Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Market Concentration under Central Clearing
Agostino Capponi
Department of Industrial Engineering & Operations Research
Columbia University
[email protected]
Joint work with W. Allen Cheng and Sriram Rajan
2014 Issac Newton Institute
Monitoring Systemic Risk: Data, Models and Metrics
Cambridge, September 23, 2014
Conclusions
Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Talk Outline
1
Introduction
2
The Model
3
Model Calibration
4
Market Concentration
5
Conclusion
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Centrally Cleared Financial Network
CM6 CM1 CM5 CCP CM2 CM4 CM3 A Network with 1 CCP and 6 Clearing Members
Financial reform mandates central clearing
CCPs may reduce counterparty risk
(see Pirrong (2011))
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Goal
CM6 dV1 (t) = ?
CM1 CM5 CCP CM2 CM4 CM3 Develop model for asset values of clearing members
Calibrate model to real data
Analyze market concentration under central clearing
z
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
The Model
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
A building block
Start with the case of only two CMs. (Trading is a gambling
game!)
I put up $2 to play CM1 CM2 I put up $3 to play CM i “puts up” $Vi for trade. Losses will not exceed this amount.
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
A building block
Fair trade means for each dollar
I win with chance 0.4 CM1 CM2 I win with chance 0.6 If each dollar’s ownership are i.i.d., realization is binomial.
3 2
3
5
2
Q(V1 (1) = 3|V1 (0) = 2, V2 (0) = 3) =
5
5
3
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
The two CM-CCP trading model
We repeat this procedure nσ 2 times and shrink the tick size
the limit, we obtain Wright-Fisher Diffusion.
V
n
. In
Proposition
d
Let Vn be the tick size. Then as n → ∞, V1 → V1∗ , where
p
dV1∗ = σ V1∗ (V − V1∗ )dW
CM1 CM2 dV1 (t) = σ V1 (t)V2 (t)dW1 (t)
σ: CCP effciency.
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
The CCP trading model
For the general case...
CM6 CM5 CM1 CCP CM4 V1 (t)
CM2 CM3 V2 (t) +V3 (t) +V4 (t) +V5 (t) +V6 (t)
Taking advantage of the centralized structure
Consistent with CCP anonymity practice
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
CM Asset Values
However, not all assets are used for trading
CCP Por&olio Futures Op0ons Swaps Margins CM1 Total Assets Opera0ng Port. Land Loans Another CCP Port. Split total assets into two portfolios Vi = ViC + Vio
Impose behavioral model of asset redistribution
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
The CCP portfolio
Joint dynamics for CCP portfolios,
OP6 CCP6 OP1 CCP1 OP5 CCP5 CCP CCP4 CCP2 OP4 CCP3 OP3 OP2 dViC = σ
q
ViC (V C − ViC )dWiC
Correlated WiC gives conservation of value
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Behavioral model
CM hedge excessive risk efficiently,
CCP Por;olio Opera&ng Port. Land Loans Another CCP Port. Hedg
e! Excessive Risk Futures Op&ons Swaps Margins dVio = θVio dWio
p
Wio = − ρWih + 1 − ρ2 Wid
| {z } |
{z
}
ex. risk
desired exposure
θ: operating vol.
ρ: ex. risk cor.
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Behavioral model
θ: operating vol.
ρ: ex. risk cor.
σ: CCP efficiency
CM i wants to hedge excessive risk
dVio = θVio dWio = − θρVio dWih + θ
| {z } |
ex. risk
dViC
p
1 − ρ2 Vio dWid
{z
}
desired exposure
q
= σ ViC (V C − ViC )dWih
Hedge equations:
θρVio = σ
q
ViC (V C − ViC ),
for all i!
Q: Do the hedge equations always admit a solution?
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Behavioral model
θ: operating vol.
ρ: ex. risk cor.
σ: CCP efficiency
κ: CCP ratio
A: Yes. Define κi :=
ViC
Vi
. Then rewrite
θρVio
q
= σ ViC (V C − ViC )
⇓
2 2
2
θ ρ (1 − κi )
Vi2
= σ 2 κi Vi
X
κj Vj
j6=i
We prove there always exists exactly one set of κi ∈ (0, 1)
⇒ All CMs have only one choice for redistribution!
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Full model
θ: operating vol.
ρ: ex. risk cor.
