Fast Valuation of Forward-Starting Basket Default. Swaps

Transcription

1 Fast Valuaton of Forward-Startng Basket Default Swaps Ken Jackson Alex Krenn Wanhe Zhang December 13, 2007 Abstract A basket default swap (BDS) s a credt dervatve wth contngent payments that are trggered by a combnaton of default events of the reference enttes. A forwardstartng basket default swap (FBDS) s a BDS startng at a specfed future tme. Exstng analytc or sem-analytc methods for prcng FBDS are tme consumng due to the large number of possble default combnatons before the BDS starts. Ths paper develops a fast approxmaton method for FBDS based on the condtonal ndependence framework. The method converts the prcng of a FBDS to an equvalent BDS prcng problem and combnes Monte Carlo smulaton wth an analytc approach to acheve an effectve method. Ths hybrd method s a novel technque whch can be vewed ether as a means to accelerate the convergence of Monte Carlo smulaton or as a way to estmate parameters n an analytc method that are dffcult to compute drectly. Numercal results demonstrate the accuracy and effcency of the proposed hybrd method. Ths research was supported n parted by the Natural Scences and Engneerng Research Councl (NSERC) of Canada. Department of Computer Scence, Unversty of Toronto, 10 Kng s College Road, Toronto, ON, M5S 3G4, Canada; krj@cs.toronto.edu Algorthmcs Inc., 185 Spadna Avenue, Toronto, ON, M5T 2C6, Canada; alex.krenn@algorthmcs.com Department of Computer Scence, Unversty of Toronto, 10 Kng s College Road, Toronto, ON, M5S 3G4, Canada; zhangw@cs.toronto.edu 1

2 Keywords: Credt dervatves; forward-startng basket default swaps; condtonal ndependence; hybrd methods. 1 Introducton The credt dervatve market has grown explosvely durng the last 10 years. Among these credt dervatves, the most sophstcated ones are the products assocated wth a portfolo of underlyng assets, such as basket default swaps (BDS) and collateralzed debt oblgatons (CDOs). Recently, new credt dervatves have emerged n more exotc forms, ncludng, for example, forward-startng BDS (FBDS), forward-startng CDOs (FCDOs), optons on tranches and leveraged super senor tranches. In ths paper, we concentrate on the valuaton of FBDS based on the condtonal ndependence framework. A basket default swap (BDS) s a credt dervatve, the underlyng assets of whch are corporate bonds or other assets subject to credt rsk. In an mth-to-default BDS, the protecton buyer pays a specfed rate (known as the premum or spread) on a specfed notonal prncpal perodcally untl the mth default occurs among the reference enttes or untl the maturty of the contract. If the mth default happens pror to the maturty of the BDS, the protecton seller pays the losses caused by the mth default only to the protecton buyer. A forward-startng BDS s a forward contract oblgatng the holder to buy or sell a BDS at a specfed future tme. For example, such a contract mght oblgate the holder to buy fve-year protecton on a second-to-default BDS wth 10 underlyng assets. Suppose that the contract starts one year later and the premum s 100 bass ponts (bps) per year. Durng the frst year, there s no payment between the buyer and seller. At the end of the frst year, f three names n the underlyng pool have defaulted, the forward contract oblgates the holder to enter a fve-year second-to-default BDS on the remanng seven reference enttes. The premum s 100 bps per year on the outstandng notonal value. In ths paper, we denote the maturty date of the forward contract, or equvalently the startng date of the BDS, by T; the maturty date of the BDS by T ; and the premum 2

