Interest Rate Models. Copyright Investment Analytics

Transcription

1 Inere Rae Model Copyrigh Invemen Analyic

2 Inere Rae Model Model ype characeriic Model axonomy One-Facor model Vaicek Ho & Lee Hull & Whie Black-Derman-oy Model wo Facor Model Fong & Vaicek Longaff & Scwarz Hull & Whie Heah-Jarrow-Moron Copyrigh Invemen Analyic Inere Rae Model Slide:

3 Inere Rae Model Ued o: Value derivaive ep. non-andard Compue hedge raio Ae porfolio rik Provide a conien framework for valuaion hedging & rik-managemen Copyrigh Invemen Analyic Inere Rae Model Slide: 3

4 ype of Model Exenion of Black-Schole Widely ued for cap/floor Black model Model of he hor erm inere rae Eay o implemen Many varieie e.g. BD HW Model of enire yield curve Mo difficul Uually implified o wo facor e.g. HJM Copyrigh Invemen Analyic Inere Rae Model Slide: 4

5 Cap Floor & Collar Very popular inrumen Grea demand for cap due o increaed inere rae volailiy Marke very liquid Ued o calculae marke view of inere rae volailiy Copyrigh Invemen Analyic Inere Rae Model Slide: 5

6 Cap Floor & Collar Cap: Limi Upide Rik / Gain Serie of inere rae call opion Cap inere rae or equiy index reurn Floor: Limi Downide Rik / Gain Serie of inere rae pu opion Collar Combine Cap & Floor Fixe inere rae or equiy index wihin a band Copyrigh Invemen Analyic Inere Rae Model Slide: 6

7 Inere Rae Cap Conrac erm Cap rike rae R x 7% erm 3 year Ree frequency quarerly Reference rae LIBOR Principal $1MM Paymen from eller o buyer: 0.5 x $1MM x MaxLIBOR - R x 0 In arrear uually ar afer 3 monh Each piece i called a caple Copyrigh Invemen Analyic Inere Rae Model Slide: 7

8 Collar Combine Floor and Cap Limi upide poenial and downide rik Sale of call & purchae of pu Premium from call offe co of pu Zero Co Collar: Special cae where Pu Premium = Call Premium Ne co i zero ypically ued o lock in gain afer marke rally Copyrigh Invemen Analyic Inere Rae Model Slide: 8

9 Collared FRN Coupon% LIBOR Copyrigh Invemen Analyic Inere Rae Model Slide: 9

10 Black Model Simple exenion of Black-Schole Originally developed for commodiy fuure Ued o value cap and floor Le F = forward price X = rike price Value of call opion: C d d 1 e r = ln F / X + σ / = σ = d 1 [ FN d σ 1 XN d ] Copyrigh Invemen Analyic Inere Rae Model Slide: 10

11 Applicaion o Cap Example: 1-year cap NP = noional principal R j = reference rae a ree period j R x = rike rae hen ge NP x Max{R j -R x 0} in arrear Bu hi i an opion on R j no F j Ue F j a an eimaor of R j and apply Black model o F j Previouly wa a forward price now a forward rae Copyrigh Invemen Analyic Inere Rae Model Slide: 11

12 Black Model for Cap Paymen: NP x Max{R j -R x 0} in arrear hee are a erie of opion: One for each R j he fuure po inere rae Called caple Le F j = forward rae from j o j+1 Value of caple j: Dicoun by 1+ F j a paid in arrear C = NP x e -r [F j Nd 1 - R x Nd ] / 1 + F j Copyrigh Invemen Analyic Inere Rae Model Slide: 1

13 Black Model - Example 8% cap on 3-m LIBOR R x = Srike = 8% Capped for period of 3m in 1-year ime f = 1-year forward rae for 3m LIBOR i 7% R f = 1-year po rae i 6.5% Yield volailiy i 0% pa See Excel workbook Swap.xl Black Model - Example Spreadhee Copyrigh Invemen Analyic Inere Rae Model Slide: 13

14 Black Model - Example Srike erm Fwd Rae Vol C = BSOp 8% 1 0 7% 0% 6.5% Holding Co Hco = R f -d for ock 0 for Fuure Cap Premium = Conver o %: C% = C x / 1 + F * x 0.5 x 1 / 1 + 7% x 0.5 Cap Premium % = % 5.18bp So co of capping $ loan would be $518 R f C/P E/A HCo Copyrigh Invemen Analyic Inere Rae Model Slide: 14

