Heat kernel methods in finance: the SABR model

Transcription

1 He erel eods ce: e SABR odel Crelo Vccro Te wor reseed s reor s bee crred ou w e suor o Reuers Fcl Sowre Pueu Frce d uder e dreco o Adre Bourere o wo e uor s rculrl reul For rers coes or roosos o collboro lese wre e crelovccro@srr

2 Absrc Te SABR odel s socsc voll odel o d closed or soluo H Kur Leews d Woodwrd [HKuLeWo] ve obed roe soluo b es o erurbve ecques A ore recse roo ws oud b Her-Lbordère [H-L 5] d [H-L 8] w e e erel eso eod Te ler reles o dee d rd eores ro Re eoer wc re los oll uow o e roessols o ce wo owever re ose rrl eresed ese resuls Te ol o s reor s o ll s d o e ese ocs udersdble w bsc owlede o clculus d ler lebr

3 Tble o coes Cer : Preseo o e reor 5 THE LINK WITH RIEMANNIAN GEOMETRY AND HEAT KERNEL EPANSIONS 7 CONTRIBUTIONS OF THIS REPORT 9 Cer : Iroduco o Re eoer REMINDERS FROM CALCULUS PROPERTIES OF DIRECTIONAL DERIVATIVES 3 CONNECTIONS 4 4 THE COVARIANT DERIVATIVE 7 5 PARALLEL TRANSPORT 9 6 GEODESICS 7 RIEMANNIAN STRUCTURES 8 CURVATURE 4 9 CURVES 6 HEAT KERNEL ON A RIEMANNIAN STRUCTURE 9 ELLIPTIC OPERATORS ON A RIEMANNIAN STRUCTURE 3 CHANGE OF COORDINATES IN A RIEMANNIAN STRUCTURE 33 3 CHAIN RULE FOR PARTIAL DERIVATIVES 34 4 CHANGE OF COORDINATES IN THE KOLMOGOROFF BACKWARD EQUATION 35 Cer 3: Te Pocré l-le 38 3 POINCARÉ HALF-SPACE 38 3 GEODESICS OF THE POINCARÉ HALF-PLANE: VERTICAL LINES GEODESIC DISTANCE FOR VERTICAL LINES 4 34 GEODESICS OF THE POINCARÉ HALF-PLANE: SEMICIRCLES 4 35 GEODESIC DISTANCE FOR SEMICIRCLES 43 3

4 36 PROOF OF THE UNIQUENESS OF THE GEODESICS GEODESIC CONNECTING TWO GIVEN POINTS 46 Cer 4: Te SABR odel 49 4 RIEMANNIAN METRIC FOR THE SABR 49 4 HEAT KERNEL EPANSION FOR THE SABR 5 Reereces 58 4

5 Preseo o e reor Te Blc-Scoles-Mero BSM odel roduced [Blc-Scoles] d [Mero] s oe o e os used cl odels or rc oos However s bee e re o crcs soe eve cosder o be e cuse o e Ocober 987 re crs [Boucud] Te crc o e BSM odel s s er-sled d urelscll sle odel: deed e BSM odel ssues e voll drv sse s cos d e rce o e ler ollows lo-orl rocess cs re cordced b rel re d Aoer crc s e BSM led voll wc s obed o e ssuo e voll be cos s deed o-cos uco o sre d e o ur Ts eeders cordco es e use o BSM led volles coceull lwed Moreover sce led voll ves rse o dere dc o e sse we us e BSM odel oe s ssu e se uderl sse ollows dere dcs oe or ec vll oo o d s c s ver rd o us Fll sce s e oo o e uderl sse deeres e led voll o e ler e BSM rewor s e e oo ecs e uderl wle rel s ecl e oose s rue Te BSM odel s lso crczed o rccl rouds Suose or sce we re us e BSM odel or rc brrer oo sce e ler s wo sres oe or e oo sel d oe or e brrer we woder wc oe o e wo BSM led volles us be used or us be used cobo o e wo A odel crcuves e ler cosseces s e locl voll odel roduced 994 [Dure] d [Der-K] Ts odel ssues e voll o e uderl sse s o loer cos bu deeds o e d o e vlue o e uderl Te locl voll odel reses owever cossec w rel re d: s observed [HKuLeWo] redcs dc bevor o sles d sews wc s ecl oose o 5

