Saddlepoint approximations for expectations and an application to CDO pricing

Transcription

1 Saddlepoin approximaions for expecaions and an applicaion o CDO pricing Xinzheng Huang a, and Cornelis. W. Ooserlee b a ABN AMRO Bank NV, Gusav Malherlaan 0, 08PP, Amserdam, he Neherlands b CWI - Naional Research Insiue for Mahemaics and Compuer Science, Science Park 3, 098 XG Amserdam, and Delf Universiy of Technology, Delf, he Neherlands June 7, 0 Absrac We derive wo ypes of saddlepoin approximaions for expecaions in he form of EX K) +, where X is he sum of n independen random variables and K is a known consan. We esablish error convergence raes for boh ypes of approximaions in he i.i.d. case. The approximaions are furher exended o cover he case of laice variables. An applicaion of he saddlepoin approximaions o CDO pricing is presened. Inroducion We consider he saddlepoin approximaions of EX K) +, where X is he sum of n independen random variables X i, i =,...,n, and K is a known consan. The expecaion is frequenly encounered in finance and insurance. I plays an inegral role in he pricing of he Collaeralized Deb Obligaions CDO) Yang e al. 006) and Anonov e al. 005)). In opion pricing, EX K) + is he payoff of a call opion Rogers & Zane, 999). In insurance, EX K) + is known as he sop-loss premium. The expecaion is also closely conneced o EX X K, which corresponds o he expeced shorfall, also known as he ail condiional expecaion, of a credi or insurance porfolio. I plays an increasingly imporan role in risk managemen in financial and insurance insiuions. In his aricle we derive wo ypes of saddlepoin expansions for he quaniy EX K) +. The firs ype of approximaions is based on Esscher iling and he Edgeworh expansion. The resuling approximaions confirm he resuls in Anonov e al. 005), which are obained by a differen approach. Our conribuions are: ) We have provided he raes of convergence for he approximaion

2 formulas in he i.i.d. case. ) We presen explici saddlepoin approximaions for he log-reurn model considered in Rogers & Zane 999) and Suder 00). Wih our formulas only one saddlepoin needs o be compued, whereas he measure change approach employed in Rogers & Zane 999) and Suder 00) requires he calculaion of wo saddlepoins. 3) We have also provided he corresponding saddlepoin approximaions for laice variables. The laice case is largely ignored in he lieraure so far, even in applicaions where laice variables are highly relevan like, for example, he pricing of CDOs. Our main conribuion is he second ype of saddlepoin approximaions. They are derived following he approach in Lugannani & Rice 980) and Daniels 987) where he Lugannani-Rice formula o ail probabiliies was derived. The higher order version of he approximaions disinguishes iself from all exising saddlepoin approximaions by is remarkable simpliciy, high accuracy and fas convergence. The applicaion of he approximaions for laice variables o he valuaion of CDO s leads o almos exac resuls. The wo expecaions we have discussed are relaed as follows, EX X K = EX K)+ PX K) + K..) Also closely relaed funcions are EK X) + and EX X < K. The connecions are well known and we pu hem here only for compleeness. EK X) + = EX K) + EX + K, EX X < K = EX EX {X K} ) /PX < K). For simpliciy of noaion, we define C := EX K) +..) The aricle is organized as follows. In secion we recall he saddlepoin approximaions for densiies and ail probabiliies. Secion 3 reviews he exising lieraure for calculaing C and relaed quaniies by he formulas in secion. In secions 4 and 5 we derive wo ypes of formulas for he saddlepoin approximaions o C. Secion 6 gives he corresponding formulas for he laice variables. Numerical resuls are presened in secion 7, including in paricular an applicaion o CDO pricing. Densiies and ail probabiliies Daing back o Esscher 93), he saddlepoin approximaion has been recognized as a valuable ool in asympoic analysis and saisical compuing. I has found a wide range of applicaions in finance and insurance, reliabiliy heory, physics and biology. The saddlepoin approximaion lieraure so far mainly focuses on he approximaion of densiies Daniels, 954) and ail probabiliies Lugannani & Rice 980) and Daniels 987)). For a comprehensive exposiion of saddlepoin approximaions, see Jensen 995).

3 We sar wih some probabiliy space Ω, F, P). Le X i, i =...n be n independenly and idenically disribued coninuous random variables all defined on he given probabiliy space and X = n i= X i. Suppose he momen generaing funcion MGF) of X is analyic and given by M ) for in some open neighborhood of zero. The MGF of he sum X is hen simply he produc of he MGF of X i, i.e., M) = M )) n. Le κ) = log M) be he Cumulan Generaing FuncionCGF) of X. The densiy and ail probabiliy of X can be represened by he following inversion formulas f X K) = PX K) = expκ) K)d,.) expκ) K) d τ > 0)..) Throughou his paper we adop he following noaion: fn) = gn) + Ohn)) means fn) gn))/hn) is bounded as n approaches some limiing value. When appropriae we delee he Ohn)) erm and wrie fn) gn), denoing gn) as an approximaion o fn). φ ) and Φ ) denoe, respecively, he pdf and cdf of a sandard normal random variable, κ ) = log M ) be he CGF of X. µ := EX and µ = EX are he expecaion of X and X under P, T represens he saddlepoin ha gives κ T) = K/n or κ T) = K, λ r := κ r) T)/κ T) r/ is he sandardized cumulan of order r evaluaed a T, and λ,r := κ r) T)/κ T) r/, Z := T κ T) and Z := T κ T), W := sgnt) KT κt) and W := sgnt) KT/n κ T) wih sgnt) being he sign of T. I is obvious ha µ = nµ, Z = nz, W = nw, λ 3 = λ,3 / n and λ 4 = λ,4 /n. In he sequel we should wrie formulas in erms of X i.e., formulas wih subscrip such as Z, W, ec) when deriving he approximaions and sudying he order of he approximaion errors. In fac he i.i.d. assumpion is only necessary for he sudy of he error convergence raes. The approximaions are however readily applicable when he random variables X i are no idenically disribued. For his reason, we should delee he error erms once he order of 3