15 Black Model - Equivalen Formulaion in erm of Price Cap = Pu opion on price Equivalen of call opion on rae Ueful if know price volailiy raher han yield vol. F = 1 / 1 + f x i forward price F = 1 / 1 + 7% x 0.5 = X = 1/1 + R x x i rike price X = 1 / 1 + 8% x 0.5 = Require price volailiy Oher parameer a before Copyrigh Invemen Analyic Inere Rae Model Slide: 15

16 Black Model - Price Example Srike erm Fwd Price Vol C = BSOp % 6.5% R f C/P E/A Cap Premium % = % 5.18bp So co of capping $ loan would be $518 NOE: Premium already expreed a % of FV hi ime we are price a pu opion HCo Copyrigh Invemen Analyic Inere Rae Model Slide: 16

17 Lab: Cap Floor & Collar pricing - Black Model Excel Workbook Swap.xl Black Model - Workhee See lab wrieup wrien oluion & oluion preadhee Pricing a 1 year cap on 3-m LIBOR Quarerly ree o 4 caple Given price volailiy o ue price formulaion Back ou forward rae from po rae Copyrigh Invemen Analyic Inere Rae Model Slide: 17

18 Soluion: Cap Floor & Collar Pricing - Black Model Copyrigh Invemen Analyic Inere Rae Model Slide: 18

19 Limiaion of Black Model Problem: Unbiaedne: empirically fale Opion on R j no ame a opion on F j Dicoun rae: fixed - bu F j variable Rae boh ochaic and fixed! If applied o price he addiional problem Aume price can be any poiive number Bu can exceed value of fuure cah flow Copyrigh Invemen Analyic Inere Rae Model Slide: 19

20 Swapion Opion on a wap Righ o ener a wap a known fixed rae Receiver Swapion: righ o receive fixed Payer Swapion: righ o pay fixed Eenially a bond opion wih rike = noional When exercied will exchange floaing for fixed Fixed paymen correpond o a bond Copyrigh Invemen Analyic Inere Rae Model Slide: 0

21 Swapion: Applicaion Anicipaory financing manage fuure borrowing co e.g. bidding for a conrac; if ge i will wan o wap lock in oday rae e.g. opion o build a plan o buy wapion Change erm of exiing wap Cancelable puable wap Copyrigh Invemen Analyic Inere Rae Model

22 Example Swapion Sraegie Floaing rae borrower doen believe rae will fall. How can he reduce funding co? Sell floor = ell receiver wapion Borrower wan o delay deciion o lock in rae a 9% for 1 year Buy payer wapion Speculaor believe rae will rie nex year Buy payer wapion Sell receiver wapion Copyrigh Invemen Analyic Inere Rae Model Slide:

23 Swapion: Creaing Swap Varian Exendible Swap: Fixed payer can exend life of wap = payer wap + payer wapion e.g. uncerain abou erm of financing Puable Swap: can cancel wap = payer wap + receiver wapion e.g. financing need diappear Cancelable Swap: counerpary can cancel = payer wap - receiver wapion e.g. credi raing woren Copyrigh Invemen Analyic Inere Rae Model

24 Puable Swap Company LIBOR Fixed Swap Counerpary LIBOR Inermediary Fixed Receiver wapion - exercied if rae fall Copyrigh Invemen Analyic Inere Rae Model Slide: 4

25 Swapion & Ae Swap Anoher applicaion: callable bond Ae wap: iue fixed-rae bond wap o floaing If iuer call bond mo corporae are callable he i lef wih wap Swapion: allow wap o be cancelled Payer wapion Copyrigh Invemen Analyic Inere Rae Model

26 Swapion Arbirage wih Callable Bond Srong demand for receiver wapion From wap buyer paying fixed concerned abou rae falling Supply: from iuer of callable bond Arbirage Iuer ell receiver wapion Swapion premium > exra yield on callable bond Nb mach erm of call proviion o erm of wapion e.g. callable afer year ell -year European wapion Copyrigh Invemen Analyic Inere Rae Model Slide: 6

27 Swapion Arbirage Example Acion Swap Counerpary Rae rie: no acion; wapion no exercied Rae fall: wapion exercied; bond called LIBOR 7.71% Iuer Proceed Iue callable bond 7.81% yield Inveor Sell 7.71% receiver wapion 40bp Inermediary Funding Co Pay in wap LIBOR Receive in wap 7.71% Pay on bond 7.81% Swapion premium 0.4% Ne Funding Co LIBOR - 30bp Copyrigh Invemen Analyic Inere Rae Model Slide: 7

