On the Edge of Completeness

Transcription

1 On he Edge of Compleeness May 2000 Jean-Paul LAURENT Professor, ISFA Acuarial School, Universiy of Lyon, Scienific Advisor, BNP Paribas Correspondence

2 On he Edge of Compleeness: Purpose and main ideas! Purpose: risk-analysis of exoic credi derivaives: "credi coningen conracs, baske defaul swaps. pricing and hedging exoic credi derivaives.! Main ideas: disinguish beween credi spread volailiy and defaul risk. dynamic hedge of exoic defaul swaps wih sandard defaul swaps.! Reference paper: On he edge of compleeness, wih Angelo Arvaniis, RISK, Ocober 1999.

3 On he Edge of compleeness: Overview! Trading credi risk : closing he gap beween supply and demand! Modelling credi derivaives: he sae of he ar! A new approach o credi derivaives modelling: closing he gap beween pricing and hedging disenangling defaul risk and credi spread risk

4 Trading credi risk: Closing he gap beween supply and demand! From sone age o he new millennium: Technical innovaions in credi derivaives are driven by economic forces. Transferring risk from commercial banks o insiuional invesors: "Securiizaion. "Defaul Swaps : porfolio and hedging issues. "Credi Coningen Conracs, Baske Credi Derivaives. The previous means end o be more inegraed.

5 Trading credi risk: Closing he gap beween supply and demand! Securiizaion of credi risk: Credi risk seller credis SPV senior deb junior deb Invesor 1 Invesor 2! simplified scheme: No residual risk remains wihin SPV. All credi rades are simulaneous.

6 Trading Credi Risk: Closing he gap beween supply and demand! Financial inermediaries provide srucuring and arrangemen advice. Credi risk seller can ransfer loans o SPV or insead use defaul swaps! good news : low capial a risk for invesmen banks! Good imes for modelling credi derivaives No need of hedging models credi pricing models are used o ease risk ransfer need o assess he risks of various ranches

7 Trading Credi Risk: Closing he gap beween supply and demand! There is room for financial inermediaion of credi risk The ransfers of credi risk beween commercial banks and invesors may no be simulaneous. Since a one poin in ime, demand and offer of credi risk may no mach. "Meanwhile, credi risk remains wihin he balance shee of he financial inermediary. I is no furher required o find cusomers wih exac opposie ineres a every new deal. "Residual risks remain wihin he balance shee of he financial inermediary.

8 Credi risk managemen wihou hedging defaul risk! Emphasis on: porfolio effecs: correlaion beween defaul evens posing collaeral compuaion of capial a risk, risk assessmen! Main issues: capial a risk can be high wha is he compeiive advanage of invesmen banks bank Credi risk seller Defaul swap Credi derivaives rading book Defaul swap Defaul swap Invesor 1 Invesor 2

9 Credi risk managemen wih hedging defaul risk! Trading agains oher dealers enhances abiliy o ransfer credi risk by lowering capial a risk bank Credi risk seller Defaul swap Credi derivaives rading book Defaul swap Defaul swap Invesor 1 Invesor 2 Defaul swaps Credi derivaives dealer Repos Bond dealer

10 New ways o ransfer credi risk : credi coningen conracs! Anaomy of a general credi coningen conrac A credi coningen conrac is like a sandard defaul swap bu wih variable nominal (or exposure) However he periodic premium paid for he credi proecion remains fixed. The proecion paymen arises a defaul of one given single risky counerpary.! Examples "cancellable swaps "quano defaul swaps "credi proecion of vulnerable swaps, OTC opions (sandalone basis) "credi proecion of a porfolio of conracs (full proecion, excess of loss insurance, parial collaeralizaion)

11 New ways o ransfer credi risk : Baske defaul derivaives! Consider a baske of M risky bonds muliple counerparies! Firs o defaul swaps proecion agains he firs defaul! N ou of M defaul swaps (N < M) proecion agains he firs N defauls! Hedging and valuaion of baske defaul derivaives involves he join (mulivariae) modelling of defaul arrivals of issuers in he baske of bonds. Modelling accuraely he dependence beween defaul imes is a criical issue.

12 Modelling credi derivaives: he sae of he ar! Modelling credi derivaives : Where do we sand?! Financial indusry approaches Plain defaul swaps and risky bonds credi risk managemen approaches! The Noah s arch of credi risk models firm-value models risk-inensiy based models Looking desperaely for a hedging based approach o pricing.

