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description
Yield-X: The Market WorksThe workings of the Yield-X yield curves,
products and Calm margining system.
Rogan Etheredge
07 February 2005
Quant Financial Research 06 Feb 2005 2
Programme
• Introduction to Yield-X
• Yield curves
• Products
• Margining
Quant Financial Research 06 Feb 2005 3
Yield-X Products• jBonds: liquid RSA bonds (in Govi)• jCarries: 1 and 13 weeks on jBonds• jFutures: futures on jBonds• jTRIs and jGOVI: futures on Total Return Indices• jSwaps: bond lookalike swaps• jRods: swaps against overnight interest rate• jNotes: futures on notional swaps• jFRAs: futures on FRAs / Jibar• jOptions: options on all futures
Quant Financial Research 06 Feb 2005 4
Physical Futures
jBonds
jCarries
jSwaps
jRods
jFutures
jGOVI
jNotes
jFRAs
Implicit mark-to-market Explicit mark-to-market
jOptions
Underlying: R194, R153, R201, R157, R203, R186
Real and notional interest rate swaps on Jibar and RODI
Yield-X Products
Quant Financial Research 06 Feb 2005 5
Explicit mark-to-market
Loss Profit
Market and Position
Totalmargin
Quant Financial Research 06 Feb 2005 6
Implicit mark-to-market
Loss Profit
Market Position
Totalmargin
Unrealisedloss
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Trading on Yield-X• Screen-based central order book
• Yield to three decimal places (1/10th basis point)
• Central counterparty:– settlement guaranteed– margining
• Report-only trades:– derivatives fall within risk positions and are guaranteed– jBonds only if both Clearing members agree
Quant Financial Research 06 Feb 2005 8
Mark-to-market
Closing yields
Yield curves
Market yields
Markto
market
Futures and
options
Quant Financial Research 06 Feb 2005 9
Margining
All positionsAll instruments: full offset.
Estimate maximum loss which position can suffer:next 24 hours, 99.95% confidence.
Loss is potential exposure of the Exchange:take the amount in margin now!
Margin earns interest at wholesale rates;
re-estimated daily; returned when position is closed.
Quant Financial Research 06 Feb 2005 10
Daily cashflows
• Explicit mark-to-markets
• Change in margin
• Settlements of spot bonds
• Net interest payments on jSwaps and jRods
• Conversion payments
• Interest on margin
• Booking fees
Quant Financial Research 06 Feb 2005 11
Yield curve methodology
• BEASSA
• Perfect fit
• Available on website daily to 30 years
Quant Financial Research 06 Feb 2005 12
Four curves
• Yield-X Curve– RODI, 3 mth Jibar, jFRAs, jNotes
• jBonds curve– RODI, 1 and 13 week jCarries, jBonds
• jSwaps curve– RODI, 3 mth Jibar, jSwaps
• jRods curve– RODI, jRods
Quant Financial Research 06 Feb 2005 13
Spreads
• Market price paramount• Instruments in generating set have zero spreads• Others have spreads calculated so that the Curve +
Spread prices them back to the market• Instruments without market prices priced off their
curve, using their latest recorded spread
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Use of curves
• Market information
• Pricing illiquid instruments
• Resets
• Close-out prices of jFRAs and jNotes
• Margining
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jBonds• R194, R153, R201, R157, R203, R186• Quoted on yield to maturity• Bond pricing formula ("BPF")• Spot: t+3• Forward: settlement date by agreement• On-screen through central order book:
