CDO modellng from a practtoner s pont of vew: What are the real problems? Jens Lund jens.lund@nordea.com 7 March 2007
Brdgng between academa and practce The speaker Traxx, standard CDOs and conventons Gaussan copula model CDO behavour Correlaton smle Compound base correlatons Some base correlaton ssues What s a good model? Interpolaton and extrapolaton, non standard tranches/portfolos Market nformaton Hedge ratos Implementaton consderatons MC strateges, how to smulate Rsk numbers for all market data Fast recursve technques, condtonal ndependence Other model proposals What makes a good prattoner? Concluson 2
Jens Lund Head of Product Development, Nordea Markets Background: Nov 1996: M.Sc. n statstcs, Unversty of Copenhagen Feb 2000: Ph.D. n statstcs, The Royal Veternary and Agrcultural Unversty Mar 2000 onwards: wth Nordea, Product Development Has done a lot of the credt modellng work n Nordea Team: 5 members, varous degrees of experence, manly Ph.D. n natural scence, lookng for more people 2 assocated programmers helpng wth nterface to tradng system Responsble for all dervatves modellng and calculatons (NPV, rsk, ) Scrptng language for descrpton of all dervatves Interest rates, credt dervatves, nflaton, equty, 3
Traxx standard portfolo/cds Traxx Europe 125 lqud names Underlyng ndex CDSes for sectors 5Y, 7Y & 10Y maturty 5 standard CDO tranches, frst to default baskets, optons US ndex CDX 3m, act/360, last 20 date roll, CDS pay accrued fee Index composton adjusted every 6m Index CDS trades at a fxed spread wth accrued fee --- Traded wth upfront premum (but quoted on spread) Together wth last 20 date roll ths ensures lqudty and (mnus counterparty rsk) perfect nettng of trades. 4
Traxx, dstrbuton of 5y spreads 5y spread pr. name 3.00% TDC 2.72% 2.50% 2.00% 1.50% 1.00% 0.50% 0.00% 1 7 13 19 25 31 37 43 49 55 61 67 73 79 85 91 97 103 109 115 121 TDC 2 Nov 2005: 293bp -> 268bp -> 290bp, Md March at approx 270bp 5
Traxx average spreads, 5y mean = 37bp Mean spread and hazard 1.80% 1.60% 1.40% 1.20% Spread Hazard 1.00% 0.80% 0.60% 0.40% 0.20% 0.00% 0 1 2 3 4 5 6 7 8 9 10 Tenor Y 6
Average market mpled survval probablty Survval probablty 102.00% 100.00% 98.00% 96.00% 94.00% 92.00% 90.00% 88.00% 0 1 2 3 4 5 6 7 8 9 10 Tenor Y (20 dec) 7
Standardzed CDO tranches Traxx Europe 3%, 6%, 9%, 12%, 22% US ndex CDX has ponts 3%, 7%, 10%, 15%, 30% Has done a lot to provde lqudty n structured credt Relable prcng nformaton avalable Quotaton: bp runnng fee Equty tranche: 500bp runnng, quoted on upfront payment! Due to tmng rsk of events Mezzanne 78% Super senor 3% equty 100% 22% 12% 9% 6% 3% 8
Reference Gaussan copula model N credt names, = 1,,N t Default tmes: T ~ F ( t) = 1 exp λ ( u) du 0 λ curves bootstrapped from CDS quotes T correlated through the copula: F (T ) = Φ(X ) wth X = (X 1,,X N ) t ~ N(0,Σ) Σ correlaton matrx, varance 1, constant correlaton ρ Could take X = ρ M + 1 ρ Z In model: correlaton ndependent of product to be prced Σ = 1 O ρ ρ O 1 9
CDO behavour Structure: 125 name, Traxx RR almost all 40% Avg CDS = 37bp Corr = 25% Start 11-oct-2005 Spreads wth corr = 25% 0.2bp 78% Super senor 100% 5y structure, ends 20-dec-2010 Premum: 3m, act/360 Valuaton 10-oct-2005 8bp 35bp 87bp 242bp 500bp runnng + 20.55% upfront Mezzanne 3% equty 22% 12% 9% 6% 3% 10
Correlaton dependence Far spreads as functon of correlaton 50.00% 3.00% 3%->6% 40.00% 2.50% 6%->9% 9%->12% 30.00% Equty 0%->3% Upfront payment 2.00% 12%->22% 20.00% 1.50% 10.00% 1.00% 0.00% 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 0.50% -10.00% -20.00% 0.00% 0.00% 20.00% 40.00% 60.00% 80.00% 100.00% 11
CDO behavour depends on Number of names Spreads of the underlyng names Tranchng: Sze of tranche Smaller tranches are more leverage/exposed to changes Order of tranche Correlaton Recovery rate 12
