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Overview
Düsseldorf, 30.11.2015
Negative Rates, SABR PDE and Approximation
PresenterJörg Kienitz Director Deloitte & Touche and Adj. Ass. Prof. Univers ity of Cape Town
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DISCLAIMER
This publication reflects the authors personal opinions that are not necessarily shared by Deloitte.
This publication contains general information only, and none of Deloitte & Touche GmbHWirtschaftsprüfungsgesellschaft or Deloitte Touche Tohmatsu Limited (“DTTL”), any of DTTL’smember firms, or any of the foregoing’s affiliates (collectively the “Deloitte Network”) are, by means ofthis publication, rendering accounting, business, financial, investment, legal, tax, or other professionaladvice or services.
This publication is not a substitute for such professional advice or services, nor should it be used as abasis for any decision or action that may affect your finances or your business. Before making anydecision or taking any action that may affect your finances or your business, you should consult aqualified professional adviser. No entity in the Deloitte Network shall be responsible for any losswhatsoever sustained by any person who relies on this publication.
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3© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft
Negative Rates- Markets, Models, Numerics -
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The good old times - Markets
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 4
Interest rate World
• Rates are significantly positive
• Volatilities are at „normal“ levels
• Quotes are in log-normal volatility or premium
• There was a simple to code approach for SABR to a model which gives a very well fit of the market observed volatility structures
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SABR
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 5
SDE
The (popular) SABR model is specified by the following system of StochasticDifferential Equations:
The function C is the local volatility. For the standard SABR we have
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SABR - Approximation Formula
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 6
Black Formula with SABR Volatility
An approximation formula for Log-normal (Black) volatilities in the SABR setting:
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SABR – Approximation Formula
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 7
General Bachelier Formula with SABR Volatility
An approximation for the Bachelier (Normal) volatilities in the SABR setting:
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SABR – The SABR Parameters
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 8
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SABR – Die SABR Parameter (Implied Bachelier)
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 9
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InputSABR - Calibration
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 10
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SABR - Calibration
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 11
Output
Usually β fix or calibrated using CMS quotes
Number of Iterations Differences to Market
SABR Parameters
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Anything to keep an eye on?SABR - Calibration
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 12
• „Smart Parameters“ (Nice starting values for localsolvers) help to stabilize yourSABR calibration and reducesthe number of iterations whichleverages the calibrationspeed.
• The ATM should be fit perfectly
• Calibration to all EUR Swaption < 0.5 sec. (even in VBA)
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The age of low or even negative ratesSABR
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 13
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Caps and Floors - ICAPMarket Quotes
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 14
Negative strikes
(Very) high volatilities
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Models – An Example for a Possible Pitfall
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 15
The SABR Approximation Formula
• After August 2007 severalmarket practices andusances changedsignificantly.
• Market parameters such as forward swap ratesbecame very small andthe correspondingvolatilites became veryhigh.
• OIS Discounting andcredit/liquidity issues andappearance of multitenored curves
• We consider the impacton the standard SABR model
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Which models can now be used?Negative Rates
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 16
• We have to handle curves
• Generally the negative rates do not have that impact on curvesconstruction and is a standard process these days
• The appearance of significant basis spreads have much more impact
• We still need to price options
• Caps/Floors
• Swaptions
• CMS Caps/Floors
• CMS Spread Options
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Negative Rates – Models
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 17
Pricing Options with negative Strikes
• The displaced diffusion or shifted log-normal model
• The Bachelier (or Normal) model
As we have already seen there are market quotes
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Negative Rates – Models
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 18
Pricing Options with negative Strikes
• The displaced diffusion or shifted log-normal model
To derive the implied volatility of Black-Scholes or Shifted-LogNormal type theimplementation from „Let‘s be rational“ by Peter Jäckel can be used.A reference implementation is provided (www.jaeckel.org)
• The Bachelier (or Normal) model
To derive the implied volatility of Bachelier type the implementation form „Fast and Accurate Analytic Basis Point Volatility“ by Fabien Le Floc'h can be used.A reference implementation is provided(http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2420757)
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Negative Rates - Markets
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 19
Implications for a banks business
• Zero Strike Floors (Implicit in many bonds)
• Options with negative strikes
• Model Choice (Construction of Volatility Surfaces/Hedges/Exotics/...)
