Dynamic Convex Risk Measures: a Survey

Dynamic convex risk measures: a survey

Based on joint work with Beatrice Acciaio and Hans F¨ollmer

Irina Penner Humboldt University of Berlin Email: [email protected]

Workshop Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 2

Outline • Static risk measures: a recap • Dynamic framework – Conditional risk measures – Time consistency – Asymptotics and bubbles – Risk measures for processes: capturing discounting ambiguity

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 1

History • Starting point: Problem of quantifying the risk undertaken by a financial institution (bank, insurance company...) • From the point of view of a supervising agency a specific monetary purpose comes into play • Axiomatic analysis of capital requirements needed to cover the risk of some future liability was initiated by Artzner, Delbaen, Eber, Heath (1997, 1999) • Coherent risk measures: Artzner, Delbaen, Eber, Heath (1997, 1999), Delbaen (2000)... • Convex risk measures: Föllmerand Schied (2002), Frittelli and Rosazza Gianin (2002)...

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 2

Monetary measures of risk The set of ﬁnancial positions: X . Usually X = L∞(Ω, F,P ). More generally: X is an ordered locally convex topological vector space. A mapping ρ : X → R is called a monetary convex risk measure if satisﬁes the following conditions for all X,Y ∈ X :

• Cash (or Translation) Invariance: If m ∈ R, then ρ(X + m) = ρ(X) − m. • (Inverse) Monotonicity: If X ≤ Y , then ρ(X) ≥ ρ(Y ). • Convexity: ρ(λX + (1 − λ)Y ) ≤ λρ(X) + (1 − λ)ρ(Y ) for λ ∈ [0, 1]. • (sometimes) Normalization: ρ(0) = 0. A convex risk measure with • Positive homogeneity: If λ ≥ 0, then ρ(λX) = λρ(X) is a coherent risk measure.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 3

Acceptance sets An important characterization of a monetary risk measure is the acceptance set A := X ∈ X ρ(X) ≤ 0 . • By cash invariance, ρ is uniquely determined through its acceptance set: ρ(X) = inf m ∈ R X + m ∈ A (1) → ρ(X) is the minimal capital requirement that has to be added to a ﬁnancial position X in order to make it acceptable. • The acceptance set of a convex risk measure is convex, solid, and such that ess inf m ∈ R m ∈ A = 0 and 0 ∈ A. • On the other hand, every set A with the above properties deﬁnes a monetary convex risk measure via (1).

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 4

Robust representation • A convex risk measure ρ : X → R has a robust representation if it can be written as ρ(X) = sup (Y (X) − α(Y )) , Y ∈X ∗ with some convex penalty function α : X ∗ → [0, ∞]. • Typically a penalty function is given by αmin(Y ) = sup (Y (X) − ρ(X)) = sup Y (X). X∈X X∈A

• If ρ is coherent, then the robust representation is of the form ρ(X) = sup Y (X) Y ∈Q for some convex set Q ⊆ X ∗.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 5

Robust representation (continued) For a convex risk measure ρ : L∞ → R the following are equivalent: • ρ has a robust representation in terms of probability measures, i.e.

ρ(X) = sup (EQ[−X] − α(Q)) Q∈M1(P )

with some penalty function α : M1(P ) → [0, ∞], where M1(P ) = Q Q probability measure on (Ω, F),Q P . • ρ is lower semicontinuous in the weak∗ topology σ(L∞,L1). • The acceptance set A is weak∗ closed in L∞.

• ρ is continuous from above, i.e. ρ(Xn) % ρ(X) for any decreasing sequence (Xn) with Xn & X.

• ρ has the Fatou property, i.e. ρ(X) ≤ lim inf ρ(Xn) for any bounded sequence (Xn) with Xn → X.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 6

Important examples

• Law-invariant risk measures • Shortfall risk • Measures of risk in a ﬁnancial market

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Law-invariant risk measures Law-invariant risk measures depend only on the law of X. • A very common law-invariant monetary risk measure is Value at Risk at level λ ∈ [0, 1]: V @Rλ(X) = inf m ∈ R P [X + m < 0] ≤ λ -unfortunately not convex. • An alternative: Average Value at Risk (Tail Value at Risk, Conditional Value at Risk, Expected Shortfall) at level λ ∈ [0, 1] 1 Z λ AV @Rλ(X) = V @Rγ (X)dγ = max EQ[−X], λ 0 Q∈Qλ

n dQ 1 o where Qλ = Q P dP ≤ λ P -a.s. .

• AV @Rλ is the smallest law-invariant coherent risk measure that dominates V @Rλ.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 8

Shortfall risk Let l : R → R be some convex and increasing loss function (or alter- natively u : R → R some concave and increasing utility function). • The set ∞ ∞ A := X ∈ L E[l(−X)] ≤ l(0) = X ∈ L E[u(X)] ≥ u(0) deﬁnes an acceptance set. • The corresponding convex risk measure ρ(X) is continuous from below. • The minimal penalty function in the robust representation of ρ is given by 1 dQ αmin(Q) = inf E l∗ λ λ>0 λ dP for all Q P .

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 9

Shortfall risk: entropic risk measure • For the exponential loss function l(x) = eγx with the risk aversion γ > 0 we obtain the entropic risk measure 1 ρ(X) = log E[e−γX ]. γ • Entropic risk measure has the robust representation 1 ρ(X) = max EQ[−X] − H(Q|P ) , Q∈M1(P ) γ i.e., the minimal penalty function of the entropic risk measure is given by 1 αmin(Q) = H(Q|P ) = E [log(dQ/dP )], γ Q where H(Q|P ) denotes the relative entropy of Q w.r.t. P .

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Static risk measures 1 - 10

Measures of risk in a ﬁnancial market Consider a ﬁnancial market model with a discounted price process given

by a semi-martingale (St)t∈[0,T ]. The set of admissible strategies is denoted by S. • The set of all ﬁnancial positions that can be hedged with some admissible strategy at no cost n ∞ R T o A := X ∈ L ∃ξ ∈ S : 0 ξtdSt ≥ −X deﬁnes an acceptance set. • The corresponding risk measure ρ is the superhedging price. • ρ is a convex risk measure if S is convex and ρ is coherent if S is a cone. • The existence of a robust representation is related to the no-arbitrage condition.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework 2 - 1

Outline • Static risk measures: a recap • Dynamic framework: Assume there is an information ﬂow described by a ﬁltered probability space

(Ω, F, (Ft)t=0,...,T ,P ), F0 = {∅, Ω}, F = FT . The time horizon T might be ﬁnite or inﬁnite.

