Estimation of Future Initial Margins Marc Henrard Advisory Partner - - PowerPoint PPT Presentation

Estimation of Future Initial Margins

Multi-Curve Interest Rate Framework

Marc Henrard

Advisory Partner - OpenGamma Visiting Professor - University College London

March 2016 2

Estimation of Future Initial Margins

1

Initial margin, multi-curve and collateral framework

2

Rational model

3

IM dynamics

4

Margin value adjustment

March 2016 3

Margins

1

Initial margin, multi-curve and collateral framework

2

Rational model

3

IM dynamics

4

Margin value adjustment

March 2016 4

Variation Margin - Initial Margin - MPR

Time in days

• 20
• 15
• 10
• 5

5 10

Value

• 10
• 5

5 10 15 Past Future IM VM

March 2016 5

Variation Margin - Initial Margin - MPR

Time in days

• 20
• 15
• 10
• 5

5 10

Cash flows / Value

• 10
• 5

5 10 15 Past Future IM

March 2016 6

Regulatory time table

2013 Mandatory clearing (USA) 2016 – 2019 Mandatory Central Clearing (Europe) EMIR Category 1: 21 June 2016 Front-loading as of 21 February 2016 2016 – 2019 Mandatory Bilateral margin Category 1: 1 September 2016 for VM and IM

March 2016 7

Cash collateral pricing formula

Nc

t = exp

(∫ t cτdτ )

Theorem (Collateral with cash price formula)

In presence of cash collateral with rate c, the quote at time t of an asset with price Vc

u at time u is

Vc

t = Nc tEQ [

(Nc

u)−1Vc u

• Ft

] for some measure Q (identical for all assets, but potentially currency-dependent). Note that the result refers to three “objects”: Vu, c and Q. This formula is also called collateral account discounting.

March 2016 8

Collateral: pseudo-discount factors

Definition (Collateral pseudo-discount factors)

The collateral (pseudo-)discount factors for the collateral rate c paid in currency X are defined by Pc(t, u) = Nc

tEQ [

(Nc

u)−1

Ft ] . In the sequel we will work with OIS discounting and use the notation Du = (Nc

u)−1.

Change of numeraire is still possible in this framework. In particular we will introduce a different measure, called M, and use the notation (Dt)−1EQ [ DuVc

u

• Ft

] = (ht)−1EM [ huVc

u

• Ft

]

March 2016 9

Multi-curve framework with collateral

I The value of a j floating coupon in currency X with collateral at rate c is an asset for each tenor j, each fixing date θ and each collateral rate c.

Definition (Forward index rate with collateral)

The forward curve Fc,j

t (θ, u, v) is the continuous function such

that, Pc(t, v)δFc,j(t, u, v) is the quote at time t of the j-Ibor coupon with fixing date θ, start date u, maturity date v (t ≤ t0 ≤ u = Spot(t0) < v) and accrual factor δ collateralised at rate c.

March 2016 10

Margins

1

Initial margin, multi-curve and collateral framework

2

Rational model

3

IM dynamics

4

Margin value adjustment

March 2016 11

Rational model

Macrina, A. (2014), Crépey, S. et al. (2015) Formulas for the discounting and forward in the rational model are Pc(t, u) = Pc(0, u) + b1(u)A(1)

t

Pc(0, t) + b1(t)A(1)

t

Lc,j(t; u, v) = Lc,j(0; u, v) + b2(u, v)A(1)

t

+ b3(u, v)A(2)

t

Pc(0, t) + b1(t)A(1)

t

where A(i)

t

is a martingale in a M-measure with A(i)

t

= exp ( aiX(i)

t

− 1

2a2 i t

) − 1 where j is an Ibor index and v − u = Tenor(j). Note: The Lc,j(t; u, v) in the rational model corresponds to the Pc(t, v)Fc,j(t, u, v) in the standard multi-curve framework.