28 Copyrigh Invemen Analyic Inere Rae Model Slide: 8 Black Model for Swapion Widely ued for European Swapion i he wapion mauriy dae R i he forward wap rae R x i he rike [ ] d d R R d d N R d N R e S X X r σ σ σ = + = = / ln

29 Swapion Example wo year wapion on 1 year emiannual payer wap Srike rae i 6% Forward wap rae volailiy i 0% Copyrigh Invemen Analyic Inere Rae Model Slide: 9

30 Swapion Example Forward wap rae R 1 = ΣR DF i + DF DF i i forward dicoun facor from ar of wap mauriy of wapion = o coupon dae i Hence R = [1- DF ] / ΣR DF i R = / =6.6% Srike Fwd Swap erm Rae Vol R f C/P E/A S = BSOp 6% % 0% 5% S = 0.95% HCo Copyrigh Invemen Analyic Inere Rae Model Slide: 30

31 Shor-Rae Model Specify a laice of inere rae r uu E.g. 3 year quarerly. r 0.5 r u r ud hen d = 3m N = r d Rik neural probabiliy: 1/ d d r dd Copyrigh Invemen Analyic Inere Rae Model Slide: 31

32 Shor-Rae Model Here r i a fuure hor erm inere rae NO a forward inere rae Laice model he evoluion of hor-erm inere rae r d r u r d d r uu r ud r dd Copyrigh Invemen Analyic Inere Rae Model Slide: 3

33 Model Characeriic Equilibrium Model Sar wih economic aumpion Derive hor rae proce Curren yield curve i an oupu Fi o oberved yield curve i only approximae No Arbirage Model Deigned o be conien wih curren erm rucure Curren erm rucure i an inpu ypically conien wih volailiy erm rucure alo o ome exen Copyrigh Invemen Analyic Inere Rae Model Slide: 33

34 One-Facor Model General Form: dr = m r d + σ r dz Io Proce: m: drif facor σ: hor rae volailiy dz: ε ; ε ~ N01 Model characeriic All rae move in ame direcion bu no by ame amoun Many differen hape poible including invered Mean reverion can be buil in Copyrigh Invemen Analyic Inere Rae Model Slide: 34

35 Deirable Model Feauure Conien wih curren erm rucure Conien wih yield volailiie No negaive inere rae Allow erm rucure o move freely Hiorically 80% parallel 11% wi Compuaionally racable Copyrigh Invemen Analyic Inere Rae Model Slide: 35

36 Model axonomy Expeced Mean Volailiy Fi erm Sr. change in r Reverion of r Yield Vol mr r Vaicek a[m - r] ye conan no no CIR a[m - r] ye σ r no no Brennan & a[b + L - r] ye frl no no Schwarz Ho & Lee g no conan ye no BD frσ limied fime ye ye Hull & a[m - r] ye fime ye ye Whie Copyrigh Invemen Analyic Inere Rae Model Slide: 36

37 Vaicek Model Model form Conan mean dr α i rae of mean reverion alo conan Shor rae volailiy i conan Bond pricing and yield ln[ A ] R = + ln[ A ] = R = Lim R = r R 1 e α 1 σ α B α = α r r d + σdz r R P B σ 3 4α A e rb Copyrigh Invemen Analyic Inere Rae Model Slide: 37 = 1 = 1 e α 1 e α α

38 Copyrigh Invemen Analyic Inere Rae Model Slide: 38 Vaicek Opion Pricing Cloed form oluion Jamhidian d N P d N XP p d N XP d N P c = = p p p d d XP P d σ σ σ = + = 1 1 ln α ν σ α 1 p e = α σ ν α 1 e =

39 Vaicek Spo Rae Volailiy Volailiy erm rucure Deermined by σ and α σ σ R = 1 e α Negaive exponenial decay Higher mean reverion dampen volailiy α Copyrigh Invemen Analyic Inere Rae Model Slide: 39

40 Volailiy erm Srucure 1.0% Volailiy erm Srucure 10.0% Spo Rae Volailiy 8.0% 6.0% 4.0%.0% α = 0.01 α = 0.1 α = % Spo Mauriy Copyrigh Invemen Analyic Inere Rae Model Slide: 40

41 Vaicek Example -year call opion on 10-year ZCB r = rbar = 5% Mean reverion parameer α = 0.1 σ = 1% rike X = 0.65 Copyrigh Invemen Analyic Inere Rae Model Slide: 41

