Table of Contents for All Volumes
VOLUME I Foundations and Vanilla Models Partt I Foundations Par 1
Intr In troduct oduction ion to Ar Arbit bitra rage ge Pr Prici icing ng Th Theor eory y .............
1.1 1.1 1.22 1. 1.33 1. 1.44 1. 1.55 1. 1.66 1. 1.77 1. 1.88 1. 1.99 1.
The Se The Setu tup p . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. Tra radi ding ng Gai Gains ns and and Arbi Arbitr trag agee . . . . . . . . . . . . . . . . . . . . . . . . . . Equi Eq uiv val alen entt Mar Marti ting ngal alee Mea Measu sure ress and and Ar Arbi bitr trag agee . . . . . . . . . . Deriv De rivat ativ ivee Secu Securit rity y Prici Pricing ng and and Com Compl plet etee Marke Markets ts . . . . . . . Girsa Gi rsano nov’ v’ss Theore Theorem m .. .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. Stoch St ochast astic ic Diffe Differen renti tial al Equat Equatio ions ns . . . . . . . . . . . . . . . . . . . . . . Expl Ex plic icit it Tra Tradi ding ng Strat Strateg egies ies and and PDEs PDEs . . . . . . . . . . . . . . . . . . Kolmo Kol mogor gorov ov’s ’s Equ Equat atio ions ns an and d th thee Fey eynm nman an-K -Kac ac Th Theo eorem rem . Blac Bl ackk-Sc Scho hole less and and Exten Extensi sion onss . . . . . . . . . . . . . . . . . . . . . . . . . 1.9. 1. 9.11 Ba Basi sics cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9. 1. 9.22 Al Alte tern rnat ativ ivee Deriv Derivat atio ion n ........................ 1.9. 1. 9.33 Ex Extten ensi sion onss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9. 1. 9.3. 3.11 Dete De term rmin inis isti ticc Pa Para rame mete ters rs an and d Di Divi vide dend ndss 1.9. 1. 9.3. 3.22 Stoc St ocha hast stic ic In Inte tere rest st Ra Rate tess . . . . . . . . . . . . . 1.10 Op 1.10 Opti tion onss with with Early Exerc Exercis isee Rights Rights . . . . . . . . . . . . . . . . . . . . 1.10. 1. 10.11 Th Thee Markov Markovian ian Case Case . . . . . . . . . . . . . . . . . . . . . . . . . .
3 3 7 8 10 12 14 16 18 21 21 25 27 27 28 30 32
XI I
2
Contents
1.10.22 Some 1.10. Some Genera Generall Bound Boundss . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 1. 10.3 .3 Ea Earl rly y Exerc Exercis isee Premi Premiaa . . . . . . . . . . . . . . . . . . . . . . . . .
34 36
Fini Fi nite te Di Diffe ffere renc nce e Me Meth thods ods . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43 43 45
2.1 1-Di 2.1 1-Dime mens nsion ional al PDE PDEs: s: Prob Problem lem Form ormul ulat atio ion n ............. 2.22 Fi 2. Fini nite te Differe Differenc ncee Di Discr scret etiza izati tion on . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 2. 2.1 Di Discr scret etiz izat atio ion n in x-Direction. Dirichlet Boundary Cond Co ndit ition ionss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. 2. 2.22 Ot Othe herr Bound Boundar ary y Cond Condit itio ions ns . . . . . . . . . . . . . . . . . . . . 2.2. 2. 2.33 Ti Time me-D -Dis iscr cret etiz izat atio ion n .. . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. 2. 2.44 Fi Fini nite te Diffe Differe renc ncee Sche Scheme me . . . . . . . . . . . . . . . . . . . . . . 2.33 St 2. Stab abil ilit ity y. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. 2. 3.11 Ma Matr trix ix Meth Method odss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. 2. 3.22 Von Neum Neuman ann n Analy Analysi siss . . . . . . . . . . . . . . . . . . . . . . . . 2.44 No 2. Nonn-Eq Equi uidi dist stan antt Discre Discreti tiza zati tion on . . . . . . . . . . . . . . . . . . . . . . . . 2.55 Sm 2. Smoot oothi hing ng and and Conti Contin nui uitty Corre Correct ctio ion n .. . . . . . . . . . . . . . . . . 2.5. 2. 5.11 Cr Cran ankk-Ni Nico cols lson on Os Osci cill llat atio ion n Rem Remed edie iess . . . . . . . . . . . . 2.5.2 2. 5.2 Co Con nti tin nui uitty Correct Correctio ion n ......................... 2.5. 2. 5.33 Gr Grid id Shif Shifti ting ng.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.66 Co 2. Con nvec ecti tion on-D -Dom omin inat ated ed PDEs PDEs . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. 2. 6.11 Up Upwi wind ndin ingg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6. 2. 6.22 Ot Othe herr Tec Techn hniq ique uess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.77 Op 2. Opti tion on Exampl Examples es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 2. 7.1 Co Con nti tin nuo uous us Barri Barrier er Opti Option onss . . . . . . . . . . . . . . . . . . . . 2.7. 2. 7.22 Di Disc scre rete te Barr Barrie ierr Opti Option onss . . . . . . . . . . . . . . . . . . . . . . . 2.7. 2. 7.33 Co Coup upon on-P -Paayi ying ng Se Secu curi riti ties es an and d Div Divid iden ends ds . . . . . . . . 2.7. 2. 7.44 Se Secu curi riti ties es with with Ear Early ly Exe Exerc rcis isee . . . . . . . . . . . . . . . . . . 2.7. 2. 7.55 Pat athh-De Depe pend nden entt Opti Option onss . . . . . . . . . . . . . . . . . . . . . . 2.7. 2. 7.66 Mu Mult ltip iple le Exer Exerci cise se Rig Righ hts. . . . . . . . . . . . . . . . . . . . . . . 2.88 Spe 2. Specia ciall Is Issu sues es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8. 2. 8.11 Me Mesh sh Refi Refine neme men nts for for Mul Multi tipl plee Even Events. ts. . . . . . . . . . . 2.8. 2. 8.22 An Anal alyt ytic icss at the the Last Last Time Time Ste Step p................ 2.8. 2. 8.33 An Anal alyt ytic icss at the the Fir First st Tim Timee Ste Step p ............... 2.99 Mu 2. Mult ltii-Di Dime mens nsio iona nall PDEs: PDEs: Pro Probl blem em Form ormul ulat atio ion n . ........ 2.10 2. 10 Tw Two-D o-Dim imen ensio siona nall PDE PDE with with No Mixed Mixed Deri Deriv vat ativ ives es . . . . . . . 2.10 2. 10.1 .1 Th Thet etaa Me Meth thod od . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10. 2. 10.22 Th Thee Al Alte tern rnat atin ingg Di Dire rect ctio ion n Im Impl plic icit it (A (ADI DI)) Me Meth thod od 2.10 2. 10.3 .3 Bo Boun unda dary ry Co Cond ndit itio ions ns and and Ot Othe herr Iss Issue uess . . . . . . . . . 2.11 2. 11 Tw Two-D o-Dim imen ensio siona nall PDE with with Mixe Mixed d Deriv Derivat ativ ives es . . . . . . . . . . 2.11 2. 11.1 .1 Or Orth thog ogon onal aliz izat atio ion n of the the PDE . . . . . . . . . . . . . . . . . . 2.11. 2. 11.22 Pre Predi dict ctoror-Co Corre rrect ctor or Sche Scheme me . . . . . . . . . . . . . . . . . . . . 2.12 2. 12 PD PDEs Es of Ar Arbi bitr trary ary Order Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45 47 49 50 52 52 53 56 58 58 58 59 60 61 62 63 63 65 67 68 69 70 72 72 75 76 78 79 80 81 84 85 85 88 91
Contents
3
Mon Mo nte Ca Carl rlo o Me Meth thods ods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.11 Fun 3. unda dame men nta tals ls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. 3. 1.11 Ge Gene nera rati tion on of of Rand Random om Sam Sampl ples es . . . . . . . . . . . . . . . . 3.1. 3. 1.1. 1.11 Inver erse se Tra ran nsf sfor orm m Met Meth hod . . . . . . . . . . . . 3.1. 3. 1.1. 1.22 Acc ccep epttan ance ce-R -Rej ejec ecti tion on Me Mettho hod d. .. .... .. 3.1. 3. 1.1. 1.33 Com ompo posi sittio ion n ........................ 3.1.2 3. 1.2 Co Corre rrela late ted d Gaus Gaussi sian an Samp Sample less . . . . . . . . . . . . . . . . . . 3.1. 3. 1.2. 2.11 Chol Ch oles esky ky De Deco comp mpos osit itio ion n ............. 3.1. 3. 1.2. 2.22 Eige Ei gen nval alue ue De Deco comp mpos osit itio ion n ............ 3.1.3 3. 1.3 Pri Princ ncip ipal al Com Compon ponen ents ts Ana Analy lysi siss (PCA (PCA)) . . . . . . . . . . 3.22 Ge 3. Gene nerat ration ion of Samp Sample le Paths Paths . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.11 Exam 3.2. Example ple:: Asian Asian Bask Basket et Opt Option ionss in in Blac Black-S k-Sch choles oles Econom Econ omy y . . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . .. . . 3.2.22 Dis 3.2. Discret cretizat ization ion Sc Schem hemes, es, Con Conve vergen rgence, ce, and St Stabi abilit lity y 3.2. 3. 2.33 Th Thee Eu Eule lerr Sche Scheme me . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. 3. 2.3. 3.11 Line Li near ar-D -Dri rift ft SDE SDEss . . . . . . . . . . . . . . . . . . . 3.2. 3. 2.3. 3.22 LogLo g-Eu Eule lerr Sche Scheme me . . . . . . . . . . . . . . . . . . . 3.2. 3. 2.44 Th Thee Impli Implici citt Euler Euler Sche Scheme me . . . . . . . . . . . . . . . . . . . . . 3.2. 3. 2.4. 4.11 Impl Im plic icit it Diff Diffus usio ion n Ter Term m .............. 3.2.5 3. 2.5 Pre Predi dict ctoror-Co Corre rrect ctor or Sche Scheme mess . . . . . . . . . . . . . . . . . . . 3.2.6 3. 2.6 It Itoo-T Tay aylor lor Exp Expan ansi sion onss an and d Hi High gher er-Or -Orde derr Sc Sche heme mess . 3.2. 3. 2.6. 6.11 Ordi Or dina nary ry Tayl ylor or Ex Expa pans nsio ion n of OD ODEs Es . . . 3.2. 3. 2.6. 6.22 ItoIt o-T Tayl ylor or Exp Expan ansi sion onss . . . . . . . . . . . . . . . . 3.2.6.3 3.2. 6.3 Milste Mil stein in Sec Second ond-Ord -Order er Dis Discret cretizat ization ion Sche Sc heme me . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.7 3. 2.7 Ot Othe herr Secon Secondd-Ord Order er Sche Scheme mess . . . . . . . . . . . . . . . . . . 3.2. 3. 2.88 Bi Bias as vs. vs. Monte Monte Carl Carloo Error Error . . . . . . . . . . . . . . . . . . . . 3.2.9 3. 2.9 Sa Samp mplin lingg of Con Conti tin nuo uous us Proc Process ess Ext Extrem remes es . . . . . . . 3.2.10 3.2. 10 PCA and Bridge Bridge Constr Construct uction ion of Browni Brownian an Moti Mo tion on Pa Path thss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.10.1 3.2. 10.1 Bro Browni wnian an Brid Bridge ge and Qua Quasi-R si-Rand andom om Sequ Se quen ences ces . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 3. 2.10.2 0.2 PC Cons Constr truc ucti tion on . . . . . . . . . . . . . . . . . . . . 3.3 Sen Sensit sitivit ivity y Com Compu putat tation ionss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 3. 3.1 Fi Fini nite te Differ Differen ence ce Estim Estimat ates es . . . . . . . . . . . . . . . . . . . . 3.3. 3. 3.1. 1.11 Blac Bl ackk-Sc Scho hole less Del Delta ta . . . . . . . . . . . . . . . . . . 3.3. 3. 3.1. 1.22 Gene Ge nera rall Case Case . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 3. 3.2 Pa Path thwi wise se Estim Estimat atee . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. 3. 3.2. 2.11 Blac Bl ackk-Sc Scho hole less Del Delta ta . . . . . . . . . . . . . . . . . . 3.3. 3. 3.2. 2.22 Gene Ge nera rall Case Case . . . . . . . . . . . . . . . . . . . . . . . 3.3. 3. 3.2. 2.33 Sens Se nsit itiv ivit ity y Path Path Ge Gene nera rati tion. on. . . . . . . . . . . 3.3.3 3. 3.3 Li Lik keli elihood hood Rati Ratioo Method Method . . . . . . . . . . . . . . . . . . . . . . 3.3. 3. 3.3. 3.11 Blac Bl ackk-Sc Scho hole less Del Delta ta . . . . . . . . . . . . . . . . . . 3.3. 3. 3.3. 3.22 Gene Ge nera rall Case Case . . . . . . . . . . . . . . . . . . . . . . . 3.3. 3. 3.3. 3.33 Eule Eu lerr Sche Scheme mess . . . . . . . . . . . . . . . . . . . . . .
