Liquidity, risk measures, and concentration of measure · 2015. 11. 18. · Liquidity, risk measures, and concentration of measure Liquidity risk & risk measures Literature on liquidity

Liquidity, risk measures, and concentration of measure

Liquidity, risk measures, and concentrationof measure

Daniel Lacker

Division of Applied Mathematics, Brown University

November 17, 2015


Liquidity risk & risk measures

Section 1




Liquidity vs liquidity risk

I Some sources/definitions of illiquidity: shortage ofcounterparties, search/transaction costs, misc. marketfrictions...

I Liquidity risk: Difficult to scale positions. Risk and even priceare sensitive to volume. Leverage is risky in illiquid markets.

I “Liquidity” is a broad concept, with many sources/definitions.But the effects on liquidity risk are not as varied.



Why risk measures?

Risk measures axiomatize risk independently of model details.Likewise, we model liquidity risk separately from particular sourcesof illiquidity.

Risk measures are to risk as probability measures are torandomness:

I Probabilistic models often hide the omega, i.e., the source ofrandomness.

I Risk measures often hide the source of risk.



Coherent risk measures

A coherent risk measure (Artzner et al. ’99) is a functional

ρ : L1(Ω,F ,P)→ (−∞,∞]

satisfying

1. Normalization: ρ(0) = 0.

2. Monotonicity: ρ(X ) ≤ ρ(Y ) if X ≤ Y a.s.3. Cash additivity: ρ(X + c) = ρ(X ) + c for c ∈ R.4. Convexity: ρ(tX + (1− t)Y ) ≤ tρ(X ) + (1− t)ρ(Y ) for

t ∈ (0, 1).5. Positive homogeneity: ρ(λX ) = λρ(X ) for λ ≥ 0.



Convex risk measures

A convex risk measure (Föllmer/Schied ’02) is a functional

ρ : L1(Ω,F ,P)→ (−∞,∞]

satisfying

1. Normalization: ρ(0) = 0.

2. Monotonicity: ρ(X ) ≤ ρ(Y ) if X ≤ Y a.s.3. Cash additivity: ρ(X + c) = ρ(X ) + c for c ∈ R.4. Convexity: ρ(tX + (1− t)Y ) ≤ tρ(X ) + (1− t)ρ(Y ) for

t ∈ (0, 1).5. Positive homogeneity: ρ(λX ) = λρ(X ) for λ ≥ 0.

Idea: Liquidity risk ⇒ ρ(λX ) 6= λρ(X ) in general.



Liquidity risk profiles

DefinitionThe liquidity risk profile of a loss X (and a given ρ) is thefunction (ρ(λX ))λ≥0.

Goal:Quantify/study liquidity risk in terms of liquidity risk profiles.Find systematic bounds when explicit computations areunavailable.



Basic interpretations of liquidity risk profiles (ρ(λX ))λ≥0

I ρ(λX ) ≥ λρ(X ) for λ ≥ 1 (leverage is risky)I ρ(λX ) ≤ λρ(X ) for λ ≤ 1

I Conservative bound on risk-per-unit X (may be ∞):

Cρ(X ) = limλ↑∞

1

λρ(λX ) = sup

λ>0

1

λρ(λX ).

I Note ρ(λX ) ≤ λCρ(X ).I Marginal risk is always well-defined (Barrieu/El Karoui ’05):

Mρ(X ) = limλ↓0

1

λρ(λX ) =

d

dλρ(λX )|λ=0.

I Mρ is a coherent risk measure vanishing liquidity risk.




I ρ(λX ) ≥ λρ(X ) for λ ≥ 1 (leverage is risky)I ρ(λX ) ≤ λρ(X ) for λ ≤ 1I Conservative bound on risk-per-unit X (may be ∞):


1

λρ(λX ) = sup

λ>0

1

λρ(λX ).

I Note ρ(λX ) ≤ λCρ(X ).

I Marginal risk is always well-defined (Barrieu/El Karoui ’05):

Mρ(X ) = limλ↓0

1

λρ(λX ) =

d

dλρ(λX )|λ=0.





I ρ(λX ) ≥ λρ(X ) for λ ≥ 1 (leverage is risky)I ρ(λX ) ≤ λρ(X ) for λ ≤ 1I Conservative bound on risk-per-unit X (may be ∞):


1

λρ(λX ) = sup

λ>0

1

λρ(λX ).

I Note ρ(λX ) ≤ λCρ(X ).I Marginal risk is always well-defined (Barrieu/El Karoui ’05):

Mρ(X ) = limλ↓0

1

λρ(λX ) =

d

dλρ(λX )|λ=0.




