advanced ois discounting - jan römanjanroman.dhis.org/...advanced_ois_discounting_ion...• a...

Advanced OIS Discounting:

Building Proxy OIS Curves When OIS

Markets are Illiquid or Nonexistent

November 6, 2013

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Agenda

1. OIS discounting basics: review of the standard curve

stripping approach

2. What if there is no OIS curve?

1. Simultaneous calibration of discounting and projection curves

2. Assumptions behind the curve stripping approaches

3. Examples

4. Conclusion

5. Q&A

Curve construction: single currency case

• Stripping the OIS curve: typically done using Overnight Indexed Swaps (OIS) • Overnight Index Swap (OIS) - a fixed/float interest rate swap with the floating leg based on

published overnight rate index

• Stripping the projection curves (e.g. 3M curve) given the OIS curve: • From instruments indexed on the 3M Libor

3M Cash

3M FRA/Fut

Swaps 3M vs. Fixed

O/N rate

OIS swaps

OIS curve (discounting)

3M curve (projection)

Curve construction: cross-currency case

• Assume the domestic and foreign curves for all needed tenors have been

already stripped

• Strip the implied foreign basis curve:

• Cross currency basis swaps: e.g. DOM3M vs. FOR3M

FX Forwards

Cross currency basis swaps

FOR curve (discounting)

Domestic Float Leg

3M DOM Index

Projection Curve

DOM3M Swap Curve

Discount Curve

DOM OIS Curve

Foreign Float Leg

3M FOR Index + Spread

Projection Curve

FOR3M Swap Curve

Discount Curve

Implied DOM3M/FOR3M Basis Curve

Single Currency Curve Construction

Selection of available curves and instruments in the most liquid markets:

Currency Overnight Rate Standard Curve

Forward Curves

Basis Curves

USD FedFunds Effective rate 3M USD Libor MuniSwaps 1M vs. 3M 3M vs. 6M 3M vs. 12M 3M Prime/Libor BS

EUR EONIA 6M Euribor 1M Euribor 3M Euribor

6M vs. 12M 3M vs. 6M

JPY MUTAN 6M JPY Libor 1M vs. 6M 3M vs. 6M

GBP SONIA 6M GBP Libor 3M GBP Libor 3M vs. 1M 12M vs. 6M

CHF TOIS 6M CHF Libor

3M CHF Libor 1M CHF Libor

12M vs. 6M

CAD Bank of Canada Overnight Repo Rate (CORRA)

6M CAD-BA 6M vs. 3M 3M vs. 1M

AUD RBA 3M BBSW 6M BBSW 1M BBSW

6M vs. 1M BBSW 3M vs. 6M BBSW

Curve construction: market instruments

• Curves stripping is based in general on market instruments such as • Swaps (Libor 3M vs. Fixed)

• Basis swaps aka Tenor basis swaps (e.g. Libor 3M vs. Libor 6M)

• Cross-currency basis swaps (e.g. USD Libor 3M vs. GBP Libor 3M)

• The building blocks of these instruments are • Fixed cashflows: Fixed * YF * Notional

• Libor payments: Libor * YF * Notional

• To price these we only need the elementary bits • Discount Factors: DF = PV(1 unit of currency)

• Forwards: FWD = PV(Libor) / DF

• Then • PV(Fixed cashflow) = Fixed * YF * Notional * DF

• PV(Floating cashflow) = YF * Notional * FWD * DF

Curve construction: market instruments

• Once we know the Discount Factors for all maturities and the Forwards for all

maturities and tenors i.e. • the Discount Curve 𝑡 → DF(𝑡)

• the Forward Curves 𝑡 → FWD𝛿(𝑡) for all tenors 𝛿

we are able to price all linear instruments

• In practice, the Discount Factors and the Forwards are stripped from market

instruments for a set of maturities and tenors. For other maturities or tenors

the values are obtained by interpolation. • Practical issues: how is this interpolation performed?

