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# The ABCD of Interest Rate Basis Spreads

Presented at QuantLib User Meeting, London, July 12, 2016

We show that forward rates can be modeled as abcd parametric tenor basis spreads over the underlying overnight rate curve. This is possible for both continuously and simply compounded forward rates, with a simple approximation for converting between the corresponding basis. Increasing interest-rate tenor dominance, as empirically observed, is recovered and can be structurally enforced using a robust methodology improvement based on relative basis between the most liquid tenors. The smoothness requirement is moved from forward rate curves to tenor basis curves, properly dealing with the market evidence of jumps in forward rates. In the case of continuously compounded tenor basis, pseudo-discount factors are also available. An implementation of this methodology is available in the QuantLib open-source project.

July 12, 2016

## Transcript

1. ### The ABCD of Interest Rate Basis Spreads Ferdinando M. Ametrano

ferdinando@ametrano.net Luigi Ballabio luigi.ballabio@gmail.com Paolo Mazzocchi mazzocchip@live.it QuantLib User Meeting, London, July 12, 2016

Rationale for ABCD Smooth Basis Modeling 3. Relation Between Continuous and Simple Basis 4. Multiple Tenor Analysis 5. Tenor Dominance 6. Conclusions 2/55
3. ### Multiple Rate Curves For Each Currency • Different forwarding curves

for different tenors (e.g. ON, 1M, 3M, 6M, 1Y) • A single discounting curve (depending on collateralization practices, usually ON) 3/55
4. ### Forward Rates and Pseudo-Discounts 1 + () = () (

+ ) = () + • Calculation of forward rates as pseudo-discount ratios is an obsolete legacy • For a given tenor in a multi-curve world , < are not specified 4/55
5. ### The Problem with Pseudo-Discounts • Forward smoothness depends on discount

factors of maturity below the forward tenor, e.g. for 6M 17 = 6 (1) 6 (7) = − () 1 0 − () 7 0 • Instruments of 6M tenor (e.g. 0x6, 1x7, 2x8, etc.) provide no information about 6 (1) 5/55
6. ### Synthetic Deposits Patch • Estimate , < to maximize forward

rate smoothness = − [ +()] 0 = () ∙ − () 0 instantaneous forward rates on the ON curve () (parametric) instantaneous basis • A polynomial basis obtained as exact-fit of market instruments is suggested by Ametrano, Mazzocchi (2014) EONIA Jumps and Proper Euribor Forwarding https://speakerdeck.com/nando1970/eonia-jumps-and-proper-euribor-forwarding 6/55

9. ### Direct Forward Rate Bootstrapping: the Problem of Non-Smooth Forwards Direct

forward rate bootstrapping gets rid of the synthetic deposit problem, but • Jumps in overnight rates propagate to longer tenor ibor rates, e.g. turns of year • Additive forward rate jumps originating from the same ON jump have different magnitude for different ibor tenors • Increased jump attenuation for longer tenor, subject to convoluted estimation procedures 9/55

14. ### The Legacy (Benchmark) EUR6M Curve • Use OIS for discounting

• Never ask a forward curve for discount factors 14/55
15. ### EUR6M/OIS6M Basis The Empirical Observation 6 = 6 − 6

0,00% 0,05% 0,10% 0,15% 0,20% 0,25% 0,30% 0,35% Legacy Curve Basis Legacy Curve Basis 15/55

Rationale for ABCD Smooth Basis Modeling 3. Relation Between Continuous and Simple Basis 4. Multiple Tenor Analysis 5. Tenor Dominance 6. Conclusions 16/55
17. ### Rebonato’s abcd Parameterization = + − + • + is

the value in = 0; • is the long term value; • For a bounded (), it must be > 0; • The hump max is at: = 1 − (relevant only if positive) 17/55
18. ### EUR6M/OIS6M Simple Basis from ABCD 6M FRA/Swap Calibration • A=0.2038%,

B=0.001, C=0.210, D=0.0452% 6 = 6 − 6 0,00% 0,05% 0,10% 0,15% 0,20% 0,25% 0,30% 0,35% Legacy Curve Basis Calibrated Simple Basis 18/55
19. ### Fitting to Market Instrument, Not to the Legacy Benchmark •

The legacy benchmark curve is something to be improved upon, displayed as comparison, not something we want to exactly reproduce • The quality of the fit has to be judged by market instruments repricing errors 19/55
20. ### EUR6M from Simple Basis {ABCD + K} Exact-Fit Correction Exact

fit is possible with a term structure of correction factors 6 , , , , : 6 bootstrapped on , linearly interpolated 6 = 6 6 6 = 6 6 + 6 6 ≈ 1 goodness of fit 20/55
21. ### On Calibration and Exact Fit • Different market instrument sets

can be used for ABCD calibration and K exact fit • ABCD does not need to be recalibrated in real- time, can be very unfrequently updated during the day • K are easily recalibrated real-time 21/55
22. ### What Did We Achieve? • Market instruments are re-priced by

both legacy curve and ABCD+K • Forward rates are different, ABCD+K being better 22/55
23. ### EUR6M/OIS6M Simple Basis from ABCD 6M FRA/Swap Calibration • A=0.2038%,