• IT Systems (Implementation of new models, adjusting existing models)
• Regulations
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20© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft
SABR, No-Arb SABR, Free Boundary SABR- SDE, Parameters, Numerics -
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SABR – Negative Rates
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 21
Choosing Parametrizations
Taking again the SABR model:
To be able to handle negative rates we may choose:
Displaced SABR
Normal SABR
Free SABR
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SABR – Negative Rates
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 22
Methodology
To be able to use the SABR model in a negative rates setting we have tomake sure that:
• Efficient and Fast Pricing is possible (e.g. for calibration)
• Monte Carlo Methods for pricing are available (e.g. when combined with a term structure model or a hybrid model)
What are the numerical techniques we have to apply?
• PDE
• Monte Carlo
• Approximation
• Integration
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SABR – Integration Formulas (Antonov et al.)
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 23
2D Integration 2012 but zero correlation
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SABR – Integration Formulas (Antonov et al.)
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 24
1D Integration 2013 but zero correlation
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ExampleSABR – Antonov et al. Integration
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 25
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ExampleSABR – Antonov et al. Integration
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 26
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SABR – Integration Formulas (Antonov et al.)
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft
General Case• Projection onto a Bachelier, CEV or Zero Correlation SABR model
• The projection is not available for all values for the correlation
• No-arbitrage problems (negative values for the density), but much betterthan for the standard formulas
• The integration limits need to be adjusted wrt to the parameters
• Small values of the CEV parameter lead to wrong results
• Small values for the forward swap rates combined with small values of theCEV parameter can lead to wrong numbers very fast
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SABR – Integration Formulas (Antonov et al.)
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 28
Projection for a non-zero correlation cases
The authors propose several models for mimicking the original model.
They suggest the zero correlation SABR with parameters
We have for the model parameters:
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Normal SABR – Integration Formulas (Korn et al.)
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 29
2D Integration C(F)=1 but non-zero correlation
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Normal SABR
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 30
Implied Bachelier Volatilities - alpha
2,94% 0,05 0,508
3,24% 0,05 0,508
2,49% 0,05 0,508
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Normal SABR
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 31
Implied Bachelier Volatilities - rho
2,94% 0,555 0,508
2,94% -0,439 0,508
2,94% 0,605 0,508
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Normal SABR
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 32
Implied Bachelier Volatilities - nu
2,94% 0,05 0,38
2,94% 0,05 0,614
2,94% 0,05 0,508
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Normal SABR – Integration Formula (Antonov)
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 33
1D Integration 2015 and non-zero correlation
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 34
SABR becomes technical
• PDE method for calculating the SABR density numerically
• Efficient schemes (DO NOT USE Crank-Nicolson)
• Non-Standard grids and other tricks speed up the calculation
• Density using the PDE and option prices via numerical integration
• (New) approximation formulas for calibration
• Implementation details(http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2402001)
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 35
The PDE
• First, we consider the Forward Equation for the reduced density
• This Forward Equation for the density is then given by
• By introducing a function this simplifies to
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 36
The Boundary Conditions I
• Conservation (Probability Mass is equal to 1)
leads to consider
and finally requires
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 37
The Boundary Conditions II
• Martingale Property (The forward is preserved)
leads to the observation that
PDE PDE
Similar to qL
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 38
The Final PDE
Continuous part of thedistribution
Lower boundary (from modelor the modellers decision)
Upper boundary (fromdiscretization)
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 39
Finite Difference Grid
Start
Initial Condition
EndBoundary
Boundary
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 40
SABR becomes technical
Explicit
Solution is directlygiven
Implicit
Solution needs to becalculated bysolving a system ofequations
• Implicit is slow but accurate
• Explicit is fast but may not suitable
• Mixing Implicit and Explicit using the Theta-Scheme (1/2 for Crank-Nicolson)
• How to choose lower and upper values fordetermning the grid
Standard schemes are tempting to apply but oftenmore efficient/sophisticated methods are available.