4 he approximaion errors has been esablished, and wrie he formulas in erms of X i.e., Z, W, ec) for boh generaliy and noaional simpliciy. The saddlepoin approximaion for densiies is given by he Daniels 954) formula: f X K) = φ ) T nw ) + λ,4 nz n 8 5λ,3 + O n ) 4 φw) T + λ ) 4 Z 8 5λ 3 =: f D..3) 4 For ail probabiliies, wo ypes of disinc saddlepoin expansions exis. The firs ype of expansion is given by PX K) = e n Z W ) Φ ) nz ) + O n e W + Z PX K) = P nλ,3 6 P λ ) 3 6 Z3 ΦZ) =: P,.4) ) Z 3 + φ nw ) λ,3 6 nz n ) + O n ) + φw) λ 3 6 Z ) =: P,.5) in he case T 0. For T < 0 similar formulas are available, see Daniels 987). The second ype of expansion is obained by Lugannani & Rice 980), wih PX K) = Φ nw ) + φ nw ) n ) ) + O n 3 Z W ΦW) + φw) Z =: P 3, W PX K) = P 3 + φ { nw ) n 3 λ,4 8 5λ,3 4 λ,3 Z P 3 + φw) Z 3 Z + W 3 Z + O λ4 8 5λ 3 4 n 5 ) } ).6) ) λ 3 Z Z 3 + W 3 =: P 4..7) Widely known as he Lugannani-Rice formula, P 3 is mos popular among he four ail probabiliy approximaions for boh simpliciy and accuracy. A good review of saddlepoin approximaions for he ail probabiliy is given in Daniels 987). 3 Measure change approaches Before we derive he formulas for EX K) +, we would like o briefly review an exising approach o approximaing he quaniy. Usually he saddlepoin 4

5 expansions for densiies or ail probabiliies are employed afer a suiable change of measure. An inversion formula similar o hose for densiies and ail probabiliies also exiss for EX K) +, which is given by E X K) + = Yang e al. 006) rewrie he inversion formula o be E X K) + = = expκ) K) d τ > 0). 3.) expκ) log K)d exp κ) K)d, 3.) where κ) = κ) log. The righ-hand side of 3.) is hen in he form of.) and he Daniels formula.3) can be used for approximaion. I should be poined ou, however, ha in his case always wo saddlepoins exis. This approach is seleced as a compeior o our approximaion formulas laer in our numerical experimens. Bounded random variables Suder 00) considers he approximaion of he expeced shorfall, in wo models of he associaed random variable. The firs model deals wih bounded random variables. Wihou loss of generaliy, we only consider he case in which X has a nonnegaive lower bound. Define he probabiliy measure Q on Ω, F) by QA) = X/µdP for A F, A hen EX X K = PX K) = {X K} XdP = µ X PX K) {X K} µ dp µ QX K). 3.3) PX K) Hence he expeced shorfall is ransformed o be a muliple of he raio of wo ail probabiliies. The MGF of X under probabiliy Q is given by M Q ) = e X X µ dp = M ) = M)κ ) µ µ as κ ) = log M) = M )/M). I follows ha κ Q ) = log M Q ) = κ) + log κ )) logµ). 3.4) For more general cases, see Suder 00), secion.6.. The saddlepoin approximaion for ail probabiliies can be applied for boh probabiliies P and Q in 3.3). A disadvanage of his approach is ha wo saddlepoins need o be deermined, as he saddlepoins under he wo probabiliy measures are generally differen. 5

6 Log-reurn model The second case in Suder 00) deals wih Ee X X K raher han wih EX X K. The expeced shorfall Ee X X K can also be wrien as a muliple of he raio of wo ail probabiliies. Define he probabiliy measure Q on Ω, F) by QA) = A ex /M)dP for A F, hen Ee X X K = e X dp = PX K) {X K} M) PX K) {X K} e X M) dp = M) QX K). 3.5) PX K) The MGF and CGF of X under probabiliy Q are given by M Q ) = e X ex M + ) dp =, M) M) κ Q ) = κ + ) κ). This also forms he basis for he approach used in Rogers & Zane 999) for opion pricing where he log-price process follows a Lévy process. Jus like he case of bounded random variables, wo saddlepoins need o be deermined for he expecaion. 4 Classical saddlepoin approximaions In he secions o follow we give, in he spiri of Daniels 987), wo ypes of explici saddlepoin approximaions for EX K) +. For each ype of approximaion, we give a lower order and a higher order version. The approximaions o EX X K hen simply follow from.). In conras o Suder 00) and Rogers & Zane 999), no measure change is required and only one saddlepoin needs o be compued. Following Jensen 995), we call his firs ype of approximaions he classical saddlepoin approximaions. Approximaion formulas for EX K) + of his ype already appeared in Anonov e al. 005), however wihou any discussion on he error erms. They are obained by means of applicaion of he saddlepoin approximaion o 3.), i.e., on he basis of he Taylor expansion of κ) K around = T. Here we provide a saisically-oriened derivaion ha employs Esscher iling and he Edgeworh expansion. Raes of convergence for he approximaions are readily available wih our approach in he i.i.d. case. Anoher advanage of our approach is ha i leads o explici saddlepoin approximaions in he log-reurn model from Suder 00), which is no possible wih he approach in Anonov e al. 005). For now we assume ha he saddlepoin = T which solves κ ) = K is posiive. The expecaion EX K) + is reformulaed under an exponenially 6