6 observed e re Ideed we e rce o e uderl decreses e locl voll odel redcs e sle ss o er rces we e rce creses e odel redcs e sle ss o lower rces I rel sse rces d re sles ove e se dreco Ts cordco bewee e odel d e relce eds o desblze e Del d Ve edes wc oe eror worse e BSM edes A clss o odels vods ll ese robles s o socsc voll odels s odels o duso wc e voll s ssued o be socsc Eles o ese odels re e Hull-We wo-cor odel e Heso odel d e SABR odel I s reor we wll cocere o e ler Te SABR odel s bee roduced b H Kur Leews d Woodwrd [HKuLeWo] We wll o dscuss dels e resos w s ood odel orou dscusso o s subec c be oud e roduco o [ReMcW] we ol s ere e SABR odel ves que cosse edes w eble couol cos s s ecelle corose bewee recso d esess o use To ob soluo o e oo rc roble e SABR odel e e c s w sse o socsc derel equos we c ssoce dereble old M o deso suc e Kolooro bcwrd equo vered b s robbl des s e equo o M I rculr or e SABR odel d M s e Pocré le Sce e ler s ver suded ecl obec e soluo o e oo rc roble c be ceved w o uc dcul We observe ule e Heso odel e bove soluo s o ec: [HKuLeWo] d [HLeWo] e uors ve oud roe soluo b us erurbve eods e Vrd s eore A ore owerul ecque s o e e erel eso: e soluo o e e equo s eressed s ower seres wose coeces c be coued us eoerc oro o M Her-Lbordère s used e ler e rs order [H-L 5] d [H-L 8] o ob ore ccure soluo o [HKuLeWo] d [HLeWo] Pulo s used e eso e secod order [Pulo] o ob eve ore recse soluo I c e e erel eso eod c be led o eerl cl odel volv sse o socsc derel equos I rereses ew ecque or solv rc equos ce wc ul ow ece e cse were closed or soluo 6

7 esed could ol be suded w couoll cosl rocedures Te e erel eso eod ves cse se-closed or soluo e soluo wre s ower seres wc c be obed w eoerc ecques Te socsc derel roble rsles e o eoerc roble or e old M Two reereces or ese ecques re e boos [Avrd] d [H-L 8] were besdes e SABR re lso reseed lcos o ul-sse oos d o e LMM odel THE LINK WITH RIEMANNIAN GEOMETRY AND HEAT KERNEL EPANSIONS A Re srucure o IR s dereble uco ssoces w elee o IR osve dee squre r o order Sce ere s equvlece bewee sclr roducs d osve dee rces s es Re srucure s w o dee dereble er erc c ce ec o For sce le us ssoce e de r w ever o o IR Sce e de r ves e usul Euclde erc s cse we ve e sdrd erc we use clculus Te ol o Re eoer s o cosder eerl erc d dee ccordl e oerors o dervve derel erl ec Oe o e dereces bewee e eerl cse d e Euclde oe s e r o e erc s o ecessrl cos For sce we wll see e erc or e SABR odel s e Pocré le deed o b e r Te SABR cosss o sse o wo correled socsc derel equos 7

8 F ω F A u ωf β u ωdw u ω A ω α ν A u ωdz u ω corrw Z ρ were F βαν d ρ re u coss To sow e l w Re eoer d e erel esos we cosder eerl sse o correled socsc derel equos ω α b u uω du σ uuω dw u ω corr ρ were Le us ve Euroe oo C o uldesol sse wose dc s descrbed b e reced equos Le T be e ur o C d le e o T be uco T T Te r vlue o C veres e equo C T αt Tαd were s e robbl des o e sse d α α α Le us se τ T d τ α : Tα e veres e Kolooro bcwrd equo w D d w l codo b τα α τ D ρ τσ τασ τα α δ α α α δ s e Drc uco deed b δz or z d δz oerwse Ts equo s e equo Moreover we ve e r ρ τσ τασ τα 8

9 s osve dee I e ucos σ d ρ re dereble e s dereble d us e bove sse o socsc derel equos deeres Re srucure w resec o wc e Kolooro bcwrd equo s e equo I [HKuLeWo] d [HLeWo] H Kur Leews d Woodwrd ve oud roe soluo b es o erurbve ecques A ore owerul eod e ler d v ore ccure roo s o e e erel eso e reseo o wc s e ol o s reor CONTRIBUTIONS OF THIS REPORT As lred sd ew owerul ecques re vlble or solv oo rc robles ecques e ro dvced Re eoer Peole wor ce d oell eresed ese resuls ve lle or o owlede o Re eoer wc s ver colced d rd ecl eld or wc oe s o sed lo o e d eor o ser ol e os eleer cs Ts reor s be ele roduco could llow o-secls o red e ore dvced ers d boos [H-L 5] [H-L 8] [Pulo] [Avrd] Te rr ol s o v reseo o e oc could be rorous d colee d e se e es o udersd d o oo bsrc Ts would o ve bee ossble we d ollowed e usul roc o rese s subec or sce [Lee] Ideed Re eoer s bee eloed ul ow ol ecs d scs lcos Ule ese wo elds e Re olds ecouered cl odel re uc sler d reduce o e sce IR Ts old s rvl ro e derel eoerc o o vew d s lle or o eres o ec or scs So we oe s rrowed o cl lcos e sdrd eerl w o rese Re olds s useless colco I s reor we ve rewre ll e deos d eores or e secl cse o e old IR Ts ccolses our ol sce reduces e dcul o e subec o s ls reduces o sle cr 9

