text: In order to not express any speciﬁc preference about the form of the Gaussian components through the introduction of priors, we use uninformative priors by setting the values of hyperparameters m , β , a and b to appropriate values as shown 0 0 0 0 in Section . Having deﬁned the parametric form of observed data, latent variables and parameters distributions, our goal is to approx- | { } imate the posterior distribution p(Y X) and the model evidence p(X), where Y = Z, (cid:36), µ, τ is the set with distribution | q(Y ), which contains all model latent variables and parameters. Based on the Bayes rule, which states that p(X)p(Y X) = p(X, Y ) the logarithm of distribution p(X) can be expressed as (cid:90) (cid:90) p(X, Y ) − ln p(X) = q(Y ) ln dY (10a) q(Y ) | − p(Y X) q(Y ) ln dY q(Y ) L || = (q) + KL(q p) (10b) || | L where KL(q p) is the Kullback-Leibler divergence between q(Y ) and p(Y X) distributions and (q) is the lower bound of || | ln p(X). Since KL(q p) is a non negative quantity, equals to zero only if q(Y ) is equal to p(Y X), maximization of ln p(X) || || is equivalent to minimizing of KL(q p). For minimizing KL(q p) and estimating p(X) we exploit the EM algorithm, as shown in Section . (cid:81) By making the assumption, based on variational inference, that the distribution q(Y ) can be factorized over M disjoint sets ∗ || such as q(Y ) = M q (Y ), as shown in [1], the optimal solution q (Y ) corresponds to the minimization of KL(q p) is i=1 i i j j given by ∗ E C ln q (Y ) = (cid:54) [ln p(X, Y )] + (11) j j i=j E (cid:54) C where (cid:54) [ln p(X, Y )] is the expectation of the joint distribution over all variables Y for j = i and is a constant. i=j j P(X, Y ) is modeled through (12). ∗ In the following, we present the analytical solution for the optimal distributions q (Y ) for model parameters and latent ∗ ∗ ∗ ∗ | j j variables, i.e. the optimal distributions q (Z), q ((cid:36)), q (τ) and q (µ τ). 3. OPTIMAL MODEL PARAMETER DISTRIBUTIONS According to (5), (6), (8) and (9), the joint distribution of all random variables can be factorized as follows | | p(X, Z, (cid:36), µ, τ) =p(X Z, µ, τ)p(Z (cid:36)) (12) | p((cid:36))p(µ τ)p(τ) X corresponds to the set of the observed variables. All proofs are given in Appendix . ∗ 3.1. Optimal q (Z) distribution ∗ Using (11) and the factorized form of (12) the distribution of the optimized factor q (Z) is given by a Multinomial distri- bution of the form ##LTLine## ##LTLine##

text: In

text: order

text: to

text: not

text: express

text: any

text: speciﬁc

text: preference

text: about

text: the

text: form

text: of

text: the

text: Gaussian

text: components

text: through

text: the

text: introduction

text: of

text: priors,

text: we

text: use

text: uninformative

text: priors

text: by

text: setting

text: the

text: values

text: of

text: hyperparameters

text: m

text: β

text: a

text: and

text: b

text: to

text: appropriate

text: values

text: as

text: shown

text: in

text: Section

text: Having

text: deﬁned

text: the

text: parametric

text: form

text: of

text: observed

text: data,

text: latent

text: variables

text: and

text: parameters

text: distributions,

text: our

text: goal

text: is

text: to

text: approx-

text: imate

text: the

text: posterior

text: distribution

text: p(Y

text: X)

text: and

text: the

text: model

text: evidence

text: p(X),

text: where

text: Y

text: Z,

text: µ,

text: τ

text: is

text: the

text: set

text: with

text: distribution

text: q(Y

text: which

text: contains

text: all

text: model

text: latent

text: variables

text: and

text: parameters.

text: Based

text: on

text: the

text: Bayes

text: rule,

text: which

text: states

text: that

text: p(X)p(Y

text: X)

text: p(X,

text: Y

text: the

text: logarithm

text: of

text: distribution

text: p(X)

text: can

text: be

text: expressed

text: as

text: p(X,

text: Y

text: ln

text: p(X)

text: q(Y

text: ln

text: dY

text: q(Y

text: p(Y

text: X)

text: q(Y

text: ln

text: dY

text: q(Y

text: L

text: (q)

text: KL(q

text: p)

text: L

text: where

text: KL(q

text: p)

text: is

text: the

text: Kullback-Leibler

text: divergence

text: between

text: q(Y

text: and

text: p(Y

text: X)

text: distributions

text: and

text: (q)

text: is

text: the

text: lower

text: bound

text: of

text: ln

text: p(X).

text: Since

text: KL(q

text: p)

text: is

text: a

text: non

text: negative

text: quantity,

text: equals

text: to

text: zero

text: only

text: if

text: q(Y

text: is

text: equal

text: to

text: p(Y

text: X),

text: maximization

text: of

text: ln

text: p(X)

text: is

text: equivalent

text: to

text: minimizing

text: of

text: KL(q

text: p).

text: For

text: minimizing

text: KL(q

text: p)

text: and

text: estimating

text: p(X)

text: we

text: exploit

text: the

text: EM

text: algorithm,

text: as

text: shown

text: in

text: Section

text: By

text: making

text: the

text: assumption,

text: based

text: on

text: variational

text: inference,

text: that

text: the

text: distribution

text: q(Y

text: can

text: be

text: factorized

text: over

text: M

text: disjoint

text: sets

text: such

text: as

text: q(Y

text: M

text: q

text: (Y

text: as

text: shown

text: in

text: the

text: optimal

text: solution

text: q

text: (Y

text: corresponds

text: to

text: the

text: minimization

text: of

text: KL(q

text: p)

text: is

text: i

text: i

text: j

text: j

text: given

text: by

text: E

text: C

text: ln

text: q

text: (Y

text: [ln

text: p(X,

text: Y

text: j

text: j

text: i=j

text: E

text: C

text: where

text: [ln

text: p(X,

text: Y

text: is

text: the

text: expectation

text: of

text: the

text: joint

text: distribution

text: over

text: all

text: variables

text: Y

text: for

text: j

text: i

text: and

text: is

text: a

text: constant.

text: i=j

text: j

text: P(X,

text: Y

text: is

text: modeled

text: through

text: In

text: the

text: following,

text: we

text: present

text: the

text: analytical

text: solution

text: for

text: the

text: optimal

text: distributions

text: q

text: (Y

text: for

text: model

text: parameters

text: and

text: latent

text: j

text: j

text: variables,

text: i.e.

text: the

text: optimal

text: distributions

text: q

text: (Z),

text: q

text: q

text: (τ)

text: and

text: q

text: (µ

text: τ).

text: OPTIMAL

text: MODEL

text: PARAMETER

text: DISTRIBUTIONS

text: According

text: to

text: and

text: the

text: joint

text: distribution

text: of

text: all

text: random

text: variables

text: can

text: be

text: factorized

text: as

text: follows

text: p(X,

text: Z,

text: µ,

text: τ)

text: =p(X

text: Z,

text: µ,

text: τ)p(Z

text: τ)p(τ)

text: X

text: corresponds

text: to

text: the

text: set

text: of

text: the

text: observed

text: variables.