How can this information be used for risk assessment?

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework 2 - 2

Outline • Static risk measures: a recap • Dynamic framework: Assume there is an information ﬂow described by a ﬁltered probability space

(Ω, F, (Ft)t=0,...,T ,P ), F0 = {∅, Ω}, F = FT . The time horizon T might be ﬁnite or inﬁnite. How can this information be used for risk evaluation? – Conditional risk measures – Time consistency – Asymptotics and bubbles – Risk measures for processes: capturing discounting ambiguity

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Conditional risk measures 3 - 1

Conditional convex risk measure ∞ ∞ A conditional convex risk measure ρt is a map L (FT ) → L (Ft) with the following properties for all X,Y ∈ L∞:

∞ • Conditional Cash Invariance: ∀ mt ∈ L (Ft):

ρt(X + mt) = ρt(X) − mt

• (Inverse) Monotonicity: X ≤ Y ⇒ ρt(X) ≥ ρt(Y )

∞ • Conditional Convexity: ∀ λ ∈ L (Ft), 0 ≤ λ ≤ 1:

ρt(λX + (1 − λ)Y ) ≤ λρt(X) + (1 − λ)ρt(Y )

• Normalization: ρt(0) = 0.

φt := −ρt is a conditional monetary utility function.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Conditional risk measures 3 - 2

Conditional coherent risk measure A conditional convex risk measure with

∞ • Conditional positive homogeneity: ∀ λ ∈ L (Ft), λ ≥ 0:

ρt(λX) = λρt(X)

is called a conditional coherent risk measure.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Conditional risk measures 3 - 3

Acceptance sets The conditional version of acceptance set is given by

∞ At := X ∈ L ρt(X) ≤ 0 .

Again, ρt is uniquely determined through its acceptance set:

∞ ρt(X) = ess inf m ∈ L (Ft) X + m ∈ At

→ ρt(X) is the minimal conditional capital requirement that has to be added to a ﬁnancial position X in order to make it acceptable at time t.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Conditional risk measures 3 - 4

Robust representation (cf. Detlefsen and Scandolo (2005))

For a conditional convex risk measure ρt the following are equivalent:

• ρt is continuous from above;

• ρt has the robust representation

min ρt(X) = ess sup EQ[−X|Ft] − αt (Q) , Q∈Qt

min where the penalty function αt is given by

min αt (Q) = ess sup (EQ[−X|Ft] − ρt(X)) = ess sup EQ[−X|Ft] ∞ X∈L X∈At for Q ∈ Qt := Q P Q = P on Ft .

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Conditional risk measures 3 - 5

Example: conditional Average Value at Risk The conditional version of Average Value at Risk (Tail Value at Risk,

Conditional Value at Risk, Expected Shortfall) at level λt can be deﬁned as

AV @Rλt (X) = max EQ[−X|Ft], Q∈Qλt

n dQ 1 o ∞ where Qλ = Q ∈ Qt ≤ P -a.s. for some λt ∈ L (Ft) s.t. t dP λt 0 < λt ≤ 1 P -a.s..

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Conditional risk measures 3 - 6

Example: conditional entropic risk measure

γtx • The exponential loss function lt(x) = e with the risk aversion ∞ γt ∈ L (Ft), γt > 0 deﬁnes the acceptance set

∞ −γtX At := X ∈ L E[e |Ft] ≤ 1 . • The corresponding conditional risk measure 1 −γtX ρt(X) = log E[e |Ft] γt is called a conditional entropic risk measure. • It has the robust representation with the minimal penalty function

min 1 1 αt (Q) = Hbt(Q|P ) = EQ[log(dQ/dP )|Ft], γt γt

where Hbt(Q|P ) denotes the conditional relative entropy of Q ∈ Qt w.r.t. P .

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: time consistency 4 - 1

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: time consistency 4 - 2

Time consistency

• Dynamic framework: a ﬁltered probability space (Ω, F, (Ft),P ).

• A dynamic risk measure is a sequence (ρt)t=0,1,... such that each ρt is a conditional risk measure w.r.t. Ft. • We obtain for each X ∈ L∞ a sequence of risk assessments

(ρt(X))t=0,1,..., called risk process of X. The question arises: How are the risk assessments at diﬀerent times interrelated? → Several notions of time consistency.

(Artzner et al. (2007), Delbaen (2006), Riedel (2004), Detlefsen and Scandolo (2005), Cheridito et al. (2006), Tutsch (2006), Weber (2006), Roorda and Schumacher (2007), Kl¨oppel & Schweizer (2007), Bion-Nadal (2008), Delbaen et al. (2008)...))

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: time consistency 4 - 3

Several notions of time consistency ∞ Assume that we have a sequence of benchmark sets (Yt)t=0,1,... in L .

Then a dynamic risk measure (ρt) is called acceptance (resp. rejection)

consistent with respect to (Yt), if

ρt+1(X) ≤ ρt+1(Y )(resp. ≥) ⇒ ρt(X) ≤ ρt(Y )(resp. ≥).

∞ for all t and for any X ∈ L and Y ∈ Yt+1. → If a ﬁnancial position is always preferable tomorrow to some element of the benchmark set, then it should also be preferable today. The bigger the benchmark set, the stronger is the resulting notion of time consistency. (Tutsch (2006))

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: time consistency 4 - 4

Several notions of time consistency

We call a dynamic convex risk measure (ρt) • (strongly) time consistent, if the sequence of benchmark sets is ∞ given by Yt = L (FT ) for all t. In this case acceptance and rejection consistency coincide. • (middle) acceptance (resp. (middle) rejection) consistent, if the ∞ sequence of benchmark sets is given by Yt = L (Ft) for all t. • weakly acceptance (resp. weakly rejection) consistent, if the sequence of benchmark sets is given by Yt = R for all t.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: time consistency 4 - 5

Time consistency

• Weak time consistency • Acceptance and rejection consistency • Strong time consistency

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: weak time consistency 5 - 1

Weak time consistency

A dynamic convex risk measure (ρt) is weakly acceptance (resp. weakly rejection) consistent, if and only if

ρt+1(X) ≤ 0 (resp. ≥) ⇒ ρt(X) ≤ 0 (resp. ≥). for all t and all X ∈ L∞. → If some position is accepted (or rejected) for any scenario tomorrow, it should be already accepted (or rejected) today. (Weber (2005), Tutsch (2006), Roorda and Schumacher (2007, 2008))

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: weak time consistency 5 - 2

Weak acceptance consistency

Let (ρt)t=0,1,... be a dynamic convex risk measure such that each ρt is continuous from above. Then the following properties are equivalent:

1. (ρt) is weakly acceptance consistent.

