Survival Probability and Intensity Derived from Credit Default Swaps

Transcription

1 Survvl Proly nd Ineny Derved from Cred Deful Swp A Dreced Reerch Projec Sumed o he Fculy of he WORCESER POLYECHNIC INSIUE n prl fulfllmen of he requremen for he Profeonl Degree of Mer of Scence n Fnncl Mhemc y Y Ln Approved: Decemer 2 Profeor Mrcel Bl, Advor Profeor Bogdn Vernecu, Hed of Deprmen

2 Arc h projec dcue he neny nd urvvl proly derved from Cred Deful Swp (CDS). We ulze wo model, he reduced neny model nd he Shf Squre Roo Dffuon (SSRD) model. In he reduced neny model, we ume deermnc neny nd mplemen compuer mulon o derve he compny urvvl proly nd neny from he CDS mrke quoe. In he SSRD model, he nere re nd neny re oh ochc nd correled. We dcu he mpcon of correlon on he nere re nd neny. We lo conduc Mone Crlo mulon o deermne he dynmc of ochc nere re nd neny.

3 Acknowledgemen I would lke o expre my grude o ll hoe who gve me he uppor o complee h projec. A he very fr, I would lke o hnk my upervor, Prof. Mrcel Bl, who guded me hroughou ll my reerch on h projec. And hnk o Mn Hung. Whou h help I would no een le o cce o lo of ueful fnncl mrke d o fnh my reerch. 2

4 le of Conen Inroducon CDS Pyoff Runnng CDS Poponed Pyoff Runnng CDS (PRCDS) Poponed Pyoff Runnng CDS CDS Forwrd Re Poon Proce me Homogeneou Poon Proce me Inhomogeneou Poon Proce Cox Proce... 4 Reduced Ineny Model Aumpon Mehodology Shfed Squre Roo Dffuon (SSRD) Model Aumpon CIR++Shor Re Model CIR++ Ineny Model Mehodology Lck of Correlon Ce n SSRD Model Numercl Scheme Smulon Reul Correlon Ce n SSRD Model Dcrezon Scheme of Shor Re nd Inene

5 5.6.2 Correlon Effec on Inere Re nd Sochc Ineny Mone Crlo Smulon Concluon Reference nd D

6 L of le Fgure 5..Mury de & correpondng CDS quoe n p of IBM on Oc. 28h, Fgure 5.2. Clron wh pecewe conn neny of IBM on Oc. 28h, Fgure 5.3.Mury de & correpondng CDS quoe n p of IBM on Dec. 2nd, Fgure 5.4. Clron wh pecewe conn neny of IBM on Dec. 2nd, Fgure 5.5.Mury de & correpondng CDS quoe n p of Dell on Aug. 22nd, Fgure 5.6. Clron wh pecewe conn neny of Dell on Aug. 22nd, Fgure 5.7.Mury de & correpondng CDS quoe n p of Dell on Dec. 2nd, Fgure 5.8. Clron wh pecewe conn neny of Dell on Dec. 2nd, Fgure 5.9.Mury de & correpondng CDS quoe n p of Prml on Dec. 8h,

7 L of Chr Fgure.. Schemc of CDS f no deful occur... 7 Fgure.2. Schemc of CDS when deful hppen... 7 Fgure 2.. melne of he pyoff... 8 Fgure 5.. Pecewe conn neny clred on CDS quoe of IBM, Oc. 28h, Fgure 5.2. Survvl Proly from clron on CDS quoe of IBM, Oc. 28h, Fgure 5.3. Pecewe conn neny clred on CDS quoe of IBM, Dec. 2nd, Fgure 5.4. Survvl Proly from clron on CDS quoe of IBM, Dec. 2nd, Fgure 5.5. Pecewe conn neny clred on CDS quoe of Dell, Aug. 22nd, Fgure 5.6. Survvl Proly from clron on CDS quoe of Dell, Aug. 22nd, Fgure 5.7. Pecewe conn neny clred on CDS quoe of Dell, Dec. 2nd, Fgure 5.8. Survvl Proly from clron on CDS quoe of Dell, Dec. 2nd, Fgure 5.9. he dynmc of x n hor re model from Prml CDS d on Dec. 8h, Fgure 5.. he dynmc of y n neny model from Prml CDS d on Dec. 8h, Fgure 5.. he deermnc funcon n he CIR++ hor re model Fgure 5.2. he deermnc funcon n he CIR++ neny model Fgure 5.3. Compron of he neny wh dfferen correlon o nere re Fgure 5.4. Mone Crlo mulon of nere re n SSRD model Fgure 5.5. Mone Crlo mulon of ochc neny n SSRD model