text: All

text: proofs

text: are

text: given

text: in

text: Appendix

text: Optimal

text: q

text: (Z)

text: distribution

text: Using

text: and

text: the

text: factorized

text: form

text: of

text: the

text: distribution

text: of

text: the

text: optimized

text: factor

text: q

text: (Z)

text: is

text: given

text: by

text: a

text: Multinomial

text: distri-

text: bution

text: of

text: the

text: form

text: ##LTLine##

text: ##LTLine##


text: Fig. 2. First row: updating of our model to new observed data. Second row: updating of model presented in [29].  as   1, if k = c o = (26) k 0, otherwise where we recall that c represents the index of the closest component and k is the index of k-th component. When the new observed sample is successfully modeled, the parameters for the Gaussian components are updated using the following the leader [6] approach described as (cid:0) (cid:1) ← 1(cid:0) − (cid:1) (cid:36) (cid:36) + o (cid:36) (27a) k k N k − k ← (cid:18)x µ (cid:19) µ µ + o new k (27b) k k k (cid:36) N + 1 k − ← (cid:36) N(x µ )2 − σ2 σ2 σ2 + o k new k k (27c) k k k ((cid:36) N + 1)2 (cid:36) N + 1 k k − where σ2 is equal to τ 1. k k When the new observed sample cannot be modeled by the existing components, a new component is created with mixing coefﬁcient (cid:36) , mean value µ and standard deviation σ , the parameters of which are given as new new new − 1 (2(cid:15))2 1 (cid:36) = , µ = x , σ2 = (28) new N new new new 12 From (28), we see that the mixing coefﬁcient for the new component is equal to 1/N since it models only one sample (the new observed one), its mean value equals the value of the new sample and its variance the variance the Uniform distribution, − whose the lower and upper bounds are x (cid:15) and x +(cid:15) respectively. When a new component is created the values for the new new − parameters for all the other components remain unchanged except mixing coefﬁcients (cid:36), which are normalized to sum N 1. N After each adaptation of the system to new observed samples, either they modeled by the trained model or not, the mixing coefﬁcients of the components are normalized to sum to one. ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Fig.

label: paragraph

text: First

label: caption

text: row:

text: updating

text: of

text: our

text: model

text: to

text: new

text: observed

text: data.

text: Second

text: row:

text: updating

text: of

text: model

text: presented

text: in

label: equation

text: as

text: if

text: k

text: c

text: o

text: k

text: otherwise

text: where

text: we

text: recall

text: that

text: c

text: represents

text: the

text: index

text: of

text: the

text: closest

text: component

text: and

text: k

text: is

text: the

text: index

text: of

text: k-th

text: component.

text: When

text: the

text: new

text: observed

text: sample

text: is

text: successfully

text: modeled,

text: the

text: parameters

text: for

text: the

text: Gaussian

text: components

text: are

text: updated

text: using

text: the

text: following

text: the

text: leader

text: approach

text: described

text: as

text: o

text: k

text: k

text: N

text: k

text: k

text: µ

text: µ

text: µ

text: o

text: new

text: k

text: k

text: k

text: k

text: N

text: k

text: N(x

text: µ

text: σ2

text: σ2

text: σ2

text: o

text: k

text: new

text: k

text: k

text: k

text: k

text: k

text: N

text: N

text: k

text: k

text: where

text: σ2

text: is

text: equal

text: to

text: τ

text: k

text: k

text: When

text: the

text: new

text: observed

text: sample

text: cannot

text: be

text: modeled

text: by

text: the

text: existing

text: components,

text: a

text: new

text: component

text: is

text: created

text: with

text: mixing

text: coefﬁcient

text: mean

text: value

text: µ

text: and

text: standard

text: deviation

text: σ

text: the

text: parameters

text: of

text: which

text: are

text: given

text: as

text: new

text: new

text: new

text: µ

text: x

text: σ2

text: new

text: N

text: new

text: new

text: new

text: From

text: we

text: see

text: that

text: the

text: mixing

text: coefﬁcient

text: for

text: the

text: new

text: component

text: is

text: equal

text: to

text: since

text: it

text: models

text: only

text: one

text: sample

text: (the

text: new

text: observed

text: one),

text: its

text: mean

text: value

text: equals

text: the

text: value

text: of

text: the

text: new

text: sample

text: and

text: its

text: variance

text: the

text: variance

text: the

text: Uniform

text: distribution,

text: whose

text: the

text: lower

text: and

text: upper

text: bounds

text: are

text: x

text: and

text: x

text: respectively.

text: When

text: a

text: new

text: component

text: is

text: created

text: the

text: values

text: for

text: the

text: new

text: new

text: parameters

text: for

text: all

text: the

text: other

text: components

text: remain

text: unchanged

text: except

text: mixing

text: coefﬁcients

text: which

text: are

text: normalized

text: to

text: sum

text: N

text: N

text: After

text: each

text: adaptation

text: of

text: the

text: system

text: to

text: new

text: observed

text: samples,

text: either

text: they

text: modeled

text: by

text: the

text: trained

text: model

text: or

text: not,

text: the

text: mixing

text: coefﬁcients

text: of

text: the

text: components

text: are

text: normalized

text: to

text: sum

text: to

text: one.

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ) 301( R )mn( L  5 300 6.75 The inset to Fig. 6 shows the resistance versus WAL (HLN model) C2 200 magnetic ﬁeld H normal to the wafer at 1.88 K, for C4 the same bilayer and reference ﬁlm contacts. All the 6.50 100 resistances are increasing with ﬁeld, but while the bilayer 0 resistance tends to saturate at high ﬁelds, that of the 6.25 0 20 40 60 80 100 T (K) reference ﬁlm increases quite linearly with H. The later Pristine 10nm Bi Se on FS indicates ﬂux-ﬂow resistance in a superconductor as the 6.00 2 3 C8 -100Vg 1T C8 -100Vg 0T mechanism for the linearly increasing resistance. If we 0 10 20 30 40 50 go back now to the inset of Fig. 5, we see that this T (K) eﬀect is stronger in the reference NbN ﬁlm than in the ∼ FIG. 7: (Color online) Resistance versus temperature of a bilayer. The peak MR at 1.9 K in all contacts is due 10 nm thick Bi Se reference ﬁlm on FS with and without 2 3 a magnetic ﬁeld of 1T. The inset shows the phase coherence to maximum ﬂux ﬂow which results from compensation length L calculated for another 10 nm Bi Se ﬁlm assuming φ 2 3 between increased vortex generation and pinning eﬀect weak anti-localization (WAL) and using the HLN model [24]. with decreasing temperature. Another interesting result in the inset to Fig. 5, is that in the C2 BL contact ∼ T data of Fig. 7 was used in order to calculate the there is a clear knee in the MR data at T 2.1 K which magnetoresistance of this ﬁlm and the result has already indicates two diﬀerent MR behaviors below and above been presented in the main panel of Fig. 6. We found it. We attribute the MR above this temperature to that MR versus T curves measured on the Bi Se ﬁlm the stronger superconducting NbN islands as seen in 2∓ 3 were insensitive to the gate voltage used here ( 100 Vg Fig. 1, while below it, where the weak-links (the darker ± or 2 MV/cm) to within the noise of the measurements. areas in Fig. 1) also become superconducting due to Steinberg et al. [22] did observe signiﬁcant gating eﬀects proximity coupling via the Bi Se , the MR increases 2 3 in a 20 nm thick Bi Se ﬁlm, but their maximum further on lowering the temperature. In the following we 2 3 electric ﬁeld was about 40 times higher than what we shall see more supporting evidence for this interpretation. used here. Thus, the presently used maximum gate 3.2. R and MR of reference Bi Se ﬁlms voltage was insuﬃcient to have any signiﬁcant eﬀect on 2 3 the MR of the bare 10 nm Bi Se ﬁlm. Fig. 8 shows the 2 3 resistance versus magnetic ﬁeld of this ﬁlm at 1.85 K. For understanding the MR of the bilayers one needs The inset shows the data obtained on the pristine ﬁlm, in addition to the MR of the 3 nm NbN reference ﬁlm, while the main panel shows the data of a one week old also the MR behavior of a second reference ﬁlm of 10 nm ﬁlm which was kept under dry air. Clear aging eﬀect was Bi Se . Such ﬁlms were prepared and Fig. 7 presents observed when the measurements were repeated after 2 3 resistance versus temperature results of a 10 nm Bi Se one week, as can be seen from the increased resistance 2 3 ﬁlm under 0 and 1 T magnetic ﬁeld and -100 V gate of about 25%. Besides this, both sets of measurements ∼ voltage. The resistivity of this ﬁlm is 6 mΩcm or 6kΩ show a linear R versus H behavior at low ﬁelds up to per square, which corresponds to an electron density about 0.1 T with saturation at higher ﬁelds. − of about 1017 cm 3 [21]. Thus the electron doping of this ﬁlm is quite low, which results from the Se rich The linear MR in Bi Se is generally attributed to 2 3 target used in its deposition process [16]. The R versus weak anti-localization (WAL) [22, 23]. If we assume that ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

font: default

text: R

text: )mn(

text: L

text: The

text: inset

text: to

text: Fig.

