4. Conditional risk measures and their robust representation

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1 4. Conditional risk measures and their robust representation We consider a discrete-time information structure given by a filtration (F t ) t=0,...,t on our probability space (Ω, F, P ). The time horizon T can be finite or infinite, and we assume F 0 = {, Ω} and F T = F. As the set X of all financial positions we take the space L = L (Ω, F, P ). The subspace L t = L (Ω, F = t, P ) consists of those positions whose outcome only depends on the history up to time t. All inequalities and equalities applied to random variables are meant to hold P -a.s.. Definition. A map ρ t : L L t will be called a monetary conditional risk measure if it satisfies the following properties for all X, Y L : i) Conditional cash invariance: ii) Monotonicity: ρ t (X + X t ) = ρ t (X) X t for any X t L t, X Y ρ t (X) ρ t (Y ), iv) Normalization: ρ t (0) = 0. A conditional risk measure will be called convex if it satisfies iii) Conditional convexity: λ L t, 0 λ 1 ρ t (λx + (1 λ)y ) λρ t (X) + (1 λ)ρ t (Y ). A convex conditional risk measure will be called coherent if it satisfies in addition v) Conditional positive homogeneity: λ L t, λ 0 ρ t (λx) = λρ t (X). For t = 0 the space L t reduces to the real line, and in this case we recover our previous definition of a (unconditional) convex risk measure. To any conditional convex risk measure ρ t we associate its acceptance set (1) A t := {X L ρ t (X) 0}. One easily checks that A t has the following properties: i) conditional convexity, 1

2 ii) X A t, Y X Y A t, iii) ess sup{x L X At } = 0, and 0 A t. Note that ρ t is uniquely determined by its acceptance set since (2) ρ t (X) = ess inf{y L t X + Y At }. A conditional convex risk measure can thus be viewed as the conditional capital requirement needed at time t to make a financial position acceptable at that time. Conversely, one can use acceptance sets to define conditional convex risk measures: If a given acceptance set A t L satisfies the above conditions then the functional ρ t : L L t defined via (2) is a conditional convex risk measure. By M 1 (P ) we denote the set of all probability measures on (Ω, F) which are absolutely continuous with respect to P. As we have shown in section 2, an unconditional convex risk measure which has the Fatou property admits a robust representation of the form (3) ρ(x) = sup (E Q X] α(q)) Q M 1 (P ) with some penalty function α : M 1 (P ) R {+ }. In fact we can take the minimal penalty function (4) α min (Q) = sup E Q X]. X L, ρ ( X) 0 In this section we are going to prove a robust representation for conditional convex risk measures which is analogous to (3). Here the penalty function will depend on the history up to time t, and so it will be given by a map α t from M 1 (P ) to the set of F t -measurable random variables with values in R {+ }. In analogy to (4), we are going to take the minimal penalty function, defined as the worst conditional loss over all positions which are acceptable at time t. To this end we choose versions of the conditional expectations with respect to Q M 1 (P ) which are well defined P - a.s.: (5) E Q X F t ] := Z 1 t E P ( X) Z T F t ] I {Zt >0}, where Z t denotes a Radon-Nikodym density of Q with respect to P on the σ-field F t. In this way, the essential supremum of conditional expected losses, taken over all X A t, is well defined in terms of P. Theorem. For a convex conditional risk measure ρ t the following properties are equivalent: 1. ρ t has the representation (6) ρ t (X) = ess sup Q M1 (P )(E Q X F t ] αt min (Q)), 2