7 iled probabiliy measure, E X K) + = K x K)fx)dx = e nw K x K)e Tx K) fx)dx, 4.) where κ T) = K and fx) = fx)exptx κt)). The same exponenial iling is also applied in Robinson 98) and Daniels 987) for he approximaion of ail probabiliies. The MGF associaed wih fx) is given by M) = MT + )/MT). I immediaely follows ha he mean and variance of a random variable X wih densiy f ) are given by E X = K and V ar X) = κ T) = nκ ξ = x K)/ nκ T) and fx)dx = gξ)dξ, Eq. 4.) reads E X K) + = e nw nκ T) 0 T). By wriing ξe nz ξ gξ)dξ. 4.) For ξ wih a densiy funcion, gξ) can be approximaed uniformly by a normal disribuion such ha gξ) = φξ) + On ). The inegral in 4.) hen becomes ) ξe nz ξ gξ)dξ = ξe nz ξ φξ) + O n dξ 0 0 nz exp = ) π 0 { = nz e nz π ξe ξ+ nz ) ) dξ + O n Φ nz ) } ) + O n. 4.3) Insering 4.3) in 4.) leads o he following approximaion E X K) + { =e nw nκ T) Tnκ π T)e nz Φ nz ) } ) + O n. 4.4) By deleing he error erm in 4.4) and represening he remaining erms in quaniies relaed o X, we obain he following approximaion, { } E X K) + κ e W T) π Tκ T)e Z ΦZ) =: C. 4.5) Higher order erms ener if gξ) is approximaed by is Edgeworh expansion, 7

8 e.g., gξ) = φξ) + λ,3 6 n ξ3 3ξ) + On ). Then E X K) + = C + On ) + e nw κ T)λ,3 ξe Zξ φξ)ξ 3 3ξ)dξ 6 = C + On ) + e nw κ T)λ,3 6 0 e Z π 0 e ξ+z) = C + On ) + e n Z W κ ) { 6 Φ nz )n Z 4 + 3nZ ) φ nz )n 3 Z 3 + } nz ) T)λ,3 ξ 4 + 3ξ ) dξ 4.6) Deleing he error erm in 4.6), we ge he higher order version of he approximaion as follows, C := C + e Z W κ T) λ 3 6 { ΦZ)Z 4 + 3Z ) φz)z 3 + Z) }. 4.7) The approximaions C and C are in agreemen wih he formulas given by Anonov e al. 005). Negaive saddlepoin We have assumed ha he saddlepoin is posiive when deriving C and C in 4.5) and 4.7), or, in oher words, µ < K. If he saddlepoin T equals 0, or equivalenly, µ = K, i is sraighforward o see ha C and C boh reduce o he following formula, κ EX µ) + 0) = π =: C ) In case ha µ > K, we should work wih Y = X and EY {Y K} insead since EX {X K} = µ + E X { X K} = µ + EY {Y K}. The CGF of Y is given by κ Y ) = κ X ). The saddlepoin ha solves κ Y ) = K is T > 0, so ha C and C can be applied o Y. Noe ha κ r) Y ) = )r κ r) X ), where he superscrip r) denoes he r-h derivaive. Transforming back o X, we find he following saddlepoin approximaion o EX K) + in he case of a negaive saddlepoin, C W = µ K + e C = C e Z W { κ T)/π) + Tκ T)e Z ΦZ) }, 4.9) κ T) λ 3 6 { ΦZ)Z 4 + 3Z ) + φz)z 3 + Z) }. 4.0) 8

9 Log-reurn model revisied We now show how o deal wih he log-reurn model in Suder 00) wihou dealing wih wo probabiliy measures simulaneously. We work wih E e X {X K} which equals E e X X K PX K). Replace x in 4.) by e x and make he same change of variables, E e X {X K} = e W 0 e K+ξ nκ T) e Zξ gξ)dξ. Afer approximaing gξ) by he sandard normal densiy, we obain E e X {X K} e W Ż +K+ π 0 e ξ+ż) dξ = e W Ż +K+ Φ Ż), 4.) where Ż = T ) κ T). Equaion 4.) is basically e K P, where P is given by.4), wih Z replaced by Ż. I is easy o verify ha his approximaion is exac when X is normally disribued. A higher order approximaion would be E e X {X K} e W Ż +K+ { ΦŻ) λ ) 3 6 nż3 + λ } 3 6 n φż)ż ). 5 The Lugannani-Rice ype formulas The second ype of saddlepoin approximaions o EX K) + can be obained wih he same change of variable as was employed in secion 4 of Daniels 987), where he Lugannani-Rice formula o ail probabiliy was derived. As a resul we shall call he obained formulas Lugannani-Rice ype formulas. In his secion we derive he approximaion formulas by means of he Lauren expansion, wihou he analysis of he raes of error convergence in he i.i.d case. An alernaive lenghy) derivaion, including he analysis of he convergence, is presened in an appendix. We look a K = nx for fixed x and le κ T) = x, so ha κ T) = nκ T) = nx = K. We follow he Bleisein approach employed in Daniels 987) o approximae κ ) x over an inerval conaining boh = 0 and = T by a quadraic funcion. Here, T need no be posiive any more. Since nx = K we have W = κ T) Tx, wih W aking he same sign as T. Le w be defined beween 0 and W such ha Then we have w W ) = κ ) x κ T) + Tx. 5.) w W w = κ ) κ T), 5.) 9

