Improving he Jarrow-Yildirim Inflaion Model Rober Hardy May 19, 2013 1 Inroducion The mos liquid inflaion markes are hose of he US, UK, France and Eurozone. Each is suppored by a regular supply of governmen-issued inflaion-linked bonds, and mos rading desks provide swaps and vanilla opions o some exen. The Eurozone marke is he mos developed in erms of derivaive producs. Roughly in order from he mos- o leas-liquid, here are prices in zero-coupon (ZC) swaps, year-on-year (YoY) swaps and opions, zero-coupon opions; i is also possible o ge prices for some firs-order exoics such as YoY digials and YoY range-accruals. Wihin all markes he YoY vanilla opion flows are concenraed on he zero-percen srike, wih he YoY 0% floors being he mos raded produc, mainly because YoY swap rades wih cliens end o have he coupons of he inflaion leg floored a 0%. Oher YoY srikes will rade less frequenly, perhaps as hedges o srucured producs such as inflaion-linked MTNs, and in periods where he marke anicipaes higher levels of inflaion here may be increased amouns of rading of he highersrike (eg 4% or 5%) caps. All hings considered, i is fair o say ha smile modelling is less imporan for inflaion markes han i is for he raes markes, say. In fac i is only he UK inflaion marke wih is LPI produc for pension funds which migh presen he need for a smile-enabled mone-carlo model, and even here one can find a reasonable workaround wih approximaion formulas. In he auhor s experience he main prioriy for inflaion modelling has been o produce a single model which he rading desk can use o: generae he convexiy adjusmens for YoY swap raes, calibrae o he erm srucure of YoY 0% srike vols, calibrae o he erm srucure of ZC 0% srike vols, generae paymen-delay adjusmens for producs like pay-as-you-go swaps. The benefi of having a single model o generae hese volailiy-dependen prices or price adjusmens is wofold: rading and risk-conrol eams prefer o have a single model which can saisfacorily explain he prices seen in he marke, and secondly i offers he rading desk more hedging sraegies such as hedging ZC opions wih YoY or hedging he YoY swap adjusmens wih YoY opions. I goes wihou saying ha a single model which achieves all hese requiremens can be used as he basis for pricing and risk managing he few pah-dependen producs ha ge requesed from ime o ime even hough i is no smile enabled we can be confiden ha i correcly reflecs he core volailiy and correlaion levels of he marke (raders end o see he spread beween ZC and YoY opion volailiies as reflecing he correlaions amongs he erm srucure of YoY opions). 1
The earlies arbirage-free model for inflaion was presened in an aricle by Jarrow and Yildirim, and is based on he FX analogy. Since i is in fac nohing oher han he well known HJM crosscurrency model, i was easy for rading houses o code up a new inflaion wrapper for he FX model and voila! he JY model became he sandard approach for dealing wih inflaion-linked derivaive producs. In recen years he JY model has seen is populariy fade, as marke-model approaches have become developed, and i is easy o undersand why: he JY model seems o suffer from over-paramerizaion, i diffuses he raher esoeric real-yield process, i is no obvious how o calibrae. In conras, he marke models ake a more inuiive slimmed-down approach and diffuse he inflaion process direcly, and i ends o be obvious how hey should be calibraed. The fac is however ha wih a small amoun of work, he JY model can be modified o produce a new model ha is perfecly able o acheive all of he requiremens lised above. In his aricle we presen he mahemaics behind a re-facoring of he JY model which produces a very saisfacory inflaion model. Furhermore, we show ha a marginally reduced version of his new model can be implemened very quickly as a wrapper around an exising implemenaion of he JY model, which means ha you can have his beer version up and running wih a minimal amoun of effor. 