text: shows

text: the

text: resistance

text: versus

text: WAL

text: (HLN

text: model)

text: magnetic

text: ﬁeld

text: H

text: normal

text: to

text: the

text: wafer

text: at

text: K,

text: for

text: the

text: same

text: bilayer

text: and

text: reference

text: ﬁlm

text: contacts.

text: All

text: the

text: resistances

text: are

text: increasing

text: with

text: ﬁeld,

text: but

text: while

text: the

text: bilayer

text: resistance

text: tends

text: to

text: saturate

text: at

text: high

text: ﬁelds,

text: that

text: of

text: the

text: T

text: (K)

text: reference

text: ﬁlm

text: increases

text: quite

text: linearly

text: with

text: H.

text: The

text: later

text: Pristine

text: Bi

text: Se

text: on

text: FS

text: indicates

text: ﬂux-ﬂow

text: resistance

text: in

text: a

text: superconductor

text: as

text: the

text: mechanism

text: for

text: the

text: linearly

text: increasing

text: resistance.

text: If

text: we

text: go

text: back

text: now

text: to

text: the

text: inset

text: of

text: Fig.

text: we

text: see

text: that

text: this

text: T

text: (K)

text: eﬀect

text: is

text: stronger

text: in

text: the

text: reference

text: NbN

text: ﬁlm

text: than

text: in

text: the

text: FIG.

text: (Color

text: online)

text: Resistance

text: versus

text: temperature

text: of

text: a

text: bilayer.

text: The

text: peak

text: MR

text: at

text: K

text: in

text: all

text: contacts

text: is

text: due

text: nm

text: thick

text: Bi

text: Se

text: reference

text: ﬁlm

text: on

text: FS

text: with

text: and

text: without

text: a

text: magnetic

text: ﬁeld

text: of

text: The

text: inset

text: shows

text: the

text: phase

text: coherence

text: to

text: maximum

text: ﬂux

text: ﬂow

text: which

text: results

text: from

text: compensation

text: length

text: L

text: calculated

text: for

text: another

text: nm

text: Bi

text: Se

text: ﬁlm

text: assuming

text: φ

text: between

text: increased

text: vortex

text: generation

text: and

text: pinning

text: eﬀect

text: weak

text: anti-localization

text: (WAL)

text: and

text: using

text: the

text: HLN

text: model

text: with

text: decreasing

text: temperature.

text: Another

text: interesting

text: result

text: in

text: the

text: inset

text: to

text: Fig.

text: is

text: that

text: in

text: the

text: BL

text: contact

text: T

text: data

text: of

text: Fig.

text: was

text: used

text: in

text: order

text: to

text: calculate

text: the

text: there

text: is

text: a

text: clear

text: knee

text: in

text: the

text: MR

text: data

text: at

text: T

text: K

text: which

text: magnetoresistance

text: of

text: this

text: ﬁlm

text: and

text: the

text: result

text: has

text: already

text: indicates

text: two

text: diﬀerent

text: MR

text: behaviors

text: below

text: and

text: above

text: been

text: presented

text: in

text: the

text: main

text: panel

text: of

text: Fig.

text: We

text: found

text: it.

text: We

text: attribute

text: the

text: MR

text: above

text: this

text: temperature

text: to

text: that

text: MR

text: versus

text: T

text: curves

text: measured

text: on

text: the

text: Bi

text: Se

text: ﬁlm

text: the

text: stronger

text: superconducting

text: NbN

text: islands

text: as

text: seen

text: in

text: 2∓

text: were

text: insensitive

text: to

text: the

text: gate

text: voltage

text: used

text: here

text: Vg

text: Fig.

text: while

text: below

text: it,

text: where

text: the

text: weak-links

text: (the

text: darker

text: or

text: MV/cm)

text: to

text: within

text: the

text: noise

text: of

text: the

text: measurements.

text: areas

text: in

text: Fig.

text: also

text: become

text: superconducting

text: due

text: to

text: Steinberg

text: et

text: al.

text: did

text: observe

text: signiﬁcant

text: gating

text: eﬀects

text: proximity

text: coupling

text: via

text: the

text: Bi

text: Se

text: the

text: MR

text: increases

text: in

text: a

text: nm

text: thick

text: Bi

text: Se

text: ﬁlm,

text: but

text: their

text: maximum

text: further

text: on

text: lowering

text: the

text: temperature.

text: In

text: the

text: following

text: we

text: electric

text: ﬁeld

text: was

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text: used

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text: Thus,

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text: 5 (cid:0) (cid:1) i ∗ − ρ˙ = Γ ρ + Γ ρ + Ω ρ Ω ρ 11 2 22 3 33 2 p 12 p 21 i ∗ − (cid:0) (cid:1) + (Ω ρ Ω ρ ) ; 2 c1 13 c1 31 − i − ∗ ρ˙ = Γ ρ + Γ ρ + Ω ρ Ω ρ 22 2 22 4 44 2 p 21 p 12 i ∗ − + (Ω ρ Ω ρ ) ; 2 c2 24 c2 42 − i − ∗ ρ˙ = Γ ρ + (Ω ρ Ω ρ ) ; 33 3 33 2 c1 31 c1 13 (cid:18) (cid:19) − i − ∗ ρ˙ = Γ ρ + (Ω ρ Ω ρ ) ; 44 4 44 2 c2 42 c2 24 −Γ i − ρ˙ = 2 + i∆ ρ + Ω (ρ ρ ) 12 2 p 12 2 p 11 22 (cid:18) (cid:19) − i − ∗ (Ω ρ Ω ρ ) ; 2 c1 32 c2 14 −Γ i − ρ˙ = 3 + i∆ ρ + Ω (ρ ρ ) 13 2 c1 13 2 c1 11 33 (cid:18) (cid:19) − i Ω ρ ; 2 p 23 −Γ ρ˙ = 4 + i(∆ + ∆ ) ρ 14 2 p c2 14 (cid:18) (cid:19) i − − + (Ω ρ Ω ρ Ω ρ ) ; 2 c2 12 p 24 c1 34 ρ˙ = −Γ(cid:0)2 + Γ3 + i(∆ − ∆ ) ρ(cid:1) 23 2 c1 p 23 (cid:18) (cid:19) i − ∗ − + Ω ρ Ω ρ Ω ρ ; 2 c1 21 p 13 c2 43 −Γ + Γ i − ρ˙ = 2 4 + i∆ ρ + Ω (ρ ρ ) 24 2 c2 24 2 c2 22 44 (cid:18) (cid:19) − i ∗ Ω ρ ; 2 p 14 −Γ + Γ − ρ˙ = 3 4 + i(∆ + ∆ ∆ ) ρ 34 2 p c2 c1 34 i − ∗ + (Ω ρ Ω ρ ) ; 2 c2 32 c1 14 FIG. 4. (Color online) Eﬀect of Doppler averaging on probe The positive decay terms in ρ˙ii (i = 1, 2, 3, 4) equations dispersion at room temperature. (a) Ω = 0, Ω = 6 Γ ; are due to the fact that there is an increase in population c1 c2 2 (b) Ω = 6 Γ , Ω = 0; (c) Ω = 6 Γ , Ω = 4 Γ ; (d) Ω = of lower levels due to the decay of atoms from higher c1 2 c2 c1 2 c2 2 c1 6 Γ2, Ωc2 = 6 Γ2; (e) Ωc1 = 6 Γ2, Ωc2 = 8 Γ2; (f) Ωc1 = 6 Γ2, levels. Ω = 12 Γ . Solid curves are for all atoms, dashed curves are c2 2 for atoms moving with positive velocities, and dotted curves REFERENCES are for atoms moving with negative velocities. ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: i