XI II II
93 93 95 96 97 99 100 1011 10 1022 10 103 104 104 106 1088 10 1100 11 1100 11 1111 11 1122 11 113 114 1155 11 1166 11 117 119 1200 12 122 126 126 1288 12 129 129 1299 12 1311 13 1333 13 1333 13 1344 13 1366 13 136 1377 13 1388 13 1388 13
XIV
Contents
3.3.3. 3.3. 3.44 Some So me Rem Remar arks ks . . . . . . . . . . . . . . . . . . . . . . 13 1399 3.44 Vari 3. arian ance ce Reduc Reducti tion on Tec Techn hniq ique uess . . . . . . . . . . . . . . . . . . . . . . . . 140 3.4.1 3. 4.1 Vari arian ance ce Redu Reduct ctio ion n and Effic Efficie ienc ncy y . . . . . . . . . . . . . . 141 3.4.2 3. 4.2 An Anti tith thet etic ic Vari Variat ates es . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1411 3.4. 3. 4.2. 2.11 Thee Gaus Th Gaussi sian an Cas Casee . . . . . . . . . . . . . . . . . . 14 1411 3.4. 3. 4.2. 2.22 Gene Ge nera rall Case Case . . . . . . . . . . . . . . . . . . . . . . . 14 1433 3.4. 3. 4.33 Co Con ntr trol ol Var Varia iate tess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1433 3.4. 3. 4.3. 3.11 Basi Ba sicc Idea Idea . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1433 3.4. 3. 4.3. 3.22 NonNo n-Li Line near ar Con Contr trol olss . . . . . . . . . . . . . . . . . 14 1455 3.4.4 3. 4.4 Im Impor porta tanc ncee Sampl Samplin ingg . . . . . . . . . . . . . . . . . . . . . . . . . 146 3.4. 3. 4.4. 4.11 Basi Ba sicc Idea Idea . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1466 3.4. 3. 4.4. 4.22 Dens De nsit ity y For Form mul ulat atio ion n . . . . . . . . . . . . . . . . . 14 1477 3.4. 3. 4.4. 4.33 Impo Im port rtan ance ce Sa Samp mpli ling ng an and d SDE SDEss . . . . . . . 14 1499 3.4. 3. 4.4. 4.44 More Mo re on SD SDE E Pat Path h Sim Simul ulat atio ion n . . . . . . . . 15 1500 3.4.4. 3.4 .4.55 Rare Ra re Ev Even entt Si Sim mul ulat ation ion an and d Li Line neari arizat zation ion 152 152 3.55 So 3. Some me Note Notess on Bermud Bermudan an Secu Securit rity y Pricin Pricingg . . . . . . . . . . . . . . 156 3.5. 3. 5.11 Ba Basi sicc Idea Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1566 3.5.2 3. 5.2 Pa Param ramet etri ricc Low Lower Bou Bound nd Meth Methods ods . . . . . . . . . . . . . . 157 3.5.3 3. 5.3 Pa Param ramet etri ricc Low Lower Bou Bound nd:: An Exam Exampl plee . . . . . . . . . . 15 1588 3.5. 3. 5.44 Re Regr gres essi sion on-B -Bas ased ed Lo Lower Boun Bound d . . . . . . . . . . . . . . . . . 15 1599 3.5. 3. 5.55 Up Uppe perr Bo Boun und d Metho Methods ds . . . . . . . . . . . . . . . . . . . . . . . . 16 1600 3.5.6 3. 5.6 Co Confi nfide denc ncee Interv Interval alss . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1611 3.5.7 3. 5.7 Ot Othe herr Me Meth thods ods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 3.A Appe Appendi ndix: x: Consta Constant ntss for Φ−1 Al Algor gorit ithm hm . . . . . . . . . . . . . . . . 16 1633 4
Fund undame ament ntals als of In Inter terest est Rat Rate e Modeli Modeling ng . . . . . . . . . . . . . . .
4.11 Fixe 4. Fixed d In Incom comee Not Notat atio ions ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 4. 1.1 Bon Bonds ds and and Forw Forwar ard d Rates Rates . . . . . . . . . . . . . . . . . . . . . 4.1.2 4. 1.2 Fut utur ures es Rates Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 4. 1.3 An Ann nui uitty Fact Factors ors and and Par Par Rates Rates . . . . . . . . . . . . . . . . . 4.22 Fi 4. Fixe xed d In Incom comee Probabi Probabilit lity y Measur Measures es . . . . . . . . . . . . . . . . . . . . 4.2.1 4. 2.1 Ri Risk sk Neutr Neutral al Measu Measure re . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 T -F -Forw orward ard Measu Measure re . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. 4. 2.33 Sp Spot ot Meas Measur uree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 4. 2.4 Term ermin inal al and and Hybri Hybrid d Measur Measures es . . . . . . . . . . . . . . . . . 4.2. 4. 2.55 Sw Swap ap Meas Measur ures es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.33 Mu 4. Mult ltii-Cu Curr rren ency cy Mark Market etss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. 4. 3.11 No Nota tati tion onss and FX FX For Forw war ards ds.. . . . . . . . . . . . . . . . . . . . 4.3. 4. 3.22 Ri Risk sk Neut Neutra rall Measu Measure ress . . . . . . . . . . . . . . . . . . . . . . . . 4.3. 4. 3.33 Ot Othe herr Me Meas asur ures es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.44 Th 4. Thee HJM Analy Analysi siss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 4. 4.1 Bon Bond d Price Price Dynam Dynamic icss . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 4. 4.2 For orw ward Rat Ratee Dynam Dynamic icss . . . . . . . . . . . . . . . . . . . . . . . 4.4. 4. 4.33 Sh Shor ortt Rate Rate Proces Processs . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.55 Ex 4. Exam ampl ples es of HJM Mod Model elss . . . . . . . . . . . . . . . . . . . . . . . . . . . .
165 165 165 165 1677 16 168 169 170 172 1733 17 174 1755 17 1766 17 1766 17 1777 17 1788 17 179 179 180 1811 18 1822 18
Contents
4.5.11 4.5. 4.5.2 4. 5.2 4.5.3 4. 5.3 5
XV
The Gauss The Gaussia ian n Model Model . . . . . . . . . . . . . . . . . . . . . . . . . . Gaus Ga ussi sian an HJM Mod Models els wi with th Ma Mark rkov ovia ian n Sh Short ort Ra Rate te Log-N Lo g-Norm ormal al HJM HJM Models Models . . . . . . . . . . . . . . . . . . . . . .
182 182 185 187
Fixe Fi xed d In Inco come me In Inst stru rume men nts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
189 189 189 192 193 194 195 198 199 1999 19 201 201 2033 20 204 205 206 2077 20 208 2088 20 2099 20 2100 21 211 211 213 2144 21 214 214 215 215 2166 21 2166 21 2177 21 2177 21 2177 21 218 218 219 2200 22
5.1 5.1 5.22 5. 5.33 5. 5.44 5. 5.5 5.66 5. 5.77 5. 5.88 5. 5.99 5. 5.10 5. 10 5.11 5.12 5.13
5.14 5. 14
5.15
5.16
5.A
Fixed Fixe d Incom Incomee Market Marketss and Part Partici icipa pan nts . . . . . . . . . . . . . . . . Cert Ce rtifi ificat cates es of Deposi Depositt and Libor Libor Rates Rates . . . . . . . . . . . . . . . . . Forw orward ard Rate Rate Agreem Agreemen ents ts (FRA) (FRA) . . . . . . . . . . . . . . . . . . . . . . Eurod Eu rodoll ollar ar Fut Futur ures es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FixedFix ed-for for-Fl -Float oating ing Swaps Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Libor Li bor-in -in-A -Arre rrears ars Sw Swap apss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Avera Av eragin gingg Swaps Swaps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Caps Ca ps and and Floors Floors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Digit Di gital al Caps Caps an and d Fl Floors oors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Europ Eu ropean ean Swap Swapti tion onss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10. 5. 10.11 Ca Cash sh-S -Set ettl tled ed Swap Swapti tion onss . . . . . . . . . . . . . . . . . . . . . . . CMS Swaps Swaps,, Cap Capss and Floors Floors . . . . . . . . . . . . . . . . . . . . . . . . . Bermu Ber mudan dan Swapt Swaption ionss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exotic Exot ic Swaps Swaps and Str Struct ucture ured d Not Notes es . . . . . . . . . . . . . . . . . . . . 5.13. 5. 13.11 Li Libor bor-Ba -Based sed Exot Exotic ic Swap Swapss . . . . . . . . . . . . . . . . . . . . . 5.13. 5. 13.22 CM CMSS-Bas Based ed Exoti Exoticc Swaps Swaps . . . . . . . . . . . . . . . . . . . . . 5.13. 5. 13.33 Mu Mult ltii-Ra Rate te Exotic Exotic Swap Swapss . . . . . . . . . . . . . . . . . . . . . . 5.13. 5. 13.44 Ra Rang ngee Ac Accru cruals als . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.13. 5. 13.55 Pa Path th-D -Depe epend nden entt Sw Swap apss . . . . . . . . . . . . . . . . . . . . . . . . Calla Ca llabl blee Li Libor bor Exotics Exotics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14. 5. 14.11 De Defin finit ition ionss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14. 5. 14.22 Pri Pricin cingg Callab Callable le Libor Libor Exoti Exotics cs . . . . . . . . . . . . . . . . . . 5.14. 5. 14.33 Type ypess of Callab Callable le Libor Libor Exot Exotic icss . . . . . . . . . . . . . . . . 5.14. 5. 14.44 Ca Call llab able le Snowb Snowbal alls ls . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.14. 5. 14.55 CL CLEs Es Accre Accreti ting ng at Coupo Coupon n Rate Rate . . . . . . . . . . . . . . . 5.14. 5. 14.66 Mu Mult ltii-T Tran ranch ches es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TARN ARNss and Other Other Trad rade-L e-Lev evel el Feat Features ures . . . . . . . . . . . . . . . . 5.15. 5. 15.11 Kn Knoc ock-o k-out ut Swap Swapss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15. 5. 15.22 TAR ARNs Ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15. 5. 15.33 Gl Glob obal al Cap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15. 5. 15.44 Gl Glob obal al Floor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.15. 5. 15.55 Pri Pricin cingg and and Trad Tradee Repr Represe esen nta tati tion on Ch Chall allen enge gess . . . . Volat olatilit ility y Der Deriv ivati ative vess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16.1 5.16 .1 Volat olatilit ility y Sw Swaps aps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.16.2 5.16 .2 Volat olatilit ility y Swaps Swaps with a Sho Shout ut . . . . . . . . . . . . . . . . . . 5.16. 5. 16.33 Mi Minn-Ma Max x Vol olat atil ilit ity y Sw Swap apss . . . . . . . . . . . . . . . . . . . . . 5.16.4 Forwar orward d Starting Starting Options Options and Other Forw Forward ard Volat olatili ility ty Contra Contracts cts . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendi Appe ndix: x: Day Day Countin Countingg Rules Rules and Other Other Trivi Triviaa . . . . . . . . 5.A. 5. A.11 Li Libor bor Rate Rate Defini Definiti tion onss . . . . . . . . . . . . . . . . . . . . . . . . 5.A. 5. A.22 Sw Swap ap Paym Paymen ents ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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XVI
Contents
Part II Vanill anilla a Models 6
Yield Yie ld Cur Curv ve Const Constru ructi ction on and Risk Man Manage agemen mentt . . . . . .
6.11 Nota 6. Notati tion onss and Probl Problem em Defin Definit ition ion . . . . . . . . . . . . . . . . . . . . . 6.1.1 6. 1.1 Di Disco scoun untt Curves Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 6. 1.2 Ma Matr trix ix Form Formul ulat atio ion n .. .. .. . . .. .. . . . . . .. .. .. .. .. 6.1. 6. 1.33 Co Cons nstr truc ucti tion on Pr Prin inci cipl ples es and and Yi Yiel eld d Cur Curv ves . . . . . . . . 6.2 Yie Yield ld Cur Curve ve Fit Fittin tingg with with N -Kn -Knot ot Splin Splines es . . . . . . . . . . . . . . . 6.2.1 C 0 Yie Yield ld Cu Curv rves: es: Bootstrap Bootstrappin pingg . . . . . . . . . . . . . . . . 6.2. 6. 2.1. 1.11 Piec Pi ecew ewis isee Lin Linea earr Yie Yield ldss . . . . . . . . . . . . . . 6.2. 6. 2.1. 1.22 Piec Pi ecew ewis isee Fla Flatt For Forw war ard d Rat Rates es . . . . . . . . . 6.2.2 C 1 Yie Yield ld Cu Curv rves: es: Hermite Hermite Splines Splines . . . . . . . . . . . . . . . 6.2.3 C 2 Yield Curves: Twice Differentiable Cubic Spline Spl iness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 C 2 Yield Curves: Twice Differentiable Tension Spline Spl iness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.33 No 6. Nonn-Pa Para rame metr tric ic Opt Optim imal al Yield Yield Cur Curv ve Fitti Fitting ng . . . . . . . . . . . 6.3.1 6. 3.1 No Norm rm Spec Specifi ifica cati tion on and and Opt Optim imiz izat atio ion n ........... 6.3. 3.22 Choice of λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3. 6. 3.33 Ex Exam ampl plee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.44 Ma 6. Mana nagin gingg Yield Yield Curve Curve Ri Risk sk . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4. 6. 4.11 Par ar-P -Poi oin nt Appr Approa oacch . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 6. 4.2 For orw ward Rat Ratee Approac Approach h . ... .. .. .. .. . .. .. . . .. .. 6.4. 6. 4.33 Fro rom m Ri Risk skss to to He Hedg dgin ing: g: Th Thee Jac Jacob obia ian n Ap Appr proa oacch . . 6.4.4 6. 4.4 Cu Cum mul ulat ativ ivee Shi Shift ftss and and othe otherr Comm Common on Tric Tricks ks . . . . . 6.5 Vario arious us Topi Topics cs in Discoun Discountt Curve Curve Constru Constructi ction on.. . . . . . . . . . 6.5.1 6. 5.1 Cu Curv rvee Overl Overlay ayss and and Tur Turnn-of of-Y -Year ear Effe Effect ctss . . . . . . . . 6.5.2 6. 5.2 Cr Cross oss-C -Cur urren rency cy Cur Curv ve Const Constru ruct ction ion . . . . . . . . . . . . . 6.5. 6. 5.2. 2.11 Basi Ba sicc Prob Proble lem m ...................... 6.5.2. 6.5 .2.22 Sepa Se parat ration ion of Di Disc scou oun nt an and d Forw orward ard Rate Ra te Curves Curves . . . . . . . . . . . . . . . . . . . . . . . . 6.5. 6. 5.2. 2.33 Cros Cr osss-Cu Curr rren ency cy Ba Basi siss Sw Swap apss . . . . . . . . . . 6.5. 6. 5.2. 2.44 Modifi Mod ified ed Cu Curv rvee Co Cons nstr truc ucti tion on Al Algo gori rith thm m 6.5.3 6. 5.3 Ten enor or Basis Basis and and Multi Multi-I -Ind ndex ex Curv Curvee Group Group Constr Con struct uction ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.A 6. A Ap Appen pendi dix: x: Spline Spline Theory Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.A. 6. A.11 He Herm rmit itee Sp Spli line ne Theory Theory . . . . . . . . . . . . . . . . . . . . . . . . 6.A.2 C 2 Cu Cubi bicc Sp Spli line ness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.A.3 C 2 Expon Exponen entia tiall Tens ension ion Spline Spliness . . . . . . . . . . . . . . . . 7
Vani anilla lla Model Modelss wit with h Local Local Vola olatil tilit ity y . . . . . . . . . . . . . . . . . . .