– included in Risk Positions; – settlement guaranteed
• Report only:– both Clearing Members must explicitly agree
• Guaranteed forwards subject to yield resets• Settlement margins
Quant Financial Research 06 Feb 2005 16
jCarries• On all jBonds• 1st leg settled on t+3
– priced at current market price, using BPF
• 2nd leg settled 1week or 13 weeks after t+3• Quoted on carry rate as a simple yield between settlement
dates– quoted rate converted to ccr to find price of 2nd leg
• jCarries are on-screen through the central order book:– included in risk positions– settlement of both legs guaranteed
• 2nd leg subject to yield resets• Settlement margins
Quant Financial Research 06 Feb 2005 17
jFutures
• On all jBonds• Expire 1st business Thursday of Feb, May, Aug and Nov• Physically settled on t+3 of expiry date• Quoted on yield to maturity• Priced using BPF for settlement date• Settlement guaranteed• Settlement margins• Safex positions will be carried over
to Yield-X
Quant Financial Research 06 Feb 2005 18
jSwaps
• Bond lookalike swaps on all jBonds– j194, j153, j201, j157, j203, j186
• Expire on parent bonds' (mid) redemption dates• Semi-annual interest payments on same days as parent bonds'• Floating rate is 3-mth Jibar, compounded at mid interest period• Quoted on fixed rate: yield for the period• Standard fixed rate, reset on interest payment dates (or when market
has moved) to market rate, rounded to 25 bps• Dealt positions are converted to the standard rate• Conversion payments: PV off jSwaps Curve (+ spread) of difference
in fixed interest payments• Interest payments net through the Exchange• Guaranteed by the Exchange
Quant Financial Research 06 Feb 2005 19
jSwap Interest Calculations
Next interest date,di,ti
Last interest date,di-1,ti-1
Floating interest payments at f (known or implied)
Fixed interest payments at x
3-mth Jibar: fi1
Mid-periodreset date,
Di,Ti
3-mth Jibar: fi2
1 1 2100 1 1 1i i i i i iL f T t f t T
1100 i iX x t t
Quant Financial Research 06 Feb 2005 20
jSwap Equation
V N f T t f t T e x t t e ez ti i
z t
i
nz ti i n n
1 11 1 0 2 1 1 11
1 1
Quant Financial Research 06 Feb 2005 21
jRods
• Swaps of floating RODI against fixed 3-mth rate• Expire monthly on last business day of each month• Quarterly interest payments on last business day of month• Floating rate is jRODI (RODI converted from nacm to nacd),
compounded daily• Quoted on fixed rate: yield for the period• Standard fixed rate, reset on interest payment dates (or when market
has moved) to market rate, rounded to 25 bps• Dealt positions are converted to the standard rate• Conversion payments: PV off jRods Curve (+ spread) of difference in
fixed interest payments• Interest payments net through the Exchange• Guaranteed by the Exchange
Quant Financial Research 06 Feb 2005 22
jRod Interest Calculations
m-j: 0 1 2 3 … j 1 m
Next interest date,di,ti
Last interest date,di-1,ti-1
Floating interest payments at f (known or implied)
Fixed interest payments at x
1100 i iX x t t
1
0
100 1 1365
mj
j
fL
jRODIs: fj
Quant Financial Research 06 Feb 2005 23
The jRod Equation
V N e x t t e ef
j
lZ T
i iz t
i
nz tj i i n n
1 3650
1
11
1 1
Quant Financial Research 06 Feb 2005 24
jNotes• Futures on notional swaps• Expire 1st business Thursday of Feb, May, Aug