2. At+1 ⊆ At for all t. 3. The inequality min min EQ[αt+1(Q)|Ft] ≤ αt (Q) holds for all Q P and all t. min In particular, the penalty process (αt (Q)) is a Q-supermartingale min for all Q ∈ Q0, where Q0 = Q P |α0 (Q) < ∞ . (Tutsch (2006), Roorda and Schumacher (2007, 2008))

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Acceptance and rejection consistency 6 - 1

Time consistency

• Weak time consistency • Acceptance and rejection consistency • Strong time consistency

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Acceptance and rejection consistency 6 - 2

Acceptance and rejection consistency

• A dynamic convex risk measure (ρt)t=0,1,... is rejection (resp. acceptance) consistent if and only if

ρt(−ρt+1) ≤ ρt (resp. ≥) ∀ t.

• Another equivalent characterization of rejection consistency is

ρt( ρt(X) − ρt+1(X) ) ≤ 0 ∀ t, ∀ X, | {z } adjustment at t+1 or, equivalently,

X ∈ At ⇒ −ρt+1(X) ∈ At ∀ t, ∀ X

( → “prudent”, “stay on the safe side”).

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Acceptance and rejection consistency 6 - 3

Step by step

Consider a conditional convex risk measure ρt restricted to the space ∞ L (Ft+1), i.e. just looking one step ahead. The corresponding “one-step” acceptance set is given by

∞ At,t+1 := X ∈ L (Ft+1) ρt(X) ≤ 0

and the minimal “one-step” penalty function by

min αt,t+1(Q) := ess sup EQ[−X|Ft],Q P. X∈At,t+1 In the same way we deﬁne for s ≥ t

∞ At,s := X ∈ L (Fs) ρt(X) ≤ 0 ,

min and denote the corresponding penalty function by αt,s (Q).

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Acceptance and rejection consistency 6 - 4

Acceptance and rejection consistency

Let (ρt)t=0,1,... be a dynamic convex risk measure such that each ρt is continuous from above. Then the following properties are equivalent:

1. (ρt) is rejection consistent (resp. acceptance consistent).

2. At ⊆ At,t+1 + At+1 resp. At ⊇ At,t+1 + At+1 for all t. 3. The inequality

min min min αt (Q) ≤ αt,t+1(Q) + EQ[ αt+1(Q) | Ft ]( resp.≥)

holds for all t and all Q P .

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Acceptance and rejection consistency 6 - 5

Acceptance and rejection consistency

min • In case of acceptance consistency, the penalty process (αt (Q)) is min a Q-supermartingale for all Q ∈ Q0 = Q P |α0 (Q) < ∞ . • In case of rejection consistency, The process

t−1 X min ρt(X) − αk,k+1(Q), t = 0, 1,... k=0 is a Q-supermartingale for all X ∈ L∞ and all Q P for which it is integrable.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Acceptance and rejection consistency 6 - 6

Sustainability

Let (ρt)t=0,1,... be any dynamic risk measure, and let X = (Xt)t=0,1,... be a bounded adapted process. We interpret X as a cumulative investment process, and we call X sustainable with respect to the

dynamic risk measure (ρt), if

ρt(Xt − Xt+1) ≤ 0 for all t = 0, 1,....

• The adjustment Xt+1 − Xt at time t + 1 is acceptable at time t

with respect to the risk measure ρt.

• A dynamic risk measure (ρt) is rejection consistent iﬀ for each X

the risk process (ρt(X)) is sustainable with respect to (ρt).

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Acceptance and rejection consistency 6 - 7

Sustainability (continued)

Let (ρt)t=0,1,... be a dynamic convex risk measure such that each ρt is continuous from above, and let X be any bounded adapted process. Then the following properties are equivalent:

1. The process X is sustainable with respect to the risk measure (ρt).

min 2. EQ [Xt|Ft−1] ≤ Xt−1 + αt−1,t(Q) Q-a.s. for all Q P and all t. In particular, the process

t−1 X min Xt − αk,k+1(Q), t = 0, 1,... k=0 is a Q-supermartingale for all Q P under which it is integrable.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: strong time consistency 7 - 1

Time consistency

• Weak time consistency • Acceptance and rejection consistency • Strong time consistency

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: strong time consistency 7 - 2

Time consistency

A dynamic convex risk measure (ρt)t=0,1,... is called (strongly) time consistent, if and only if for all X,Y ∈ L∞ and t, s ≥ 0 one of the following four equivalent conditions holds:

1. ρt+1(X) ≥ ρt+1(Y ) ⇒ ρt(X) ≥ ρt(Y )

2. ρt+1(X) = ρt+1(Y ) ⇒ ρt(X) = ρt(Y )

3. ρt+s(X) = ρt+s(Y ) ⇒ ρt(X) = ρt(Y )

4. Recursiveness: ρt = ρt(−ρt+s) (Artzner et al. (2007), Riedel (2004), Delbaen (2006), Detlefsen and Scandolo (2005), Kl¨oppel and Schweizer (2007), Cheridito et al. (2006), Frittelli & Rosazza Gianin (2004), Bion-Nadal (2008), Delbaen et al. (2008)...)

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: strong time consistency 7 - 3

Equivalent characterizations

Let (ρt)t=0,1,... be a dynamic convex risk measure such that each ρt is continuous from above. Then the following conditions are equivalent:

1. (ρt) is (strongly) time consistent.

2. At = At,t+s + At+s ∀ t, s.

min min min 3. αt (Q) = αt,t+s(Q) + EQ[αt+s(Q)|Ft] ∀ t, s, Q P .

min min 4. EQ[ρt+s(X) + αt+s(Q)|Ft] ≤ ρt(X) + αt (Q) ∀X, t, s, Q P .

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: strong time consistency 7 - 4

Equivalent characterizations: coherent case

Let (ρt)t=0,1,... be a dynamic convex risk measure such that each ρt is ∗ min continuous from above. Assume that Q := Q ≈ P α0 (Q) = 0 6= ∅, and that the initial risk measure ρ0 is coherent. Then the following conditions are equivalent:

1. (ρt) is time consistent. 2. The representation

ρt(X) = ess sup EQ[−X|Ft] (2) Q∈Q∗ holds for all X and all t, and the set Q∗ is stable. 3. The representation (2) holds for all X and all t, and the process ∗ (ρt(X)) is a Q-supermartingale for all Q ∈ Q .