7.11 Gene 7. General ral Fram Framew ewor ork k .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 7.1. 7. 1.11 Mod Model el Dyna Dynami mics cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 7. 1.2 Vola olati tili litty Smile Smile and and Impl Implie ied d Densi Densitty . . . . . . . . . . . .
227 228 2288 22 2300 23 2300 23 2322 23 232 2333 23 2344 23 236 238 241 243 243 246 2477 24 248 2499 24 250 2522 25 2544 25 256 256 2577 25 2577 25 258 258 2600 26 2611 26 263 268 268 271 272 275 275 276 2766 27 2766 27
Contents
7.1. 1.33 Choice of ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CEV CE V Mod Model el . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. 7. 2.11 Ba Basi sicc Proper Properti ties es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 7. 2.2 Ca Call ll Option Option Pric Pricin ingg . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 7. 2.3 Re Regu gula lariz rizat atio ion n .. .. .. .. .. .. .. .. .. ... .. .. .. .. .. 7.2.4 7. 2.4 Di Disp spla laced ced Diff Diffus usion ion Model Modelss . . . . . . . . . . . . . . . . . . . . Quad Qu adrat ratic ic Vola Volati tilit lity y Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 7. 3.1 Ca Case se 1: Tw Twoo Rea Reall Root Rootss to to the the Le Left ft of S (0) . . . . . . 7.3. 7. 3.22 Ca Case se 2: One One Rea Reall Root Root to the the Lef Leftt of S (0) . . . . . . . 7.3. 7. 3.33 Ex Exte tens nsio ions ns an and d Oth Other er Roo Roott Con Config figur urat atio ions. ns. . . . . . . Finite Fin ite Differ Differenc encee Solutio Solutions ns for for General General ϕ . . . . . . . . . . . . . . . 7.4. 7. 4.11 Mu Mult ltip iple le λ and T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 7. 4.2 Forw orward ard Equ Equat ation ion fo forr Call Call Opti Option onss . . . . . . . . . . . . . Asympt Asy mptoti oticc Expans Expansion ionss for for Gener General al ϕ . . . . . . . . . . . . . . . . . . 7.5.1 7. 5.1 Exp Expan ansi sion on aro aroun und d Di Disp splac laced ed Lo Log-N g-Norm ormal al Proc Proces esss . 7.5. 7. 5.22 Ex Expa pans nsio ion n arou around nd Gau Gauss ssia ian n Proce Process ss . . . . . . . . . . . . Extens Ext ension ionss to to TimeTime-Depe Depend nden entt ϕ . . . . . . . . . . . . . . . . . . . . . . 7.6.1 7. 6.1 Se Sepa parab rable le Case Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 7. 6.2 Sk Skew ew Aver Averagi aging ng . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6. 7. 6.2. 2.11 Exam Ex ampl ples es . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6. 7. 6.2. 2.22 A Cavea eatt Ab Abou outt th thee Pr Proc oces esss Dom omai ain n .. 7.6.33 Sk 7.6. Skew ew and and Con Conve vexit xity y Ave Averagi raging ng by by SmallSmall-Nois Noisee Expans Exp ansion ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.4 7. 6.4 Nu Nume meri rical cal Examp Example le . . . . . . . . . . . . . . . . . . . . . . . . . . .
277 2788 27 2788 27 2800 28 2822 28 2833 28 285 2855 28 2899 28 2899 28 2900 29 291 291 2933 29 293 2966 29 2977 29 2977 29 298 3022 30 3044 30
Vani anilla lla Models Models wit with h Stochas Stochastic tic Vola olatil tilit ity y I. . . . . . . . . . . . .
313 313 313 315 316 322 3222 32 3255 32 328 330 334 337 3411 34 343 345 350 350 353 354 356
7.22 7.
7.33 7.
7.4
7.5
7.6
8
XVII XV II
8.1 8.1 8.22 8. 8.33 8. 8.44 8.
8.5 8.5 8.66 8. 8.7 8.8 8.99 8.
8.A
Model Defini Model Definiti tion on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Model Mod el Pa Param ramet eters ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Ba sic Proper Properti ties es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fou ourie rierr In Inte tegra grati tion on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4. 8. 4.11 Ge Gene nera rall Theor Theory y . .. .. .. .. .. .. .. .. ... .. .. .. .. .. 8.4. 8. 4.22 Ap Appl plic icat atio ions ns to SV SV Model Model . . . . . . . . . . . . . . . . . . . . . 8.4.3 8. 4.3 Nu Nume meri rical cal Impl Implem emen enta tati tion on . . . . . . . . . . . . . . . . . . . . . 8.4.4 8. 4.4 Re Refin finem emen ents ts of of Nume Numeri rical cal Im Impl plem emen enta tati tion on . . . . . . . 8.4.55 Fouri 8.4. ourier er In Integ tegrat ration ion for Arb Arbitr itrary ary Eur European opean Pa Payo yoffs ffs Inte In tegra grati tion on in Vari Varian ance ce Domain Domain . . . . . . . . . . . . . . . . . . . . . . . CEVCE V-T Type Stoc Stocha hast stic ic Vola Volati tili litty Models Models and and SAB SABR R ...... Numeric Num erical al Example Examples: s: Volat Volatili ility ty Smil Smilee Statics Statics . . . . . . . . . . . . Numeric Num erical al Example Examples: s: Volat Volatili ility ty Smil Smilee Dynamic Dynamicss . . . . . . . . . Hedg He dgin ingg in Stocha Stochast stic ic Vol Volat atil ilit ity y Models Models . . . . . . . . . . . . . . . . 8.9.1 8. 9.1 He Hedg dgee Const Constru ruct ction ion,, Delta Delta and and Veg Vegaa . . . . . . . . . . . . 8.9.2 8. 9.2 Mi Mini nim mum Vari Varian ance ce Delt Deltaa Hedgi Hedging ng . . . . . . . . . . . . . . 8.9.3 8. 9.3 Mi Mini nim mum Vari Varian ance ce Hed Hedgin ging: g: an an Examp Example le . . . . . . . . Appendi Appe ndix: x: General General Volat Volatilit ility y Processes . . . . . . . . . . . . . . . . .
305 309
XVIII XV III
9
Con onte ten nts
Vani anilla lla Models Models wit with h Stochas Stochastic tic Vola olatil tilit ity y II . . . . . . . . . . . .
9.1 Fouri ourier er Int Integra egratio tion n with with Time Time-De -Depend penden entt Param Paramete eters rs . . . . 9.2 Asy Asympt mptoti oticc Exp Expansi ansion on wit with h Tim Time-D e-Depen ependen dentt Volat olatili ility ty . . 9.33 Av 9. Avera eragin gingg Methods Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 9. 3.1 Vola olati tili litty Avera Averagin gingg . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.2 9. 3.2 Sk Skew ew Avera Averagin gingg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.3 9. 3.3 Vola olati tili litty of Vari Varian ance ce Avera Averagin gingg . . . . . . . . . . . . . . . . 9.3.4 9. 3.4 Ca Cali libr brat atio ion n by Para Parame mete terr Avera Averagin gingg . . . . . . . . . . . . 9.44 PD 9. PDE E Me Meth thod od . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4. 9. 4.11 PD PDE E For orm mul ulat atio ion n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.2 9. 4.2 Ra Rang ngee for Stoch Stochast astic ic Vari Varian ance ce . . . . . . . . . . . . . . . . . . 9.4.3 9. 4.3 Di Discr scret etiz izin ingg Stocha Stochast stic ic Vari Varian ance ce . . . . . . . . . . . . . . . . 9.4.4 9. 4.4 Bou Bound ndary ary Co Cond ndit ition ionss for for St Stoch ochast astic ic Vari arian ance ce.. . . . . 9.4. 9. 4.55 Ra Rang ngee for Unde Underl rlyi ying ng . . . . . . . . . . . . . . . . . . . . . . . . . 9.4.6 9. 4.6 Di Discr scret etiz izin ingg the Unde Underly rlyin ingg . . . . . . . . . . . . . . . . . . . . 9.55 Mo 9. Mon nte Carlo Carlo Method Method.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 9. 5.1 Exa Exact ct Sim Simul ulat ation ion of Vari Varian ance ce Proce Process ss . . . . . . . . . . . 9.5.2 9. 5.2 Bi Biase ased d Tay aylo lor-T r-Type ype Sc Sche heme mess fo forr Var arian iance ce Proc Process ess 9.5. 9. 5.2. 2.11 Eule Eu lerr Sche Scheme mess . . . . . . . . . . . . . . . . . . . . . . 9.5. 9. 5.2. 2.22 High Hi gher er-O -Ord rder er Sc Sche heme mess . . . . . . . . . . . . . . . 9.5. 9. 5.33 Mo Mome men nt Ma Matc tchi hing ng Sc Sche heme mess fo forr Var aria ianc ncee Pr Proce ocess. ss. . 9.5. 9. 5.3. 3.11 LogLo g-no norm rmal al Ap Appr prooxi xima mati tion on . . . . . . . . . . . 9.5. 9. 5.3. 3.22 Tru runc ncat ated ed Gau Gauss ssia ian n ................. 9.5. 9. 5.3. 3.33 Quad Qu adra rati ticc-Ex Expo pone nen nti tial al . . . . . . . . . . . . . . . 9.5. 9. 5.3. 3.44 Summ Su mmar ary y of of QE QE Alg Algor orit ithm hm . . . . . . . . . . . 9.5. 9. 5.44 Br Broa oadi diee-Ka Kay ya Sc Sche heme me fo forr the the Un Unde derl rlyi ying ng . . . . . . . 9.5.5 9. 5.5 Ot Othe herr Sche Scheme mess for the the Underl Underlyi ying ng . . . . . . . . . . . . . . 9.5. 9. 5.5. 5.11 Tayl ylor or-T -Typ ypee Sche Scheme mess . . . . . . . . . . . . . . . . 9.5. 9. 5.5. 5.22 Simp Si mpli lifie fied d Broa Broadi diee-Ka Kay ya . . . . . . . . . . . . . . 9.5. 9. 5.5. 5.33 Mart Ma rtin inga gale le Cor Corre rect ctio ion n ............... 9.A 9. A Ap Appe pend ndix ix:: Proof Proof of Propo Proposi siti tion on 9.3. 9.3.44 . . . . . . . . . . . . . . . . . . . 9.B 9. B Ap Appen pendi dix: x: Coeffic Coefficie ien nts for for Asymp Asympto toti ticc Expan Expansi sion on . . . . . . . .
359 359 359 362 366 3677 36 369 3700 37 372 3777 37 3777 37 3788 37 379 3811 38 3822 38 3833 38 383 3844 38 385 3855 38 3855 38 3866 38 3866 38 3877 38 3888 38 3900 39 3900 39 392 3922 39 3922 39 3922 39 3933 39 3977 39
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VOLUME II Term Structure Models Part III II I Term Structure Structure Models 10 One One-F -Fact actor or Short Short Rate Rate Models I . . . . . . . . . . . . . . . . . . . . . . .
10.1 The The One-Fact One-Factor or Gaussian Gaussian Short Short Rate Rate Model . . . . . . . . . . . . 10.1. 10 .1.11 Th Thee Ho Ho-L -Lee ee Model Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1 10 .1.1 .1.1 .1 No Nota tati tion onss and and Firs Firstt Step Stepss . . . . . . . . . . . . 10.1. 10. 1.1.2 1.2 Fi Fitt ttin ingg the Term Term Str Struc uctu ture re of Disc Di scou oun nt Bo Bond ndss . . . . . . . . . . . . . . . . . . . . . 10.1.1.3 10.1 .1.3 Ana Analysi lysiss and and Com Compar parison ison wit with h HJM HJM Appr Ap proac oach h. . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. 10 .1.22 Th Thee Mean-R Mean-Rev evert ertin ingg GSR Model Model . . . . . . . . . . . . . . . 10.1 10 .1.2 .2.1 .1 Th Thee Vas Vasic icek ek Model Model . . . . . . . . . . . . . . . . . . 10.1 10 .1.2 .2.2 .2 Th Thee Gen Gener eral al On Onee-F Fac acto torr GS GSR R Mod Model el . . . 10.1. 10. 1.2.3 2.3 Ti Time me-S -Sta tati tion onari aritty an and d Ca Capl plet et Hu Hump mp . . . 10.1. 10 .1.33 Eu Europ ropea ean n Option Option Pricin Pricingg . . . . . . . . . . . . . . . . . . . . . . 10.1. 10. 1.3.1 3.1 Th Thee Jams Jamshi hidi dian an De Decom compos posit ition ion . . . . . . . 10.1. 10. 1.3.2 3.2 Ga Gaus ussia sian n Sw Swap ap Ra Rate te Ap Appr prox oxim imat atio ion n ... 10.1. 10 .1.44 Sw Swap apti tion on Calib Calibrat ratio ion n .. .. .. .. .. .. .. ... .. .. .. .. 10.1. 10 .1.55 Fi Fini nite te Differ Differen ence ce Method Methodss . . . . . . . . . . . . . . . . . . . . . 10.1 10 .1.5 .5.1 .1 PD PDE E an and d Sp Spat atia iall Bo Boun unda dary ry Co Cond ndit itio ions ns . 10.1.5.2 10.1 .5.2 Det Determ ermini ining ng Spa Spatia tiall Bou Bounda ndary ry Cond Co ndit ition ionss fr from om PDE . . . . . . . . . . . . . . . . 10.1 10 .1.5 .5.3 .3 Up Upwi wind ndin ingg . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. 10 .1.66 Mo Mon nte Carlo Carlo Si Sim mul ulat ation ion . . . . . . . . . . . . . . . . . . . . . . . 10.1. 10. 1.6.1 6.1 Ex Exact act Disc Discret retiza izati tion on . . . . . . . . . . . . . . . . . 10.1. 10. 1.6.2 6.2 Ap Appr prox oxim imat atee Discr Discret etiz izat ation ion . . . . . . . . . . . 10.1. 10. 1.6.3 6.3 Us Usin ingg oth other er Me Measu asures res fo forr Sim Simul ulat atio ion n... 10.2 Th Thee Affin Affinee One One-F -Fact actor or Model . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. 10 .2.11 Ba Basic sic Defini Definiti tion onss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 10 .2.1 .1.1 .1 SD SDE E . .. ... .. .. .. .. .. .. ... .. .. .. .. .. 10.2. 10. 2.1.2 1.2 Re Regu gular larit ity y Issue Issuess . . . . . . . . . . . . . . . . . . . . 10.2. 10. 2.1.3 1.3 Vol olat atil ilit ity y Sk Skew ew . . . . . . . . . . . . . . . . . . . . . . 10.2. 10. 2.1.4 1.4 Ti Time me-D -Depe epend nden entt Param Paramet eters ers . . . . . . . . . . 10.2. 10 .2.22 Di Disco scoun untt Bon Bond d Pri Prici cing ng an and d Ext Exten ende ded d Tran ransf sform orm . . 10.2. 10. 2.2.1 2.1 Co Cons nsta tan nt Param Paramet eter erss . . . . . . . . . . . . . . . . 10.2. 10. 2.2.2 2.2 Pie Piece cewi wise se Co Cons nsta tan nt Para Parame mete ters rs . . . . . . . 10.2. 10 .2.33 Di Disco scoun untt Bond Calib Calibrat ration. ion. . . . . . . . . . . . . . . . . . . . . 10.2 10 .2.3 .3.1 .1 Ch Chan ange ge of Var Varia iabl bles es . . . . . . . . . . . . . . . . . 10.2. 10. 2.3.2 3.2 Al Algo gorit rithm hm fo forr ω (t) . . . . . . . . . . . . . . . . . . . 10.2. 10 .2.44 Eu Europ ropea ean n Option Option Pricin Pricingg . . . . . . . . . . . . . . . . . . . . . .