Concentration inequalities

In general, computing liquidity risk profiles is hard!

Goal:Systematically study concentration inequalities of the form

ρ(λX ) ≤ γ(λ), ∀λ ≥ 0,

where

I ρ is a convex risk measure.

I X ∈ L1 is a loss.I γ : [0,∞)→ [0,∞] is nondecreasing and convex, called a

shape function.

Capital required to cover a loss of λX is no more than γ(λ).



Centered concentration inequalities

We may study centered concentration inequalities of the form

ρ(λ(X − EX )) ≤ γ(λ), ∀λ ≥ 0.

Think of selling X for the “price” EX .



Literature on liquidity risk quantification

I Liquidity-adjusted risk measures (Acerbi-Scandolo ’08,Weber et al. ’13, Jarrow-Protter ’05)

I Distinguish portfolio content/value

X 7→ ρ(V (X )), ρ coherent, V marks to market.

I Set-valued risk measures (Jouini et al. ’04, Hamel et al.’11, etc.)

I State capital requirements in terms of multiple numeraires forwhich exchange is costly

Both approaches go beyond convex risk measures, which wereintroduced precisely for the modeling of liquidity risk!


Tail bounds and integral criteria for shortfall concentration inequalities

Section 2

Tail bounds and integral criteria for shortfallconcentration inequalities



Entropic risk measure

Entropic risk measure: ρexp(X ) = logE[eX ].

Classical Chernoff (tail) bound

If ρexp(λX ) ≤ γ(λ), ∀λ ≥ 0, then

P(X > t) ≤ e−γ∗(t), ∀t ≥ 0,

where γ∗(t) := supλ≥0(tλ− γ(λ)).



Shortfall risk measures

Definition (Föllmer/Schied ’02)

A loss function is ` : R→ [0,∞), nondecreasing and convex, andsatisfying `(0) = 1 < `(x) ∀x > 0. The shortfall risk measure is

ρ`(X ) = inf {m ∈ R : E[`(X −m)] ≤ 1} .

Entropic risk measure: If `(x) = ex then ρ`(X ) = logE[eX ].

General tail boundIf ρ`(λX ) ≤ γ(λ), ∀λ ≥ 0, then

P(X > t) ≤ 1/`(γ∗(t)), ∀t ≥ 0,

where γ∗(t) := supλ≥0(tλ− γ(λ)).



Proof of tail boundRecall

ρ`(X ) = inf {m ∈ R : E[`(X −m)] ≤ 1} .The inf is attained, so ρ`(λX ) ≤ γ(λ) implies

E[`(λX − γ(λ))] ≤ 1.

Note x 7→ `(λx − γ(λ)) is nondecreasing. By Markov’s inequality,

P(X > t) ≤ P[`(λX − γ(λ)) ≥ `(λt − γ(λ))

]≤ 1/`(λt − γ(λ)).

Optimize over λ:

P(X > t) ≤ infλ≥0

1/`(λt − γ(λ)) = 1/`(γ∗(t)).



Integral criteria for concentration

QuestionWe saw that a necessary condition for concentration,ρ`(λX ) ≤ γ(λ) ∀λ ≥ 0, is the tail bound P(X > t) ≤ 1/`(γ∗(t))∀t > 0. When is it sufficient?

Classical caseLet EX = 0. The following are well known to be equivalent, up toa (universal, i.e. X -independent) change in c > 0:

1. logE[exp(λX )] ≤ cλ2, ∀λ ≥ 0 (i.e. X is subgaussian)2. P(X > t) ≤ exp(−ct2), ∀t ≥ 03. E[exp(c|X+|2)]




TheoremLet EX = 0. Consider the statements

1. ρ`(λX ) ≤ γ(cλ), ∀λ ≥ 02. P(X > t) ≤ 1/`(γ∗(ct)), ∀t ≥ 03. E[`(γ∗(cX+))] 0,`′′′ ≥ 0, and limx→∞ x2`′′(x)/`(γ∗(x)) = 0, then 3⇒ 1.Note all implications hold up to a “universal” change in c.




TheoremMore generally, if EX is not necessarily 0,

1. ρ`(λ(X − EX )) ≤ γ(cλ), ∀λ ≥ 02. P(X − EX > t) ≤ 1/`(γ∗(ct)), ∀t ≥ 03. E[`(γ∗(cX+))] 0,`′′′ ≥ 0, and limx→∞ x2`′′(x)/`(γ∗(x)) = 0, then 3⇒ 1.Note all implications hold up to a “universal” change in c .