2 The JY model The JY model is based on real and nominal economies, each wih is own yield curve, which are conneced by a spo process for he inflaion index. The inflaion index is analogous o he FX spo process and dicaes he curren nominal price of real asses. The JY model specifies he dynamics of he real and nominal discoun facors and he spo index process (respecively B r (; T ), B n (; T ) and I()), under he risk-neural measure P n, as follows: db n (; T ) B n (; T ) = r n() d + σ Bn (; T ) dw n, db r (; T ) B r (; T ) = [r r() σ I ()σ Br (; T )ρ ri ()] d + σ Br (; T ) dw r, di() I() = [r n() r r ()] d + σ I () dw I, where ( ) W n, W r, W I is a Brownian moion under Pn wih correlaion marix 1 ρ nr ρ ni ρ nr 1 ρ ri ρ ni ρ ri 1 Gaussian dynamics for he raes are specified: T σ Bk (; T ) = σ k () e s λ k(u) du ds, k = n, r wih σ n, σ r, λ n and λ r being deerminisic funcions (he shor-rae vols and he mean reversions). I is no obvious how o fully calibrae he JY model, bu below is an ouline of he approach his auhor found o be mos pracicable and useful (paricularly because i gives good-qualiy risks): 1. calibrae he erm srucure of nominal volailies, he σ n, in order o correcly price libor caps a a given srike (depending on he nominal vol hedge o be used), 2
2. calibrae he erm srucure of real volailies, he σ r, in order o produce he correc convexiy adjusmen for YoY swap raes, 3. calibrae he erm srucure of CPI volailies, he σ I, in order o correcly price a chosen se of YoY opions (usually being he 0% floors). This calibraion recipe sill leaves a number of parameers unconsrained: he wo mean reversions λ n and λ r and he hree correlaions ρ ni, ρ ri and ρ nr. I is possible o use hisorical series o pu a figure on ρ nr and in principle he same applies o ρ ni and ρ ri bu esimaion is raher more difficul because of he low number of hisorical index daa poins. Insead i was preferred o se ρ ni = ρ ri = 0 because i means ha he sep where σ I is calibraed o YoY opions will no disurb he previous calibraion o he convexiy adjusmens (which depend on σ r, σ n, ρ ni and ρ ri ). For he mean reversions a pragmaic soluion works bes, a compromise based on hisorical analysis and mahemaical simpliciy, which for he Eurozone marke mean choosing λ n = λ r = 10%. 3 Refacoring he JY model The key o improving he calibraion and he dynamics of he JY model is o no diffuse he real-yield curve bu insead o diffuse an inflaion curve, as we show in his secion. To moivae he new erms we inroduce, we recall ha in he JY model he ime- value of he inflaion index wih mauriy T is given by: I(; T ) = I() B r(; T ) B n (; T ) = I() T e fn(;s) fr(;s) ds, where f n (; s) and f r (; s) represen he ime- values of he insananeous forward raes wih mauriy s in he nominal and real economies. Clearly he spread f n (; s) f r (; s) defines an implied curve of insananeous inflaion forward raes, which we wrie as f i (; s) and which our new model diffuses (raher han f r (; s) in he JY model). To his end we define a process ZC(; T ) which represens he coninuously-compounded inflaionary growh beween imes and T : ZC(; T ) := e T fi(;s) ds. The new model keeps he same processes as JY for he nominal discoun facors and he inflaion spo index, bu replaces he process of real discoun facors wih he ZC process. The specificaion is: db n (; T ) B n (; T ) = r n() d + σ Bn (; T ) dw n, dzc(; T ) ZC(; T ) = r i() d + µ(; T ) d + σ ZC (; T ) dw i, di() I() = r i() d + σ I () dw I, where i can be checked ha for no-arbirage he drif erm mus be given as µ(; T ) = σ ZC (; T ) 2 ρ ii σ I ()σ ZC (; T ) + [ρ ni σ I () ρ ni σ ZC (; T )] σ Bn (; T ) and where ( ) W n, W i, W I is a Brownian moion under Pn wih correlaion marix 1 ρ ni ρ ni ρ ni 1 ρ ii ρ ni ρ ii 1 3
Gaussian dynamics for he nominal and inflaion raes are specified: T σ Bn (; T ) = σ n () σ ZC (; T ) = σ i () T e s λn(u) du ds, e s λi(u) du ds, wih σ n, σ i, λ n and λ i being deerminisic funcions (he shor-rae vols and he mean reversions). In his new model we have I(; T ) = I()ZC(; T ), from which i follows ha he forward-index erms I(; T ) have dynamics given by: di(; T ) I(; T ) = [ρ niσ I () ρ ni σ ZC (; T )] d + σ I () dw I () + σ ZC (; T ) dw i (), and herefore he Black-Scholes volaiilies of inflaion opions (boh YoY and ZC) are only dependen on he parameers σ I and σ i and he correlaion ρ ii. The convexiy adjumen is given by: E ( ) I(T2 ) = I(; T 2) I(T 1 ) I(; T 1 ) ( ) T1 exp [ρ ii σ I (s) σ ZC (s; T 1 )] [σ ZC (s; T 2 ) σ ZC (s; T 1 )] ds ( ) T1 exp [ρ ni σ I (s) ρ ni σ ZC (s; T 1 )] [σ Bn (s; T 2 ) σ Bn (s; T 1 )] ds The convexiy adjusmen erms facor nealy ino wo pars: one depending only on he inflaion parameers, and he oher which also depends on he nominal vol erms. This second erm arises from here being a paymen delay on he denominaor erm. 4 Calibraion of he new model The new specificaion gives a much