• How are the curves represented? In terms of Discount Factors, Forwards directly, etc.?

When the OIS market is iliquid

• Liquid OIS markets exist for the G5 currencies (USD, EUR, GBP, JPY, CHF),

AUD, CAD among others

• What if there is no OIS market or the market is not liquid enough? • This means we don’t dispose of an OIS curve

• Therefore we can’t strip the projection curves in the usual manner

• One possible idea is to turn to a cross-currency market which is liquid

enough and try to simultaneously strip the projection curve and the implied

discounting curve from • local swaps

• cross-currency basis swaps

• For example, say we look at HUF • Local swaps: HUF3M vs. Fixed

• Cross-currency swaps: HUF3M vs. EUR3M

• What are the collateral assumptions behind this procedure?

Simultaneous stripping of disc. and proj. curves

• Vanilla swap HUF3M vs. Fixed: • Quarterly HUF3M (3M BUBOR)

• Annually Fixed

• Cross-currency basis swap EUR3M vs. HUF3M • EUR Floating Leg: Quarterly EUR3M (EURIBOR3M)

• HUF Floating Leg: Quarterly HUF3M + spread

Vanilla swap HUF3M vs. Fixed

Cross-currency basis swap EUR3M vs. HUF3M

Discounting Curve Projection Curve

EUR Leg EONIA curve EUR3M curve

HUF Leg HUFdisc curve HUF3M curve


Fixed Leg HUFdisc curve N/A

Floating Leg HUFdisc curve HUF3M curve


• Stripping equations • Assume EUR curves are already stripped (e.g. from EONIA swaps and vanilla EUR3M swaps)

• Written for (say) the 1Y point:

𝐾𝛿𝐷4 = 𝛿1𝐹1𝐷1 + 𝛿2𝐹2𝐷2+ 𝛿3𝐹3𝐷3+ 𝛿4𝐹4𝐷4

𝛿1(𝐹1 + 𝑠)𝐷1 + 𝛿2(𝐹2 + 𝑠)𝐷2+ 𝛿3(𝐹3+𝑠)𝐷3+ 𝛿4(𝐹4+𝑠)𝐷4= EURLeg /𝑋(0)

• 𝐷𝑖 = Discount Factors

• 𝐹𝑖 = HUF3M Forwards

• 𝛿𝑖 = Year Fractions

• 𝑋(0) = spot EURHUF exchange rate

• 𝐾 = Quoted 1Y Par Swap rate

• 𝑠 = Quoted 1Y EUR3M/HUF3M basis spread


• Stripping equations • Assume EUR curves are already stripped (e.g. from EONIA swaps and vanilla EUR3M swaps)

• Written for (say) the 1Y point:

𝐾𝛿𝐷4 = 𝛿1𝐹1𝐷1 + 𝛿2𝐹2𝐷2+ 𝛿3𝐹3𝐷3+ 𝛿4𝐹4𝐷4

𝛿1(𝐹1 + 𝑠)𝐷1 + 𝛿2(𝐹2 + 𝑠)𝐷2+ 𝛿3(𝐹3+𝑠)𝐷3+ 𝛿4(𝐹4+𝑠)𝐷4= EURLeg /𝑋(0)

• The unknowns here are 𝐷4 and 𝐹4

• The greyed-out 𝐷s and 𝐹s are computed by interpolation

• This can be handled by a solver

• 𝐷𝑖 = Discount Factors

• 𝐹𝑖 = HUF3M Forwards

• 𝛿𝑖 = Year Fractions

• 𝑋(0) = spot EURHUF exchange rate

• 𝐾 = Quoted 1Y Par Swap rate

• 𝑠 = Quoted 1Y EUR3M/HUF3M basis spread

Simultaneous stripping: in practice

• In practice things might need to be done differently • The payment dates of the local vanilla swaps and those of the cross-currency basis swaps

might be misaligned (due to differing conventions)