13 Modelling credi derivaives : Where do we sand? Plain defaul swaps! Saic arbirage of plain defaul swaps wih shor selling underlying bond plain defaul swaps hedged using underlying risky bond bond srippers : allow o compue prices of risky zerocoupon bonds repo risk, squeeze risk, liquidiy risk, recovery rae assumpions! Compuaion of he P&L of a book of defaul swaps Involves he compuaion of a P&L of a book of defaul swaps The P&L is driven by changes in he credi spread curve and by he occurrence of defaul.

14 Modelling credi derivaives: Where do we sand? Credi risk managemen! Assessing he varieies of risks involved in credi derivaives Specific risk or credi spread risk "prior o defaul, he P&L of a book of credi derivaives is driven by changes in credi spreads. Defaul risk "in case of defaul, if unhedged, "dramaic jumps in he P&L of a book of credi derivaives.

15 Modelling credi derivaives: Where do we sand? The Noah s arch of credi risk models! firm-value models : Modelling of firm s asses Firs ime passage below a criical hreshold! risk-inensiy based models Defaul arrivals are no longer predicable Model condiional local probabiliies of defaul λ() d τ : defaul dae, λ() risk inensiy or hazard rae [ ] () d P [ + d[ λ = τ, τ >! Lack of a hedging based approach o pricing. Misundersanding of hedging agains defaul risk and credi spread risk

16 A new approach o credi derivaives modelling based on an hedging poin of view! Rolling over he hedge: Shor erm defaul swaps v.s. long-erm defaul swaps Credi spread ransformaion risk! Credi coningen conracs, baske defaul swaps Hedging defaul risk hrough dynamics holdings in sandard defaul swaps Hedging credi spread risk by choosing appropriae defaul swap mauriies Closing he gap beween pricing and hedging! Pracical hedging issues Uncerainy a defaul ime Managing ne residual premiums

17 Long-erm Defaul Swaps v.s. Shor-erm Defaul Swaps Rolling over he hedge! Purpose: Inroducion o dynamic rading of defaul swaps Illusraes how defaul and credi spread risk arise! Arbirage beween long and shor erm defaul swap sell one long-erm defaul swap buy a series of shor-erm defaul swaps! Example: defaul swaps on a FRN issued by BBB counerpary 5 years defaul swap premium : 50bp, recovery rae = 60% Credi derivaives dealer If defaul, 60% Unil defaul, 50 bp Clien

18 Long-erm Defaul Swaps v.s. Shor-erm Defaul Swaps Rolling over he hedge! Rolling over shor-erm defaul swap a incepion, one year defaul swap premium : 33bp cash-flows afer one year: 33 bp Credi derivaives Marke dealer 60% if defaul! Buy a one year defaul swap a he end of every yearly period, if no defaul: Dynamic sraegy, fuure premiums depend on fuure credi qualiy fuure premiums are unknown Credi derivaives dealer?? bp 60% if defaul Marke

19 Long-erm Defaul Swaps v.s. Shor-erm Defaul Swaps Rolling over he hedge! Risk analysis of rolling over shor erm agains long erm defaul swaps Credi derivaives dealer?? bp 50 bp Marke + Clien! Exchanged cash-flows : Dealer receives 5 years (fixed) credi spread, Dealer pays 1 year (variable) credi spread.! Full one o one proecion a defaul ime he previous sraegy has eliminaed one source of risk, ha is defaul risk

20 Long-erm Defaul Swaps v.s. Shor-erm Defaul Swaps Rolling over he hedge! negaive exposure o an increase in shor-erm defaul swap premiums if shor-erm premiums increase from 33bp o 70bp reflecing a lower (shor-erm) credi qualiy and no defaul occurs before he fifh year Credi derivaives dealer 70 bp 50 bp Marke + Clien! Loss due o negaive carry long posiion in long erm credi spreads shor posiion in shor erm credi spreads

21 Rolling over he hedge : porfolio of homogeneous loans P! Consider a porfolio of of homogeneous loans same uni nominal, non amorising τ i : defaul ime of counerpary i same defaul ime disribuion (same hazard rae λ()): P [ [ [ ] τ, + d τ > = λ()d i F : available informaion a ime Condiional independence beween defaul evens [ τ τ [, + d[ F ] = P τ [, + d[ i j i { τ [, d[ } i + [ F ] P[ τ [, d[ F ], + i equal o zero or o λ 2 ()(d) 2, i.e no simulaneous defauls. Remark ha indicaor defaul variables 1 are { τ i [, + d[ } (condiionally) independen and equally disribued. j

