An Implementation of the Displaced Diffusion, Stochastic Volatility Extension of the LIBOR Market Model

Transcription

1 Maser Thess Deparmen o Busness Sudes Auhor: Chrsan Sørensen Advsor: Elsa Ncolao An Implemenaon o he Dsplaced Duson, Sochasc Volaly Exenson o he LIBOR Mare Model A Comparson o he Sandard Model Handelshøjsolen, Aarhus Unverse July 007

2 Inroducon...3. Problem saemen...3. Delmaon Srucure o he paper...5 Noaon and denons...6. The neres rae cap...8. Exoc caps Change o numerare Relave prces Marngales Changng numerare The Grsanov heorem The Blac ormula The sandard LIBOR mare model No arbrage drs Volaly Desrable properes o he volaly speccaon Model and emprcal mpled volales Deren approaches o oban a non consan volaly Consan elascy o varance models Dsplaced duson models Jump duson models Regme swchng models Sochasc volaly models The exended LIBOR mare model No arbrage drs Comparson o he models Implemenaon ssues Choosng he measure/numerare Samplng he orward raes dsrbuon The covarance marx...44

3 8... Low dscrepancy numbers Condonally deermnsc dr erms Payo and expeced value Daa Calbraon The yeld curve Nelson-Segel Esmaon Correlaon Volaly Calbraon o he sandard model Calbraon o he exended model...70 Convergence...8. Convergence o calbraon seup Convergence o prcng seup...86 Prcng Concludng remars Concluson Apprasal o he exended model Suggesons or urher wor...03 The nroducon maes up or he mandaory absrac.

4 Inroducon The LIBOR mare model arbued manly o Brace, Gaare & Musela 997 and Jamshdan 997 s an neres rae model, whch models drecly a se o nonoverlappng dscreely compounded orward raes. I has become ncreasngly popular as allows or a lexble ramewor or prcng o a varey o neres rae producs. The dynamcs o he model resuls n log normally dsrbued orward raes, whch s he man assumpon o he Blac ormula. Ths means ha he LIBOR mare model can recover exacly he caple prces compued by he Blac ormula, addonally aords a nancal nerpreaon o he Blac mpled volaly. Ths has ncreased he accepance o he model as he Blac ormula s now he mare sandard. However gven log normally dsrbued orward raes he model can only produce mpled volales ha are consan across sres. As we wll show ha he mpled volales observed n he mare are no consan across sres he orward raes canno be log normally dsrbued. Researchers and praconers have hereore searched or exensons o he LIBOR mare model ha could remedy hs decency o he sandard model. Several models have been developed ha are able o reproduce a non-la volaly. Bu no consensus has been reached on he bes model.. Problem saemen In hs paper we wll analyze one o hese exensons proposed by Josh & Rebonao 003: The dsplaced duson, sochasc volaly exenson o he LIBOR mare model. The model has wo dsnc eaures o nduce a non-la volaly. The dsplaced duson eaure resuls n a sew n mpled volales and he sochasc volaly resuls n a smle. The paper wll analyze he derences encounered when gong rom he sandard o he exended LIBOR mare model. The analyss wll concenrae on a heorecal comparson o he sochasc derenal equaons assumed or he orward raes n he wo models, a comparson o he calbraon o he models boh n erms o procedure and o mare daa and an analyss o he derences concerned wh 3

5 prcng, boh mplemenaon ssues and derences n he prces produced by he models. As he exenson consss o wo separable eaures we wll o a large exend rea he wo eaures ndependenly, as we wll also analyse he calbraon and prcng derences or each eaure boh separable and n combnaon. We wll show ha he model s an nuve exenson o he sandard model, whch allows a nancal nerpreaon o he parameers. The model aords a consderably beer o mare daa han he sandard model, bu hs mprovemen n comes a he cos o a subsanal ncrease n calbraon and compuaon me, due o he need or repeaed samplngs o he sochasc volaly. We wll also show ha he models resul n derng prces and he analyss wll show ha he wo separable eaures o he model are oen acng n opposng drecon and ha he prcng derences wll der or changng characerscs o he producs.. Delmaon The emprcal par o he analyss s conduced based on one radng day, as an analyss across deren radng days would exen he scope o he paper consderably. The ssue o day coun convenons has no been addressed ully n he analyss. We assume he 30/360 day coun convenon or boh he producs we calbrae o and he producs we prce. We are aware ha he producs ha we use or calbraon o he models are no quoed wh hs parcular day coun convenon, bu a prelmnary analyss has showed ha he assumpon has only lle nluence, and as he scope o he analyss s a general comparson o he models, we are comorable wh he assumpon. In addon o hese delmaons some o he ssues we address, ncludng esmaon o he correlaon srucure and boosrappng caple volales rom cap volales, could each be nvesgae n greaer deph. Bu n order o manan ocus on he comparson o he models we resrc our reamen o hese necessary bu secondary ssues o he pon where we oban a worng soluon. Bu n a praccal mplemenaon o he models hese would be ssues o be addressed urher. We asses ha hs delmaon has a very small mpac on he comparson as he models are based on he same mehods. 4

6 .3 Srucure o he paper The paper s based prmarly on Josh & Rebonao 003 and ollows he mplemenaon n Rebonao 00 supplemened by Brgo & Mercuro 006 and Gaare, Bacher & Masymu 006. The paper s srucured n hree pars. The rs par esablshes he oundaon or he ollowng pars by nroducng noaon and denons ncludng he producs ha we consder eher as calbraon nsrumens or n he prcng comparson. Ths par also ncludes a bre secon nroducng he mahemacal conceps and resuls employed laer n he paper. Fnally he rs par ncludes a presenaon o he Blac ormula, as many conceps are drecly lned o hs ormula and he ormula s ulzed n he calbraon. The second par s devoed o an analyss o he models. Frs we esablsh he sandard model and he connecon o he Blac ormula. We hen urn o an analyss o he volaly, comparng he volaly surace observed n he mare o he volaly surace obanable rom he sandard model. Beore urnng o he dsplaced duson, sochasc volaly model we gve a bre overvew o some o he possble model classes whch nroduce non-la volaly. Havng esablshed he models we address he mplemenaon ssues concerned wh he models. The las par s an emprcal par conanng he resuls o he mplemenaon. Frs we wll presen he mare daa ha are used n he mplemenaon. Then we descrbe he calbraon procedures used o oban he model parameers o he yeld curve, he correlaon and he volaly. Nex we address he ssue o convergence o he model, and nally we analyze he prcng derences beween he models. We end he paper wh a concluson, an apprasal o he exenson and suggesons or urher research. 5