and Nov• Swaps are for 2, 5 and 10 years from expiry date• Quarterly interest payments on 1bThu 2, 5, 8, 11
cycle• Floating rate is 3-mth Jibar• Quoted on fixed rate: yield for the period• Marked-to-market at closing fixed rate• Cash settled on expiry off the Yield-X Curve
Quant Financial Research 06 Feb 2005 25
jNote Interest Calculations
Next interest date,di,ti
Last interest date,di-1,ti-1
Floating interest payments at f (known or implied)
Fixed interest payments at x
3-mth Jibar: fi
1100 i i iL f t t
1100 i iX x t t
Quant Financial Research 06 Feb 2005 26
jFRAs• Futures on FRAs / Jibar• Expire 1st business Thursday of Feb, May, Aug
and Nov• FRA interest payment date is 3 months from
expiry on 1bThu 2, 5, 8, 11 cycle• Floating rate is 3-mth Jibar• Quoted on fixed rate: yield for the period• Marked-to-market at closing fixed rate• Cash settled on expiry off the Yield-X
Curve / 3 mth Jibar
Quant Financial Research 06 Feb 2005 27
jFRA Interest Calculations
Interest date,d1,t1
Contact expiry date,d0,t0
Floating interest payment at f (known or implied)
Fixed interest payment at x
3-mth Jibar: f1
1 1 0100L f t t
1 0100X x t t
Quant Financial Research 06 Feb 2005 28
jOptions
• On all futures contracts• American calls and puts• Expire on underlying future's expiry date• Strike prices in same units as future's quote• Each option on one futures contract• Fully margined: premium not paid up-front, but realised over the life of the
option through the mark-to-market process• Naked options traded on price• Options with delta traded on volatility• Early exercise allowed but never optimal• Automatic exercise of options more than 2.5% in
the money on expiry• Long options may be abandoned• Pricing via Modified Black Option Formula
Quant Financial Research 06 Feb 2005 29
Volatilities and Skews
• Time structure: each expiry has its own atm vol• Skew: Moneyedness grid defines additive skew
for various degrees of moneyedness:Vol = Atm Vol + S[Moneyedness]
• Grid and atm vol found by weighted linear regressions of traded volatilities:
– weight by vega * number of options– atm vol also weighted by time since trade– historical trades also brought forward,
with decay factor
• Manual override possible
Quant Financial Research 06 Feb 2005 30
Yield-X Margining
• Comprehensive
• All-embracing
• Logical
• Margining
Calm
Quant Financial Research 06 Feb 2005 31
Two prerequisites
• Logical: all interest rate instruments may be valued by reference to a single construct:
The yield curve
• Technical: the existence of reliable yield curves
Quant Financial Research 06 Feb 2005 32
Margining Factors Program
Statistical analysis of historical data to create
margining factors
Margin Calculator
Calculation of value-at-risk of positions
Historicalmarket
data
Mark-to-market data for the day
Positionsof
participants
Marginsof
participants
Margining
factors
The two stages of margining
Quant Financial Research 06 Feb 2005 33
Data
• Zero coupon yield curves from 08 May 2002: 685 days– Data cleaned– Extend backwards– Yield-X Curve when available
• 1d, 1w, 1m, 3m-12m, 18m-60m, 6y-30y
Quant Financial Research 06 Feb 2005 34
Historical Yields
6.0
7.0
8.0
9.0
10.0
11.0
12.0
13.0
14.0
08-May-02 08-Nov-02 08-May-03 08-Nov-03 08-May-04
Yie
ld (
cc
r)
1d
60m
30.00y
Historical yields
Quant Financial Research 06 Feb 2005 35
Yield changes
• Compare today’s yields with yesterday’s forwards one day