8 Inroducon In 997 one em from JP Morgn Che nvened he cred deful wp (CDS). A CDS conrc eween wo counerpre. I w degned o hf he rk o hrd pry enurng proecon gn deful. Deful occur when compny fl o mke pymen owed o ome eny. he uyer of he CDS mke ere of pymen o he eller nd n exchnge receve cern ch moun f cred nrumen deful. CDS cn e ued for hedgng, peculon, nd rrge. he pred of CDS he nnul moun h he proecon uyer py o he proecon eller over he lengh of he conrc. A hown n fgure., CDS purched me nd regulr premum pymen re mde me, 2, 3... If no deful occur, hen he uyer connue pyng premum, nd o on unl he end of he conrc me. Proecon uyer Proecon eller Fgure.. Schemc of CDS f no deful occur However, f deful hppen me, he proecon eller py he uyer for he lo, nd he uyer op pyng premum, llured n fgure.2. Proecon uyer Proecon eller Fgure.2. Schemc of CDS when deful hppen 7

9 2 CDS Pyoff [] Bed on he deful me of compny nd he correpondng proecon pymen de, he pyoff of CDS dvded no wo pr: he premum leg nd he proecon leg. We defne e of prmeer efore our nly. Fr melne wh eqully pced nervl creed, hown n fgure 2., nd he nervl wdh re, [, ]. We lo defne B() rdu u D (, ) he dcoun fcor, where B () e he nk ccoun numerre, r B( ) he nnneou hor nere re, nd he fr deful me of compny. We lo defne ( ) he fr h follow he fr deful me, LGD he proecon pymen when deful hppen, R he premum pymen n exchnge for he proecon gn he deful proly, nd { } n ndcor funcon Fgure 2.. melne of he pyoff 2. Runnng CDS For Runnng CDS (RCDS), he proecon pymen re R exchnged pecfc me or when deful hppen n exchnge for ngle proecon pymen. he moun LGD pd when deful hppen. he premum leg gven y: D (, )( ) R D (, ) R. ( ) { } { } he proecon leg gven y: D (, ) L. { } GD herefore he dcouned pyoff for RCDS gven y: D(, )( ) R D(, ) R D(, ) L. (2.) RCDS, ( ) { } { } { } GD 8

10 2.2 Poponed Pyoff Runnng CDS (PRCDS) In h ce, he proecon pymen L GD pd he fr fer he deful me,.e. ( ). he dcouned pyoff gven y: D(, ) R D (, ) L PRCDS, { } { } GD. (2.2) 2.3 Poponed Pyoff Runnng CDS 2 here noher poponed pymen form n he CDS. In h ce, one more R pymen mde compred o he poponed pyoff runnng CDS. he dcouned pyoff he conrc nl me gven y: PR2 CDS D (, ), { } { } (, ) R D L GD. (2.3) 2.4 CDS Forwrd Re he CDS forwrd re R, () defned h vlue of R h mke he vlue of he dcouned CDS pyoff equl o zero me, whch deermned y: CDS, (, R, (), LGD) E () G. In he ove equon, () he dcoun CDS pyoff me, formuled n econ 2 y (2.), (2.2) or (2.3). G F ( u, u ) denoe he nformon on he deful free mrke up o me nd he exc deful me f deful hppen. o mplfy he compuon, eer o wch he flron o he deful free mrke y ung he followng equon: CDS R L E G E F. (2.4) { }, (,, (), GD) () () Q{ F } ( u, u ) denoe he gm lger of he deful me efore me. 9

11 3 Poon Proce In h projec, we ue Poon proce o decre he deful me of compny. he deful me cn e vewed he fr jump of Poon proce. Bed on he nure of he neny funcon, he Poon proce cn e clfed me homogeneou Poon proce, me nhomogeneou Poon proce, nd Cox proce. We wll recll ome mporn fc ou hee procee n he followng ex. 3. me Homogeneou Poon Proce [] A me homogeneou Poon proce defned proce wh onry ndependen ncremen nd nl vlue of zero. he me eween wo conecuve jump re ndependenly nd denclly drued n exponenl rndom vrle wh men, where conn n me. If we defne he deful me he fr jump nd Q{} denoe he proly of n even, hen: Q{ [, d) } d. (3.) hu, he proly of deful hppenng n d nervl knowng h deful h no ken plce o fr d. he urvvl proly o me herefore gven y: Q{ } exp( ). (3.2) Alo, he proly of defulng me eween me nd : Q { } exp( ) exp( ). (3.3) 3.2 me Inhomogeneou Poon Proce [] In h ce we conder h he neny () deermnc nd me vryng. We ume he neny o e pove nd pecewe connuou funcon n me. We defne: () ( u) du. (3.4)