Application to optionsThink of X = (X1, . . . ,Xn) as asset prices. Let Ck(x) = (x − k)+and Pk(x) = (x − k)− denote call and put payoffs.

TheoremFix a “good” pair (`, γ), and fix kci , k

pi > 0. Suppose

ρ`(λ(Ckci (Xi )− E[Ckci (Xi )])) ≤ γ(λ),ρ`(λ(Pkpi

(Xi )− E[Pkpi (Xi )])) ≤ γ(λ), ∀i = 1, . . . , n, λ ≥ 0.

Then there exists c > 0 such that, for every f satisfyingf (x) ≤ cf + |x | for some cf ∈ R,

ρ`(λ(f (X1, . . . ,Xn)− E[f (X1, . . . ,Xn)])) ≤ γ(cλ), ∀λ ≥ 0.

Message: Liquidity risk of linear growth options controlled(uniformly) by liquidity risk of calls and puts.


Duals of (uniform) concentration inequalities

Section 3




Penalty functions

Theorem (Classical risk measure duality)

The following are equivalent for a risk measure ρ:

1. ρ has a penalty function, i.e. α : P∞ → [0,∞] such that

ρ(X ) = supQ∈P∞

(EQ [X ]− α(Q)

), ∀X ∈ L1,

where P∞ is the set of Q � P with dQ/dP ∈ L∞.2. ρ has the Fatou property: If Xn → X and ∃Y ∈ L1 with|Xn| ≤ Y , then ρ(X ) ≤ lim infn→∞ ρ(Xn).

The minimal penalty function is

α(Q) = supX∈L1

(EQ [X ]− ρ(X )).



Penalty function examples

Shortfall risk measures (Föllmer/Schied ’02)

If ` is a loss function, the minimal penalty of ρ` is

α`(Q) =

{inft>0

1t

{1 + EP

[`∗(t dQdP

)]}if Q � P

∞ otherwise.

Entropic risk measure

If `(x) = ex above so ρ(X ) = logE[eX ], then the minimal penaltyfunction is relative entropy,

Q 7→ H(Q|P) =

{EQ [log(dQ/dP)] if Q � P∞ otherwise.



Dual inequalities

Concentration inequalities have useful dual forms:

TheoremIf ρ has a penalty function α, then

ρ(λX ) ≤ γ(λ) for all λ ≥ 0,

if and only if

γ∗(EQ [X ]) ≤ α(Q) for all Q ∈ P∞.

Recall γ∗(t) = supλ≥0(λt − γ(λ)).



Derivation of the dual inequality

ρ(λX ) ≤ γ(λ), ∀λ ≥ 0,

if and only if

supQ∈P∞

(EQ [λX ]− α(Q)

)≤ γ(λ), ∀λ ≥ 0,




ρ(λX ) ≤ γ(λ), ∀λ ≥ 0,

if and only if

EQ [λX ]− α(Q) ≤ γ(λ), ∀λ ≥ 0, ∀Q ∈ P∞,

if and only if

λEQ [X ]− γ(λ) ≤ α(Q), ∀λ ≥ 0, ∀Q ∈ P∞,




ρ(λX ) ≤ γ(λ), ∀λ ≥ 0,

if and only if

EQ [λX ]− α(Q) ≤ γ(λ), ∀λ ≥ 0, ∀Q ∈ P∞,

if and only if

supλ≥0

(λEQ [X ]− γ(λ)

)≤ α(Q), ∀Q ∈ P∞,

if and only ifγ∗(EQ [X ]) ≤ α(Q), ∀Q ∈ P∞.



Uniform concentration inequalitiesA simple but important extension:

Corollary

If ρ has a penalty function α, and X ⊂ L1, then

ρ(λX ) ≤ γ(λ) for all λ ≥ 0, X ∈ X ,

if and only if

γ∗(

supX∈X

EQ [X ])≤ α(Q) for all Q ∈ P∞.

Example: If X = {X − EP [X ] : |X | ≤ 1 a.s.}, then

supX∈X

EQ [X ] = ‖Q − P‖TV .



Example: transport inequalitiesSuppose (Ω, d) is a metric space and

X ={f − EP [f ] : f : Ω→ R is 1-Lipschitz

}.

Kantorovich duality Wasserstein distance:

supX∈X

EQ [X ] = infM

∫Ω2

d(x , y)M(dx , dy) =: W1(Q,P),

where the inf is over couplings M of Q and P.

