[PPT] - Calibration and Pricing with the LGM Model HSH NORDBANK Dr. Werner PowerPoint Presentation

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ASPECTS OF PRICING IRREGULAR SWAPTIONS WITH QUANTLIB

Calibration and Pricing with the LGM Model

HSH NORDBANK

Dr. Werner Kürzinger

HSH-NORDBANK.DE

Düsseldorf, November 30th, 2017

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The content of this presentation reflects the personal view and opinion of the author only. It does not express the view or opinion of HSH Nordbank AG on any subject presented in the following.

Disclaimer

1 November 30th, 2017

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European and Bermudan Swaptions: Appearance Regular European Swaptions (Black76) Irregular European Swaptions (LGM) Calibration

Payoff Matching

European und Bermudan Swaptions: Appearance

November 30th, 2017 4

hedges of callable loans and bonds
EPE and ENE profiles of swaps for the computation of XVA

… Irregular swaptions appear in a natural way for example by constructing:

callable zero bonds
EPE and ENE profiles of irregular swaps

…

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European and Bermudan Swaptions: Appearance Regular European Swaptions (Black76) Irregular European Swaptions (LGM) Calibration

Payoff Matching

Regular European Swaptions

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adjustment: modified following
settlement style: cash

Broker quotes of implied volatilities follow typically a swap market model. Swap market model solution for an European swaption is the well known Black76 pricing formula. QuantLib engine: BlackSwaptionEngine

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European and Bermudan Swaptions: Appearance Regular European Swaptions (Black76) Irregular European Swaptions (LGM) Calibration

Payoff Matching

Irregular European Swaptions

November 30th, 2017 8

notification period <> spot period
exercise fees

…

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Need of a model to calibrate against swaption prices. Choice: Linear Gauss Markov model (LGM model):

model is technically comfortable
multicurve extension is easy to implement
parameterization is connected to the Hull-White model (HW model)

Irregular European Swaptions in the LGM Model

November 30th, 2017 9

can be found in front office systems

Alternatives:

HW model in the HJM framework

…

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LGM model, process of „state variable“: with LGM model, numeraire:

Swap and Swaption in the LGM Model: Basics

dW t dX ) ( = α ) ( = X

November 30th, 2017 10 Hagan: Evaluating and Hedging Exotic Swap Instruments via LGM

LGM model, numeraire: with

)) ( ) ( ) ( exp( ) ( 1 ) , (

2 2 1

t t H x t H t D x t N ζ + = ds s t

t

) ( ) (

2

∫

= α ζ

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Connection to the HW model:

Swap and Swaption in the LGM Model: Basics κ

κ

/ ) 1 ( ) (

*t

e t H

−

− = ) 1 ( ) (

* * 2 2

− =

t

e t

κ

σ ζ

November 30th, 2017 11 Hagan: Evaluating and Hedging Exotic Swap Instruments via LGM

The HW short rate volatility is chosen to be constant for simplicity.

) 1 ( 2 ) (

* * 2

− =

t

e t

κ

κ σ ζ

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Swap is characterized by its fixed leg. Assume a given basis- or coupon spread on the floating leg . The transformed spread is then given by (the nominal is assumed to be constant):

Swap and Swaption in the LGM Model: Swap

n

A t D S S ) (

∑

= τ

float i

S ,

November 30th, 2017 12 Hagan: Evaluating and Hedging Exotic Swap Instruments via LGM

with

fix i i float i float i

A t D S S ) (

1 , ,

∑

=

= τ

) (

1 ,

∑

=

m i i fix i fix

t D A τ

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Times: , fixed rate: , spread: Parameters: and Discount curve: Swaption value (payer):

Swap and Swaption in the LGM Model: Swaption ) ( i

i

t H H = ) ( ex

ex

t ζ ζ = ) ( i

i

t D D =

n i ex

t t t t ,..., ,..., , 0 S R

State variable has to fulfill (defines „break even“ point of state variable): We focus on an option on a bullet swap with nominal=1 here.

