary set, we nally conclude that the equality holds a.s. The minimal super-hedging price is given in the sense of (conditional) essential in mum. A generalized concept and existence of conditional essential supremum (resp. conditional essential in mum) of a family of vector-valued random variables with respect to a random partial order are discussed in [22,23]. Here we use the classical case with a natural partial order for a family of real-valued random variables (see Appendix A).  Proof. For any θ T− , θ T− ∈ L (R, F T− ), de ne Due to the convexity of g, it holds That implies that there exists θ T− ∈ L (R, F T− ) such that g(θ T− ) ∈ P T− (h T ) where g(θ T− ) is the lower bound of any pair g(θ T− ) and g(θ T− ) from the set P T− (h T ).
for some sequence θ n T− ∈ L (R, F T− ). Moreover, it holds Proof. The rst equality (3.6) is a direct consequence of Lemma 3.6. In order to obtain (3.7), we rst prove that inf Observe that Dom g is an upper set, i.e. an interval. Since P T− (h T ) ≠ ∅, there exists a strategy a T− ∈ Dom g hence Dom g contains the interval [a T− , ∞). Thus we can say that Dom g T− admits a non empty interior on which g T− is convex hence continuous. It follows that We deduce that inf holds. Actually, it is not very clear how to solve the optimization problem with the essential in mum. Now it has been transferred into a classical one just with in mum according to Theorem 3.7 so that we can know how to deal with it. Before characterizing the optimal solutions and studying the existence of optimal strategies, we rst recall the concept of immediate pro t (IP) as introduced in [18] and give a weak version of fundamental theorem of asset pricing to build a basic principle for the hedging and pricing.

Weak fundamental theorem of asset pricing
Let us extend the acceptable set A t to A t,t+s ⊆ L (R, F t+s ) by the same axiomatic conditions in De nition 2.1.
In what follows, all acceptable sets are supposed to be closed. The risk measure ρ t is de ned on L (R, F t+s ) for some s ≥ instead of L (R, F T ), the risk measure function is ρ t (X) = ess inf{Y ∈ L (R, F t )|X + Y ∈ A t,t+s } and the corresponding acceptable set is First we consider the general one-step model from t to t + , super-hedging the contingent claim h t+ at time t means that there exists some P t ∈ L (R, F t ) and strategy θ t ∈ L (R, F t ) such that P t + θ t ∆S t+ − h t+ is acceptable with respect to the acceptable set A t,t+ . Similarly we can express the set of all super-hedging prices as The minimal super-hedging price at time t for this one-step model is For the contingent claim h T we de ne recursively where P * t+ can be regarded as the contingent claim h t+ . Let us recall the concept of immediate pro t as introduced in [18], which means that it is possible to super-replicate contingent claim zero with a negative price.

De nition 4.1. Absence of Immediate Pro t (AIP) holds if
for any ≤ t ≤ T.
It is obvious that (AIP) property automatically holds at time T since P T ( ) = L (R+, F T ). Next we characterize (AIP) for general model with t ≤ T − .

Theorem 4.2. (Weak Fundamental theorem of asset pricing) (AIP) property holds if and only if
Proof. For the backward recursion starting from P * T = h T = , the set of super-hedging prices for contingent claim zero at time T − is and the minimal super-hedging price is P * T− = ess inf where g(x) = xS T− + ρ T− (xS T ) for the case h T = . Now it is easy to see that

Example 4.3. For the classical one-step super-hedging problem, i.e., a contingent claim h T can be superreplicated at time T − means that there exist some P T− ∈ L (R, F T− ) and strategy θ T− ∈ L (R, F T− ) such that P T− + θ T− ∆S T − h T ≥ almost surely. In this case the acceptable set A T− is as follows:
This also implies that ρ T− (X) = − ess inf F T− X. Then from Theorem 4.2 AIP property can be expressed as the same equivalent condition: Thus the second equivalent condition of (AIP) in [18,Theorem 3.4] is one of the special cases in our paper when taking the worst-case risk measure ρ T− (X) = − ess inf F T− X.
Remark 4.4. The condition (4.11) implies (4.12) trivially. Actually, the risk at time T − of position X ∈ L (R, F T ) given by ρ T− (X) = − ess inf F T− X is the worst-case (maximum) one. Indeed, from (2.1), we can easily see that such that we can get by taking X = S T that