12 By nverng he funcon, we cn on he deful me y ung ndrd exponenl rndom vrle : ( ). he proly of deful occurrng n he nex d me nervl : Q{ [, d) } ( ) d. (3.5) We cn ely ge he urvvl proly up o me : Q{ } exp( ( )) exp( ( ) d). (3.6) Smlrly, he urvvl proly eween me nd : Q { } exp( ( )) exp( ( )). (3.7) 3.3 Cox Proce [] Dfferen from he prevou wo procee, he Cox proce ume me vryng nd ochc neny. We cn ll ge mlr formul n he compuon of he urvvl proly. he cumuled neny o me cn e expreed : () d. (3.8) he proly h he compny wll deful n he nex d nervl : Q [, d), F d, (3.9) where F conn he deful free mrke nformon up o me hown n econ 2.4. he proly h he deful me of he compny greer hn : Q Q Q d E Q d F ( ) ( ) ( ) ( ) ( ) ( ). (3.) he cumuled neny deful me ( ) n exponenl rndom vrle whch ndependen of F, hu

13 ( ) ( ) ( ) ( ) d E Q d F E Q d E e. (3.) 4 Reduced Ineny Model here re everl model h del wh CDS o explore he neny nd urvvl proly n del, uch rucurl model nd he reduced form model. hey oh re ued o model cred rk. Srucurl model re ed on he complee knowledge of deled nformon e. hey were developed y Blck, Schole, nd Meron []. In conr, he reduced form model nvened y Jrrow nd urnull ed on he nformon e vlle o he mrke [2]. In h chper, we mnly focu on he reduced form neny model o deermne he neny nd he urvvl proly y ung he CDS forwrd re mrke quoe. 4. Aumpon [] In he reduced form model, we ume he deful me he me nhomogeneou Poon proce, whch men he neny () deermnc nd pecewe conn n me: () for [, ), nd he cumuled neny funcon me : () ( u) du d. n In h neny model, we lo ume h he nere re nd he deful me re ndependen. We re only concerned ou he vlue of he runnng CDS he conrc nl me. A for he poponed runnng CDS, we cn derve he formul n mlr mnner. 4.2 Mehodology [] o clcule he CDS vlue, we need o pply he fler chnge equon (2.4) for he purpoe of mplfcon. 2

14 CDS R L E F { }, (,, (), GD ) RCDS () Q{ F } { } E D (, )( ( ) ) R, { } (, ), { } { } (, ) D R D L GD F Q{ F } { } R, ED(, )( ( ) ){ } F + R, ED(, ){ } F Q{ F } LGDE { } (, ) D F () he dcouned CDS pyoff me defned n equon (2.). RCDS Conderng he uon nl me, CDS (, R, L ),, GD { } R, ED(, )( ( ) ) { } +, (, ) RE D { } Q{ } (4.2) LGDE { } (, ) D Snce we ume ndependence eween he deful me nd he nere re, ED(, ) E D(, ) E (4.) { } { }, (4.3) nd he proecon leg erm of he runnng CDS formul (4.) : L E[ D(, )] L E[ D(, ) ] GD { } GD { } { [, d )} L ED [ (, )] E[ ] L P(, Q ) ( [, d)) GD { [, d)} GD (4.4) Applyng equon (3.5) nd (3.6), we on: Q{ [, d)} Q{ [, d) } Q{ } ( ) dexp( ( ) d). (4.5) hu, pu (4.5) ck no (4.4), we cn rewre he equon (4.4) : L P(, u) ( u)exp( ( ) d) du GD u. (4.6) 3