3 dates by T, = 1,...,n, where T = T 0 < T 1 <... < T n = T. Fgure 1 llustrates possble scenaros for an mth-to-default FBDS. Whether the BDS starts or not s determned by the number of enttes left n the basket at T: f less than m names survve tll T, the contract termnates wthout any payments as shown n case (a); f at least m enttes survve tll T, the BDS starts, and the cash flows are the same as those for a BDS startng at T, as shown n cases (b) and (c). (a) Less than m enttes survve tll T 0 T 0 = T (b) At least m enttes survve tll T and the mth default does not occur n [T, T Spreads 0 T 0 = T T T T n = T 1 2 (c) At least m enttes survve tll T and the mth default occurs n [T, T Spreads 0 T 0 = T T 1 T 2 T n = T Termnal Default Payment Fgure 1: Cash flows for an mth-to-default FBDS In ths paper, f all the underlyng names n the reference pool have dentcal loss-gvendefaults, dentcal default correlatons and dentcal rsk-neutral default probabltes, the pool s called completely homogeneous. If the loss-gven-defaults are the same, the pool s termed homogeneous. Otherwse, the pool s referred to as nhomogeneous or heterogeneous. Prcng and hedgng of FBDS has become an actve research area. The most common approach s Monte Carlo smulaton. Such methods are flexble, but are computatonally expensve. Therefore, more effcent analytcal or sem-analytcal approaches are beng de- 3

4 veloped by researchers. Zhang [13 developed a condtonal squared method for FBDS: frst condtonal on the common factor, then condtonal on the outstandng pool at T. For a completely homogeneous pool, the method performs effectvely. However, for a homogeneous or nhomogeneous pool, the total number of scenaros grows exponentally wth the orgnal pool sze. Therefore, the method s extremely tme consumng: for some m, t costs more than a Monte Carlo smulaton. A smlar condtonal squared method was ntroduced for a reset tranche and a forward-startng tranche by Bahet, Mashal and Nald [4. Recently, a more effcent method for prcng a FCDO, that converts t to an equvalent CDO prcng problem, was developed by Andersen [1 and Jackson and Zhang [8, ndependently. However, due to the contract property, these effcent methods for FCDOs can not be appled to the valuaton of FBDS drectly. The object of ths paper s to develop a fast approxmaton method for the valuaton of FBDS. The method converts part of the FBDS valuaton to an equvalent BDS valuaton problem, thereby avodng the large number of possble default combnatons. Ths transformaton approach s a generc method applcable to any model based on the condtonal ndependence framework. To complete the prcng, our new method combnes Monte Carlo smulaton wth the analytc approach to obtan an accurate and effcent hybrd method. The Monte Carlo method generates a coarse approxmaton for an mportant parameter n the effcent analytc method. Ths parameter can not easly be computed drectly. Alternatvely, the analytc method can be vewed as a means to accelerate the convergence of Monte Carlo smulaton. We beleve ths s the frst tme that such a hybrd method has been proposed and appled n the quanttatve fnance area. The rest of the paper s organzed as follows. Secton 2 derves the prcng equatons for FBDS. Secton 3 descrbes the method to convert a FBDS nto an equvalent BDS. Secton 4 revews the wdely used Gaussan factor copula model. Secton 5 ntroduces a valuaton method for BDS based on the condtonal ndependence framework. Secton 6 presents two numercal examples. Secton 7 dscusses method senstvty. Secton 8 concludes the paper. 4

5 2 Prcng equatons We assume the underlyng pool contans K nstruments wth loss-gven-default N k for name k. Assume that the recovery rates are constant, and the nterest rate process s ndependent of the default process of the basket. Let d denote the expected value of the rsk-free dscount factor correspondng to T. Suppose no replacement of the underlyng assets n the pool and a constant premum or spread 1 s. Wthout loss of generalty, we assume that the default payment happens at the nearest premum date followng (or equal to) the termnal default tme, f t occurs before the contract maturty; and no accrued nterest s pad out at the termnal default tme. Let τ k denote the default tme of the kth name and set τ k =, f name k never defaults. The termnal default tme τ, whch trggers the default payment, can be expressed as a functon of ndvdual default tmes τ k. We denote the loss of the FBDS at the termnal default tme by g(n k ), τ = τ k (T, T L = 0, otherwse where g( ) s a payoff functon. Let B T denote the set of names left n the basket at T. We also denote the number of names n B T by B T and the probablty dstrbuton of B T s composton by P(B T ). The event m B T K s the event that the BDS assocated wth the FBDS actually starts. In general, the valuaton of a FBDS balances the expectaton of the present values of the premum payments (premum leg) aganst the default payments (default leg), such that EV prem = EV def. Throughout the paper, E denotes the rsk-neutral expectaton wth respect to the rsk-neutral probablty P. To compute the expectaton numercally, we ntroduce the termnal default probablty Π (k) = P(τ = τ k (T, T, m B T K) 1 Ths assumpton asssts us to compute a far spread. If we are nterested only n computng the value of the contracts, the restrcton can be relaxed and a nonconstant spread consdered. 5