10 and = 0 w = 0, = T w = W. Differeniae boh sides of 5.) once and wice o obain w dw d W dw d = κ ) κ T), ) dw + w W ) d w d d = κ ). In he neighborhood of = T or, equivalenly, w = W ) we have dw d = κ T). Noe ha µ = EX = κ 0). In he neighborhood of = 0 or, equivalenly, w = 0), we have dw d = κ 0) if T = 0, 5.3) dw d = κ T) κ 0) = x µ if T 0. W W Hence, in he neighborhood of = 0 we have w. Moreover, d dw w, κ ) d dw µ w. 5.4) The inversion formula for EX K) + can hen be formulaed as: E X K) + = e n w W w) d dw τ > 0). 5.5) dw Taking he firs hree erms of he Lauren expansion of d dw where a w = 0 gives d dw A w + A w + A 3, 5.6) A = A = γ γ d dw dw w = d dw dw = γ γ w d, 5.7) d. 5.8) The pah of inegraion, γ, races ou a circle around 0 in a counerclockwise manner. Since w and have poles of order and a = 0, respecively, we obain A = lim 0 w = w 0) = x µ W, 5.9) d A = lim 0 d = ) A 3 can now be chosen such ha he approximaion 5.6) is exac a T, where we have dw d = κ T). This leads o A 3 = T κ T) x µ ) W = x µ ) W TZ W 3. 5.) 0

11 We subsiue 5.6) in 5.5) o ge E X K) + A e n w W w) dw w + A 3 e n w W w) dw. 5.) Afer ye anoher change of variables, y = nw, he firs erm becomes A e n w W dw w) w = A τ+i n e y dy nwy y. 5.3) The inegral in 5.3) is precisely he inversion formula of EY W) +, where Y is a sandard Gaussian disribued variable. By basic calculus we find The second erm in 5.) is given by A 3 EY W) + = φw) W ΦW). 5.4) e n w W w) dw = A 3 e y nw ) dye nw n = A 3 A e nw = 3 φw). 5.5) πn n Adding up 5.3) and 5.5) we obain he higher order version of he Lugannani- Rice ype saddlepoin approximaion o he expecaion E X K) +, C 4 := µ K) ΦW) φw) + φw) + µ K) W TZ W ) This is a very compac approximaion formula ha only involves κ T), and no cumulans of higher order. In his sense he complexiy of he calculaion of C 4 is comparable o C. In he appendix we will show however ha he order of error convergence of C 4 is O n 5 denoe by C 3, is given by ). A lower order version of he approximaion, which we will C 3 := µ K) ΦW) φw). 5.7) W C 3 is an exremely nea formula requiring only he knowledge of W. More precisely, we don need) o compue κ T). The order of error convergence of C 3 is shown o be O n 3. Remark. Ineresingly, Marin 006) gives an approximaion formula for EX K) +, decomposing he expecaion o one erm involving he ail probabiliy and anoher erm involving he probabiliy densiy, E X K) + µ K)PX K) + K µ f X K). T

12 Marin 006) suggess approximaing PX K) by he Lugannani-Rice formula P 3 in.6) and f X K) by he Daniels formula f D in.3). In he i.i.d. case, his leads o an approximaion C M := nµ x)p 3 + nx µ )f D /T wih a rae of convergence n / as he firs erm has an error of order n / and he second erm has an error of order n 3/. We propose o replace P 3 by is higher order version, P 4 in.7). This gives he following formula, E X K) + C 3 + µ K)φW) W 3 λ 3 Z Z 3 ). 5.8) Equaion 5.8) is simpler han C M as λ 4 is no included. I has a rae of convergence of order n 3/. However compared o C 4, Equaion 5.8) conains a erm of λ 3 and is cerainly more complicaed o evaluae. Noe furher ha if we neglec in C M he erms of he higher order sandard cumulans λ 3 and λ 4 in f D we ge precisely C 3 as given in 5.7). For hese reasons, C 4 is o be preferred. Zero saddlepoin Daniels 987) noed ha if he saddlepoin equals T = 0, or in oher words, µ = K, he approximaions o ail probabiliy P o P 4 all reduce o PX K) = λ 30) 6 π. We would like o show ha, under he same circumsances, C 3 and C 4 also reduce o he formula C 0 in 4.8). To show ha C 3 C 0 when T = 0, we poin ou ha lim C κ 0) κ T) 3 = lim T ΦW)) φw) T. T 0 T 0 T W Noe ha when T 0, κ 0) κ T) T κ 0), T ΦW)) 0 and T W κ 0) see 5.3)). This implies ha lim T 0 C 3 = C 0. Similarly we also have lim T 0 C 4 = C 0. 6 Laice variables So far we have considered approximaions o coninuous variables. Le us now urn o he laice case. This case is largely ignored in he lieraure, even in applicaions in which laice variables are highly relevan. For example, in he pricing of CDOs, he random variable concerned is essenially he number of defauls in he pool of companies and is hus discree. Suppose ha ˆX only akes ineger values k wih nonzero probabiliies pk).