15 By umng he pecewe conn neny re : ( ) for [, ), he ove formul (4.6) ecome: GD, (4.7) L P(, u)exp( ( u )) du where j () d defned he cumulve neny. he premum leg of he runnng CDS n formul (4.) : R E[ D(, )]( ) E[ ] R E[ D(, )] E[ ], ( ) { }, { } R P(, )( ) Q{ [, d)} R P(, ) Q{ }, ( ), (4.8) Ung he dcrezon of he defned pecewe conn ecome: defned ove, (4.8) R P(, u)( u ) exp( ( u )) du R P(, ) exp( ). (4.9),, herefore we on dcrezed cheme of runnng CDS pyoff me n he reduced neny model: CDS (, R, L ), GD, R P(, u)( u )exp( ( u )) dur P(, ) exp( ) L P(, )exp( ( u )) du GD (4.) 5 Shfed Squre Roo Dffuon (SSRD) Model In h chper, we conder he uon wh ochc neny nd ochc nere re. h model w propoed y Brgo nd Alfon n 23 []. he Cox Ingeroll Ro model ppled n SSRD o decre he dynmc of nere re nd neny. 4

16 5. Aumpon Snce he neny ochc n h model, he deful me of compny cn e vewed he fr jump of Cox proce. We denoe he ochc neny nd he ochc nere re r. 5.2 CIR++Shor Re Model [] We cn wre he hor re he um of wo pr, deermnc funcon nd Mrkovn proce x : r x (, ). (5.) Accordng o he CIR model, he dynmc of hor re x cn e wren : dx k( x ) d x dw, (5.2) where he prmeer vecor ( k,,, x ). he zero coupon ond prce derved from he CIR model : CIR x( ) d P (,, x, ) E e F A(,, )exp( B(,, )) x), (5.3) where: 2hexp ( kh)( )/2 A (,, ) 2 h( kh)(exp ( ) h ) 2(exp ( ) h ) B (,, ), 2 h( k h)(exp ( ) h ) 2 2 k /, And h k

17 5.3 CIR++ Ineny Model [] By ung n pproch mlr o h n econ 5.2, we cn epre he neny no deermnc funcon nd ochc proce, y (, ), (5.4) nd he dynmc of y cn e wren : dy ( y ) d y dz, (5.5) where he prmeer vecor (,,, y ). If he correlon eween he hor re nd neny, hen he wo ochc procee hve he followng relonhp: dw dz d. 5.4 Mehodology [] We compue he dcouned CDS pyoff me n he SSRD model: CDS R L E F { }, (,, (), GD) RCDS() Q{ F } { } ED(, )( ( ) ) R, { } (, ), { } { } (, ) D R D L GD F Q{ F } he premum leg of he ove formul cn e wren, { } ED(, )( ( ) ) R, { } (, ), D R { } F Q{ F } { } R, E (, )( ( ) ){ [, )} (, ) D d D { } F E[exp( d) F] E[exp( { }, R E E D(, )( ( ) ){ [, d)} (, ) D { } F F d) F] 6

18 { } R, E D(, )( ( ) ) E { [, d)} F F (, ) E D E { } F F E[exp( d) F] R { }, E (, )( ( ) ) { [, ) } D Q d F F E D(, ) Q{ F } F E[exp( ) ] d F R { }, E D(, )( ( ) )exp( ) (, ) exp( ) udu d F E D udu F E[exp( ) ] d F R ( ) E exp( r du)exp( du) d F E exp( r du)exp( du) F { }, ( ) u u u u { } R, ( ( ) ) E exp( ( u u) ) exp( ( u u) ) r du d F E r du F Smlrly, we cn lo clcule he proecon leg, { } { } E { } (, ) (, ) D L GD F E D L GD { [, d)} F Q{ F} E exp( ) d F { } E E D(, ) LGD { [, d)} F F E exp( ) d F { } L GDE D(, ) E{ [, d)} F F E exp( d) F { } LGDE D(, ) Q{ [, d) F } F Eexp( ) d F { } LGDE D(, )exp( ) udu d F E exp( ) d F { } LGDE exp( ru du)exp( ) udu d F Eexp( d) F { } LGD E exp( ( ru u) du) F d (5.6) herefore we cn rewre he CDS erm funcon of nere re r nd neny : 7

19 CDS (, R (), L ),, GD { } R, ( ( ) ) E exp( ( u u) ) exp( ( u u) ) r du F d E r du F { } LGD E exp( ( ru u) du) F d { } R, ( ( ) ) E exp( ( ru u) du), exp( F d R E ( ru u) du) F LGD E exp( ( ru u ) du) F d (5.7) 5.5 Lck of Correlon Ce n SSRD Model Under he uncorreled umpon, we cn epre he expecon on he nere re nd he neny n (5.7), hu CDS (, R (), L ),, GD { } R, ( ( ) ) E exp( rudu) F E exp( udu) F d R E exp( r du) F E exp( du) F, u u LGD E exp( ru du) F E exp( udu) F d (5.8) mk We know h E exp( rdu u ) F P (, ), (5.9) nd from (5.4) n he CIR++ neny model we know h: E exp( udu) F E exp( ( yu ( u, )) du) F CIR exp( ( u, ) du) E exp( yudu) F exp( ( u, ) du) P (,, y, ) (5.) Nex, o mplfy he ove formul, le u fr revew ome fc n he CIR++ model. In he Cox proce we hve 8