6 We also defne the survval ndcator functon by 1 = 1 {τ>t, m B T K}. Its probablty dstrbuton satsfes Π = P( 1 = 1) = P ( m B T K ) Under the assumptons above, the value of the default leg satsfes K k=1 Π (k) (1) EV def = K g(n k ) k=1 n =1 d ( Π (k) Π (k) ) n 1 = =1 d K k=1 g(n k ) ( Π (k) Π (k) ) 1 (2) Smlarly, the value of the premum leg satsfes n EV prem = E [sn T 1 T j d j = s T d E [ N T 1 j=1 =1 (3) where N T s the sum of the notatonal values of all names n B T, and T = T T 1. Therefore, the far spread can be computed by s = EV n def DV01 = =1 d K k=1 g(n k) ( Π (k) Π (k) ) 1 n =1 T d E [ (4) N T 1 where DV01 n =1 T d E [ N T 1. Alternately, f the spread s set, the value of the FBDS s the dfference between the two legs: V fwd = EV prem EV def 3 FBDS to BDS Defne ˆΠ (k) = P(τ = τ k (T, T ). Then ˆΠ (k) = P(τ = τ k (T, T, m B T K) + P(τ = τ k (T, T, B T < m) = Π (k) + P(τ = τ k (T, T, B T < m) 6

7 However, f B T < m, the mth default could never happen. Therefore, P(τ = τ k (T, T, B T < m) = 0 Hence, we obtan Π (k) ndependent of B T. Most mportantly, = (k) ˆΠ. Therefore, we can compute Π (k) ˆΠ (k) through (k) ˆΠ, whch s s defned n the same form as a smlar probablty used to value BDS n [6 and [9 wth T = 0. Therefore, we can use the method for BDS to compute the key probablty Π (k) wth modfed default probabltes. That s, nstead of usng the probablty of name k defaultng before tme t, we use the probablty of name k defaultng n (T, t. The startng pool of the BDS assocated wth the FBDS s random n the orgnal problem; after the transformaton, the startng pool n the equvalent BDS contans all the K names, whch becomes certan. Therefore, we avod the large combnatoral problem due to the consderaton of all the possble startng pools n the orgnal formulaton of the problem. To llustrate the transformaton, we compute Π (k) for a smple frst-to-default FBDS and ts equvalent BDS. We assume τ k are ndependent of each other and follow an exponental dstrbuton wth constant ntensty λ k over tme. Then, P(τ k (t, t + dt) = λ k exp( λ k t)dt for an nfntesmal dt. Wthout the transformaton, the event τ = τ k (t, t + dt for the frst-to-default FBDS s equvalent to the event (τ k (t, t + dt) ( (τ j (0, T) (τ j > t + dt), j k ). Therefore, P(τ = τ k (t, t + dt) = P(τ k (t, t + dt) j k ( P(τj (0, T) + P(τ j > t + dt) ) = λ k exp( λ k t)dt j k(1 exp( λ j T) + exp( λ j (t + dt))) (5) Wth the transformaton, the modfed probablty ˆπ k (t) P(τ k t) P(τ k T) satsfes ˆπ k (t) = exp( λ k T) exp( λ k t) 7