10 o eleer clculus d ler lebr Ts wor s orl oe sce ere s o suc roc boo or ccou o Re eoer Tere s bee rece es row eres Re eoerc ecques or solv cl robles [Avrd] [Frell] [H-L 8] so e ul o s reor oes beod e SABR odel Te reor s orzed s ollows: Cer reses e bsc oos o Re eoer Cer 3 res e Pocré le wc s e sce odel used or e SABR d ll Cer 4 les ll ese oos or e cse o e SABR

11 Iroduco o Re eoer REMINDERS FROM CALCULUS Le be url uber le M IR d le : M R be uco Le { e e } be e cocl bse o IR s or { } e s e vecor wose - cooe s d e oer re zero Le M e - rl dervve o deoed or s e dervve o we resrced o e sr le ss rou d v e s dreco s o { e α IR} α : Le d - rl dervves or ever { } e rde o s e vecor uco d s deoed rd Le v v v be vecor o v deoed D v or v I s well ow ro bsc clculus IR o or e drecol dervve o lo D s e dervve o we resrced o { v α IR} D v v v α : s e drecol dervve o lo v s equl o e sclr roduc bewee v d rd Le { } d suose e dervve o ds - rl dervve Te we dee e secod rl dervve o s d we deoe

12 Le : M IR be suc ds - rl dervve d le Φ be e vecor uco wose cooes re e Te dverece o Φ s deed s dv Φ Le : M IR d secod rl dervves e Llce oeror or Llc o s deed s : s e Llc o s e dverece o e rde o Now le ψ b b d or { } be ucos ro M o IR e oeror sed uco : M IR d secod rl dervves o e uco b ψ s clled secod order ler derel oeror Ts oeror s sd ellc or o-zero M we ve I rculr e Llc s ellc PROPERTIES OF DIRECTIONAL DERIVATIVES I s seco we wll eed e coce o drecol dervves o ore eerl coe Frs we e v s o ecessrl cos vecor bu sed s dereble uco v : M IR I rculr v s o ecessrl o or Tus we dee

13 D v : v rd were v rd deoes e sclr roduc bewee ese wo vecors Le Y be uco ro IR o D v Y s e uco D Y D Y s v v IR d le v rd Y v rd Y D Y v d wre s ler cobo e cocl bss e e } o Le us cll vecor eld o v Y Y be s sclr coordes e { v rd Y D Y e IR dereble uco ro IR IR o IR d le us deoe Vec IR e se o ll vecor elds We wll deoe D IR IR e se o dereble ucos ro IR o IR I rculr Vec IR D IR IR I : IR IR d : IR IR re ucos e s usul s e uco sed IR o I d re dereble e so s I rculr or rel α we deoe α e uco sed IR o α We rove e drecol dervve veres e ollow roeres: or ever Vec IR Y D IR IR d IR : IR dereble ucos we ve uco d D Y D Y D Y or ever Vec IR Y Y D IR IR d α α IR we ve D α α D Y α D Y Y αy 3 or ever Vec IR Y D IR IR or ever : IR IR dereble IR we ve D rd Y Y D Y 3

14 Te rs wo roeres re obvous cosequeces o e ler o sclr roducs o vecors d ler o e rde w resec o sclrs Le us rove e rd We ve d us e Lebz rule us D Y D Y D Y rd Y rd Y Y rd D rd Y Y rd rd Y Y rd Y rd Y rd Y rd Y DY 3 CONNECTIONS We ow dee oeror wc eerlzes dervves b e ree roeres o e reced seco s os Le d be url ubers coeco s uco : Vec IR D IR IR D IR IR suc us e oo Y sed o Y we ve: or ever Vec IR Y D IR IR d : IR IR dereble ucos we ve Y Y Y or ever Vec IR Y Y D IR IR d α α IR we ve α α Y α Y Y αy 4

15 uco d 3 or ever Vec IR Y D IR IR or ever : IR IR dereble IR we ve rd Y Y Y Le us bss e e } o { IR d or le us deoe E e uco sed elee o IR o e F lso bss { e e } o IR d or deoe E e uco sed elee o IR o Le us ve coeco d { } { } we ve s e E E ler cobo o elees o e e } so ere es rel ubers Γ deed { o IR or suc E E Γ e Te coeces Γ re clled e Crsoel sbols o e coeco Ideed sce I s e sdrd drecol dervve e e Crsoel sbols re ll zero E s cos uco s rde s zero us e drecol dervve o s zero d ereore s coordes bse re zero E Le be vecor eld Tus or IR belos o IR us s ler cobo o e e Tereore ere es ucos ro IR o IR suc e Tus b recll e we ve E s E s e vecor eld sed elee o IR o E I e se w ve dereble uco Y ro IR o IR ere es Y Y ucos ro IR o IR suc 5