401 401 402 4022 40 4022 40 403 405 407 4077 40 4099 40 412 4144 41 414 416 4177 41 418 4199 41 420 420 4211 42 4211 42 421 423 424 425 4255 42 4255 42 426 4266 42 4277 42 427 428 430 4311 43 4311 43 4322 43 4333 43
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10.2. 10. 2.55 Swap Swapti tion on Calib Calibrat ration ion . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 10 .2.5 .5.1 .1 Ba Basi sicc Probl Problem em . . . . . . . . . . . . . . . . . . . . . . 10.2.5 10. 2.5.2 .2 Ca Cali libr brat ation ion Algo Algorit rithm hm . . . . . . . . . . . . . . . 10.2.6 10.2 .6 Qua Quadrat dratic ic One-Fact One-Factor or Model . . . . . . . . . . . . . . . . . . . 10.2.7 10.2 .7 Num Numeric erical al Methods Methods for the Affine Affine Short Short Rate Rate Model Mod el . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
435 435 4355 43 436 437
11 One One-F -Fac actor tor Short Short Rate Rate Models Models II . . . . . . . . . . . . . . . . . . . . . .
439 439 439 439 4411 44 4411 44 442 445
437
11.11 Log-N 11. Log-Norm ormal al Short Short Rate Rate Models Models . . . . . . . . . . . . . . . . . . . . . . . 11.1. 11. 1.11 Th Thee BlackBlack-De Derm rman an-T -Toy oy Mode Modell . . . . . . . . . . . . . . . . . 11.1. 11. 1.22 Bl Blac ackk-Kar Karasi asins nski ki Model Model . . . . . . . . . . . . . . . . . . . . . . . 11.1. 11. 1.33 Is Issu sues es in Log-No Log-Norm rmal al Model Modelss . . . . . . . . . . . . . . . . . . . 11.1.4 11.1 .4 San Sandma dmannnn-Son Sonder derman mann n Tran Transfor sformat mation ion . . . . . . . . . 11.2 Oth Other er Sh Short ort Rate Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 11.2 .1 Po Powe wer-T r-Type ype Models Models and Empirical Empirical Model Estima Est imatio tion n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445 11.2. 11. 2.22 Th Thee Black Black Shado Shadow w Rate Rate Mod Model el . . . . . . . . . . . . . . . . . 44 4466 11.2.3 Span Spanned ned and Unspanne Unspanned d Stochastic Stochastic Volatili Volatility ty:: thee Fong and Vasic th Vasicek ek Model . . . . . . . . . . . . . . . . . . . 448 11.3 Numerical Methods for General One-Factor Short Rate Models Mode ls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 11.3. 11. 3.11 Fi Fini nite te Differe Differenc ncee Me Meth thods ods . . . . . . . . . . . . . . . . . . . . . 45 4500 11.3. 11. 3.22 Ca Cali libr brat atio ion n to Init Initia iall Yield Yield Curve Curve . . . . . . . . . . . . . . . 451 11.3.2 11. 3.2.1 .1 Forw orward ard Ind Induc ucti tion on . . . . . . . . . . . . . . . . . . . 452 11.3.2 11. 3.2.2 .2 Forw orward ard-f -from rom-B -Bac ackw kward ard In Indu duct ctio ion n . . . . . 453 453 11.3.2 11. 3.2.3 .3 Yi Yiel eld d Cu Curv rvee an and d Vola olati tili litty Ca Cali libr brat atio ion n . 455 11.3.2 11. 3.2.4 .4 Th Thee Dyb Dybvi vigg Param Paramet eteri erizat zation ion . . . . . . . . . 457 11.3 11 .3.2 .2.5 .5 Li Link nk to to HJM Mode Models ls.. . . . . . . . . . . . . . . . . 45 4588 11.3.2 11. 3.2.6 .6 Th Thee Hagan Hagan an and d Woodw Woodward ard Parame Pa rameter terizat ization ion . . . . . . . . . . . . . . . . . . . . 459 11.3. 11. 3.33 Mo Mon nte Carlo Carlo Simul Simulat atio ion n . . . . . . . . . . . . . . . . . . . . . . . 46 4622 11.3.3 11. 3.3.1 .1 SD SDE E Discret Discretiza izati tion on . . . . . . . . . . . . . . . . . . 462 11.3.3 11. 3.3.2 .2 Pra Pract ctic ical al Issu Issues es with with Mon Monte Carlo Carlo Meth Me thods ods . . . . . . . . . . . . . . . . . . . . . . . . . . . 464 11.A Appe Appendi ndix: x: Mark Markov ov-F -Func unctio tional nal Models Models . . . . . . . . . . . . . . . . . . 466 11.A. 11. A.11 St Stat atee Process Process and and Numera Numeraire ire Mapp Mappin ingg . . . . . . . . . . 46 4666 11.A.2 11.A .2 Libo Liborr MF Paramete Parameteriza rizatio tion n . . . . . . . . . . . . . . . . . . . . 467 11.A.3 11.A .3 Sw Swap ap MF Par Parame ameter terizat ization ion . . . . . . . . . . . . . . . . . . . . 469 11.A. 11. A.44 No Nonn-Pa Param ramet etri ricc Calibr Calibrat atio ion n . . . . . . . . . . . . . . . . . . . . 470 11.A.5 11.A .5 Num Numeric erical al Implemen Implementat tation ion . . . . . . . . . . . . . . . . . . . . . 471 11.A.6 11.A .6 Com Commen ments ts and Compariso Comparisons ns . . . . . . . . . . . . . . . . . . . 472
Contents
12 Mul Multiti-F Fact actor or Short Short Rate Models . . . . . . . . . . . . . . . . . . . . . . .
12.1 The The Gau Gaussia ssian n Mode Modell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1.1 12.1 .1 Dev Develop elopmen mentt from Sep Separab arabili ility ty Con Condit dition ion.. . . . . . . . 12.1. 12. 1.1.1 1.1 Me Mean an-R -Rev evert ertin ingg Sta State te Vari Variab able less . . . . . . . 12.1. 12. 1.1.2 1.2 Fur urth ther er Chan Changes ges of Vari Variab able less . . . . . . . . . 12.1. 12 .1.22 Cl Class assic ical al Devel Develop opme men nt . . . . . . . . . . . . . . . . . . . . . . . . 12.1.2.1 12.1 .2.1 Dia Diagona gonaliza lizatio tion n of Mean Rev Reversi ersion on Matrix Mat rix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. 12 .1.33 Co Corre rrela lati tion on Stru Struct ctur uree . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. 12 .1.44 Th Thee Two-F Two-Fact actor or Gaus Gaussia sian n Model Model . . . . . . . . . . . . . . . 12.1 12 .1.4 .4.1 .1 So Some me Basi Basics cs . . . . . . . . . . . . . . . . . . . . . . . . 12.1. 12. 1.4.2 4.2 Vari arian ance ce an and d Corr Correla elati tion on St Stru ruct ctur uree . . . . 12.1. 12. 1.4.3 4.3 Vol olat atil ilit ity y Hu Hump mp . . . . . . . . . . . . . . . . . . . . . 12.1. 12. 1.4.4 4.4 An Anot othe herr For Form mul ulat atio ion n of the Two-F Tw o-Fact actor or Model . . . . . . . . . . . . . . . . . . . 12.1.5 12.1 .5 Mul Multiti-F Fact actor or Stati Statisti stical cal Gaussia Gaussian n Model Model . . . . . . . . . 12.1. 12 .1.66 Sw Swap apti tion on Pricin Pricingg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.1. 12. 1.6.1 6.1 Jam Jamsh shid idian ian De Decom compos posit ition ion . . . . . . . . . . . 12.1. 12. 1.6.2 6.2 Ga Gaus ussia sian n Sw Swap ap Ra Rate te Ap Appr prox oxim imat atio ion n ... 12.1. 12 .1.77 Ca Cali libr brat ation ion via via Benchm Benchmar ark k Rates Rates . . . . . . . . . . . . . . . 12.1. 12 .1.88 Mo Mon nte Carlo Carlo Si Sim mul ulat ation ion . . . . . . . . . . . . . . . . . . . . . . . 12.1. 12 .1.99 Fi Fini nite te Differ Differen ence ce Method Methodss . . . . . . . . . . . . . . . . . . . . . 12.2 12 .2 Th Thee Affi Affine ne Mod Model el . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. 12 .2.11 In Intr trodu oduct ction ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. 12 .2.22 Ba Basic sic Model Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. 12 .2.33 Re Regu gula larit rity y Issues Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. 12 .2.44 Di Disco scoun untt Bon Bond d Prices Prices . . . . . . . . . . . . . . . . . . . . . . . . . 12.2. 12 .2.55 So Some me Concre Concrete te Models Models . . . . . . . . . . . . . . . . . . . . . . . . 12.2. 12. 2.5.1 5.1 Fon ong-V g-Vasi asice cek k Model Model . . . . . . . . . . . . . . . . . 12.2. 12. 2.5.2 5.2 Lo Long ngst staff aff-S -Sch chw war artz tz Mode Modell . . . . . . . . . . . . 12.2. 12. 2.5.3 5.3 Mu Mult lti-F i-Fact actor or CIR CIR Models Models . . . . . . . . . . . . . 12.2. 12 .2.66 Br Brief ief Note Notess on Option Option Pricin Pricingg . . . . . . . . . . . . . . . . . . 12.3 12 .3 Th Thee Qu Quad adrat ratic ic Gaussi Gaussian an Model Model . . . . . . . . . . . . . . . . . . . . . . . 12.3. 12 .3.11 Qu Quad adra rati ticc Gauss Gaussian ian Mode Models ls are Affin Affinee . . . . . . . . . . . 12.3. 12 .3.22 Th Thee Bas Basics. ics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3. 12 .3.33 Pa Para rame mete teri rizat zation ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3. 12. 3.3.1 3.1 Sm Smile ile Gene Generat ratio ion n ... .. .. ... .. . . .. .. . . 12.3. 12. 3.3.2 3.2 Qu Quad adrat ratic ic Ter Term m .. .. .. .. ... .. . . .. .. . . 12.3. 12. 3.3.3 3.3 Li Line near ar Term Term . . . . . . . . . . . . . . . . . . . . . . . . 12.3. 12 .3.44 Sw Swap apti tion on Pricin Pricingg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.3.4.1 12.3 .4.1 St State ate Vect Vector or Dist Distrib ributi ution on Unde Underr the the Annuit Ann uity y Meas Measure ure . . . . . . . . . . . . . . . . . . . . 12.3. 12. 3.4.2 4.2 Ex Exact act Pri Prici cing ng of Eu Europ ropean ean Sw Swap apti tion onss . . 12.3. 12. 3.4.3 4.3 Ap Appr prox oxim imat atio ions ns fo forr Eu Europ ropean ean Sw Swap apti tion onss 12.3. 12 .3.55 Ca Cali libr brat ation ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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473 473 474 474 475 479 4811 48 482 484 485 4855 48 486 487 488 491 4966 49 496 500 5011 50 5044 50 505 506 506 5077 50 508 5099 50 511 511 512 513 5144 51 514 515 5166 51 518 5188 51 5199 51 521 5222 52 522 5233 52 5244 52 527
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Contents
12.3.6 Spa 12.3.6 Spanne nned d Stochast Stochastic ic Volat Volatilit ility y . .. .. .. . .. .. . . .. .. 12.3. 12. 3.77 Nu Nume meric rical al Method Methodss . . . . . . . . . . . . . . . . . . . . . . . . . . . 12.A Appe Appendi ndix: x: Quadrat Quadratic ic Forms Forms of Gauss Gaussian ian Vect Vectors ors . . . . . . . .
