Advanced OIS Discounting:

Advanced OIS Discounting: Building Proxy OIS Curves When OIS Markets are Illiquid or Nonexistent November 6, 2013

About Us Our Presenters: Ion Mihai, Ph.D. Quantitative Analyst imihai@numerix.com Jim Jockle Chief Marketing Officer jjockle@numerix.com Follow Us: Twitter: LinkedIn: @nxanalytics @jjockle http://linkd.in/numerix http://linkd.in/ionmihai http://linkd.in/jimjockle 2

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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 O/N rate OIS swaps OIS curve (discounting) Stripping the projection curves (e.g. 3M curve) given the OIS curve: From instruments indexed on the 3M Libor 3M Cash 3M FRA/Fut 3M curve (projection) Swaps 3M vs. Fixed

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: FX Forwards Cross currency basis swaps FOR curve (discounting) Cross currency basis swaps: e.g. DOM3M vs. FOR3M Domestic Float Leg 3M DOM Index Foreign Float Leg 3M FOR Index + Spread Projection Curve DOM3M Swap Curve Projection Curve FOR3M Swap Curve Discount Curve DOM OIS 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 AUD RBA 3M BBSW 6M BBSW 1M BBSW 6M vs. 3M 3M vs. 1M 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 FWD δ (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 Discounting Curve Fixed Leg HUF disc curve N/A Cross-currency basis swap EUR3M vs. HUF3M Discounting Curve Projection Curve Floating Leg HUF disc curve HUF 3M curve Projection Curve EUR Leg EONIA curve EUR3M curve HUF Leg HUF disc curve HUF 3M curve

Simultaneous stripping of disc. and proj. curves Stripping equations Assume EUR curves are already stripped (e.g. from EONIA swaps and vanilla EUR3M swaps) Written for (say) the 1Y point: KδD 4 = δ 1 F 1 D 1 + δ 2 F 2 D 2 + δ 3 F 3 D 3 + δ 4 F 4 D 4 δ 1 (F 1 + s)d 1 + δ 2 (F 2 + s)d 2 + δ 3 (F 3 +s)d 3 + δ 4 (F 4 +s)d 4 = EURLeg /X(0) D i = Discount Factors F i = HUF3M Forwards δ i = Year Fractions X(0) = spot EURHUF exchange rate K = Quoted 1Y Par Swap rate s = Quoted 1Y EUR3M/HUF3M basis spread

Simultaneous stripping of disc. and proj. curves Stripping equations Assume EUR curves are already stripped (e.g. from EONIA swaps and vanilla EUR3M swaps) Written for (say) the 1Y point: KδD 4 = δ 1 F 1 D 1 + δ 2 F 2 D 2 + δ 3 F 3 D 3 + δ 4 F 4 D 4 δ 1 (F 1 + s)d 1 + δ 2 (F 2 + s)d 2 + δ 3 (F 3 +s)d 3 + δ 4 (F 4 +s)d 4 = EURLeg /X(0) The unknowns here are D 4 and F 4 The greyed-out Ds and Fs are computed by interpolation This can be handled by a solver D i = Discount Factors F i = HUF3M Forwards δ i = 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 HUF disc A projection curve HUF 3M HUF3M vs. Fixed Vanilla swaps HUF3M vs EUR3M Cross currency basis swaps HUF disc curve (discounting) HUF 3M curve (projection) 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 HUF disc the true HUF OIS discounting curve? Is HUF 3M the true HUF 3M projection curve?

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 c d Foreign collateral earns c f A pricing theory can be constructed rigorously with the help of a replication argument 1 1 see for example Piterbarg (2010)

Pricing with collateral The price of a collateralized domestic derivative V is given by with domestic collateral: V t = E t e T t c d s ds V T with foreign collateral: V t = E t e T t [c d s +h s ]ds V T The price of a collateralized foreign derivative V f is given by with domestic collateral: V f t = E f t e T t [c f s h s ]ds V f T with foreign collateral: V f t = E f t e T t c f s ds V f T The fact that the spread when computing from the foreign point of view is h follows from the Dom-For parity condition

Dom-For parity with collateral 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: Time T HX T H Time t V d t V f t Dom For This implies that the drift of the FX rate X (in the domestic measure E) is r d,f = c d c f + h