16 6 Y E Y We ve e ollow resul: Teore Le be coeco o IR le IR Vec IR IR D Y d le E Y E Y Te Γ e Y rd Y Y Proo B us roeres d o coecos we ve E E Y Y E E Y B us roer 3 o coecos d e decooso us Crsoel sbols we ve E rd Y E E Y E Y E E e rd Y E e Y Γ us Y s equl o Γ e rd Y E e Y Γ e rd Y e e Y Fll we ve e rd Y e e rd Y e e rd Y e e rd Y d relc e de w e de we ve e cl

17 Rer Le e ollow or Γ be e r Γ Tus e reced equo c be wre rd Y Γ Y Y e Rer Le or { } d le us deoe E : E Te we ve Y Y Γ Y e 4 THE COVARIANT DERIVATIVE A wce dereble curve o s ervl o rel ubers We deoe or I we ve e secod ucos s e vecor IR s wce dereble uco : I IR were I e cooes o s or We deoe e rs dervve o d Le be url uber d le us deoe D e se o ll dereble V : I IR Ler cobos o elees o w s or vecor elds o Le D IR V d le V D IR IR D re deed e se we s V s eedble o V or ever I we ve V V 7

18 Gve bss e e } o { IR we deoe E or e elee o D sed I o e As or vecor elds o IR we ve V D ere es ucos V V ro I o IR suc V V E Le E be e elee o IR IR D sed IR o e e E s eedble o E Suose ve curve : I IR d coeco Le e e } be { bss o IR d le E : IR IR or be e uco sed A covr dervve lo w resec o s uco ver e ollow roeres: Der : D IR IR D IR or ever V V D d α IR we ve α IR Der α V α V α Der V α Der V or ever V D d : IR IR dereble uco we ve Der V V Der V 3 or ever V D d or ever eeso V o V we ve Der V E V IR o e Teore Tere ess oe d ol oe covr dervve Der lo w resec o I V V E s elee o D e 8

19 9 e V V V Der Γ Proo B us roer o covr dervves we ve E V Der E V Der V Der B roer we ve E Der V E V E V Der E Der V e V Sce E s eeso o E we ve b roer 3 E E Der E Γ E Γ e Ts les V Der us be equl o Γ e V e V Γ e V e V wc roves s esece d uqueess Rer Te reced equo c be wre e ollow r or e V V V Der Γ 5 PARALLEL TRANSPORT A derel uco o curve wose covr dervve lo curve s zero s clled rllel lo Sce e e re lerl deede V s rllel lo d ol veres e ollow ler d ooeeous sse o ordr derel equos

20 V V Γ or wc c be wre e r or V V Γ or We ve e ollow resul: Teore Le be coeco le : I IR be curve le I d v IR ere ess uque dereble uco V rllel lo w resec o d suc V v Proo See Teore 4 o Lee 997 Te Le e e Crsoel sbols re Γ or e e Crsoel sbols or vecor w cooes I we deoe Γ e equo or e rllel rsor s wc ds e soluo or rel cos c [ Γ ] V Γ V V d c e Γ d V c e Γ 6 GEODESICS equl o Le Te ccelero o w resec o s Der We ve Der s

21 Γ e Γ e We s s eodesc s ccelero s zero Ts es curve s eodesc d ol s dervve s rllel lo e curve Te equos or eodesc re e ollow Γ or or e r or Γ or A eodesc s e e soluo o sse o derel equos ve bove I s url o s weer ere s esece d uqueess o soluos o suc derel equos Ts queso s swered e ollow: Teore Le be coeco o elee I d eodesc ree o er coo do IR d : I IR suc Proo See Teore 4 o Lee 997 v IR Te ere ess oe ervl I o IR d v Moreover wo suc eodescs Rer Le : I IR be eodesc d suose ere ess I suc Te s cos s or ever Ideed le d le ψ : I IR be e uco sed I o Te ψ s eodesc d veres e se l codos s e ψ d ψ us b e reced eore d ψ do cocde

22 7 RIEMANNIAN STRUCTURES Le M be subse o suc or ever M e uco IR Re erc o M s dereble uco : M IR IR IR IR IR IR u v u v s sclr roduc e s serc d osve dee Le us bss e e } o { Te e r : IR d or M le us deoe : e e s serc d osve dee Ts les s verble d we deoe e elee o - row d - colu o s Ele Le be e de r or ever srucure o IR wc s clled e Euclde erc IR Te dees Re Ele Le IH be e osve l-sce o IR s { IR : > } IH We dee o IH e Re erc suc δ e s e r corresod o e o s equl o e de r dvded b w s Re erc s clled e Pocré l-sce IH We wll use e oo u v or u v I Y d Z re vecor elds we wll deoe Y Z e uco sed IR o Y Z

23 vecor elds Le be coeco or Y d Z d or ever M we ve IR we wll s s coble w or ever D Y Z Y Z Y Z we recll D Y Z deoes e dervve o Y Z lo e dreco We s s serc or ever vecor elds d Y we ve We ve e ollow: Y D Y D Y Y Teore Le be Re erc o subse o IR e ere ess oe d ol oe coeco wc s serc d coble w Te Crsoel sbols o s coeco re ve b e orul Γ l l Proo See Teore 54 o Lee 997 l l l Te coeco o e reced eore s clled e Lev-Cv coeco Ele Le be e Euclde erc o re zero d e e Crsoel sbols re zero IR Sce s cos e s dervves Ele Le us coue e Crsoel sbols o e Pocré l-sce Γ l l l l l IH We ve δ l δl δ δ δ δ δ l l l 3