8 The default ntensty for ˆπ k (t) s stll λ k. However, these ˆπ k (t) are for the equvalent BDS. In the BDS, the event τ = τ k (t, t + dt s equvalent to the event (τ k (t, t + dt) (τ j > t + dt, j k). Therefore, P(τ = τ k (t, t + dt) = P(τ k (t, t + dt) j k(p(τ j > t + dt)) = λ k exp( λ k t)dt j k(1 ˆπ j (t + dt)) whch s equal to (5). Therefore, Π (k) = T T P(τ = τ k (t, t+dt) n the orgnal FBDS s the same as that n the equvalent BDS, whch supports the correctness of the transformaton. Once Π (k) s known, the computaton of EV def s straghtforward followng (2). To compute EV prem or DV01, we need to compute the expectaton E [ N T 1. Suppose we know the correlaton ρ between N T and 1, then E [ N T 1 can be computed from ρ = E[ N T 1 E [ NT E [ 1 var ( N T ) var ( 1 ) (6) where E [ 1 = Π and var ( 1 ) = Π ( 1 Π ). Once Π (k) s known, the computaton of Π defned n (1) s straghtforward, snce the term P ( m B T K ) can be computed by the pool loss dstrbuton methods for CDOs, e.g., [3, [10, [5 and [7. Smlarly, the terms E [ N T and var ( NT ) are easy to compute wth known pool loss dstrbuton. Therefore, E [ N T 1 can be computed by E [ N T 1 = E [ NT Π + ρ var ( N T ) Π ( 1 Π ) (7) Hence, we can compute the premum leg value and complete the valuaton. The only unknown varable n (7) s the correlaton coeffcent ρ. We propose to use Monte Carlo smulaton to approxmate ρ. As we show later, the spread s not senstve to the value of the correlaton coeffcents. Therefore, a rough approxmaton only to the ρ s needed. Ths s an mportant property of ths applcaton whch contrbutes to the 8

9 effectveness of our hybrd method. To compute the value of ˆΠ (k) for FBDS, we need to compute the jont dstrbuton of K correlated random varables 1 {T<τk T }. One effectve approach s the condtonal ndependence framework. In the next secton, we revew the market-standard Gaussan factor copula model one example of the condtonal ndependence framework. 4 Gaussan factor copula model Due to ther tractablty, Gaussan factor copula models are wdely used to specfy a jont dstrbuton for default tmes consstent wth ther margnal dstrbuton. A one factor model was frst ntroduced by Vascek [12 to evaluate the loan loss dstrbuton, and the Gaussan copula was frst appled to mult-name credt dervatves by L [11. After that, the model was generalzed by Andersen, Sdenus and Basu [3, Andersen and Sdenus [2, Hull and Whte [5, and Laurent and Gregory [10, to name just a few. Assume the rsk-neutral (cumulatve) default probabltes π k (t) = P(τ k t), k = 1, 2,..., K are known. To generate the dependence structure of default tmes, we ntroduce random varables U k satsfyng U k = β k X + σ k ε k, for k = 1, 2,...,K (8) where X s the systematc rsk factor reflectng the health of the macroeconomc envronment; ε k are dosyncratc rsk factors, whch are ndependent of each other and also ndependent of X; the constants β k and σ k, satsfyng β 2 k + σ2 k = 1, are assumed to be known. The random varables X and ε k are assumed to follow zero-mean unt-varance dstrbutons, so the correlaton between U and U j s β β j. The default tmes τ k and the random varables U k are connected by a percentle-to- 9