3 where the penalty function α min t is given by (7) α min t (Q) = ess sup X At E Q X F t ]. In fact the representation (6) also holds if we replace M 1 (P ) in (6) by the smaller set P t := {Q M 1 (P ) Q = P on Ft, E P α t (Q)] < }. 2. ρ t has the following Fatou-property: ρ t (X) lim inf n ρ t(x n ) P a.s. for any bounded sequence (X n ) L which converges P a.s. to X L, 3. ρ t is continuous from above, i.e. for any sequence (X n ) L and X L. X n X = ρ t (X n ) ρ t (X) P a.s. Proof. 1) 2): Lebesgue s dominated convergence theorem for conditional expectations, applied to P and to the random variables ( X n )Z T appearing in the version (5) of E Q X n F t ], yields for each Q M 1 (P ). Thus E Q X n F t ] E Q X F t ] P a.s. E Q X F t ] α min t (Q) lim inf n (ess sup Q M1 (P )(E Q X n F t ] αt min (Q)) = lim inf ρ t (X n ) P a.s., and so the representation (6) implies ρ t (X) lim inf ρ t (X n ). 2) 3): The Fatou property yields lim inf ρ t (X n ) ρ t (X). On the other hand we have lim sup ρ t (X n ) ρ t (X) by monotonicity, and this implies the convergence in 3). 3) 1) Since X + ρ t (X) A t for any X L by conditional cash-invariance, the definition of αt min implies hence αt min (Q) E Q X F t ] ρ t (X), ρ t (X) E Q X F t ] αt min (Q) for any Q M 1 (P ). This yields the inequality (8) ρ t (X) ess sup Q M1 (P )(E Q X F t ] αt min (Q)). 3

4 In order to prove the equality in (6), both for M 1 (P ) and for the smaller set P t, it is therefore enough to show that (9) E P ρ t (X)] E P ess sup Q Pt (E Q X F t ] αt min (Q)]). To this end, consider the map ρ : L R defined by ρ(x) := E P ρ t (X)]. It is easy to check that ρ is a convex risk measure. Property 3) implies that ρ is continuous from above; equivalently, it has the Fatou property. Thus ρ has the robust representation ρ(x) = where the penalty function α(q) is given by α(q) = sup (E Q X] α(q)), Q M 1 (P ) sup E Q X]. X L, ρ(x) 0 Next we prove that Q = P on F t if α(q) <. Indeed, take A F t and λ > 0. Then λp A] = E P ρ t (λi A )] = ρ(λi A ) E Q λi A ] α(q), hence P A] QA] + 1 λ α(q) for any λ > 0. Thus α(q) < implies P A] QA] for any A F t, and hence P = Q on F t. Moreover, (10) E P αt min (Q)] α(q) holds for every Q M 1 (P ). Indeed, E P αt min (Q)] = sup E P Y ] by equation (12) below, and so the definition of the penalty function α(q) implies (10) since ρ(x) 0 for all X A t. Thus we have shown that α(q) < implies Q P t. We can now conclude the proof of inequality (8) as follows, using inequality (10) in the third step: E P ρ t (X)] = ρ(x) = sup Q M 1 (P ),α(q)< (E Q X] α(q)) sup Q P t (E Q X] α(q)) sup E P E Q X F t ] αt min (Q)] Q P t E P ess supq Pt (E Q X F t ] α min t (Q) ] ). 4

5 This shows that equality (6) is valid if the essential supremum is only taken over the set P t. In view of inequality (8), this coincides with the essential supremum over all Q M 1 (P ), The following lemma was used in the proof of the Theorem. Lemma. For Q M 1 (P ) and 0 s t, (11) E Q αt min (Q) F s ] = ess sup E Q Y F s ], and in particular (12) E Q αt min (Q)] = sup E Q Y ]. Proof. First we show that the family { EQ X F t ] X At } is directed upward for any Q P t. Indeed, for X, Y A t we can take Z := XI A + Y I A c, where A := {E Q X F t ] E Q Y F t ]} F t. Conditional convexity of ρ t implies Z A t, and clearly we have E Q Z F t ] = max (E Q X F t ], E Q Y F t ]). Thus the family is directed upward. It follows that its essential supremum can be computed as the supremum of an increasing sequence within this family, i.e., there exists a sequence (X Q n ) in A t such that By monotone convergence we get α min t (Q) = lim n E Q X Q n F t ] P a.s. E Q α min t (Q) F s ] = lim n E Q EQ X Q n F t ] Fs ] ess sup E Q Y F s ]. The converse inequality follows directly from the definition of α min t (Q). This shows (11), and (12) can be viewed as the special case s = 0. Example: The entropic conditional risk measure. Suppose that preferences are characterized by an exponential utility function u(x) = 1 exp( βx) with β > 0. At time t the conditional expected utility of a financial position X L is then given by the F t - measurable random variable U t (X) = E1 e βx F t ]. 5