13 The inversion formula of E ˆX K) + can hen be formulaed as E ˆX K) + = k=k+ τ+iπ = = k K)pk) = τ iπ τ+iπ τ iπ k=k+ expκ) K) expκ) K) k K) me m d m= τ+iπ τ iπ e e d τ > 0). ) expκ) k)d For K > µ, we proceed by expanding he wo erms in he inegrand separaely. According o a runcaed version of Wason s Lemma see Lemma 4.5. and 4.5. in Kolassa, 006), for an inegrand in he form of exp nα T) ) j=0 T)j, he change in he conour of inegraion for from τ ±i o τ ±iπ leads o a negligible difference which is exponenially small in n. Blackwell & Hodges 959) declare furher ha he inegral over he range τ +iy where y > log n/ n is negligible. This means ha we are able o incorporae he formulas for coninuous variables C and C in he approximaions for he laice variables. We find, for laice variables, he following approximaions corresponding o C and C in 4.5) and 4.7), respecively, T e T Ĉ = C e T ), 6.) T e T Ĉ = C e T ) + e W + Z {φz) Z ΦZ)} Te T T e T Te T) κ T) e T ) 3. 6.) For he approximaions o E ˆX ˆX K, we also need he laice version for he ail probabiliy or is higher order version P ˆX K) e W + Z ΦZ) T e T =: ˆP 6.3) P ˆX K) e W + Z { ΦZ) T e T λ3 + φz) 6 Z ) + Z T Ze T ) λ 3 6 Z3 T ) e T } =: ˆP. 6.4) Recall ha he Lugannani-Rice formula for laice variables reads P ˆX Ẑ K) ΦW) + φw) =: W ˆP 3, 6.5) where Ẑ = e T ) κ T). Similar laice formulas can also be obained for C 3 and C 4, which will be denoed by Ĉ3 and Ĉ4, respecively. 3

14 We firs wrie down he inversion formula of he ail probabiliy of a laice variable, Q ˆX K) = k=k Q ˆX = k) = τ+iπ expκ Q ) K) τ iπ e d. 6.6) Combining 6.6) wih Lemma from Appendix A), we obain E ˆX{ ˆX K} = τ+iπ τ iπ κ expκ) K) ) e d. By he same change of variables as in secion 5, we have E ˆX{ ˆX K} = = τ+iπ τ iπ τ+iπ τ iπ κ )e w Ww d e dw dw e w Ww µ w + κ ) d e dw µ w dw. As in Appendix A, since lim 0 e =, his leads o Ĉ 3 = µ K) ΦW) φw) C ) W Including higher order erms we obain Ĉ 4 = Ĉ3 + φw) e T Ẑ e T ) + µ K) W ) A higher order version of ˆP 3 can be derived similarly, P ˆX Ẑ K) ΦW) + φw) + λ ) 4 8 5λ 3 4 e T λ 3 e T + e T ) Ẑ Ẑ3 W + W 3 =: ˆP ) This can be used o esimae E ˆX ˆX K. The raes of convergence of Ĉ o Ĉ4 in he i.i.d. case are idenical o heir non-laice counerpars and shall no be elaboraed furher. 7 Numerical resuls 7. Exponenial and Bernoulli variables By wo numerical experimens we evaluae he qualiy of he various approximaions derived in he earlier secions. The approach proposed by Yang e al. 4

15 006) is used as a compeior o our approximaion formulas. Since heir approach employs he saddlepoin approximaion o densiies, he approximaions for coninuous variables need no be modified for laice variables. Their firs order approximaion o C will be denoed by C Y and he second order approximaion will be denoed by C Y. The calculaion of C Y resp. C Y ) requires he nd resp. 3rd and 4h) derivaives of he funcion κ) log. As a resul, he complexiy of he calculaion of C Y and C Y is comparable o ha of C and C, respecively. In our firs example X = n i= X i where X i are i.i.d. exponenially disribued wih densiy px) = e x. The CGF of X reads κ) = n log ). The saddlepoin o κ ) = K is given by T = n/k. Moreover, we have κ T) = K n, λ 3 = n, λ 4 = 6 n. The exac disribuion is available as X Gamman, ). The ail probabiliy is hen given by γn, K) PX K) =, Γn) and EX {X K} = n γn +, K), Γn + ) where Γ and γ are he gamma funcion and he incomplee gamma funcion, respecively. We firs fix n = 00. For differen levels K, from 07 o 45, we calculae EX K) +. The expecaion decreases from 4.50 o as K increases. The ail probabiliy EX 45) is , indicaing ha we have enered he ail of he disribuion. The relaive errors of he various approximaions are illusraed in Figure. Then we fix he raio K/n =.5 and se n = 0 i for i =,...8. The expecaion decreases from 0.70 o as n increases. The ail probabiliy EX 47) is The relaive errors of he various approximaions are shown in Figure. In he second example we consider he sum of Bernoulli random variables. This is paricularly relevan for CDO pricing because he number of defauls in an underlying porfolio can be modeled by a sum of Bernoulli random variables. Consequenly, by he resuls in his example we are able o esimae, a leas parially, he performance of various approximaions for CDO pricing. We se X = n i= X i where X i are i.i.d. Bernoulli variables wih PX i = ) = PX i = 0) = p = 0.5. Is CGF is given by κ) = n log p + pe ). Here he saddlepoin o κ ) = K equals T = log and κ T) = Kn K), λ 3 = n K p) n K)p n K, λ 4 = n 6nK + 6K. nkn K) nkn K) 5

16 Relaive Error C C C 3 C 4 C Y C Y K Figure : Relaive Errors of various saddlepoin approximaions for E P n i= Xi K) + for fixed n and differen K. X i is exponenially disribued wih densiy fx) = e x x 0). n=00, K ranges from 07 o Relaive Error C 0 6 C C C 4 C Y C Y n Figure : Relaive Errors of various saddlepoin approximaions for E P n i= Xi K) + for differen n. X i is exponenially disribued wih densiy fx) = e x x 0). n = 0 i for i =,... 8, K =.5n. 6

