Dynamic Generalized AVaR

introduce a generalization of AVaR, calledutility based shortfall risk measure.

[Cheridito & Li, 09] use a convenient representation for AV aRwhich has an immediate generalization to a convex risk measure, calledgeneralized AV aR (gAV aR) here. This convex risk measure gives then raise to a variational preference by multiplying the robust representation with−1.

As shown in [Cheridito & Stadje, 09] as well as [Artzner et al., 07] the natural dynamic extension of AV aR, and hence of gAV aR, just in terms of conditional expectations is not time-consistent, cp. [Artzner et al., 99]’s Definition 5.5. We thus define a time-consistent dynamic version ofgAV aR, called dyn gAV aR: In a first approach as in [Cheridito & Stadje, 09] recur-sively in terms of the definition of time-consistency. Thereafter, recurrecur-sively in terms of the penalty function as in [Maccheroni et al., 06b] by compos-ing one period ahead penalties directly achievcompos-ing the robust representation.

By Corollary 4.8 in [Cheridito et al, 06], both approaches induce the same consistent dynamic convex risk measure or, equivalently, the same time-consistent dynamic variational preference.

Consider again the underlying filtered reference space (Ω,(F_t)t≤T,P0).

Set Lⁱ_t := Lⁱ(Ω,Ft,P⁰|Ft) for i ∈ {0} ∪ N∪ {∞}, t ≤ T. To start with, we first consider the static convex risk measure gAV aR for some end pe-riod payoff X_T ∈ L^∞_T as in [Cheridito & Li, 09], T < ∞. Later this static risk measure will serve as dyn gAV aR₀ in the definition of the dynamic con-vex risk measure (dyn gAV aR_t)t≤T. We obtain robust representations for gAV aR in terms of a penalty α^min, serving as α₀^min in the penalty function (α^min_t )t≤T for (dyn gAV aR_t)t≤T.

Definition 3.5.18. For (θ, β, p)∈]0,∞[×]1,∞[×[1,∞[, define the risk mea-sure gAVaR forX_T ∈L^∞_T , called generalized Average Value at Risk (gAVaR), by virtue of

gAV aR^β,p_θ (X_T) := min

s∈R

1 θ

(s−X_T)⁺

β p −s

, where k · k_p := (E^P0|FT[| · |^p])^p1 denotes the usual p-norm.

For ease of notation, we do not explicitly state the parameters but just write gAV aRinstead of gAV aR^β,p_θ when these are obvious. We have:

Proposition 3.5.19. (a) For (θ, β, p) ∈ ]0,1[×{1} ×[1,∞[, gAV aR^β,p_θ is a coherent risk measure for X_T ∈L^∞_T with robust representation in terms of minimal penalty α^min by virtue of

α^min(Q) =

( 0 if k_dP^dQ|FT

0|FT

kq ≤ ¹_θ,

∞ else, for Q ∈ M, where q:= _p−1^p and k_dP^dQ|FT

0|FT

k_q =

E^P0|FT[|_dP^dQ|FT

0|FT

|^q]¹_q . (b) For θ ∈]0,1[, β =p= 1, we have k_d^dQ|FT

P0|FT

k∞ = ess sup|_d^dQ|FT

P0|FT

| and hence the robust representation becomes

gAV aR^1,1_θ (X_T) = sup

Q∈M

E^Q|FT [−X_T]

0≤ dQ|FT

dP0|F_T

≤ 1 θ

= AV aRθ(XT),

which again shows AV aR to be a coherent risk measure.

(c) For (θ, β, p) ∈ ]0,∞[×]1,∞[×[1,∞[, gAV aR^β,p_θ is a convex risk mea-sure for X_T ∈ L^∞_T with minimal penalty α^{gAV aR}(Q) := ck_dP^dQ|FT

0|FT

k^d_q, where q:= _p−1^p , d:= _β−1^β and c=θ^d−1β^1−dd⁻¹. Hence

gAV aR^β,p_θ (X_T) = sup

Q∈M

(

E^Q|FT[−X_T]−c

dQ|F_T

dP0|F_T

d

q

) . Proof. cp. [Cheridito & Li, 09].

[Cheridito & Stadje, 09] recursively achieve a time-consistent dynamic version of AV aR for end period payoff X_T. Mimicking this approach by virtue of the definition of time-consistency for dynamic convex risk measures, i.e. ρ_t = ρ_t(−ρ_t+1) or ,equivalently, π_t = π_t(π_t+1) for dynamic variational preferences,²³ we obtain a time-consistent dynamic version of gAV aR^β,p_θ .