March 2016 12

Rational model - spread

Let’s define Lc(t; Ti−1, Ti) = 1 δi ( Pc(t, Ti−1) − Pc(t, Ti) ) with δi the accrual factor for the period [Ti−1, Ti]. The dynamics for this quantity is described by Lc(t; Ti−1, Ti) = Lc(0; Ti−1, Ti) + (b1(Ti−1) − b1(Ti))/δiA(1)

t

Pc(0, t) + b1(t)A(1)

t

Lc,j(t; u, v) = Lc(0) + (b1(u) − b1(v))/δiA(1)

t

Pc(0, t) + b1(t)A(1)

t

+(Lc,j(0)−Lc(0)) + (b2−·b1·)A(1)

t

+ b3(u, v)A(2)

t

Pc(0, t) + b1(t)A(1)

t

March 2016 13

Rational model - calibration

Calibration one-factor model, term structure.

10 20 30 40 50 60 70 80 90

P1YxP10Y P2YxP3Y P2YxP4Y P2YxP5Y P2YxP7Y P2YxP10Y P3YxP4Y P3YxP5Y P3YxP7Y P3YxP10Y P4YxP4Y P4YxP5Y P4YxP7Y P4YxP10Y P5YxP4Y P5YxP5Y P5YxP7Y P5YxP10Y P7YxP3Y P7YxP4Y P7YxP5Y P7YxP7Y P7YxP10Y P10YxP1Y P10YxP2Y P10YxP3Y P10YxP4Y P10YxP5Y P10YxP7Y

Black vol (%)

Market Calibrated

March 2016 14

Rational model - calibration

Calibration one-factor model, smile.

Moneyness (%)

• 1
• 0.5

0.5 1 Black vol (%) 35 40 45 50 55 60 Market 5Yx5Y Calibrated 5Yx5Y Market 5Yx10Y Calibrated 5Yx10Y

March 2016 15

Margins

1

Initial margin, multi-curve and collateral framework

2

Rational model

3

IM dynamics

4

Margin value adjustment

March 2016 16

Initial margin

Initial margin are usually computed as Value-at-Risk (VaR) or Expected Shortfall (ES) using historical or Monte-Carlo approaches. Direct brute force calculation of future IM and MVA might be expensive. Historical VaR usually implies full revaluation Some simplifying methods neglect stochastic interactions between IM and market. Nested Monte Carlo iterations are very expensive. Our approach: Characterize the initial margin process in terms of the dynamics of the underlying processes and a conditional risk measure.

March 2016 17

Abstract formulation

Initial Margin

The initial margin process {IMt}0≥t≤T associated to the portfolio is a process such that IMt = λt(Zt,t+δ) where Zt,t+δ are the cash flows given default associated with the portfolio between times t, t + δ (loss given default) δ is the margin period of risk (time to “close out”) {λt}0≤t≤T is a family of conditional risk measures

March 2016 18

Risk measure

Conditional risk measures: λt maps any F measurable r.v. to an Ft-measurable r.v. with finite expectation. Examples: If P∗ denotes the subjective probability observed by a CCP: VaRα,P∗

t

[X] = ess inf {Θ : P∗[X ≤ Θ|Ft] ≤ α a.s., Θ is Ft-measurable} ESα,P∗

t

[X] = VaRα,P∗

t

[X] +

1 1−αEP∗ [

(X − VaRα,P∗

t

[X])+|Ft ] Notation: ¯ λt(X) = −λt(−X)

March 2016 19

Risk measure

Properties: We use several properties satisfied by VaRα,P∗

t

and ESα,P∗

t

; Let X, Y be two F−measurable r.v. and let Θ be an Ft-measurable r.v. with EP∗ [Θ] < ∞. We assume: Normalization: λt(0) = 0 P-a.s. Monotonicity : X ≤ Y(P − a.s.) ⇒ λt(X) ≤ λt(Y). (P-a.s.) Conditional positive homogeneity: if Θ ≥ 0(P − a.s.) ⇒ λt(ΘX) = Θλt(X). Conditional translation: λt(X + Θ) = λt(X) + Θ