10 percentle transformaton, such that π k (t) = P(τ k t) = P(U k u k (t)) where each u k (t) can be vewed as a default barrer. Thus the dependence among default tmes s captured by the common factor X. In the Gaussan factor copula model, we assume X and ε k follow standard normal dstrbutons. Consequently, each U k also follows a standard normal dstrbuton. Hence we have u k (t) = Φ 1 (π k (t)). (9) where Φ s the standard normal cumulatve dstrbuton functon. Condtonal on a partcular value x of X, the condtonal rsk-neutral default probabltes are defned as π k (t, x) P(τ k t X = x) = P(U k u k (t) X = x) (10) Substtutng (8) and (9) nto (10), we have π k (t, x) = P [ β k x + σ k ε k Φ 1 (π k (t)) [ Φ 1 (π k (t)) β k x = Φ σ k (11) Ths s an example of a condtonal ndependence framework: the default events of the names are assumed to be condtonally ndependent. Thus, the problem of correlated names s reduced to the problem of ndependent names. By (2) and (3), the mean values of the default leg and premum leg for a FBDS can be evaluated as EV def = E x [V def dφ(x) = n ( K d g(n k ) ( Π (k) (x) Π (k) 1 (x))) dφ(x) (12) =1 k=1 10

11 EV prem = s = s = = n =1 =1 T d ( E [ N T Π + ρ var ( N T ) Π ( 1 Π ) ) n ( [ T d E x NT Π (x) dφ(x) + ρ s n =1 var x (N T ) Π (x) ( 1 Π (x) ) ) dφ(x) T d ( E x [ NT Π (x) + ρ var x (N T ) Π (x) ( 1 Π (x) )) dφ(x) (13) E x [V prem dφ(x) where E x denotes the rsk-neutral expectaton wth respect to the rsk-neutral probablty P x, condtonal on X = x. For smplcty, we denote the ntegrand of (13) by E x [V prem. However, t s essental for computatonal effcency that we use the uncondtonal ρ n (13), rather than the condtonal ρ (x), as would be expected from the defnton of E x [V prem. To approxmate the ntegrals (12) and (13), we use a quadrature rule, such as the Gaussan-Legendre rule or the Gaussan-Hermte rule. Thus, for example, the ntegral (12) reduces to EV def M w m E xm [V def (14) m=1 where the w m and x m are the quadrature weghts and nodes, respectvely. Therefore, the man challenge n prcng a FBDS les n computng P x ( B T ), P x (N T ), Π (k) (x) and Π (x), condtonal on a gven value x of X. 4.1 Condtonal forward default probabltes Condtonal on a gven x, to compute the dstrbuton of (k) ˆΠ, we need to specfy the dstrbuton of 1 {T<τk T }, whch s equal to the condtonal dstrbuton of 1 {uk (T)<U k u k (T )}. To ths end, we ntroduce condtonal forward default probabltes ˆπ k (t, x) = π k (t, x) π k (T, x), for t T (15) so that the condtonal dstrbuton of 1 {T<τk T } satsfes P x (1 {T<τk T } = 1) = ˆπ k (T, x). Armed wth the condtonal forward default probabltes, the condtonal termnal default 11

12 (k) probablty ˆΠ for a FBDS can be computed usng the methods developed for BDS dscussed n the next secton. Wth the condtonal default probabltes π k (T, x), the condtonal dstrbuton of N T and B T can also be computed usng the methods for CDOs or by brute force to explore all the possble combnatons. 4.2 Condtonal forward default ntenstes Besdes the condtonal forward default probabltes, to compute the condtonal probablty ˆΠ (k) (x) by the methods for BDS, we need to ntroduce the condtonal forward default ntenstes. Assume the condtonal forward default dstrbuton that name k defaults n (T, t follows the Cox process P x (T < τ k t) = 1 exp ( Λ k (t, x)) (16) where Λ k (t, x) = t T λ k (u, x)du (17) and λ k ( ) s the condtonal forward default ntensty of the kth name. We know P x (T < τ k t) = ˆπ k (t, x) (18) where ˆπ k (t, x) s gven by (15). If we assume Λ k (t, x) s lnear between premum dates T, then (17) mples that λ k (t, x) s a pecewse constant functon, such that λ k (t, x) = λ k (T, x), for t (T 1, T Combnng ths result wth (17), we have Λ k (T, x) = Λ k (T 1, x) + λ k (T, x) T 12

