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Negative Rates in a Multi Curve FrameworkCap Pricing and Volatility Transformation
Mattias Jonsson Ulrica Samark
May 11, 2016
Background to Negative Rates
Background to Negative Rates
Various reasons for negative rates:
Fight the deflation (SWE, EUR).
Protect its economy against the Eurozone dept crisis (DEN).
Make the currency weaker (SUI).
Background to Negative Rates
Background to Negative Rates
Problems caused by negative rates:
Construction of interest rate curves.
Quotation of volatility surface.
Interpolation and extrapolation of volatilities.
Outline
1 Introduction to CapsCap ContractMarket Conventions
2 Objectives
3 TheoryShifted SABRMulti Curve FrameworkVolatility Transformation
4 Methods for Calibration
5 Results
6 Conclusions
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Cap ContractIntroduction to Caps
Provides its holder an insurance against an increase of theinterestQuoted in volatilitiesComposed by caplets
Source: Nordea
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Market ConventionsIntroduction to Caps
Maturity ≤ 2 years
3 month caplets
3M EURIBOR
Maturity > 2 years
6 month caplets
6M EURIBOR
All maturities
OIS discount curve
Normal cap volatilities
Shifted Black cap volatilities (3%)
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Market ConventionsIntroduction to Caps
Maturity ≤ 2 years
3 month caplets
3M EURIBOR
Maturity > 2 years
6 month caplets
6M EURIBOR
All maturities
OIS discount curve
Normal cap volatilities
Shifted Black cap volatilities (3%)
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Market ConventionsIntroduction to Caps
Maturity ≤ 2 years
3 month caplets
3M EURIBOR
Maturity > 2 years
6 month caplets
6M EURIBOR
All maturities
OIS discount curve
Normal cap volatilities
Shifted Black cap volatilities (3%)
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Objectives
1 Calibrate shifted SABRAbility to price caps that are not quoted in the market
2 Volatility TransformationTo get 6 month cap volatilities
3 Analysis of Calibration MethodsComparison of the proposed methods in terms of fit tomarket data
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Objectives
1 Calibrate shifted SABRAbility to price caps that are not quoted in the market
2 Volatility TransformationTo get 6 month cap volatilities
3 Analysis of Calibration MethodsComparison of the proposed methods in terms of fit tomarket data
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Objectives
1 Calibrate shifted SABRAbility to price caps that are not quoted in the market
2 Volatility TransformationTo get 6 month cap volatilities
3 Analysis of Calibration MethodsComparison of the proposed methods in terms of fit tomarket data
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Shifted SABR modelTheory
dXt = VtXβs dWt , X0 = x ,
dVt = νVtdZt , V0 = α,
where dWtdZt = ρdt, Xt = Ft + sand X0 = x = f + s.
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Shifted SABR modelTheory
dXt = VtXβs dWt , X0 = x ,
dVt = νVtdZt , V0 = α,
where dWtdZt = ρdt, Xt = Ft + sand X0 = x = f + s.
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Shifted SABR modelTheory
σSABRshift (x ,Ks) =α log( x
Ks)∫ x
Ks
dx ′
C(x ′)
·(
ζ
x(ζ)
)·{
1 +
[2γ2 − γ2
1 + 1/x2av
24α2C (xav)2+
1
4ρανγ1C (xav) +
2 − 3ρ2
24ν2
]T + . . .
},
where Ks is the strike for the shifted process Xt .
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Multi Curve FrameworkTheory
Spread between EURIBOR 3M and OIS 3M (since 2007)
One fixing rate (OIS), several funding rates (EURIBOR1M, EURIBOR 3M, ...)
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Volatility TransformationTheory
STEP-BY-STEPALGORITHM
Volatility on forwardcurve is transformedto the OIS curve.
These aretransformed from3M to 6Mvolatilities.
Volatility from OIScurve is transformedback to the forwardcurve.
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Volatility TransformationTheory
STEP-BY-STEPALGORITHM
Volatility on forwardcurve is transformedto the OIS curve.
These aretransformed from3M to 6Mvolatilities.
Volatility from OIScurve is transformedback to the forwardcurve.
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Methods for Calibration
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
BootstrapMethods for Calibration
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
BootstrapMethods for Calibration
Forward prices: Capmi (K ) − Capmi−1(K ) i = 1, . . . ,M
Back out ”market” caplet volatilities by inverting Black’sformula.
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
RebonatoMethods for Calibration
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
RebonatoMethods for Calibration
Fit a functional form to get caplet volatilities
σ(s) =[a + b(T − s)
]exp
[− c(T − s)
]+ d .
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Global SABRMethods for Calibration
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Global SABRMethods for Calibration
Specifies a set of nodes for the parameter term structure,φ(T ) = (α(T ), β, ρ(T ), ν(T )), in the SABR model.
Optimizes the node values
F(φ(T )
)=∑T
∑K
(Capm(T ,K ) − CapSABR(T ,K )
)2
.
Knots for Global SABR method
Param. Nbr Knots Placement Interpolation
α 6 0, 1, 3, 5, 7, 10 ”spline”ρ 3 0, 5, 10 ”pchip”ν 6 0, 1, 3, 5, 7, 10 ”spline”
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Global SABRMethods for Calibration
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Results
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Results
Bootstrap
Captures ATMbest
Fails toreconstruct afew OTM caps
Lowest meanerror
Rebonato
Can’t captureATM
Fails toreconstruct capprices
Highest meanerror
Global SABR
Captures ATM
Fails toreconstructsome OTMcaps
Mid meanerrors
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Conclusions
The calibration of shifted SABR works well.
The developed algorithm for volatility transformationworks fine.
The best performing method is Bootstrap.
NegativeRates in a
Multi CurveFramework
MattiasJonsson,
Ulrica Samark
Introductionto Caps
Cap Contract
MarketConventions
Objectives
Theory
Shifted SABR
Multi CurveFramework
VolatilityTransformation
Methods forCalibration
Results
Conclusions
Further Research
Develop a framework for estimating correlations betweenforward rates.
Refine node selection algorithm in global SABR.
Investigate if normal volatilities could be used, and developa similar transformation technique.
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