20 ( ) d Q( ) E e. (5.) Suung he neny formul no (5.), we on: ( ) d y (, ) d (, ) d y d Q( ) E e E e e E e. (5.2) Menwhle, we hve he d from mrke quoe h: mk Q( ) e () mk. (5.3) o clre he mrke d no he model, we eque (5.2) nd (5.3). he deermnc funcon (, ) n he CIR++ neny model cn e derved : y d CIR (, ) (, ) d ln ( ) ln (,, mk E e mk P y, ). (5.4) hrough (5.4), we cn mplfy he expreon of (5.) elow: CIR E exp( udu) F exp( ( u, ) du) P (,, y, ) exp ( ) ln P (, y,, ) ( ) ln P (, y,, ) P ( y,,, ) CIR CIR CIR mk mk CIR P (,, y, ) CIR mk mk P y CIR P (,, y, ) exp ( ) ( ) (,,, ) exp ( ) Alo, mk mk () d d E exp( udu) F E exp( udu) F E exp( udu) d d d exp mk ( ) mk ( ) d ()exp () () mk mk mk (5.5) (5.6) Afer pung (5.9), (5.5) nd (5.6) no (5.8), dcree compuon cheme of he CDS pyoff oned: 9

21 CDS (, R (), L ),, GD mk { } R, ( ) mk mk mk P d mk, mk mk GD mk mk mk ( ) (, ) ( )exp ( ) ( ) R P mk (, )exp ( ) () L P (, ) ( )exp ( ) () d (5.7) Comprng he CDS pyoff formul from he deermnc neny model (4.) wh he one from he ochc neny model (5.7), we fnd h he wo expreon re conen,.e. under he no correlon condon, he umpon of deermnc neny nd ochc neny led o excly he me reul [] Numercl Scheme [] In he prevou chper we hve derved he runnng CDS formul for pecewe conn neny n n negrl form. Here we develop no dcree form whch convenen for compuer mulon. CDS, (, R, L ) GD R ( u ) P(, u)exp( ( u )) dur P(, ) exp( ) L P(, )exp( ( u )) du GD R ( ) P(, )exp( ( ))( ) R P(, ) exp( ) L P(, ) exp( ( ))( ) GD 2 (, )exp( ) (, ) exp( ) GD (, )exp( ) R P R P L P where, nd () d ( ). k 2 (5.8) A we menoned n econ 2.4, he CDS forwrd re R, () mke he dcouned pyoff formul (5.8) equl o zero. herefore, o ge he nene, we cn plug he mrke quoe of he CDS forwrd re for dfferen mure no (5.8) nd olve for.

22 For exmple, we ume he rng me of he CDS conrc, nd le he end of he conrc e yer, 2 yer, 3 yer ec. We lo e up he dcree me qurerly. For he mrke quoe R, (), we cn ge he fr yer nene, 2, 3, 4 y olvng he equon: CDS (, R, L ; ). (5.9) MK,y,y GD For he econd yer, y ung he mrke CDS d nd he fr yer nene derved from (5.9), we cn olve for he nene of he econd yer: CDS (, R, L,,,, ; ). (5.2) MK,2y,2y GD herefore, for he n h yer nene, we cn ju plug n he CDS forwrd re nd olve he equon ervely Smulon Reul Below we preen ome numercl exmple, ed on horcl IBM, nd Dell CDS d [3] [6]. () IBM CDS Clron, Oc. 28h, 28 Recovery Re=4% Mury (yr) Mury (de) R(,) le 5.. Mury de & correpondng CDS quoe n p of IBM on Oc. 28h, 28 2

23 De Ineny Survvl Proly le 5.2. Clron wh pecewe conn neny of IBM on Oc. 28h, 28 2% Pecewe Conn Ineny.8%.6%.4%.2% %.8%.6% Oc-8 Oc-9 Oc- Oc- Oc-2 Oc-3 Oc-5 Oc-8 Fgure 5.. Pecewe conn neny clred on CDS quoe of IBM, Oc. 28h, 28 22

