DELFT UNIVERSITY OF TECHNOLOGY

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1 DELFT UNIVERSITY OF TECHNOLOGY REPORT 09-0 Saddlepoin Approximaions for Expecaions X. Huang, and C.. Ooserlee ISSN Repors of he Deparmen of Applied Mahemaical Analysis Delf 009

2 Copyrigh c 009 by Deparmen of Applied Mahemaical Analysis, Delf, The Neherlands. No par of he Journal may be reproduced, sored in a rerieval sysem, or ransmied, in any form or by any means, elecronic, mechanical, phoocopying, recording, or oherwise, wihou he prior wrien permission from Deparmen of Applied Mahemaical Analysis, Delf Universiy of Technology, The Neherlands.

3 Saddlepoin approximaions for expecaions Xinzheng Huang a,b,, and Cornelis.. Ooserlee a,c a Delf Insiue of Applied Mahemaics, Delf Universiy of Technology, Mekelweg 4, 68CD, Delf, he Neherlands b Group Risk Managemen, Rabobank, Croeselaan 8, 35CB, Urech, he Neherlands c CI - Naional Research Insiue for Mahemaics and Compuer Science, Kruislaan 43, 098 SJ, Amserdam, he Neherlands January, 009 Absrac e derive wo ypes of saddlepoin approximaions o expecaions in he form of E(X K) + and EX X K, where X is he sum of n independen random variables and K is a known consan. e 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. Inroducion e consider he saddlepoin approximaions of E(X K) + and EX X K, where X is he sum of n independen random variables X i, i =,..., n, and K is a known consan. These wo expecaions can be frequenly encounered in finance and insurance. In opion pricing, E(X K) + is he payoff of a call opion (Rogers & Zane, 999). I also plays an inegral role in he pricing of he Collaeralized Deb Obligaions (CDO) (Yang e al., 006; Anonov e al., 005). In insurance, E(X K) + is known as he sop-loss premium. The erm EX X K corresponds o he expeced shorfall, also known as he ail condiional expecaion, of a credi or insurance porfolio, which plays an increasingly imporan role in risk managemen in financial and insurance insiuions. e derive wo ypes of saddlepoin expansions for he wo quaniies. The firs ype of approximaion formulas for E(X K) + already appeared in Anonov e al. (005). e here provide a simpler and more saisically-oriened derivaion ha requires no knowledge of complex analysis. The second ype of approximaions is obained by wo disinc approaches. The resuling formulas disinguish hemselves from all exising approximaion formulas by heir remarkable simpliciy. e also 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. The laice case is largely ignored, even in applicaions where laice variables are more relevan, for example, he pricing of CDOs. Corresponding auhor; X.Huang@udelf.nl

4 The wo quaniies are relaed as follows, EX X K = E(X K)+ P(X K) + K, () E(X K) + = E X {X K} KP(X K), () EX X K = E X {X K}. P(X K) (3) I is also sraighforward o exend our resuls o he funcions E(K X) + and EX X < K. The connecions are well known and we pu hem here only for compleeness. For simpliciy of noaion, we define E(K X) + = E(X K) + EX + K, EX {X<K} = EX EX {X K}, EX X < K = ( EX EX {X K} ) /P(X < K). C := E(X K) +, S := EX X K, J := E (4) X {X K}. 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; Daniels, 987). For a comprehensive exposiion of saddlepoin approximaions, see Jensen (995). e sar wih some probabiliy space (Ω, F, P). Le X i, i =... n be n independen coninuous random variables all defined on he given probabiliy space and X = n i= X i. Suppose ha for all i, he momen generaing funcion (MGF) of X i is analyic and given by M Xi, he MGF of he sum X is hen simply he produc of he MGF of X i, i.e., M() = n M Xi (), i= for in some open neighborhood of zero. Le κ() = log M() be he Cumulan Generaing Funcion(CGF) of X. The densiy and ail probabiliy of X can be represened by he following inversion formulas f X (K) = P(X K) = Throughou his paper we adop he following noaion: exp(κ() K)d, (5) exp(κ() K) d (τ > 0). (6) φ( ) and Φ( ) denoe, respecively, he pdf and cdf of a sandard normal random variable,

