Liquidity Risk Es-ma-on in Condi-onal Vola-lity Models Darolles S. 1,3, Francq C. 1,2, Le Fol G. 1,3, Zakoian J.M. 1,2 1 CREST, Laboratoire de Finance Assurance 2 Université Lille 3 3 Université Paris-‐Dauphine We are grateful to the Agence NaGonale de la Recherche (ANR), which supported this work via the Project ECONOM\&RISK (ANR 2010 blanc 1804 03).
Mo-va-on In risk management,
liquidity o?en associated with simple transac-on costs
Explains why liquidity adjusted risk measures are of the form market risk measure + liquidity term where
market risk measure : obtained from historical market prices liquidity term : obtained from bid-‐ask spreads data
Mo-va-on But Market risk measures obtained from historical market prices already include a liquidity component
Why ? Liquidity has a direct impact on historical price variaKons
Our objec-ve in this paper is to extract this liquidity component from global risk measures computed from historical prices:
global risk measure = market risk measure + liquidity term
where global risk measure : obtained from historical market prices market risk measure : obtained from historical market prices
Main contribu-on Our liquidity risk measure: -‐ is defined as a intrinsec characteris-c of a given asset and allows simple liquidity rankings
-‐ takes into account the dynamic properKes of prices (-me varying market risk) -‐ can be defined for different condi&onal risk measures (VaR, Expected ShorUall, …) -‐ can be computed when only historical market prices are available
Agenda
1. Global risk and global risk-‐parameter 2. AddiKve decomposiKon of global risk 3. Inference 4. Empirical ApplicaKons
1. Global risk and Global risk-‐parameter st 1 STEP VolaKlity modeling GARCH(1,1) governs the returns process
ε t = σ t (θ 0 )ηt , ηt i.i.d . Eηt2 = 1 σ t2 (θ 0 ) = ω0 + a0ε t2−1 + b0σ t2−1
ʹ′ • θ 0 = ( ω 0 , a 0 , b 0 ) is a vola&lity-‐parameter
• captures the volaKlity persistence in asset returns • this model can be easily generalized to more complex condiKonal volaKlity models
1. Global risk and Global risk-‐parameter nd 2 STEP Global risk measure The condiGonal VaR of ε at level α is
t
Pt −1 [ε t < −VaRtG (α )] = α With the previous specificaKon of εt, the global risk VaRtG (α ) = −σ t (θ0 ) Fη−1 (α ) depends on
• the dynamics of the GARCH process through σt (θ0) • the (constant) lower tail of the innovaKon process
1. Global risk and Global risk-‐parameter rd 3 STEP Global risk-‐parameter (Francq, Zakoian (2012)) A0 (scale stability) There exists a funcGon H such that for any θ, for any K > 0, and any sequence (xi)
Kσ(x1, x2, … ; θ )= σ(x1, x2, …;θ*); where θ* = H(θ;K)
We can then concentrate in a single global risk-‐parameter θ0,α the 2 dimensions of risk θ0G,α = H (θ0 ,−Fη−1 (α ))
and obtain the global risk as VaRtG (α ) = σ t (θ0G,α )
1. Global risk and Global risk-‐parameter rd 3 STEP Global risk-‐parameter (Francq, Zakoian (2012)) In our GARCH(1,1) example … θ 0G,α = (K 2ω0 , K 2 a0 , b0 ) K = − Fη−1 (α ) … but A0 is also saKsfied for more complex GARCH specificaKon (power-‐transformed asymmetric GARCH model)
2. Addi-ve decomposi-on of global risk We need the following assumpKon to idenKfy both global and market risks from returns