13 from whch, we obtan λ k (T, x) = 1 ( ) Λ k (T, x) Λ k (T 1, x) T (19) From (16) and (18), we know Λ k (T 1, x) = ln ( 1 ˆπ k (T 1, x) ) Λ k (T, x) = ln ( 1 ˆπ k (T, x) ) Substtutng these expressons for Λ k (T 1, x) and Λ k (T, x) nto (19), we obtan λ k (T, x) = 1 T ln ( 1 ˆπk (T 1, x) 1 ˆπ k (T, x) ), for = 1, 2,..., n (20) 5 Termnal default probabltes Avalable methods for BDS nclude the convoluton technque by Laurent and Gregory [10 and the recursve method based on the order statstcs of ndvdual default tmes by Iscoe and Krenn [6. Here we revew the recursve method n [6 and use t n our numercal examples. satsfy In a frst-to-default BDS, the condtonal probabltes ˆΠ (k) (x) = P x (τ = τ k (T 1, T ) ˆΠ (k) (x) = (k) (k) ˆΠ (x) ˆΠ 1 (x) = λ k (T, x) K k=1 λ (x) k(t, x)( Π 1 Π (x) ) (21) where λ k ( ) s the condtonal forward default ntenstes defned n (20); and Π (x) = K ( k=1 1 ˆπk (T, x) ). For the mth-to-default BDS, Iscoe and Krenn [6 derve the recursve relaton between the mth-to-default and the (m 1)st-to-default contracts: (m 1)P m (B) = j k P m 1 (B [j ) (K m + 1)P m 1 (B) 13

14 where P m (B) = P(τ = τ k (T 1, T ) for the mth-to-default BDS; and B [j s the set of names obtaned by excludng name j from B. Nave mplementaton of ths recurson causes the recalculaton of the same probabltes. To avod the recalculaton, use m 1 ( ) K v 1 P m (B) = ( 1) m v 1 m v 1 v=0 J B: J =v P 1 (B [J ) where J s a subset of B and B [J = B \ J. Here, for smplcty, we gve the recurson for the uncondtonal probabltes, but a smlar recurson s also vald for the condtonal probabltes. 6 Numercal results Based on the methods descrbed above, we propose the followng steps for prcng FBDS: 1. Convert π k (T ) to condtonal default probabltes π k (T, x) usng (11) and then compute the condtonal forward default probabltes ˆπ k (T, x) usng (15); 2. Compute the condtonal dstrbuton Π (k) (x) by the recursve method n Secton 5 and P x (m B T K) and P x (N T ) by the methods for CDOs as well as Π (x) usng (1), E x [ NT and varx (N T ) from P x (N T ); 3. Run a Monte Carlo smulaton to approxmate the ρ n (6); 4. Evaluate E x [V def and E x [V prem by (12) and (13), respectvely; 5. Approxmate EV def and EV prem usng a quadrature rule (see (14)); 6. Complete the computaton usng the prcng equatons (4). The numercal experments are based on two FBDS: one s a homogeneous pool; the other s an nhomogeneous pool. The contracts are 5-year BDS startng one year later wth quarterly premum payments,.e., T = T 0 = 1, T = , for = 1,...,20. The contnuously compounded nterest rates are 4% for each T. The recovery rate of the 14