24 % Survvl Proly 98% 96% 94% 92% 9% 88% 86% Oc-8 Oc-9 Oc- Oc- Oc-2 Oc-3 Oc-4 Oc-5 Oc-6 Oc-7 Oc-8 Fgure 5.2. Survvl proly from clron on CDS quoe of IBM, Oc. 28h, 28 le 5. gve he CDS forwrd mrke re of IBM for dfferen mure (.5 yer o yer) on Oc. 28. We clre hee d no he dcree equon (5.8) o clcule he neny nd deful proly whn yer. le 5.2 gve he correpondng numercl reul of he neny nd deful prole. hen we plo he mulon reul of he nene nd urvvl prole d n Fgure 5. nd Fgure 5.2. () IBM CDS Clron, Dec. 2ed, 2 Recovery Re=4% Mury (yr) Mury (de) R(,) le 5.3. Mury de & correpondng CDS quoe n p of IBM on Dec. 2nd, 2 23

25 De Ineny Survvl Proly % % % % % % % le 5.4. Clron wh pecewe conn neny of IBM on Dec. 2nd, 2.8% Pecewe Conn Ineny.6%.4%.2% %.8%.6%.4%.2% Dec- Dec-2 Dec-3 Dec-4 Dec-5 Dec-6 Dec-8 Dec-2 Fgure 5.3. Pecewe conn neny clred on CDS quoe of IBM on Dec. 2nd, 2 24

26 % Survvl Proly 98% 96% 94% 92% 9% 88% Dec-Dec-2Dec-3 Dec-4Dec-5 Dec-6Dec-7Dec-8 Dec-9Dec-2 Dec-2 Fgure 5.4: Survvl proly from clron on CDS quoe of IBM on Dec.2nd, 2 le 5.3 gve he CDS forwrd mrke re of IBM for dfferen mure ( yer o yer) on Dec. 2. le 5.2 gve he correpondng numercl reul of he neny nd deful prole. We lo how he neny nd proly curve n Fgure 5.3 nd 5.4. (c) Dell CDS Clron, Aug 22nd, 28 Recovery Re=4% Mury (yr) Mury (de) R(,) le 5.5. Mury de & correpondng CDS quoe n p of Dell on Aug. 22nd, 28 25

27 De Ineny Survvl Proly % % % % % % % % le 5.6. Clron wh pecewe conn neny of Dell on Aug. 22nd, 28 2% Pecewe Conn Ineny.5% %.5% Aug-8 Aug-9 Aug- Aug- Aug-2 Aug-3 Aug-5 Aug-8 Fgure 5.5. Pecewe conn neny clred on CDS quoe of Dell on Aug. 22nd, 28 26

28 % Survvl Proly 98% 96% 94% 92% 9% 88% 86% Aug-8 Aug-9 Aug- Aug-Aug-2 Aug-3Aug-4 Aug-5 Aug-6 Aug-7 Aug-8 Fgure 5.6. Survvl proly from clron on CDS quoe of Dell on Aug. 22nd, 28 Smlr o he clron of IBM, le 5.5 gve he CDS forwrd mrke re of Dell for dfferen mure (.5 yer o yer) on Aug. 28. le 5.6 how he correpondng neny nd deful prole. Fgure 5.5 nd 5.6 lne ll he d n me. (d) Dell CDS Clron, Dec. 2nd, 2 Recovery Re=4% Mury (yr) Mury (de) R(,) le 5.7. Mury de & correpondng CDS quoe n p of Dell on Dec. 2nd, 2 27

29 De Ineny Survvl Proly % % % % % % % le 5.8. Clron wh pecewe conn neny of Dell on Dec. 2nd, 2 4.5% Pecewe Conn Ineny 4% 3.5% 3% 2.5% 2%.5% % Dec- Dec-2 Dec-3 Dec-4 Dec-5 Dec-6 Dec-8 Dec-2 Fgure 5.7. Pecewe conn neny clred on CDS quoe of Dell on Dec.2nd, 2 28