Pricing with collateral 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 Domestic rate Foreign rate FX rate drift Classical theory r d r f r d r f Domestic collateral c d c f,d = c f h r d,f = c d c f + h Foreign collateral c d,f = c d + h c f r d,f = c d c f + h 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

Examples: 2 curves stripping We consider the HUF market: Local swaps: HUF3M vs. Fixed Cross-currency swaps: HUF3M vs. EUR3M The simultaneously strip these two sets of instruments to produce two curves A discounting curve HUF disc A projection curve HUF 3M HUF3M vs. Fixed Vanilla swaps HUF3M vs EUR3M Cross currency basis swaps HUF disc curve (discounting) HUF 3M curve (projection)

1BD 2W 2M 4M 6M 8M 10M 1Y 2Y 3Y 5Y 7Y 9Y 11Y 15Y 25Y 35Y 50Y Examples: EUR market data EUR: market quotes 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% EONIA: OIS par swap rates EUR3M: par swap rates EONIA EUR3M 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%

2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y Examples: HUF market data 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% HUF market quotes HUF3M par swap rates EUR3M vs. HUF3M xccy basis swap spreads HUF3M Swap Rates HUF3M Flatter Curve 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 0-20 EUR HUF basis spreads 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y 25Y 30Y -40-60 -80-100 -120

Examples: two curves stripping - results Stripped HUF curves under Base Scenario and Flatter HUF3M swap curve HUF disc discounting curve HUF 3M forwarding curve HUF disc Zero Rates 4.50% 4.00% 3.50% 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% 0 Y 5 Y 10 Y 15 Y 20 Y 25 Y HUF 3M 3M forwards 7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% 0 Y 5 Y 10 Y 15 Y 20 Y 25 Y Base Scenario Flatter HUF3M Base Scenario Flatter HUF3M

Examples: 3 curves stripping 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 Cross-currency swaps: HUF3M vs. EUR3M HUF6M vs. Fixed Vanilla swaps HUF3M vs. EUR3M Cross currency basis swaps HUF3M vs. HUF6M tenor basis swaps HUF disc curve (discounting) HUF 3M curve (projection) HUF 6M curve (projection)

Examples: 3 curves stripping Vanilla swap HUF6M vs. Fixed Discounting Curve Projection Curve Fixed Leg HUF disc curve N/A Floating Leg HUF disc curve HUF 6M curve Tenor basis swap HUF6M vs. HUF3M Discounting Curve Projection Curve 3M Floating Leg HUF disc curve HUF 3M curve 6M Floating Leg HUF disc curve HUF 6M curve Cross-currency basis swap EUR3M vs. HUF3M Discounting Curve Projection Curve EUR Leg EONIA curve EUR3M curve HUF Leg HUF disc curve HUF 3M curve

Examples: HUF market data 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% HUF market quotes HUF6M par swap rates EUR3M vs. HUF3M xccy basis swap spreads HUF6M Swap Rates 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y 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 0-20 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y 25Y 30Y -40-60 -80-100 -120

Examples: HUF market data HUF market quotes HUF6M vs. HUF3M basis spreads 30.00 25.00 20.00 15.00 10.00 5.00 0.00 HUF6M vs 3M basis spreads 1Y 2Y 3Y 4Y 5Y 6Y 7Y 8Y 9Y 10Y 12Y 15Y 20Y 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 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 HUF disc discounting curve HUF 3M forwarding curve HUF 6M forwarding curve HUF disc Zero Rates 4.50% 4.00% 3.50% 3.00% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% 0 Y 5 Y 10 Y 15 Y 20 Y 25 Y 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 HUF disc discounting curve HUF 3M forwarding curve HUF 6M forwarding curve HUF 3M 3M forwards HUF 6M 6M forwards 7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% 0 Y 5 Y 10 Y 15 Y 20 Y 25 Y 7.00% 6.00% 5.00% 4.00% 3.00% 2.00% 1.00% 0.00% 0 Y 5 Y 10 Y 15 Y 20 Y 25 Y Base Scenario Steeper 3M vs 6M Base Scenario Steeper 3M vs 6M

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 crosscurrency 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

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Contact Us Our Presenters: Ion Mihai, Ph.D. Quantitative Analyst imihai@numerix.com Jim Jockle Chief Marketing Officer jjockle@numerix.com Follow Us: Twitter: LinkedIn: @nxanalytics @jjockle http://linkd.in/numerix http://linkd.in/ionmihai http://linkd.in/jimjockle 33

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