A1 (Iden-fica-on assump-on) For an infinitely liquid asset, the innovaGons of the GARCH(1,1) process are Gaussian
We define the market risk-‐parameter as θ0M,α = H (θ0 ,−Φ−1 (α )) and the corresponding market risk is
( )
VaRtM (α ) = σ t θ0M,α
2. Addi-ve decomposi-on of global risk Interpreta-on of A1 (Iden-fica-on assump-on) Usual way used to include liquidity shocks (see Duffie, Pan (1997), An Overview of value at risk) ε t = σ t (θ0 )ηt + Jump, ηt i.i.d . Gaussian
2. Addi-ve decomposi-on of global risk Interpreta-on of A1 (Iden-fica-on assump-on) Usual way used to include liquidity shocks (see Duffie, Pan (1997), An Overview of value at risk) ε t = σ t (θ0 )ηt + Jump, ηt i.i.d . Gaussian One step further (see Meddahi, Mykland (2012), Fat Tails or Many Small Jumps ?) ε t = σ t (θ0 ) χt + Jump, χt i.i.d . Student
2. Addi-ve decomposi-on of global risk Interpreta-on of A1 (Iden-fica-on assump-on) Usual way used to include liquidity shocks (see Duffie, Pan (1997), An Overview of value at risk) ε t = σ t (θ0 )ηt + Jump, ηt i.i.d . Gaussian One step further (see Meddahi, Mykland (2012), Fat Tails or Many Small Jumps ?) ε t = σ t (θ0 ) χt + Jump, χt i.i.d . Student Our approach ε t = σ t (θ 0 ) χt = σ t (θ 0 )ηt + σ t (θ0 )(−ηt + χt )
2. Addi-ve decomposi-on of global risk Interpreta-on of A1 (Iden-fica-on assump-on) Usual way used to include liquidity shocks (see Duffie, Pan (1997), An Overview of value at risk) ε t = σ t (θ0 )ηt + Jump, ηt i.i.d . Gaussian One step further (see Meddahi, Mykland (2012), Fat Tails or Many Small Jumps ?) ε t = σ t (θ0 ) χt + Jump, χt i.i.d . Student Our approach « LIQUIDITY » ε t = σ t (θ 0 ) χt = σ t (θ 0 )ηt + σ t (θ0 )(−ηt + χt )
2. Addi-ve decomposi-on of global risk −1 −1 A2 ( Consistency a ssump-on) α ) for a sufficient ( ) 0 > Φ α > F η ( small α
Defini-on The liquidity risk-‐parameter is (for a small enough) θ L = H (θ ,− F −1 (α ) + Φ −1 (α ))
0,α
0
η
and the corresponding liquidity risk is
( )
VaRtL (α ) = σ t θ0L,α
Proposi-on Under A0-‐A2, VaRtG (α ) = VaRtM (α ) + VaRtL (α )
3. Inference Two-‐step approach
st 1 STEP θ ˆ n : Gaussian QML esKmator of θ0 (does not require to know the distribuKon of ηt)
2nd STEP ξ n ,α : nonparametric esKmator of the innovaKon quanKle funcKon ξ α , obtained from ηˆt = ε t σ t (θˆn ) Final STEP
(
θˆnG,α = H θˆn ,−ξ n,α
)
(
θˆnM,α = H θˆn ,−Φ −1
)
(
θˆnG,α = H θˆn ,−ξ n,α + Φ −1
)
3. Inference AsymptoKc distribuKon follows from the joint distribuKon of θˆ, ξ n ,α In the general case of power-‐transformed asymmetric GARCH model
(
)
ε t = σ t (θ 0 )ηt , ηt i.i.d . Eηt2 = 1 q
σ t (θ 0 ) = ω0 + ∑ α 0i + (ε δ
i =1
x + = max(x,0), x − = min( x,0)
)
+ δ t −i
(
+ α 0i − − ε
p
) +∑β
− δ t −i
j =1
δ σ 0 j t− j
3. Inference We use the following technical assumpKons (same as those required for the Gaussian QML)
3. Inference ∂σ t (θ ) 1 1 ∂σ tδ (θ ) Let Dt (θ ) = = σ t (θ ) ∂θ δ σ tδ (θ ) ∂θ 1
If η1 admits a conKnuous and strictly posiKve density f in a neigborhood of ξ α , we have ⎛ κ 4 − 1 −1 ⎞ ⎛ n θˆn − θ 0 ⎞ J λ δ θ ⎜ α 0 ⎟ ⎜ ⎟
(
⎜ n (ξ n ,α ⎝
)
→ N (0, Σα ) Σα = ⎜ 4 ⎜ λ δθ ' − ξα )⎟⎠ ⎝ α 0
where pα κ 4 −1 2 ( λ = ξ + , p = E η α α α 1 1{η