17 In his specific case, X is binomially disribued wih ) n PX = k) = p k p) n k, k which means ha C as defined in.) can also be calculaed exacly. Similar o he exponenial case, we firs fix n = 00. For differen levels K from 6 o 30 we calculae EX K) +. The expecaion decreases from 0.4 o as K increases. The ail probabiliy EX 30) is Then we fix he raio K/n = 0. and se n = 0 i for i =,...8. The expecaion decreases from 0.98 o as n increases. The ail probabiliy EX 56) is The relaive errors of he various approximaions are presened in Figures 3 and 4, respecively. Noe ha he saddlepoin approximaions in he Bernoulli case are based on he formulas Ĉ-Ĉ4 for laice variables, derived in secion 6. In summary all approximaions work quie well in our experimens in he sense ha hey all produce small relaive errors, also in he case ha he expecaion is very small. The error convergence raes of he approximaions C -C 4 shown in Figure and Figure 4 confirm he derived heoreical convergence raes. The higher order Lugannani-Rice ype formulas, C 4 and is laice siser, are clearly he winners. They produce almos exac approximaions and have he highes error convergence rae. Moreover, he calculaion of C 4 requires he same informaion as C and Ĉ. The performance of C Y and C Y is in general comparable o C and C 3 bu inferior o C. 7. CDO ranche pricing In his secion we show how he saddlepoin approximaions can be used for he CDO ranche pricing. The value and paymens of a CDO are derived from a porfolio of fixedincome underlying asses, for example bonds. CDO securiies are spli ino differen risk classes, or ranches, and he pricing of he CDOs involves deermining he fair spread of he ranches. Deails of he CDOs can be found in Bluhm & Overbeck 007) and Hull & Whie 004). Here we focus on he calculaion of he fair spread of a CDO ranche. Le us denoe by m = m, m =,,... he paymen daes, and le L i m ) be he loss due o obligor i up o m and L m ) = L i m ) he porfolio loss. Then he fair spread of a CDO ranche wih a lower aachmen poin K and an upper aachmen poin K is given by m d0, m) EL K,K m ) EL K,K m ) s = m d0, m) K K EL K,K m ), where d0, m ) denoes he discoun facor from ime m o 0 and EL K,K m ) := EminL m, K ) EminL m, K ) 7

18 0 0 Relaive Error 0 3 C C C 3 C 4 C Y 0 4 C Y K Figure 3: Relaive Errors of various saddlepoin approximaions for E P n i= Xi K) + for fixed n and differen K. X i is Bernoulli disribued wih px i = ) = 0.5. n = 00, K ranges from 6 o Relaive Error C C C C 4 C Y C Y n Figure 4: Relaive Errors of various saddlepoin approximaions for E P n i= Xi K) + for differen n. X i is Bernoulli disribued wih px i = ) = 0.5. n = 0 i for i =,... 8, K = n/5. 8

19 represens he ranche loss a m. As EminX, K) := EX EX K) +, we obain m s = d0, m) EL m K ) + EL m K ) + EL m K ) + +EL m K ) + m d0, m)k K EL m K ) + + EL m K ) +. So we see ha he pricing of a CDO ranche can be reduced o he calculaion of EL K) + for a number of paymen daes and wo aachmen poins, which is exacly wha we have been working on in he previous secions. For simpliciy of noaion from now on we omi he subscrip ime index. Le D i be he defaul indicaor of obligor i. Assuming a consan recovery rae, λ, he loss due o obligor i is given by L i = λd i. Wih D = D i he number of defauls in he porfolio, hen we have EL K) + = E Li K) + = λe Di K/λ) + = λed K/λ) +. 7.) The quaniy K/λ is in general no an ineger. Consequenly we need o make an adjusmen before we can apply he saddlepoin approximaions for laice variables. We have, denoing by x he neares ineger ha is greaer han or equal o x, ED K/λ) + = k K/λ)PD = k) k K/λ = k K/λ )PD = k) + K/λ K/λ) k K/λ k K/λ PD = k) = EX K/λ ) + + K/λ K/λ)PD K/λ ). 7.) For example, for he aachmen poin 3% of he itraxx index wih a noional 5), and a recovery λ = 0.6, we have EL 3% 5) + = 0.6ED 3.75/0.6) + = 0.6 ED 7) PD 7). Boh he expecaion and he ail probabiliy in 7.) can be approximaed by he saddlepoin approximaions based on he same saddlepoin. Finally we subsiue 7.) in 7.). Now we consider he approximaion of 7.) in he indusrial sandard Gaussian copula model. In his model, A i, he sandardized asse reurn of counerpary i is normally disribued and can be decomposed as A i = ρy + ρǫ i, where Y is a sysemaic facor which affecs all counerparies and ǫ i is a specific risk which only affecs obligor i; ρ is called he asse correlaion. The counerpary defauls a ime if A i < c wih p = PA i < c) being he defaul probabiliy. Noe ha boh c and p are ime-dependen. We consider a homogeneous porfolio of 5 counerparies, alhough he saddlepoin approximaions can handle well inhomogeneous porfolios. An applicaion of saddlepoin approximaions o inhomogeneous credi porfolios can be found in Yang e al. 006) for CDO pricing and Huang e al. 007) for 9