42 Vaicek Example Copyrigh Invemen Analyic Inere Rae Model Slide: 4

43 Problem wih Vaicek Rae can become negaive Wih non-zero probabiliy Volailiy i conan Empirical evidence ugge ha volailiy i correlaed wih he level of inere rae Copyrigh Invemen Analyic Inere Rae Model Slide: 43

44 Cox-Ingeroll-Ro Volailiy increae wih r ½ dr = α r r d + σ rdz Drawback No erm rucure conien No conien wih volailiy erm rucure Curve end o be monoonically increaing decreaing or very lighly humped Copyrigh Invemen Analyic Inere Rae Model Slide: 44

45 No Arbirage Model Conien wih erm and volailiy rucure Example: Ho and Lee 1986 Black Derman oy 1990 Hull and Whie 1993 Copyrigh Invemen Analyic Inere Rae Model Slide: 45

46 Ho and Lee Originally developed in binomial framework Coninuou ime limi of hor rae proce: dr = θ d + σdz θi ime-dependen drif reflecing: Slope of iniial forward rae curve Volailiy of hor rae proce f 0 θ = + σ Nb: volailiy i conan Copyrigh Invemen Analyic Inere Rae Model Slide: 46

47 Copyrigh Invemen Analyic Inere Rae Model Slide: 47 Ho and Lee Bond Pricing Forward dicoun funcion B = - r B e A P = 1 0 ln 0 0 ln B P B P P LnA σ =

48 Ho & Lee Opion Pricing c = P0Nd 1 XP0Nd p = XP0N-d P0N- d 1 d 1 ln P0 XP0 σ + P = 1 σ P d = d σ P σ P = σ Copyrigh Invemen Analyic Inere Rae Model Slide: 48

49 Ho-Lee Model Calibraion Ue marke daa o calibrae model Find volailiy which minimize um of quare of difference beween model opion price and marke price: Min N i= 1 P i P i Pˆ i Copyrigh Invemen Analyic Inere Rae Model Slide: 49

50 Ho-Lee Example Given volailiie of 3-monh 7% cap Back ou price uing Black76 Calibrae a Ho-lee model 18.5% Cap Volailiy erm Srucure 18.0% 17.5% 17.0% 16.5% 16.0% 15.5% 15.0% 14.5% 14.0% Mar-01 May-01 Jul-01 Sep-01 Nov-01 Jan-0 Mar-0 May-0 Jul-0 Sep-0 Nov-0 Jan-03 Mar-03 May-03 Jul-03 Sep-03 Nov-03 Jan-04 Mar-04 May-04 Jul-04 Copyrigh Invemen Analyic Inere Rae Model Slide: 50

51 Ho-Lee Soluion Calibraion volailiy: 17.6% Cap 7% Black 76 Black76 Ho-Lee Ho-Lee 17.6% Mauriy erm Volailiy Spo Rae Spo DF Fwd Rae Caple Price Cap Price Caple Price Cap Price Difference 11-Jan Mar % 6.35% % Jun % 6.54% % Sep % 6.83% % Dec % 7.11% % Mar % 7.33% % Jun % 7.47% % Sep % 7.56% % Dec % 7.64% % Mar % 7.7% % Jun % 7.75% % Sep % 7.78% % Dec % 7.81% % Mar % 7.84% % Jun % 7.86% Copyrigh Invemen Analyic Inere Rae Model Slide: 51

52 Hull & Whie Model Shor Rae Proce Model Like Ho-Lee bu wih mean reverion Like Vaicek fied o erm rucure α i mean reverion parameer dr = [ θ αr] d + σdz Drif proce θ = f 0 + αf 0 σ + 1 e α α Copyrigh Invemen Analyic Inere Rae Model Slide: 5

53 Hull-Whie Volailiy HW i ame a Vaicek wih imedependan mean reverion parameer α Volailiy rucure i funcion of volailiy and mean reverion of hor rae σ R = σ α 1 e α Copyrigh Invemen Analyic Inere Rae Model Slide: 53

54 Copyrigh Invemen Analyic Inere Rae Model Slide: 54 Hull-Whie Dicoun Funcion r B e A P = 1 1 e B = α α ln ln ln 3 = e e e P B P P A α α α σ α

55 Copyrigh Invemen Analyic Inere Rae Model Slide: 55 Hull-Whie Opion Pricing Modified verion of Black-Schole 1 d N KP d N P c = 1 d N P d N KP p = P P P d d KP P d σ σ σ = + = 1 1 / ln P e e = α α α σ σ

56 Hull-Whie Calibraion Calibrae wr mean reverion and volailiy parameer Min N Where C * i α σ C i α σ C α σ α σ 1 i C i i he marke price of opion I C* i i he model price of opion I Copyrigh Invemen Analyic Inere Rae Model Slide: 56