6 The set A t := { X L Ut (X) U t (0) } = { X L Ee βx F t ] 1 } satisfies the necessary conditions for an acceptance set. The induced convex conditional risk measure ρ t is the conditional entropic measure given by ρ t (X) = ess inf { Y L t = ess inf { Y L t } X + Y A t Ee βx F t ] e βy }, i.e., ρ t (X) = 1 γ log Ee γx F t ]. It is easy to see that ρ t has the Fatou property and thus admits a robust representation of the form (6). In analogy to the unconditional case, the minimal penalty function takes the form α t (Q) = 1 β Ĥt(Q P ), where Ĥt(Q P ) denotes the conditional relative entropy of Q with respect to P, given F t : Ĥ t (Q P ) := E Q log Z ] T Ft Z t = E P ZT Z t log Z T Z t F t ] I {Zt >0}; note that Z t > 0 Q-a.s., and that Z t = 1 for Q P t. This is the reason why ρ t is called the conditional entropic risk measure. In the coherent case we obtain the following representation result: Corollary. Let ρ t be a coherent conditional risk measure which has the Fatou property. Then ρ t has the representation (13) ρ t (X) = ess sup Q Q t E Q X F t ], where Q t := { Q M 1 (P ) Q = P on Ft, α min t (Q) = 0 P a.s. }. Proof. In the coherent case the penalty function αt min (Q) can only take the values 0 or. Indeed, for A := {αt min (Q) > 0}, X A t and any λ > 0 we have λi A X A t due to conditional positive homogeneity of ρ t, hence 6

7 α min t (Q) = ess sup X At E Q X F t ] ess sup X At E Q λi A X F t ] = λi A αt min (Q), and the lower bound converges to on A as λ. Thus αt min (Q) = on A. But for Q P t we have E P αt min (Q)] <, and this implies P A] = 0. Thus P t coincides with Q t, and the representation (6), where we are free to take P t instead of M 1 (P ), reduces to (13). Example: Conditional Value at Risk. For any α (0, 1), the acceptance set A t = {X L P X < 0 F t ] α} defines a monetary conditional risk measure, called conditional value at risk at level α: α (X) Ft = ess inf{m t L t P X + m t < 0 F t ] α} = ess sup{c t L t P X < c t F t ] α} = q α (X) Ft, where q α (X) Ft is a conditional quantile of X, i.e., a quantile of the conditional distribution of X under P, given the σ-field F t. Conditional Value at Risk is conditionally positively homogenous, but it is not conditionally convex. However, we do obtain a coherent conditional risk measure by taking the average up to some level λ (0, 1]. The resulting risk measure α (X) Ft := 1 λ λ 0 α (X)dα is called conditional Value at Risk at level λ. In analogy to the unconditional case it can be shown that the following representation holds: α (X) Ft = ess sup{e Q X F t ] Q = P on F t, dq dp 1 λ }. Remark. The representations (6) and (13) can be formulated in terms of equivalent probability measures Q P if ρ t is sensitive with respect to P in the sense that P ρ t ( ɛi A ) > 0] > 0 for all ɛ > 0 and for any A F such that P A] > 0. More precisely, let M e 1(P ) denote the set of all probability measures on (Ω, F) which are equivalent to P, and let ρ t be sensitive with respect to P. Then we can replace M 1 (P ) by M e 1(P ) in (6), in the definition of P t in the Theorem, and in the definition of Q t in the Corollary. 7