Prces n the market have a correlaton smle In practce: Correlaton depends on product, 10-oct-2005, 5Y Traxx Europe Tranche Maturty Far coupons Equty upfront: 29.2% 3-6%: 0.97% 6-9%: 0.28% 9-12% 0.13% 12-22%: 0.07% Compound correlaton 30.00% 25.00% 20.00% 15.00% 10.00% 5.00% 0.00% 0%->3% 3%->6% 6%->9% 9%->12% 12%->22% Tranche 13
Compound correlatons The correlaton on the ndvdual tranches Mezzanne tranches have low correlaton senstvty and even non-unque or non-exstent correlaton for gven spreads! No way to extend to, say, 2%-5% tranche or bespoke tranches What alternatves exsts? 14
Base correlatons Base corrlaton 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 0% 5% 10% 15% 20% 25% Short Detachment pont Long 15
Steep base correlaton curves Base smle Corr Far coupons 3% 5% 33.94% 6% 30% -0.65% 9% 45% -0.13% 12% 50% 0.41% 22% 55% 0.31% Negatve spreads!! Base correlatons depends on prevous ponts Somewhat contradctng the whole dea of base tranches! 16
Are base correlatons a real soluton? No, t s merely a convenent way of descrbng prces on CDO tranches An ntermedate step towards better models that exhbt a smle No general extenson to other products No smle dynamcs Interpolaton ssues Correlaton smle modellng, versus Models wth a smle and correlaton dynamcs Base correlaton s NOT a model!! Nevertheless: they are used a lot! 17
Why have models? How to use them? We do see prces on the standard tranches n the market, so why have a model at all? Interpolaton Non standard tranches, e.g. 2%-4% Extrapolaton Attachment/detachment ponts below 3% Bespoke portfolos Other products: CDS -> CDO, CDO -> CDO 2, etc. Usually: map expected losses to fnd corr for other tranches Rsk numbers 18
Delta hedges Delta rsk: how much does the NPV change when the underlyng credt spreads wden by 1bp? CDO tranches typcal traded wth ntal credt hedge,.e. only correlaton rsk left! Convenently quoted as amount of underlyng ndex CDS to buy n order to hedge credt rsk,.e. deltacdo/deltacds Splt out on ndvdual names or just consder ndex? Base correlaton: fnd by long/short strategy n the same way as NPV! 19
Deltas n dfferent models Deltas dffer between models: Tranche 0%-3% 3%-6% 6%-9% 9%-12% 12%-22% Compound corr 22.1 9.1 2.7 1.2 0.6 Base corr 22.1 6.1 2.0 0.9 0.5 RFL 25.9 7.5 0.4 0.1 0.1 Agreement on delta amounts requres model agreement Non-unque deltas when spread & correlaton s connected,.e. n models wth smle dynamcs 20
Last 20 schedule and date roll conventon End date wll be 20th of Mar, Jun, Sep or Dec. If we have passed any of these dates we roll to the next date, so e.g. end date 21st Jun wll roll to 20st Sep, etc. Frst perod wll be long f we would otherwse get less than 1m to frst date n scheudle! Stub/long perod n the begnnng. Intermedate ponts are rolled Followng. Usually n the credt market start and end dates can fall on nonbusness days. Always start protecton the day after the trade day, even f a non busness day. 21
Rsk ladders CDS curve most often bootstrapped from yearly quotes Rsk on the yearly quotes, 1Y, 2Y, 3Y, 4Y, 5Y, 6Y, etc. However: trades end every quarter Rsk mght move around when crossng 20 Mar, Jun, Sep, Dec Example: Date QTR before roll Y before roll QTR after roll Y after roll 4Y 100 150 4.25Y Get rsk on a quarterly ladder, even though the curve s stll bootstrapped from yearly quotes. Be aware how your rsk changes on rolls. 200 4.50Y 200 4.75Y 5Y 100 50 22
Implementaton strateges Key: effcency, flexblty and fast + accurate rsk! Copula type models: Monte Carlo Recursve/FFT technques 23
Implementaton of Gaussan copula by MC Monte Carlo smulaton of X~N(0,Σ) Smulate Y~N(0,I) Fnd A such that AA =Σ X=AY How to fnd A? Cholesky decomposton Egenvalue decomposton: A=Psqrt(λ) The latter s better, n partcular wth Sobol sequences Smulaton: Smulate default tme T =F -1 (Φ(X )) for all names, and prce. Do t, say, 10000 tmes. 24 Can prce any dervatve, smple.
