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Network Structure and Counterparty Credit Risk

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Abstract and Figures

In this paper we offer a novel type of network model, which is capable of capturing the precise structure of a financial market based, for example, on empirical findings. With the attached stochastic framework it is further possible to study how an arbitrary network structure and its expected counterparty credit risk are analytically related to each other. This allows us, for the first time, to model and to analytically analyse the precise structure of a financial market. It further enables us to draw implications for the study of systemic risk. We apply the powerful theory of characteristic functions and Hilbert transforms, which have not been used in this combination before. We then characterise Eulerian digraphs as distinguished exposure structures and we show that considering the precise network structures is crucial for the study of systemic risk. The introduced network model is then applied to study the features of an over-the-counter and a centrally cleared market. We also give a more general answer to the question of whether it is more advantageous for the overall counterparty credit risk to clear via a central counterparty or classically bilateral between the two involved counterparties. We then show that the exact market structure is a crucial factor in answering the raised question.
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max(Y, 0)Y
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Y φYΛ=
{e1
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4(1+t2)3+3t5
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16
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4
95
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u v w
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2
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(1eit)(1+eit)
t2{φY}2(tsin(t))
t2limt0{φY}(t)
E(max[Y; 0]) =tφmax[Y;0](0)
i=1
6
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Xa1Xa2v
v
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w
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γvγ(v)γ(v)=γ+(v)γ(v)
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Dk=(V, Ak)X±PΛv
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E(λΛv±Xλ)=0γ(v)=0
E(max[λΛv±Xλ; 0])=1
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vV γ(v)=0
Dk=(V, Ak)γ(v)=0vV
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max
λΛv
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=1
2t[ {φY(t)}](0).
Y=λΛvXλ
ΛvXλλΛv
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wV{v}k yv,w (C)=kCx(k)
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yv,w (C)=yw,v (C)
v w max[yv,w (C); 0]zb(D)=
vVwUC
vmax[yv,w (C); 0]D=(V, ̂
A)
M
MD=
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A)X±P
kC
E(max[Yv,w (C); 0]) v w
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v,w φYv,w φmax[Yv,w ;0]
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i+t
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2[1+1
1+t2]+i
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1+t2}(t){1
1+t2}(0)]=
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2i+2t{1
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2i+2t](0)
i=1
2
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w
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v w
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X2
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2
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v,w N(0,Kσ)
{φY}(0)=0
φmax[Y;0](t)=1
2[1+φY(t)]+i
2[ {φY}(t){φY}(0)]
=1
2+et22
2
2+it
2
π
1
1+t2
1
1+t2
{1
1+t2}(ω)=2i 1
(1+t2)(ωt),i+i1
(1+t2)(ωt), ω=
i
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1+ω2ω t
{φY}(t)=2
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2
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N>2v(N1)
kC
(N1)σK
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2N(N1)
2K
D=(V, ̂
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M
kC
vV U(k)
vv k C
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v)=wU(k)
vx(k)
v,w Rv
k yv,k
v k
vmax[yv,k (U(k)
v); 0]
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v) Dk