∑

=

        − − − − −         − − − −         − =

n i ex ex i i i ex ex n n ex Swaption

H H y N D S R H H y N D y N D NPV

1 * * *

) ( ) ( ) ( ζ ζ τ ζ ζ ζ

( ) ( )

∑

=

− − − − − − − − − − =

n i ex i i i i ex n n n

H H H H y D S R H H H H y D D

1 2 * 2 *

) ( 2 / 1 ) ( exp ) ( ) ( 2 / 1 ) ( exp ζ τ ζ

*

y

November 30th, 2017 13 Hagan: Evaluating and Hedging Exotic Swap Instruments via LGM

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The model is going to be fixed via 2 loops:

Inner: Find state variable by solving the break even equation.
Outer: Fixing of parameters

and by fit to swaption prices

Swap and Swaption in the LGM Model: Calibration

*

y

) (t H H = ) (t ζ ζ =

November 30th, 2017 14 Hagan: Methodology for Callable Swaps and Bermudan „Exercise into“ Swaptions

Outer: Fixing of parameters

and by fit to swaption prices

f proper calibration instruments.

) ( i

i

t H H = ) ( ex

ex

t ζ ζ =

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European and Bermudan Swaptions: Appearance Regular European Swaptions (Black76) Irregular European Swaptions (LGM) Calibration

Payoff Matching

Options on Irregular Swaps: Calibration

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Payoff Matching The original irregular underlying swap is replicated by a regular swap. The parameters nominal, tenor and strike are used to fit the NPV, delta and gamma of the

riginal swap.

The resulting swap is used to find the Black76 volatility. The NPV can be found pricing the

Options on Irregular Swaps: Calibration

November 30th, 2017 17 Hagan: Methodology for Callable Swaps and Bermudan „Exercise into“ Swaptions

swaption of the replicating regular swap via Black76.

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Basket Cash flows of the underlying swap (interest and nominal payments ) are replicated by a set of n regular swaps (the „basket“) with weights . The begin date is identical for all swaps: The end date is for the i-th swap:

Options on Irregular Swaps: Calibration

,

t T

i Begin =

) (

,

t t n i t T

n i End

− + =

i

C

i i

N N −

−1 i

R

November 30th, 2017 18 Hagan: Methodology for Callable Swaps and Bermudan „Exercise into“ Swaptions

The fixed rate of the i-th swap consists of the individual fair rate plus a global parameter lambda : The i-th swap has fixed rate . Lambda is fixed by the condition, that the initial nominal of the original swap is equal to the weighted initial nominal of the basket components. The result is a vector of weights and an „add on“ lambda . For each of the corresponding swaptions a Black76 volatility can be found straightforward. The NPV proxy could be the weighted sum of the NPVs of the basket swaptions (talk by A. Miemiec, 2013). QuantLib engine: HaganIrregularSwaptionEngine

,

n

n i End

λ

i

C λ

i

r N λ +

i

r

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Basket + LGM with HW Parameterization The result from basket calibration is a set of n swaps with fixed rates and weights . Since all basket swaps are regular, we can price the swaptions via Black76. The LGM model in the HW parameterization has two parameters, choose the mean reversion for example at 1,5%.

Options on Irregular Swaps: Calibration

λ +

i

r

i

C

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Ansatz The remaining parameter (HW short rate volatility) could be fixed via a least squares approach: The result is the short rate volatility in the HW model.

( )

{ }

∑ =

−

n i LGM i B i i

NPV NPV C

1 2 76 2

min

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European and Bermudan Swaptions: Appearance Regular European Swaptions (Black76) Irregular European Swaptions (LGM) Calibration

Payoff Matching

Irregular Bermudan Swaptions (HW)

[ ]

t

dW t dt r t t dr ) ( ) ( ) ( σ κ θ + − =

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QuantLib engine/model: TreeIrregularSwaptionEngine/GeneralizedHullWhite

Hagan: Evaluating and Hedging Exotic Swap Instruments via LGM

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European and Bermudan Swaptions: Appearance Regular European Swaptions (Black76) Irregular European Swaptions (LGM) Calibration

Payoff Matching

Example 1: Regular European Swaption

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Nominal=1,5 Mio. EUR Single curve, SABR volatility

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„Irregular“ European swaption, 2% vs. 6M Start/end: 8.8.2024/8.8.2034 Exercise: 6.8.2024 Evaluation date: 4.1.2016 Nominal=1,5 - 4,16 Mio. EUR, nominal is increasing 12% p.a.