57 Hull-Whie Example Given volailiie of 3-monh 7% cap Back ou price uing Black76 Calibrae a Ho-lee model 18.5% Cap Volailiy erm Srucure 18.0% 17.5% 17.0% 16.5% 16.0% 15.5% 15.0% 14.5% 14.0% Mar-01 May-01 Jul-01 Sep-01 Nov-01 Jan-0 Mar-0 May-0 Jul-0 Sep-0 Nov-0 Jan-03 Mar-03 May-03 Jul-03 Sep-03 Nov-03 Jan-04 Mar-04 May-04 Jul-04 Copyrigh Invemen Analyic Inere Rae Model Slide: 57

58 Hull-Whie Soluion Cap 7% Black 76 Black76 Hull Whie Hull Whie α -1.16% σ 69.53% Mauriy erm Volailiy Spo Rae Spo DF Fwd Rae Caple Price Cap Price Caple Price Cap Price Difference 11-Jan Mar % 6.35% % Jun % 6.54% % Sep % 6.83% % Dec % 7.11% % Mar % 7.33% % Jun % 7.47% % Sep % 7.56% % Dec % 7.64% % Mar % 7.7% % Jun % 7.75% % Sep % 7.78% % Dec % 7.81% % Mar % 7.84% % Jun % 7.86% Copyrigh Invemen Analyic Inere Rae Model Slide: 58

59 Black-Derman-oy Model Simple one-facor model All rae ochaic Volailiie can change over ime No negaive inere rae Fi he yield curve and yield volailiy erm rucure Eay o ue & implemen Copyrigh Invemen Analyic Inere Rae Model Slide: 59

60 Black-Derman-oy Model Inpu: Curren po yield curve ZCB price Spo rae volailiie Oupu: A binomial hor rae ree ha mache: erm rucure of zero coupon yield erm rucure of po rae volailiy hi can hen be ued o price: Bond Derivaive Any inere rae coningen claim Copyrigh Invemen Analyic Inere Rae Model Slide: 60

61 Model Form & Aumpion Model : dln r = + σ ' r d dz ln θ + σ σ Shor rae are log-normally diribued: Yield are alway poiive Volailiy of log of hor rae depend only on ime Probabiliy up and down move i 50% Expeced reurn on all ae i equal A ingle hor rae hold in each fuure period Copyrigh Invemen Analyic Inere Rae Model Slide: 61

62 he Shor Rae ree r r u r d P P u P = [0.5 P u P d ] P d 1 + r Noe: Need P u and P d bu no r u or r d Copyrigh Invemen Analyic Inere Rae Model Slide: 6

63 he Shor-Rae ree Example 1-Perdiod ZCB 10% r u r d P P = [ ] P = Copyrigh Invemen Analyic Inere Rae Model Slide: 63

64 Example: Year ZCB 10% P % 9.8% P u P d r uu r ud r dd P u = [ ] P u = P = [ ] P = 8.35 P d = [ ] P d = Copyrigh Invemen Analyic Inere Rae Model Slide: 64

65 Vanilla Swap r uu Swap paymen in arrear: r r u r d r ud C uu = NP[r u -c] C ud = NP[r u -c] C dd = NP[r d -c] C u = NP[r-c] C d = NP[r-c] C = 0 r dd C C u C d C uu C ud C dd o value work back adding cah flow a curren node: V u = C u + [0.5V uu + V ud ] 1 + r u Copyrigh Invemen Analyic Inere Rae Model Slide: 65

66 Cap r C r u r d C u C d r uu r ud r dd C uu C ud C dd Cap paymen in arrear: C uu = C ud = NP x Max[r u -r x 0] C dd = NP x Max[r d -r x 0] C u = C d = 0 o value work back adding cah flow a curren node: V u = C u + [0.5V uu + V ud ] 1 + r u Copyrigh Invemen Analyic Inere Rae Model Slide: 66

67 Valuing Oher Securiie Same principle apply for any fixed income ecuriie Bond bond opion wapion Embedded opion e.g. callable bond CMO Cap floor collar FRA wap forward FRN Fuure Can even calculae convexiy bia ince know cah flow a every node Copyrigh Invemen Analyic Inere Rae Model Slide: 67