528 5288 52 528
13 Th The e Quasi-G Quasi-Gaus aussia sian n Model Model . . . . . . . . . . . . . . . . . . . . . . . . . . . .
533 533 5333 53 535 536
13.1 One-F One-Fact actor or Quasi-Gau Quasi-Gaussia ssian n Model . . . . . . . . . . . . . . . . . . . . . 13.1. 13. 1.11 De Defin finit ition ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1. 13. 1.22 Loc Local al Vol Volat atil ilit ity y. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 13.1. 13. 1.33 Sw Swap ap Rate Rate Dy Dyna nami mics cs . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.4 Appro Approximat ximatee Local Volatil Volatility ity Dynam Dynamics ics for Swap Swap Rate Ra te . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.4 13. 1.4.1 .1 Si Simp mple le Appr Approx oxim imat ation ion . . . . . . . . . . . . . . . 13.1.4 13. 1.4.2 .2 Ad Adv van ance ced d Appro Approxi xima mati tion. on. . . . . . . . . . . . . 13.1. 13. 1.55 Li Line near ar Local Vola Volati tili litty . . . . . . . . . . . . . . . . . . . . . . . . 13.1.6 13.1 .6 Lin Linear ear Local Local Volat Volatilit ility y for a Swa Swapt ption ion Stri Strip p ...... 13.1. 13. 1.77 Vola olati tili litty Ca Cali libr brat atio ion n . . .. .. .. .. .. .. .. . .. .. .. .. 13.1. 13. 1.88 Me Mean an Rever Reversio sion n Calib Calibrat ration. ion. . . . . . . . . . . . . . . . . . . . 13.1 13 .1.8 .8.1 .1 Eff Effec ects ts of of Mean Mean Rev Rever ersi sion on . . . . . . . . . . . . 13.1.8 13. 1.8.2 .2 Ca Cali libr brat atin ingg Mean Mean Revers Reversion ion to Volat olatili ility ty Ratios Ratios . . . . . . . . . . . . . . . . . . . . . 13.1.8 13. 1.8.3 .3 Ca Cali libr brat atin ingg Mean Mean Revers Reversion ion to Inter In ter-T -Tempo emporal ral Correla Correlatio tions ns . . . . . . . . . . 13.1.8.4 13.1 .8.4 Fin Final al Comm Commen ents ts on Mean Rev Reversi ersion on Cali Ca libr brat ation ion . . . . . . . . . . . . . . . . . . . . . . . . . 13.1. 13. 1.99 Nu Nume meric rical al Method Methodss . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.1.9 13. 1.9.1 .1 Di Direc rectt Inte Integr grat ation ion . . . . . . . . . . . . . . . . . . . 13.1.9 13. 1.9.2 .2 Fi Fini nite te Diff Differe erenc ncee Method Methodss . . . . . . . . . . . . 13.1.9 13. 1.9.3 .3 Mo Mon nte Carl Carloo Simul Simulat ation ion . . . . . . . . . . . . . . 13.1.9 13. 1.9.4 .4 Si Sing nglele-St Stat atee Appr Approx oxim imat ation ionss . . . . . . . . . . 13.2 One-Factor Quasi-Gaussian Model with Stochastic Volat olatili ility ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2. 13. 2.11 De Defin finit ition ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.2. 13. 2.22 Sw Swap ap Rate Rate Dy Dyna nami mics cs . . . . . . . . . . . . . . . . . . . . . . . . . 13.2. 13. 2.33 Vola olati tili litty Ca Cali libr brat atio ion n . . .. .. .. .. .. .. .. . .. .. .. .. 13.2. 13. 2.44 Me Mean an Rever Reversio sion n Calib Calibrat ration. ion. . . . . . . . . . . . . . . . . . . . 13.2. 13. 2.55 No Nonn-Zer Zeroo Correla Correlati tion on . . . . . . . . . . . . . . . . . . . . . . . . . 13.2. 13. 2.66 PD PDE E and Monte Monte Carlo Carlo Met Method hodss . . . . . . . . . . . . . . . . 13.3 Mul Multiti-F Fact actor or Quas Quasi-G i-Gaus aussian sian Model Model . . . . . . . . . . . . . . . . . . . . 13.3. 13. 3.11 Ge Gene neral ral Mult Multi-F i-Fact actor or Model Model . . . . . . . . . . . . . . . . . . . 13.3.2 13.3 .2 Local and St Stoch ochast astic ic Volat olatili ility ty Par Parame ameter terizat ization ion . 13.3. 13. 3.33 Sw Swap ap Rate Rate Dyna Dynami mics cs and and Appro Approxi xima mati tion onss . . . . . . . 13.3. 13. 3.44 Vola olati tili litty Ca Cali libr brat atio ion n . . .. .. .. .. .. .. .. . .. .. .. .. 13.3.5 Mean Revers Reversions, ions, Correlat Correlations, ions, and Numerical Numerical Schem Sc hemes es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13.A Appe Appendi ndix: x: Density Density App Approx roxima imatio tion n .. .. .. .. .. .. .. .. ... ..
537 538 538 541 543 544 5466 54 5466 54 548 551 553 5544 55 554 556 559 559 563 5633 56 564 566 5677 56 5677 56 568 568 5688 56 570 5722 57 5777 57 578 579
Con onte ten nts
13.A.1 13.A.1 13.A.2 13.A .2 13.A. 13 .A.33 13.A.4 13.A .4 13.A. 13 .A.55
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Simplified Simpli fied Forw Forward ard Measu Measure re Dynami Dynamics cs . . . . . . . . . . Effectiv Effec tivee Volat olatili ility ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . Thee Forw Th Forwar ard d Equat Equation ion for for Call Call Option Optionss . . . . . . . . . Asympt Asy mptoti oticc Exp Expansi ansion on . . . . . . . . . . . . . . . . . . . . . . . . Prooff of Th Proo Theor eorem em 13.1.1 13.1.144 . . . . . . . . . . . . . . . . . . . . . .
579 580 581 582 5833 58
14 Th The e Libor Libor Mark Market et Model Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . .
585 585 586 5866 58 5877 58 587 587 588 5911 59 592 594 5977 59 5977 59 598 5999 59 6000 60 6033 60 604 6055 60 6077 60 6088 60 608 609 6100 61 6133 61 614 6155 61 615 615 6166 61 6177 61 6199 61 6200 62 6200 62 6211 62 6211 62 624 6266 62 627 629
14.1 14 .1 Intr Introdu oduct ction ion and and Setup Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.1. 14 .1.11 Mo Moti tiv vat ation ion and and Histori Historical cal Notes Notes . . . . . . . . . . . . . . . . 14.1. 14 .1.22 Ten enor or Struc Structu ture re . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 LM Dynamics Dynamics and Measures Measures . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2. 14 .2.11 Se Sett ttin ingg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2. 14 .2.22 Pro Proba babi bili litty Measur Measures es . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 14 .2.3 .3 Li Link nk to HJM HJM An Anal alys ysis is . . . . . . . . . . . . . . . . . . . . . . . . . 14.2.4 14.2 .4 Sep Separab arable le Dete Determi rminis nistic tic Volat olatilit ility y Func Functio tion n ...... 14.2. 14 .2.55 St Stoch ochast astic ic Vola Volati tili litty . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2. 14 .2.66 Ti Time me-D -Depe epend nden ence ce in Model Model Param Paramet eter erss . . . . . . . . . 14.3 14 .3 Co Corre rrela lati tion on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3. 14 .3.11 Em Empi piri rical cal Prin Princi cipa pall Compo Compone nen nts Anal Analys ysis is . . . . . . . 14.3 14 .3.1 .1.1 .1 Ex Exam ampl ple: e: USD USD For Forw war ard d Rat Rates es . . . . . . . . 14.3. 14 .3.22 Co Corre rrela lati tion on Esti Estima mati tion on and and Smooth Smoothin ingg . . . . . . . . . . 14.3 14 .3.2 .2.1 .1 Ex Exam ampl ple: e: Fit Fit to to USD USD Data Data . . . . . . . . . . . 14.3. 14 .3.33 Ne Negat gativ ivee Eig Eigen env valu alues es . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3. 14 .3.44 Co Corre rrela lati tion on PCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.3 14 .3.4 .4.1 .1 Ex Exam ampl ple: e: USD USD Data Data . . . . . . . . . . . . . . . . . 14.3 14 .3.4 .4.2 .2 Poor Ma Man’ n’ss Cor Corre rela lati tion on PC PCA A ......... 14.4 14 .4 Pri Prici cing ng of Eur Europe opean an Option Optionss . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4. 14 .4.11 Ca Capl plet etss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4. 14 .4.22 Sw Swap apti tion onss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4. 14 .4.33 Sp Spre read ad Option Optionss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.4. 14. 4.3.1 3.1 Term Corr Correla elati tion on . . . . . . . . . . . . . . . . . . . . 14.4 14 .4.3 .3.2 .2 Sp Spre read ad Opt Optio ion n Pric Pricin ingg . . . . . . . . . . . . . . . 14.5 14 .5 Ca Calib librat ration ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5. 14 .5.11 Ba Basic sic Princ Princip iple less . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.5.2 14.5 .2 Par Parame ameter terizat ization ion of of λk (t) . . . . . . . . . . . . . . . . . . . . 14.5. 14 .5.33 In Inte terpo rpola lati tion on on the the Whole Whole Gr Grid id.. . . . . . . . . . . . . . . . 14.5.4 14.5 .4 Con Constr struct uction ion of λk (t) from λk (t) . . . . . . . . . . . . . . 14.5 14 .5.4 .4.1 .1 Co Cov var aria ianc ncee PCA PCA . . . . . . . . . . . . . . . . . . . . 14.5. 14. 5.4.2 4.2 Co Corre rrela lati tion on PCA PCA . . . . . . . . . . . . . . . . . . . . 14.5. 14. 5.4.3 4.3 Di Disc scus ussio sion n and and Re Recom comme mend ndat ation ion . . . . . . 14.5. 14 .5.55 Ch Choi oice ce of Calibr Calibrat atio ion n Instr Instrum umen ents ts . . . . . . . . . . . . . . 14.5.6 14.5 .6 Cal Calibr ibrati ation on Objective Objective Func Functio tion n .. .. ... .. .. .. .. .. 14.5. 14 .5.77 Sa Samp mple le Calibr Calibrat atio ion n Algori Algorith thm m. . .. . .. . .. .. .. .. .. 14.5. 14 .5.88 Spe Speed ed-U -Up p Throu Through gh Sub Sub-Pr -Prob oble lem m Spli Splitt ttin ingg . . . . . . . 14.5.9 14.5 .9 Cor Correla relatio tion n Calib Calibrat ration ion to Spread Spread Opt Option ionss . . . . . . .
XXIV XX IV
Con onte ten nts
14.5.10 Volat 14.5.10 olatili ility ty Skew Skew Calibrat Calibration ion . . . . . . . . . . . . . . . . . . . . 14.66 Mo 14. Mon nte Carlo Carlo Si Sim mul ulat ation ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6. 14. 6.11 Eu Euler ler-T -Type ype Sche Scheme mess . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6 14 .6.1 .1.1 .1 An Anal alys ysis is of Com Compu puta tati tion onal al Eff Effor ortt . . . . . . 14.6 14 .6.1 .1.2 .2 Lo Long ng Time Time Step Stepss . . . . . . . . . . . . . . . . . . . . 14.6 14 .6.1 .1.3 .3 No Note tess on th thee Cho Choic icee of of Num Numer erai aire re . . . . . 14.6. 14. 6.22 Ot Othe herr Si Sim mul ulat atio ion n Sc Sche heme mess . . . . . . . . . . . . . . . . . . . . . 14.6.2.1 14.6 .2.1 Spec Specialial-Purpo Purpose se Sc Schem hemes es wit with h Drif Driftt Predi Pre dict ctoror-Co Corre rrect ctor or . . . . . . . . . . . . . . . . . . 14.6.2 14. 6.2.2 .2 Eu Eule lerr Sc Sche heme me wi with th Pre Predi dict ctoror-Co Corre rrect ctor or . 14.6.2 14. 6.2.3 .3 La Laggi gging ng Pre Predi dict ctoror-Co Corre rrect ctor or Sc Sche heme me . . . 14.6.2 14. 6.2.4 .4 Fur urth ther er Re Refin finem emen ents ts of Dr Drif iftt Est Estim imat atio ion n 14.6.2.5 14.6 .2.5 Bro Browni wnianan-Brid Bridge ge Sc Schem hemes es and and Oth Other er Idea Id eass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.6.2 14. 6.2.6 .6 Hi High gh-O -Ord rder er Sche Scheme mess . . . . . . . . . . . . . . . . . 14.6. 14. 6.33 Ma Mart rtin ingal galee Discret Discretiz izat ation ion . . . . . . . . . . . . . . . . . . . . . . 14.6.3 14. 6.3.1 .1 De Defla flate ted d Bon Bond d Pri Price ce Di Disc scret retiza izati tion on . . . . 14.6.3 14. 6.3.2 .2 Co Comm mmen ents ts an and d Alte Altern rnat ativ ives es . . . . . . . . . . 14.6. 14. 6.44 Vari arian ance ce Reduct Reductio ion n . .. .. .. .. .. .. .. .. .. .. .. .. .. 14.6.4 14. 6.4.1 .1 An Anti tith thet etic ic Sampl Samplin ingg . . . . . . . . . . . . . . . . . 14.6.4 14. 6.4.2 .2 Co Con ntr trol ol Vari Variat ates es . . . . . . . . . . . . . . . . . . . . 14.6.4 14. 6.4.3 .3 Im Impor porta tanc ncee Sampl Samplin ingg . . . . . . . . . . . . . . . . 15 Th The e Libor Libor Mark Market et Model Model II . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15.1 Inter Interpolat polation ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.1.1 15.1 .1 Bac Back k Stub, Stub, Simple Simple Interpola Interpolatio tion n ... .. . .. .. . . .. .. 15.1.2 15.1 .2 Bac Back k Stub, Stub, Arbitra Arbitrage-F ge-Free ree Inter Interpolat polation ion . . . . . . . . . 15.1. 15. 1.33 Bac Back k Stub, Stub, Gauss Gaussian ian Model Model . . . . . . . . . . . . . . . . . . . . 15.1. 15. 1.44 Fro ron nt St Stub ub,, Ze Zero ro Vola Volati tili litty . . . . . . . . . . . . . . . . . . . . . 15.1.5 15.1 .5 Fron rontt Stub, Stub, Exogenous Exogenous Volat Volatilit ility y ............... 15.1. 15. 1.66 Fro ron nt Stub, Stub, Simpl Simplee In Inte terpo rpolat lation ion . . . . . . . . . . . . . . . 15.1. 15. 1.77 Fro ron nt Stub, Stub, Gaussi Gaussian an Model Model . . . . . . . . . . . . . . . . . . . 15.2 Adv Advanc anced ed Swapt Swaption ion Pricin Pricingg via Mark Markov ovian ian Project Projection ion . . . . 15.2.1 15.2 .1 Adv Advanc anced ed Form Formula ula for for Swap Swap Rate Rate Volat Volatilit ility y...... 15.2. 15. 2.22 Ad Adv van ance ced d For Form mul ulaa for Swap Swap Rate Rate Skew Skew . . . . . . . . . 15.2. 15. 2.33 Sk Skew ew and and Smile Smile Cali Calibr brat ation ion in LM Models Models . . . . . . . 15.33 Ne 15. Nearar-Mar Mark kov LM Models Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.4 Sw Swap ap Market Market Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5 Ev Evolvi olving ng Sepa Separat ratee Discou Discount nt and Forw orward ard Rat Ratee Curv Curves es.. . . . 15.5. 15. 5.11 Bas Basic ic Ideas Ideas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15.5. 15. 5.22 HJM Exten Extensio sion n . .. .. .. .. .. .. .. .. .. ... .. .. .. .. 15.5 15 .5.3 .3 Ap Appl plic icat atio ions ns to LM Mode Models ls . . . . . . . . . . . . . . . . . . . 15.5. 15. 5.44 De Dete term rmin inist istic ic Spread Spread . . . . . . . . . . . . . . . . . . . . . . . . . . 15.66 SV Models 15. Models with with No Nonn-Zer Zeroo Correla Correlati tion on.. . . . . . . . . . . . . . . . . . 15.7 Mul Multiti-Stoc Stochas hastic tic Volat Volatili ility ty Extensi Extensions ons . . . . . . . . . . . . . . . . . .