24 δ δ δ δ δ δ δ δ δ δ δ δ Ts les Γ or d re dsc wo b wo Te o-zero Crsoel sbols re us: Γ Γ or Γ or Γ or Now coue e rces Γ We ve Γ s Γ s es e dol r wose dol vecor s For e r Γ s equl o d e oer re zero es e r wose - d - eres re So e equos or e eodescs re e ollow: or 8 CURVATURE Te Re esor s deed s: 4

25 5 Γ Γ Γ Γ Γ Γ r r r l r l r l l R l e Rcc esor s l l R R e sclr curvure s l l l R R Ele For e Euclde erc ll e ree ucos deed bove re zero sce e Crsoel sbols re zero Ele Te o-zero cooes o e Re esor or e Pocré l-le IH re R R R R R R Te cooes o e Rcc esor re R R R R Fll e sclr curvure s R

26 9 CURVES A dssble curve s uco :[b] R wc s e r d le dervves ll os d s dereble w o-zero dervve uless os el os B Rer 6 o cos eodesc s dssble curve sce s dervve s lws ozero Le [c d] be ervl o rel ubers d le :[cd] [ b] be dereble d verble uco wose verse s lso dereble rculr s oooc Te s clled rererzo o Le be ve Re srucure o IR We cll seed o w resec o e uco sed [ b] o u seed s seed s cosl equl o Te le o w resec o s deed s b L : We ve e ollow b d d I rculr we s dssble curve s b d Teore Le L L ψ be dssble curve d ψ rererzo o e :[b] R Proo Tere ess ervl [c d] d dereble d verble uco :[cd] [b] wose verse s lso dereble d suc ψ ψ e We ve d L ψ ψ ψ c d c ψ d d b d L 6

27 were e eule equl ollows ro e rules o ero b subsuo We lso ve Prooso Le :[b] R be dssble curve d le λ be e le o Le be e verse o e uco sed [ b] o e le o resrced o [ ] Te ψ : s u seed rererzo o Moreover :[ λ] [ b] s uco suc λ b d e rererzo s u seed e Proo See Eercse 6 o Lee 997 Te ls eore ells dssble curve ds oe d ol oe rerzo w u seed Ts rerzo s clled e rerzo b rc le For eodescs we ve sroer resul: Teore Le cos :[b] R be eodesc e e uco sed [ b] o s Proo See Le 55 o Lee 997 Te ls resul les :[b] R s eodesc e ere ess cos suc L b Rer Le be o-cos eodesc d le :[b] R ψ be rererzo o we w o deere weer ψ s eodesc oo We ve ψ d ψ ψ ψ 7

28 Tus ψ ψ ψ ψ Γ Γ [ Γ ] were e ls equl ollows ro e c s eodesc We ve sce we ve suosed s o-cos s les ψ s eodesc oo d ol or ever s d ol ere es coss α d β suc α β Le :[b] R be dssble curve we s s z curve or ever dssble curve ψ :[cd] R w e se edos e { b} { ψ c ψ d} we ve L L ψ We ve e ollow: Teore Gve wo os d o M ere ess z curve wose edos re d A z curve w u seed rerzo s eodesc w resec o e Lev-Cv coeco Proo See Le 6 d Teore 66 o Lee 997 Gve wo os d o M we cll eodesc dsce bewee d e le o z curve o e 8

29 HEAT KERNEL ON A RIEMANNIAN STRUCTURE Le b b d or { } ollow derel oeror o order : be ucos ro M o IR d cosder e D : b Ts oeror s ellc d ol or o-zero M we ve s d ol e r d us lso e r s osve dee or o-zero Ts les s Re erc o M e e oeror deed bove s ellc Now le be uco ro equo IR IR IR o IR d cosder e D were D oeres o cosdered s uco o s clled e erel equo Te uco s clled e erel or e e erel equo deed bove s soluo o s equo d veres lso e ollow codos: s o clss s o clss C w resec o C w resec o 3 veres l δ For M le us deoe ds e eodesc dsce bewee d w resec o e Re erc We dee e Se world uco w resec o s ds S : 9

30 Now le us deoe C e r wose er or { } s S we dee e V Vlec-Moree deer w resec o s C de V : de de Now or le : M IR be deed b A d cosder e A : b de de coeco wose Crsoel sbols re A A Le be curve e rllel rsor o w resec o s coeco s ve b see lso 335 o Avrd We ve e ollow crucl Pr e A du Teore Le be e e erel or e e erel equo ve bove Le us M d le :[ ] M be z curve suc d Te ere es ucos or IN suc ds e ollow soc eso or de V S Pr e 4π Proo See [H-L 8] d [Avrd] We ve d 3