68 Conrucing he ree Chooe hor rae ree o ha: Calculaed ZCB price po rae agree wih marke price po rae ZCB yield volailiy agree wih inpu volailiy Procedure Gue a new column of hor rae Calculae he price of he nex period ou ZCB Compare: calculaed ZCB price wih marke price yield volailiy from price ree wih inpu volailiy Modify gue and repea unil mache Copyrigh Invemen Analyic Inere Rae Model Slide: 68

69 Filling a New Column in he Shor Rae ree Gue wo number: r min : he hor rae a he boom node σ r : he hor rae volailiy in he column Noe: hi i NO he yield volailiy we are rying o mach Shor Rae Volailiy: σ r = 0.5 * Ln r u / r d Ue r u = r d expσ r o ep up he column node by node Copyrigh Invemen Analyic Inere Rae Model Slide: 69

70 Example Col 4 Col Nex node up: r u = x exp x 0.17 = r min Gue r min = σ r = 0.17 Copyrigh Invemen Analyic Inere Rae Model Slide: 70

71 Example: Uing 3-Year ZCB o Calibrae Shor Rae ree We know: Shor rae ree for fir wo period Marke price of he 3-year ZCB Volailiy of he 3-year ZCB yield 18% Copyrigh Invemen Analyic Inere Rae Model Slide: 71

72 Example: Uing 3-Year ZCB o Calibrae Shor Rae ree Before: Price Rmin σ Marke Model Yield Vol Marke Model % 18.50% Shor Rae 0.00% Spo Rae 0.00% 14.3% 6.9% 10.00% 0.00% 7.1% 0.00% 9.79% 4.78% 0.00% 0.00% Copyrigh Invemen Analyic Inere Rae Model Slide: 7

73 Example: Uing 3-Year ZCB o Calibrae Shor Rae ree Gue: r min = 9.76% σ = 17.19% Price Rmin 9.76% σ 17.19% Marke Model Yield Vol Marke Model % 18.00% Shor Rae 19.41% Spo Rae 19.41% 14.3% 15.41% 10.00% 13.76% 1.00% 13.76% 9.79% 10.75% 9.76% 9.76% Copyrigh Invemen Analyic Inere Rae Model Slide: 73

74 Checking he Yield Volailiy Compue he ZCB yield po rae y u = [100/75.067] 1/ -1 = % y 3 = [100/ ] 1/3-1 = 1% y d = [100/81.51] 1/ -1 = % Compue yield volailiy & compare wih marke daa: 0.5ln y u / y d = 0.5ln / = 18% Copyrigh Invemen Analyic Inere Rae Model Slide: 74

75 Lab: Black-Derman-oy Workhee: BD Opion Pricing Required: Conruc hor rae ree emiannual Price a ZCB mauring May 005 Price 1% of 05 Price a call opion on he 1% of 05 rike 100 mauriy May 1999 Compue opion dela See Noe & Soluion hen: Exercie for he BD model See inrucion in folder Copyrigh Invemen Analyic Inere Rae Model Slide: 75

76 Limiaion of One-Facor Model hee are one-facor model Yield curve evoluion quie limied However imple o eimae & ue Have o re-calibrae frequenly Parameer abiliy queionable Markovian Propery Evoluion of hor rae doe nor depend on previou behavior Copyrigh Invemen Analyic Inere Rae Model Slide: 76

77 Copyrigh Invemen Analyic Inere Rae Model Slide: 77 Volailiy Proce in One- Facor Model Volailiy proce i conrained Spo rae volailiy a ime depend on: S-mauriy and -mauriy yield volailiie Slope of volailiy curve a mauriy <= <= Reul i ha volailiy curve end o loe definiion over ime [ ] R R R R R σ σ σ σ σ σ + = [ ] + = R R R R R σ σ σ σ σ σ

78 Volailiy Proce Example 19.0% Cap Volailiy erm Srucure 18.5% 18.0% 17.5% 17.0% 16.5% 16.0% =0 =1 = 15.5% 15.0% 14.5% 14.0% Mar-01 May-01 Jul-01 Sep-01 Nov-01 Jan-0 Mar-0 May-0 Jul-0 Sep-0 Nov-0 Jan-03 Mar-03 May-03 Jul-03 Sep-03 Nov-03 Jan-04 Mar-04 May-04 Jul-04 Copyrigh Invemen Analyic Inere Rae Model Slide: 78

79 wo Facor Model Limiaion of one-facor model Yield curve end o be monoonic or lighly humped wo-facor model are richer Epecially volailiy proce Example Longaff & Schwarz Hull & Whie Heah Jarrow Moron Copyrigh Invemen Analyic Inere Rae Model Slide: 79