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v)=0zm(Dk)
zm(Dk)=
vVyv,k (U(k)
v)=2
vV
max[yv,k (U(k)
v); 0].
K
kC(K1)
zm(D)=zm(Dk)+zb(DDk),
DDk=(V, ̂
AAk)
M
D=(V, A)X±P
Ak±±X(k)
v,w
±PYv,k (U(k)
v)max[Yv,k (U(k)
v); 0]φYv
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v); 0] =t[φmax[Yv;0](t)](0)
i,
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v)=wU(k)
v±X(k)
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vV
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v); 0]=
vV
t[φmax[Yv;0](t)](0)
i
Dk
k
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vX(1)
v,w Xv,w
X(1)
v,w Yv(U(k)
v)=wU(k)
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et2
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πtN1σ
2
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v); 0] =1
it
1
21+et2
2(N1)σ2+itN1σ
2
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2πσ
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G2N(N1)
2
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A)N2
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v±X(k)
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v)]=0γ(v)=0k
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v); 0]=1
2t[ {φYv,k (t)](0)γ(v)=0k
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XL(0,1)l, m, n {1,2,3}mn
v1v2
v3
e1
e2
e3
v1v2
v3
a1
a2
a3
G D
D=(V, {a1, a2, a3})
G
1
1+t2=i
i+ti
i+tYvl=XmXnXmXn
Emax[Yvl(U(1)
vl); 0] =1
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2tt
1+t2(0)=1
2
vll{1,2,3}D
E(Zm(D))=E(Zm(D1))=
3
l=1Emax[Yvl(U(1)
vl); 0]=31
2=3
2
5
2
v1v2
v3
v1v2
v3
v1v2
v3
v1v2
v3
v1v2
v3
v1v2
v3
v1v2
v3
v1v2
v3
8=23G
G=(V, E )
1
1+t2
1
1+t2
1
1+t2
1
1+t2+it
1+t2
1
1+t2
φYvl(t)=1
(1+t2)2=1
1+t2
1
1+t2Yvl=Xm+Xnvl
{φYvl}(t)l{1,2,3}t(3+t2)
2(1+t2)2
Emax[Yvl,1(U(1)
vl); 0] =1
2tt(3+t2)
2(1+t2)2(0)=3
4
GE(Zm(G))=9
4
1
8(23
2+65
2)=9
4
kC D K N
X±P
E(Zm(D)) =E(Zm(Dk))+E(Zb(DDk)) <E(Zb(D))
vV
t[N1φv,k (t)](0)
i+
vV
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v
t[K1φv,w (t)](0)
i<
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v
t[Kφv,w (t)](0)
i,
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21+φX(t)M+i
2{φX(t)M}(t){φX(t)M}(0)
max[Y, 0]Y=M
j=1±XjXjP M N
N1K K 1X Xj
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v
t[N1φv,k (t)](0)
i+(N1)t[K1φv,w (t)](0)
i<(N1)t[Kφv,w (t)](0)
i
(N1)(t[Kφv,w (t)](0)t[K1φv,w (t)](0)) >t[N1φv,K (t)](0).
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2t[ {φX(t)N1}](0)<(N1)
2t[ {φX(t)K}](0)t[ {φX(t)K1}](0)
P
G=(V, E )N
XN(0,1)
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1
2N12
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2
K2
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.
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4(N1)N>2
G=(V, E )XL(0,1)
φX(t)1
1+t2
1
it[Mφv,(t)](0)=1
2t[ {φX(t)M}](0)=Γ(1
2+M)
πΓ(M)=M
22M2M
M
K=
N
v
kΓ(t)=
0xt1exdx
N K K =5
N=18
GD
XL(0,1)
E(Zb(GD))=7.5
E(Zm(GD))=8.875
K=2
N=6
M=N1=2
1
it[2φv,k (t)]=3π
4
π=3
4
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a1,...,amCn1,...,nmN
φY(z)0z
{φY(t)}(ω)=2i
m
j=1φY(z)
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ωz, ω.
{φY(t)}(ω)=2i
m
j=11
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zaj
nj1
∂znj1[(zaj)njφY(z)
ωz]i lim
zωφY(z)
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R,  ]0,[C
CRC
CφY(z)
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a1,...,amCn1,...,nmNφY(z)0z
t=ω R 
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C
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m
j=1φY(z)
ωz, aj.
C
φY(z)
ωzdz =ω
R
φY(t)
ωtdt +C
φY(z)
ωzdz +R
ω+
φY(t)
ωtdt +CR
φY(z)
ωzdz.
R0
P V
−∞
φY(t)
ωtdt =lim
Rlim
0ω
R
φY(t)
ωtdt +R
ω+
φY(t)
ωtdt.
Cω+eiθπθ0
C
φY(z)
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0i0
π
φY(ω+eiθ)eiθ
ω(ω+eiθ)=πiφY(ω)=πiφY(z)
ωz, ω.
CRReiθ0θπ
φY(z)0zCR
φY(z)
ωzdz =lim
Riπ
0
φY(Reiθ)Reiθ
ωReiθ=0.
P V
−∞
φY(t)
ωtdt =2πi
m
j=1φY(z)
ωz, aj+πiφY(z)
ωz, ω,
{φY(t)}(ω)=2i
m
j=1φY(z)
ωz, aj+iφY(z)
ωz, ω.
φX(t)=1
12it=1
1+4t2+2it
1+4t2X
Γ(α, β)α=1β=2α>0
β>0
φX(z) z
φX(z)
ωzi
2
{φX(t)}(ω)=iφX(z)
ωz, ω=2ω
1+4ω2i1
1+4ω2
(X)=XφX(t)=
1
1+4t22it
1+4t2
i
2
{φX(t)}(ω)=2i φX(z)
ωz,i
2+iφX(z)
ωz, ω=2ω
1+4ω2+i1
1+4ω2
XP φX
φXφXφXD
φXX φX(t)=φX(t)
tRX φX
X
X
(X)=XφX
φX(t)=η(t)+iν(t)
φX(t)=η(t)iν(t)
tRη=Re(φX)ν=Im(φX)
φX=φXη ν
φ(t)
φ(t)=η(t)+iν(t)=η(t)+i{η}(t)tR
φXXXP
φX(t)=φX(t)+i{φX}(t).
