ASPECTS OF PRICING IRREGULAR SWAPTIONS WITH QUANTLIB Calibration and Pricing with the LGM Model HSH NORDBANK Dr. Werner Kürzinger Düsseldorf, November 30th, 2017 HSH-NORDBANK.DE
Disclaimer 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. November 30th, 2017 1
Contents European and Bermudan Swaptions: Appearance Regular European Swaptions (Black76) Irregular European Swaptions (LGM) Calibration - - Payoff Matching Payoff Matching - Basket - Basket + LGM with HW Parameterization Irregular Bermudan Swaptions (HW) Numerical Examples Implementation Summary Literature November 30th, 2017 2
Contents European and Bermudan Swaptions: Appearance Regular European Swaptions (Black76) Irregular European Swaptions (LGM) Calibration - - Payoff Matching Payoff Matching - Basket - Basket + LGM with HW Parameterization Irregular Bermudan Swaptions (HW) Numerical Examples Implementation Summary Literature November 30th, 2017 3
European und Bermudan Swaptions: Appearance Swaptions (options on interest rate swaps) serve as building blocks in the context of: - callable fixed rate bonds - callable zero bonds - callable loans (i.e. German law: §489 BGB) - - hedges of callable loans and bonds 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 … November 30th, 2017 4
Contents European and Bermudan Swaptions: Appearance Regular European Swaptions (Black76) Irregular European Swaptions (LGM) Calibration - - Payoff Matching Payoff Matching - Basket - Basket + LGM with HW Parameterization Irregular Bermudan Swaptions (HW) Numerical Examples Implementation Summary Literature November 30th, 2017 5
Regular European Swaptions „Regular“ could be interpreted that the swaption has the features of the broker. Conventions depend on currencies. Typical conventions for EUR swaption volatilities are: - fix basis: 30/360, fix frequency: 12m - float basis: Actual/360, float frequency: 3m (maturity=1 year), 6m (maturity>1 year) - calendar: TARGET - 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 November 30th, 2017 6
Contents European and Bermudan Swaptions: Appearance Regular European Swaptions (Black76) Irregular European Swaptions (LGM) Calibration - - Payoff Matching Payoff Matching - Basket - Basket + LGM with HW Parameterization Irregular Bermudan Swaptions (HW) Numerical Examples Implementation Summary Literature November 30th, 2017 7
Irregular European Swaptions Irregularities are then in principle all features of the underlying swap or the swaption, that do not match the conventions of the broker: - step-up coupon on fixed side - (step-up) spread on the float side - nominal structure - notification period <> spot period - exercise fees … November 30th, 2017 8
Irregular European Swaptions in the LGM Model 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) - can be found in front office systems Alternatives: - HW model in the HJM framework … November 30th, 2017 9
Swap and Swaption in the LGM Model: Basics LGM model, process of „state variable“: ( 0 ) 0 = α ( ) = dX t dW X with LGM model, numeraire: LGM model, numeraire: 1 ( , ) exp( ( ) 2 ( ) ( )) = + ζ N t x H t x 1 H t t 2 ( ) D t t ∫ ( ) 2 ( ) ζ = α t s ds with 0 Hagan: Evaluating and Hedging Exotic Swap Instruments via LGM November 30th, 2017 10
Swap and Swaption in the LGM Model: Basics Connection to the HW model: * t ( ) = ( 1 − − κ ) / κ H t e 2 σ σ ( ( ) ) ( ( 2 2 * * * * 1 1 ) ) ζ ζ = = κ κ t t − − t t e e 2 κ The HW short rate volatility is chosen to be constant for simplicity. Hagan: Evaluating and Hedging Exotic Swap Instruments via LGM November 30th, 2017 11
Swap and Swaption in the LGM Model: Swap Swap is characterized by its fixed leg. S , Assume a given basis- or coupon spread on the floating leg . i float The transformed spread is then given by (the nominal is assumed to be constant): n ∑ ∑ ( ( ) ) = = τ τ S S S S D D t t A A , , i float i float i fix = 1 i with m ∑ = τ ( ) A D t , fix i fix i 1 i = Hagan: Evaluating and Hedging Exotic Swap Instruments via LGM November 30th, 2017 12
Swap and Swaption in the LGM Model: Swaption , 0 ,..., ,..., R t t t t S Times: , fixed rate: , spread: ex i n ( ex ) ( i ) ζ = ζ H = t H t Parameters: and ex i ( i ) D = D t Discount curve: i Swaption value (payer): − * * * − − ( − ) ζ − − ( − ) ζ y y H H y H H n ∑ ( ) = − 0 − τ − 0 NPV D N D N n ex R S D N i ex 0 Swaption n i i ζ ζ ζ = 1 i ex ex ex * y State variable has to fulfill (defines „break even“ point of state variable): ( ) n ( ) ∑ * 2 * 2 = exp − ( − ) − 1 / 2 ( − ) ζ − τ ( − ) exp − ( − ) − 1 / 2 ( − ) ζ D D y H H H H R S D y H H H H 0 0 0 0 0 n n n ex i i i i ex = 1 i We focus on an option on a bullet swap with nominal=1 here. Hagan: Evaluating and Hedging Exotic Swap Instruments via LGM November 30th, 2017 13
Swap and Swaption in the LGM Model: Calibration The model is going to be fixed via 2 loops: * y Inner: Find state variable by solving the break even equation. - ( i ( t ) ) ζ ζ = = ζ ζ ( ex ( t ) ) H = H = H H t t Outer: Fixing of parameters Outer: Fixing of parameters and and by fit to swaption prices by fit to swaption prices - - i ex of proper calibration instruments. Hagan: Methodology for Callable Swaps and Bermudan „Exercise into“ Swaptions November 30th, 2017 14
Contents European and Bermudan Swaptions: Appearance Regular European Swaptions (Black76) Irregular European Swaptions (LGM) Calibration - - Payoff Matching Payoff Matching - Basket - Basket + LGM with HW Parameterization Irregular Bermudan Swaptions (HW) Numerical Examples Implementation Summary Literature November 30th, 2017 15
Options on Irregular Swaps: Calibration In the case of a „regular“ swaption one can work with the implied volatilities and the corresponding Black76 NPVs. What would be the right volatility (and the NPV) in the case of an „irregular“ swaption? November 30th, 2017 16
Options on Irregular Swaps: Calibration 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 original swap. The resulting swap is used to find the Black76 volatility. The NPV can be found pricing the swaption of the replicating regular swap via Black76. Hagan: Methodology for Callable Swaps and Bermudan „Exercise into“ Swaptions November 30th, 2017 17
Options on Irregular Swaps: Calibration Basket − Cash flows of the underlying swap (interest R and nominal payments N N ) are − 1 i i i replicated by a set of n regular swaps (the „basket“) with weights C . i Begin = T t The begin date is identical for all swaps: , 0 i i ( ) = + − T t t t The end date is for the i-th swap: , , 0 0 0 0 End End i i n n n n r The fixed rate of the i-th swap consists of the individual fair rate plus a global parameter i + λ λ r lambda : The i-th swap has fixed rate . i Lambda is fixed by the condition, that the initial nominal of the original swap is equal to N 0 the weighted initial nominal of the basket components. λ The result is a vector of weights C and an „add on“ lambda . For each of the corresponding i 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 Hagan: Methodology for Callable Swaps and Bermudan „Exercise into“ Swaptions November 30th, 2017 18
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