In each case (ρt) is a dynamic coherent risk measure.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: strong time consistency 7 - 5

Supermartingale properties

Let (ρt)t=0,1,... be a time consistent dynamic convex risk measure such that each ρt is continuous from above. Then • the process

Q min Vt (X) := ρt(X) + αt (Q), t = 0, 1,... ∞ is a Q-supermartingale for all X ∈ L and all Q ∈ Q0. Q • (Vt (X)) is a Q-martingale if Q is a “worst case” measure for X at time 0, i.e. if min ρ0(X) = EQ[−X] − α0 (Q). In this case Q is a “worst case” measure for X at any time t.

• The converse holds if T < ∞ or if limt→∞ ρt(X) = −XP -a.s.: If Q (Vt (X)) is a Q-martingale, then Q is a “worst case” measure for X at any time t.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: strong time consistency 7 - 6

Supermartingale properties

Let (ρt)t=0,1,... be a time consistent dynamic convex risk measure such

that each ρt is continuous from above. • For Q P we can view the process

Q min Ft (X) := EQ[−X|Ft] − αt (Q), t = 0, 1,...

as a risk evaluation of the position X at time t, using the speciﬁc model Q. • We have the following maximal inequality for the excess of the Q required capital ρt(X) over the risk evaluation Ft (X): for c > 0 Q Q ρ0(X) − F0 (X) Q sup ρt(X) − Ft (X) ≥ c ≤ . t≥0 c

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: strong time consistency 7 - 7

Supermartingale properties

Let (ρt)t=0,1,... be a time consistent dynamic convex risk measure such

that each ρt is continuous from above. Then the penalty process min (αt (Q)) is a Q-supermartingale for all Q ∈ Q0 with the Riesz decomposition

" ∞ # min X min min αt (Q) = EQ αk,k+1(Q) Ft + lim EQ αs (Q)|Ft s→∞ k=t | {z } | {z } Q-martingale, “bubble” Q-potential and the Doob-decomposition

" ∞ # t−1

min X min Q X min αt (Q) = EQ αk,k+1(Q) Ft + Mt − αk,k+1(Q). k=0 k=0

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Asymptotics and bubbles 8 - 1

Outline • Static risk measures: a recap • Dynamic framework: – Conditional risk measures – Time consistency – Asymptotics and bubbles – Risk measures for processes: capturing discounting ambiguity

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Asymptotics and bubbles 8 - 2

Asymptotics

Let (ρt)t=0,1,... be a time consistent dynamic convex risk measure, and assume that T = ∞. Then for every Q ∈ Q0 there exist

min ρ∞(X) := lim ρt(X) and α∞(Q) := lim αt (Q) Q-a.s.. t→∞ t→∞ ∞ ∞ The functional ρ∞ : L → L is normalized, monotone, conditionally convex and conditionally cash invariant with respect to Ft for any t ≥ 0. We call (ρt)t=0,1,...

• asymptotically safe under the model Q, if ρ∞(X) ≥ −XQ -a.s. ( → covers the ﬁnal loss)

• asymptotically precise under the model Q, if ρ∞(X) = −XQ -a.s. ( → in this case ρ∞ is a risk measure itself). Not every time consistent dynamic risk measure is asymptotically precise or asymptotically safe.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Asymptotics and bubbles 8 - 3

Asymptotic safety

For a time consistent dynamic risk measure (ρt)t=0,1,... and a model

Q ∈ Q0 the following properties are equivalent:

1. (ρt)t=0,1,... is asymptotically safe under Q.

min 1 2. α∞(Q) = lim αt (Q) = 0 Q-a.s. and in L (Q), t→∞ min i.e., (αt (Q)) is a Q-potential. 3. The martingale M Q in the Riesz decomposition of the process min (αt (Q)) vanishes, i.e., there is no bubble.

4. No model R ∈ Q0 with R Q admits bubbles. In particular, every time consistent coherent dynamic risk measure is

asymptotically safe under each Q ∈ Q0.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Asymptotics and bubbles 8 - 4

A counterexample • Let Ω := (0, 1], P := λ, and let −t −t t Ft := σ ((k2 , (k + 1)2 ]|k = 0,..., 2 − 1), t = 0, 1,....

• Take A ∈ F := σ(∪t≥0Ft) such that P [A] > 0 and c P [A ∩ Jt,k] 6= 0 for any dyadic interval Jt,k. • For any t ≥ 0 we ﬁx the same acceptance set ∞ At := X ∈ L X ≥ −IA .

• The corresponding conditional convex risk measure ρt is given by

∞ ρt(X) = − ess sup m ∈ L (Ft) m ≤ X + IA .

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Asymptotics and bubbles 8 - 5

A counterexample (continued)

• ρt is normalized and continuous from above with the penalty function given by min αt (Q) = EQ[ IA | Ft ].

• The dynamic risk measure (ρt)t=0,1,... is time consistent, since

∞ At = At+s = At+s + L+ (Ft+s) = At+s + At,t+s

S∞ ∞ • ρ∞(X) = − ess sup m ∈ t=0 L (Ft) m ≤ X + IA

• (ρt)t=0,1,... is not asymptotically safe:

ρ∞(−IA) = 0 6≥ IA,

• The penalty function is a Q-martingale (bubble) with

min min α∞ (Q) = lim αt (Q) = IA 6= 0. t→∞

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Asymptotics and bubbles 8 - 6

Asymptotic precision A simple suﬃcient condition for asymptotic precision is the following:

Let (ρt)t=0,1,... be a time consistent dynamic convex risk measure, and suppose that there exists a worst case measure QX ≈ Q for each X ∈ L∞, i.e., min X ρ0(X) = EQX [−X] − α0 (Q ) X for some Q ≈ Q. Then (ρt)t=0,1,... is asymptotically precise under Q.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 1

Examples

• Dynamic entropic risk measure • Hedging under constraints • Dynamic average value at risk • Recursive construction

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 2

Example: Dynamic entropic risk measure

• Let (γt)t=0,1,... denote the process of strictly positive adapted risk 1 ∞ aversion parameters s.t. γt, ∈ L (Ft) for each t. γt • The corresponding dynamic risk measure 1 −γtX ρt(X) = log E[e |Ft], , t = 0, 1,... γt is a dynamic entropic risk measure. • The penalty process of the entropic risk measure is given by

min 1 αt (Q) = Hbt(Q|P ), , t = 0, 1,..., γt

where Hbt(Q|P ) the conditional relative entropy of Q P w.r.t. P .