23As we assumeT being finite, time-consistency of dynamic risk measures is by Propo-sition 4.5 in [Cheridito et al, 06] equivalent to “one-step time consistency” as applied in this article.

As in [Cheridito & Stadje, 09], we now define a time-consistent version for the more general risk measure gAV aR in terms of the definition of time-consistency:

Definition 3.5.20. We recursively define the dynamic convex risk measure called dynamic generalized average value at risk, (dyn gAV aRt)t≤T, as fol-lows: Let X_i ∈ F_i, i≤T, then we set for all t

dyn gAV aR_t(X_j) := −X_j, dyn gAV aR_t(X_t+1) := ess inf

s∈L^∞_t

1 θ E

(s−X_t+1)⁺

p F_t^β_p

−s

, dyn gAV aR_t(X_z) := dyn gAV aR_t(−dyn gAV aR_t+1(X_z)) for j ≤t, t+ 1 < z≤T.²⁴

Remark 3.5.21. In terms of Definition 3.5.20, for an adapted payoff process (X_t)t≤T and a stopping timeτ ≤T, the termdyn gAV aR_t(X_τ)is well defined for t≤T.

Remark 3.5.22. From [Cheridito & Stadje, 09], we see that the natural dy-namic generalization

gAV aRt(XT) := ess inf

s∈L^∞_t

1 θ E

(s−XT)⁺

p Ft

^β_p

−s

is not time-consistent. But in these terms our definition becomes dyn gAV aRt(Xz) = gAV aRt(−dyn gAV aR_t+1(Xz)).

Proposition 3.5.23. (dyn gAV aR_t)t≤T is a time-consistent dynamic convex risk measure, i.e. satisfies for t < T

dyn gAV aRt=dyn gAV aRt(−dyn gAV aRt+1).

24The last term is well-defined as dyn gAV aR_t+1(X_z) is Ft+1-measurable. A special case is of course z = T, in which case we are back in the setting of [Cheridito & Stadje, 09]

but for gAVaR instead of AVar.

In our optimal stopping approach time-consistency can be written as: For (X_t)_t≤T, and a stopping time τ ≤T we obtain for t < T

dyn gAV aR_t(X_τ) = dyn gAV aR_t X_τI^{τ≤t}

−dyn gAV aR_t+1(X_τ)I^{τ≥t+1}

= −X_τI^{τ≤t}

+dyn gAV aR_t(−dyn gAV aR_t+1(X_τ)I^{τ≥t+1}).

Proof. Being a dynamic time-consistent convex risk measure is immediate by Definition 3.5.20 in terms of the static convex risk measure gAV aR as the recursion formula is just the definition of time-consistency.

Our special form of time-consistency follows immediately as we have al-ready seen in the theoretical section. Nevertheless, we make it explicit here:

As τ ≤ T, X_τ is F_T-measurable, i.e. at time T we know when we have stopped the process. Writing Xτ = XτI^{τ≤t} +XτI^{τ≥t+1} we obtain with conditional cash invariance

dyn gAV aR_t(−dyn gAV aR_t+1(X_τ))

= dyn gAV aR_t(−dyn gAV aR_t+1(X_τI^{τ≤t}+X_τI^{τ≥t+1}))

= dyn gAV aR_t(−dyn gAV aR_t+1(X_τI^{τ≤t})

| {z }

=−XτI{τ=t}

−dyn gAV aR_t+1(X_τ)I{τ≥t+1})

= dyn gAV aR_t(X_τI^{τ≤t}−dyn gAV aR_t+1(X_τ)I^{τ≥t+1})

= −XτI^{τ≤t}+dyn gAV aRt(−dyn gAV aRt+1(Xτ)I^{τ≥t+1}).

By [F¨ollmer & Penner, 06], Theorem 4.5, (dyn gAV aR_t)t≤T then of course possesses a robust representation in terms of a minimal penalty satisfying the no-gain condition by Proposition 3.2.15.

Definition 3.5.24. We say that the dynamic variational preference(π^aR_t )t≤T

is obtained by dynamic generalized average value at risk(dyn gAV aR_t)t≤T if

it is of the form

π^aR_t :=−dyn gAV aR_t.