text: ρ˙

text: Γ

text: ρ

text: Γ

text: ρ

text: Ω

text: ρ

text: Ω

text: ρ

text: p

text: p

text: i

text: (Ω

text: ρ

text: Ω

text: ρ

text: i

text: ρ˙

text: Γ

text: ρ

text: Γ

text: ρ

text: Ω

text: ρ

text: Ω

text: ρ

text: p

text: p

text: i

text: (Ω

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text: Ω

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text: ρ˙

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text: Ω

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text: ρ˙

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text: −Γ

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text: Ω

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text: ρ

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text: ρ

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text: ρ

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text: Ω

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text: i(∆

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text: −Γ

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text: i(∆

text: ρ

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text: i

text: (Ω

text: ρ

text: Ω

text: ρ

text: FIG.

text: (Color

text: online)

text: Eﬀect

text: of

text: Doppler

text: averaging

text: on

text: probe

text: The

text: positive

text: decay

text: terms

text: in

text: ρ˙ii

text: (i

text: equations

text: dispersion

text: at

text: room

text: temperature.

text: (a)

text: Ω

text: Ω

text: Γ

text: are

text: due

text: to

text: the

text: fact

text: that

text: there

text: is

text: an

text: increase

text: in

text: population

text: (b)

text: Ω

text: Γ

text: Ω

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text: Ω

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text: (d)

text: Ω

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text: decay

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text: (e)

text: (f)

text: levels.

text: Ω

text: Γ

text: Solid

text: curves

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text: for

text: all

text: atoms,

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text: with

text: positive

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text: and

text: dotted

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text: REFERENCES

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text: terminology of the present paper), including eﬀects such as wrong way risk (meant as pro- cyclicality of the margins), credit dependence between members and left tailed distributions of their P&Ls. Ghamami (2015) proposes a static one-period model where a CCVA can be priced by Monte-Carlo. Brigo and Pallavicini (2014) extend the bilateral counterparty risk dynamic setup of their previous papers to centrally cleared trading. However, they ignore the default fund and the credit risk dependence issues that are inherent to the position of a clearing member. 1.2 Contributions and Outline This paper develops an XVA (costs) analysis of centrally cleared trading, parallel to the one that has been developed in the last years for bilateral transactions (see e.g. Cr´epey, Bielecki, and Brigo (2014, Parts II and III) or Brigo, Morini, and Pallavicini (2013)). A dynamic framework incorporates the sequence of cash ﬂows involved in the waterfall of resources of the clearing house. As compared with Arnsdorf (2012) and Ghamami (2015), our CCVA accounts not only for the central clearing analog of the CVA, which is the cost for a member of its losses on the default fund in case of other members’ defaults, but also for the cost of funding its margins (MVA) and for the cost of the capital (KVA) that is implicitly required from members through their default fund contributions (and for completeness and reference we also compute a DVA term). The framework of this paper can be used by a clearing house to ﬁnd the right bal- ance between initial margins and default fund in order to minimize the CCVA (subject to the regulatory constraints), hence optimize its costs to the members for a given level of resilience. A clearing house can also use it to analyze the beneﬁt for a dealer to trade centrally as a member rather than on a bilateral basis, or to help its members manage their CCVA (regarding the question for instance of how much of these costs they could consider passing to their clients). The paper is organized as follows. Sect. presents our clearing house setup. The wa- terfall of resources of the CCP is described in Sect. . The CCVA analysis is conducted in Sect. . Sect. introduces the common shock model that is used for the default times of the members of the clearing house. Sect. provides an executive summary of the centrally cleared XVA analysis of this paper and recalls for comparison purposes the bilateral CSA XVA methodology of Cr´epey and Song (2016). Sect. designs an experimental framework used in the numerics of Sect. . Sect. concludes. Regulatory formulas are recalled in Sect. . Pro(cid:82)ofs o(cid:82)f all lemmas are deferred to Sect. . 1.3 Basic Notation and Terminology ± ± b = ; x = max( x, 0); δ represents a Dirac measure at a point a; λ denotes a (a,b] R a the Lebesgue measure on . Unless otherwise stated, a ﬁltration satisﬁes the usual condi- + tions; a price process is a special semimartingale in a c`adl`ag version; all inequalities between random quantities are meant almost surely or almost everywhere, as suitable; all the cash ﬂows are assumed to be integrable whenever required; by “martingale” we mean local mar- tingale unless otherwise stated, but true martingale is assumed whenever necessary. This means that we only derive local martingale properties. Usually in applications one needs true martingales, but this is not a real issue in our case, where even square integrability follows from additional assumptions postulated when dealing with BSDEs, which are our main pricing tool in this paper. 3

text: terminology

text: of

text: the

text: present

text: paper),

text: including

text: eﬀects

text: such

text: as

text: wrong

text: way

text: risk

text: (meant

text: as

text: pro-

text: cyclicality

text: of

text: the

text: margins),

text: credit

text: dependence

text: between

text: members

text: and

text: left

text: tailed

text: distributions

text: of

text: their

text: P&Ls.

text: Ghamami

text: proposes

text: a

text: static

text: one-period

text: model

text: where

text: a

text: CCVA

text: can

text: be

text: priced

text: by

text: Monte-Carlo.

text: Brigo

text: and

text: Pallavicini

text: extend

text: the

text: bilateral

text: counterparty

text: risk

text: dynamic

text: setup

text: of

text: their

text: previous

text: papers

text: to

text: centrally

text: cleared

text: trading.

text: However,

text: they

text: ignore

text: the

text: default

text: fund

text: and

text: the

text: credit

text: risk

text: dependence

text: issues

text: that

text: are

text: inherent

text: to

text: the

text: position

text: of

text: a

text: clearing

text: member.

label: section

text: Contributions

text: and

text: Outline

text: This

text: paper

text: develops

text: an

text: XVA

text: (costs)

text: analysis

text: of

text: centrally

text: cleared

text: trading,

text: parallel

text: to

text: the

text: one

text: that

text: has

text: been

text: developed

text: in

text: the

text: last

text: years

text: for

text: bilateral

text: transactions

text: (see

text: e.g.

text: Cr´epey,

text: Bielecki,

text: and

text: Brigo

text: Parts

text: II

text: and

text: III)

text: or

text: Brigo,

text: Morini,

text: and

text: Pallavicini

text: A

text: dynamic

text: framework

text: incorporates

text: the

text: sequence

text: of

text: cash

text: ﬂows

text: involved

text: in

text: the

text: waterfall

text: of

text: resources

text: of

text: the

text: clearing

text: house.