7 Noaon and denons To aclae he analyss he ollowng secon wll conan cenral noaon and denons. We wll also nroduce producs whch are undamenal o he analyss. Frs we consder some conceps lned o he erm srucure o neres raes. The erm srucure o neres raes s a represenaon o neres raes as a uncon o me o maury. The enor srucure denes he possble maury mes as T, T,, T N and he enor s dened as τ T T. The dscoun raes, meanng he prce a me o a zero-coupon bond payng un o currency a maury me, T, s denoed by maury T s denoed by R,T. The dscoun rae and he spo neres rae are relaed by P,T. The spo neres rae a me or P, T R, T m T m where m s he compoundng requency o he spo rae For sem-annual compoundng he compoundng requency would be. We also consder a se o spannng orward raes. Gven a se o zero-coupon bonds, we dene orward raes as gven by, T, τ P, T P, T τ τ Spannng orward raes wll reer o a se o consecuve non-overlappng orward raes 6

8 I s he neres rae ha we can conrac or a me or a loan beween uure me T and T. T and T are called he rese and he maury o he h orward rae. The conceps o rese and maury can be llusraed by seeng ha 3 T, T, τ R T, T τ So he orward rae or a gven perod can change as me passes, bu a he rese me, s xed reses and equal o he spo rae or he same me perod. The neres s due or paymen a he end o he perod,.e. a he maury me o he orward rae. As we wor wh he 30/360 day coun convenon he enor wll be consan a 0,5 year, and we wll oen suppress he dependence on enor and wre,t or equvalenly bu lgher. The orward raes wll be expressed wh a compoundng requency equal o he enor,.e. semannual compoundng. The mng convenon and he relaons beween he orward raes, dscoun raes and spo raes can be llusraed by he ollowng gure. Fgure : Illusraon o mng convenons or orward raes, dscoun raes and spo raes P,T R,T T T P,T R,T P,T 3 R,T 3 T 3 The LIBOR mare model models he dynamcs o he se o orward raes,.e. descrbes, T, τ as me elapses, eepng T and τ xed. So wha we model or each orward rae s he neres rae or a gven uure perod o me. As we have seen hese orward raes deermne he shor spo rae a her rese, and we have hereby modelled he uure shor spo raes. Bu a any me we could also exrac he curren 7

9 values o he orward raes ha has no ye rese, and rom hem calculae he whole yeld curve by nverng Equaon. Consder or example a produc whose payo s deermned n one years me as some uncon o he our year spo rae. We would hen model he orward raes or he nex ve years. These raes are evolved one year o he me where he payo s deermned. A hs me we can calculae he our year spo rae rom he orward rae ha has jus rese he one perod spo rae and he ones ha has no ye rese. Ths shows ha he LIBOR mare model can be used or prcng o a varey o neres rae producs. A a laer me we wll also need o be able o calculae he swap rae. The equlbrum swap rae or an neres rae swap maurng a n, can be shown o be gven by: 4 R s P0, n τ n P0,. The neres rae cap The LIBOR mare model s calbraed o a se o neres rae caps. An neres rae cap s a collecon o opons, caples, each payng some uure neres rae s above a ceran level, mang a suable nsrumen as nsurance agans ncreasng neres raes or a borrower payng loang neres rae on hs deb. The properes o he caple can be llusraed by an example: A borrower pays a loang rae on hs deb as he neres rae or he perod T -τ o T s deermned a me T -τ as τ -spo EURIBOR wh compoundng equal o τ. The neres rae paymen mus be pad a T. Hs neres paymen a me T assumng no amorzaon o he deb s hereore L τ R T τ, τ 5 where L s he prncpal o he deb 8

10 He s hereore concerned abou ncreasng neres raes as hs would lead o an ncrease n hs uure neres paymen. He could hereore buy an neres caple or he perod, T -τ o T. The caple wll reurn a payo he spo EURIBOR a me T -τ s greaer han he cap rae/sre, R K. The payo wll be pad a me T. The payo a me T rom he caple s gven by: NP τ max R T ì τ, τ R K,0 6 where NP s R T R K he noonal prncpal conraced or τ, τ s he spo τ - perod EURIBOR one perod beore maury s he cap rae Boh raes expressed wh a compoundng requency equal o he enor The holder o he caple hus receves an neres paymen on he noonal prncpal equal o he posve derence beween he cap rae and he spo rae. The cap can hereore be seen as a call opon on he orward rae reseng a T -τ, and pus an upper bound on he ne neres rae pad. In algnmen wh he mng sandard or he orward raes llusraed above, he sze o he payo s deermned as he orward rae reses a T -τ, whle he amoun s receved one perod laer a he me o maury, T. The caple only provdes nsurance or hs parcular neres paymen. Insead he borrower could purchase a cap, whch s a collecon o caples coverng he perod rom now o maury. The prce a me o a cap maurng a me T n s gven by 7 C, T c, T c, T s he value o n where n he caple maurng a T We see ha he rs caple maures a T and he las a he maury o he cap. There s no caple maurng a T as he neres rae underlyng hs cap has already rese. 9

11 A 0 year sem-annual cap s hereby made up by 9 consecuve caples, he rs payng n years me based on he orward rae reseng n hal a year, he las payng n 0 years me based on he orward rae reseng n 9,5 year. Each caple s sad o be n-he-money ITM, a-he-money ATM or ou-o-hemoney OTM he underlyng orward rae s currenly hgher, equal o or lower han he cap rae respecvely. A cap s dened as ATM or a cap rae equal o he swap rae or a swap ha has pay os a he same me as he cap.. Exoc caps The prcng derences beween he sandard and he exended LIBOR mare model wll be analysed n he conex o wo exoc caps: The Rache and he Scy cap. 3 Boh producs are pah dependen as he payos o he underlyng caples are no only a uncon o he orward rae maurng a he maury me o he caple, bu also a uncon o he orward raes ha has already rese. For boh producs he payo uncon n Equaon 6 s bascally unalered, bu he sre o he underlyng caples are made pah dependen. Denng he vecor o sres as gven by: R K, wh elemens R K beng he sre o he h caple, he sres are 8 R K g The sres are a uncon o all orward raes. The reason ha pah dependen caps are a neresng producs n a comparson o he sandard and he exended model s ha hey depend on he correlaon beween orward raes. As wll be shown laer correlaon beween orward raes wll depend on he volaly o he orward raes, and s hereore expeced ha gong rom he I.e. a orward swap wh he same me o maury and enor as he cap. The me o he sar o he swap equal o he enor. 3 Hull 003 0