• Log of ratio
• Basic unit: Trading day
Quant Financial Research 06 Feb 2005 36
Yield changes – 1d ratesLPR1
-16.0%
-14.0%
-12.0%
-10.0%
-8.0%
-6.0%
-4.0%
-2.0%
0.0%
2.0%
4.0%
6.0%
08-May-02 08-Nov-02 08-May-03 08-Nov-03 08-May-04
1d
Quant Financial Research 06 Feb 2005 37
Yield changes – 60m ratesLPR15
-4.0%
-3.0%
-2.0%
-1.0%
0.0%
1.0%
2.0%
3.0%
4.0%
08-May-02 08-Nov-02 08-May-03 08-Nov-03 08-May-04
60m
Quant Financial Research 06 Feb 2005 38
Yield changes – 30y ratesLPR40
-12.0%
-10.0%
-8.0%
-6.0%
-4.0%
-2.0%
0.0%
2.0%
4.0%
6.0%
8.0%
08-May-02 08-Nov-02 08-May-03 08-Nov-03 08-May-04
30.00y
Quant Financial Research 06 Feb 2005 39
30 day exponential weighting30 day Exponential Weights
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
13-Jun-03 13-Jul-03 12-Aug-03 11-Sep-03 11-Oct-03 10-Nov-03 10-Dec-03 09-Jan-04 08-Feb-04 09-Mar-04
Quant Financial Research 06 Feb 2005 40
750 day exponential weighting750 day Exponential Weights
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
13-Jun-03 13-Jul-03 12-Aug-03 11-Sep-03 11-Oct-03 10-Nov-03 10-Dec-03 09-Jan-04 08-Feb-04 09-Mar-04
Quant Financial Research 06 Feb 2005 41
Double exponential weightingDouble Exponential Weights
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
13-Jun-03 13-Jul-03 12-Aug-03 11-Sep-03 11-Oct-03 10-Nov-03 10-Dec-03 09-Jan-04 08-Feb-04 09-Mar-04
w1
750 day
Combined
Quant Financial Research 06 Feb 2005 42
1d Vols: alpha = 0.7
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
45.0%
50.0%
08-May-02 08-Nov-02 08-May-03 08-Nov-03 08-May-04
xVol30
xVol750
xVol
Vols – 1d
Quant Financial Research 06 Feb 2005 43
60m Vols: alpha = 0.7
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
08-May-02 08-Nov-02 08-May-03 08-Nov-03 08-May-04
xVol30
xVol750
xVol
Vols – 60m
Quant Financial Research 06 Feb 2005 44
30.00y Vols: wt = 0.7
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
30.0%
35.0%
40.0%
45.0%
50.0%
08-May-02 08-Nov-02 08-May-03 08-Nov-03 08-May-04
yVol30
yVol750
yVol
Vols – 30y
Quant Financial Research 06 Feb 2005 45
Corrs: 1d vs 30.00y; Wt = 0.7
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
08-05-02 08-11-02 08-05-03 08-11-03 08-05-04
Corr30
Corr750
Corr
Correlations: 1d vs 30y
Quant Financial Research 06 Feb 2005 46
Principal Components Analysis
• “The curse of dimensionality”
• PCA reduces it:– Find variance / covariance matrix for latest day– Find its eigenvectors: the components– First n eigenvectors: the principal components– Eigenvalues are variances of components– First n account for most of the variance
Quant Financial Research 06 Feb 2005 47
Volatilities: Bondsbd (Swaps): 04-Jun-04(67.3%, 84.1%, 91.8%, 97.2%, 98.9%)
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
PCs
Ycs
Vols: Curves and Components
Quant Financial Research 06 Feb 2005 48
The First Component: 68%
-1.5%
-1.0%
-0.5%
0.0%
0.5%
1.0%
1.5%
0.0 5.0 10.0 15.0 20.0 25.0 30.0
Term
Yie
ld %
ag
e c
ha
ng
e
PC1
PC2
PC3
PC4
PC5
First component
Quant Financial Research 06 Feb 2005 49
The Second Component: 16%
-1.5%
-1.0%
-0.5%
0.0%
0.5%
1.0%
1.5%
0.0 5.0 10.0 15.0 20.0 25.0 30.0
Term
Yie
ld %
ag
e c
ha
ng
e
PC1
PC2
PC3
PC4
PC5
Second component
Quant Financial Research 06 Feb 2005 50
The Third Component: 8.5%
-1.5%
-1.0%
-0.5%
0.0%
0.5%
1.0%
1.5%
0.0 5.0 10.0 15.0 20.0 25.0 30.0