Corollary (see Bobkov/Götze ’99)

If ρ has a penalty function α, then

ρ(λ(f − EP f )) ≤ γ(λ) for all λ ≥ 0, f 1-Lipschitz,⇔ γ∗ (W1(Q,P)) ≤ α(Q) for all Q ∈ P∞.



Tail bounds revisited

ρ`(f ) = inf{m ∈ R : EP [`(f − c)] ≤ 1

},

α`(Q) = inft>0

1

t

{1 + EP

[`∗(tdQ

dP

)]}.

TheoremThen the following are equivalent up to a change in c :

1. ρ`(λ(f − EP f )) ≤ γ(λ) for all λ ≥ 0, f 1-Lipschitz.2. γ∗(W1(Q,P)) ≤ α`(Q) for all Q ∈ P∞.3. P(Ar ) ≥ 1− 1/`(γ∗(cr)) for all r > 0, A with P(A) ≥ 1/2.


Tensorization & time consistency

Section 4




Tensorization

Question #1

Suppose we know liquidity risk profiles of X and Y .What we say about the liquidity risk of X + Y ?

Question #2

Suppose we know liquidity risk profiles of f (X ) and g(Y ), for somecollection of functions f and g .What can we say about the liquidity risk of combinations of X andY , i.e. h(X ,Y ) for some h?

∃ nice answers for law invariant risk measures, related to timeconsistency!



Law invariance & time consistency

Assume: ρ(X ) = ρ(Y ) whenever Xd= Y .

Define ρ̃ as ρ on distributions, i.e. ρ̃(P ◦ X−1) = ρ(X ).

For σ-fields G ⊂ F and X ∈ L1, let

ρ(X |G)(ω) := ρ̃(P(X ∈ · | G)(ω)).

Say ρ is acceptance consistent (c.f. Weber ’06) if

ρ(ρ(X |G)) ≥ ρ(X ), ∀G, X .

Note: if (Ft)t≥0 is any filtration then

ρ(ρ(X |Ft)|Fs) ≥ ρ(X |Fs), for s < t.



Time consistency & tensorization

Proposition

If X and Y are independent, and if ρ is acceptance consistent, then

ρ(X + Y ) ≤ ρ(X ) + ρ(Y ).

Proof.Cash additivity and independence imply

ρ(X + Y |X ) = X + ρ(Y |X ) = X + ρ(Y ).

Acceptance consistency implies

ρ(X + Y ) ≤ ρ (ρ(X + Y |X ))= ρ (X + ρ(Y ))

= ρ(X ) + ρ(Y ).




Proposition

If X and Y are independent, and if ρ is acceptance consistent, then

ρ(X + Y ) ≤ ρ(X ) + ρ(Y ).Proof.Cash additivity and independence imply

ρ(X + Y |X ) = X + ρ(Y |X ) = X + ρ(Y ).

Acceptance consistency implies

ρ(X + Y ) ≤ ρ (ρ(X + Y |X ))= ρ (X + ρ(Y ))

= ρ(X ) + ρ(Y ).




Corollary

Suppose X and Y are independent and ρ is acceptance consistent.Then

ρ(λ(X + Y )) ≤ ρ(λX ) + ρ(λY ).

We could have just used convexity to get

ρ(λ(X + Y )) ≤ 12

(ρ(2λX ) + ρ(2λY )) ,

without extra assumptions...

But for more general combinations h(X ,Y ), we really needacceptance consistency!




Proposition

Suppose (Xi )ni=1 are i.i.d. and ρ is acceptance consistent. If

ρ(λ(f (X1)− E[f (X1)])) ≤ γ(λ), λ ≥ 0,

for all 1-Lipschitz functions f , then

ρ(λ(f (X1, . . . ,Xn)− E[f (X1, . . . ,Xn)])) ≤ nγ(λ), λ ≥ 0,

for each n and each 1-Lipschitz function f .

Message: Concentration for µ = Law(X1) plus acceptanceconsistency concentration for product measures µn.




How do we extend this to different classes of f = f (X1, . . . ,Xn)?

Classically1, tensorization proofs operate on dual inequalities,exploiting the chain rule for relative entropy:

H (ν(dx)K νx (dy) | µ(dx)Kµx (dy)) = H(ν|µ) +∫Eν(dx)H(K νx |Kµx ),

whenever ν(dx)K νx (dy) and µ(dx)Kµx (dy) are in P(E × F ) for

some standard Borel spaces E and F .

Is there a substitute for the chain rule in the risk measure setting?

1Marton ’96, Talagrand ’96, survey of Gozlan/Léonard ’10



Law invariance & information divergences

Assume: (Ω,F ,P) nonatomic, ρ(X ) = ρ(Y ) whenever X d= Y .