12 sandard o he exended model wll aec he prce and hedgng parameers o Rache caps. For he Rache cap he sre s gven by RK T c 9 where c s a consan margn For each caple he sre s gven by he orward rae as has rese one perod beore he orward rae underlyng he caple plus a consan spread. For a borrower payng loang neres rae he Rache cap wll resul n he suaon, ha he borrower wll always now hs maxmum neres paymen a he nex dae o paymen, bu all subsequen paymens wll be unnown. For he Scy cap he sre s gven by RK max T, RK c 0 where c s a consan margn For each caple he sre s gven by he mnmum o he orward rae as has rese one perod beore and he cap rae one perod beore,.e. he capped neres rae or he perod beore, plus a consan spread. The scy cap wll hereore resul n a suaon where he maxmum paymen or he nex perod wll be he paymen o he curren perod plus a spread. The produc characerscs can be llusraed by he ollowng graphs showng he evoluon o he cap rae n he wo producs or he same orward rae evoluon.

13 Fgure : Realzed cap raes or Rache and Scy cap or a gven orward rae pah Rache Cap Scy Cap 0% 0% 8% 8% 6% 6% 4% 4% % % 0% Tme 0% Tme Forward rae Caprae Forward rae Caprae The graphs show ha he cap raes evolve dencally when he neres rae s allng or rsng slghly, bu when he neres rae rses more han slghly he ncrease n he cap rae rom perod o perod s resrced or he Scy, as he cap rae can only go up by he margn. For he Rache he cap rae s always se a he value o he prevous orward rae plus he margn. Thereby he Rache allows larger jumps n he cap rae han he Scy. From hs s evden ha he Scy cap mus be more valuable han he Rache or he same margn.

14 3 Change o numerare When developng he Lbor mare model he procedure o changng numerare/changng measure s employed. A bre explanaon o he procedure and he conceps o numerare and measure s hereore warraned a hs sage. The goal o hs secon s o explan he conceps n an nuve maer, whch exacly enables he undersandng o he subsequen secons. In obanng hs goal no aemps wll be made o presen he ull and rgorous mplcaons o he conceps. For a ull presenaon o he conceps see e.g. Nec Relave prces The procedure o changng numerare s closely lned o he concep o relave prces. Gven he prce, P, o asse and he prce, N, o anoher raded asse he relave prce o asse wh respec o he oher asse s dened as Z P N The relave prce s hereby he prce o an asse expressed as uns o he oher asse, he numerare. Any asse wh a srcly posve prce n any sae o he world can be used as numerare. 4 I s apparen ha he modellng o Z s equvalen o modellng P. Bu as wll be shown laer, by choosng careully he asse used as numerare one wll oban nce modellng properes. 3. Marngales The nex concep o be nroduced s a marngale. Gven a probably space Ω Φ,,Q 5 he sochasc process X s a marngale under Q sases Q E X Φ X, T > T 4 Rebonao For a descrpon o he probably space concep see Nec 000 3

15 The me expecaon under he probably measure Q o he value o he process a some laer me T s equal o he value o he process a me,.e. he process s drless. Q To smply noaon we wll denoe he me expecaon by X E n he ollowng. 3.3 Changng numerare The probably measure assgns probables o he possble saes o he world. In he real world he probably measure s gven by Q, and a process ha s a marngale under Q would be a marngale n he physcal world. Bu we could also consder probably measures deren rom he physcal one. These measures would descrbe probables n hese worlds. For nsance n he well nown rs ree world he probables are gven by he rs ree measure Q 0. T In he absence o arbrage s possble o nd or each chosen numerare an equvalen probably measure, under whch relave prces are marngales; hs measure s called he equvalen marngale measure. Changng numerare/changng measure s he procedure o gong rom one numerare o anoher and changng he measure so ha he relave prces are marngales. The mahemacal represenaon o hs reveals a powerul resul. Denng Q N as he equvalen marngale assocaed wh he numerare N any asse prcng problem can be saed as he ollowng wh V denong he me value o he asse. 3 V E N QN V N E V T, N T QN V T N T T > I shows ha he prce o any asse can be expressed as he expeced relave prce o he asse under he equvalen marngale measure assocaed wh he numerare mes he curren prce o he numerare. 4

16 In opon prcng we now he opon value a maury gven he sae o he world hrough he payo speccaon. As we oen ae he curren value o he numerare as gven he opon prcng problem hereore becomes a problem o evaluang he expecaon under he equvalen marngale measure o he payo relave o he uure value o he numerare. The numerare should hereore be chosen n order o smply he evaluaon o he expecaon as much as possble. An example o he concep can be cas n he ramewor o he Blac-Scholes model. Blac & Scholes 973 dscovered ha he prcng o a European call opon could be grealy smpled one chooses as numerare he rolled up money mare accoun. The rolled up money mare accoun s he asse correspondng o nvesng un o currency a me 0 a he prevalng nsananeous spo rae r and rollng he nvesmen over, always a he nsananeous spo rae. The value o he accoun a me s gven by 4 B exp r s ds 0 Under hs numerare he equvalen marngale measure s he rs ree measure, Q 0, under whch all asses grow a he rs ree rae. Inserng hs n Equaon 3 gves us he ollowng expresson or he value o he call opon a me 0, C0 5 C0 B0 E C0 E Q0 0 C0 exp Q0 0 0 r C T B T C T exp r s ds 0 0 s ds Q E C T 0 The second equaly s obaned usng B0 and he las s obaned by he ac ha n he Blac-Scholes world neres raes are deermnsc and can hereore be aen ousde he expecaon. The choce o numerare smples he evaluaon o he 5