Price representation
In this section, the study is restricted to bounded non-negative contingent claims. The main purpose is to give the speci c expression of minimal super-hedging prices in the sense of risk management. Notice that the risk measure ρ t is based on the space L and its dual representation is not used in the previous content. Next we give a new risk measure de ned on the space L ∞ under which the minimal superhedging price of a bounded contingent claim is just the risk of its opposite payo .
Let us recall the general axiomatic de nition of conditional coherent risk measure ρ t : L ∞ (R, F T ) → L (R, F t ) (see De nition 1,2 and 3 in [24]): F t ) is said to be a conditional coherent risk measure if it satis es the following properties: -Normalization: ρ t ( ) = ; -Conditional translation invariance: for all X ∈ L ∞ (R, F T ) and m t ∈ L ∞ (R, F t ),
Let us de ne recursively (ρ t ) ≤t≤T for some bounded position Y ∈ L ∞ (R, F T ) based on the given dynamic risk measure (ρ t ) ≤t≤T asρ Actually, it can be proved thatρ t are conditional coherent risk measures de ned in De nition 5.1 for all ≤ t ≤ T and (ρ t ) t is time-consistent, that is for all X, Y ∈ L ∞ (R, F T ) and ≤ t ≤ T,ρ t+ (X) =ρ t+ (Y) implies ρ t (X) =ρ t (Y) (see Section 5 in [24]). Then the pricing problem is naturally equivalent to measure the risk of contingent claim under the conditional coherent risk measureρ t , that is which is the time-consistent price process.
Lemma 5.2. Assume the condition (AIP) holds, thenρ t are conditional coherent risk measures for all ≤ t ≤ T on L ∞ . Moreover, (ρ t ) t is time-consistent whenever the underlying dynamic risk measure (ρ t ) t is or not.
Proof. Indeed,ρ T (·) trivially satis es the conditions in the De nition 5.1 such thatρ T (·) is a conditional coherent risk measure. And all the other properties except normalization forρ t with ≤ t ≤ T − are easy to be inherited from ρ t by the induction. Here we just need to prove the normalization. Assumeρ t+ ( ) = , thenρ = as (AIP) implies that ρ t (∆S t+ ) and ρ t (−∆S t+ ) are both non-negative. The time-consistency can be easily deduced from the de nition of (ρ t ) t . Next we can give the expression of P * t in the sense of robust representation for conditional coherent risk measureρ t . First let us give the following sets of probability measures for all ≤ t ≤ T as: Proof. From Lemma 5.2ρ t is a conditional coherent risk measure. And the lower semi-continuity ofρ t is inherited from the underlying risk measure ρ t . Thus the following robust representation (see [24]) can be obtainedρ

B. Measurable subsequences
First, let us recall the existence of convergent subsequences of the random sequence from L (R d ), see [7,Lemma 2.1.2]. The technical constructions of these convergent subsequences can be found in the proof of this lemma.

Lemma 2.1.2([7])
Let η n ∈ L (R d ) be such that η := lim inf |η n | < ∞. Then there areη k ∈ L (R d ) such that for all ω the sequence ofη k (ω) is a convergent subsequence of the sequence of η n (ω). It is worth noting that the subsequenceη k is random due to the fact that The more detailed results about the random convergent subsequence can be found in [25,Section 6.3]. Let (K, d) be a compact metric space and N be the set of all natural numbers.