• Or simply the quoted maturities for one set of instruments are different from the quoted

maturities for the other set of instruments

• Thus performing a bootstrap might not be the best solution

• Instead, we could use a global solver on all quoted instruments at once • This is slower but produces more stable results and is clear of the problems above

• We could still perform intermediate passes using the quotes up to some fixed maturities in

order to find good initial guesses for the later passes

Simultaneous stripping: in practice

• The simultaneous stripping produces two curves • A discounting curve HUFdisc

• A projection curve HUF3M

• By construction, if we use these two curves as discounting curve and

projection curve, respectively, then we will price at par both

• the vanilla swaps HUF3M vs. Fixed,

• and the cross-currency basis swaps HUF3M vs. EUR3M

• Is HUFdisc the “true” HUF OIS discounting curve?

• Is HUF3M the “true” HUF 3M projection curve?

HUF3M vs. Fixed Vanilla swaps

HUF3M vs EUR3M Cross currency

basis swaps

HUFdisc curve (discounting)

HUF3M curve (projection)

Pricing with collateral

• How does curve stripping fit into a general model for derivatives pricing? • So far we have only considered “linear” instruments and defined formally Discount Factors

and Forwards. Can these be used to price something else but swaps?

• Is this backed by a theory where the curves get back their usual meaning?

• Consider an economy with two currencies: domestic (Dom) and foreign (For).

Assume collateral can be posted in any of the two currencies • The choice of the collateral currency holds for the whole lifetime of the derivative (i.e. there is

no option to switch collateral)

• Domestic collateral earns 𝑐𝑑

• Foreign collateral earns 𝑐𝑓

• A pricing theory can be constructed rigorously with the help of a replication

argument1

1see for example Piterbarg (2010)


• The price of a collateralized domestic derivative 𝑉 is given by

► with domestic collateral: 𝑉 𝑡 = 𝔼𝑡 𝑒− 𝑐𝑑 𝑠 𝑑𝑠

𝑇𝑡 𝑉 𝑇

► with foreign collateral: 𝑉 𝑡 = 𝔼𝑡 𝑒− [𝑐𝑑 𝑠 +ℎ 𝑠 ]𝑑𝑠

𝑇𝑡 𝑉 𝑇

• The price of a collateralized foreign derivative 𝑉𝑓 is given by

► with domestic collateral: 𝑉𝑓 𝑡 = 𝔼𝑡𝑓𝑒− [𝑐𝑓 𝑠 −ℎ 𝑠 ]𝑑𝑠

𝑇𝑡 𝑉𝑓 𝑇

► with foreign collateral: 𝑉𝑓 𝑡 = 𝔼𝑡𝑓𝑒− 𝑐𝑓 𝑠 𝑑𝑠

𝑇𝑡 𝑉𝑓 𝑇

• The fact that the spread when computing from the foreign point of view is – ℎ follows from the

Dom-For “parity” condition

• Domestic-foreign “parity” condition: • fix the collateral currency;

• then, computing the price of a contingent claim through For or through Dom yields the same

result:

• This implies that the drift of the FX rate X (in the domestic measure 𝔼) is

𝑟𝑑,𝑓 = 𝑐𝑑 − 𝑐𝑓 + ℎ

Dom-For parity with collateral

𝑉𝑓 𝑡 𝑉𝑑 𝑡

𝐻 𝐻𝑋 𝑇

Dom For

Time 𝑡

Time 𝑇


• Once the collateral currency has been chosen, pricing under a CSA is in

some way the same as pricing in the classical theory, as long as the right

curves are used

• We have a pricing theory that is consistent and extends the formal swap

pricing theory based on stripping

• Stripping produces the initial term structures of the rates i.e. today’s values of

the curves (discounting, forwarding)