22 Rolling over he hedge : porfolio of homogeneous loans Denoe by N() he ousanding amoun of he porfolio (i.e. he number of non defauled loans) a ime. By law of large numbers, Since we ge, 1 N The ousanding nominal decays as Assume zero recovery; Toal defaul loss and +d: N()-N(+d) Cos of defaul per ousanding loan: + 1{ } λ()d () τ i [, d[ ( + d) N( ) = 1 τ i [ + d[ ( + d) N( ) = λ()d N() N, N { } N () = N( 0) exp λ() s N () N( + d) N() 0 ds = λ()d

23 Rolling over he hedge : porfolio of homogeneous loans Cos of defaul per ousanding loan = λ()d is known a ime. Insurance diversificaion approach holds Fair premium for a shor erm insurance conrac on a single loan (i.e. a shor erm defaul swap) has o be equal o λ()d. Relaes hazard rae and shor erm defaul swap premiums.! Expanding on rolling over he hedge Le us be shor in 5 years (say) defaul swaps wrien on all individual loans. " p 5Y d, periodic premium per loan. Le us buy he shor erm defaul swaps on he ousanding loans. "Corresponding premium per loan: λ()d. Cash-flows relaed o defaul evens N()-N(+d) perfecly offse

24 Rolling over he hedge : porfolio of homogeneous loans Ne (premium) cash-flows beween and +d: Where N () = N( 0) exp λ() s "Payoff similar o an index amorising swap. A incepion, p 5Y mus be such ha he risk-neural expecaion of he discouned ne premiums equals zero: Pricing equaion for he long-erm defaul swap premium p 5Y : T E exp ( ) r s ds N( ) p5y 0 0 "where r() is he shor rae a ime. Premiums received when selling long-erm defaul swaps: Premiums paid on hedging porfolio : 0 ds N() λ()d N() [ p () Y λ ]d ( λ( ) ) = 0 d 5 N() p d 5Y

25 Rolling over he hedge : porfolio of homogeneous loans! Convexiy effecs and he cos of he hedge Ne premiums paid N() [ p () Y λ ]d! Wha happens if shor erm premiums λ() become more volaile? "Ne premiums become negaive when λ() is high. " Meanwhile, he ousanding amoun N() ends o be small, miigaing he losses. "conrarily when λ() is small, he dealer experimens posiive cash-flows p 5Y - λ() on a larger amoun N().! The more volaile λ(), he smaller he average cos of he hedge and hus he long erm premium p 5Y. 5

26 Hedging exoic defaul swaps : main feaures! Exoic credi derivaives can be hedged agains defaul: Consrains he amoun of underlying sandard defaul swaps. Variable amoun of sandard defaul swaps. Full proecion a defaul ime by consrucion of he hedge. No more disconinuiy in he P&L a defaul ime. Safey-firs crieria: main source of risk can be hedged. Model-free approach.! Credi spread exposure has o be hedged by oher means: Appropriae choice of mauriy of underlying defaul swap Compuaion of sensiiviies wih respec o changes in credi spreads are model dependen.

27 Hedging Defaul Risk in Credi Coningen Conracs! Credi coningen conracs clien pays o dealer a periodic premium p T (C) unil defaul imeτ, or mauriy of he conrac T. dealer pays C(τ) o clien a defaul ime τ, if τ T. Credi derivaives dealer C(τ) if defaul p T (C) unil defaul Clien! Hedging side: Dynamic sraegy based on sandard defaul swaps: A ime, hold an amoun C() of sandard defaul swaps λ() denoes he periodic premium a ime for a shor-erm defaul swap

28 Hedging Defaul Risk in Credi Coningen Conracs! Hedging side: Credi derivaives dealer λ() C() unil defaul C(τ) if defaul Marke Amoun of sandard defaul swaps equals he (variable) credi exposure on he credi coningen conrac.! Ne posiion is a basis swap : Credi derivaives dealer λ() C() unil defaul p T (C) unil defaul Marke+Clien! The clien ransfers credi spread risk o he credi derivaives dealer

29 Closing he gap beween pricing and hedging! Wha is he cos of hedging defaul risk?! Discouned value of hedging defaul swap premiums: T E ( ) exp r + λ ( s) ds λ( ) C( ) 0 0 d Discouning erm Premium paid a ime on proecion porfolio! Equals he discouned value of premiums received by he seller: T E ( ) exp r + λ ( s) ds pt d 0 0

