CONT-Central Clearing of Interest Rate Swaps a Comparison of Offerings

8/13/2019 CONT-Central Clearing of Interest Rate Swaps a Comparison of Offerings

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http://slidepdf.com/reader/full/cont-central-clearing-of-interest-rate-swaps-a-comparison-of-offerings 2/25Electronic copy available at: http://ssrn.com/abstract=1783798


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F t

N 0 = 1) T 0

T C T 1,T 2 T N

T (fl)1 ,T

(fl)2 T

(fl)M :

F t = Et

N T i>t

B−1T iC ∆i −

M T (fl)j >t

B−1T (fl)j

LT (fl)j−1 ,T

(fl)j τ j

,

∆i τ j

LT (fl)j−1 ,T

(fl)j

T (fl)j−1

Et

t

Bt

T N = T (fl)M

P T t t T

t ≤ T 0

F t = C

N i=1

P T it ∆i − P T 0t + P T (fl)M

t .

T (fl)S F t

F t = C

N T i>t

P T it ∆i − (1 + LT (fl)S−1,T

(fl)S τ S )P

T (fl)S

t + P T (fl)M

t ,

S T (fl)S −1 < t ≤ T

(fl)S



F t

F t

F t − F t− = −C ∆i, t T i

F t − F t− = LT (fl)j−1 ,T

(fl)j τ j , t T

(fl)j

T i,

C ∆i, T (fl)j

−LT (fl)j−1 ,T

(fl)j τ j

F 0.

F 0,

V 0 =ti≤T

E

B−1ti (F ti − F ti−1)

+ F 0 + F 0,

E0

V 0 =

E

{ˆ T

0

B−1

s dF

s} + 2F 0

Bt dBt = rtBtdt rt t

F T = 0 E{´ T 0

B−1s dF s} =

E{´ T 0

rsF sB−1s ds}+ E{F T B−1T } −F 0B−10 = E{

´ T 0

rsF sB−1s ds}−F 0

V 0 = E{

ˆ T 0

rsF sB−1s ds} + F 0.

V 0 =´ T 0

rsF sB−1s ds + F 0

rs,F s Bs

V 0

V t = E{´ T

0 rsF sB

−1s ds} ∀t > 0



drt = (Θ(t) − art)dt + σdW t,

Θ(t)

a σ

a → 0

F t

V 0

V i0

C ∆i T i

V i0 = ˆ T i

0

E{rsC ∆iP T is B−1s }ds + C ∆iP T i0

= C ∆iP T i0

ˆ T i0

dsET i{rs} + C ∆iP T i0 .

ET i

P T it

V i0 = C ∆iP T i0

σ2e−2aT i

−4eaT i + 3e2aT i + 1

4a3

− σ2T i

2a2 + T iRT i + 1

a → 0

V i0 → C ∆iP T i0 (RT iT i −

1

6σ2T 3i + 1),

RT i T i, P T i0 = e−RT iT i .

V i0

C ∆iP T i0



RT iT i

T i

σT 32i ,

V i0,fl i

T i T (fl)i

τ i = T i − T i−1

V i0,fl = E{

ˆ T i0

rsB−1s LT i−1,T iτ ids} + LT i−1,T i0 τ iP T i0 ,

LT i−1,T iτ i = 1P T iT i−1

− 1

T i−1

s = T i−1,

V i0,fl = P T i0

−

σ2e−2aT i

4a3 +

σ2e−aT i

a3 −

3σ2

4a3 +

σ2T i

2a2 − T iRT i

+ P T i−1

0

3σ2e−2aT i

4a3 +

σ2e−2aT i−1

a3

− σ2e−aT i−1

a3 +

σ2e−aτ i

2a3

− σ2

e−2aτ i

2a3 − 3σ

2

e−a(T i+T i−

1)

2a3 + 3σ

2

4a3 − σ

2

T i2a2

+ T iRT i

+ L

T i−1,T i0 τ iP T i0

a → 0

V i0,fl = P T i0 L

T i−1,T i0 τ i(RT iT i −

1

6σ2T 3i + 1 ) +

1

2P T i−10 σ2T 2i−1τ i

1 −

2τ i

T i−1

,

LT i−1,T i0

V i0

T i−1

t = 0



T

C V 0 = V i0 −V i0,fl

F 0

T = 10

σ

N 0 V 0

V 0 = N 0

N i=1

V i0 −M i=1

V i0,fl

,



σ a

a = −0.003

σ = 74



12

14

16

18

20

22

24

26

28

30

1212010

18

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0

50

100

150

200

Mean Reversion Rate years1

S h o r t R a t e O U

V o l B a s i s P t s

IDCG Swap Maturity10Y



20

25

30

35

40

45

50

55

1212010

60

0.0 0.1 0.2 0.3 0.4 0.5 0.6

0

50

100

150

200

Mean Reversion Rate years1

S h o r t R a t e O U

V o l B a s i s P t s

IDCG Swap Maturity30Y

a σ

a = 0.107

σ = 158



•



•

S t

S T = BT (N i=1

B−1T iC ∆i −

M j=1

B−1T (fl)j

LT (fl)j−1 ,T

(fl)j τ j −

ˆ T 0

rtB−1t S tdt).

C i T i

S T = BT (N i=1

B−1T iC i −

ˆ T 0

rtB−1t S tdt)}

= BT (

N i=1

B−1T iC i +

ˆ T 0

S tdB−1t )

= BT

N i=1

B−1T iC i + BT (S T B

−1T − S 0 −

ˆ T 0

B−1t dS t)

= B

T

N i=1

B−1

T iC i +

X −

BT

S 0 −

BT ˆ T

0

B−1

t dS

t.

BT

S 0 +

ˆ T 0

B−1u dS u =N i=1

B−1T iC i.

S t

F t S 0 =E0{

N i=1 B−1T i

C i} = F 0

t

t



S 0 +ˆ t0 B

−1u dS u =

N T i≤t

B−1T i C i + Et{

N T i≤t

B−1T i C i}.

LHS = S 0 + S tB−1t −

S 0B−10 +´ t0

ruB−1u S udu = S tB−1t +´ t0

ruB−1u S udu. Et{N

T i≤tB−1T i

C i} =

B−1t Et{N

T i≤tBtB−1T i

C i} = B−1t F t

S tB−1t +

ˆ t0

ruB−1u S udu = B−1t (

N T i≤t

BtB−1T iC i + F t).

Bt

S t =N

T i≤t

BtB−1T iC i + F t − Bt

ˆ t

0

ruB−1u S udu.

t

´ t

0 B−1

u dS u = N T i≤t

B−1T iC i+

Et{N

T i≤t B−1T i C i}−S 0 =

N

T i≤t B−1T i C i + B

−1t F t−S 0

0 t

t

t 0

0

t

S u − S 0





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CONT-Central Clearing of Interest Rate Swaps a Comparison of Offerings