80 Fong & Vaicek wo facor Shor rae r Shor rae variance v dr dv wih = [ α r = [ γ v dz 1 r] d v] d dz = + ξ ρd vdz vdz Cloed formula for bond opion price See alo Selby & Srickland Copyrigh Invemen Analyic Inere Rae Model Slide: 80 1

81 Longaff & Schwarz Characeriic One of few model ha provide cloed form oluion when volailiy i ochaic Equilibrium Model r = ax + by; a <> b v = a x + b y Sae variable x & y follow ochaic DE : dx = γ δx d + xdz 1 dy = η θy d + ydz Copyrigh Invemen Analyic Inere Rae Model Slide: 81

82 Longaff & Schwarz Explici formula for: Dicoun bond price Derivaive Volailiy erm rucure Volailiy erm rucure wo facor Allow greaer variey of rucure han one facor model Copyrigh Invemen Analyic Inere Rae Model Slide: 8

83 Longaff & Schwarz Calibraion Very rich model Lo of poible calibraion crieria: Qualiy of fi o marke yield curve Sabiliy of coefficien Mach oberved correlaion beween rae Copyrigh Invemen Analyic Inere Rae Model Slide: 83

84 Longaff & Schwarz Pro & Con Advanage Many differen hape of erm rucure Complex volailiy erm rucure poible Correlaion beween rae can vary Shor rae no perfecly correlaed wih i volailiy If i were all rae would be perfecly correlaed a BD Cloe form oluion Diadvanage Complexiy Parameerizaion Calibraion Copyrigh Invemen Analyic Inere Rae Model Slide: 84

85 Hull & Whie -Facor Model Model form dr du = [ θ + = bud u αr] d + σ dz u i random mean reverion level Characeriic Similar o original HW model ime dependen funcion θ Exra flexibiliy of drif erm due o u + σ dz Richer erm rucure evoluion and volailiy rucure 1 1 Copyrigh Invemen Analyic Inere Rae Model Slide: 85

86 Copyrigh Invemen Analyic Inere Rae Model Slide: 86 H-W Volailiy erm Srucure [ ] C B C B R σ ρσ σ σ σ + + = 1 1 e B = α α b e b b e b C b α α α α α =

87 Copyrigh Invemen Analyic Inere Rae Model Slide: 87 Hull-Whie Opion Pricing Same modified Black-Schole model More complex formula for σ P = b b b P e b V e U e e e b UV e b V e U e B α α α α α α α α α α ρσ σ α α σ α σ σ [ ] 1 e e b U = α α α α [ ] 1 b b e e b b V = α

88 Hull-Whie Pro and Con Advanage More complex erm and volailiy rucure Cloed form oluion Imporan characeriic like mean reverion and yield correlaion can be modeled Calibrae well wih cap & floor Diadvanage Rae can in heory become negaive Copyrigh Invemen Analyic Inere Rae Model Slide: 88

89 Heah-Jarrow-Moron Model HJM framework: very general cla of model Model forward rae no hor rae Enire forward curve move in he ree Much more general han ju hor-rae moving Copyrigh Invemen Analyic Inere Rae Model Slide: 89

90 he Forward Rae ree Le f = forward rae a ime beween and +1 f00 f01 f0 f u 11 f u 1 f d 11 f uu 0 f ud 0 f du 0 f d 1 f dd 0 Copyrigh Invemen Analyic Inere Rae Model Slide: 90

91 HJM Arbirage Queion: Wha are allowable movemen in forward curve? HJM how ha movemen canno be arbirary Oherwie ree i no arbirage free Copyrigh Invemen Analyic Inere Rae Model Slide: 91

92 HJM Arbirage Rericion HJM worked rericion on evoluion of forward curve o preven arbirage Baically f move o f u or f d f u -f and f d - f depend only on volailiy of forward rae Once volailiie and iniial forward curve are known enire ree i deermined. Copyrigh Invemen Analyic Inere Rae Model Slide: 9

93 Copyrigh Invemen Analyic Inere Rae Model Slide: 93 General Form of HJM Sochaic Differenial Equaion Volailiy funcion σ i can depend on enire forward rae curve Drif erm = + = n i i i dz f d df 1 σ α = = n i i i du f u f 1 σ σ α

94 HJM in erm of Bond Reurn HJM can be reaed in erm of pure dicoun bond price reurn P P = r d + n i= 1 v i dz i Volailiie of forward rae and bond price: v i = σ i u du Copyrigh Invemen Analyic Inere Rae Model Slide: 94