30 % Survvl Proly 95% 9% 85% 8% 75% 7% Dec- Dec-2 Dec-3 Dec-4 Dec-5 Dec-6 Dec-7 Dec-8 Dec-9 Dec-2 Fgure 5.8. Survvl proly from clron on CDS quoe of Dell, on Dec. 2nd, 2 le 5.7 gve he CDS forwrd mrke re of Dell for dfferen mure ( yer o yer) on Dec. 2. le 5.8 how he correpondng neny nd deful prole. Fgure 5.7nd 5.8 how he chnge of neny nd deful prole n me. Bed on he mulon reul, we cn nlyze he compny nformon for IBM nd Dell n dfferen me perod eprely. For IBM we fnd h he nene dd no chnge drmclly whn he yer perod. he urvvl prole derved from 28 nd 2 hve mlr curve, whch lo men h he compny howng edy performnce nd u. For Dell, he yer urvvl proly n %;however, he yer urvvl proly drop o 72.45% n 2. By comprng he d n 28 nd 2, demonre declne for Dell n une performnce. 5.6 Correlon Ce n SSRD Model In h econ, we conder more generl ce, n whch we cn pu correlon fcor eween he hor re nd ochc neny. We wll dcu he effec on he ochc hor re nd neny procee n he dfferen correlon cenro. o mule he dynmc of he hor re nd ochc neny, we fr need o derve he prmeer n he hor re CIR++ model nd he neny CIR++ model eprely. he prmeer ( k,,, x ) n he CIR++ hor re model cn e derved from he mrke quoe of he nere re produc, uch cp, floor nd zero coupon ond. Smlrly, he 29

31 prmeer (,,, y ) n he ochc neny model cn e oned y he mrke prce of he cred deful wp produc. In h projec we wll no demonre deled procedure nd cheme on he prmeer dervon. Ined, we ju ue he d gven n he Brgo nd Mercuro ook. [] Recovery Re=25% Mury (yr) Mury (de) R(,) le 5.9. Mury de & correpondng CDS quoe n p of Prml on Dec. 8h, 23 Accordng o he clron o he Prml CDS d nd Cp prce, Alfon ge he elow prmeer o decre he dynmc of hor re nd nene n he CIR++ model on Dec. 8h, 23 [] : : k.52895,.3994,.335, x :.58337,.49846, , y We wll ue he ove d o mule he correled hor re nd ochc nene nd conduc he compron nd nly n he followng econ Dcrezon Scheme of Shor Re nd Inene We fr dcreze he dynmc of x nd y n he CIR++ model y ung he prmeer vecor nd. Brgo nd Alfon hve propoed n mplc Euler cheme nd derved he correpondng explc cheme for he proce n 23 []. he mplc Euler cheme re: 3

32 2 x x ( )( ) ( ) k kx x W W 2 2 y y ( )( ) ( ) y y Z Z 2 dwdz d (5.2) o on n explc expreon, we need o olve for he ove equon: ( W W ) x 2( k ( )) where ( W ) 4( ( )( ))( ( )) W x k k 2 (5.2) Smlrly, ( Z Z ) y 2( ( )) where * 2 2 * 2 2 ( Z ) 4( ( )( ))( ( )) Z y 2 (5.22) We know he ndrd Brownn moon W h norml druon N(, ), nd he ncremen W W hve norml druon N(, ). o ge wo correled ndrd Brownn moon, we need o genere wo ndependen ndrd Brownn moon phw nd V fr, nd hen le correlon of. Z W V. hu he wo Brownn moon W nd 2 Z hve o conruc he ph n fgure 5.9 nd 5. elow, we fr genered wo correled ndrd Brownn moon y he mehod menoned n he ove prgrph. hen we pply he numercl formul n (5.2) nd (5.22) o dcreze ech me ep of x nd y. Here we le he me ep o e qurer of yer, whch.25 n he mulon. 3

33 3.5 x -3 dynmc of x Fgure 5.9. he dynmc of x n hor re model from Prml CDS d on Dec. 8h, 23.2 dynmc of y Fgure 5.. he dynmc of y n neny model from Prml CDS d on Dec. 8h, 23 32

34 In he nex ep we need o derve he deermnc funcon (, ) (5.) nd (5.4). [] For (, ) n he CIR++ model, we hve: nd (, ) n equon CIR mk CIR (, ) f (, ) f (,, ) where : f P (, ) mk mk mk mk 2 2 (, ) P (, ) P (, ) CIR h h h f (,, ) 2k x 2 h( kh)(exp{ h} ) 2 h( kh)(exp{ h} ) h k 2 2 exp{ } 4 exp{ } 2 (5.23) herefore, we cn on (, ) y uung he mrke d of he zero coupon ond for dfferen mure nd ( k,,, x ) no (5.23)..35 ph funcon for hor re model Fgure 5.. he deermnc funcon n he CIR++ hor re model 33