31 R Q 6 were R s e sclr curvure ELLIPTIC OPERATORS ON A RIEMANNIAN STRUCTURE Le be Re erc o M our ol s seco s o dee Re loue o e oo o Llc For rs we wll dee Re loues o e rde d o e dverece Le M le : M R be uco d rl dervves d le rd e be e rde o we recll {e e} s e cocl bss o IR Te rde o w resec o s deed s rd rd e e rd I rculr e sdrd rde s e oe corresod o e Euclde erc Le : M IR be suc ds - rl dervve d le Φ be e vecor uco wose cooes re e Te dverece o Φ w resec o s deed s dv de Φ Φ dv de e 3

32 dv de Φ de As or e rde e sdrd dverece s e dverece w resec o e Euclde erc Le : M R be uco d rl secod dervves e Llce- Belr oeror o w resec o s deed s e s s s rd : dv dverece o e rde o de rd dv de de de rd dv de de were s usul deoes e elee o Now le D be e derel oeror deed b D : b For D e ellc oeror deed s bove d le Q : M IR be deed b were e de A Q : A A de A were deed bove Te c be roved 3

33 33 [ ] de de Q A A D We observe s eresso s equl o e Llce-Belr oeror d ol A or ever CHANGE OF COORDINATES IN A RIEMANNIAN STRUCTURE Le IR IR : be becve d dereble uco suc M M d s usul le us deoe e cooes o Le IR e Jcob o deoed Jc s e squre r o order wose elee s Le M :[b] be curve d le us cosder e curve Te cooes o re d e dervve o ec cooe s rd d d so e le o w resec o s equl o b d L b d Le us deoe * e r uco wose elee s

34 s e roduc o e colu o Jc es es e colu o Jc Te we ve b L d L * s e le o w resec o s equl o e le o w resec o * Tus we c s we e ce o coordes M b es o e e erc ces ro o * As observed bove * s e roduc o e rsose o Jc es es Jc s * Jc Jc Suose we ce e coordes ro o Ts es s ced o Le L : Jc sce Jc L we ve e erc becoes d s verse s L L L L 3 CHAIN RULE FOR PARTIAL DERIVATIVES Le M R le M IR : be wce dereble uco d le : M M be wce dereble d becve uco We deoe e cooes o Le be rel vrbles d suose s ve s uco o s Le lso : We c cosder s rel vrbles d eress s uco o s B e c rule o rl dervves we ve or 34

35 35 ereore We ve d so we ve 4 CHANGE OF COORDINATES IN THE KOLMOGOROFF BACKWARD EQUATION Le be url uber d le IR M M IR : le D be e derel oeror deed b : b D d le ver e rl derel equo

36 36 D were IR M d D cs o s uco o Le M M : be wce dereble d becve uco wose secod rl dervves re zero d le us cosder s uco o d B e resul ve e reced rr we ve d Tus D b b Le L be e Jcob r o w resec o e vrbles s L B eleer rules o e roduc o rces we ve s e er o e r L L We observe sce s becve we c cosder s uco o us se

37 37 : L L γ d : rd b b β we ve D γ β Sce e dervve o w resec o does o ce we cosder s uco d we ve γ β

38 3 Te Pocré l-le I s cer we sud e Pocré l-le Re erc o e osve l-le wc wll resul o be e erc or e SABR odel Te cer s orzed s ollows: er roduc e Pocré odel or eerc deso d coued e equos or e eodescs we sud dels e cse We d wo clsses o soluos o ese equos el e vercl les d e osve secrcles w ceer o e s We coue e eodesc dsce or e d sow e re e ol eodescs A vercl le s equo ± be or b d rel coss w b > osve secrcle s equo r r q c cos q 3 POINCARÉ HALF-SPACE Le IH be e osve l-sce o IR s { IR : > } IH We dee o IH e Re erc suc δ e s e r corresod o e o s equl o e de r dvded b Ts es e sclr roduc s equl o e sdrd Euclde sclr 38

39 roduc dvded b Tus we ve e - elee o e verse o s equl o δ As see beore e equos or e eodescs re e ollow: or s Now le d le us se : : Te e sse o derel equos wc s equvle o 3 GEODESICS OF THE POINCARÉ HALF-PLANE: VERTICAL LINES Frs suose or ever s es ere ess cos suc or ever Moreover e secod equo becoes To solve le us observe d d 39

40 d us e secod equo s vered d ol Te ere ess cos b d suc b d ereore ere s oer cos c suc b us be osve c I cocluso e eresso or e curve s be ubers Ts curve s vercl le c be Sce > e were b d c re rel Now we suose s curve s rerzed b rc le B Prooso 9 we c ve e seed o e curve s equl o Tus sce bc e we ve we se equl o so ± b b c e b e c c c c ± c Tus e eodescs re be or coss d 33 GEODESIC DISTANCE FOR VERTICAL LINES Le us e wo os d w e se coorde U o rererzo ere s ol oe eodesc o e d oud bove o ese wo os e sr le see corsed bewee d We wll rove suos wou loss o eerl < e le o e rc o curve o e s equl o lo Ideed le us e curve wc we cll d le be suc d Sce e curve s rerzed b rc le e le o e rc o corsed bewee d s 4