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Details – Transforming Variables (gain efficiency)No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 41
f 1
alpha 0,35
beta 0,25
rho -0,1
nu 1
nd 2 ybar zbar
T 1 -1,33333333 -1,97706584
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 42
Discretization
Lower bound (z.B. 0 or –Displacement)
Upper bound
Between lower and upperbound
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 43
The Discrete Operator
See LeFloch (2014)
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 44
Probabilities at the Boundaries
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Problems with Crank-NicolsonNo-Arb SABR – Efficient Schemes
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 45
10 time steps
80 time steps
100 time steps
120 time steps
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And with an efficient schemeNo-Arb SABR – Efficient Schemes
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10 time steps80 time steps100 time steps120 time steps
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 47
Standard SchemesImplicit Euler
Crank Nicolson
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 48
Efficient Schemes, e.g. Lawson SwayneLawson-Swayne
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 49
Efficient SchemesLawson-Swayne
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And with an efficient schemeNo-Arb SABR – Pricing via Integration
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 50
K
Integrating
with respect to the density (blue)
• Choose an integration scheme
• Just use the representation of thePDE with appropriate handling of theleft tail (red)
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Option Pricing FormulasNo-Arb SABR – Pricing via Integration
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We assume that the FDM Scheme applied uses steps and a size of
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No-Arb SABR – PDE Ansatz
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 52
SABR becomes technical
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No-Arb SABR – Approximation Formulas
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 53
Old Results from Libor Market Models
• Andersen/Brotherton-Ratcliffe show an approximation in the context of Libor Market Models with Stochastic Volatility
• The results can be carried over to the SABR model
• This approximation can be used in calbration
To be determined forSABR!
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No-Arb SABR – Approximation Formulas
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 54
Old Results from Libor Market Models
LogNormal Volatility Approximation Bachelier Volatility Approximation
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Benchmarking Approximation FormulasComparison – Approximation vs Integration/PDE
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 55
• Comparison of approximation formulas
• Comparison to Bachelier volatility
• Comparison to Call prices
• Benchmark against Integration/PDE Solutions
• Integration/PDE leads to density and prices
• Implied Bachelier Vol Solver is used (-> see later)
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Hagan, Andersen/Brotherton-RatcliffeApproxformulas vs PDE Solution
Comparison –Approximation Formulas for SABR
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 56
forward 1beta 0,25v0/alpha 0,35gamma/nu 1rho 0,35tau 2
displacement 0
Values from
Hagan, P.
ICBI Global Derivatives 2014
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Hagan, Andersen/Brotherton-RatcliffeApproxformulas vs PDE Solution
Comparison – Normal Vol Approximation vs PDE
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 57
forward 0,0005beta 0v0/alpha 0,5gamma/nu 1rho -0,3tau 2
displacement 0
forward 0,0005beta 0v0/alpha 0,5gamma/nu 1rho 0,3tau 2
displacement 0
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Comparison – Approximations H vs AB
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forward (f0) 0,0005beta 0v0/alpha 0,5gamma/nu 1rho 0,9T 2
forward (f0) 0,0005beta 0v0/alpha 0,5gamma/nu 1rho -0,9T 2
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Comparison – Approximations H vs AB
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 59
forward (f0) 0,0005beta 0v0/alpha 0,5gamma/nu 1rho -0,6T 20
forward (f0) 0,0005beta 0v0/alpha 0,5gamma/nu 1rho 0,6T 20
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ConclusionComparison – Approximations H vs AB
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 60
• Numerical experiments suggest that the correlation determines whichapproximation to apply (Hagan approximation or Andersen/Brotherton-Ratcliffeapproximation)
• The calculation time does not differ between the methods
• We need to validate the calibration procedure(Do the parameters run to the boundaries?)