631 6311 63 6322 63 6333 63 6344 63 6366 63 6366 63 637 638 638 640 641 6433 64 644 645 646 6477 64 647 648 648 651 651 652 653 655 656 657 6600 66 661 662 664 6666 66 668 670 670 672 6733 67 674 6777 67 681 681 683
Contents
15.7.11 15.7. 15.7. 15 .7.22 15.7. 15 .7.33 15.7. 15 .7.44 15.7.5
Introdu Intr oduct ction ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Setu Se tup p . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. Pricin Pri cingg Caplet Capletss and Swap Swapti tion onss . . . . . . . . . . . . . . . . . Spre Sp read ad Option Optionss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Another Anoth er Use of Multi Multi-Dimen -Dimensional sional Stoch Stochastic astic Volat olatili ility ty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
XXV XX V
683 6844 68 6855 68 6866 68 687
XXVI XX VI
Con onte ten nts
VOLUME III Products and Risk Management Part IV Produc Products ts 16 Singl Single-Rat e-Rate e Vanill Vanilla a Derivativ Derivatives es . . . . . . . . . . . . . . . . . . . . . . . .
16.11 Europ 16. Europea ean n Sw Swap apti tion onss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1. 16. 1.11 Sm Smil ilee Dy Dyna nami mics cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1. 16. 1.22 Ad Adju just stab able le Backbo Backbone. ne. . . . . . . . . . . . . . . . . . . . . . . . . . 16.1.3 16.1 .3 Stoc Stochas hastic tic Volat Volatili ility ty Swapt Swaption ion Grid Grid . . . . . . . . . . . . . 16.1.4 16.1 .4 Cal Calibr ibrati ating ng Stochast Stochastic ic Volat olatilit ility y Model to Swap Sw apti tion onss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.1. 16. 1.55 So Some me Other Other Inte Interpo rpolat lation ion Rul Rules es . . . . . . . . . . . . . . . . 16.22 Ca 16. Caps ps and Floors Floors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2. 16. 2.11 Bas Basic ic Proble Problem m . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 16.2. 16. 2.22 Se Setu tup p an and d Norms Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2. 16. 2.33 Ca Cali libr brat atio ion n Procedu Procedure re . . . . . . . . . . . . . . . . . . . . . . . . . 16.33 Ter 16. ermi mina nall Sw Swap ap Rate Rate Mod Models els . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3. 16. 3.11 TS TSR R Ba Basic sicss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3. 16. 3.22 Li Line near ar TSR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.3. 16. 3.33 Exp Expon onen enti tial al TSR Model Model . . . . . . . . . . . . . . . . . . . . . . . 16.3. 16. 3.44 Sw Swap ap-Y -Yiel ield d TSR Model Model . . . . . . . . . . . . . . . . . . . . . . . 16.4 Libo Libor-in r-in-Arr -Arrears ears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.55 Li 16. Libor bor-w -wit ithh-De Dela lay y . . .. . .. . .. . .. . .. . .. . .. . .. . .. . . .. . .. . 16.5. 16. 5.11 Sw Swap ap-Y -Yiel ield d TSR Model Model . . . . . . . . . . . . . . . . . . . . . . . 16.5. 16. 5.22 Ot Othe herr Ter Termi mina nall Swap Swap Rate Rate Models Models . . . . . . . . . . . . . 16.5.3 Appro Approximat ximations ions Inspired Inspired by Term Term Structure Structure Models Mod els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.5. 16. 5.44 Ap Appl plic icat atio ions ns to Averag Averagin ingg Swap Swapss . . . . . . . . . . . . . . . 16.66 CM 16. CMS S an and d CMS-Li CMS-Link nked ed Cash Cash Flows Flows . . . . . . . . . . . . . . . . . . . . 16.6. 16. 6.11 Th Thee Replic Replicat atio ion n Method Method for for CMS. . . . . . . . . . . . . . . 16.6.2 Ann Annuit uity y Mapping Mapping Funct Function ion as a Condition Conditional al Expect Exp ected ed Valu Valuee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6. 16. 6.33 Sw Swap ap-Y -Yiel ield d TSR Model Model . . . . . . . . . . . . . . . . . . . . . . . 16.6. 16. 6.44 Li Line near ar and Othe Otherr TSR Model Modelss . . . . . . . . . . . . . . . . . 16.6. 16. 6.55 Th Thee Quasi-G Quasi-Gau aussi ssian an Model Model . . . . . . . . . . . . . . . . . . . . 16.6. 16. 6.66 Th Thee Li Libor bor Mark Market Model Model . . . . . . . . . . . . . . . . . . . . . . 16.6.7 16.6 .7 Cor Correct recting ing Non-A Non-Arbi rbitra trage-F ge-Free ree Meth Methods ods . . . . . . . . . 16.6.8 Impac Impactt of Annuit Annuity y Mapping Mapping Funct Function ion and Mean Rev Re vers ersion ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.6. 16. 6.99 CD CDF F and and PDF PDF of CM CMS S Rat Ratee in Forw orward ard Me Measu asure re . 16.6. 16. 6.10 10 SV Model Model fo forr CMS Rate . . . . . . . . . . . . . . . . . . . . . .
691 691 6911 69 6922 69 6933 69 696 697 6999 69 700 7000 70 701 702 703 7033 70 705 708 709 710 713 714 7155 71 715 7166 71 7177 71 718 720 722 722 724 725 728 729 7300 73 734
Cont Co nten ents ts
XXVII XXV II
16.6.11 Dynami 16.6.11 Dynamics cs of CMS Rate Rate in Forw Forward ard Measu Measure re . . . . . 16.6.12 16.6 .12 Cas Cash-S h-Sett ettled led Swapt Swaption ionss . . . . . . . . . . . . . . . . . . . . . . . 16.7 Qu 16.7 Quan anto to CMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7. 16 .7.11 Ov Overv erview iew . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.7.2 16.7 .2 Mode Modelin lingg the Join Jointt Distrib Distribut ution ion of Swap Rate Rate and Forw Forward ard Exchan Exchange ge Rate . . . . . . . . . . . . . . . . . . . 16.7.3 16.7 .3 Norm Normaliz alizing ing Const Constan antt and Fina Finall Form Formula ula . . . . . . . . 16.8 Eur Eurodoll odollar ar Futu Futures res . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8.1 16.8 .1 Fund undamen amental tal Results Results on Fut Future uress . . . . . . . . . . . . . . . . 16.8. 16 .8.22 Mo Moti tiv vat ation ionss an and d Pla Plan n . .. .. .. .. .. .. .. .. .. .. .. .. 16.8 16 .8.3 .3 Pr Prel elim imin inar arie iess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.8. 16 .8.44 Exp Expan ansi sion on Arou Around nd the the Fut Futur ures es Valu Valuee . . . . . . . . . . . 16.8.5 16.8 .5 Forw orward ard Rate Rate Varia ariance ncess . . . . . . . . . . . . . . . . . . . . . . . 16.8. 16 .8.66 Forw orward ard Rate Rate Correl Correlat ation ionss . . . . . . . . . . . . . . . . . . . . . 16.8. 16 .8.77 Th Thee Form ormul ulaa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16.9 16 .9 Co Con nve vexi xitty and Momen Moment Exp Explo losio sions ns . . . . . . . . . . . . . . . . . . . .
735 738 740 7400 74
17 Mult Multi-Rat i-Rate e Vanill Vanilla a Derivativ Derivatives es . . . . . . . . . . . . . . . . . . . . . . . . .
759 759 759 761 762 7622 76 7644 76 7666 76 767 770 770 771
17.1 Introd Introduct uction ion to Multi-R Multi-Rate ate Vani Vanilla lla Deriv Derivati ative vess . . . . . . . . . . 17.2 Marg Margina inall Distrib Distribut ution ionss and Referenc Referencee Measure Measure . . . . . . . . . . 17.3 17 .3 De Depen pende denc ncee St Stru ruct ctur uree vi viaa Co Copu pula lass . . . . . . . . . . . . . . . . . . . . . 17.3. 17 .3.11 In Intr trodu oduct ction ion to to Gaussi Gaussian an Copu Copula la Method Method . . . . . . . . 17.3. 17 .3.22 Ge Gene neral ral Copul Copulas as . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.3. 17 .3.33 Ar Arch chim imed edea ean n Co Copu pulas las . . . . . . . . . . . . . . . . . . . . . . . . . 17.3. 17 .3.44 Ma Maki king ng Copu Copula lass from from Other Other Copu Copulas las . . . . . . . . . . . . 17.4 Cop Copula ula Methods Methods for CMS Spread Spread Option Optionss . . . . . . . . . . . . . . 17.4. 17 .4.11 No Norm rmal al Model Model for the the Spread Spread . . . . . . . . . . . . . . . . . . 17.4. 17 .4.22 Ga Gaus ussi sian an Copu Copula la for for Spread Spread Opti Option onss . . . . . . . . . . . . 17.4.3 Spread Volatili olatility ty Smile Smile Modeling Modeling with the the Power Power Gaus Ga ussi sian an Copul Copulaa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.4. 17 .4.44 Co Copu pula la Impl Implie ied d From From Sprea Spread d Option Optionss . . . . . . . . . . . 17.5 Rat Rates es Observed Observed at Diff Differen erentt Tim Times es.. . . . . . . . . . . . . . . . . . . . . 17.6 Num Numeric erical al Methods Methods for Copulas Copulas . . . . . . . . . . . . . . . . . . . . . . . 17.6. 17 .6.11 Nu Nume meri rical cal Inte Integra grati tion on Method Methodss . . . . . . . . . . . . . . . . . 17.6.2 17.6 .2 Dim Dimens ension ionali ality ty Red Reduct uction ion for CMS Sp Spread read Opt Option ionss 17.6.3 Dimen Dimensionali sionality ty Reduct Reduction ion for for Other Other Multi-Rate Multi-Rate Deriv Der ivati ative vess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.6.4 17.6 .4 Dim Dimens ension ionali ality ty Red Reduct uction ion by Cond Conditi itioni oning ng . . . . . . . 17.6. 17 .6.55 Di Dime mens nsio iona nali litty Red Reduc ucti tion on by Measu Measure re Ch Chan ange ge . . . 17.6. 17 .6.66 Mo Mon nte Carlo Carlo Me Meth thods ods . . . . . . . . . . . . . . . . . . . . . . . . . 17.7 Lim Limita itatio tions ns of th thee Cop Copula ula Method Method . . . . . . . . . . . . . . . . . . . . . 17.8 St Stoch ochast astic ic Volat Volatilit ility y Modeli Modeling ng for for Multi Multi-Rat -Ratee Option Optionss . . . . 17.8. 17 .8.11 Me Measu asure re Chan Change ge by by Drift Drift Adju Adjust stme men nt . . . . . . . . . . . 17.8. 17 .8.22 Me Measu asure re Chan Change ge by by CMS Cap Caple lett Calib Calibrat ration ion . . . . . 17.8. 17 .8.33 Im Impa pact ct of Corr Correla elati tion onss on the the Sprea Spread d Smil Smilee . . . . . .