• As seen above, the collateral is reflected in the curves that are used for

discounting

Domestic rate Foreign rate FX rate drift

Classical theory 𝑟𝑑 𝑟𝑓 𝑟𝑑 − 𝑟𝑓

Domestic collateral 𝑐𝑑 𝑐𝑓,𝑑 = 𝑐𝑓 − ℎ 𝑟𝑑,𝑓 = 𝑐𝑑 − 𝑐𝑓 + ℎ

Foreign collateral 𝑐𝑑,𝑓 = 𝑐𝑑 + ℎ 𝑐𝑓 𝑟𝑑,𝑓 = 𝑐𝑑 − 𝑐𝑓 + ℎ

Examples: 2 curves stripping

• We consider the HUF market: • Local swaps: HUF3M vs. Fixed


• The simultaneously strip these two sets of instruments to produce two curves • A discounting curve HUFdisc

• A projection curve HUF3M


HUF3M vs EUR3M Cross currency

basis swaps



Examples: EUR market data

• EUR: market quotes – EONIA: OIS par swap rates

– EUR3M: par swap rates

EONIA EUR3M

1BD 0.09%

1W 0.09%

2W 0.10%

1M 0.10%

2M 0.10%

3M 0.11% 0.22%

4M 0.11%

5M 0.12%

6M 0.13%

7M 0.13%

8M 0.13%

9M 0.14%

10M 0.15%

11M 0.15%

1Y 0.15%

18M 0.19%

2Y 0.24% 0.41%

30M 0.31%

3Y 0.39% 0.58%

4Y 0.61% 0.82%

5Y 0.85% 1.06%

6Y 1.07% 1.28%

7Y 1.27% 1.48%

8Y 1.44% 1.66%

9Y 1.60% 1.82%

10Y 1.75% 1.96%

11Y 1.88% 2.09%

12Y 1.99% 2.20%

15Y 2.23% 2.43%

20Y 2.41% 2.59%

25Y 2.48% 2.63%

30Y 2.50% 2.65%

35Y 2.53% 2.66%

40Y 2.55% 2.68%

50Y 2.60% 2.73%

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

1B

D

2W 2M

4M

6M

8M

10

M 1Y

2Y

3Y

5Y

7Y

9Y

11

Y

15

Y

25

Y

35

Y

50

Y

EONIA

EUR3M

Examples: HUF market data

• HUF market quotes – HUF3M par swap rates

– EUR3M vs. HUF3M xccy basis swap spreads

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

2Y

3Y

4Y

5Y

6Y

7Y

8Y

9Y

10

Y

12

Y

15

Y

20

Y

HUF3M Swap Rates

HUF3M Flatter Curve

-120

-100

-80

-60

-40

-20

0

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y 25Y 30Y

EUR HUF basis spreads

HUF3M Swap Rates

HUF3M Flatter Curve

2Y 3.49% 4.34%

3Y 3.68% 4.36%

4Y 3.88% 4.38%

5Y 4.03% 4.40%

6Y 4.22% 4.42%

7Y 4.44% 4.44%

8Y 4.62% 4.46%

9Y 4.79% 4.48%

10Y 4.91% 4.50%

12Y 5.00% 4.54%

15Y 5.01% 4.60%

20Y 4.74% 4.70%

HUFEUR basis

1Y -78

2Y -83

3Y -90

4Y -94

5Y -97

6Y -99

7Y -100

8Y -99

9Y -98

10Y -96

12Y -91

15Y -81

20Y -61

25Y -40

30Y -20

Examples: two curves stripping - results

• Stripped HUF curves under Base Scenario and Flatter HUF3M swap curve – HUFdisc discounting curve

– HUF3M forwarding curve

0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

0 Y 5 Y 10 Y 15 Y 20 Y 25 Y

HUFdisc Zero Rates

Base Scenario Flatter HUF3M

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

0 Y 5 Y 10 Y 15 Y 20 Y 25 Y

HUF3M 3M forwards

Base Scenario Flatter HUF3M

• If the market doesn’t directly quote 3M swaps?