41 Frs le e eodesc be be e ro be we ob or Sce s cse s cres ro < we ob < lo lo b b lo us lo b Now le e eodesc be be s decres us < les < d e s cse lo Te uco b lo lo b b lo 34 GEODESICS OF THE POINCARÉ HALF-PLANE: SEMICIRCLES Now le be o-ecessrl zero We observe e rs equo ves 3 D wc s rue d ol ere ess o-zero cos r suc r As beore we ose s 4 r Le r s s soluo o e reced equo Se ζ : e r r r r ζ ζ ζ 4

42 We ow recll e ollow resul ro clculus: s ecve d dereble rel uco e D[ ] Tus D[ ] [ ζ ] z dz ζ z D z dz ζ ζ z [ z ] dz ζ ζ ζ ζ z z dz Now se equl o θ : rccos z s z cosθ d d z sθ dθ Tus e reced erl s sθ dθ cosθ cos θ cosθ dθ Observe sθ cosθ cos θ s θ sθ D lo cosθ sθ cos θ cosθ us Tereore sθ cosθ cos sθ dθ lo z lo θ cosθ z ζ ζ lo ζ ζ e ζ ζ e ζ ζ e ζ e ζ ζ e ζ e ζ e ζ e ζ e ζ e Sce ζ α e ζ > 4

43 ζ e e cos r cos Tereore r r cos d us r c or soe cos c I cocluso e eodesc s Sce > e we ve r > We ow observe sce we ve e equl r r c cos cos c r r cos Sce oreover or ] - [ e uco ssues ll e vlues ]- [ d e uco / cos ll e vlues ] [ we ve e eodesc we ve oud s e osve secrcle w ceer c d rdus r 35 GEODESIC DISTANCE FOR SEMICIRCLES Le us e wo os d w Frs we sow u o rererzo ere s ol oe secrcle o ese wo os Ts s equvle o 43

44 rove ere s ol oe o e s w e se dsce ro d Ideed le c be suc e dsce ro c o s equl o ro c o e e equl s e uque soluo c c c Now le us coue e eodesc dsce o wo os oed b secrcle We sw lo ζ ζ lo r r Le be e secrcle d le be suc d We suose < so e le o e rc o e eodesc o d s lo r r lo r r lo r r r r I c be roved e ler eresso s equl o rccos 36 PROOF OF THE UNIQUENESS OF THE GEODESICS Now we rove vercl les d secrcles re e ol eodescs or e Pocré l-le For we ve o sow ve rel ubers u u v v ere ess curve 44

45 wc s vercl le or secrcle d suc u u v We observe sce e -coorde s osve we us ve u > uco v B Rer 9 or ever rel uber α e cooso o eodesc w e α s sll eodesc so e wo clsses o eodescs we ve oud oeer w er dervves re w d b rbrr coss d ± α be ± α ± be ψ r r α c cos α r r s α ψ cos α cos α w r > d c coss Le α v d e be e d we ve soluo or u b u b bα α v u : u v u ue s eodesc d veres e l codo or s e curve Now le v we ve o d coss r α c d w r > suc r r α c u u v v cos α cos α cos α αr r s α Aer lo bu sle clculos we d e soluo o s sse s ve b r v v v v u v u v v α c u v u v v u v v rcs v v v v 45

46 37 GEODESIC CONNECTING TWO GIVEN POINTS Le us ve wo os z d z o e Pocré l le d le us d e eodesc coec e s eodesc IH ψ :[ τ ] IH or soe τ > suc ψ z d ψ τ z Frs le us suose d cosder e secrcle ψ r α c r cos α w ceer e o c o d rdus r deed b c d c c r Frs we wll d e vlue suc ψ z d we wll dee ew curve b so z : ψ To d we ve o solve e ollow sse o equos r α c r cos α Ts ves r c rccos rcs α α or ever α T s 46

47 r ψ rccos α B os r τ ψ rccos τ z α sce r ψ rccos z α s les τ rccos α r rccos α r d ves α lo τ r r r r b us e relo rccos lo or s s ossble sce r Now suose Te eodesc co z d z s vercl le o e or I lo α α e ψ e so we se α e ψ α e : ψ We deere α order o ve τ z s ves 47

48 ατ lo lo l τ α lo 48

49 49 4 Te SABR odel 4 RIEMANNIAN METRIC FOR THE SABR Le C b d σ be ucos ro IR o IR w C erble le F d α be rel coss d le P ρ be serc d verble r o order wose dol eres re equl o A socsc voll odel s deed s e ollow sse o wo correled socsc derel equos: u u u u u u Z W corr dz A du A b A dw F C A F F ρ ω ω σ ω α ω ω ω ω ω Se : ρ ρ e verse o e r P s e r ρ ρ ρ rculr s les ρ For {} we deoe ρ e elee o P Se : C σ : σ σ z : z : d le be soluo o e equo z z b σ σ ρ For {} se : σ σ ρ