X φX
φXf=F{φX}
F1{f}(t)=1
2π
−∞ f(x)eixt dx =φX(t)
φXf
PX
f+(x)=f(x)+(x)f(x)xR
(x)=
1, x >0
0, x =0
1, x <0
f+2f(t)
f+X
f+
F1{f+}=F1{f}+F1{f}=φX+F1{f}=φX
F1{f+}=η+iνF1(f)=iν
i⋅ ⋅f ν i(x)1
πx
ν(t)=φX(x)1
πx =1
π
−∞
φX(t)
xtdx ={φX}(t),
φX(t)=φX(t)+i{φX}(t),
φXXP
P
φX(t)=φX(t)φX(t)=φX(t)i{φX}(t).
n φX
{φn
X}=iφn
X, n N,
P
ΛvmN
vXiΓ(α, β)i{1,2,...,m}
Xi (1βit)α
Y=m
i=1Xi (1βit)αm YΓ(mα, β)
i(1βit)αm
αβm
v
Yv=λΛvXλ
λΛvΛ+
vΛ
vh(λ)=v t(λ)=v
ΛvΛv=Λ+
vΛ
vΛ+
v=γ+(v)
Λ
v=γ(v)
0=E(Yv)=EλΛv±Xλ0=λΛ+
vE(Xλ)+λΛ
vE(Xλ)
0
0=Λ+
vE(Xλ)+Λ
vE(Xλ) E(Xλ)=E(Xλ)
0=[γ+(v)γ(v)]E(Xλ)
γ(v)=0E(Xλ)>0γ(v)=0
E(λΛv±Xλ)=0
max[λΛv±Xλ; 0]=max[Yv; 0]
Emax[λΛv±Xλ; 0]=1
2E(Yv)+1
2t[ {φYv}(t)
{φYv}(0)](0)E(Yv)=0γ(v)=0
γ(v)=0γ+(v)=Λ+
v=Λ
v=γ(v)
φYv=λΛ+
vφXλλΛ
vφX
λ
{φYv}
{φYv}(0)=0
φXXP φY
Y=λΛvXλ{φY(t)}(t)
{φY}(0)=0φmax[Y;0](t)=1
2[1+φY(t)]+i
2[ {φY(t)}(t)]
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... Investor behavior and information disclosure strategy not only have an effect on the formation and contagion of counterparty credit risk respectively, but also often have more complex influence on it under interaction. In credit asset market, investors' adoption of different information disclosure strategies varies in accordance with the credit asset status and behavior preference [16,17,27] . This will lead to the serious distortion and asymmetry of information in the actual situation of credit asset and investor credit, and affect the behavior choices of other investors, easily inducing investor behavior and psychology cognition deviation. ...
... This is not only convenient for analyzing the formation process of complex systems, but also convenient for studying evolution characteristics of risk contagion of complex systems under different influence factors. As a typical complex system, credit asset market can be abstracted as a network, and counterparties are abstracted as nodes in credit asset market and the credit contract relationship among counterparties are abstracted as edges [16] . Therefore, with the help of epidemic model, the formation of counterparty credit association network and evolution characteristics of counterparty credit risk contagion can be effectively analyzed under interaction of investor behavior and information disclosure strategy. ...
... And it has impact on level of investor risk cognition. What's more, Investor influence θ (0 ≤ θ < 1) is included, that is, the degree of association of investor in credit asset network [16,15] . The association of investor and the significant influence on investor behavior associated with them become greater with the increase of θ . ...
... Investor Influence . The influence of investors is related to the topological structure of the investors' networks [34,42,43]. In this paper, the investor influence is measured by the edge weight and the investor's degree , which represents the weighted average of the edge weight between the investor and the connected investors: ...