30 Case sudy: defaulable ineres rae swap! Consider a defaulable ineres rae swap (wih uni nominal) We are defaul-free, our counerpary is defaulable (defaul inensiy λ()). We consider a (fixed-rae) receiver swap on a sandalone basis.! Recovery assumpion, paymens in case of defaul. if defaul a ime τ, compue he defaul-free value of he swap: PV τ and ge: 0 δ 1 recovery rae, (PV τ )+ =Max(PV,0), (PV τ τ )- =Min(PV,0) τ In case of defaul, "we receive defaul-free value PV τ "minus δ ( PV ) + ( PV ) = PV ( 1 δ )( PV ) + "loss equal o (1-δ)(PV τ ) +. τ + τ τ τ

31 Case sudy: defaulable ineres rae swap! Defaulable and defaul-free swap Presen value of defaulable swap = Presen value of defaul-free swap (wih same fixed rae) Presen value of he loss. To compensae for defaul, fixed rae of defaulable swap (wih given marke value) is greaer han fixed rae of defaul-free swap (wih same marke value). Le us remark, ha defaul immediaely afer negoiaing a defaulable swap resuls in a posiive jump in he P&L, because recovery is based on defaul-free value.! To accoun for he possibiliy of defaul, we may consiue a credi reserve. Amoun of credi reserve equals expeced Presen Value of he loss This accouns for he expeced loss bu does no hedge agains realized loss.

32 Case sudy: defaulable ineres rae swap! Using a hedging insrumen raher han a credi reserve Consider a credi coningen conrac ha pays (1-δ)(PV τ )+ a defaul ime τ (if τ T), where PV τ is he presen value of a defaul-free swap wih same fixed rae han defaulable swap. Such a credi conrac + a defaulable swap synhesises a defaul-free swap (a a fixed rae equal o he iniial fixed rae): A defaul, we receive (1-δ)(PV τ )+ +PV τ -(1-δ)(PV τ )+ = PV τ The upfron premium for his credi proecion is equal o he Presen Value of he loss (1-δ)(PV τ )+ given defaul: E T 0 exp + 0 ( ) ( )( ) + r λ ( u) du λ( ) 1 δ PV d

33 Case sudy: defaulable ineres rae swap Inerpreing he cos of he hedge! Average cos of defaul on a large porfolio of swaps Large number of homogeneous defaulable receiver swaps: " Same fixed rae and mauriy; iniial nominal value N(0)=1 " independen defaul daes and same defaul inensiy λ(). Ousanding nominal amoun: Nominal amoun defauled in [, +d [: Cos of defaul in [, +d [: (N()-N(+d)) (1-δ)(PV ) + Where PV : presen value of receiver swap wih uni nominal. Aggregae cash-flows do no depend on defaul risk. N () N ( + d) = λ() dexp λ() sds Aggregae cash-flows are hose of an index amorising swap Sandard discouning provides previous slide pricing equaion N( ) = exp λ( s) ds 0 0

34 Case sudy: defaulable ineres rae swap Inerpreing he cos of he hedge! Randomly exercised swapion: Assume for simpliciy no recovery (δ=0). Inerpre defaul ime as a random ime τ wih inensiy λ(). A ha ime, defauled counerpary exercises a swapion, i.e. decides wheher o cancel he swap according o is presen value. PV of defaul-losses equals price of ha randomly exercised swapion! American Swapion PV of American swapion equals he supremum over all possible sopping imes of randomly exercised swapions. "The upper bound can be reached for special defaul arrival daes: "λ()=0 above exercise boundary and λ()= on exercise boundary

35 Case sudy: defaulable ineres rae swap! Previous hedge leads o (small) jumps in he P&L: Consider a 5,1% fixed rae defaulable receiver swap wih PV=3%. Buy previous credi coningen conrac a marke price. "Due o credi proecion, we hold a synheic defaul-free 5,1% swap. "Toal PV remains equal o 3%. Assume ha defaul immediae defaul: τ=0 +. Clearly a 5,1% defaul free swap has PV>3%, hus occurring a posiive jump in P&L.! Jumps in he P&L due o exra defaul insurance: To hedge he previous credi coningen conrac: A ime 0, we hold an amoun of shor erm defaul swap ha is equal o he Presen Value of a defaul-free 5,1% swap This amoun is greaer han 3%, he curren Presen Value.