20 he calculaion of he porfolio Value a Risk. We choose o work wih a homogeneous porfolio only because we can obain he exac soluion by binomial expansion in his case. For simpliciy we consider only hree paymen daes and ake he following defaul probabiliies, p ) = , p ) = 0.005, p 3 ) = Furher we assume an asse correlaion ρ = 0.3 and a consan recovery rae λ = 0.4. The homogeneiy assumpion allows us o calculae he exac ranche losses and spreads by he binomial disribuion, which can be used as benchmarks o evaluae he performance of he saddlepoin approximaions. For all sandard aachmen poins of he itraxx index, i.e., 3%, 6%, 9%, % and %, we calculae EL K) + = ELY ) K + dpy ), by approximaing he inegral by he Gauss-Legendre quadraure wih 50 nodes in he inerval Y 5, 5. In Table we presen he esimaes derived from he saddlepoin approximaions Ĉ4 and ˆP 3. In parenhesis are he relaive errors of he approximaions wih respec o he exac resuls obained wih he binomial disribuion. AP p )= p )=0.005 p 3 )=0.05 3% 6.96e e-05) e-0.06e-05).7946e e-06) 6% e-05.05e-05).59e e-06) 9.609e-0.5e-06) 9%.6686e e-06) 4.67e-03.7e-06) 5.373e e-07) % 3.798e e-06).5707e e-06) 3.055e-0.3e-06) %.5578e-0.6e-05) 7.445e e-07) e e-07) Table : The saddlepoin approximaions o EL K) + for hree paymen daes and a variey of aachmen poins AP) and heir relaive errors. Suppose ha d0, ) =.05, d0, ) =., d0, 3 ) =. and =. The saddlepoin approximaion o he spreads of various ranches in basis poins) are shown in Table. The resuls confirm he high accuracy of he saddlepoin approximaions. 8 Conclusions We have derived wo ypes of saddlepoin approximaions o EX K) +, where X is he sum of n independen random variables and K is a known consan. For each ype of approximaion, we have given a lower order as well as a higher order version. We have also esablished he error convergence raes for he approximaions in he i.i.d. case. The approximaions have been furher exended o cover he case of laice variables. Numerical examples, including in paricular an applicaion of he saddlepoin approximaions o CDO pricing, 0

21 Tranche SA Benchmark 3%, 6% %, 9% %, % %, % %, 00% Table : The saddlepoin approximaions SA) o he spreads in basis poins) of various ranches. show ha all hese approximaions work very well. The higher order Lugannani- Rice ype formulas o EX K) + are paricularly aracive because of heir remarkable simpliciy, exremely high accuracy and fas convergence. Acknowledgmen: The auhors would like o hank an anonymous referee for poining ou an elegan derivaion of he saddlepoin approximaion formula C 4. A Error Convergence of he Lugannani-Rice ype formulas In his secion we presen an alernaive derivaion of he Lugannani-Rice ype saddlepoin approximaions o EX K) +. An analysis of he error convergence of he approximaion formulas is also provided here. In his alernaive derivaion o Equaion 5.6), insead of direcly wih EX K) +, we firs work on he saddlepoin approximaions o E X {X K}, which is relaed o EX K) + in he following way, EX K) + = E X {X K} KPX K). A.) To sar, we derive he following inversion formula for E X {X K}. Lemma. Le κ) = log M) be he cumulan generaing funcion of a coninuous random variable X. Then E X {X K} = κ ) expκ) K) d τ > 0). A.) Proof. We sar wih he case ha X has a nonnegaive lower bound. Employing he same change of measure as in 3.3), we have E X {X K} = µqx K), where QX K) = expκ Q ) K) d τ > 0).

22 Subsiuing κ Q ), which is given by 3.4), we find E X {X K} = µ = exp κ) + log κ ) log µ K d κ ) expκ) K) d. In he case ha X has a negaive lower bound, a, wih a > 0, we define Y = X + a so ha Y has a nonnegaive lower bound. Then, he CGF of Y and is firs derivaive are given by κ Y ) = κ) + a and κ Y ) = κ ) + a, respecively. Since E X {X K} = E Y a){y a K} = E Y {Y a K} apy a K), and E Y {Y a K} = κ ) expκ) K) d + apy a K), we are again led o A.). For unbounded X, we ake X L = maxx, L), where L < /τ is a consan. Since X L is bounded from below, we have E X L {XL K} = = κ X L ) expκ X L ) K) d, M X L ) exp K) d, A.3) where M X L τ) = M τ) + L LeτL xe τx )dpx). For L < /τ, M X L τ) increases monoonically as L decreases and approaches M τ) as L. Noe also ha E X {X K} = E XL {XL K} for all L < K. Now ake he limi of boh sides of A.3) as L. Due o he monoone convergence heorem, we again obain E X {X K} = = M ) exp K) d κ ) expκ) K) d. We apply he same change of variables as in secion 5. Based on Lemma Le w be defined beween 0 and W such ha w W ) = κ ) x κ T) + Tx. Then we have w W w = κ ) κ T),