17 expecaon as he dynamcs o he underlyng under he rs ree measure resuls n a closed orm soluon o he expecaon. Inserng he closed orm soluon n Equaon 5 provdes us wh he Blac-Scholes ormula. The rolled up money mare accoun s hereby a very convenen numerare gven he assumpons o Blac and Scholes as yelds a closed orm soluon or he call opon. The use o he rolled up money mare accoun s especally useul n he Blac- Scholes world as hey assume deermnsc neres raes. When valung neres rae producs we do no assume deermnsc neres raes, and he aracveness o he rolled up money mare accoun s reduced. Insead we consder he ermnal measure, 6 whch s he marngale measure correspondng o choosng as numerare he zero coupon bond maurng a he same me as he asse under consderaon. Snce he prce o he numerare s a maury and denong he ermnal measure correspondng o he zero coupon bond maurng a T n by Q n hs leads o he ollowng represenaon o he general prcng ormula n Equaon 3 Qn 6 V P, T E V T n n The ermnal measure can also be dened n erms o a orward rae as he marngale measure usng as numerare he zero coupon bond maurng a he maury o he orward rae. Ths hghlghs ha opposed o he rs ree measure he ermnal measure depends on he maury o he orward rae. Somemes we wll use he saemen under he ermnal measure or a collecon o orward raes. By hs we mean under he ermnal measures o each orward rae. Under he ermnal measure all relave prces are marngales. Bu as he orward raes are no raded asses, we canno conclude ha hey are marngales. However can be shown 7 hough ha under he ermnal measure each orward rae s a marngale and ha a orward rae wll no be a marngale under he ermnal measure 6 Also called he orward measure 7 Rebonao 00 6

18 o anoher orward rae; or a gven ermnal measure one and only one orward rae wll be a marngale. A cenral resul can be obaned by combnng he above wh he resul ha any duson wh deermnsc dr and volaly leads o a log normal dsrbuon. 8 Under he orward measure he orward raes have zero drs and hereore he orward raes wll be log normally dsrbued under he orward measure as long as he volaly s deermnsc. 9 Addonally can be proved ha under a specc measure only one orward rae can be log normally dsrbued. We hereore have ha under a ceran ermnal measure, only he orward rae ha pays o a he maury o he numerare wll be log normally dsrbued, any oher orward rae wll no. 3.4 The Grsanov heorem Gven ha we can choose deren numerares and hereby change he measure we need o examne how he dynamcs o a process change when we go rom one measure o anoher. The Grsanov heorem saes ha W s a P-brownan moon,.e. a Brownan moon under P, and Q s a probably measure equvalen o P 0, hen here exss a process γ such ha ~ 7 W W γ sds 0 s a Q-brownan moon. Tha s, under Q he P-brownan moon plus a dr s a Brownan moon. We see ha he consequence o changng measure s ha he dr o he process changes; under P he process W % has a dr, under Q does no. Obvously he change o measure can also conver a non-drng process under P o a drng process under Q. An mporan mplcaon o Grsanov s heorem s ha he change o measure does no change he volaly o a process. The process 8 Rebonao 00 9 And he orward raes ollow a duson 0 I assgns zero probables o he same saes o he world 7

19 W % has he same volaly under boh P and Q. The same carres over o he correlaon beween wo processes,.e. when changng measure he processes under consderaon are ransormed n a way ha changes he drs o he processes whle he volaly and correlaons reman he same. 8

20 4 The Blac ormula The opon prcng ormula developed by Blac and Scholes n her semnal arcle Blac & Scholes 973 was orgnally developed or he prcng o opons under he assumpon o consan neres rae and volaly. Bu he use o he ormula was soon expanded o nclude non-consan bu sll deermnsc neres raes and volales and opons on oher deren asses. One o he developmens was acheved by Blac hmsel Blac 976 by he dervaon o a ormula or prcng opons wren on commodes. To crcumven he predcably o he spo prce o commodes Blac modeled he opons as wren on he uure prce o he commody raher han he spo prce as s he case or he Blac-Scholes ormula. As neres raes are also predcable, a verson o he Blac ormula was soon ormulaed o prce neres rae caples. The man assumpon underlyng he dervaon o he ormula s ha orward raes are log normally dsrbued wh a varance o T T a maury, T. Blac The Blac ormula or he prce o a caple wh enor, τ, and maurng a Tτ wres: c Blac P, T τ [, T, T τ N d R N d ] K NPτ 8 d d ln where [, T, T τ R ] T Nx s he sandard cumulave normal dsrbuon R s he cap rae K d Blac Blac Blac T s he Blac volaly K T Blac All he parameers and varables o he ormula bu he volaly are drecly observable n he mare. The volaly, Blac, s no drecly observable, and he mare pracce s o quoe he prces o caples as he mpled volaly o he opon,.e. he number ha pu no he ormula as Blac wll gve he mare prce o he Due o e.g. harves perods n he case o agrculural commodes 9

21 caple. As he mare quoes prces or a range o maures and cap raes we can oban a marx o mpled volales, and he mpled volales are dened by Blac T, RK, j where T denes he rese me o he underlyng orward rae and R K, j denes he cap rae. When ploed n cap rae/rese me space we oban he mpled volaly surace. I we consder only one cap rae he mpled volaly as a uncon o maury descrbes he erm srucure o volaly. As we have saed beore, a se o orward raes canno be smulaneously log-normal, would hereore seem ha he Blac ormula s nernally nconssen used or prcng caples a deren maures, as assumes log normaly o he orward raes. Bu we add he qualer ha each orward rae should be log-normally dsrbued under s own ermnal measure, and he prcng s done n each o hese measures, he assumpon o log-normaly wll be recovered. And as he prcng s ndependen o he measure, we wll recover he same prce. However powerul he Blac ormula s, has wo major drawbacs. Frs o all does no provde us wh he dynamcs o he orward raes. The ormula only assumes ha each orward rae s log normally dsrbued under s own ermnal measure. Ths assumpon s sucen or he prcng o caples as hey only depend on one orward rae. Ths means ha he payo depends only on he dsrbuon o he orward rae a maury, he possble pahs leadng o hs dsrbuon s rrelevan, and also allows us o consder each orward rae ndvdually under s own ermnal measure. The value o exoc producs could depend on mulple orward raes a pons n me deren rom he maury. Ths means ha we need a seup n whch we can oban he jon dsrbuons o mulple orward raes under one common measure and we need margnal dsrbuons no jus he ermnal ones. In order o oban hs we need he dynamcs o he orward raes. A varey o dynamcs would resul n log normal orward raes and hereby rean he valdy o he Blac ormula. We now rom Secon 3.3 ha among ohers any duson wh deermnsc dr and volaly wll produce a log normal dsrbuon. 0

22 Second under he assumpon o log normaly he Blac mpled volaly s lned drecly o he varance o he orward rae a maury. Ths means ha caples o he same maury bu wh deren sres should exhb he same Blac volaly as he same orward rae s underlyng he caples. So when we assume log normal orward raes we also assume a la volaly surace across sres. As we wll see laer hs s no he case. Caples derng n sre wll also der n Blac mpled volaly. Ths s evdence agans he assumpon o log normaly o orward raes, and dynamcs leadng o log normaly would no be conssen wh he mare.