text: As

text: compared

text: with

text: Arnsdorf

text: and

text: Ghamami

text: our

text: CCVA

text: accounts

text: not

text: only

text: for

text: the

text: central

text: clearing

text: analog

text: of

text: the

text: CVA,

text: which

text: is

text: the

text: cost

text: for

text: a

text: member

text: of

text: its

text: losses

text: on

text: the

text: default

text: fund

text: in

text: case

text: of

text: other

text: members’

text: defaults,

text: but

text: also

text: for

text: the

text: cost

text: of

text: funding

text: its

text: margins

text: (MVA)

text: and

text: for

text: the

text: cost

text: of

text: the

text: capital

text: (KVA)

text: that

text: is

text: implicitly

text: required

text: from

text: members

text: through

text: their

text: default

text: fund

text: contributions

text: (and

text: for

text: completeness

text: and

text: reference

text: we

text: also

text: compute

text: a

text: DVA

text: term).

text: The

text: framework

text: of

text: this

text: paper

text: can

text: be

text: used

text: by

text: a

text: clearing

text: house

text: to

text: ﬁnd

text: the

text: right

text: bal-

text: ance

text: between

text: initial

text: margins

text: and

text: default

text: fund

text: in

text: order

text: to

text: minimize

text: the

text: CCVA

text: (subject

text: to

text: the

text: regulatory

text: constraints),

text: hence

text: optimize

text: its

text: costs

text: to

text: the

text: members

text: for

text: a

text: given

text: level

text: of

text: resilience.

text: A

text: clearing

text: house

text: can

text: also

text: use

text: it

text: to

text: analyze

text: the

text: beneﬁt

text: for

text: a

text: dealer

text: to

text: trade

text: centrally

text: as

text: a

text: member

text: rather

text: than

text: on

text: a

text: bilateral

text: basis,

text: or

text: to

text: help

text: its

text: members

text: manage

text: their

text: CCVA

text: (regarding

text: the

text: question

text: for

text: instance

text: of

text: how

text: much

text: of

text: these

text: costs

text: they

text: could

text: consider

text: passing

text: to

text: their

text: clients).

text: The

text: paper

text: is

text: organized

text: as

text: follows.

text: Sect.

text: presents

text: our

text: clearing

text: house

text: setup.

text: The

text: wa-

text: terfall

text: of

text: resources

text: of

text: the

text: CCP

text: is

text: described

text: in

text: Sect.

text: The

text: CCVA

text: analysis

text: is

text: conducted

text: in

text: Sect.

text: Sect.

text: introduces

text: the

text: common

text: shock

text: model

text: that

text: is

text: used

text: for

text: the

text: default

text: times

text: of

text: the

text: members

text: of

text: the

text: clearing

text: house.

text: Sect.

text: provides

text: an

text: executive

text: summary

text: of

text: the

text: centrally

text: cleared

text: XVA

text: analysis

text: of

text: this

text: paper

text: and

text: recalls

text: for

text: comparison

text: purposes

text: the

text: bilateral

text: CSA

text: XVA

text: methodology

text: of

text: Cr´epey

text: and

text: Song

text: Sect.

text: designs

text: an

text: experimental

text: framework

text: used

text: in

text: the

text: numerics

text: of

text: Sect.

text: Sect.

text: concludes.

text: Regulatory

text: formulas

text: are

text: recalled

text: in

text: Sect.

text: all

text: lemmas

text: are

text: deferred

text: to

text: Sect.

text: Basic

text: Notation

text: and

text: Terminology

text: b

text: x

text: max(

text: x,

text: δ

text: represents

text: a

text: Dirac

text: measure

text: at

text: a

text: point

text: a;

text: λ

text: denotes

text: a

text: (a,b]

text: R

text: a

text: the

text: Lebesgue

text: measure

text: on

text: Unless

text: otherwise

text: stated,

text: a

text: ﬁltration

text: satisﬁes

text: the

text: usual

text: condi-

text: tions;

text: a

text: price

text: process

text: is

text: a

text: special

text: semimartingale

text: in

text: a

text: c`adl`ag

text: version;

text: all

text: inequalities

text: between

text: random

text: quantities

text: are

text: meant

text: almost

text: surely

text: or

text: almost

text: everywhere,

text: as

text: suitable;

text: all

text: the

text: cash

text: ﬂows

text: are

text: assumed

text: to

text: be

text: integrable

text: whenever

text: required;

text: by

text: “martingale”

text: we

text: mean

text: local

text: mar-

text: tingale

text: unless

text: otherwise

text: stated,

text: but

text: true

text: martingale

text: is

text: assumed

text: whenever

text: necessary.

text: This

text: means

text: that

text: we

text: only

text: derive

text: local

text: martingale

text: properties.

text: Usually

text: in

text: applications

text: one

text: needs

text: true

text: martingales,

text: but

text: this

text: is

text: not

text: a

text: real

text: issue

text: in

text: our

text: case,

text: where

text: even

text: square

text: integrability

text: follows

text: from

text: additional

text: assumptions

text: postulated

text: when

text: dealing

text: with

text: BSDEs,

text: which

text: are

text: our

text: main

text: pricing

text: tool

text: in

text: this

text: paper.


text: Σ 45 52 56 61 73 108 176 367 1053 ω (9.20) 1.80 4.60 (1.00) 6.80 (0.80) 13.80 (8.80) (7.20) Table 2: Positions ω in the swap of the nine members with CDS spreads Σ , in the respective i i ω = ωcsa or ωccp meaning, when the reference member 0 is the name with CDS spread 61 i i i | | bp and the second smallest α in Table 1. i Example 7.2 Table 3 is the analog of Table 2 when the member with spread 367 bp (name (cid:80) with the second largest credit spread in Table 1, with corresponding entries emphasized in bold in Table 2) is taken as reference member 0 (prototype of a risky name). In this case | | ν = (cid:54) ω = 5.14. 0 i=0 i Σ 45 52 56 61 73 108 176 367 1053 i ω (1.05) 0.20 0.52 (0.11) 0.77 (0.09) 1.57 (1.00) (0.82) i Table 3: Analog of Table 2 when the reference member 0 is the name with CDS spread 367 bp (name with the second largest credit spread Σ ) in Table 1. i 7.4 Margins ∈ CCP setup The initial margin IMi posted by each member i N is set through (3.4), using as risk measure ρ the risk-neutral value at risk of some level a “close to 1” . Since the pricing function P(cid:63) in(cid:40)(7.1) is decreasing in S, therefore IMi can be proxied, at each simulated time ζ in (6.4) or (6.9), by √ IMζi = Nom × |ωi| × PP(cid:63)(cid:63)((ζζ,, SSζζ)eσ−√δP(cid:48)Φ(cid:63)(−ζ1,(1S−ζae)σ+(κδ−(cid:48)Φσ−221)(δa(cid:48)))+−(κ−Pσ(cid:63)22(ζ)δ,(cid:48)S),ζ), ωωii ≥≤ 00, (7.2) (cid:48) where Φ is the standard normal cdf and where we recall that δ = δ +h is the margin period of risk. − For instance, the reference member 0, with ωccp = 1, is long one unit in the swap with 0 mark-to-market proﬁle shown in Figure 4, hence the exposure of the CCP to member − 0 is the opposite proﬁle. Accordingly (recalling that Figure√ 4 shows ( Pt)), the CCP asks initial margins to the member 0 based on P(cid:63)(ζ, Sζeσ ≤δ(cid:48)Φ−1(1−a)+(µ−σ22 )δ(cid:48))− P (ζ, S ), consistent with the second line in (7.2) in case ω 0. (cid:63) ζ i=0 Consistently with a “cover two” EMIR rule (see Sect. ), the default fund contributions are set as the sum of the two largest exposures of the clearing members (exposures in the sense of their EADs as explained in Sect. ), allocated between them proportionally to their initial margins. − ≥ (cid:54) CSA setup The initial margin Ic 0 required by the member 0 from the member i = 0 (cf. (6.5)) is given by the right-hand side formula in (7.2) valued at some quantile level a (possibly diﬀerent from the one used in the CCP setup). For instance, if ωcsa = +2, meaning that the member 0 has a “double Figure 4 i exposure” with regard to counterparty i, then the member 0 asks the counterparty i 24 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: (cid:34)(cid:32) (cid:33) (cid:35) (cid:90) Likewise, we have (cid:82) − (cid:48) Et Pv+δ(cid:48) + (cid:48) e sv+δ rududDs − Pv− − IMv = gva,δ(cid:48) × (1 − a)e−κtSt, (7.5) [v,v+δ ] (cid:18) (cid:19) (cid:88) where gva,δ(cid:48) = − eσ√δ(cid:48)Φ−1(1−a) − e−σ√δ(cid:48) φ(Φ1−−1a(a)) βv−+1δ(cid:48)e−σ22 δ(cid:48) d βTlhleκTl−1. (7.6) l=l (cid:48) v+δ Based on (7.3) through (7.4), explicit formulas for the EADs follows. Figure 5 shows the time-0 EADs of the nine CCP members for their positions in the swap corresponding to the choice of the name of examples 7.1 or 7.2 as reference member. Figure 5: Time-0 EADs in basis points (IM quantile level a = 70%, liquidation period δ = 5 days). The two largest EADs, in red, size the default fund. The reference member EAD is in green. The corresponding positions ω of the member are displayed at the bottom. i Left: Reference member with Σ =61 bps and ν = 53.00. Right: Reference member with 0 0 Σ =367 bps and ν = 5.14. 0 0 7.6 XVA Data The following numerical values are used in the sequel: 1 2 R¯ = 1, λ¯ = Σ , λ = 0, k = 10%, h = 1 day, µ = , m = 104, (7.7) 2 0 T¯ where m is the number of simulations used for estimating the expectations in (6.4) and (6.9). The level of 10% used for k is consistent with reference orders of magnitude for a hurdle rate. Moreover, in a CCP setup, unless otherwise stated, we set R = 0, δ = 5 days , a = 70%, T = 1 month, Y = 1 year, (7.8) E(cid:63) = 25%Kccp, c = 30 bp, 26 ##LTFigure## ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