De nition 6.3.1([25])
An N-valued, F-measurable function is called a random time. A strictly increasing sequence (τ k ) ∞ k= of random times is called a measurably parameterised subsequence or simply a measurable subsequence.
Step 2: By the normalization procedureX k := X |L k | andL k := L k |L k | , we get thatX k +L k ∈ A t on the set Λ t . Applying [25,Proposition 6.3.3] to the sequence (L k ) ∞ k= , there is a F t -measurably parameterised subsequence (σ i ) ∞ i= such that the subsequence (Lσ i ) ∞ i= converges to someL. As |L k | = for any k ≥ , we can see that |L| = . Actually,L = − a.s. asLσ i < for large enough i.
Step 4: On the other hand,X k = X |L k | trivially converges to zero as L k diverges to −∞. Finally, we deduce that lim k (X k +L k ) = − ∈ A t on the set Λ t if A t is closed. This is contradicted with the third condition: A t ∩ L (R, F t ) = L (R+, F t ) in the De nition 2.1. Thus, the assumption ρ t (X) = −∞ with a positive probability is impossible, that is ρ t (X) > −∞ with probability one.
Since we assume that ρ t (X) < +∞ for any X ∈ L (R, F T ), then it holds ρ t (X) ∈ L (R, F t ). By Lemma 2.3, we know that ρ t (X) = limn ↓ Yn a.s. where Yn ∈ L (R, F t ) satisfying X + Yn ∈ A t . As the set A t is closed, F t -decomposable and contains 0, we deduce that X + ρ t (X) ∈ A t for any X ∈ L (R, F T ). Now let us prove the lower semi-continuity of ρ t . Consider a sequence Xn ∈ L (R, F T ) which converges to X . Denote αn := ρ t (Xn), then Xn + αn ∈ A t . Our goal is to prove the inequality ρ t (X ) ≤ lim inf αn a.s. Let us divide it into the following three cases: a) As for the case where lim inf αn = +∞, the inequality ρ t (X ) ≤ lim inf αn holds trivially. Thus we may assume w.l.o.g. that lim inf αn < +∞. b) Let us consider the case where lim inf αn = −∞. Suppose that the F t -measurable set Γ t := {ω ∈ Ω : lim inf αn = −∞} has a positive probability. Obviously, −∞ is an accumulation point of (αn) ∞ n= on the set Γ t . For convenience, denote α∞ := lim inf αn. Again, [25, Proposition 6.3.4 (i)] implies that there is a F tmeasurably parameterised subsequence (µ k ) ∞ k= such that the subsequence (β k ) ∞ k= := (αµ k ) ∞ k= diverges to −∞ on the set Γ t of positive probability. Let (Z k ) ∞ k= := (Xµ k ) ∞ k= be the corresponding subsequence of the sequence Xn. Then we can see that Z k +β k ∈ A t on the set Γ t as (Z k (ω)+β k (ω)) Γt = (X µ k (ω) (ω)+α µ k (ω) (ω)) Γt = ∞ p≥k (Xp(ω) + αp(ω)) µ k =p Γt . Then, using the normalization procedureZ k := Z k |β k | andβ k := β k |β k | , we get that Z k +β k ∈ A t on the set Γ t . By passing once again to a measurably parameterised subsequence, we may assume thatβ k converges to − according to the similar statements in the above Step 2 and Step 3. Note that Z k = Xµ k converges to X and β k diverges to −∞ such thatZ k converges to zero, we nally get that lim k (Z k +β k ) = − ∈ A t on the set Γ t if the set A t is closed. This contradicts with the third condition in the De nition 2.1. Thus, α∞ = lim inf αn > −∞ with probability one. c)Combining the cases a) and b), we can assume w.l.o.g. that α∞ ∈ L (R, F t ) and X + α∞ ∈ A t . It follows that ρ t (X ) ≤ α∞ = lim inf ρ t (Xn) a.s.
At last, if the set A t is closed, the acceptable set A t can be represented as A t = {X ∈ L (R, F T )|ρ t (X) ≤ }. Indeed, it is clear that ρ t (X) ≤ for all X ∈ A t . Reciprocally, if ρ t (X) ≤ , we get that X = −ρ t (X) + a t where a t ∈ A t . Finally we can deduce that X ∈ A t since ≤ −ρ t (X) ∈ A t and A t + A t ⊆ A t .