742 743 744 745 7477 74 7488 74 748 751 7533 75 7544 75 755
774 7755 77 778 779 7800 78 783 785 787 7911 79 793 795 796 797 798 799
XXVIII Con Conten tents ts
17.8.44 Co 17.8. Conn nnec ecti tion on to Term Term Str Struc uctu ture re Models Models . . . . . . . . . . . 17.9 CMS Spread Spread Optio Options ns in Term Term Struc Structur turee Models . . . . . . . . . 17.9. 17. 9.11 Li Libor bor Mark Market Model Model . . . . . . . . . . . . . . . . . . . . . . . . . . 17.9. 17. 9.22 Qu Quad adrat ratic ic Gauss Gaussian ian Model Model . . . . . . . . . . . . . . . . . . . . . 17.A Appendix: Implied Correlation in Displaced Log-Normal Models Mode ls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.A. 17. A.11 Pre Prelim limin inari aries es . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17.A. 17. A.22 Im Impl plie ied d Lo Log-N g-Norm ormal al Corre Correlat lation ion . . . . . . . . . . . . . . . . 17.A.3 17.A .3 A Few Numerical Numerical Results Results . . . . . . . . . . . . . . . . . . . . . . 18 Cal Callab lable le Libor Libor Exo Exotic ticss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
800 800 802 802 8044 80 805 805 8066 80 807
809 18.11 Mod 18. Model el Calib Calibrat ration ion for for Callab Callable le Libor Libor Exoti Exotics cs . . . . . . . . . . . . 809 18.1. 18. 1.11 Ri Risk sk Fact Factors ors for for CLEs . . . . . . . . . . . . . . . . . . . . . . . . . 81 8100 18.1. 18. 1.22 Mod Model el Choice Choice and and Calibr Calibrat ation ion . . . . . . . . . . . . . . . . . . 81 8133 18.2 Valu aluati ation on The Theory ory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814 18.2 18 .2.1 .1 Pr Prel elim imin inar arie iess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 8144 18.2. 18. 2.22 Re Recu curs rsion ion for for Callab Callable le Libor Libor Exoti Exotics cs . . . . . . . . . . . . 81 8155 18.2. 18. 2.33 Ma Margi rgina nall Exerci Exercise se Valu Valuee Decom Decompos posit ition ion . . . . . . . . . 816 18.3 Mon Monte te Carlo Valu Valuati ation on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 817 18.3.1 18.3 .1 Reg Regress ression ion-Bas -Based ed Valu aluati ation on of CLE CLEs, s, Basi Basicc Sch Scheme eme 817 18.3. 18. 3.22 Re Regre gressi ssion on for Unde Underly rlyin ingg . . . . . . . . . . . . . . . . . . . . . 81 8199 18.3. 18. 3.33 Valu aluin ingg CLE as a Ca Canc ncela elabl blee Note Note . . . . . . . . . . . . . . 821 18.3. 18. 3.44 Us Usin ingg Regres Regressed sed Vari ariab able less for for Deci Decisio sion n Only Only . . . . . 822 18.3.5 18.3 .5 Reg Regress ression ion Valu aluati ation on wit with h Bounda Boundary ry Opt Optimi imizat zation ion 824 18.3. 18. 3.66 Lo Low wer Boun Bound d via Regres Regressio sion n Sche Scheme me . . . . . . . . . . . . 825 18.3.7 18.3 .7 Ite Iterat rativ ivee Improv Improveme emen nt of Lower Lower Bound Bound . . . . . . . . . . 827 18.3. 18. 3.88 Up Upper per Bound Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 830 18.3 18 .3.8 .8.1 .1 Ba Basi sicc Ideas Ideas . . . . . . . . . . . . . . . . . . . . . . . . . 83 8300 18.3 18 .3.8 .8.2 .2 Ne Nest sted ed Si Sim mul ulat atio ion n (N (NS) S) Al Algo gori rith thm m . . . . 83 8311 18.3.8 18. 3.8.3 .3 Bi Bias as and and Comp Comput utat ation ional al Cost Cost of of NS Algor Al gorit ithm hm . . . . . . . . . . . . . . . . . . . . . . . . . . 834 18.3.8.4 18.3 .8.4 Con Confide fidence nce In Inter terv vals and Prac Practic tical al Usage Us age . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 836 18.3.8 18. 3.8.5 .5 No Nonn-An Analy alyti ticc Exerc Exercis isee Val Value uess . . . . . . . . . 837 18.3.8 18. 3.8.6 .6 Im Impr prov ovem emen ents ts to NS Al Algor gorit ithm hm . . . . . . . 839 18.3.8 18. 3.8.7 .7 Ot Othe herr Upper Upper Bou Bound nd Al Algor gorit ithm hmss . . . . . . . 841 18.3. 18. 3.99 Re Regre gressi ssion on Vari Variab able le Choice Choice . . . . . . . . . . . . . . . . . . . . 84 8422 18.3.9 18. 3.9.1 .1 St Stat atee Vari Variab able less Appr Approac oach h . . . . . . . . . . . . . 842 18.3.9 18. 3.9.2 .2 Ex Expl plan anat atory ory Vari Variab ables les . . . . . . . . . . . . . . . 843 18.3.9 18. 3.9.3 .3 Ex Expl plan anat atory ory Vari ariab ables les wi with th Co Con nve vexi xitty . 84 8466 18.3.10 18.3 .10 Reg Regress ression ion Implem Implemen entat tation ion . . . . . . . . . . . . . . . . . . . . 848 18.3.10.1 Autom Automated ated Explanatory Explanatory Variable Variable Selec Se lecti tion on . . . . . . . . . . . . . . . . . . . . . . . . . . . 848 18.3.1 18. 3.10.2 0.2 Su Subop bopti tima mall Poin Point Exclu Exclusi sion on . . . . . . . . . . 850 18.3.1 18. 3.10.3 0.3 Tw Twoo Step Step Regress Regression ion.. . . . . . . . . . . . . . . . . 851
Con onte ten nts
18.3.10.44 Robust 18.3.10. Robust Implemen Implementatio tation n of Regression Regression Algor Al gorit ithm hm . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4 Valu aluati ation on with Low–Di Low–Dimen mension sional al Models . . . . . . . . . . . . . . . 18.4. 18 .4.11 Si Sing nglele-Ra Rate te Calla Callabl blee Libor Libor Exotic Exoticss . . . . . . . . . . . . . . 18.4.2 18.4 .2 Cal Calibr ibrati ation on Targe Targets ts for the Local Projection Projection Meth Me thod od . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18.4. 18 .4.33 Re Revi view ew of Suit Suitab able le Local Local Models Models . . . . . . . . . . . . . . . 18.4. 18 .4.44 De Defin finin ingg a Suit Suitab able le Ana Analog log fo forr Core Core Sw Swap Rat Rates es . . 18.4. 18 .4.55 PD PDE E Method Methodss for Pat Pathh-De Depen pende den nt CLEs CLEs . . . . . . . . 18.4 18 .4.5 .5.1 .1 CL CLEs Es Ac Accr cret etin ingg at at Cou Coupon pon Ra Rate te . . . . . . 18.4. 18. 4.5.2 5.2 Sn Snow owba balls lls . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Ber Berm muda udan n Swapt Swaption ionss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19.1 19.1 19.2 19 .2 19.3 19 .3 19.4
Definit Defin ition ionss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Local Loc al Projection Projection Meth Method od . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Smile Sm ile Calib Calibrat ration ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amorti Amo rtizin zing, g, Accretin Accreting, g, Other Other Non-Sta Non-Standa ndard rd Swapt Swaption ionss . . . 19.4.1 19.4 .1 Rel Relati ations onship hip Between Between Non-Stan Non-Standar dard d and Stand St andard ard Swap Rates Rates . . . . . . . . . . . . . . . . . . . . . . . . . 19.4. 19 .4.22 Sa Same me-T -Ten enor or Approa Approach ch.. . . . . . . . . . . . . . . . . . . . . . . . . 19.4.3 19.4 .3 Rep Represe resent ntati ative ve Swapt Swaption ion Appro Approac ach h ............. 19.4. 19 .4.44 Ba Bask sket et Approa Approacch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4.5 Super-R Super-Replicat eplication ion for for Non-Stand Non-Standard ard Bermu Bermudan dan Swap Sw apti tion onss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4. 19 .4.66 Ze Zeroro-Co Coupo upon n Berm Bermud udan an Swap Swapti tion onss . . . . . . . . . . . . . 19.4. 19 .4.77 Am Ameri erican can Swap Swapti tion onss . . . . . . . . . . . . . . . . . . . . . . . . . . 19.4. 19. 4.7.1 7.1 Am Ameri erica can n Swap Swapti tion onss vs. Hig HighhFrequ requenc ency y Bermuda Bermudan n Swapti Swaptions ons . . . . . . . 19.4 19 .4.7 .7.2 .2 Th Thee Pro Proxy Li Libor bor Ra Rate te Met Metho hod d ........ 19.4. 19. 4.7.3 7.3 Th Thee Libo Libor-a r-as-E s-Ext xtrara-St Stat atee Me Meth thod od . . . . . 19.4. 19 .4.88 Mi Midd-Co Coupo upon n Exe Exerci rcise se . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Fl 19.5 Flex exii-Sw Swap apss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5. 19 .5.11 Pu Purel rely y Global Global Bound Boundss . . . . . . . . . . . . . . . . . . . . . . . . . 19.5. 19 .5.22 Pu Purel rely y Loc Local al Bound Boundss . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5. 19 .5.33 Ma Margi rgina nall Exerci Exercise se Val Value ue Deco Decompo mposit sition ion . . . . . . . . . 19.5. 19 .5.44 Na Narro rrow w Band Limit Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6 Mon Monte te Carlo Valu Valuati ation on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6. 19 .6.11 Re Regre gressi ssion on Method Methodss . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.6. 19 .6.22 Pa Para rame metr tric ic Bound Boundary ary Meth Methods ods . . . . . . . . . . . . . . . . . 19.6. 19. 6.2.1 2.1 Sa Samp mple le Exerc Exercise ise Str Strat ategi egies es for for Berm Ber mud udan an Swap Swapti tion onss . . . . . . . . . . . . . . . . 19.6 19 .6.2 .2.2 .2 So Some me Nume Numeri rica call Tes Tests ts . . . . . . . . . . . . . . . 19.6. 19. 6.2.3 2.3 Ad Addi diti tion onal al Comm Commen ents ts . . . . . . . . . . . . . . . . 19.7 Oth Other er Topi opics cs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
XXIX XX IX
852 852 856 856 856 857 859 861 8622 86 864 867 8677 86 868 870 872 874 8755 87 876 8799 87 882 886 8877 88 888 8899 88 890 8911 89 892 8933 89 8933 89 895 896 897 8977 89 8988 89 898 9011 90 904 904
XXX XX X
Contents
19.7.1 19.7 .1 Robust Robust Bermuda Bermudan n Swaptio Swaption n Hedging Hedging wit with h European Euro pean Swapt Swaption ionss . . . . . . . . . . . . . . . . . . . . . . . . . . 19.7. 19. 7.22 Ca Carry rry and and Exe Exerci rcise se . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.7. 19. 7.33 Fas astt Pri Pricin cingg vi viaa Exe Exerci rcise se Pre Premi miaa Rep Repres resen enta tati tion on . 19.A Append Appendix: ix: Forwa Forward rd Volatili Volatility ty and Correlation . . . . . . . . . . . 19.B 19. B Appen Appendi dix: x: A Pri Prime merr on Momen Momentt Ma Matc tchi hing ng.. . . . . . . . . . . . . . 19.B. 19. B.11 Bas Basic icss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.B. 19. B.22 Exa Examp mple le 1: Asia Asian n Optio Option n in BSM BSM Model Model . . . . . . . . 19.B. 19. B.33 Exa Examp mple le 2: Bask Basket et Opt Option ion in BSM BSM Model Model . . . . . . . 20 TARNs, Volat Volatilit ility y Swaps, and Other Derivativ Derivatives es . . . . . .
20.11 TAR 20. ARNs Ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1. 20. 1.11 De Defin finit ition ionss an and d Exampl Examples es . . . . . . . . . . . . . . . . . . . . . . 20.1.2 20.1 .2 Valu aluati ation on and Risk with Globally Globally Calibra Calibrated ted Models Mod els . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.1. 20. 1.33 Loc Local al Projecti Projection on Method Method . . . . . . . . . . . . . . . . . . . . . . 20.1.4 20.1 .4 Volat olatili ility ty Smile Smile Effects . . . . . . . . . . . . . . . . . . . . . . . . 20.1. 20. 1.55 PD PDE E fo forr TAR ARNs Ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2 Volat olatilit ility y Sw Swaps aps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2. 20. 2.11 Loc Local al Projecti Projection on Method Method . . . . . . . . . . . . . . . . . . . . . . 20.2. 20. 2.22 Sh Shou outt Op Opti tion onss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20.2. 20. 2.33 Mi Minn-Ma Max x Vol olat atil ilit ity y Sw Swap apss . . . . . . . . . . . . . . . . . . . . . 20.2.4 20.2 .4 Imp Impact act of Volat olatilit ility y Dyn Dynami amics cs on Volat olatili ility ty Sw Swaps aps 20.3 Forw orward ard Swapt Swaption ion Straddl Straddles es . . . . . . . . . . . . . . . . . . . . . . . . . . 21 Out Out-of -of-Mod -Model el Adjus Adjustme tmen nts . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21.11 Adju 21. Adjust stin ingg th thee Mod Model el . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1. 21. 1.11 Ca Cali libr brat atio ion n to Co Coupo upons ns . . . . . . . . . . . . . . . . . . . . . . . 21.1. 21. 1.22 Ad Adju just sters ers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1. 21. 1.33 Pa Path th Re-W Re-Weig eigh hti ting ng . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1. 21. 1.44 Pro Proxy xy Model Model Method Method . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.5 21.1 .5 Asse Asset-B t-Based ased Adjus Adjustm tmen ents ts . . . . . . . . . . . . . . . . . . . . . . 21.1.6 21.1 .6 Map Mappin pingg Func Functio tion n Adjustm Adjustmen ents ts . . . . . . . . . . . . . . . . 21.22 Ad 21. Adju just stin ingg th thee Ma Mark rket et . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.33 Ad 21. Adju just stin ingg th thee Trad radee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3. 21. 3.11 Fee Adjus Adjustm tmen ents ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 21.3 .2 Fee Adjus Adjustme tment nt Impa Impact ct on on Exotic Exotic Der Deriv ivati ative vess . . . . 21.3. 21. 3.33 St Stri rike ke Adju Adjust stme men nt . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part V Risk Managemen Managementt
904 9077 90 908 912 9133 91 913 914 9166 91 919 919 9199 91 921 9222 92 923 925 927 9288 92 9299 92 9322 93 934 939 945 946 9466 94 948 9500 95 955 957 959 959 960 9611 96 962 9633 96
Con onte ten nts
22 In Intr troduct oduction ion to Risk Manag Manageme ement nt . . . . . . . . . . . . . . . . . . . . .
22.1 Risk Risk Managem Managemen entt and Sensi Sensitiv tivit ity y Computa Computatio tions ns . . . . . . . . 22.1. 22 .1.11 Ba Basic sic Info Inform rmat ation ion Flow. Flow. . . . . . . . . . . . . . . . . . . . . . . . 22.1. 22 .1.22 Ri Risk sk:: Th Theor eory y and Practi Practice ce . . . . . . . . . . . . . . . . . . . . . 22.1. 22 .1.33 Exa Examp mple le:: the BlackBlack-Sc Scho holes les Model Model . . . . . . . . . . . . . . 22.1.4 22.1 .4 Exam Example ple:: Blac Black-S k-Sch choles oles Model Model wit with h Time-D Tim e-Depen ependen dentt Par Parame ameter terss . . . . . . . . . . . . . . . . . . . 22.1. 22 .1.55 Ac Actu tual al Risk Compu Computa tati tion onss . . . . . . . . . . . . . . . . . . . . . 22.1. 22 .1.66 Wh What at abo about ut Θprm and Θnum ? . . . . . . . . . . . . . . . . . . 22.1.7 22.1 .7 A Note on Trad Trading ing P&L and and the Compu Computat tation ion of Implied Implied Volat Volatili ility ty . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 22 .2 P&L Analy Analysis sis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2. 22 .2.11 P&L Predic Predictt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2. 22 .2.22 P&L Explai Explain n. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2. 22. 2.2.1 2.1 Wat ater erfa fall ll Explai Explain n ................... 22.2 22 .2.2 .2.2 .2 Bump Bu mp-a -and nd-R -Res eset et Exp Expla lain in . . . . . . . . . . . . 22.3 Valu alue-at e-at-Ri -Risk sk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.A Appe Appendi ndix: x: Alternat Alternativ ivee Proof of Lem Lemma ma 22.1.1 . . . . . . . . . . . 23 Payoff Smoothing Smoothing and and Rel Relat ated ed Methods Methods . . . . . . . . . . . . . .