36 Case sudy: defaulable ineres rae swap! Alernaive hedging approach: "Fixed rae of defaul-free swap wih 3% PV = 5% (say) Consider a credi coningen conrac ha pays a defaul ime: Presen value of a defaul free 5% swap minus recovered value on he 5,1% defaulable swap. a defaul ime, holder of defaulable swap + credi conrac receives: "recovery value on 5,1% defaulable swap + PV of defaul free 5% swap - recovered value on 5,1% defaulable swap " = PV of defaul free 5% swap Assume credi conrac has a periodic annual premium denoed by p. Prior o defaul ime, defaulable swap + credi conrac pays: " Defaul-free swap cash-flows wih fixed rae = 5,1%-p p mus be equal o 10bp = 5,1%-5%, oherwise arbirage wih 5% defaul-free swap.

37 Case sudy: defaulable ineres rae swap! Credi coningen conrac ransforms 5,1% defaulable swap ino a 5% defaul free swap wih he same PV. If defaul occurs immediaely, no jump in he hedged P&L. To hedge he defaul paymen on he credi coningen conrac, we mus hold defaul swaps providing paymens of: PV of defaul free 5% swap - recovery on 5,1% defaulable swap: "PV τ (5%) - δ PV τ (5.1%) + - PV τ (5.1%) - PV (5.1%) is close o PV τ τ (5%) (here 3%=PV of defaulable swap). Required paymen on hedging defaul swap close o (1- δ)pv τ (5.1%)+ "Plain defaul swap pays 1- δ a defaul ime.! Nominal amoun of hedging defaul swap almos equal o PV τ (5.1%)+

38 Hedging Defaul risk and credi spread risk in Credi Coningen Conracs! Purpose : join hedge of defaul risk and credi spread risk! Hedging defaul risk only consrains he amoun of underlying sandard defaul swap. Mauriy of underlying defaul swap is arbirary.! Choose mauriy o be proeced agains credi spread risk PV of credi coningen conracs and sandard defaul swaps are sensiive o he level of credi spreads Sensiiviy of sandard defaul swaps o a shif in credi spreads increases wih mauriy Choose mauriy of underlying defaul swap in order o equae sensiiviies.

39 Hedging credi spread risk! Example: dependence of simple defaul swaps on defaulable forward raes. Consider a T-mauriy defaul swap wih coninuously paid premium p. Assume zero-recovery (digial defaul swap). PV (a ime 0) of a long posiion provided by: T PV = E ( ) exp r + λ ( s) ds ( λ( ) p) d 0 0 where r() is he shor rae and λ() he defaul inensiy. Assume ha r(.) and λ(.) are independen. B(0,): price a ime 0 of a -mauriy defaul-free discoun bond f(0,): corresponding forward rae B(0, ) = E exp r( u) du = 0 exp 0 f (0, u) du

40 Hedging credi spread risk B( 0, ) Le be he defaulable discoun bond price and f ( 0, ) he corresponding insananeous forward rae: B( 0, ) = E exp ( r + λ) ( u) du = exp f (0, u) du 0 0 Simple expression for he PV of he T-mauriy defaul swap: T 0 ( ) PV( T) = B(0, ) f (0, ) f(0, ) p d The derivaive of defaul swap presen value wih respec o a shif of defaulable forward rae is provided by: f ( 0, ) PV () = PV () PV ( T ) + B (0,) f B ( 0, ) "PV()-PV(T) is usually small compared wih.

41 Hedging credi spread risk Similarly, we can compue he sensiiviies of plain defaul swaps wih respec o defaul-free forward curves f(0,). And hus o credi spreads. Same approach can be conduced wih he credi coningen conrac o be hedged. " All he compuaions are model dependen. Several mauriies of underlying defaul swaps can be used o mach sensiiviies. "For example, in he case of defaulable ineres rae swap, he nominal amoun of defaul swaps (PV τ ) + is usually small. "Single defaul swap wih nominal (PV τ ) + has a smaller sensiiviy o credi spreads han defaulable ineres rae swap, even for long mauriies. "Shor and long posiions in defaul swaps are required o hedge

42 Explaining hea effecs wih and wihou hedging! Differen aspecs of carrying credi conracs hrough ime. Assume hisorical and risk-neural inensiies are equal.! Consider a shor posiion in a credi coningen conrac.! Presen value of he deal provided by: T PV ( u) E u exp r + λ ( s) ds pt λ( ) C( ) u u! (afer compuaions) Ne expeced capial gain: u ( ) ( ) = d [ ( + ) ()] = ( () + λ() ) () + ( λ() () ) E PV u du PV u r u u PV u du u C u p du! Accrued cash-flows (received premiums): p T du By summaion, Incremenal P&L (if no defaul beween u and u+du): ( ) rupvudu () () + λ() u Cu () + PVu () du T