95 HJM Model Characeriic Calibraed o iniial forward rae curve Rae evolve randomly horough ime Volailiie and correlaion a pecified by he volailiy funcion Generaliy Handle very general volailiy erm rucure Mehodology Mulinomial ree Mone-Carlo imulaion Copyrigh Invemen Analyic Inere Rae Model Slide: 95

96 HJM Model ype Volailiy Specificaion σ = σ Give he Ho-Lee model σ = σe -k- Hull-Whie model σ = σf Forward model forward rae can explode Bound: σ = σ min{fm} Marke Model Specify volailiy ec in erm of imple inere rae Ueful for LIBOR baed conrac Copyrigh Invemen Analyic Inere Rae Model Slide: 96

97 HJM Implemenaion Pricing a bond opion wih HJM -mauriy opion on S-mauriy bond Mehodology Chooe dicree poin on forward curve Calibraed from marke daa Evolve hee hrough ime unil opion mauriy Find mauriy value of opion from bond price a ime : P = exp f u du Copyrigh Invemen Analyic Inere Rae Model Slide: 97

98 rinomial ree for -facor Gauian HJM Model Pure dicoun bond P u + p u P p m P m + p d P d + Copyrigh Invemen Analyic Inere Rae Model Slide: 98

99 Copyrigh Invemen Analyic Inere Rae Model Slide: 99 Gauian HJM -Facor rinomial ree Sae equaion Pricing Generae pure dicoun bond price a each node Hence opion payoff a each node Non-Recombining Hence 3 N node afer n ep! { } v P P P u + = + exp 1 α { } v v P P P m + = + exp 1 α { } v v P P P d = + exp 1 α

100 HJM and Mone-Carlo Mehod ree mehodology very compuaionally inenive Hence Mone-Carlo ypically preferred Baic idea: Generae large number of erm rucure pah Alo volailiy erm rucure pah Evaluae bond or derivaive for each ieraion Copyrigh Invemen Analyic Inere Rae Model Slide: 100

101 Copyrigh Invemen Analyic Inere Rae Model Slide: 101 HJM Opion Formula European call opion Implemen hi via Mone-Carlo imulaion = max0 exp K P r d E c τ [ ] = = M j j j Y KP Y P M c 1 max0 1 = = u v u u v Y i i N i i j 1 1 exp τ ε τ τ

102 Variance Reducion echnique: Aniheic Variae hee are ued o peed convergence of he Mone Carlo approximaion and he mo popular are he following Aniheic Variae Ue boh u and 1-u o double ample ize cheaply d:= A log a covariance beween V{z} and V{1-z} i negaive he overall variance will be ubanially reduced [ V {} u + V { u} ] Copyrigh Invemen Analyic Inere Rae Model Slide: 10 V e 1 = e e 1

103 Variance Reducion echnique: Conrol Variae Conrol Variae Correc Mone Carlo eimae of exoic value wih vanilla MC error Vˆ E = V EMC + V BSMC V BS Variae Conrol Variae correlaion Reduce eimae variance when conrol and variae are correlaed Baed on cancellaion of hared eimaion error No benefi if conrol i uncorrelaed wih variae If negaive correlaed may increae eimae variance! Copyrigh Invemen Analyic Inere Rae Model Slide: 103

104 Quai-Random Number Low Dicrepancy Sequence Deerminiic equence generaed by number heory Halon Sobel Faure Sequence appear random bu no clumpy Behavior i ideal for fa convergence Copyrigh Invemen Analyic Inere Rae Model Slide: 104

105 Random v. Sobel Pakov 1997 Random poin in he uni quare Sobol poin in he uni quare

106 Pricing Error for a European Call Faure Peudo 1 Peudo -0.0 Copyrigh Invemen Analyic Inere Rae Model Slide: 106

107 Copyrigh Invemen Analyic Inere Rae Model Slide: 107 Seniiviy Facor wih MCS Approximae uing finie difference raio Dela Gamma Vega hea P P P C P P C P C + P P P C P C P P C P C + + σ σ σ σ σ σ + C C C C C C +

108 HJM Model Pro and Con Model ypically have recombining ree Eimaion fiing can be hard wo facor and muli-facor model can be developed Powerful mehodology - indury andard Copyrigh Invemen Analyic Inere Rae Model Slide: 108

109 Summary: Inere Rae Model One-Facor Model Srengh Simple ofen eay o calibrae Weaknee Range of erm & volailiy rucure limied wo-facor Model Srengh Flexible powerful & wide range of behavior Weaknee Complex compuaionally demanding Copyrigh Invemen Analyic Inere Rae Model Slide: 109