σ: CCP efficiency
κ: CCP ratio
CM6 So...
dV1 (t) = θ 1− ρ 2 (1− κ1 (t))V1 (t)dW1d (t)
CM1 CM5 CCP CM2 CM4 CM3 dVi = dViC + dVio
q
p
= θVio d(−ρWih + 1 − ρ2 Wid ) + σ ViC (V C − ViC )dWih
p
= θ 1 − ρ2 (1 − κi )Vi dWid
z
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Model Calibration
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
The Model Calibration
θ: operating vol.
ρ: ex. risk cor.
σ: CCP efficiency
κ: CCP ratio
ξ: risk prem. & cap. raising
Table : Calibration Data over 209 Weeks
Model Var.
ViC
Vio
Proxy
Marginsi
Enterprise Valuei
Data Source
DTCC
Bloomberg
Method
Duffie et al.(2014)
Mkt Cap+STD+0.5LTD
We calibrate
dVi = ξVi dt + θ
p
1 − ρ2 (1 − κi )Vi dWi
⇓
∆(Vio
+
ViC )
= ξ(Vio + ViC )∆t + θ
p
1 − ρ2 Vio ∆Wi
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Margin Model of Duffie et al. (2014),
Based on
∆Position Value ≈ Position Notional×CDS duration×∆CDS spread.
Three components
Initial margin: worst historical loss + short charge for
jump-to-default risk
Variation margin buffer: historical daily change of current
portfolio
Velocity drag: accounts for limits on speed of collateral
circulation
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Results
θ: operating vol.
ρ: ex. risk cor.
σ: CCP efficiency
ξ: risk prem. & cap. raising
Consider ξ ≈ 14%. ρ ≈ 7%. (Minton et al. (2009))
3 moment conditions from our model → GMM for (θ, σ)
Table : Parameter estimates for ρ = 7%, in percent
ξ
10
14
18
20
24
θ̂
12.4
17.9
23.7
26.7
32.7
σ̂
54.1
78.4
104.2
117.4
144.2
s.e.θ̂
7.7
7.3
7.6
7.8
8.1
s.e.σ̂
35.1
30.3
27.2
26.0
24.1
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Results
θ: operating vol.
ρ: ex. risk cor.
σ: CCP efficiency
ξ: risk prem. & cap. raising
Table : Parameter estimates for ρ = 7%, in percent
ξ
10
14
18
20
24
θ̂
12.4
17.9
23.7
26.7
32.7
σ̂
54.1
78.4
104.2
117.4
144.2
s.e.θ̂
7.7
7.3
7.6
7.8
8.1
s.e.σ̂
35.1
30.3
27.2
26.0
24.1
High ξ → high θ. (Losers raise capital faster).
High θ → high σ. (Higher risk means more efficient CCP).
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Results
θ: operating vol.
ρ: ex. risk cor.
σ: CCP efficiency
ξ: risk prem. & cap. raising
Table : Parameter estimates for ρ = 7%, in percent
ξ
10
14
18
20
24
θ̂
12.4
17.9
23.7
26.7
32.7
σ̂
54.1
78.4
104.2
117.4
144.2
s.e.θ̂
7.7
7.3
7.6
7.8
8.1
s.e.σ̂
35.1
30.3
27.2
26.0
24.1
√
ξ = 14%: σCCP ≈ 78.4 × 19 = 341.7%
341.7
$1 of CCP port. assets → 17.9×0.07
≈ $272.7 of operating
port. (Banks too aggressive?)
κi = ViC /Vi ≈ 1/273.7 ≈ 0.37%. Empirical average 0.56%
z
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Market Concentration
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Empirical Observation
Enterprise Value Herfandahl Index (ICE Clear Credit CMs)
Can we explain the increasing trend in market concentration?
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Market Concentration
θ: operating vol.
ρ: ex. risk cor.
σ: CCP efficiency
κ: CCP ratio
Proposition
p
Let dVi = θ 1 − ρ2 (1 − κi )Vi dWi and supi6=j ρ(Wi , Wj ) ≤
Then the Herfandahl index is a submartingale.
√1 .
2
(≈ ρi ≤ 84% to systematic factor)
The Herfandahl index increases when market is “normal”.
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Policy Implications
Accumulation of systemic risk when market is normal
We show can be circumvented with capital injection. But
leads to increase in leverage (Adrian and Shin (2011))
Systemic Risk!.