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 3

Dynamic entropic risk measure (continued) The time consistency properties of the dynamic entropic risk measure are

completely determined by the adapted process of risk aversion (γt):

1. (ρt)t=0,1,... is time consistent if γt = γ ∈ R for all t;

2. (ρt)t=0,1,... is rejection consistent if γt ≥ γt+1 for all t;

3. (ρt)t=0,1,... is acceptance consistent if γt ≤ γt+1 for all t.

Moreover, the converse holds, if γt ∈ R for all t, or if the ﬁltration (Ft)t=0,1,... is rich enough in the sense that for all t and for all B ∈ Ft

such that P [B] > 0 there exists A ⊂ B such that A/∈ Ft and P [A] > 0. In fact, the entropic risk measure with constant risk aversion γ ∈ [0, ∞]

is the only time consistent dynamic risk measure such that ρ0 is law invariant.(Kupper and Schachermayer (2009))

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 4

Examples

• Dynamic entropic risk measure • Hedging under constraints • Dynamic average value at risk • Recursive construction

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 5

Hedging under constraints (cf. F¨ollmerand Kramkov (1997)) • Financial market with d risky assets. Their discounted prices are

given by the adapted process (St)t=0,...,T . • The set of allowed investment strategies S is subject to some convex trading constraints. • The set of all ﬁnancial positions that that can be hedged at time t by means of some admissible strategy at no cost n ∞ PT o At := X ∈ L ∃ξ ∈ S : k=t+1 ξk(Sk − Sk−1) ≥ −X deﬁnes a convex acceptance set.

• The corresponding conditional convex risk measure ρt(X) is the superhedging price of X at time t.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 6

Hedging under constraints (continued) Under the no arbitrage assumption the dynamic risk measure

(ρt(X))t=0,...,T is

• normalized, i.e. ρt(0) = 0 for all t

• continuous from above, i.e. each ρt has a robust representation. The minimal penalty functions can be identiﬁed in terms of the upper variation process of Q w.r.t. S • (strongly) time consistent • a minimal process hedging X under constraints.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 7

Examples

• Dynamic entropic risk measure • Hedging under constraints • Dynamic average value at risk • Recursive construction

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 8

Dynamic Average Value at Risk

• Let (λt)t=0,1,... denote the adapted process of level parameters s.t.

0 < λt ≤ 1 P -a.s. for each t. • The corresponding dynamic Average Value at Risk is given by

AV @Rλt (X) = max EQ[−X|Ft], t = 0, 1,..., Q∈Qλt

n dQ 1 o where Qλ = Q ∈ Qt ≤ P -a.s. for each t. t dP λt • The dynamic Average Value at Risk with constant parameter λ < 1 is in general neither weakly acceptance nor weakly rejection

consistent. Moreover, there exists no sequence of parameters (λt) such that the corresponding dynamic Average Value at Risk is time consistent.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 9

Dynamic Average Value at Risk

• Let (λt)t=0,1,... denote the adapted process of level parameters s.t. 0 < λt ≤ 1 P -a.s. for each t. • The corresponding dynamic Average Value at Risk is given by

AV @Rλt (X) = max EQ[−X|Ft], t = 0, 1,..., Q∈Qλt

n dQ 1 o where Qλ = Q ∈ Qt ≤ P -a.s. for each t. t dP λt • The dynamic Average Value at Risk with constant parameter λ < 1 is in general neither weakly acceptance nor weakly rejection

consistent. Moreover, there exists no sequence of parameters (λt) such that the corresponding dynamic Average Value at Risk is time consistent. • What can one do?

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 10

Examples

• Dynamic entropic risk measure • Hedging under constraints • Dynamic average value at risk • Recursive construction

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 11

Recursive construction

Suppose that T < ∞ and let (ρt)t=0,...,T be a dynamic convex risk measure. Consider a new risk measure (ρet)t=0,...,T deﬁned recursively by

ρeT (X) := ρT (X) = −X ∞ ρet(X) := ρt(−ρet+1(X)), t = 0,...,T − 1,X ∈ L .

• Then (ρet) is again a dynamic convex risk measure and it is (strongly) time consistent by deﬁnition. (cf. Cheridito et al. (2006), Cheridito and Kupper (2006), Drapeau (2006))

• If the original risk measure (ρt) is rejection (resp. acceptance) consistent, then (ρet) lies below (resp. above) (ρt).

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Examples 9 - 12

Recursive construction (continued)

The recursively constructed dynamic convex risk measure (ρet)t=0,...,T has the following properties:

• Sustainability w.r.t. (ρt):

ρt(ρet(X) − ρet+1(X)) ≤ 0 ∀ t, ∀ X,

( → the adjustments ρet(X) − ρet+1(X) are acceptable at any time)

• ρT (X) = −X ( → the ﬁnal loss is covered)

∞ Moreover, for each X ∈ L the risk process (ρet(X))t=0,...,T is the smallest bounded adapted process with these both properties.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Risk measures for processes 10 - 1

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Dynamic framework: Risk measures for processes 10 - 2

Risk measures for processes Dynamic framework: Multiperiod information structure:

(Ω, F, (Ft)t=0,...,T ,P ),T ≤ ∞, F0 = {∅, Ω}, F∞ = σ(∪t≥0Ft).

• So far we have discussed risk assessment for discounted payoffs taking into account the new information but not the timing of the payment. • Now we will consider cash flow processes that model the evolution of financial values. Risk assessment in this framework accounts not only the amounts but also the timing of a payment.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Risk measures for processes 11 - 1

Risk measures for processes • Setup and deﬁnition • Transforming processes into random variables • Translating the results from Part I: – Robust representation – Cash additivity and subadditivity – Time consistency, supermartingales, asymptotics and bubbles... • Examples

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Risk measures for processes: setup and deﬁnition 12 - 1

Risk measures for processes: setup •R ∞ denotes the set of all bounded adapted processes on ∞ (Ω, F, (Ft)t=0,...,T ,P ). A process X ∈ R is understood as a cumulated cash-ﬂow (value process).

∞ ∞ • For 0 ≤ t ≤ s ≤ T , we deﬁne the projection πt,s : R → R as

πt,s(X)r = 1{t≤r}Xr∧s, r ∈ T ∞ ∞ ∞ ∞ and denote Rt,s := πt,s(R ), Rt := Rt,T .