Example 2: Irregular European Swaption

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Nominal=1,5 - 4,16 Mio. EUR, nominal is increasing 12% p.a. Single curve, SABR volatility

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Code is in „prove of concept“ state. Extensions are accommodated within class HaganIrregularSwaptionEngine:

translating HW parameters to LGM parameters
LGM implementation (together with A. Miemiec)
modified Excel (*.qlo) and new Python (*.i) interface

Implementation

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modified Excel (*.qlo) and new Python (*.i) interface

Next steps could be:

extension to piecewise constant HW short rate volatilities (for Bermudans)
separate LGM model (model is already implemented in ORE)
extension to other calibration schemes

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Swaptions are basic components in many settings.
Quotation, calibration and evaluation can be performed separately and eventually in

different models.

Calibration is not unique for nonstandard instruments: There is an additional degree
f freedom in pricing besides (for example) the choice of the mean reversion for

Bermudans.

Summary

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Bermudans.

QuantLib provides many possibilities for pricing of complex swaptions. We presented
ne possible scheme to compute the NPV and the HW model parameters of an irregular

European swaption with a QuantLib prototype.

Results can be extended to the multicurve- and the Bermudan case easily.

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Björk: Arbitrage Theory in Continuous Time, third edition (2009)
Hagan: Evaluating and Hedging Exotic Swap Instruments via LGM
Hagan: Methodology for Callable Swaps and Bermudan „Exercise into“ Swaptions
Miemiec: Pricing of Accreting Swaptions using QuantLib, talk given at QuantLib User

Meeting 2013

Literature

November 30th, 2017 27

ASPECTS OF PRICING IRREGULAR SWAPTIONS WITH QUANTLIB

Calibration and Pricing with the LGM Model

Disclaimer

Contents

Contents

European und Bermudan Swaptions: Appearance

Contents

Regular European Swaptions

Contents

Irregular European Swaptions

Irregular European Swaptions in the LGM Model

Swap and Swaption in the LGM Model: Basics

dW t dX ) ( = α ) ( = X

)) ( ) ( ) ( exp( ) ( 1 ) , (

t t H x t H t D x t N ζ + = ds s t

) ( ) (

∫

= α ζ

Swap and Swaption in the LGM Model: Basics κ

/ ) 1 ( ) (

e t H

− = ) 1 ( ) (

− =

e t

σ ζ

) 1 ( 2 ) (

− =

e t

κ σ ζ

Swap and Swaption in the LGM Model: Swap

A t D S S ) (

∑

= τ

S ,

A t D S S ) (

∑

= τ

) (

∑

=

t D A τ

Swap and Swaption in the LGM Model: Swaption ) ( i

t H H = ) ( ex

t ζ ζ = ) ( i

t D D =

t t t t ,..., ,..., , 0 S R

∑

( ) ( )

∑

y

Swap and Swaption in the LGM Model: Calibration

y

) (t H H = ) (t ζ ζ =

) ( i

t H H = ) ( ex

t ζ ζ =

Contents

Options on Irregular Swaps: Calibration

Options on Irregular Swaps: Calibration

Options on Irregular Swaps: Calibration

Options on Irregular Swaps: Calibration

( )

{ }

∑ =

−

NPV NPV C

min

Contents

Irregular Bermudan Swaptions (HW)

[ ]

dW t dt r t t dr ) ( ) ( ) ( σ κ θ + − =

Contents

Example 1: Regular European Swaption

Example 2: Irregular European Swaption

Implementation

Summary

Literature