Rsk numbers n MC prcng Nave rsk numbers: V = V ( λ + ε) V ( λ ) λ ε For credt rsk we can exchange dfferentaton and ntegraton: V = E[ g( τ )] = g( τ ) f ( τ λ, K, λ ) dτ 1 N V ( λ,, λ + εs,, λ ) = ε g( τ ) f ( τ λ,, λ + εs,, λ ) dτ ε g( ) log f ( 1, K, + s, K, N ) ε = 0 f ( τ ) d ε K K K K N 1 N 1 ε = 0 ε = 0 = = τ τ λ λ ε λ τ E[ g( τ ) log f ( τ λ1, K, λ + εs, K, λn ) ε 0] ε = Calculate dervatve and prce n same smulaton, but for a dfferent payout functon Speed up of factor 5x125=625. Also mproves stablty! 25
Recursve/FFT mplementaton Wrte X~N(0,Σ) as Only vable for dervatves that depend on the number of defaults or the cumulatve loss (perhaps dscretsed f RR or notonals are not equal) Condtonal on M: X s are ndependent For gven horzon T, the default ndcators 1 are ndependent Calculate dstrbuton of number of defaults recursvely n N = #names Bnomal expresson Fnd loss dstrbuton by ntegratng over M Fast and no MC nose X 1 2 = am + a Z ( X C ( T )) a= ρ 26
Recursve buld-up of loss dstrbuton Condtonal on M, gven a tme horzon t: ndependence and p s the probablty name has survved up to tme t. # ssuers n p(n ssuers, k defaults) = p(n-1,k)p n + (1-p n )p(n-1,k-1) 2 1 0 p 1 p 2 (1-p 1 ) p 2 +p 1 (1-p 2 ) (1-p 1 ) (1-p 2 ) p 1 1-p 1 1 0 1 2 Next: ntegrate over M # defaults 27
28 What s the survval probablty? Let, wth X ~H The model matches quantles: F (T ) = H (X ) Ths means the condtonal survval probabltes are: 2 1 X am a Z = + ) 1 )) ( ( ( ) )) ( ( ( ) )) ( ( ( ) ( 2 1 1 1 M a am t F H Z P M t F H X P M t X H F P M t T P = = =
The search for better copulas has started... Better means descrbng the observed prces n the market for Traxx produces a correlaton smle has a reasonable low number of parameters one can have a vew on and nterpret has a plausble dynamcs for the correlaton smle constant parameters can be used on a range of tranches / products maturtes (portfolos) Effcent prcng and rsk numbers Often start from Gaussan model descrbed as a 1 factor model Computatonal effcency! 1 2 = + X am a Z 29
What makes a good practtoner? Has a lot of common sense Understands the dfference between up and down, elbow and head Understands products and markets IT knowledge Excel, Vsual Basc, system functonalty,, C++ Make thngs operatonal, streamlne repettve processes, can mplement the math so t works, etc. Mathematcal sklls Requred to develop models and make effcent mplementaton Numercal analyss, analyss, algebra, stochastc models, etc. 30
Concluson Models wll have to be developed further Smle descrpton and dynamcs Delta amounts and other relevant rsk numbers Bespoke tranches Computatonal effcency Wll have to go through a couple of teratons Market and products are changng over tme Practcal challenges Manage market data & nformaton Provde smooth nfrastructure for all the numbers/trades/etc. 31