Term
Yie
ld %
ag
e c
ha
ng
e
PC1
PC2
PC3
PC4
PC5
Third component
Quant Financial Research 06 Feb 2005 51
The Fourth Component: 5%
-1.5%
-1.0%
-0.5%
0.0%
0.5%
1.0%
1.5%
0.0 5.0 10.0 15.0 20.0 25.0 30.0
Term
Yie
ld %
ag
e c
ha
ng
e
PC1
PC2
PC3
PC4
PC5
Fourth component
Quant Financial Research 06 Feb 2005 52
The Fifth Component: 1.5%
-1.5%
-1.0%
-0.5%
0.0%
0.5%
1.0%
1.5%
0.0 5.0 10.0 15.0 20.0 25.0 30.0
Term
Yie
ld %
ag
e c
ha
ng
e
PC1
PC2
PC3
PC4
PC5
Fifth component
Quant Financial Research 06 Feb 2005 53
Construct historical components1 n 1 n
1 1
1 n1 1 n
n
m n
* =Yieldchanges
Historicalcomponent
changes
Eigenmatrix
Quant Financial Research 06 Feb 2005 54
PC1 Vols: alpha = 0.7
0.0%
20.0%
40.0%
60.0%
80.0%
100.0%
120.0%
140.0%
08-May-02 08-Nov-02 08-May-03 08-Nov-03 08-May-04
xVol30
xVol750
xVol
Vols - PC1
Quant Financial Research 06 Feb 2005 55
pc3
-25.0%
-20.0%
-15.0%
-10.0%
-5.0%
0.0%
5.0%
10.0%
15.0%
20.0%
25.0%
08
-Ma
y-0
2
08
-Ju
n-0
2
08
-Ju
l-0
2
08
-Au
g-0
2
08
-Se
p-0
2
08
-Oct
-02
08
-No
v-0
2
08
-De
c-0
2
08
-Ja
n-0
3
08
-Fe
b-0
3
08
-Ma
r-0
3
08
-Ap
r-0
3
08
-Ma
y-0
3
08
-Ju
n-0
3
08
-Ju
l-0
3
08
-Au
g-0
3
08
-Se
p-0
3
08
-Oct
-03
08
-No
v-0
3
08
-De
c-0
3
08
-Ja
n-0
4
08
-Fe
b-0
4
08
-Ma
r-0
4
08
-Ap
r-0
4
pc3
13 Sep 02: +100 pts
13 Jun 03:-150 pts
11 Sep 03:-100 pts
17 Oct 03:-150 pts
12 Dec 03:-50 pts15 Aug 03:
-100 pts
14 Jun 02: +100 pts
PC3: Yield Changes
Quant Financial Research 06 Feb 2005 56
PC3 Vols: wt = 0.7
0.0%
10.0%
20.0%
30.0%
40.0%
50.0%
60.0%
70.0%
80.0%
08-May-02 08-Nov-02 08-May-03 08-Nov-03 08-May-04
yVol30
yVol750
yVol
Vols – PC3
Quant Financial Research 06 Feb 2005 57
Corrs: PC1 vs PC3; Wt = 0.7
-1.00
-0.80
-0.60
-0.40
-0.20
0.00
0.20
0.40
0.60
0.80
1.00
08-05-02 08-11-02 08-05-03 08-11-03 08-05-04
Corr30
Corr750
Corr
Correlations: PC1 vs PC3
Quant Financial Research 06 Feb 2005 58
PC1Vol = 77.6%; Skew = 0.07; Kurt = 11.2
0
2
4
6
8
10
12
14
16
-32.7% -28.8% -24.9% -21.0% -17.1% -13.1% -9.2% -5.3% -1.4% 2.5% 6.4% 10.4% 14.3% 18.2% 22.1% 26.0% 29.9% 33.9%
PC1 distribution
Quant Financial Research 06 Feb 2005 59
PC1: Smoothed distribution
Quant Financial Research 06 Feb 2005 60
ReproBars
• Choose 5 percentile ranges:– 5%, 25%, 40%, 25%, 5%– Find expected value in each range from
smoothed distribution
• Get a discrete distribution with just 5 points
Quant Financial Research 06 Feb 2005 61
PC5: ReprobarsPC5: 5-bar repro
-4.0% -3.3% -2.7% -2.0% -1.3% -0.6% 0.03% 0.7% 1.4% 2.1% 2.7% 3.4%
Quant Financial Research 06 Feb 2005 62
Construction of scenario curves
• All 5*5*5*5*5 = 3125 combinations of 5 abscissae/ordinates from 5 PC distributions
• Each combination is a set of 5 independent component changes, and has its own probability
• Apply the changes to the base curve to get 3125 scenario curves, each with its own probability.
Quant Financial Research 06 Feb 2005 63
Valuation of positions
• Value position 3125 times, to get 3125 values, each with its own probability
• Smooth the resulting distribution
• Find EWCV at .05% over 1 trading day
Quant Financial Research 06 Feb 2005 64
Expected worst case valueP&L Distribution
-4 -3 -2 -1 0 1 2 3 4
Cuttoff:.05%
ETL: expected loss, given that
loss is more than .05%
Quant Financial Research 06 Feb 2005 65
Margin
"On the one day in 2,000 when things are really bad, just how bad are they likely to be?"