We break this “trade off” with a self-funded tax/subsidy
system
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Tax and Subsidy System
θ: operating vol.
ρ: ex. risk cor.
σ: CCP efficiency
µ: policy rate
Consider tax rate of γ1 and subsidy rate of γ2 :
p
dVi = θ 1 − ρ2 Vio dWid − γ1 ViC dt + γ2 V C − ViC dt
Self funding condition:
−γ1 (t)V C (t) + γ2 (t)(N − 1)V C (t) = 0.
Policy rate µ(t) = γ1 (t) + γ2 (t) > 0. Then
p
VC
o
d
C
2
dVi = θ 1 − ρ Vi dWi − µ Vi −
dt
N
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Policy Implications
θ: operating vol.
ρ: ex. risk cor.
σ: CCP efficiency
µ: policy rate
Proposition
p
VC
o
d
C
2
Let dVi = θ 1 − ρ Vi dWi − µ Vi −
dt. Then there
N
exists a process µ such that the Herfandahl index is a
supermartingale.
Larger µ contributes a stronger downward “drift” to the
Herfandahl index ⇒ The Herfandahl index is “pulled” down
z
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Conclusion
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Conclusions
We give an analytically tractable model for clearing member
assets value dynamics
Demonstrated calibration to real data and discussed
interpretations
Analyzed market concentration and discussed systemic risk
implications
z
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
References
Adrian , T., and Shin, H., Financial Intermediary Balance
Sheet Management, Federal Reserve Bank of New York Staff
Reports, no. 532, 2011
Duffie, D., Scheicher, M., and Vuillemey, G., Central Clearing
and Collateral Demand, Journal of Financial Economics
(forthcoming), 2014.
Minton, B., Stulz, R., and Williamson, R., How Much Do
Banks Use Credit Derivatives to Hedge Loans?, Journal of
Financial Services Research, 35(1), 1-31, 2009.
Pirrong, C., The Economics of Central Clearing: Theory and
Practice, ISDA Discussion Paper Series, 1, 2011.
Capponi, A., Cheng, W., and Rajan, S., Market Concentration
under Central Clearing, Working paper
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Talk Outline
Introduction
The Model
Model Calibration
Market Concentration
Conclusions
Thank You
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Appendix
Other Results
Proposition
Vi > Vj if and only if ViC > VjC .
Moreover, we expect this relationship to be superlinear as proven
in the 2 CM case.
1
0.9
0.8
0.7
0.6
κ1
0.5
0.4
0.3
0.2
δ
δ
δ
δ
0.1
=
=
=
=
0.1
1
10
10 0
0
1
2
3
4
5
6
7
8
9
10
V1
V2
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Appendix
The Data
Figure : Margins v.s. Enterprise Value
$7.0B
$6.0B
margins, $B
$5.0B
$4.0B
$3.0B
$2.0B
$1.0B
$0.0B
$-1.0B
B
$0.0
0.0B
$20
0.0B
$40
.0B
0
$60
0.0B
$80
.0B
00
$10
enterprise values, $B
.0B
00
$12
0.0B
0
$14
$16
00.0
B
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Appendix
GMM Conditions
Hedge equation
θρVio = σ
q
ViC (V C − ViC )
⇓
1
log θ + log ρ + log Vio = log σ + (log ViC log(V C − ViC ) + ε
2
Full model
dVi = ξVi dt + θ
p
1 − ρ2 (1 − κi )Vi dWid
Full model (ii)
dVi = ξVi dt + θ(1 −
κi )Vi dWio
q
+ σ ViC (V C − ViC )dWiC .
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Appendix
Limiting procedure
We have U time periods, (tu , tu+1 ), each of length
∆t := tu+1 − tu .
Each time period is sectioned into m equal periods of length
∆m t.
Each CM can rebalance at the beginning of each smaller
period of length ∆m t but can only redistribute at the
beginning of each larger period of length ∆t.
∆m t → 0, then ∆t → 0.
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Appendix
Limiting procedure
In practice, the time needed to trade a security with assets already
posted to a counterparty is much shorter than the time needed to
transfer assets to a counterparty. Our time frames are designed to
capture this empirical feature. Indeed, a portion of empirically
observed margins is used to account for the time lag in transferring
collateral (Duffie et al. (2014)).
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