23 5 The sandard LIBOR mare model The sandard LIBOR mare model resolves he rs decency o he Blac ormula by provdng dynamcs or a se o orward raes under one common measure. The orward raes are modelled as d μ, d dz Q 9 where d s a n vecor o percenage ncremens o orward raes μ, s a n vecor o dz Q s a n vecor o he probably measure Q dr erms, whch depend on boh correlaed brownan moons under s a n n dagonal marx wh he,' h elemen,, and beng he nsananeous volales o he ' h orward raes The correlaon beween he orward raes are gven by he correlaon marx, Γ, wh elemens: 0 ρj E dz dz j Somemes wll be more llusrave o wor wh he dynamcs o a sngle orward rae n sead o he ull marx ormulaon above. For he h orward rae he model reads: d μ, d dz The model speccaon s rom Rebonao 00

24 Inegraon leads o he ollowng expresson or he orward rae a some laer me, T. T T T exp μ, u u du [ dz u ] du Ths represenaon wll prove useul n some o he laer analyss. Bu or now suces o say ha he equaon hghlghs ha evolvng he orward raes narrows down o evaluaon o wo negrals; a dr negral and a volaly negral. By choosng hese dynamcs some modellng choces have been made. Frs o all he dynamcs o he orward raes s descrbed by a pure duson model and all uncerany s descrbed by a number o Brownan moons. Ths rules ou he possbly o jumps n he process. 3 Second by choosng o model he orward raes as geomerc Brownan moons he model nsures ha he orward raes canno ae negave values, and also he volaly o he percenage ncremens are consan regardless o he level o he orward raes. The sandard model resrcs he volaly o be deermnsc. I does no necessarly need o be consan, bu mus only be a uncon o me. As he speccaon o he volaly s a ey ssue o he model, and some addonal analyss s necessary or hs subjec a ull secon, ollowng he curren, s devoed o volaly. The man reason or he populary o he LIBOR mare model s he recovery o Blac prces. The model s capable o exacly reproducng he Blac prces o caples. The caple value depends on one orward rae only and each caple can be consdered n solaon. For each caple we can hereore change he measure o he ermnal measure. Under hs measure he orward rae wll be a marngale and hereore he dynamcs reduce o a drless duson wh deermnsc volaly. By nvong he resul rom Secon 3.3 ha any duson wh deermnsc dr and volaly resuls n a log normal dsrbuon we see ha he dynamcs o he sandard model produce log normally dsrbued orward raes. As he LIBOR mare model 3 Research explorng he possbly o jumps are beng conduced see e.g. Glasserman & Kou 003 3

25 mplcly assumes he same dsrbuon o he underlyng he LIBOR mare model wll produce he same prces as he Blac ormula. The dsrbuon under he ermnal measure o he orward raes n he LIBOR mare model n erms o he volaly s gven by T 3 ln T N 0, u du The varance o he orward rae a maury s he negraed squared nsananeous volaly o he rese o he orward rae. As he varance a maury n he Blac ormula s T T, he sandard LIBOR mare model can recover he Blac prce Blac o any caple as long as one chooses; 4 Blac T, RK, j T u du T We see ha he Sandard model resuls n log normal orward raes and hereby he model can only produce a la volaly surace across sres. 5. No arbrage drs We have seen ha he orward raes are drless under he ermnal measure. Bu when usng he sandard LIBOR mare model we wll choose one measure 4 and consder all he orward raes under hs measure. As a resul o he Grsanov heorem changng he measure rom he orward measure o one common measure or all he orward raes wll change he drs o he orward raes processes whle leavng unalered he volaly and correlaon. Ths means ha he change o measure wll no aec he volaly or correlaon and can hereore by obaned n one measure and appled drecly n anoher. Bu he drs wll depend on he measure and only he orward rae maurng a he same me as he numerare wll be drless. We wll now show he drs or he remanng orward raes. 4 The opmal choce o measure s dscussed n secon 8. 4

26 5 As saed n Secon 3.3 a necessary and sucen condon o no arbrage s ha he relave prces o all raded asse should be marngales under he equvalen marngale measure assocaed wh he chosen numerare. As orward raes are no raded asses we canno ulze hs resul drecly or he orward raes. Bu an applcaon o he resul and some sochasc calculus 5 leads o he ollowng dr erms necessary o preven arbrage beween he model orward raes when choosng as numerare he zero-coupon bond maurng a he payo me o orward rae j,.e. P,T j 5 j <j >j j j or 0 or or μ τ τ ρ μ τ τ ρ μ As expeced he orward rae whch pays a he same me as he chosen numerare, s drless,.e. a marngale. The oher orward raes has non zero drs. These drs ensure ha no arbrage opporunes exs beween orward raes under he equvalen marngale measure assocaed wh he choce o numerare. We see ha he drs can be expressed as uncons o nsananeous volales, correlaons and orward raes, whch are all quanes ha we are already usng n he duson erm. Addonally, as he expressons conan he orward raes, he drs are sae dependen. By loong a he expresson or a orward rae a some laer me usng as numerare, P,j, whch maures aer he maury o he orward rae, we see why hs consues a problem. 6 du u dz u du u u u u u u T T T j exp τ τ ρ 5 For a dervaon see Appendx : No arbrage drs

27 The sae dependency means ha he orward raes are no longer log normally dsrbued, and he samplng s made much more dcul. The problem arses as he negral semmng rom he sae dependen drs mus be evaluaed numercally. In a Mone Carlo smulaon, hs means ha a ne sepped algorhm s necessary n order o evaluae and updae he dr erm requenly enough as he orward rae changes. As urns ou deren approxmaons some very accurae exs, so ha we can approxmae he sae dependen drs by a deermnsc one, hereby recoverng he log normaly o he orward raes. Ths wll be explored n deal n Secon

