ASPECTS OF PRICING IRREGULAR SWAPTIONS WITH QUANTLIB Calibration and Pricing with the LGM Model

Transcription

1 ASPECTS OF PRICING IRREGULAR SWAPTIONS WITH QUANTLIB Calbraton and Prcng wth the LGM Model HSH NORDBANK Dr. Werner Kürznger Düsseldorf, November 30th, 2017 HSH-NORDBANK.DE

2 Dsclamer The content of ths presentaton reflects the personal vew and opnon of the author only. It doesnot expresstheveworopnonofhsh NordbankAG on anysubjectpresentedn the followng. November 30th,

3 Contents European and Bermudan Swaptons: Appearance Regular European Swaptons(Black76) Irregular European Swaptons(LGM) Calbraton - Payoff Matchng - Basket - Basket + LGM wth HW Parameterzaton Irregular Bermudan Swaptons(HW) Numercal Examples Implementaton Summary Lterature November 30th,

4 Contents European and Bermudan Swaptons: Appearance Regular European Swaptons(Black76) Irregular European Swaptons(LGM) Calbraton - Payoff Matchng - Basket - Basket + LGM wth HW Parameterzaton Irregular Bermudan Swaptons(HW) Numercal Examples Implementaton Summary Lterature November 30th,

5 European und Bermudan Swaptons: Appearance Swaptons(optons on nterest rate swaps) serve as buldng blocks n the context of: - callable fxed rate bonds - callable zero bonds - callable loans(.e. German law: 489 BGB) - hedgesofcallableloansandbonds - EPE andene proflesofswapsforthecomputatonofxva Irregularswaptonsappearn a naturalwayforexamplebyconstructng: - callable zero bonds - EPE andene proflesofrregularswaps November 30th,

6 Contents European and Bermudan Swaptons: Appearance Regular European Swaptons(Black76) Irregular European Swaptons(LGM) Calbraton - Payoff Matchng - Basket - Basket + LGM wth HW Parameterzaton Irregular Bermudan Swaptons(HW) Numercal Examples Implementaton Summary Lterature November 30th,

7 Regular European Swaptons Regular could be nterpreted that the swapton has the features of the broker. Conventons depend on currences. Typcal conventons for EUR swapton volatltes are: - fx bass: 30/360, fx frequency: 12m - float bass: Actual/360, float frequency: 3m (maturty=1 year), 6m (maturty>1 year) - calendar: TARGET - adjustment: modfed followng - settlement style: cash Broker quotes of mpled volatltes follow typcally a swap market model. Swap market model soluton for an European swapton s the well known Black76 prcng formula. QuantLb engne: BlackSwaptonEngne November 30th,

8 Contents European and Bermudan Swaptons: Appearance Regular European Swaptons(Black76) Irregular European Swaptons(LGM) Calbraton - Payoff Matchng - Basket - Basket + LGM wth HW Parameterzaton Irregular Bermudan Swaptons(HW) Numercal Examples Implementaton Summary Lterature November 30th,

9 Irregular European Swaptons Irregulartesarethenn prncpleall featuresoftheunderlyngswaportheswapton, thatdo not match the conventons of the broker: - step-upcouponon fxedsde - (step-up) spreadon thefloatsde - nomnal structure - notfcaton perod <> spot perod - exercse fees November 30th,

10 Irregular European Swaptons n the LGM Model Need ofa model tocalbrateaganstswaptonprces. Choce: Lnear GaussMarkovmodel (LGM model): - model s techncally comfortable - multcurve extenson s easy to mplement - parameterzaton s connected to the Hull-Whte model (HW model) - canbefoundn front offcesystems Alternatves: - HW model n thehjm framework November 30th,

11 Swap and Swaptonn thelgm Model: Bascs LGM model, process of state varable : dx ) = α( t dw wth X ( 0) = 0 LGM model, numerare: N( t, x) = 1 D( t) exp( H ( t) x H ( t) ζ ( t )) 2 wth ζ ( t) t 2 = α ( s) ds 0 Hagan: Evaluatng and Hedgng Exotc Swap Instruments va LGM November 30th,