• If we have quotes for 3M vs. 6M tenor basis swaps then we can

simultaneously strip three curves: • Local swaps: HUF6M vs. Fixed

• Local tenor swaps: HUF6M vs. HUF3M




HUF3M vs. EUR3M Cross currency

basis swaps



HUF3M vs. HUF6M tenor basis swaps



Vanilla swap HUF6M vs. Fixed

Tenor basis swap HUF6M vs. HUF3M

Cross-currency basis swap EUR3M vs. HUF3M


EUR Leg EONIA curve EUR3M curve

HUF Leg HUFdisc curve HUF3M curve


3M Floating Leg HUFdisc curve HUF3M curve

6M Floating Leg HUFdisc curve HUF6M curve


Fixed Leg HUFdisc curve N/A

Floating Leg HUFdisc curve HUF6M curve


• HUF market quotes – HUF6M par swap rates

– EUR3M vs. HUF3M xccy basis swap spreads

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y

HUF6M Swap Rates

-120

-100

-80

-60

-40

-20

0

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y 25Y 30Y

EUR HUF basis spreads

HUF6M Swap Rates

2Y 3.49%

3Y 3.68%

4Y 3.88%

5Y 4.03%

6Y 4.22%

7Y 4.44%

8Y 4.62%

9Y 4.79%

10Y 4.91%

12Y 5.00%

15Y 5.01%

20Y 4.74%

HUFEUR basis

1Y -78

2Y -83

3Y -90

4Y -94

5Y -97

6Y -99

7Y -100

8Y -99

9Y -98

10Y -96

12Y -91

15Y -81

20Y -61

25Y -40

30Y -20


• HUF market quotes – HUF6M vs. HUF3M basis spreads

0.00

5.00

10.00

15.00

20.00

25.00

30.00

1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y

HUF6M vs 3M basis spreads

Base scenario Steeper 3M vs. 6M

Base scenario

Steeper 3M vs. 6M

1Y 9.90 25.00

2Y 5.90 23.00

3Y 5.20 21.00

4Y 4.00 19.00

5Y 3.10 17.00

6Y 2.40 15.00

7Y 1.90 13.00

8Y 1.50 11.00

9Y 1.30 9.00

10Y 1.00 7.00

12Y 1.00 7.00

15Y 1.00 7.00

20Y 1.00 7.00

Examples: 3 curves stripping - results

• Stripped HUF curves under Base Scenario and Steeper HUF3M vs. HUF6M

basis swap curve – HUFdisc discounting curve



0.00%

0.50%

1.00%

1.50%

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

0 Y 5 Y 10 Y 15 Y 20 Y 25 Y

HUFdisc Zero Rates

Base Scenario Steeper 3M vs 6M

Examples: 3 curves stripping - results

• Stripped HUF curves under Base Scenario and Steeper HUF3M vs. HUF6M

basis swap curve – HUFdisc discounting curve



0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

0 Y 5 Y 10 Y 15 Y 20 Y 25 Y

HUF6M 6M forwards


0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

0 Y 5 Y 10 Y 15 Y 20 Y 25 Y

HUF3M 3M forwards


Summary

• In the aftermath of the great financial crisis the market has turned

towards OIS discounting as the new standard

• The G5 currencies and a few others have liquid OIS markets which

provide the quotes form which the OIS curves in those currencies can

be stripped

• But for the other currencies the OIS markets are more often than not

inexistent or illiquid. This renders impossible the stripping of the OIS

curve.

• There is no magical solution, but one can contemplate using

available cross-currency quotes to infer a number of curves. This

requires the simultaneous stripping of single-currency and cross-

currency instruments

• Discounting intimately tied to collateral hence a there is a trade-off:

not having a curve at all vs. using curves based on a different

collateral assumption

Q&A

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Click the Q&A Button on the Green WebEx Toolbar located

at the top of your screen to reveal the Q&A Window where

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Ion Mihai, Ph.D.

Quantitative Analyst

[email protected]

Jim Jockle

Chief Marketing Officer

[email protected]

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