B=0.001, C=0.210, D=0.0452% 6 = 6 − 6 0,00% 0,05% 0,10% 0,15% 0,20% 0,25% 0,30% 0,35% Calibrated Simple Basis Calibrated Simple Basis 23/55
24. ### EUR6M ABCD Errors Exact-Fit with ∈ [1 − , 1

+ ] 24/55

Rationale for ABCD Smooth Basis Modeling 3. Relation Between Continuous and Simple Basis 4. Multiple Tenor Analysis 5. Tenor Dominance 6. Conclusions 25/55
26. ### The Case for Continuous Basis • Simple basis is observed

in the market • Continuous basis is preferred for derivative pricing (Schlenkrich and Miemiec 2015) • Continuous basis also provides well defined pseudo- discounts = − [ +()] 0 = ∙ − 0 • Continuous basis will also allow to obtain tenor basis dominance by construction, as explained later on 26/55
27. ### Relation between Simple and Continuous Basis (1/2) • Simple basis:

= − • Continuous basis: = − 1 + () = +() + = 1 + () () () + • Therefore taking the logarithm: () + = ln 1 + () 1 + () () 27/55
28. ### Relation between Simple and Continuous Basis (2/2) Using a first

order approximation: + ≈ − () () That is: () ≈ + () () 28/55
29. ### If the Continuous Basis Is abcd, Then the Simple Basis

Is ABCD Too (1/2) If = + − + , then + ≈ () () ≈ ( ) + ( ) − + ( ) 29/55
30. ### If the Continuous Basis Is abcd, Then the Simple Basis

Is ABCD Too (2/2) Approximating = , with = 0 or = (), the simple basis has ABCD parameters: • ≈ − + 2 + − + + 2 1 • ≈ 1 − − 1 • ≈ • ≈ 30/55
31. ### If the Simple Basis Is ABCD, Then the Continuous Basis

Is abcd Too If = + + , then is abcd too. Inverting the equations from the previous slide: • = 1 1−− − + − 1−− • = 1−− • = • = 31/55
32. ### EUR6M/OIS6M from abcd FRA/Swap Calibration • a=0.1930%, b=0.0010, c=0.211, d=0.0458%

0,00% 0,05% 0,10% 0,15% 0,20% 0,25% 0,30% 0,35% Legacy Curve Basis Simple Basis from Calibrated Continuous Basis 6 = 6 − 32/55

34. ### Continuous Basis {abcd + k} Exact-Fit Correction Exact fit is

possible with a term structure of correction factors : boostrapped on , linearly interpolated = ∙ = ∙ ∙ − 0 ≈ 1 goodness of fit 34/55
35. None
36. ### EUR6M/OIS6M from abcd FRA/Swap Calibration • a=0.1930%, b=0.0010, c=0.211, d=0.0458%

0,00% 0,05% 0,10% 0,15% 0,20% 0,25% 0,30% 0,35% Simple Basis from Calibrated Continuous Basis Simple Basis from Calibrated Continuous Basis 6 = 6 − 36/55
37. ### EUR6M abcd Errors Exact-Fit with ∈ [0.9984, 1.0005] 37/55

Rationale for ABCD Smooth Basis Modeling 3. Relation Between Continuous and Simple Basis 4. Multiple Tenor Analysis 5. Tenor Dominance 6. Conclusions 38/55
39. ### 0,00% 0,05% 0,10% 0,15% 0,20% 0,25% 0,30% 0,35% 0,40% 0,45%

0,50% 1M ABCD 3M ABCD 6M ABCD 1Y ABCD 0,00% 0,05% 0,10% 0,15% 0,20% 0,25% 0,30% 0,35% 0,40% 0,45% 0,50% 1M abcd 3M abcd 6M abcd 1Y abcd 39/55
40. ### Basis Maximum Location • The time corresponding to the maximum

continuous spread is roughly the same for all tenors. • If one considers the basis spread as a measure of uncertainty, then it is expected to be the same for all tenors. 40/55