12 Swap and Swaptonn thelgm Model: Bascs Connecton to the HW model: H ( t) = (1 e κ*t ) / κ ζ ( t) 2 σ = ( e 2κ 2* κ * t 1) The HW short rate volatlty s chosen to be constant for smplcty. Hagan: Evaluatng and Hedgng Exotc Swap Instruments va LGM November 30th,

13 Swap and Swaptonn thelgm Model: Swap Swap s characterzed by ts fxed leg. S, float Assumea gvenbass-orcouponspreadon thefloatngleg. The transformed spread s then gven by(the nomnal s assumed to be constant): n S = τ S D( t ) = 1, float, float A fx wth A fx = m = 1 τ, fx D( t ) Hagan: Evaluatng and Hedgng Exotc Swap Instruments va LGM November 30th,

14 Swap and Swapton n the LGM Model: Swapton t, t 0,..., t,..., t R S Tmes:, fxedrate:, spread: ex Parameters: and Dscount curve: Swapton value(payer): H = H t ) ζ = ζ t ) n ( D = D ( t ) ex ( ex NPV Swapton = y D0 N ζ * ex D n N y * ( H n ζ H ex 0 ) ζ ex n * y ( H H 0) ζ ex τ ( R S) D N = 1 ζ ex State varable has to fulfll(defnes break even pont of state varable): D y * n * 2 * 2 ( y ( H n H 0) 1/ 2( H n H 0) ζ ex ) τ ( R S) D exp( y ( H H 0) 1/ 2( H H 0 ζ ex ) 0 = Dn exp ) = 1 Wefocuson an optonon a bulletswapwthnomnal=1 here. Hagan: Evaluatng and Hedgng Exotc Swap Instruments va LGM November 30th,

15 Swap and Swaptonn thelgm Model: Calbraton The model sgongtobefxedva 2 loops: y * - Inner: Fnd state varable by solvng the break even equaton. H = H (tt ) ζ = ζ (tt ) ( - Outer: Fxng of parameters and ex by ft to swapton prces ofpropercalbratonnstruments. ( ex Hagan: Methodology for Callable Swaps and Bermudan Exercse nto Swaptons November 30th,

16 Contents European and Bermudan Swaptons: Appearance Regular European Swaptons(Black76) Irregular European Swaptons(LGM) Calbraton - Payoff Matchng - Basket - Basket + LGM wth HW Parameterzaton Irregular Bermudan Swaptons(HW) Numercal Examples Implementaton Summary Lterature November 30th,

17 Optons on Irregular Swaps: Calbraton In thecaseofa regular swaptononecanworkwththempledvolatltesandthe correspondng Black76 NPVs. Whatwouldbetherghtvolatlty(andtheNPV) n thecaseofan rregular swapton? November 30th,

18 Optons on Irregular Swaps: Calbraton Payoff Matchng The orgnal rregularunderlyngswapsreplcatedbya regularswap. The parametersnomnal, tenorandstrkeareusedtoft thenpv, deltaandgammaofthe orgnal swap. The resultngswapsusedtofnd theblack76 volatlty. The NPV canbefoundprcngthe swapton of the replcatng regular swap va Black76. Hagan: Methodology for Callable Swaps and Bermudan Exercse nto Swaptons November 30th,

19 Optons on Irregular Swaps: Calbraton Basket Cash flows of the underlyng swap(nterest R and nomnal payments replcatedbya setofn regularswaps(the basket ) wthweghts. C N 1 N ) are The begndatesdentcalforall swaps: TBegn, = t 0 The end datesforthe-thswap: T ( ) = t + t t End, 0 0 n n The fxedrate ofthe-thswapconsstsofthendvdual far rate r plus a global parameter lambda λ: The -thswaphasfxedrate + λ. r Lambda sfxedbythecondton, thatthentalnomnal oftheorgnal swapsequalto the weghted ntal nomnal of the basket components. The resultsa vectorofweghts C andan addon lambda λ. Foreachofthecorrespondng swaptons a Black76 volatlty can be found straghtforward. The NPV proxycouldbetheweghtedsumofthenpvs ofthebasketswaptons(talk bya. Memec, 2013). QuantLb engne: HaganIrregularSwaptonEngne N 0 Hagan: Methodology for Callable Swaps and Bermudan Exercse nto Swaptons November 30th,