beer relaionship beween he parameers of he model and he prices of he asses ha are raded in he marke and his immediaely improves he prospecs for a beer calibraion. The correspondence is: he prices of nominal opions (eg libor caps or floors) are deermined by σ n and λ n (as hey are in he JY model), he Black-Scholes volailiies of inflaion opions are deermined by σ I, σ i, ρ ii and λ i, he YoY convexiy adjusmens can be weaked wih ρ ni and ρ ni. Imporanly, he way ha σ i and σ I affec he Black-Scholes volailiies of ZC and YoY opions is quie differen, so we now have a mechanism for adjusing he spread beween ZC and YoY opions: if we pu more inflaion volailiy ino he model wih σ i we will end o increase he ZC-YoY volailiy spread, whereas if we use σ I o increase he inflaion volailiy we will end o decrease he ZC-YoY volailiy spread. Taking all his ino consideraion, he following scheme for calibraion of his new model has been found o work very well in pracice i will fi he YoY volailiies exacly and has been able o generae a good qualiy fi o ZC volailiies and YoY convexiy adjusmens. 1. sar he calibraion loop wih λ i = 0.1, ρ ni = 0, ρ ii = 0 and ρ ni = 0, and σ i = 0.005, 4
2. calibrae he erm srucure of nominal volailies, he σ n, and adjus he level of mean reversion λ n in order o correcly price he nominal hedge insrumens (eg libor caps and swapions a a given srike), 3. calibrae he erm srucure of he σ I, in order o correcly hi he Black-Scholes volailiies of he YoY opions a a given srike (again, depending on he inflaion vol hedge o be used), 4. increase (decrease) σ i in order o ge generally higher (lower) levels of BS volailiies for he inflaion ZC opions, and reurn o sep 3 5. increase (decrease) λ i in order o pu more (less) curvaure ino he shape of he BS volailiies of he ZC opions, and reurn o sep 3. 6. increase (decrease) ρ ni in order o widen (narrow) he convexiy adjusmen, and reurn o sep 3 Seps 4, 5 and 6 can obviously be encapsulaed in a minimizaion rouine. The wo remaining parameers which we have no ye addressed, ρ ii and ρ ni have a more suble effec on he shapes of he calibraed insrumens, bu wihin he calibraion loop we are suggesing here hey can be used o change he way marke movemens in he YoY opion volailiies generae moves in he ZC volailiies and in he convexiy adjusmens. In oher words hey give some degree of conrol o he rader o choose how marke moves in YoY ge carried across ino he ZC and convexiy markes. 5 A quick implemenaion of (a slighly-resriced version of) he new model A he cos of losing one degree of freedom, i is possible o map he new model back ono he JY model. This allows us o obain an almos immediae implemenaion of he new scheme wihin an implemenaion of JY. Namely, we insis ha we mus always have λ n = λ i and herefore lose he flexibiliy o calibrae o nominal swapions as well as caps, for example; his is a reasonable compromise. In oher words, if we have a se {λ n, λ i ; σ n (), σ i (), σ I (), ρ ni (), ρ ni (), ρ ii ()} of mean reversions and erm-srucure values for he parameers of he new model and furhermore have λ n = λ i, hen wih he following definiions: λ r := λ i, σ r () := σ n () 2 + σ i () 2 2ρ ni ()σ n ()σ i (), ρ nr () := 1 σ r () (σ n() ρ ni ()σ i ()), ρ ri () := 1 σ r () (ρ ni()σ n () ρ ii ()σ i ()), we have anoher se {λ n, λ r ; σ n (), σ r (), σ I (), ρ nr (), ρ ni (), ρ ri ()} of mean reversions and ermsrucure parameers which we can use in a JY model o generae exacly he same volailiy disribuions. This means ha by wriing a simple wrapper a he fron and back of an exising implemenaion of JY, we can very quickly build an implemenaion of his improved model. A lile more work on a basic spreadshee calibraion rouine will hen be enough o have a workable model which he rading desk can experimen wih. None of he inernals of he pricing engines needs o be re-plumbed. 5
6 Beer PnL Explain In he JY model a bump on he nominal vol parameer σ n will impac he valuaion of a book of inflaion opions poenially in wo ways: 1. i will affec Black-Scholes volailiy, 2. i will affec he convexiy adjusmen. The new model is in a much beer siuaion since a bump in he nominal vol will affex only he convexiy adjusmen, and here i only causes a change hrough he paymen delay componen of he adjusmen; in he JY model he change in he convexiy adjusmen will be due boh o he change of paymen delay and he change in implied inflaion volailiy. 6