28 6 Volaly Speccaon o he volaly n he Lbor mare model s a ey ssue whch gves rse o some concepual dscussons. We have hereore devoed an enre secon o hs subjec. The secon wll sar by dscussng he properes ha we would le he erm srucure o volaly rom he model o exhb. Ths dscusson leads o he selecon o a parameerzaon o he volaly. Nex he secon wll urn o a comparson beween he deren shapes o he mpled volaly suraces obanable rom he sandard Lbor Mare Model and he emprcally observed. Fnally he secon wll dscuss deren approaches o generae a non-consan volaly surace. 6. Desrable properes o he volaly speccaon As Rebonao 00 llusraes, he mos ypcal shape o he erm srucure o volaly, s a hump shape. Gong rom shor o longer maures he mpled volaly s rs ncreasng unl reaches a maxmum around -3 years aer whch decreases. Alhough here are varaons n he mpled volaly, he mpled volaly exhbs he same qualave shape. The rs mplcaon o hs s ha he volaly speccaon mus be able o generae he humped shape. Second as long as we only consder deermnsc volaly and gven ha we have no vew on he uure volaly we should consder mehomogenous volaly speccaons. The erm me-homogenous descrbes he propery ha he erm srucure o volaly s consan over me, meanng ha he volaly o a 4 year caple n years me s he same as he volaly o a year caple oday. Mahemacally: T 7 u du u du or all, and h. h T h T < T h 7

29 The nerpreaon o he or all, and h senence s ha he me homogeney.e. he same volaly or he same remanng maury apples or any par o caples and and or any lengh o remanng me o maury h. The me homogeney condon ranslaes no a requremen ha he volaly s a uncon o resdual maury T- only. Rebonao 999 proposes a now wdespread ormulaon o he nsananeous volaly uncon: [ a b T ] e c T d 8 where a,b,c and d are parameers o be deermned As s only a uncon o remanng me o maury s me-homogenous and hs speccaon s capable o producng a hump 6 shaped nsananeous erm srucure o volaly whch carres over o he erm srucure o boh mpled caple and cap volales. The speccaon hereore ulls he desred properes. The parameer d deermnes he general level o he volaly, and he remanng parameers deermne he placemen and magnude o he hump. The speccaon allows or an analycal evaluaon o he negraed squared volaly, whch wll prove very useul n he calbraon o he sandard model. 6. Model and emprcal mpled volales Havng chosen a speccaon o he nsananeous volaly ha s capable o reproducng he hump shaped volaly we can capure he shape o he erm srucure o volaly or a gven sre. Bu as or he Blac ormula he log normaly o orward raes wll mean ha he model wll produce an mpled volaly surace ha s la across sres. 6 And many oher plausble shapes 8

30 We wll now nvesgae he emprcally observed 7 volaly suraces n order o assess hs propery. Unl around 994 he volaly suraces n all currences where n agreemen wh he log normal assumpon o he Blac ormula and he sandard volaly model by beng la across sres. Around 994, begnnng n Japanese Yen, bu soon spreadng o oher currences, he volaly surace began experencng non-la volales across sres. In hs perod he volaly was monooncally decreasng, rom low o hgh sres as llusraed n Fgure 3. Fgure 3: A ypcal volaly sew 4 3,5 Impled volaly 3, Sre The monooncally decreasng volaly was observable unl he mare crss n he lae 998. Durng and aer hs crss he shape o he non-la volaly changed no a shape smlar o he one llusraed n Fgure 4. 7 Josh & Rebonao 003 9

31 Fgure 4: A ypcal volaly smle 4 3,5 Impled volaly 3, Sre The volaly was no longer monooncally decreasng; was smlng, as had been observed earler n he equy mares. 8 I sll had a general decreasng shape rom low o hgh sres, bu or he opons wh he hghes sre, he volaly was ncreasng. Some auhors use he erm hocey sc shaped or hs volaly shape, bu we use he erm smlng or a non-la volaly shape whch s no monooncally decreasng. The separaon n me o he appearance o he sew and he smle n mpled volales sugges ha wo deren sources are causng he non-la volaly. Havng esablshed he hsory o he mpled volaly surace o neres raes, we now urn o he curren surace. 8 Appearng aer

32 Fgure 5: Impled EUR cap volaly surace Sepember ,5 6 Sre Tme o Maury We see ha he curren cap surace 9 dsplays boh a sew and a smle. The smle s mos pronounced or he shores maures and s more shallow or he longer maures Ths leads us o rejec he log normaly o he sandard LIBOR mare model, as prescrbes a consan volaly across sres. Ths does no mply ha we hereby dscard he sandard model all ogeher. The sandard model s nuve and easy o boh calbrae and mplemen and he usably o he model mus be based on a concree evaluaon o he prcng and hedgng ables o he model or each produc under consderaon. Bu he underlyng assumpon s no ullled. The sew and he smle s evdence o a dsrbuon ha s sewed and exhbs excess uross. We wll now gve a bre overvew o deren approaches o oban a non consan volaly across sres. 6.3 Deren approaches o oban a non consan volaly Several deren mehods have been proposed o accoun or he observed devaons rom la volaly. As he smlng volaly suraces were rs experenced n equy 9 Obaned by nerpolang he quoed cap marx 3

33 mares, mos o hese mehods have been developed n order o mach hese smles, and subsequenly been adoped or neres raes. The mehods nclude among ohers: - Consan elascy o varance models - Dsplaced duson models - Jump duson models - Regme swchng models - Sochasc volaly models 6.3. Consan elascy o varance models The consan elascy o varance CEV models ae he approach ha he volaly o absolue changes n he orward raes s no proporonal o he level o he orward rae, as Equaon 9 suggess. Sll, hgher levels o he orward raes lead o a larger volaly, bu he scalng s no proporonal. Omng he dr erm he CEV model 0 wres: β 9 d dz For a geomerc Brownan moon he volaly o he ncremens scale perecly wh he level o he orward rae resulng n a consan volaly o percenage ncremens, whch n urn leads o a log normal dsrbuon. In he CEV model he parameer β allows a deren scalng o he volaly. For 0<β< he volaly o he ncremens wll scale less han proporonally wh he level o he orward rae. Ths means ha he volaly o percenage ncremens wll be larger or lower levels o he orward rae han or hgher. Ths leads o a sewed dsrbuon o he logarhm o he orward rae ha n urn leads o monooncally decreasng mpled volaly as a uncon o sre. We see ha condonal on he evoluon o he orward rae, he volaly s nown wh cerany, no addonal sochasc varables has been nroduced. The sae o he world a me s deermned unquely by he evoluon o he orward rae up o me. 0 Andersen & Andreasen 000 3