20 Optons on Irregular Swaps: Calbraton Basket + LGM wth HW Parameterzaton The resultfrombasketcalbratonsa setofn swapswthfxedrates r + λandweghts C. Snce all basket swaps are regular, we can prce the swaptons va Black76. The LGM model n the HW parameterzaton has two parameters, choose the mean reverson for example at 1,5%. Ansatz The remanng parameter(hw short rate volatlty) could be fxed va a least squares approach: mn { ( ) } n 2 B76 LGM 2 C NPV NPV = 1 The resultstheshortrate volatltyn thehw model. November 30th,

21 Contents European and Bermudan Swaptons: Appearance Regular European Swaptons(Black76) Irregular European Swaptons(LGM) Calbraton - Payoff Matchng - Basket - Basket + LGM wth HW Parameterzaton Irregular Bermudan Swaptons (HW) Numercal Examples Implementaton Summary Lterature November 30th,

22 Irregular Bermudan Swaptons(HW) HW model (Hull-Whte ext. Vascek): [ θ ( t) κ( t) r] dt + σ ( t dwt dr = ) Evaluaton of Bermudans wth a tree: pecewse constant HW short rate volatltes needed QuantLb engne/model: TreeIrregularSwaptonEngne/GeneralzedHullWhte Hagan: Evaluatng and Hedgng Exotc Swap Instruments va LGM November 30th,

23 Contents European and Bermudan Swaptons: Appearance Regular European Swaptons(Black76) Irregular European Swaptons(LGM) Calbraton - Payoff Matchng - Basket - Basket + LGM wth HW Parameterzaton Irregular Bermudan Swaptons (HW) Numercal Examples Implementaton Summary Lterature November 30th,

24 Example 1: Regular European Swapton Regular European swapton, 2% vs. 6M Start/end: / Exercse: Evaluaton date: Nomnal=1,5 Mo. EUR Sngle curve, SABR volatlty November 30th,

25 Example 2: Irregular European Swapton Irregular European swapton, 2% vs. 6M Start/end: / Exercse: Evaluaton date: Nomnal=1,5-4,16 Mo. EUR, nomnal s ncreasng 12% p.a. Sngle curve, SABR volatlty November 30th,

26 Implementaton Code sn proveofconcept state. Extensons are accommodated wthn class HaganIrregularSwaptonEngne: - translatng HW parameters to LGM parameters - LGM mplementaton(together wth A. Memec) - modfed Excel (*.qlo) and new Python (*.) nterface Next steps could be: - extenson to pecewse constant HW short rate volatltes(for Bermudans) - separate LGM model (model s already mplemented n ORE) - extenson to other calbraton schemes November 30th,

27 Summary - Swaptons are basc components n many settngs. - Quotaton, calbratonandevaluatoncanbeperformedseparatelyandeventuallyn dfferent models. - Calbraton s not unque for nonstandard nstruments: There s an addtonal degree of freedom n prcng besdes (for example) the choce of the mean reverson for Bermudans. - QuantLb provdes many possbltes for prcng of complex swaptons. We presented onepossbleschemetocomputethenpv andthehw model parametersofan rregular European swapton wth a QuantLb prototype. - Results can be extended to the multcurve- and the Bermudan case easly. November 30th,

28 Lterature - Björk: Arbtrage Theory n Contnuous Tme, thrd edton(2009) - Hagan: Evaluatng and Hedgng Exotc Swap Instruments va LGM - Hagan: Methodology for Callable Swaps and Bermudan Exercse nto Swaptons - Memec: PrcngofAccretngSwaptonsusngQuantLb, talk gvenatquantlbuser Meetng 2013 November 30th,