15 Name Notonal Credt Ratng β k C C C C C C C C C C Table 1: Inhomogeneous FBDS pool Tme Ratng C C C C C C C C Table 2: Part of rsk-neutral cumulatve default probabltes nstruments n the pool s 15%. The pool structure of the nhomogeneous FBDS s defned n Table 1; the homogeneous pool has the same structure except that the notonal values are 100 for all names. Part of the rsk-neutral cumulatve default probabltes for dfferent credt ratngs are lsted n Table 2. The rsk-neutral default probabltes for all tme ponts can be found at Table 3 lsts the premums for the mth-to-default FBDS (m = 1,..., 4) computed by the condtonal squared method of Zhang [13 (column Analytc ), and our fast approxmaton method descrbed above (column Approxmaton ) wth 10 3 trals n the Monte Carlo smulaton to approxmate the correlaton coeffcents ρ. The table also lsts the 95% confdence nterval of the spread computed by a Monte Carlo method (column 95% CI ). The 95% confdence nterval s computed as follows: each Monte Carlo experment conssts of 10 6 trals; we repeat each Monte Carlo experment 500 tmes; then, we compute the 95% 15

16 Pool m 95% CI Analytc Approxmaton Rel Err 1 [104.66, Homogeneous 2 [35.70, Pool 3 [14.80, [6.29, [108.79, Inhomogeneous 2 [37.23, Pool 3 [15.18, [6.35, Table 3: FBDS premums (bps) Pool Method m = 1 m = 2 m = 3 m = 4 95% CI [104.66, [35.70, [14.80, [6.29, 6.46 Homogeneous 100 [104.64, [35.82, [14.92, [6.37, 6.38 Pool 1,000 [104.88, [35.87, [14.93, [6.38, ,000 [104.96, [35.89, [14.94, [6.38, % CI [108.79, [37.23, [15.18, [6.35, 6.57 Inhomogeneous 100 [108.89, [37.35, [15.29, [6.46, 6.47 Pool 1,000 [109.14, [37.42, [15.31, [6.46, ,000 [109.23, [37.44, [15.32, [6.46, 6.47 Table 4: 95% confdence nterval comparson (bps) confdence nterval from the emprcal dstrbuton of those 500 samples. The last column of Table 3 lsts the relatve errors of the spreads computed by our fast approxmaton method, usng the spreads computed by the analytc method for the exact soluton. Table 4 compares the 95% confdence nterval computed by 10 6 trals of Monte Carlo smulaton (row 95% CI ) wth those computed by our fast approxmaton method wth 100, 1,000 and 10,000 trals (rows 100, 1,000 and 10,000, respectvely). These tables demonstrate that our fast approxmaton method s accurate for the valuaton of FBDS. For the homogeneous FBDS, the runnng tme of the Monte Carlo smulaton wth 10 6 trals s about 400 tmes slower than our fast approxmaton method; for the nhomogeneous FBDS, the runnng tme of the Monte Carlo smulaton s about 20 tmes slower than our fast approxmaton method. Our fast approxmaton method s also faster than the analytc method. For example, for the frst-to-default homogeneous FBDS, the runnng tme of the analytc method s about 40 tmes slower than our fast approxmaton method. These 16

17 comparsons demonstrate that our fast approxmaton method for FBDS outperforms both the Monte Carlo method and the analytc method. 7 Senstvty analyss Snce we use a Monte Carlo method to approxmate the correlaton coeffcents ρ, the ρ are usually not exact. Therefore, a natural queston to ask s: how senstve s the FBDS spread to small changes n the ρ? If the senstvty s weak, then our approxmaton method can obtan accurate results wth a modest amount of work. Weak senstvty s a key requrement to ensure that ths knd of hybrd method s an effectve computatonal approach. The senstvty of the spread wth respect to small changes n the ρ s determned by s ρ = s DV01 DV01 ρ = (E[V def/dv01) T d var(n T ) Π (1 DV01 Π ) = E[V def DV01 2 T d var(n T ) Π (1 Π ) T d var(nt ) Π (1 = s Π ) n j=1 T j d j (E[N T Π j + ρ j var(nt ) Π j (1 Π j )) (22) The term var(n T ) n (22) s usually much smaller than E[N T, as the underlyng names usually have credt qualtes above the nvestment grade and the dfference between the nvestment grade level and the best credt level s small. For example, n our numercal experments, var(n T )/E[N T 4%. Because of the good credt qualtes, Π > 0.5. Therefore, Π (1 Π ) < Π. To obtan an ntutve feelng for the sze of s/ ρ, we omt the relatvely small terms n the denomnator and the mnor effects of Π and T d, and approxmate (22) by s s T d var(nt ) Π (1 Π ) ρ n j=1 T j d j E[N T Π j s var(nt ) ne[n T 17