(Artzner et al. (2007), Riedel (2004), Cheridito, Delbaen & Kupper (2006), Cheridito & Kupper (2006), Frittelli & Scandolo (2006), Jobert & Rogers (2008))

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Risk measures for processes: setup and deﬁnition 12 - 2

Risk measures for processes: setup We characterize risk measures for processes in the following 3 cases:

Case 1: T < ∞, T := {0,...,T }

Case 2: T = ∞, T := N0

Case 3: T = ∞, T := N0 ∪ {∞}

In case 2 and 3 we assume that each (Ω, Ft) is a standard Borel space,

and that for any sequence of atoms A0 ⊇ A2 ⊇ ..., where At ∈ Ft for

each t ∈ N0, one has ∩t∈N0 At 6= ∅.(Parthasarathy)

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Risk measures for processes: setup and deﬁnition 12 - 3

Conditional convex risk measure: a recap ∞ ∞ A map ρt : L (Ω, FT ,P ) → L (Ft) is called a conditional convex risk measure for random variables if it satisﬁes the following properties for all X,Y ∈ L∞:

∞ • Conditional cash invariance: ∀m ∈ L (Ft):

ρt(X + m) = ρt(X) − m

• Monotonicity: X ≤ Y ⇒ ρt(X) ≥ ρt(Y )

∞ • Conditional convexity: ∀λ ∈ L (Ft), 0 ≤ λ ≤ 1:

ρt(λX + (1 − λ)Y ) ≤ λρt(X) + (1 − λ)ρt(Y )

• Normalization: ρt(0) = 0.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Risk measures for processes: setup and deﬁnition 12 - 4

Conditional risk measure for processes ∞ ∞ A map ρt : Rt → L (Ft) is called a conditional convex risk measure ∞ for processes if it satisﬁes the following properties for all X,Y ∈ Rt : ∞ • Conditional cash invariance: ∀m ∈ L (Ft):

ρt(X + m1{t,t+1,...}) = ρt(X) − m

• Monotonicity: X ≤ Y ⇒ ρt(X) ≥ ρt(Y )

∞ • Conditional convexity: ∀λ ∈ L (Ft), 0 ≤ λ ≤ 1:

ρt(λX + (1 − λ)Y ) ≤ λρt(X) + (1 − λ)ρt(Y )

• Normalization: ρt(0) = 0.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Risk measures for processes: setup and deﬁnition 12 - 5

Acceptance set • In the same way as in case of random variables we deﬁne the acceptance set of a conditional convex risk measure for processes:

∞ At = X ∈ R ρt(X) ≤ 0 .

• Again, ρt is uniquely determined through its acceptance set:

∞ ρt(X) = ess inf m ∈ L (Ft) X + m1{t,t+1,...} ∈ At .

→ ρt(X) is the minimal conditional capital requirement that has to be added to the cash ﬂow X at time t in order to make it acceptable.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Transforming processes into random variables 13 - 1

Risk measures for processes • Setup • Deﬁnition • Transforming processes into random variables • Translating the results from Part I: – Robust representation – Cash additivity and subadditivity – Time consistency, supermartingales, asymptotics and bubbles... • Examples

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Transforming processes into random variables 13 - 2

Transforming processes into random variables We deﬁne

• the product space (Ω¯, F¯, P¯) as: Ω¯ = Ω × T ¯ ¯ F = σ(At × {t} At ∈ Ft, t ∈ T), P = P ⊗ µ,

where µ = (µt)t=0,...,T is any adapted sequence with µt > 0, P µt = 1. Then t∈T R∞ = L∞(Ω¯, F¯, P¯).

• the ﬁltration (F¯t)t=0,...,T on (Ω¯, F¯) by ¯ Ft = σ {Aj × {j},At × {t, t + 1,...} Aj ∈ Fj, j ≤ t} .

Then X is F¯t-measurable iﬀ X = (X0,...,Xt−1,Xt,Xt,...) with

Xs Fs-measurable, s = 0, . . . , t.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Transforming processes into random variables 13 - 3

Risk measures on the product space There is a one-to-one correspondence between • conditional convex risk measures for processes

∞ ∞ ρt : Rt → L (Ft) and • conditional convex risk measures for random variables on the product space

∞ ∞ ρ¯t : L (Ω¯, F¯, P¯) → L (Ω¯, F¯t, P¯).

The relation is given by

ρ¯t(X) = −X01{0} − ... − Xt−11{t−1} + ρt(X)1{t,t+1,...}.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Transforming processes into random variables 13 - 4

Optional random measures

Let Q be a probability measure on (Ω, (Ft), F) with Q loc P and consider the set Γ(Q) of optional random measures on T under Q: ∞ γ = (γt)t∈T s.t. γt ∈ L (Ft), γt ≥ 0 P -a.s. ∀t ∈ T and T X • T < ∞: γt = 1 P -a.s.( Γ(Q) := Γ independent of Q). t=0 ∞ X • T = N0: γt = 1 Q-a.s. t=0 ∞ X • T = N0 ∪ {∞}: γt + γ∞ = 1 Q-a.s. and γ∞ = 0 Q-a.s. on t=0 the support of the singular part of Q w.r.t. P .

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Transforming processes into random variables 13 - 5

Discounting processes

Let Q be a probability measure on (Ω, (Ft), F) with Q loc P and consider the set D(Q) of predictable discounting processes on T under Q:

D = (Dt)t∈T s.t. D decreasing, predictable,D0 = 1 and

• T < ∞: DT +1 := 0 (D(Q) := D independent of Q).

• T = N0: Dt & 0 Q-a.s.

• T = N0 ∪ {∞}: Dt & D∞ ≥ 0 Q-a.s. and D∞ = 0 Q-a.s. on the support of the singular part of Q w.r.t. P .

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Transforming processes into random variables 13 - 6

Correspondence There is a one-to-one correspondence between optional measures γ ∈ Γ(Q) and predictable discounting processes D ∈ D(Q).

The relation is given by

t−1 X X Dt = 1 − γs = γs, t = 0, 1,...,D∞ = γ∞ Q-a.s. s=0 s∈T,s≥t and

γt = Dt − Dt+1, t = 0, 1, . . . , γ∞ = D∞ Q-a.s.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Transforming processes into random variables 13 - 7

Probability measures on the product space For any probability measure Q¯ on (Ω¯, F¯) the following holds: Q¯ P¯ if and only if there exists • a probability measure Q on (Ω, F) s.t. Q P if T < ∞ and

Q loc P if T = ∞ • a random measure γ ∈ Γ(Q) (resp. a discounting factor D ∈ D(Q)) such that " # " T # X X EQ¯ [X] = EQ γtXt = EQ Dt(Xt − Xt−1) t∈T t=0 for all X ∈ R∞. In this case we write

Q¯ = Q ⊗ γ = Q ⊗ D.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Transforming processes into random variables 13 - 8

Idea of the proof We use the Itˆo-Watanabe decomposition for nonnegative supermartingales:

Proposition 1 Let (Ut) be a nonnegative P -supermartingale on some

probability space (Ω, F, (Ft),P ). Then there exist a nonnegative

P -martingale (Mt) and a predictable decreasing process (Dt) such that

M0 = U0, D0 = 1 and

Ut = MtDt ∀ t.