45. ### EUR1Y abcd Errors Exact-Fit with ∈ [0.9998, 1.0019] 45/55

Rationale for ABCD Smooth Basis Modeling 3. Relation Between Continuous and Simple Basis 4. Multiple Tenor Analysis 5. Tenor Dominance 6. Conclusions 46/55
47. ### Ensuring Tenor Dominance • It is a comforting result of

the previous section that tenor dominance is empirically retrieved • This point is so crucial that one might want to ensure it by construction • It can be achieved by calibrating basis spread between adjacent tenors < , generalizing equations = − = − 47/55
48. ### Tenor Calibration Sequence • Calibrating the basis curves from lower

to higher tenors in order (i.e., 1M over ON, 3M over 1M, etc.) is hampered by the reduced liquidity of the 1M tenor • The problem can be solved by calibrating in order: – the most liquid positive basis spread of 6M over ON – the positive spread of 1Y over 6M – the negative basis spread of 3M over 6M – the negative basis spread of 1M over 3M • All continuous basis spreads have similar c parameter 48/55
49. ### Unintended Robustness • The approach is also robust to absolute

negative basis, as observed in the summer 2016: 1M Euribor rate levels below the equivalent ON rates 49/55
50. ### Tenor Dominance by construction 0,00% 0,05% 0,10% 0,15% 0,20% 0,25%

0,30% 0,35% 1M abcd 3M abcd 6M abcd 1Y abcd 50/55

Rationale for ABCD Smooth Basis Modeling 3. Relation Between Continuous and Simple Basis 4. Multiple Tenor Analysis 5. Tenor Dominance 6. Conclusions 51/55

53. ### Bibliography • F. Ametrano, P. Mazzocchi. Eonia Jumps and Proper

Euribor Forwarding (2014). https://speakerdeck.com/nando1970/eonia-jumps-and-proper- euribor-forwarding • F. Ametrano, M. Bianchetti. Everything You Always Wanted to Know About Multiple Interest Rate Curve Bootstrapping but Were Afraid to Ask (2013). http://ssrn.com/abstract=2219548 • M. Henrard. Interest Rate Modelling in the Multi-Curve Framework (Palgrave Macmillian 2014) • R. Rebonato. Volatility and Correlation (Wiley 2004) • S. Schlenkrich, A. Miemiec. Choosing the Right Spread (Wilmott 2015). http://ssrn.com/abstract=2400911 53/55
54. ### Further Research • Other Currencies • Fixed c parameter •

Calibrate Max Location, Max Value, Long-Time Value, Short-Time Value • Delta weighted (NPV) calibration 54/55
55. ### Key Points • Forward rates can be modeled as abcd

parametric tenor basis spreads over the underlying overnight rate curve. • This is possible for both continuously and simply compounded forward rates, with a simple approximation for converting between the corresponding basis. • Increasing interest-rate tenor dominance, as empirically observed, is recovered and can be structurally enforced using a robust methodology improvement based on relative basis between the most liquid tenors. • The smoothness requirement is moved from forward rate curves to tenor basis curves, properly dealing with the market evidence of jumps in forward rates. • In the case of continuously compounded tenor basis, pseudo-discount factors are also available (legacy systems compatibility, synthetic deposits, etc.) 55/55
56. ### Synthetic Deposits: Back to the abcd Future • Synthetic deposits

, < can be obtained by integration of the abcd continuous basis: = − [ +()] 0 = ( ) ∙ − () 0 • Additional recipe to the polynomial exact-fit suggestion of Ametrano, Mazzocchi (2014) 56/55
57. ### Synthetic Deposits: The abcd Limit • Being a global fit,

abcd is often not accurate enough at the short end of the basis curve, even if { } can mitigate the issue • Key problem: lack of a liquid traded instrument for the spot tenor, e.g. 0x6 FRA 57/55
58. ### Polynomial Basis Parameterization at the Short End of the Curve

• Ametrano, Mazzocchi (2014) used polynomials for exact-fit, but best-fit is more robust • Polynomials can be used for a best-fit focused on the basis curve short end only Continuous () = ∙ =0 Simple () = ∙ =0 58/55
59. ### Polynomial Basis Parameterization • Using () ≈ + () ,

an approximate transformation between simple and continuous polynomial basis can be obtained: () ≈ ∙ • If the simple basis is polynomial, then the continuous one is a polynomial of the same order, and vice-versa 59/55
60. ### Polynomial Best-fit of the Simple Basis • Best-fit is robust

to left extrapolation, e.g. 0x6 FRA 60/55
61. ### Divide, Itera, et Impera: Polynomials Synthetic Deposits in abcd Calibration

1. Calculate the synthetic deposits using polynomial best-fit of the short end continuous basis 2. Feed these synthetic deposits (at least the 0xX FRA) in the abcd continuous basis calibration 3. Bootstrap { } 61/55
62. ### abcd Continuous Basis The Short End of the Curve •

The abcd integral over the first 6M is not accurate 62/55
63. ### abcd Continuous Basis On Polynomials Steroids • The abcd integral

over the first 6M is now accurate! 63/55