• The approximation formulas lead to better results than the PDE pricer
• PDE approach is numerically much more involved
• We need methods to stabilize the set-up
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Free Boundary SABR – (Antonov et al.)
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SABR gets technical
There is an approximation formula which numerically approximates the freeboundary SABR model with zero correlation:
For the general model a Markovian projection has to be applied to approximatethis case by means of the zero correlation model.
We have a PDE solution and approximation formulas here as well, see
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2647344
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Free Boundary SABR - Example
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 62
Calculation via Numerical Integration
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Free Boundary SABR – Approximation I
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Approximation in Free SABR Setting
For the free SABR case we find:
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Free Boundary SABR – Approximation II
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 64
Approximation in Free SABR Setting
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Free Boundary SABR - Example
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Comparison Integration, PDE, Approximation
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Free Boundary SABR - Example
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 66
Comparison Integration, PDE, Approximation
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Free Boundary SABR - Example
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 67
Comparison Integration, PDE, Approximation
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Free Boundary SABR - Example
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 68
Comparison Integration, PDE, Approximation
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Free Boundary SABR - Example
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 69
Calculation via PDE
• PDE method for calculating the SABR density numerically can be adapted tothe Free SABR case
• Non-Standard grids and other tricks speed up the calculation again
• Density using the PDE and option prices via numerical integration
• Being able to check the integration and the (new) approximation formulas
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Free Boundary SABR - Example
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 70
Calculation via PDE
alpha 0,011281809
beta 0,25
nu 0,3
rho -0,3
forward 0,005
timemat 3
Nsteps 320
Tsteps 320
nd 4
Transformation of Variables
Solving the PDE
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Free Boundary SABR – (Antonov et al.)
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SABR gets technical
• Integrals solution for correlation 0 case and Markovian projection for othercases (might be instable for large values of absolute correlation)
• PDE approach as shown for No-Arbitrage SABR
• Approximation formulas as shown for No-Arbitrage SABR
This allows to test approximation formulas and fastens the calibration.
Are the good old times back?
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Free Boundary SABR – (Antonov et al.)
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 72
Examples
We have applied the modified PDE solution to different parameter sets. The following illustrations are for different values of T and beta.
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Bachelier Implied Volatility - betaFree SABR
© 2014 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 73
4,70% 0,18 -0,05 0,384
4,70% 0,21 -0,05 0,384
4,70% 0,15 -0,05 0,384
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Bachelier Implied Volatility - rhoFree SABR
4,70% 0,18 -0,05 0,384
4,70% 0,185 0,1 0,384
4,70% 0,185 -0,13 0,384
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Bachelier Implied Volatility - nuFree SABR
4,70% 0,18 -0,05 0,384
4,70% 0,185 -0,06 0,568
4,70% 0,185 -0,06 0,088
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Bachelier Implied Volatility - alphaFree SABR
4,70% 0,18 -0,05 0,384
4,02% 0,185 -0,06 0,384
5,06% 0,185 -0,06 0,384
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Calibrating Caps with Free Boundary SABRFree Boundary SABR – (Antonov et al.)
Calibration example on EUR June data.
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 77
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Free Boundary SABR – (Antonov et al.)
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 78
Calibrating Caps with Free Boundary SABR
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Free Boundary SABR – (Antonov et al.)