XXXI XX XI
969 969 970 9700 97 9722 97 974 977 9799 97 9800 98 981 984 9855 98 9877 98 987 9888 98 989 992 995 995 995 9966 99 9966 99 9999 99 1000 1000 10011 100 1003 1004 1006 1006 1008 10099 100 10122 101 1013 1016 1016 1018 1023 1023
23.1 23 .1 Issu Issues es with Discre Discreti tiza zati tion on Schem Schemes es . . . . . . . . . . . . . . . . . . . . . 23.1. 23 .1.11 Pro Probl blem emss with with Gr Grid id Dime Dimens nsion ionin ingg . . . . . . . . . . . . . . 23.1. 23 .1.22 Gr Grid id Shif Shifts ts Relat Relativ ivee to Pa Payo yout ut . . . . . . . . . . . . . . . . . 23.1. 23 .1.33 Ad Addi diti tion onal al Commen Comments ts . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Basi Basicc Tec echni hnique quess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.1 23.2 .1 Ada Adapti ptive ve Integra Integratio tion n .. .. .. .. .. .. .. .. .. .. .. .. .. 23.2.2 23.2 .2 Add Adding ing Singula Singularit rities ies to the Gri Grid d. .. ... .. .. .. .. .. 23.2.3 23.2 .3 Sin Singul gularit arity y Rem Remov oval al . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2.4 23.2 .4 Par Partia tiall Analytic Analytical al Integra Integratio tion n . .. .. .. . .. .. .. .. .. 23.3 Pa Payo yoff ff Smoothin Smoothingg For Numeric Numerical al Integra Integratio tion n and PDEs . . 23.3.1 23.3 .1 In Introd troduct uction ion to Pa Payo yoff ff Smoothing Smoothing . . . . . . . . . . . . . . . 23.3.2 23.3 .2 Pa Payo yoff ff Smoothin Smoothingg in One Dimens Dimension ion . . . . . . . . . . . . 23.3. 23. 3.2.1 2.1 Bo Box x Smoothi Smoothing ng . . . . . . . . . . . . . . . . . . . . . 23.3. 23. 3.2.2 2.2 Ot Othe herr Smoot Smoothi hing ng Meth Methods ods . . . . . . . . . . . 23.3.3 23.3 .3 Pa Payo yoff ff Smoothin Smoothingg in Multiple Multiple Dimens Dimension ionss . . . . . . . . 23.4 Pa Payo yoff ff Smoothing Smoothing for Mon Monte te Carlo . . . . . . . . . . . . . . . . . . . . . 23.4.1 23.4 .1 Tube Monte Monte Car Carlo lo for Digital Digital Options Options . . . . . . . . . . . 23.4.2 23.4 .2 Tube Monte Monte Carlo Carlo for Barrier Barrier Option Optionss . . . . . . . . . . 23.4.3 23.4 .3 Tube Monte Monte Carlo Carlo for for Callabl Callablee Libor Exoti Exotics cs . . . . . 23.4.4 23.4 .4 Tube Monte Monte Car Carlo lo for TARN ARNss . . . . . . . . . . . . . . . . . . 23.A Appendix: Delta Continuity of Singularity-Enlarged Grid Gri d Met Method hod.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 10244 23.B Appendix: Conditional Independence for Tube Monte Carlo Car lo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026
XXXII
Cont Co nten ents ts
24 Path athwise wise Differ Differen entiat tiation ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1029
24.1 Pathwise Pathwise Differentiati Differentiation: on: Found oundations ations . . . . . . . . . . . . . . . . . . 24.1. 24. 1.11 Ca Call llab able le Libor Exoti Exotics cs . . . . . . . . . . . . . . . . . . . . . . . . 24.1.1 24. 1.1.1 .1 CL CLE E Gr Gree eeks ks . . . . . . . . . . . . . . . . . . . . . . . . 24.1.1 24. 1.1.2 .2 Kee Keepi ping ng th thee Exerc Exercise ise Ti Time me Con Const stan antt . . . 24.1.1 24. 1.1.3 .3 No Noise ise in CLE CLE Greeks Greeks . . . . . . . . . . . . . . . . 24.1.2 24.1 .2 Barr Barrier ier Options Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2 Pat Pathw hwise ise Differen Differentia tiatio tion n for PDE Based Models . . . . . . . . . 24.2.1 24.2 .1 Model and Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.2 24.2 .2 Buc Bucke keted ted Deltas Deltas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.2.3 24.2 .3 Sur Surviv vival al Den Densit sity y. . .. . .. . .. . .. . .. . . .. . .. . .. . .. . 24.3 Pat Pathw hwise ise Differe Different ntiat iation ion for Monte Monte Carlo Carlo Based Models Models . . 24.3.1 24.3 .1 Pat Pathw hwise ise Deriv Derivati ative vess of Forw Forward ard Libor Libor Rates . . . . . 24.3.2 24.3 .2 Pat Pathw hwise ise Deltas Deltas of European European Options Options . . . . . . . . . . . 24.3.2 24. 3.2.1 .1 Pa Path thwi wise se Del Delta tass of the the Num Numer erair airee . . . . . 24.3.2 24. 3.2.2 .2 Pa Path thwi wise se Delt Deltas as of the the Payoff Payoff . . . . . . . . . 24.3.3 24.3 .3 Adj Adjoin ointt Method For For Greeks Greeks Calculat Calculation ion . . . . . . . . . 24.3.4 24.3 .4 Pat Pathw hwise ise Delta Delta App Appro roxim ximati ation on for Cal Callab lable le Liborr Exot Libo Exotics ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24.4 Not Notes es on Lik Likelih elihood ood Rat Ratio io and Hybrid Hybrid Methods Methods . . . . . . . . .
1029 10299 102 10300 103 10322 103 10333 103 10344 103 1038 10388 103 1039 1042 1045 1045 1048 10488 104 10499 104 1050 1052 1054
10577 25 Import Importance ance Sampling Sampling and and Control Control Varia Variates tes . . . . . . . . . . . . 105 25.1 Importa Importance nce Sampli Sampling ng In Sho Short rt Rate Models . . . . . . . . . . . . . 25.2 Pa Payo yoff ff Smooth Smoothing ing by Impo Importa rtance nce Samplin Samplingg . . . . . . . . . . . . . 25.2. 25. 2.11 Bi Bina nary ry Option Optionss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.2 25.2 .2 TARN ARNss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.3 25.2 .3 Rem Remov oving ing the First First Digital Digital . . . . . . . . . . . . . . . . . . . . 25.2.4 25.2 .4 Smoo Smoothi thing ng All Dig Digita itals ls by One-Step One-Step Surviv Survival al Condit Con dition ioning ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2.5 Sim Simulati ulating ng Under Under the Surviv Survival al Measure Using Condit Con dition ional al Gau Gaussia ssian n Dra Draws ws . . . . . . . . . . . . . . . . . . . 25.2.6 Genera Generalized lized Trigger Trigger Products Products in Multi-F Multi-Factor actor LM Model Modelss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3 Mode Model-Ba l-Based sed Contro Controll Varia ariates tes . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.1 Low Low-Dime -Dimensiona nsionall Markov Markov Appro Approximat ximation ion for for LM models mode ls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3.2 25.3 .2 Tw Two-D o-Dime imensi nsiona onall Ext Extens ension ion.. . . . . . . . . . . . . . . . . . . . 25.3.3 25.3 .3 App Appro roxim ximati ating ng Volat Volatili ility ty Struct Structure ure . . . . . . . . . . . . . 25.3.4 25.3 .4 Mark Markov ov Appro Approxim ximati ation on as a Contro Controll Varia Variate te . . . . . 25.4 Inst Instrumen rument-Based t-Based Control Control Variates Variates . . . . . . . . . . . . . . . . . . . . 25.5 Dyn Dynami amicc Con Contro troll Varia ariates tes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.6 Con Contro troll Varia ariates tes and Ris Risk k Sta Stabil bilit ity y . . . .. . .. . .. . .. . .. . ..
1057 1059 1059 10 59 10622 106 1062 1063 1066 1068 1071 1072 1075 1076 1078 1080 1084 1087
Conten Con tents ts XXXIII
10899 26 Vega egass in Libor Mar Mark ket Models Models . . . . . . . . . . . . . . . . . . . . . . . . . 108 26.1 Basic Basic Prob Problem lem of Vega Computat Computation ionss . . . . . . . . . . . . . . . . . . 26.2 Rev Review iew of Cal Calibr ibrati ation on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3 Vega Calcul Calculati ation on Met Methods hods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3.1 26.3 .1 Dir Direct ect Vega Vega Calculat Calculation ionss . . . . . . . . . . . . . . . . . . . . . . 26.3. 26. 3.1.1 1.1 De Defin finit ition ion and and Analys Analysis is . . . . . . . . . . . . . . 26.3. 26. 3.1.2 1.2 Nu Nume meri rical cal Examp Example le . . . . . . . . . . . . . . . . . . 26.3. 26 .3.22 Wh What at is a Good Veg Vega? a? . . . . . . . . . . . . . . . . . . . . . . . . 26.3.3 26.3 .3 Ind Indirec irectt Vega Calculat Calculation ionss . . . . . . . . . . . . . . . . . . . . . 26.3. 26. 3.3.1 3.1 De Defin finit ition ion and and Analys Analysis is . . . . . . . . . . . . . . 26.3.3.2 26.3 .3.2 Num Numeric erical al Exam Example ple and Per Perfor forman mance ce Analys Ana lysis is . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.3. 26 .3.44 Hy Hybr brid id Veg Vegaa Ca Calc lcul ulat ation ionss . . . . . . . . . . . . . . . . . . . . . . 26.3. 26. 3.4.1 4.1 De Defin finit ition ion and and Analys Analysis is . . . . . . . . . . . . . . 26.3. 26. 3.4.2 4.2 Nu Nume meri rical cal Examp Example le . . . . . . . . . . . . . . . . . . 26.4 Sk Skew ew and Smile Vegas egas.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5 Vegas and Cor Correla relatio tions ns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.5.1 26.5 .1 Term Correlat Correlation ion Effects Effects . . . . . . . . . . . . . . . . . . . . . . 26.5.2 26.5 .2 Wh What at Correla Correlatio tions ns shoul should d be Kept Con Consta stant nt?? . . . . 26.5.3 26.5 .3 Vegas with with Fix Fixed ed Term Term Correlat Correlation ionss . . . . . . . . . . . . 26.5.4 26.5 .4 Num Numeri erical cal Example Example . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.6 Del Deltas tas with Bac Backbon kbonee . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.7 Vega Project Projection ionss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.8 Som Somee Not Notes es on Computin Computingg Mode Modell Vegas . . . . . . . . . . . . . . . .
1089 1091 1092 1092 10922 109 10955 109 10966 109 1099 10999 109 1102 1105 11 05 11055 110 11077 110 1107 1109 1109 1110 1112 1113 1114 1116 1118
Appendix A
Mark Ma rko ovia vian n Pr Project ojection ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123
A.1 Margina Marginall Distribu Distributio tions ns of Ito Processes Processes . . . . . . . . . . . . . . . . . A.2 App Approx roxima imatio tions ns for Con Condit dition ional al Expected Expected Value Valuess . . . . . . . . A.2.1 A.2 .1 Gau Gaussia ssian n Approxim Approximati ation on . . . . . . . . . . . . . . . . . . . . . . A.2. A. 2.22 Le Least ast-S -Squ quar ares es Projecti Projection on . . . . . . . . . . . . . . . . . . . . . . A.3 App Applica licatio tions ns to Local Stochas Stochastic tic Volat Volatili ility ty Models Models . . . . . . . A.3. A. 3.11 Ma Mark rkov ovian ian Proje Project ctio ion n onto onto an SV SV Model Model . . . . . . . . A.3. A. 3.22 Fi Fitt ttin ingg the Mark Market with with an LSV LSV Model . . . . . . . . . . A.3.3 A.3 .3 On Calculat Calculating ing Proxy Proxy Local Volat Volatilit ility y... . . .. .. . . A.4 Bask Basket et Options Options in Local Volat Volatili ility ty Models Models . . . . . . . . . . . . . . A.5 Bask Basket et Options Options Stoch ochast astic ic Volat Volatilit ility y Mode Models ls . . . . . . . . . . in St A.A Appendix: E( zn (t)zm (t)) and E( zn (t)) )).. . . . . . . . . . . . . . A.A. A. A.11 Proo Prooff of Proposit Proposition ion A.A. A.A.11 . . . . . . . . . . . . . . . . . . . . A.A. A. A.1.1 1.1 St Step ep 1. 1. Redu Reduct ction ion to Cov Covari arian ance ce.. . . . . . . A.A. A. A.1.2 1.2 St Step ep 2. Linear Linear App Appro roxi xima mati tion on . . . . . . . . . A.A. A. A.1.3 1.3 St Step ep 3. Coefficien Coefficients ts . . . . . . . . . . . . . . . . . . A.A. A. A.1.4 1.4 St Step ep 4. Order Order of Appr Approx oxim imat ation ion . . . . . . . A.A. A. A.22 Proo Prooff of Lemma Lemma A. A.A. A.22 . . . . . . . . . . . . . . . . . . . . . . . .
1123 1128 1128 1130 11 30 1131 1131 11 31 11333 113 11377 113 1139 1143 11466 114 11477 114 11477 114 11488 114 11488 114 11499 114 11499 114