50 5 d suose e r s osve dee or ever Te Re erc or e bove sse o socsc derel equos s : C C C σ σ ρ σ ρ Le us cosder e uco d C d q IR IR : : σ ξ s Jcob s C σ so becoes ρ ρ σ σ σ ρ σ ρ σ C C C C C Moreover we ve ξ q b b q b σ ξ Now cosder e ollow ce o coordes F : IR IR qξ qξ : q ρξ ξ : ρ ξ Is Jcob s ρ ρ

51 5 so becoes ρ ρ ρ ρ ρ ρ ρ d b b σ ρ b b σ ρ Now le us cosder e SABR odel e e socsc voll odel w b d z z ν σ or soe cos ν I rculr s les ν ξ d ν ρ ξ so rculr ρ ν d us coordes s ν so e secod order dervves er s z z z z ν Now le us cosder e uco : ν τ e τ τ τ ν so

52 5 z z τ τ ν z z z z τ τ ν d us z z τ τ z z z z τ τ Tus z d z s e erc s o e Pocré l-le 4 HEAT KERNEL EPANSION FOR THE SABR Le us se : F F e we ve ds rccos rccos F d us rccos F S We ve

53 53 s ds ds V see Sec 43 o [Pulo] or [59] o [Avrd] sce or z we ve cos s z z e s ds rccos s F rccos cos F F e rccos F F V We ve e δ δ de Sce or we ve de de b A d b e A δ δ δ δ d us

54 A A Now le us ve wo os z d z w Le > s see beore e eodesc suc z d z s ve b :[ ] IH u r αu u c r cos αu u w c c c r r c u rccos rcs r r α lo r r We ve Pr e so we eed o coue We ve u u A u du e u A u du d r rα s αu u u du cos αu u cos αu u e Pr e u A u du rα s αu u cos αu u e du cos αu u r α s αu u α e du e cos αu u αu u du B e ce o vrbles 54

55 v α u u we ve Pr α u e v dv e lo cos e lo cos α u cos α u u [ v] α Now le us ve wo os z d z d le > e eodesc suc z d z s ve b w u lo d α lo As beore we ve :[ ] IH αu u e u e αu u u αe We ve Pr e α αu u u A u du e e αu e u α du e Q : A A de A de δ A A δ A [ A A ] A A 55

56 We ve S e rccos 4 e F e or we ve rccos 4 e cos rccos 4 F F u F α π d we ve lso rccos 4 e cos rccos 4 F F u F α α π were F We ve d 6 Q R 4 3 were R s e sclr curvure

57 B us e roo rs order o e reced orul oe c d e ollow vlue or e BSM led voll σ BS KT K T lo K d 3 4 v T " lo P " " " Te dels o e dervo o suc orul re 433 o [H-L 5] or 64 o [H-L 8] 57

58 Reereces: [Avrd] Avrd IG Alc d Geoerc Meods or He Kerel Alcos Fce uublsed 7 [Blc-Scoles] Blc F Mro S Te Prc o Oos d Corore Lbles Jourl o Polcl Ecoo 83: [Boucud] Boucud J-P Ecoocs eeds scec revoluo Nure Vol Oc 8 8 [Der-K] Der E K I Rd o Sle Rs 7: [Dure] Dure B Prc w Sle Rs 7: [Frell] Frell S Geoerc Arbre Teor d Mre Dcs ://ersssrco/sol3/ersc?bsrc_d39 9 [Gror ] Gror A He Kerel d Alss o Molds Aerc Mecl Soce 9 [HKuLeWo] H PS Kur D Leews AS Woodwrd DE M Sle Rs Wllo Mze 8: 84-8 [HLeWo] H PS Leews AS Woodwrd DE Probbl Dsrbuo e SABR Model o Socsc Voll Mrc 5 uublsed [H-L 5] Her-Lbordere P A Geerl Asoc Iled Voll or Socsc Voll Models ://ersssrco/sol3/ersc?bsrc_d6986 [H-L 8] Her-Lbordere P Alss Geoer d Model Fce CRC Press 8 58

59 [H-L 9] Her-Lbordere P A Geoerc Aroc o e Asocs o Iled Voll Froers Quve Fce ed R Co Wle 9 [Lee] Lee Jo M Re olds Srer 997 [Mero] Mero Rober C Teor o Rol Oo Prc Bell Jourl o Ecoocs d Mee Scece 4 : [Pulo] Pulo L Asoc Iled Voll e Secod Order w Alco o e SABR Model ://ersssrco/sol3/ersc?bsrc_d43649 [ReMcW] Reboo R McK K We R Te SABR/LIBOR Mre Model Wle 9 [Vsslevc] Vsslevc D V He erel eso: user s ul ://rvor/bs/e-/