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 79
Calibrating Caps with Free Boundary SABR
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SABR
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 80
Shape of the (Bachelier) Smile
DD SABR / No Arb SABR
Implied Volatilities tend decrease for small strikes(‚SABR Knee‘)
Free SABR / Normal SABR
Implied Volatilities do not decrease for small strikes
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SABR – Sticky Absorbing SABR
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 81
A New Parametrization
The SABR model was:
To be able to handle negative rates we may choose:
Displaced SABR
Normal SABR
Free SABRSticky Absorbing SABR
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Recovering the SABR Knee (using ) SABR – Sticky Absorbing SABR
© 2014 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 82
Recovering the Knee
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Controlling the Stickyness of Rates (using)
SABR – Sticky Absorbing SABR
© 2014 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 83
Free SABR
Sticky SABR
MovingStickynessaround
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SABR
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 84
Summary
• Using new numerical methods the SABR model can be tackled
• Integration, PDE and Approximation formulas are available
• The probability density and prices can be calculated efficiently
• The Free SABR model, the Displaced and the Normal SABR canaccomodate negative rates
• The Free SABR and Normal SABR imply that the implied volatilities do not tend to 0 for very small number
• To obtain this result DD SABR or an artifical lower bound should be applied
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85© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft
Mixture of Models- ZABR, SABR -
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Rationale – Controlling the WingsMixture of Models
• Use different models / different parametrizations which are analytically tractable
• Take a convex combination
• This was first done with Shifted Log-Normal models (different displacements)
• We can apply the ZABR or the (Free Boundary) SABR model
• This methodology is for pricing Vanillas since it does not give a reasonablemodel dynamics -> See Piterbarg „Mixture of Models: A Simple Recipe for a … hangover“ (2009)
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 86
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Smile Control by Mixing?ZABR – Mixture of ZABR Models
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 87
Mixing ZABR models tocontrol the behaviour of thewings.
This allows to control thebehaviour of the left or theright wing separately.
In contrast to simply usingZABR.
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Smile Control by Mixing – Antonov ApproachSABR – Mixture of Different SABR Models
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The same mixing approach can be applied to SABR models.
In this case Antonov et al. suggest to mix free SABR with zero correlation SABR.
However, using their approach prices have to be calculated and transformed to impliedBachelier volatilties. The approach recently put forward by Kienitz only considers impliedBachelier volatilties.
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Smile Control by Mixing – Kienitz ApproachSABR – Mixture of Different SABR Models
© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 89
Here we consider the mixing of SABR models while keeping the tail behaviourto the right the same for both SABR models.
We use the approximation formula in the Free SABR context.
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© 2015 Deloitte & Touche GmbH Wirtschaftsprüfungsgesellschaft 90
Negative Rates, SABR, CMS - LiteraturePaper, Books
• Andreasen, J., Huge, B., „ZABR Expension for theMasses“, SSRN 2012
• Andersen, L., Brotherton-Ratcliffe, R., „Extended Libor –market Models with Stochastic Volatility“, SSRN 2001
• Antonov, A., Spector, M., „Advanced Analytics for theSABR Model“, SSRN 2012
• Antonov, A., Konikov, M., Spector, „SABR Spreadsits wings“, Risk 2014
• Antonov, A., „The Free Boundary SABR: Extensionsto Negative Rates“, SSRN
• Antonov A., Konikov M., Spector M., „Mixing SABR Models for Negative Rates“, SSRN
• Hagan, P., „Convexity Conundrums“, Bloomberg Working paper
• Hagan, P., „Managing Smile Risk“, Wilmott Magazine 2002
• Hagan, P. et al., „No-Arbitrage SABR“, WilmottMagazine, 2014
• Kienitz, J., „Pricing CMS Spread Options“, Global Derivatives 2011
• Kienitz, J., „Advanced Issues in Libor Market Models“, Global Derivatives 2013
Papers
Books
• Kienitz, J. „Interest Rate Derivatives Explained – Vol1“, Palgrave McMillan, 2014
• Kienitz, J., Wetterau, D., „Financial Modelling“, Wiley, 2013
• LeFloch, F. „Finite Difference Techniques for Arbitrage Free SABR“, SSRN 2014
• LeFloch, F., „Explicit SABR Calibration through simple Expansions“, SSRN 2014
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