Remark 3.5.25. By Proposition 3.5.23,(π_t^aR)_t≤T is time-consistent, i.e. for t < T, z ≤T, we have

π_t^aR(Xz) =π_t^aR(π^aR_t+1(Xz)), more elaborately for a stopping time τ ≤T

π_t^aR(X_τ) = X_τI^{τ≤t}+π^aR_t π^aR_t+1(X_τ)I^{τ≥t+1}

which shows time-consistency in terms of Proposition 3.2.16.

As the assertion in Theorem 3.4.1 can be reformulated not to make use of the robust representation of dynamic variational preferences, we can directly apply the variational Snell envelope approach²⁵ and achieve for t < T

U_t = max

X_t;π_t^aR(U_t+1)

= max{Xt;−dyn gAV aRt(Ut+1)}

= max (

X_t; ess sup

s∈L^∞_t

s−1

θ E

(s−U_t+1)⁺

p F_t^β_p

)

as Ut+1 is Ft+1-measurable. In order to achieve explicit solutions in terms of worst-case distributions as done in the theoretical section, we rather want to have the robust representation of (dyn gAV aR_t)t≤T. Hence, we end this section by establishing an alternative way to introduce a time-consistent dy-namic version of gAV aR in terms of a robust representation, i.e. we ap-propriately define a penalty (α^gAVaR_t )t≤T: Hereto, we will use the minimal penalty α^{gAV aR} of the static gAV aR as defined in Proposition 3.5.19. We apply the recursive procedure from [Maccheroni et al., 06b] in terms of one

25In [Cheridito et al, 06], Section 5.3, optimal stopping problems with general monetary risk measures are considered. In that case, the Snell envelope can only be given in this form as the risk measure does not necessarily possess a robust representation.

period ahead penalties (γ_t)t≤T to achieve a time-consistent dynamic mini-mal penalty (α^gAVaR_t )t≤T. We then show that the dynamic time-consistent variational preferences obtained by virtue of both procedures coincide.

To ease notation, we do not state this example in terms of one-period ahead penalties γt but in terms of s-period ahead penalties α^min_t,t+s, s ≥ 0, as defined in [F¨ollmer & Penner, 06], p. 76. s-period ahead penalties con-stitute a direct generalization of our one-period ahead penalties by virtue of γ_t(Q|F_t+1(·|F_t)) = α^min_t,t+1(Q). We do not rigorously introduce the theory in terms of these more general s-period ahead penalties: All assertions, in particular the no-gain condition, can be analogously stated in terms ofα^min_t,t+1. The respective results are given in [F¨ollmer & Penner, 06], Theorem 4.5.

Making use of α^{gAV aR} in Proposition 3.5.19(c), define thes-period ahead penalty at t by

α^{gAV aR}_t,t+s (Q) := α^{gAV aR}(Q|Ft+s(·|F_t)) =c E^P0

"

dQ dP0

_F

t+s

!q

F_t

#!^d_q

fors≥0,t+s≤T,Q∈ M, and the parameters as in Proposition 3.5.19(c).

Note, that we then have α^{gAV aR}_0,0+T (Q) =α^{gAV aR}(Q) =c

dQ dP0

FT

d

q

. Then, the one period ahead penalty γ_t^{gAV aR} onM|Ft+1 is defined by

γ_t^{gAV aR}(Q|_Ft+1(·|F_t)) := α^{gAV aR}_t,t+1 (Q) =c E^P0

"

dQ dP0

Ft+1

!q

F_t

#!^d_q . Given this one-step ahead penalty, we recursively define a dynamic penalty (α^{gAV aR}_t )t≤T as in Theorem 2 in [Maccheroni et al., 06b]:

Definition 3.5.26. Let F_t∈ F_t.We define the dynamic penalty (α^{gAV aR}_t )_t≤T by virtue of

α^{gAV aR}_T (Q)(ω) :=

( 0 if Q=I^{ω},

∞ else for ω∈Ω, α^{gAV aR}_t (Q)(Ft) :=

Z

α^{gAV aR}_t+1 (Q(·|F_t+1))dQ(·|F_t) +γ_t^{gAV aR}(Q(·|F_t)|_Ft+1)

if Q(F_t)>0, α^{gAV aR}_t (Q)(F_t) := ∞ if Q(F_t) = 0, for t < T.²⁶

Applying (α^{gAV aR}_t )t≤T to a robust representation, we define dynamic vari-ational preferences (π^α_t^{gAV aR})t≤T by

π^α_t^{gAV aR}(X_T) := ess inf

Q∈M

n

E^Q[X_T|F_t] +α^{gAV aR}_t (Q)o for X_T ∈L^∞_T .