43 Explaining hea effecs wih and wihou hedging! Apparen exra reurn effec : λ( u )( C( u) + PV( u) )du Bu, probabiliy of defaul beween u and u+du: λ(u)du. Losses in case of defaul: "Commimen o pay: C(u) "Loss of PV of he credi conrac: PV(u) "PV(u) consiss in unrealised capial gains or losses in he credi derivaives book ha disappear in case of defaul. Expeced loss charge: λ( u )( C( u) + PV( u) )du! Hedging aspecs: Cu ( ) + PVu ( ) If we hold shor-erm digial defaul swaps, we are proeced a defaul-ime (no jump in he P&L). Premiums o be paid: λ( u )( C( u) + PV( u) )du Same average rae of reurn, bu smooher variaions of he P&L.

44 Hedging Defaul Risk in Baske Defaul Swaps! Example: firs o defaul swap from a baske of wo risky bonds. If he firs defaul ime occurs before mauriy, The seller of he firs o defaul swap pays he non recovered fracion of he defauled bond. Prior o ha, he receives a periodic premium.! Assume ha he wo bonds canno defaul simulaneously We moreover assume ha defaul on one bond has no effec on he credi spread of he remaining bond.! How can he seller be proeced a defaul ime? The only way o be proeced a defaul ime is o hold wo defaul swaps wih he same nominal han he nominal of he bonds. The mauriy of underlying defaul swaps does no maer.

45 Real World hedging and risk-managemen issues! uncerainy a defaul ime illiquid defaul swaps recovery risk simulaneous defaul evens! Managing ne premiums Mauriy of underlying defaul swaps Lines of credi Managemen of he carry Finie mauriy and discree premiums Correlaion beween hedging cash-flows and financial variables

46 Real world hedging and risk-managemen issues Case sudy : hedge raios for firs o defaul swaps! Consider a firs o defaul swap associaed wih a baske of wo defaulable loans. Hedging porfolios based on sandard underlying defaul swaps Uncerain hedge raios if: " simulaneous defaul evens "Jumps of credi spreads a defaul imes! Simulaneous defaul evens: If counerparies defaul alogeher, holding he complee se of defaul swaps is a conservaive (and hus expensive) hedge. In he exreme case where defaul always occur alogeher, we only need a single defaul swap on he loan wih larges nominal. In oher cases, holding a fracion of underlying defaul swaps does no hedge defaul risk (if only one counerpary defauls).

47 Real world hedging and risk-managemen issues Case sudy : hedge raios for firs o defaul swaps! Wha occurs if here is a jump in he credi spread of he second counerpary afer defaul of he firs? defaul of firs counerpary means bad news for he second.! If hedging wih shor-erm defaul swaps, no capial gain a defaul. Since PV of shor-erm defaul swaps is no sensiive o credi spreads.! This is no he case if hedging wih long erm defaul swaps. If credi spreads jump, PV of long-erm defaul swaps jumps.! Then, he amoun of hedging defaul swaps can be reduced. This reducion is model-dependen.

48 On he edge of compleeness?! Firm-value srucural defaul models: Sock prices follow a diffusion processes (no jumps). Defaul occurs a firs ime he sock value his a barrier! In his modelling, defaul credi derivaives can be compleely hedged by rading he socks: Complee pricing and hedging model:! Unrealisic feaures for hedging baske defaul swaps: Because defaul imes are predicable, hedge raios are close o zero excep for he counerpary wih he smalles disance o defaul.

49 On he edge of compleeness? hazard rae based models! In hazard rae based models : defaul is a sudden, non predicable even, ha causes a sharp jump in defaulable bond prices. Mos credi coningen conracs and baske defaul derivaives have payoffs ha are linear in he prices of defaulable bonds. Thus, good news: defaul risk can be hedged. Credi spread risk can be subsanially reduced bu no compleely eliminaed. More realisic approach o defaul. Hedge raios are robus wih respec o defaul risk.

50 On he edge of compleeness Conclusion! Looking for a beer undersanding of credi derivaives paymens in case of defaul, volailiy of credi spreads.! Bridge beween risk-neural valuaion and he cos of he hedge approach.! dynamic hedging sraegy based on sandard defaul swaps. hedge raios in order o ge proecion a defaul ime. hedging defaul risk is model-independen. imporance of quaniaive models for a beer managemen of he P&L and he residual premiums.