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text: we

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text: on

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text: explicit

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text: EADs

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text: Data

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text: following

text: numerical

text: values

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text: used

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text: the

text: sequel:

text: R¯

text: λ¯

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text: essentially not impacted by ν . This illustrates the multilateral netting beneﬁt provided by 0 the CCP, especially for members with a large compression factor ν . 0 ν 2.91 4.87 5.14 6.50 6.94 10.74 29 53 66.50 0 α 0.69 (0.46) (0.44) (0.36) 0.34 0.23 0.09 (0.05) (0.04) 0 Σ 176 45 367 1053 73 56 52 61 108 0 CVA 11.07 25.06 19.34 14.06 28.37 42.69 111.38 238.22 299.37 DVA (8.76) (4.49) (30.85) (90.10) (8.08) (13.59) (28.77) (52.70) (111.33) MVA 30.38 13.63 110.50 339.69 31.41 39.34 98.46 204.72 449.68 KVA 11.17 21.16 19.40 21.14 29.26 46.28 122.20 221.63 275.87 BVA 52.62 59.85 149.24 374.90 89.04 128.31 332.05 664.57 1024.92 CVA 7.88 11.33 6.54 3.57 10.85 11.73 11.91 11.60 9.23 DVA (2.57) (0.69) (5.43) (13.03) (1.07) (0.89) (0.81) (0.90) (1.57) MVA 5.19 1.39 10.33 24.24 2.22 1.76 1.61 1.86 3.23 MLA 1.17 1.22 1.09 0.89 1.22 1.22 1.22 1.22 1.20 KVA 10.79 11.59 10.00 7.97 11.44 11.52 11.54 11.58 11.21 CCVA 25.03 25.54 27.95 36.67 25.73 26.23 26.27 26.26 24.87 Table 4: XVA numbers obtained by considering alternately each of the nine members in Table 1 as reference member 0, using the α for setting the positions of the members in i each case as explained in Sect. . (Up) Credit spread Σ , coeﬃcient α and compression 0 0 factor ν of the reference member in each case (ordered by increasing ν , i.e. decreasing | | 0 0 α ). (Middle) CSA XVA numbers. (Bottom) CCP XVA numbers. 0 Table 5 shows the percentage standard errors corresponding to the Monte Carlo esti- mates of Table 4. As we can see from the table, the standard errors are typically no more than a few percents in relative terms. Standard errors of Monte Carlo estimates are no longer shown in the sequel. ν 2.91 4.87 5.14 6.50 6.94 10.74 29 53 66.50 0 α 0.69 (0.46) (0.44) (0.36) 0.34 0.23 0.09 (0.05) (0.04) 0 Σ 176 45 367 1053 73 56 52 61 108 0 CVA 3.40 2.87 3.40 4.97 3.22 3.22 3.22 2.90 2.89 DVA 5.66 10.38 4.08 2.58 8.92 9.21 9.49 9.28 7.05 MVA 0.79 0.78 0.75 0.96 0.77 0.64 0.63 0.84 0.80 KVA 0.58 0.54 0.64 0.81 0.54 0.54 0.54 0.54 0.55 CVA 2.55 2.93 3.13 4.49 2.69 2.71 2.70 2.91 2.66 DVA 3.11 3.02 3.05 3.42 3.15 2.92 2.94 3.27 3.21 MVA 0.86 0.78 0.77 0.96 0.91 0.67 0.69 0.95 0.93 MLA 0.65 0.60 0.71 0.88 0.61 0.61 0.60 0.60 0.62 KVA 0.58 0.58 0.65 0.84 0.57 0.59 0.59 0.59 0.58 Table 5: Percentage standard errors corresponding to the Monte Carlo estimates of Table 4. 8.2 Impact of the Credit Spread of the Reference Member The CCP multilateral netting beneﬁt dominates the comparison between our CSA and CCP XVA numbers. However, in our stylized setup, we cannot see the netting beneﬁt across assets of bilateral trading. In order to compensate for this bias and obtain comparative 28 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: essentially

text: not

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text: ν

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text: the

text: multilateral

text: netting

text: beneﬁt

text: provided

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text: the

text: CCP,

text: especially

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text: large

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text: ν

text: ν

text: α

text: Σ

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text: DVA

text: MVA

text: MLA

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text: Table

text: XVA

text: numbers

text: obtained

text: by

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text: CSA

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text: percentage

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text: than

text: a

text: few

text: percents

text: in

text: relative

text: terms.

text: Standard

text: errors

text: of

text: Monte

text: Carlo

text: estimates

text: are

text: no

text: longer

text: shown

text: in

text: the

text: sequel.

text: ν

text: α

text: Σ

text: CVA

text: DVA

text: MVA

text: KVA

text: CVA

text: DVA

text: MVA

text: MLA

text: KVA

text: Table

text: Percentage

text: standard

text: errors

text: corresponding

text: to

text: the

text: Monte

text: Carlo

text: estimates

text: of

text: Table

text: Impact

text: of

text: the

text: Credit

text: Spread

text: of

text: the

text: Reference

text: Member

text: The

text: CCP

text: multilateral

text: netting

text: beneﬁt

text: dominates

text: the

text: comparison

text: between

text: our

text: CSA

text: and

text: CCP

text: XVA

text: numbers.

text: However,

text: in

text: our

text: stylized

text: setup,

text: we

text: cannot

text: see

text: the

text: netting

text: beneﬁt

text: across

text: assets

text: of

text: bilateral

text: trading.