Moreover such a decomposition is unique on {t < τ0}, where

τ0 := inf{t > 0 | Ut = 0}.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Transforming processes into random variables 13 - 9

Idea of the proof (continued) ¯ ¯ ¯ Let Q P with the density Z = (Zt)t∈T. • We apply the Itˆo-Watanabe decomposition to the nonnegative P -supermartingale  

X Ut := EP  µsZs Ft = MtDt, t ∈ T. s≥t

• The martingale (Mt) deﬁnes a consistent sequence of probability dQt measures Qt on Ft via dP = Mt, t ∈ T ∩ N0.

• By Parthasarathy’s extension theorem, the family (Qt) admits a unique extension to a probability measure Q on F∞ = σ(∪tFt), such that Q loc P with dQ Mt = , t ∈ T ∩ N0. dP Ft

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Transforming processes into random variables 13 - 10

Idea of the proof (continued)

• (Dt) deﬁnes a discounting factor with the corresponding random

measure (γt). • We have " # ∞ X X EQ¯ [X]= EP XtµtZt = EP [XtEP [Ut − Ut+1|Ft]] t∈T t=0 ∞ X = EP [Xt(MtDt − Mt+1Dt+1)] t=0 ∞ " ∞ # X X = EQ[Xtγt] = EQ Xtγt . t=0 t=0

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Transforming processes into random variables 13 - 11

Probability measures on the product space For Q¯ P¯ with the decomposition Q¯ = Q ⊗ γ = Q ⊗ D, the

conditional expectation given F¯t takes the form   ¯ X γs EQ¯ [X | Ft ] = X01{0}+...+Xt−11{t−1}+EQ  Xs Ft 1{t,t+1,...}, Dt s≥t

where the last term on the right-hand-side is well deﬁned Q-a.s. on

{Dt > 0}.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 1

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 2

Robust representation: a recap

A conditional convex risk measure ρt for random variables is continuous from above if and only if it has the robust representation

min ρt(X) = ess sup(EQ[−X] − αt (Q)), Q∈Qt where Qt = Q P Q = P |Ft ,

and the minimal penalty function αt is given by

min αt (Q) = ess sup EQ[−X|Ft] = ess sup (EQ[−X|Ft] − ρt(X)) . ∞ X∈At X∈L

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 3

Robust representation

A conditional convex risk measure ρt for processes is continuous from above if and only if it has the robust representation     X ρt(X) = ess sup ess sup EQ − γsXs Ft − αt(Q ⊗ γ) loc Q∈Q γ∈Γt(Q) t s∈T,s≥t " T # ! X = ess sup ess sup EQ − Ds∆Xs|Ft − αt(Q ⊗ D) loc Q∈Qt D∈Dt(Q) s=t where

loc Qt = Q loc P Q = P |Ft Γt(Q) = γ ∈ Γ(Q) γ0 = ... = γt−1 = 0 , Dt(Q) = D ∈ D(Q) D0 = ... = Dt = 1 ,

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 4

Robust representation

loc for Q ∈ Qt and γ ∈ Γt(Q).

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 5

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 6

Cash additivity and sub-additivity

A conditional convex risk measure for processes ρt is called ∞ • cash sub-additive if for all t, s ≥ 0 and m ∈ L (Ft)

ρt(X + m1{t+s,...}) ≥ ρt(X) − m for m ≥ 0 (resp. ≤ for m ≤ 0)

(El Karoui & Ravanelli (2008)); • cash additive at t + s for some s ≥ 1 if

∞ ρt(X + m1{t+s,...}) = ρt(X) − m, ∀ m ∈ L (Ft);

• cash additive if it is cash additive at all times.

Remark. By monotonicity and cash invariance every conditional convex risk measure on processes is cash sub-additive.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 7

Cash additivity and sub-additivity

Let ρt be a conditional convex risk measure on processes, continuous from above. Then

• ρt is cash additive at time t + s if and only if there is no discounting

up to time t + s. In this case ρt is cash additive up to time t + s.

• T < ∞ or T = N0 ∪ {∞}: ρt is cash additive if and only if it reduces to a risk measure for random variables, i.e. ρt(X) = ess sup EQ[−XT Ft] − αt(Q) . Q∈Qt

• T = N0: ρt cannot be cash additive.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 8

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 9

Time consistency • As in the case of risk measures for random variables, in the dynamic setting we obtain for each cash ﬂow process X a sequence of risk

assessments (ρt(X))t=0,1,.... ∞ (Notation: ρt(X) := ρt(πt,T (X)) for X ∈ R .) • The issue of time consistency arises. • Using the one-to-one correspondence on the product space, we can translate all the notions of time consistency and their characterizations from the ﬁrst part to the present context. • Here we concentrate only on the strong time consistency.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 10

(Strong) Time consistency

A dynamic convex risk measure for random variables (ρt)t=0,1,... is called time consistent if

∞ ρt(X) = ρt(−ρt+1(X)) ∀ t ≥ 0, ∀ X ∈ L (FT ).

A dynamic convex risk measure for processes (ρt)t=0,1,... is called time consistent if

∞ ρt(X) = ρt(Xt1{t} − ρt+1(X)1{t+1,...}) ∀ t ≥ 0, ∀ X ∈ R .

(Cheridito, Delbaen & Kupper (2006), Cheridito & Kupper (2006))

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 11

Step by step • ”One-step” acceptance set for processes:

∞ At,t+1 := X ∈ Rt,t+1 ρt(X) ≤ 0 .

∞ (X ∈ Rt,t+1 iﬀ X = (0,..., 0,Xt,Xt+1,Xt+1,...)) • ”One-step” penalty function for processes:

1 αt,t+1(Q ⊗ D) = ess sup EQ −γtXt − Dt+1Xt+1 Ft . Dt X∈At,t+1

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 12

Equivalent characterizations: a recap

Let (ρt)t=0,...,T be a dynamic convex risk measure for random variables

such that each ρt is continuous from above . Then the following conditions are equivalent:

1. (ρt) is time consistent

2. At = At,t+1 + At+1

min min min 3. αt (Q) = αt,t+1(Q) + EQ[ αt+1(Q) | Ft ] min min 4. EQ[ ρt+1(X) + αt+1(Q) | Ft] ≤ ρt(X) + αt (Q).