DefinitionFor a standard Borel space E and µ ∈ P(E ), define a risk measureρµ on L

∞(E , µ) byρµ(f ) = ρ(f (X )),

where X : Ω→ E has P ◦ X−1 = µ.

Let α(·|µ) be its minimalpenalty function,

α(ν|µ) = supf ∈L∞(E ,µ)

(∫Ef dν − ρµ(f )

).

Call α the divergence induced by ρ.



Law invariance & information divergences

Assume: (Ω,F ,P) nonatomic, ρ(X ) = ρ(Y ) whenever X d= Y .

DefinitionFor a standard Borel space E and µ ∈ P(E ), define a risk measureρµ on L

∞(E , µ) byρµ(f ) = ρ(f (X )),

where X : Ω→ E has P ◦ X−1 = µ. Let α(·|µ) be its minimalpenalty function,

α(ν|µ) = supf ∈L∞(E ,µ)

(∫Ef dν − ρµ(f )

).

Call α the divergence induced by ρ.



Chain rules

TheoremA law invariant ρ is acceptance consistent if and only if its induceddivergence α satisfies

α (ν(dx)K νx (dy) | µ(dx)Kµx (dy)) ≥ α(ν|µ) +∫Eν(dx)α(K νx |Kµx ),


some standard Borel spaces E and F .

(Same for rejectionconsistence, with inequality reversed.)

There is a third characterization in terms of acceptance sets.



Chain rules

TheoremA law invariant ρ is acceptance consistent if and only if its induceddivergence α satisfies

α (ν(dx)K νx (dy) | µ(dx)Kµx (dy)) ≥ α(ν|µ) +∫Eν(dx)α(K νx |Kµx ),


some standard Borel spaces E and F . (Same for rejectionconsistence, with inequality reversed.)

There is a third characterization in terms of acceptance sets.



Examples

Shortfall risk measureIf `(x + y) ≤ `(x)`(y), then

ρ(X ) = inf{m ∈ X : E[`(X −m)] ≤ 1}

is acceptance consistent.

Kupper & Schachermayer ’09: The only time consistent riskmeasures are entropic. ⇒ Relative entropy is the only divergencesatisfying the chain rule.


Large deviations

Section 5

Large deviations


Large deviations

Large deviations

Conservative boundsGiven Y ∈ L1, the conservative risk measure is

Cρ(Y ) = limn→∞

1

nρ(nY ).

This bounds the risk per unit of Y .

QuestionSay (Yn)

∞n=1 now depend on n but are roughly of same magnitude.

Consider the sequence of investments, of size n and compositionYn, given by nYn.

How can we control ρ(nYn), or1nρ(nYn)?


Large deviations

Large deviations

An answerLet (Xn)

∞n=1 be i.i.d with values in E and law µ, and let

Yn = F

(1

n

n∑i=1

δXi

), e.g. Yn = F̃

(1

n

n∑i=1

φ(Xi )

).

for some F ∈ Cb(P(E )).

Then

I If ρ is acceptance consistent:lim supn→∞

1nρ(nYn) ≤ supν∈P(E) (F (ν)− α(ν|µ)).

I If ρ is rejection consistent:lim infn→∞

1nρ(nYn) ≥ supν∈P(E) (F (ν)− α(ν|µ)).

When ρ(X ) = logE[eX ], this recovers Sanov’s theorem!


Large deviations

Large deviations

An answerLet (Xn)


Yn = F

(1

n

n∑i=1

δXi

), e.g. Yn = F̃

(1

n

n∑i=1

φ(Xi )

).

for some F ∈ Cb(P(E )). ThenI If ρ is acceptance consistent:

lim supn→∞1nρ(nYn) ≤ supν∈P(E) (F (ν)− α(ν|µ)).





Large deviations

Large deviations

An answerLet (Xn)


Yn = F

(1

n

n∑i=1

δXi

), e.g. Yn = F̃

(1

n

n∑i=1

φ(Xi )

).

for some F ∈ Cb(P(E )). ThenI If ρ is acceptance consistent:

lim supn→∞1nρ(nYn) ≤ supν∈P(E) (F (ν)− α(ν|µ)).




Liquidity risk & risk measuresTail bounds and integral criteria for shortfall concentration inequalitiesDuals of (uniform) concentration inequalitiesTensorization & time consistencyLarge deviations

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Liquidity, risk measures, and concentration of measure · 2015. 11. 18. · Liquidity, risk measures, and concentration of measure Liquidity risk & risk measures Literature on liquidity