Quant Financial Research 06 Feb 2005 66
Calm Results
• Assumptions:– Positions as at 28 Jan 2005– Yield curve and bond yields as at same date– Settlement day is 02 Feb 2005– Margining valuations for 31 Jan 2005– Margins calculated as the one-trading-day
Value-at-Risk (VAR) of the position at 99.95% confidence level
Quant Financial Research 06 Feb 2005 67
Physical Bonds• (NB: settlement margins are ignored. Once a deal has
been irrevocably committed to, it falls out of the risk position.)
Instrument Position MarginSpot R153 +R1m 15,806
-R1m 17,665 Spot R157 +R1m 28,470
-R1m 30,507 Spot R186 +R1m 49,029
-R1m 51,378
Quant Financial Research 06 Feb 2005 68
Forward Bonds
Instrument Settlement Position MarginR153 04-May-2005 +R1m 15,985
-R1m 17,692 02-Feb-2006 +R1m 14,344
-R1m 15,415
Quant Financial Research 06 Feb 2005 69
Carries
Instrument Settlement Position MarginR153 02-Feb-2005 -R1mR153 09-Feb-2005 +R1m 101
R153 02-Feb-2005 -R1mR153 02-Feb-2006 +R1m 4,318
Quant Financial Research 06 Feb 2005 70
Compound Spot Position
Instrument Settlement Position MarginR153 02-Feb-2005 -R1mR157 02-Feb-2005 +R2mR186 02-Feb-2005 -R1m 16,443
Quant Financial Research 06 Feb 2005 71
Positions with Futures
Instrument Settlement Position MarginMay 05 R153 -1 18,087
May 05 R153 -1 Spot R153 02-May-2005 +R1m 669
Quant Financial Research 06 Feb 2005 72
Positions with jSwaps
Instrument Settlement Position Marginj153 7.5 swap +R1m 15,743
j153 7.5 swap +R1mMay 05 R153 1 3,268
Quant Financial Research 06 Feb 2005 73
jNotes
Instrument Settlement Position MarginMay 05 jN02 +1 6,642
Aug 05 jN05 +1 13,777
Nov 05 jN10 +1 22,251
Quant Financial Research 06 Feb 2005 74
Distribution of Scenario ValuesScenario Values: May 05 jN02
(6,923 (6,091 (5,259 (4,427 (3,595 (2,762 (1,930 (1,098 (266) 566 1,399 2,231 3,063 3,895 4,727 5,560 6,392 7,224
Quant Financial Research 06 Feb 2005 75
Scenario Values: jN05Scenario Values: Aug 05 jN05
(14,
405)
(12,
757)
(11,
109)
(9,4
61)
(7,8
13)
(6,1
65)
(4,5
17)
(2,8
69)
(1,2
22)
426
2,0
74
3,7
22
5,3
70
7,0
18
8,6
66
10,
314
11,
962
13,
609
Quant Financial Research 06 Feb 2005 76
Scenario Values: jN10Scenario Values: Nov 05 jN10
(23,
417)
(20,
664)
(17,
910)
(15,
157)
(12,
403)
(9,6
50)
(6,8
96)
(4,1
43)
(1,3
89)
1,3
64
4,1
18
6,8
71
9,6
25
12,
378
15,
132
17,
885
20,
639
23,
392
Quant Financial Research 06 Feb 2005 77
jFRAs
Instrument Settlement Position MarginMay 05 jFRA +1 751
May 06 jFRA +1 1,052
Nov 06 jFRA +1 1,034
Quant Financial Research 06 Feb 2005 78
jRods
Instrument Settlement Position MarginJun 05 jRod +R1m 1,007
Jan 06 jRod +R1m 2,939
Quant Financial Research 06 Feb 2005 79
jGoviInstrument Settlement Position Margin
May 05 jGOVI +1 30,307
May 05 jGOVI +100R152 02-Feb-2005 -R14mR194 02-Feb-2005 -R39mR153 02-Feb-2005 -R55mR201 02-Feb-2005 -R14mR157 02-Feb-2005 -R30mR203 02-Feb-2005 -R3mR186 02-Feb-2005 -R17m 39,578
Questions