Remark 3.5.27. (π_t^{αgAV aR})t≤T is a time-consistent dynamic variational pref-erence. Indeed: It is a dynamic variational preference by virtue of its defi-nition in terms of a robust representation. Time-consistency of (π_t^{αgAV aR})t≤T

follows by Proposition 3.2.15 as the penalty(α^{gAV aR}_t )_t≤T is defined recursively in terms of the no-gain condition.

We have achieved two distinct time-consistent variational preferences gen-eralizing gAV aR: (π^aR_t )t≤T = (−dyn gAV aR_t)t≤T and (π_t^{αgAV aR})t≤T. We now show that these preferences coincide, i.e.

(π_t^aR)t≤T = (π^α_t^{gAV aR})t≤T,

given equality of the respective model parameters not explicitly stated here.

By Corollary 4.8 in [Cheridito et al, 06], it suffices to check thatπ₀^aR(X_T) = π₀^{αgAV aR}(X_T) forF_T-measurable random variablesX_T. However, for bothπ₀^aR as well as π^α₀^{gAV aR} we have a robust representation:

π₀^{αgAV aR}(X_T) = ess inf

Q∈M

n

E^Q[X_T] +α^{gAV aR}₀ (Q)o ,

26Intuitively, α^gAVaR_T (Q)(ω) is the penalty that only allows for the observed path (ω1, . . . , ωT).

and on the other hand we have

π₀^aR(X_T) = −dyn gAV aR₀(X_T)

= −ess inf

s∈R

1

θ E^P0[|s−X_T|^p]^β_p

−s

= −gAV aR(X_T)

= ess inf

Q∈M

E^Q|FT [X_T] +α^{gAV aR}(Q) ,

where the second equality follows by Definition 3.5.20 and time-consistency, and the last by Proposition 3.5.19. Hence, it suffices to show equality of the minimal penalties at t= 0, i.e. for all Q∈ M, we have to show

α^{gAV aR}(Q) =α^{gAV aR}₀ (Q).

Indeed: As we have seen that α^{gAV aR}_0,0+T (Q) = α^{gAV aR}(Q), it leaves to show α^{gAV aR}_0,0+T (Q) = α^{gAV aR}₀ (Q). By Theorem 4.5 in [F¨ollmer & Penner, 06], we have the no-gain condition fors-period ahead penalties reducing to

α^{gAV aR}₀ (Q) = α^{gAV aR}_0,0+T (Q) +E^Q h

α^{gAV aR}_T (Q) F₀i

. The right hand side equalsα^{gAV aR}_0,0+T (Q) as E^Q

h

α^{gAV aR}_T (Q) F₀i

= 0 by defini-tion ofα_T and the assumption thatQ∈ M: OtherwiseE^Q

h

α^{gAV aR}_T (Q) F₀i

=

∞contradicting Q∈ M.

Hence, both time-consistent dynamic variational preferences, π^aR and π^{αgAV aR}, coincide and we have

π^aR_t (Xτ) = ess inf

Q∈M

n

E^Q[Xτ|Ft] +α^{gAV aR}_t (Q) o

= X_τI^{τ≤t}

+ min

µ∈M|Ft+1

Z

π_t+1^aR(Xτ)dµ+γ_t^{gAV aR}(µ)

I^{τ≥t+1}. We have the following recursive representation of the Snell envelope of time-consistent dynamic variational preferences induced by dynamic generalized

average value at risk, (dyn gAV aR_t)t≤T: U_t = max

X_t;π^aR_t (U_t+1)

= max (

X_t; ess inf

µ∈M|Ft+1

Z

π_t+1^aR(U_t+1)dµ+γ_t^{gAV aR}(µ) )

= max (

X_t; ess inf

µ∈M|Ft+1

Z

U_t+1dµ+c

E

dµ dP0|F_t+1(·|F_t)

q

F_t

^d_q!) . This representation enables us, given an explicit structure of (X_t)t≤T, to solve the problem for an optimal stopping time τ^∗ as in Theorem 3.4.1.

Im Dokument On Dynamic Coherent and Convex Risk Measures: Risk Optimal Behavior and Information Gains (Seite 140-149)