text: In

text: order

text: to

text: compensate

text: for

text: this

text: bias

text: and

text: obtain

text: comparative

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text: A.1 CCP Setup Under centrally cleared trading, the “cover two” EMIR rule prescribes to size the de- fault fund as, at least, the maximum of the greatest and of the sum of the second and third greatest exposures “under extreme but plausible market conditions” (see European Parlia- ment (2012, article 42, paragraph 3, page 37)). This total amount is then allocated between the clearing members according to some repartition key, e.g. proportional to their initial margins. As explained in the paper, default fund contributions are “implicit capital” that the clearing members put at the disposal of the CCP. In addition, to cover their residual risk beyond the guarantee provided by the diﬀerent margin layers of the CCP, the regula- tory capital K = Kcm (cid:18)of a generic reference member is deﬁned, follow(cid:19)ing Basel Commit- tee on Banking Supervision (2014, pa(cid:80)ge 11), as: × DF C × × Kcm = max Kccp , 8% 2% DF C , (A.3) E + ∈ JiDF Ci i N (cid:88) where DFC is the default fund contribution of the reference member and where × × Kccp = RW Cap JiEADi Ratio (A.4) ∈ i N with RW = 20% and Cap = 8%. Ratio Remark A.1 Ghamami (2015) argues that the CCP regulatory capital Kcm of a member should rather be based on its expected future unfunded default fund contributions (see the remark 3.4), which represent the losses of the member beyond the level already funded via its default fund contribution. A.2 CSA Setup In the bilateral setup, the capital at risk K of the bank reduces to its regulatory capital (there is no bilateral trading analog of the default fund), which comprises a ﬁrst contribution for counterparty default losses and a second one for the volatility of the CVA (the market risk of the bank being supposed to be hedged out). Since we focus on the reference member ∈ { } 0 with n counterparties i N(cid:63) = 1, 2, . . . , n , the capital formulas below all need to be ∈ summed over i N(cid:63). A.2.1 Kccr (cid:88) The Basel II regulatory capital speciﬁed for counterparty risk is deﬁned as Kccr = Cap RWAi, Ratio ∈ i N(cid:63) where × × × RWAi = 12.5 w 1.4 EADi. i ≥ Here Cap 8% (which is the value that we use in the numerics) is a chosen capital Ratio ratio that the b(cid:18)ank(cid:18)must hold. The(cid:114)capital weight wi is gi(cid:19)ven by th(cid:19)e intern(cid:98)al ratings-based formula − − − √Φ 1 (DP ) corr − − 1 + (T i 2.5)b(DP ) w = (1 R ) Φ − i + − i Φ 1(0.999) DP − i i i i 1 corri 1 corri 1 1.5b(DPi) 34 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: CCP

text: Setup

text: Under

text: centrally

text: cleared

text: trading,

text: the

text: “cover

text: two”

text: EMIR

text: rule

text: prescribes

text: to

text: size

text: the

text: de-

text: fault

text: fund

text: as,

text: at

text: least,

text: the

text: maximum

text: of

text: the

text: greatest

text: and

text: of

text: the

text: sum

text: of

text: the

text: second

text: and

text: third

text: greatest

text: exposures

text: “under

text: extreme

text: but

text: plausible

text: market

text: conditions”

text: (see

text: European

text: Parlia-

text: ment

text: article

text: paragraph

text: page

text: This

text: total

text: amount

text: is

text: then

text: allocated

text: between

text: the

text: clearing

text: members

text: according

text: to

text: some

text: repartition

text: key,

text: e.g.

text: proportional

text: to

text: their

text: initial

text: margins.

text: As

text: explained

text: in

text: the

text: paper,

text: default

text: fund

text: contributions

text: are

text: “implicit

text: capital”

text: that

text: the

text: clearing

text: members

text: put

text: at

text: the

text: disposal

text: of

text: the

text: CCP.

text: In

text: addition,

text: to

text: cover

text: their

text: residual

text: risk

text: beyond

text: the

text: guarantee

text: provided

text: by

text: the

text: diﬀerent

text: margin

text: layers

text: of

text: the

text: CCP,

text: the

text: regula-

text: tory

text: capital

text: K

text: Kcm

text: a

text: generic

text: reference

text: member

text: is

text: deﬁned,

text: Basel

text: Commit-

text: tee

text: on

text: Banking

text: Supervision

text: as:

text: DF

text: C

text: Kcm

text: max

text: Kccp

text: DF

text: C

text: E

text: JiDF

text: Ci

text: i

text: N

text: where

text: DFC

text: is

text: the

text: default

text: fund

text: contribution

text: of

text: the

text: reference

text: member

text: and

text: where

text: Kccp

text: RW

text: Cap

text: JiEADi

text: Ratio

text: i

text: N

text: with

text: RW

text: and

text: Cap

text: Ratio

text: Remark

text: Ghamami

text: argues

text: that

text: the

text: CCP

text: regulatory

text: capital

text: Kcm

text: of

text: a

text: member

text: should

text: rather

text: be

text: based

text: on

text: its

text: expected

text: future

text: unfunded

text: default

text: fund

text: contributions

text: (see

text: the

text: remark

text: which

text: represent

text: the

text: losses

text: of

text: the

text: member

text: beyond

text: the

text: level

text: already

text: funded

text: via

text: its

text: default

text: fund

text: contribution.

text: CSA

text: Setup

text: In

text: the

text: bilateral

text: setup,

text: the

text: capital

text: at

text: risk

text: K

text: of

text: the

text: bank

text: reduces

text: to

text: its

text: regulatory

text: capital

text: (there

text: is

text: no

text: bilateral

text: trading

text: analog

text: of

text: the

text: default

text: fund),

text: which

text: comprises

text: a

text: ﬁrst

text: contribution

text: for

text: counterparty

text: default

text: losses

text: and

text: a

text: second

text: one

text: for

text: the

text: volatility

text: of

text: the

text: CVA

text: (the

text: market

text: risk

text: of

text: the

text: bank

text: being

text: supposed

text: to

text: be

text: hedged

text: out).

text: Since

text: we

text: focus

text: on

text: the

text: reference

text: member

text: with

text: n

text: counterparties

text: i

text: n

text: the

text: capital

text: formulas

text: below

text: all

text: need

text: to

text: be

text: summed

text: over

text: i

text: Kccr

text: The

text: Basel

text: II

text: regulatory

text: capital

text: speciﬁed

text: for

text: counterparty

text: risk

text: is

text: deﬁned

text: as

text: Kccr

text: Cap

text: RWAi,

text: Ratio

text: i

text: where

text: RWAi

text: w

text: EADi.

text: i

text: Here

text: Cap

text: (which

text: is

text: the

text: value

text: that

text: we

text: use

text: in

text: the

text: numerics)

text: is

text: a

text: chosen

text: capital

text: Ratio

text: ratio

text: that

text: the

text: hold.