18 Pool ρ Spreads (bps) Rel Err (%) -1 [107.37, 36.31, 15.03, 6.40 [2.26, 1.14, 0.63, 0.35 Homogeneous 0 [105.36, 35.97, 14.96, 6.38 [0.34, 0.21, 0.11, [103.42, 35.65, 14.88, 6.36 [1.50, 0.70, 0.41, [112.03, 37.93, 15.43, 6.49 [2.56, 1.26, 0.70, 0.39 Inhomogeneous 0 [109.66, 37.53, 15.34, 6.47 [0.36, 0.21, 0.11, [107.39, 37.15, 15.25, 6.45 [1.72, 0.81, 0.47, 0.29 Table 5: Senstvty result Therefore, the relatve error n the spread due to the error n ρ s s s ρ var(n T ) ne[n T 2 var(n T ) ne[n T as ρ 2. Furthermore, the relatve error due to the errors n all ρ s bounded by n =1 2 var(n T ) ne[n T = 2 var(n T ) E[N T (23) To further llustrate the weak dependence of the spread, s, on the correlaton coeffcents, ρ, we compute the spreads wth all ρ set to 1, 0 or 1, respectvely. The results are lsted n Table 5, where the values nsde each parenthess correspond to the spreads and relatve errors for m = 1, 2, 3, 4, respectvely, for the homogeneous and nhomogeneous mth-to-default FBDS consdered above. From the table, we see that the maxmum relatve error for both FBDS s smaller than 3%, whch s less than the bound gven by (23). Moreover, note that takng all the ρ to be 0 gves a farly good rough approxmaton to the spread for these two examples. 8 Concluson In ths paper, we ntroduce a fast hybrd approxmaton method for FBDS. The termnal default probabltes for the FBDS are converted to the form of the termnal default probabltes for BDS. Based on the condtonal ndependence framework, computng the condtonal termnal default probabltes for FBDS are smlarly converted to computng the equvalent 18

19 termnal default probabltes for BDS. After we obtan the termnal default probabltes, the default leg s straghtforward to compute. For the premum leg, we combne a Monte Carlo smulaton for the correlaton coeffcents ρ wth an analytc method. The transformaton method avods the large combnatoral problem assocated wth exstng analytc methods. Our approach s a generc method applcable to any model based on the condtonal ndependence framework. The method s partcularly effectve for FBDS because the spread s not very senstve on the values of the ρ. The numercal results for both homogeneous and nhomogeneous FBDS show that our method s more effectve than both Monte Carlo smulaton and exstng analytc methods. We beleve that ths hybrd approach s applcable to the valuaton of other fnancal nstruments. Moreover, we avod the large combnatoral problem by applyng a transformaton to the termnal default probabltes n FBDS. Ths technque can also be appled to other products wth smlar propertes. References [1 L. Andersen. Portfolo losses n factor models: Term structures and ntertemporal loss dependence. Workng paper, avalable at September [2 L. Andersen and J. Sdenus. Extensons to the Gaussan copula: Random recovery and random factor loadngs. Journal of Credt Rsk, 1(1):29 70, [3 L. Andersen, J. Sdenus, and S. Basu. All your hedges n one basket. Rsk, 16(11):67 72, [4 P. Bahet, R. Mashal, and M. Nald. Step t up or start t forward. The Journal of Fxed Income, 16(2):33 38, [5 J. Hull and A. Whte. Valuaton of a CDO and an n th to default CDS wthout Monte Carlo smulaton. Journal of Dervatves, 12(2):8 23,

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