23 , he inversion formula for E X {X nx} can be formulaed as: E X {X nx} = = n nκ )en w W w) e n w W µ w) = nµ en w W dw w) w nw + ne W+i W i d dw dw w + κ ) e nw W) κ ) d dw µ w d dw µ w dw dw. A.4) The firs inegral akes he value Φ nw ) = ΦW). The second inegral does no have a singulariy, because of Equaion 5.4). Hence here is no problem o change he inegraion conour from he imaginary axis along τ > 0 o one along W, as done in Equaion A.4), even if W and T are boh negaive. The major conribuion o he second inegral comes from he saddlepoin. The erms in he brackes are expanded around T and inegraed o give an expansion of he form nφ nw )b n + b3 n 3 + b5 n ). A.5) By Wason s lemma his is an asympoic expansion in a neighborhood of W. For more deails see Lemma 4.5. in Kolassa 006). Coefficien b in A.5) can be obained by only aking ino accoun he leading erms of he Taylor expansion of κ ) d dw µ w = κ ) d µ +... = x µ dw T w W Z W Therefore we are led o E X {X nx} = nµ Φ nw ) + nφ nw ) A.6) x n µ ) ) + O n 3 Z W A.7) Subracing KPX K) from A.7) wih he ail probabiliy approximaed by he Lugannani-Rice formula P 3 from.6), we see immediaely ha E X nx) + = nµ x) Φ nw ) φ nw ) ) + O n 3. A.8) nw Rewrie A.7) and A.8) in quaniies relaed o X and deleing he error erms we obain he following approximaion, E K X {X K} µ ΦW) + φw) Z µ. A.9) W E X K) + µ K) ΦW) φw) =: C 3. A.0) W 3

24 Nex, we consider he coefficien b 3 in A.5). Wrie U := κ T)T κ T). The Taylor expansion of κ )/ around T gives κ ) = κ T) T + T) U T) κ + T) U T T T A.) Furhermore, we expand expnκ ) x) in he same way as Daniels 954): expnκ ) x) =exp =exp nκ T) Tx + nκ T) T) + n 6 κ T) T)3 + n 4 κ4) nκ T) Tx + nκ T) T) ) + n 6 κ T) T)3 + n 4 κ4) T) T)4 + n 7 κ T) T) We pu A.) and A.) ogeher, and have, a he line = T +iy, n T+i T i e nκ) x κ ) nw d = ne T+i T i + n 6 κ T) T)3 + n 4 κ4) T) T)4 + n { κ T) T nw = ne π + T) U T) + T + κ T) U T T 3 7 κ e nκ T) T) ) T) A.) T) T) } +... e nκ T)y n 6 κ T)iy 3 + n 4 κ4) T)y4 + { κ T) κ T) U T T 3 n 7 κ T) y iy U T T y =nφ { κ nw ) T) κ + n 3 T) nz Z x =nφw) + n 3 nz λ, λ,3 ) d + Uλ,3 Z xλ,4 5xλ,3 + xλ,3 8Z 4Z TZ Z x Z } dy λ,3 T + U Z 3 ) + O n 5 ). A.3) Noice ha A.3) is iself a saddlepoin approximaion o EX {X K} for K > µ. However, i becomes inaccurae when T approaches zero due o he presence of a pole a zero in he inegrand. Meanwhile expanding /w in he ) } + O n 5 4

25 second inegral in A.4) around W gives ne nw = nµ e nw W+i W i W+i W i =nµ φ nw ) nw e nw W) µ w dw e nw W) W w W ) W + w W ) W dw nw ) 3 + O n ) 5. A.4) Adding A.3) and A.4) o Φ nw ) and hen subracing nx imes Equaion.7), we obain { EX nx) + =nµ x) Φ nw ) + φ } nw ) nw + nφ { nw ) n 3 + µ x TZ W 3 + O n ) } 5, A.5) which can be rewrien as: EX K) + C 3 + φw) References + µ K) TZ W 3 =: C 4. A.6) Anonov, A., Mechkov, S. & Misirpashaev, T. 005), Analyical echniques for synheic CDOs and credi defaul risk measures, Technical repor, Numerix. Blackwell, D. & Hodges, J. L. 959), The probabiliy in he exreme ail of a convoluion, The Annals of Mahemaical Saisics 3, 3 0. Bluhm, C. & Overbeck, L. 007), Srucured Credi Porfolio Analysis, Baskes & CDOs, Chapman & Hall/CRC, Boca Raon. Daniels, H. E. 954), Saddlepoin approximaions in saisics, The Annals of Mahemaical Saisics 54), Daniels, H. E. 987), Tail probabiliy approximaions, Inernaional Saisical Review 55, Esscher, F. 93), On he probabiliy funcion in he collecive heory of risk, Skandinavisk Akuarieidskrif 5, Huang, X., Ooserlee, C.W. & Weide, van der, J.A.M. 007), Higher-order saddlepoin approximaions in he Vasicek porfolio credi loss model, Journal of Compuaional Finance ), Hull, J. & Whie, A. 004), Valuaion of a CDO and an nh o Defaul CDS Wihou Mone Carlo Simulaion, Journal of Derivaives,

26 Jensen, J. 995), Saddlepoin Approximaions, Oxford Universiy Press. Kolassa, J. E. 006), Series Approximaion Mehods in Saisics, 3rd ed., Springer, New York. Lugannani, R. & Rice, S. 980), Saddlepoin approximaions for he disribuion of he sum of independen random variables, Advances in Applied Probabiliy, Marin, R. 006), The saddlepoin mehod and porfolio opionaliies, RISK December), Robinson, J., Saddlepoin Approximaions for Permuaion Tess and Confidence Inervals, Journal of Royal Saisical Sociey B 44), 9 0 Rogers, L. C. G. & Zane, O. 999), Saddlepoin approximaions o opion prices, The Annals of Applied Probabiliy 9), Suder, M. 00), Sochasic Taylor expansions and saddlepoin approximaions for risk managemen, PhD hesis, ETH Zürich. Yang, J., Hurd, T. & Zhang, X. 006), Saddlepoin approximaion mehod for pricing CDOs, Journal of Compuaional Finance 0), 0. 6