March 2016 20

Risk measure

Cashflow decomposition: Vt = 1 Dt EQ [

n

∑

i=1

DTiCTi

• Ft

] Value given default: (signs for member, assuming ”closing

• ut” exactly at δ ):

Zt,t+δ = 1 ¯ D(t, t + δ)Vt+δ

• “close out” value

+

• n

∑

i=1

1 ¯ D(t, Ti)CTi1{Ti∈[t,t+δ)}

Cashflows missed in [t, t + δ)

− Vt

• Variation margin

at t

¯ D: discounting term associated to the funding of CCP (may differ from D)

March 2016 21

Risk measure

Cashflow decomposition: Vt = 1 Dt EQ [

n

∑

i=1

DTiCTi

• Ft

] Value given default: (signs for member, assuming ”closing

• ut” exactly at δ ):

Zt,t+δ = 1 ¯ D(t, t + δ)Vt+δ

• “close out” value

+

• n

∑

i=1

1 ¯ D(t, Ti)CTi1{Ti∈[t,t+δ)}

Cashflows missed in [t, t + δ)

− Vt

• Variation margin

at t

¯ D: discounting term associated to the funding of CCP (may differ from D)

March 2016 22

Initial margin computation

IMt = λt ( 1 ¯ D(t, t + δ)Vt+δ − Vt +

n

∑

i=1

1 ¯ D(t, Ti)CTi1{Ti∈[t,t+δ)} ) Rational framework leads to complexity reduction: Profit from the explicit/semi-explicit expressions available for most common IR derivatives with rational framework under the M measure: Vt = Vt(A(1), A(2)) ; Ct = Ct(A(1), A(2)) Exploit numerically the simplify setup Introduce historically estimated change of measure between M and P∗ (CCP scenario information)

March 2016 23

Initial margin computation

IMt = κESα,P∗

t

( ht+δ ht Vt+δ − Vt +

n

∑

i=1

hTi ht Ct+δ1{Ti∈[t,t+δ)} ) Rational framework leads to complexity reduction: Profit from the explicit/semi-explicit expressions available for most common IR derivatives with rational framework under the M measure: Vt = Vt(A(1), A(2)) ; Ct = Ct(A(1), A(2)) Exploit numerically the simplify setup Introduce historically estimated change of measure between M and P∗ (CCP scenario information)

March 2016 24

Initial margin computation

IMt = κESα,P∗

t

( ht+δ ht Vt+δ − Vt +

n

∑

i=1

hTi ht Ct+δ1{Ti∈[t,t+δ)} ) Rational framework leads to complexity reduction: Profit from the explicit/semi-explicit expressions available for most common IR derivatives with rational framework under the M measure: Vt = Vt(A(1), A(2)) ; Ct = Ct(A(1), A(2)) Exploit numerically the simplify setup Introduce historically estimated change of measure between M and P∗ (CCP scenario information)

March 2016 25

Change of measure between M and P∗

Change of measure at time δ: Comparing the model and historical distribution of P&L

• 2
• 1

1 2 ×104 0.2 0.4 0.6 0.8 1 ×10-4

PnL CCP vs. model

model CCP data

• 2
• 1

1 2 ×104 0.2 0.4 0.6 0.8 1 ×10-4 PnL CCP vs. adj. model

model CCP data

Change of measure after time δ: Using independent increments assumption and iterating

March 2016 26

IM: Interest rate swaps

Assume no payments in [t, t + δ]. Then: IMt = κESα,P∗

t

( Swt − ht+δ ht Swt+δ ) Since the swap price processes satisfy Swt = c0(t) + c1(t)A(1)

t

+ c2(t)A(2)

t

P(0; t) + b0(t)A(1)

t

, the properties of the risk measure and the assumed model imply IMt = C0

t + C1 t v(Rt) + C2 t ¯

v(Rt) where C0, C1, C2, R are Ft-meas. (depend only on t, A(1), A(2)), v(x) := κESα,P∗ ( Y(1)

δ

+ xY(2)

δ

) ; Y(i)

δ := A(i) δ + 1

for i = 1, 2.