35 A for (, ), ecue of deermnc feure, we cn e up n equon for (, ) under he condon of zero correlon. he mo rgh forwrd pproch o mplemen equon (5.4) no he dcrezon cheme: CIR (, ) ln P (,, y, ) CIR where P (,, y, ) he ond prce derved from he CIR model wh prmeer vecor : P y A B y CIR * * (,,, ) (,, )exp( (,, )) ) wh: * * 2h exp ( h ) /2 * A (,, ) * * * 2 h ( h )(exph ) * 2(exp * h ) B (,, ) * * * 2 h ( h )(exp h ) 2 2 / nd * 2 2 h 2..2 P funcon for neny model Fgure 5.2. he deermnc funcon n he CIR++ neny model 34

36 5.6.2 Correlon Effec on Inere Re nd Sochc Ineny Here we chooe dfferen o explore he effec of he correlon on he nere re nd ochc neny. We conder hree ce:. Inere re nd ochc neny hve no correlon (). 2. Inere re nd ochc neny hve negve correlon ( ). 3. Inere re nd ochc neny hve pove correlon (). A menoned n he 5.6., we genere wo correled ndrd Brownn moon ph fr. We fx one Brownn moon ph nd chnge he oher ph y vryng he correlon coeffcen eween hem. hen we cn oerve he chnge n he dynmc of y hown n he elow fgure..2 dynmc of y.8.6 Rho= Rho=- Rho= Fgure 5.3. Compron of he neny wh dfferen correlon o nere re 35

37 From he mulon reul n Fgure 5.3, we fnd ou h he hree curve re que cloe o ech oher, epeclly fer yer. herefore we cn come o he concluon h he correlon doe no ffec he dynmc of y much. In uch ce, we cn gnore he correlon eween he nere re nd neny when clrng hem no he CDS Mone Crlo Smulon Mone Crlo mehod re ochc echnque ed on he ue of rndom numer nd proly c o nvege prolem. In h projec we mplemened Mone Crlo mehod o mule he dynmc of he nere re nd ochc neny. In he mulon we ue he me Prml CDS d on Dec. 8h, 23 [] nd he correpondng derved prmeer vecor nd hown n econ 5.6. A correlon of.3 ued n h mulon. We genere, mple ph o mplemen he Mone Crlo mehod..35 ochc nere re Fgure 5.4. Mone Crlo mulon of nere re n SSRD model 36

38 .2 ochc neny Fgure 5.5. Mone Crlo mulon of ochc neny n SSRD model From he mulon reul, we dcover h he ochc nere re h ome ocllon durng he fr yer, hen dply edy ncree n he followng yer. A for he ochc neny, he pek how n he fr yer, whch men h he Prml compny fce deful cr durng he fr yer. he neny ocllng u mller fer yer. h men h he compny come no relvely edy uon compred o he fr yer. 6 Concluon In h pper we nroduced he c concep of CDS, nd explned he premum pymen, proecon pymen nd he relonhp eween he wo. hen we preened reduced form (Ineny) model ed on he me nhomogeneou Poon proce. We ued he mrke nrumen CDS nd ond prce o nfer he mpled deful prole from mrke quoe. A compuon cheme w developed o clcule he correpondng neny nd urvvl proly from he mrke d for everl compne. hen we nveged he feure of he SSRD model. We howed h when here no correlon eween he nere re nd ochc nene, he SSRD model hve he me reul wh he reduced neny model. We lo muled he dynmc of he nere re nd ochc neny ung he Mone Crlo mehod. 37

39 7 Reference nd D. Dmno Brgo nd Fo Mercuro: Inere Re Model heory nd Prcce wh Smle, Inflon nd Cred (Sprnger Fnnce, 26). 2. R. Jrrow nd S. urnull, Prcng Opon of Fnncl Secure Sujec o Deful Rk, Journl of Fnnce, 5 (995): Cred Deful Swp Prce, Inernonl Bune Mchne Corporon (IBM US), Ocoer 28, 28, BLOOMBERG (cceed My 2, 29). 4. Cred Deful Swp Prce, Inernonl Bune Mchne Corporon (IBM US), Decemer 2, 2, BLOOMBERG (cceed Decemer 2, 2). 5. Cred Deful Swp Prce, Dell Inc. (DELL US), Augu 22, 28, BLOOMBERG (cceed My 2, 29). 6. Cred Deful Swp Prce, Dell Inc. (DELL US), Decemer 2, 2, BLOOMBERG (cceed Decemer 2, 2). 38