34 6.3. Dsplaced duson models Dsplaced duson models are characersed by nsead o modellng he orward rae as a duson models he orward rae plus some consan,α he dsplacemen coecen. Agan omng he dr erm he dsplaced duson reads d α 30 dz α We see ha he volaly s uncondonally deermnsc as n a geomerc Brownan moon. Bu he volaly s he volaly o he percenage ncremens o he quany α, no he orward rae sel. For a posve duson coecen he dsplaced duson wll produce a monooncally decreasng mpled volaly across sres by alerng he scalng o nsananeous volaly wh he orward rae, jus as he CEV model, and can be shown ha he CEV and he dsplaced duson model wll resul n almos dencal volaly sews. The CEV model s preerred over he dsplaced duson heorecally as ensures posve raes. The dsplaced duson ensures ha orward rae s hereore only bounded o be larger han α. α s posve, and he Bu he dsplaced duson s preerred over CEV as he CEV model does no provde racable analycal soluons or caple prces. In he dsplaced duson model he quany α s log normally dsrbued. Ths means ha all he resuls lned o he log normaly can be reaned. Ths mples.a. ha caple prces can be calculaed rom a smple exenson o he Blac ormula. 3 Rebonao 00 shows ha or reasonable values o α he probably o experencng negave raes are raher small, and he resulng derence beween a CEV and dsplaced duson model s neglgble. Dsplaced duson s hereore preerred or s racably alhough CEV s nancally more appealng. Rubnsen 983 Marrs See Appendx 5: Dsplaced duson exenson o he Blac ormula 33

35 6.3.3 Jump duson models Jump duson models nroduce an addonal source o pure uncerany as nroduces a new sochasc varable o he process. A smple example 4 could be: N d, T dz d Y 3 where N Y s a Posson process s a log normal random varable The nroducon o he jump resuls n a non log-normal dsrbuon o he orward raes, whch produces dependen on he specc jump-duson model chosen a nonla volaly across sres Regme swchng models Regme swchng models wor wh wo or more deren processes or he orward rae. Dependen on he sae o he world he evoluon o he orward rae wll be governed by one o he possble processes. A smple example o a regme swchng model could be 5 3 d d S S P S S p P S S p, T dz, T dz or or S S S S The model changes, accordng o he probably p, beween wo regmes wh deren volales. Regme swchng models can also produce non-la volales. 4 Glasserman & Kou The auhors own ormulaon 34

36 6.3.5 Sochasc volaly models Sochasc volaly models devaes rom he sandard LIBOR mare model by mang he volaly no only a uncon o me bu also a uncon o addonal sae varables. I s oen, as n Heson 993, assumed ha he squared volaly ollows a mean reverng duson: 33 d, T dz d κ, θ ν dz, The sochasc volaly models are capable o producng smlng volales as hey nroduce excess uross n he dsrbuon o he underlyng. Inuvely he orgn o he smle can be analysed n he conex o a resul by Hull & Whe 987: When he Brownan moons drvng he volaly and he Brownan moons drvng he orward raes are uncorrelaed he prce o a caple under sochasc volaly s gven by he Blac prce negraed over he volaly dsrbuon: 34 Blac, P P T φ d Blac Blac Blac The caple prce under sochasc volaly wll hence be an average o Blac prces over he possble values o he Blac volaly. ATM he Blac ormula s approxmaely lnear n Blac. A hgh volaly sae wll hereore ose he correspondng low volaly sae. Thereore he prce accordng o Equaon 34 wll no be very deren rom he prce one would oban usng he sandard Blac ormula usng he average volaly, leadng o an mpled volaly o he ATM caple approxmaely equal o he average volaly. Bu as one moves eher n or ou o he money he Blac ormula becomes convex n Blac. Ths means ha he average prce across volales wll be hgher han he prce one would oban usng he average volaly, leadng o a hgher mpled volaly han ATM. Because o he 35

37 ncreasng convexy when movng rom ATM o ITM or OTM, sochasc volaly resuls n a smlng mpled volaly. 36

38 7 The exended LIBOR mare model In Secon 6. we saw ha he smle n he volaly emerged years aer he monooncally decreasng slope. We argued ha hs could be aen as evdence ha we should model he non-la volaly by wo dsnc eaures, one gvng rse o a monooncally decreasng volaly and anoher gvng rse o a smle. Josh & Rebonao 003 ae hs approach. Ther model s a combnaon o a dsplaced duson and a sochasc volaly model. d α α μ α, d α dz 35 d d α [ a b T ] c T RSa RLa a d adza RSb RLb b d bdzb [ ln c ] RSc RLc ln c d cdzc [ ln d ] RSd RLd ln d d ddzd da db e d 36 E E E [ dzdza ] E[ dzdzb ] E[ dzdzc ] E[ dzdzd ] [ dzadzb ] E[ dzadzc ] E[ dzadzd ] [ dz dz ] E[ dz dz ] E[ dz dz ] 0 b c b d c d 0 E dz dz j ρ, 37 [ ] j Superscrps n parenhess are labels The dsplaced duson s nroduced n he orm presened above, modellng he quany α nsead o as a duson. For he sochasc volaly par o he model he roue aen s no mang he volaly sochasc by addng a random erm o he volaly speccaon sel. Insead s assumng he same unconal orm or he nsananeous volaly as n he sandard model bu he coecens or her logarhms ollow Ornsen- Uhlenbec processes, wh reverson level, RL, reverson speed, RS, and volaly. 37