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Investor heterogeneities include investor risk preference, investor risk cognitive level, information value, and investor influence. From the perspective of the stock price linkage, this article constructs an SCIR contagion model of investor risk on a single-layer network. It digs out the investor risk caused by rumors in the stock market under the stock price linkage and its contagion mechanism. The function and influence of different mechanism probabilities and investor heterogeneities on the effects of risk contagion in the stock market are explored through computer simulation. Based on the SCIR contagion model of investor risk on single-layer network, we construct an SCI 1 I 2 R contagion model of investor risk on bilayer-coupled networks. Initially, the evolution mechanisms of investor risk contagion in the stock market are compared in single-layer and bilayer-coupled networks. Thereafter, the evolution characteristics and rules of investor risk contagion under different connection modes and heterogeneous mechanism probabilities are compared on bilayer-coupled networks. The results corroborate the following. (1) In the SCIR contagion model of investor risk on a single-layer network, immune failure probability and immune probability have the “global effect”. (2) Investor heterogeneities both have “global effect” and “local effect” on investor risk contagion. (3) Compared with the investor risk contagion on a single-layer network, bilayer-coupled networks can expand the investor risk contagion and have a “global enhancement” effect. (4) Among the three interlayer connection modes of the SCI 1 I 2 R model of investor risk contagion on bilayer-coupled networks, the assortative link has the effect of “local enhancement”, while the disassortative link has the effect of “local inhibition”. (5) In the SCI 1 I 2 R model of investor risk contagion on bilayer-coupled networks, heterogeneous mechanism probabilities have “global effect” and “local effect”. The research conclusion provides a theoretical basis for regulators to prevent financial risks from spreading among different investors, which is of high theoretical value and practical significance.
... According to the latest research results on behavioral finance, counterparty risk preference is one of the important factors affecting the probability of credit risk contagion. The degree of counterparty risk preference has the dual effect of promoting and suppressing credit risk contagion [2,13,56,57]. Specifically, risk-averse counterparties help curb the spread and contagion of credit risk, whereas counterparties of risk preference promote the contagion of credit risk. ...
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Let X be an arbitrary real-valued random variable (r.v.), with the characteristic function (c.f.) f. Expressions for the c.f.'s of the r.v.'s \max(0,X) and |X| in terms of (f and) the Hilbert transform of f are given.
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We show whether central clearing of a particular class of derivatives lowers counterparty risk. For plausible cases, adding a central clearing counterparty (CCP) for a class of derivatives such as credit default swaps reduces netting efficiency, leading to an increase in average exposure to counterparty default. Further, clearing different classes of derivatives in separate CCPs always increases counterparty exposures relative to clearing the combined set of derivatives in a single CCP. We provide theory as well as illustrative numerical examples of these results that are calibrated to notional derivatives position data for major banks.
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Networks modeling bilaterally-cleared and centrally-cleared derivatives markets are shown to yield economically different price impact, volatility and contagion after an initial bankruptcy. A large bankruptcy in bilateral markets may leave a counterparty unable to expectationally prevent bankruptcy (checkmate) or make counterparties push markets and profit from contagion (hunting). In distress, bilateral markets amplify systemic risk and volatility versus centralized markets and are more subject to crises with real effects: contagion, unemployment, reduced tax revenue, higher transactions costs, lower risk sharing, and reduced allocative efficiency. Pricing distress volatility may suggest when to transition to central clearing. The model suggests three metrics for the well-connected part of a market - number of counterparties, average risk aversion, and standard deviation of total exposure - may characterize its fragility.
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This paper presents a novel method to price discretely-monitored single- and double-barrier options in Levy process-based models. The method involves a sequential evaluation of Hilbert transforms of the product of the Fourier transform of the value function at the previous barrier monitoring date and the characteristic function of the (Esscher transformed) Levy process. A discrete approximation with exponentially decaying errors is developed based on the Whittaker cardinal series (Sinc expansion) in Hardy spaces of functions analytic in a strip. An efficient computational algorithm is developed based on the fast Hilbert transform that, in turn, relies on the FFT-based Toeplitz matrix-vector multiplication. Our method also provides a natural framework for credit risk applications, where the firm value follows an exponential Levy process and default occurs at the first time the firm value is below the default barrier on one of a discrete set of monitoring dates.
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Credit risk associated with interbank lending may lead to domino effects, where the failure of one bank results in the failure of other banks not directly affected by the initial shock. Recent work in economic theory shows that this risk of contagion depends on the precise pattern of interbank linkages. We use balance sheet information to estimate the matrix of bilateral credit relationships for the German banking system and test whether the breakdownof a single bank can lead to contagion. We find that the financial safety net (institutional guarantees for saving banks and cooperative banks) considerably reduces - but does not eliminate - the danger of contagion. Even so, the failure of a single bank could lead to the breakdown of up to 15% of the banking system in terms of assets.
Central Clearing of OTC Derivatives: bilateral vs multilateral netting
  • Rama Cont
  • Thomas Kokholm
Rama Cont and Thomas Kokholm.`Central Clearing of OTC Derivatives: bilateral vs multilateral netting'. In: Statistics and Risk Modeling 31.1 (2014), pp. 3 22. url: http://dx.doi.org/10.1515/strm-2013-1161.
  • Frederick W King
Frederick W. King. Hilbert Transforms. Vol. 2. Cambridge University Press, 2009.