March 2016 27

Functions v, ¯ v

The functions v,¯ v we just defined are smooth and increasing and can be approximated by a grid with few elements.

• 30
• 20
• 10

10 20 30 R

• 70
• 60
• 50
• 40
• 30
• 20
• 10

10 20 VaR and ES, α=0.003 VaR ES

• 15
• 10
• 5

5 10 15 R

• 10
• 5

5 10 15 20 25 30 35 VaR and ES, α=0.997 VaR ES

*Values obtained with 4 · 107 MC iterations

March 2016 28

Numerical estimation (IM)

IM for a 5 years future swap with tenor of 6 months

2 4 6 8 10

t (years)

0.5 1 1.5 2 2.5 3 3.5 4 ×104

IM - ES 5 x 5 swap- M

Mean value (yellow) and 90% and 10% percentiles (red and blue)

March 2016 29

Numerical tests: speed

CPU time in prototype implementation of MC2 and the refined method Paths⋆ Grid† MC2 (s) Refined Method (s) Accel. points Initial Evolution Total factor 1250 120 28.52 5.73 0.20 5.93 4.81 1250 520 126.32 5.76 0.76 6.52 15.83 2500 120 94.21 5.72 0.23 5.95 19.37 2500 520 423.37 5.73 0.92 6.65 63.66

⋆ 1250 paths ≈ 5yrs history; 2500 ≈ 10yrs history. † 120 ≈ monthly periodicity; 520 ≈ weekly periodicity.

March 2016 30

Margins

1

Initial margin, multi-curve and collateral framework

2

Rational model

3

IM dynamics

4

Margin value adjustment

March 2016 31

Cost of IM: Funding rate

Between times u and δu, a clearing member pays on average the funding cost:

• rf

u

funding rate

IMu δu Assumption: The treasury of a clearing member funds all liquidity requirements by securing a basket of funds with best-matching maturities. rf

t := M

∑

k=1

γkL ( t, T∆k

i∗

k (t), T∆k

i∗

k (t)+1

) + A(3)

t

− rt where ∆1, . . . , ∆M ∈ R+ are maturities γ1, . . . , γM ∈ [0, 1] with ∑M

k=1 γk = 1 weights

A(3) is an idiosyncratic factor.

March 2016 32

MVA

Then, the (Follmer Schweizer) price of MVA can be modeled as MVAt = 1 Dt EQ [∫ T

t

Du rf

u IMu du

• Ft

] + Ht = 1 ht EM [∫ T

t

hu rf

u IMu du

• Ft

] + Ht where H is a martingale orthogonal to the tradable assets. IM and rf are Markovian: functions of the underlying factors. In the swap portfolio case and A(3) ≡ 0, the MVA is hedgeable and H ≡ 0.

March 2016 33

Numerical MVA

MVA for a 5 years future swap with tenor of 6 months (A(3) ≡ 0)

2 4 6 8 10

t (years)

100 200 300 400 500 600

MVA with IM - ES 5 x 5 swap- M

Mean value (yellow) and 90% and 10% percentiles (red and blue)

March 2016 34

Conclusion

Proposed a method based on explicit computation of the IM as a dynamic process by itself. Numerical implementation can become efficient as we remove one layer of numerical effort. The implementation is basically portfolio size invariant:

• nly the cash flow description (ci(t)) is portfolio

composition dependent. The approach clearly differentiate between pricing and risk measures. In particular the IM computation includes the tail used in the actual CCP margin computation. The dynamic IM can be used as a base to MVA computations.

March 2016 35

Presentation based on the paper

• C. A. Garcia Trillos, M. P. A. Henrard, A. Macrina (2016) Estimation of

Future Initial Margins in a Multi-Curve Interest Rate Framework http://ssrn.com/abstract=2682727

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