text: weight

text: wi

text: is

text: by

text: ratings-based

text: formula

text: √Φ

text: (DP

text: corr

text: (T

text: i

text: w

text: R

text: Φ

text: i

text: i

text: Φ

text: DP

text: i

text: i

text: i

text: i

text: corri

text: corri

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text: Consistent with our sign convention that all cash ﬂows and values are seen from the perspective of the clearing house, we count a margin positively when it is posted by a member and we deﬁne the variation margin VMi, initial margin IMi and default fund contribution DF Ci of the member i as the piecewise constant process reset at the respective grid times following, respectively (while the member i is alive): VMi = P i −, IMi = ρi , DF Ci = (cid:37)i , (3.1) lh lh lh lh lT lT where ρi and (cid:37)i refer to suitable risk measures as explained below. Note that (3.1) deﬁnes the level of reset of the respective cumulative amounts. Starting from VMi = P i−, IMi = ρi − 0 0 −0 0 and DF Ci = (cid:37)i , the corresponding updates at grid times are (P i − P i − −), (ρi ρi − ) −0 0 lh (lh h) lh lh h and ((cid:37)i (cid:37)i − ). lT lT T Remark 3.1 In practice, the variation margin only tracks the mark-to-market of the port- folio up to some thresholds, or free credit lines of the members, and up to minimal transfer amounts devoted to avoiding useless updates. These features, which can be important in the case of bilateral transactions, are omitted here as negligible in the case of centrally cleared transactions. (cid:90) Let (cid:82) (cid:48) − Lit,t+δ(cid:48) = Pti+δ(cid:48) + (cid:48) e st+δ rududDsi Pti−. (3.2) [t,t+δ ] In particular, at margin call times(cid:90)t = lh, we have, in view of the speciﬁcation of the variation margin by the ﬁrst identity in (3.1):(cid:82) (cid:48) − Lilh,lh+δ(cid:48) = Plih+δ(cid:48) + (cid:48) e slh+δ rududDsi VMlih, (3.3) [lh,lh+δ ] (cid:48) which is the variation-margined loss-and-proﬁt of the member i at the time horizon δ = δ + h of the margin period of risk (cumulative loss-and-proﬁt also accounting for all the contractual cash ﬂows capitalized at the risk-free rate during the margin period of risk (cid:48) [t, t + δ ]). The risk measure used for ﬁxin(cid:0)g the init(cid:1)ial margins is a univariate risk measure computed at the level of each member individually, which we write as ρi = ρ Li (cid:48) , (3.4) lh lh,lh+δ (cid:80) where ρ can be value at risk, expected shortfall, etc.. The dependence between the portfolios of the members is only reﬂected in the initial margins through the structural constraint that ∈ P i = 0. i N Remark 3.2 Historically, for computing initial margins, CCPs have been mostly using the SPAN methodology, introduced by the Chicago Mercantile Exchange in the 80s. This methodology is based, for each member, on the consideration of the most unfavorable among sixteen reference scenarios (see Kupiec and White (1996)). Nowadays, value at risk methodologies tend to become the standard. Unless defaults happen, margins do not imply any transfer of ownership and can be seen in this sense as a loan by the posting member. By contrast, default fund contributions can be consumed in case of other members’ defaults, hence they should really be viewed as capital put at the disposal of the CCP by the clearing members. The “cover two” EMIR 6

text: Consistent

text: with

text: our

text: sign

text: convention

text: that

text: all

text: cash

text: ﬂows

text: and

text: values

text: are

text: seen

text: from

text: the

text: perspective

text: of

text: the

text: clearing

text: house,

text: we

text: count

text: a

text: margin

text: positively

text: when

text: it

text: is

text: posted

text: by

text: a

text: member

text: and

text: we

text: deﬁne

text: the

text: variation

text: margin

text: VMi,

text: initial

text: margin

text: IMi

text: and

text: default

text: fund

text: contribution

text: DF

text: Ci

text: of

text: the

text: member

text: i

text: as

text: the

text: piecewise

text: constant

text: process

text: reset

text: at

text: the

text: respective

text: grid

text: times

text: following,

text: respectively

text: (while

text: the

text: member

text: i

text: is

text: alive):

text: VMi

text: P

text: i

text: IMi

text: ρi

text: DF

text: Ci

text: lh

text: lh

text: lh

text: lh

text: lT

text: lT

text: where

text: ρi

text: and

text: refer

text: to

text: suitable

text: risk

text: measures

text: as

text: explained

text: below.

text: Note

text: that

text: deﬁnes

text: the

text: level

text: of

text: reset

text: of

text: the

text: respective

text: cumulative

text: amounts.

text: Starting

text: from

text: VMi

text: P

text: i−,

text: IMi

text: ρi

text: −0

text: and

text: DF

text: Ci

text: the

text: corresponding

text: updates

text: at

text: grid

text: times

text: are

text: (P

text: i

text: P

text: i

text: (ρi

text: ρi

text: −0

text: lh

text: (lh

text: h)

text: lh

text: lh

text: h

text: and

text: lT

text: lT

text: T

text: Remark

text: In

text: practice,

text: the

text: variation

text: margin

text: only

text: tracks

text: the

text: mark-to-market

text: of

text: the

text: port-

text: folio

text: up

text: to

text: some

text: thresholds,

text: or

text: free

text: credit

text: lines

text: of

text: the

text: members,

text: and

text: up

text: to

text: minimal

text: transfer

text: amounts

text: devoted

text: to

text: avoiding

text: useless

text: updates.

text: These

text: features,

text: which

text: can

text: be

text: important

text: in

text: the

text: case

text: of

text: bilateral

text: transactions,

text: are

text: omitted

text: here

text: as

text: negligible

text: in

text: the

text: case

text: of

text: centrally

text: cleared

text: transactions.

text: Let

text: e

text: st+δ

text: rududDsi

text: Pti−.

text: [t,t+δ

text: In

text: particular,

text: at

text: margin

text: call

text: lh,

text: we

text: have,

text: in

text: view

text: of

text: the

text: speciﬁcation

text: of

text: the

text: variation

text: margin

text: by

text: the

text: ﬁrst

text: identity

text: in

text: e

text: slh+δ

text: rududDsi

text: VMlih,

text: [lh,lh+δ

text: which

text: is

text: the

text: variation-margined

text: loss-and-proﬁt

text: of

text: the

text: member

text: i

text: at

text: the

text: time

text: horizon

text: δ

text: δ

text: h

text: of

text: the

text: margin

text: period

text: of

text: risk

text: (cumulative

text: loss-and-proﬁt

text: also

text: accounting

text: for

text: all

text: the

text: contractual

text: cash

text: ﬂows

text: capitalized

text: at

text: the

text: risk-free

text: rate

text: during

text: the

text: margin

text: period

text: of

text: risk

text: [t,

text: t

text: δ

text: The

text: risk

text: measure

text: used

text: for

text: the

text: margins

text: is

text: a

text: univariate

text: risk

text: measure

text: computed

text: at

text: the

text: level

text: of

text: each

text: member

text: individually,

text: which

text: we

text: write

text: as

text: ρi

text: ρ

text: Li

text: lh

text: lh,lh+δ

text: where

text: ρ

text: can

text: be

text: value

text: at

text: risk,

text: expected

text: shortfall,

text: etc..

text: The

text: dependence

text: between

text: the

text: portfolios

text: of

text: the

text: members

text: is

text: only

text: reﬂected

text: in

text: the

text: initial

text: margins

text: through

text: the

text: structural

text: constraint

text: that

text: P

text: i

text: i

text: N

text: Remark

text: Historically,

text: for

text: computing

text: initial

text: margins,

text: CCPs

text: have

text: been

text: mostly

text: using

text: the

text: SPAN

text: methodology,

text: introduced

text: by

text: the

text: Chicago

text: Mercantile

text: Exchange

text: in

text: the

text: This

text: methodology

text: is

text: based,

text: for

text: each

text: member,

text: on

text: the

text: consideration

text: of

text: the

text: most

text: unfavorable

text: among

text: sixteen

text: reference

text: scenarios

text: (see

text: Kupiec

text: and

text: White

text: Nowadays,

text: value

text: at

text: risk

text: methodologies

text: tend

text: to

text: become

text: the

text: standard.

text: Unless

text: defaults

text: happen,

text: margins

text: do

text: not

text: imply

text: any

text: transfer

text: of

text: ownership

text: and

text: can

text: be

text: seen

text: in

text: this

text: sense

text: as

text: a

text: loan

text: by

text: the

text: posting

text: member.

text: By

text: contrast,

text: default

text: fund

text: contributions

text: can

text: be

text: consumed

text: in

text: case

text: of

text: other

text: members’

text: defaults,

text: hence

text: they

text: should

text: really

text: be

text: viewed

text: as

text: capital

text: put

text: at

text: the

text: disposal

text: of

text: the

text: CCP

text: by

text: the

text: clearing

text: members.

text: The

text: “cover

text: two”

text: EMIR