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 13

Equivalent characterizations

Let (ρt)t=0,...,T be a dynamic convex risk measure for processes such

that each ρt is continuous from above . Then the following conditions are equivalent:

1. (ρt) is time consistent

2. At = At,t+1 + At+1 ∀t

3. Dtαt(Q ⊗ D) = Dtαt,t+1(Q ⊗ D) + EQ[Dt+1αt+1(Q ⊗ D)|Ft]

for all t ≥ 0, all Q loc P and D ∈ D(Q).

4. EQ[Dt+1(Xt + ρt+1(X) + αt+1(Q ⊗ D))|Ft] ≤

Dt(Xt + ρt(X) + αt(Q ⊗ D))

for all Q loc P and D ∈ D(Q).

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 14

Supermartingale properties

Let (ρt)t=0,...,T be a time consistent dynamic convex risk measure • for random variables. Then min . the penalty process (αt (Q))t≥0 . the process min ρt(X) + αt (Q), t ≥ 0 min are Q-supermartingales for all Q P s.t. α0 (Q) < ∞. • for processes. Then min . the discounted penalty function process (Dtαt (Q))t≥0 . the process t X Dtρt(X − Xt1{t,..}) + Ds(−∆Xs) + Dtαt(Q ⊗ D), t ≥ 0 s=0

are Q-supermartingales for all Q and D s.t. α0(Q ⊗ D) < ∞.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Translating the results 14 - 15

Bubbles min The discounted penalty function process (Dtαt (Q))t=0,1,... has the Riesz decomposition

" T # X Q,D Dtαt(Q ⊗ D) = EQ Dkαk,k+1(Q ⊗ D) Ft +Nt Q-a.s., k=t | {z } Q-potential where  Q,D  0 if T < ∞, Nt := Q-a.s. lim EQ [Dsαs(Q ⊗ D) | Ft ] if T = ∞  s→∞ is a non-negative Q-martingale → “bubble”. In case T = N0 ∪ {∞}: no “bubble” ⇔ “asymptotic safety”:

ρ∞(X) := lim ρt(X) ≥ −X∞ Q-a.s. on {D∞ > 0}. t→∞

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Examples 15 - 1

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Examples 15 - 2

Separation of model- and discounting ambiguity • A simple class of risk measures for processes can be obtained using the representation X ρt(X) = ess sup ψt Xsγs , γ∈G t s≥t

where ψt is a ﬁxed conditional convex risk measure for random

variables and Gt ⊆ Γt is a ﬁxed collection of discounting factors.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Examples 15 - 3

Separation of model- and discounting ambiguity Some examples of a risk measure for processes with the representation X ρt(X) = ess sup ψt Xsγs : γ∈G t s≥t

• Gt = {δ{s}} for some s ∈ {t, t + 1,...}. In this case

ρt(X) = ψt(Xs)

∞ is a conditional convex risk measure on L (Ω, Fs,P ).

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Examples 15 - 4

Separation of model- and discounting ambiguity Some examples of a risk measure for processes with the representation X ρt(X) = ess sup ψt Xsγs : γ∈G t s=≥t

• Gt = {(γt, . . . , γT )}, i.e. a single discounting factor. In this case

T T X X ρt(X) = ψt Ds∆Xs = ψt ∆Ys = ψt(YT ) s=t s=t

i.e. on discounted cash-ﬂows ρt reduces to a risk measure for random variables.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Examples 15 - 5

Separation of model- and discounting ambiguity Some examples of a risk measure for processes with the representation X ρt(X) = ess sup ψt Xsγs : γ∈G t s≥t • Gt = (1{τ=s})s=t,t+1,... τ ∈ Θt , where Θt denotes the set of all stopping times valued in {t, t + 1 ...}. Here we obtain

T ! X ρt(X) = ess sup ψt Xsγs = ess sup ψt(Xτ ) γ∈Gt s=t τ∈Θt

i.e. ρt is deﬁned by the worst stopping of (ψt(Xs))s=t,t+1,.... (Cheridito & Kupper (2006))

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Examples 15 - 6

Entropic risk measure for processes On the product space we deﬁne the conditional entropic risk measure ∞ ∞ ρ¯t : L (Ω¯, F¯, P¯) → L (Ω¯, F¯t, P¯) by

1 −Rt·X ρ¯t(X) = · log E ¯ e F¯t Rt P

with the risk aversion parameter Rt = (r0, .., rt−1, rt, .., rt), rs > 0, Fs-mb. for all s = 0, . . . , t.

The corresponding conditional entropic risk measure for processes ∞ ∞ ρt : Rt → L (Ft) takes the form T 1 X ρ (X) = ρP − log e−rtXs µt = ρP (−ρµ (X(ω, ·))) , t t r s t t t s=t P ∞ ∞ where ρt : LT → Lt is the usual conditional entropic risk measure for µ T −t+1 random variables and ρt : R → R is its analogous “with respect to time”.

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Examples 15 - 7

Average Value at Risk for processes • On the product space the conditional AVaR at level

∞ Λt = (λ0, ..λt−1, λt, .., λt), 0 < λs ≤ 1, λs ∈ L (Fs) ∀ s is deﬁned by

¯ ¯ ¯ ¯ ¯ −1 ρ¯t(X) = ess sup{EQ¯ [−X|Ft]|Q ∈ Qt, dQ/dP ≤ Λt }

• The corresponding conditional Average Value at Risk for processes ∞ ∞ ρt : Rt → L (Ft) is given by

T n h X i γs dQ 1 o ρt(X) = ess sup EQ − Xsγs|Ft |Q ∈ Qt, γ ∈ Γt, ≤ ∀s . µt dP s λ s=t s t

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Examples 15 - 8

Average Value at Risk for processes • One can also deﬁne a “decoupled version” of conditional Average Value at Risk for processes as ! λ1,λ2 λ2 X ∞ ρt (X) := ess sup AV @Rt Xsγs ,X ∈ Rt . λ1 γ∈Γt s∈Tt

∞ λ2 Here, λ1, λ2 ∈ L (Ft) with 0 < λ1, λ2 ≤ 1, AV @Rt is the usual Average Value at Risk for random variables, and n o λ1 γs 1 Γ = γ ∈ Γt(P ) ≤ , s ≥ t . t µs λ1 • The decoupled version is less conservative than the Average Value

at Risk with λt = λ1λ2 deﬁned the previous slide, i.e.,

λ1,λ2 λ1λ2 ∞ ρt (X) ≤ ρt (X) ∀ X ∈ R .

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011 Examples 15 - 9

Thank you for your attention!

Risk Analysis in Economics and Finance, Guanajuato, February 2-4, 2011