39 As descrbed n secon 6. he coecens o he volaly uncon deermne he general level o he volaly and he locaon and magnude o he hump. Thereore he sochasc volaly par o he model can be nerpreed as allowng he level and locaon and magnude o he hump o change sochascally, whle preservng he unconal orm or he volaly. We noce ha he exended model encompasses he sandard model. The sandard model can be consdered a specal case wh all reverson levels equal o he curren level o he coecens, all volales n he Ornsen Uhlenbec processes se o 0 and α 0. Two oher neresng specal cases exs: The Dsplaced Duson model wh no sochasc volaly and he Sochasc Volaly model wh no dsplaced duson. 6 When negrang Equaon 35 we ge an expresson or he orward rae a some uure pon n me very smlar o he one presened n Equaon or he sandard model. The Ornsen Uhlenbec process s a mean reverng process. For each ncremen he dr erm wll pull he value o he coecen owards he reverson level, he srengh o he pull deermned by he reverson speed. Besdes hs deermnsc pull a sochasc erm wll aec he coecen, by a Brownan moon scaled by he volaly o he process. The mean reverng behavour o he process ensures ha he coecens do no devae oo much rom he reverson level. The concep o me homogeney s hereby mananed n a slghly deren meanng, as we aan ha he sochasc volaly lucuaes around he same reverson level n he uure as does oday. The mean reverson causes he average volaly o be more varable or shorer maures han longer. As a lower varably n he volaly wll resul n a more shallow smle, he mean reverson wll resul n he smle beng more shallow or longer maures han shorer. 6 We wll reurn o hese specal cases laer 38

40 We hereby see ha ncreasng he volales o he coecens ncreases he smle, whle ncreasng he reverson speeds maes he smle more shallow or longer maures han shor. The Ornsen Uhlenbec process allows an exac closed orm soluon 7 or he value o he coecen a uure pons n me. Ths wll prove very useul n he calbraon o he model and n he prcng process. The condons ha he Brownan moons should be par wse uncorrelaed reduces he compuaonal burden, as each coecen can be evolved ndependenly o he oher. Addonally as Brownan moons drvng he orward raes are uncorrelaed wh he Brownan moons drvng he volaly he calbraon o he model s smpled by he Hull and Whe resul presened n Secon No arbrage drs As he dervaon o he no arbrage drs holds or deermnsc as well as sochasc volaly he no arbrage drs are bascally unchanged. Tang no accoun he dsplaced duson par o he model he no arbrage drs, when usng P,T j as numerare, becomes: 38 μ μ μ α α α 0 α α j j α ρ α ρ τ α τ α τ τ or >j or <j or j We jus remember ha he volaly s sochasc no deermnsc, when negrang he drs. 7 See Appendx : Evolvng an Ornsen Uhlenbec process 39

41 7. Comparson o he models We see ha n erms o he sochasc derenal equaons descrbng he dynamcs o he orward raes, he dsplaced duson, sochasc volaly model s an nuve exenson o he sandard model. The model adds a sew by modellng α nsead o n he duson, hereby nducng he same eec as he CEV model, ha he volaly o he percenage ncremens are no consan as n he sandard model bu decreasng or an ncreasng orward rae. The sochasc volaly s nroduced by mananng he paramerc orm o he volaly, bu leng he coecens o he uncon be sochasc. Ths allows an nuve nerpreaon o he sochascy as varaons n he general level and he placemen and magnude o he hump o he volaly. We see ha gong rom he sandard o he exended model does no aler sgncanly he no arbrage drs. Alhough he derences n gong rom he sandard o he exended model a hs me seem small, we wll now address he ssue o mplemenaon, whch wll reveal some larger derences. 40

42 8 Implemenaon ssues The reamen o he mplemenaon ssues wll answer he queson: Gven he dynamcs o he models jus presened, how do we acually oban prces rom he models? An mplemenaon o he models consss o wo pars: Algorhms o calbrae he model,.e. deermnng he parameers o he model, and a prcng model. Ths secon wll nroduce conceps whch are cenral o boh pars, bu he presenaon wll be made n he conex o he prcng model, whle he calbraon procedure s dscussed n deph n Secon 0. Recallng he prcng ormula n Equaon 3 we see ha he prcng problem amouns o evaluang an expeced value under he measure nduced by he choce o numerare. The prcng procedure hereby comprses o he ollowng seps: - Choosng he measure/numerare - Sample he dsrbuon o he orward raes under he equvalen marngale measure - Calculang he payo, ang expeced value and ransorm rom relave prces o C 8. Choosng he measure/numerare We choose o prce under he ermnal measure o he las caple n he produc. Tha s we ae as numerare he zero coupon bond maurng a he maury o he produc. Ths means ha he prcng ormula consdered are Equaon 6. The prce a me, C, o a cap maurng a me T n can hereby be wren as Qn 39 C, T P, T E C T, T F n n n n We mus calculae he expeced value o he payos a maury. The expeced value mus be calculaed under he ermnal measure. And rom he expeced value we 4

43 oban he curren value o he opon by mulplyng by he curren value o he numerare, he zero coupon bond maurng a he same me as he cap. 8. Samplng he orward raes dsrbuon For he exoc producs we can no longer calculae he expecaon analycally as s possble or sandard caps, we mus resor o numercal mehods. Because o he dmenson o he problem he mos ecen mehod s Mone Carlo smulaon. The brue orce approach o evolve he orward raes would be o Euler dscreze he orward rae and volaly equaons and perorm a shor sepped Mone Carlo smulaon o he volaly and he orward raes smulaneously. Ths would be very compuaonally necen. So we urn o a smarer approach whch wll reduce he compuaonal burden consderably. The wo producs we are neresed n prcng, he Rache and he Scy, are characersed by he ac ha he payos are deermned unquely by he value o he orward raes a her own reses. We can hereore mplemen he procedure Rebonao 00 calls he very long jump approach. The very long jump approach evolves all orward raes o her respecve reses n one jump. Ths aords a consderable reducon n he compuaon me compared o he brue orce approach. No all producs exhb hs propery ha he payos can be deermned solely rom he values o he orward raes a her own reses. For nsance we consdered such an opon n Secon. Tha opons payo was deermned a maury based on he our year spo rae. In order o deermne hs payo we would need he value o he orward raes mang up he our year spo rae a he maury o he opon,.e. beore her reses. A model usng he very long jump approach would no be able o prce hs opon. Forunaely oher ecen mplemenaon echnques exs or hese producs, whch avod he brue orce approach. In shor hese echnques consss o evolvng all orward raes beween prce sensve evens 8 usng acor reducon,.e. 8 Prce sensve evens are pons n me where some elemen concernng he prce s deermned, e.g. he sre, a payo ec. 4