Construcción de la válvula

The Contagion Index of a set A of nodes can be viewed as the expected Default Impact of the set Awhen drawing the market shocks from their conditional distri- bution given that the set A has defaulted, i.e. c(k) +k<0,∀k∈A:

CI(A, c, E) =_E[DI(A,(c+ ˜)+, E)] (3.15)

where _e is drawn from the distribution of conditional on the event c(k) +k <

0,∀k ∈A.

Thus, we can compute an estimator of the Contagion Index of the setAby Monte- Carlo method:

Forj = 1..N:

• draw samples ˜i(ωj), i= 1..n from the conditional distribution P(i ∈.|c(j) +

j <0,∀j ∈A).

• run the default cascades initiated by the market shock triggering the failure of the setAof nodes and compute the Default ImpactDI(A,(c+ ˜(ωj))+, E)

The Monte-Carlo estimator of the Contagion Index CIc is then given by: c CI(A, c, E) = 1 N N X j=1 DI(A,(c+ ˜(ωj))+, E) (3.16)

Example: one factor α-stable copula model

In the following sections, we consider a one factor α-stable copula model for the market shocks,

i =σiFα−1Gα ρ1/αS+ (1−ρ)1/αZi

(3.17)

where S, Z1, ..., Zn are independent and identically distributed random variables

with marginal cumulative density function Gα, and marginal probability density

function gα.

Fα is the marginal cumulative distribution of the 0is, which we choose to be the

conditional distribution of S given S <0 in order to generate exclusively negative market shocks,

Fα(x) =

1

Gα(0)

Gα(x)1x<0+ 1x>0. (3.18)

σi is a scaling factor that allows to calibrate the magnitude of the market shock on

institution i to its marginal default probabilitypi:

σi =−

c(i)

F−1

α (pi)

(3.19)

The loss incurred by the network is a function of the correlated market shocks

i, i = 1..n, so it is a function of the common factor S and the specific factors

requires the knowledge of the joint distribution ofS and Zi, i= 1..n conditional on

the event c(k) +k<0. Since the variablesS and Zi, i= 1..nare independent, the

Zi’s fori6=kare independent of the eventc(k)+k <0. In this case, the Contagion

Index of all nodes k = 1..n in the network can be calculated simultaneously using the same draws of idiosyncratic factors and just re-drawing the common factor S

conditional on the eventc(k) +k <0. Hence, we only need to know the conditional

distribution ofS given c(k) +k<0, that has a density function

fα(x|c(k) +k <0) = 1 pk Gα   G−_α1◦Fα −c_σ(k) k −ρ1/αx (1−ρ)1/α  gα(x) (3.20) Proof. P(S ≤s|c(i) +i <0) = 1 pi P S≤s, c(i) +σi Fα−1Gα ρ1/αS+ (1−ρ)1/αZi <0 = 1 pi P(S ≤s, Zi < G−_α1Fα −c_σ(i) i −ρ1/αS (1−ρ)1/α ) = 1 pi Z s −∞ Gα( G−_α1Fα −c_σ(i) i −ρ1/αu (1−ρ)1/α )gα(u)du

Hence, the conditional probability density function of S givenc(i) +zi <0 is:

fα(s|c(i) +i <0) = 1 pi Gα   G−1 α Fα −c_σ(i) i −ρ1/α_s (1−ρ)1/α  gα(s) (3.21)

The algorithm to estimate the Contagion Index of all nodesk= 1..nin the network is:

• draw Zi(ωj), i = 1..n a copy of the idiosyncratic factors from their α-stable

distributionGα.

• Fork = 1..n:

– drawS(ωj) from the conditional distributionP(S(ωj)∈.|c(k) +k(ωj)<

0).

– compute the market shocks for all nodes i in the network:

i(ωj) = Fα−1Gα ρ1/αS(ωj) + (1−ρ)1/αZi(ωj)

(3.22)

– run the default cascades initiated by the market shock triggering the failure of node k and compute the Default ImpactDI(k, c+(ωj), E) of

k with the stressed capital levels (c(i) +i(ωj))+, i= 1..n.

Both the gaussian and the cauchy copulae belong to the family ofα-stable copulae: the gaussian corresponds toα= 2 and the Cauchy to α= 1.

Unless otherwise specified, the results presented in the below sections correspond to a scale-free network, simulated according to the preferential atachment proce- dure presented in chapter 2, with 400 nodes, average degree 10, in-exponent 2 and out-exponent 3. Exposures are iid and follow a Pareto distribution with parameter

α = 1.9. Market shocks are simulated from a Cauchy copula model with depen- dence parameter ρ = 10%. We favor the Cauchy copula to the commonly used Gaussian copula in order to generate scenarios with clusters of shocks with large magnitude, that is possible only in the presence of tail dependence as in the Cauchy copula (Embrechts et al., 2001). The nodes are ranked in descending order of their interbank liability. The first 12% are assigned a default probability of 6 basis points, the second 13% are assigned a default probability of 33 basis points and the re- maining ones are assigned a default probability of 79 basis points. In other words,

we assume that institutions are willing to lend money to other institutions with a good credit rating, that is institutions with a small default probability. Thus, an institution with a high interbank liability should be assigned a small default prob- ability. The Contagion Index is computed according to the simulation procedure described in section 3.2.4 with 1000 independent draws of the correlated market shocks.