5 µ := EX is he expecaion of X under P, T represens he saddlepoin ha gives κ (T ) = K, λ r := κ (r) (T )/κ (T ) r/ is he sandardized cumulan of order r evaluaed a T, Z := T κ (T ), := sgn(t ) KT κ(t ) wih sgn(t ) being he sign of T. The saddlepoin approximaion for densiies is hen given by he Daniels (954) formula f X (K) φ( ) T ( + λ ) 4 Z 8 5λ 3 =: f D. (7) 4 For ail probabiliies, wo ypes of disinc saddlepoin expansions exis. The firs ype of expansion is given by P(X K) e + Z P(X K) P ( λ 3 6 Z3 Φ(Z) =: P, (8) ) + φ( ) λ 3 ( Z ) =: P, (9) 6 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 P(X K) Φ( ) + φ( ) Z =: P 3, (0) ( ) λ4 P(X K) P 3 + φ( ) Z 8 5λ 3 λ 3 4 Z Z =: P 4. () The saddlepoin approximaions are asympoic approximaions bu hey are known o give accurae resuls in erms of relaive error even for relaively small n. hen X i are i.i.d. random variables, he rae of convergence of f D is n and he raes of convergence of P o P 4 are n /, n, n 3/, n 5/, respecively. idely 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 E(X K) + and EX X K, we would like o briefly review a differen approach o approximaing he wo quaniies. This usually involves a change of measure and borrows he saddlepoin expansions for densiies or ail probabiliies. An inversion formula similar o hose for densiies and ail probabiliies also exiss for E(X K) +, which is given by E (X K) + = exp(κ() K) d (τ > 0). () Yang e al. (006) rewrie he inversion formula o be E (X K) + = 3 exp(κ() log K)d. (3)

6 Take κ Q () = κ() log, where subscrip Q denoes a probabiliy measure differen from he original measure P, he righ-hand side of (3) is hen in he form of (5) and he Daniels formula (7) can be used for approximaion. I should be poined ou, however, ha in his case always wo saddlepoins exis. Moreover, he MGF of X under he new measure Q is problemaic as M Q (0),which suggess ha Q is no a probabiliy measure. Bounded random variables Suder (00) considers he approximaion of he expeced shorfall, in wo models of he associaed random variable. The firs case deals wih bounded random variables. ihou loss of generaliy, we only consider he case ha X has a posiive lower bound. Define he probabiliy measure Q on (Ω, F) by Q(A) = X/µdP for A F, hen A µ X EX X K = XdP = P(X K) {X K} P(X K) {X K} µ dp µ = Q(X K). (4) P(X 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 reads 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 ()) log(µ). (5) For bounded variables in general i is only necessary o apply a linear ransform on he random variable X beforehand so ha he new variable has a posiive lower bound and hus Q( ) is a valid probabiliy measure. The saddlepoin approximaion for ail probabiliy can be applied for boh probabiliies P and Q in (4). A disadvanage of his approach is ha wo saddlepoins need o be found as he saddlepoins under he wo probabiliy measures are generally differen. Log-reurn model The second case in Suder (00) deals wih Ee X X K raher han EX X K. The expeced shorfall Ee X X K can also be wrien o be a muliple of he raio of wo ail probabiliies. Define he probabiliy measure Q on (Ω, F) by Q(A) = A ex /M()dP for A F, hen Ee X X K = P(X K) {X K} e X dp = M() P(X K) {X K} e X M() dp = M() Q(X K). (6) P(X 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. 4

7 4 Classical saddlepoin approximaions In his and in he secions o follow we give, in he spiri of Daniels (987), wo ypes of explici saddlepoin approximaions for E(X K) +. For each ype of approximaion, we give a lower order version and a higher order version. The approximaions o EX X K hen simply follow from (). No measure change is required and only one saddlepoin need o be compued. Following Jensen (995), we call his firs ype of approximaions he classical saddlepoin approximaions. Approximaion formulas for E(X K) + of his ype already appeared in Anonov e al. (005). They are obained by means of rouine applicaion of he saddlepoin approximaion o (), i.e., on he basis of he Taylor expansion of κ() K around = T. Here we provide a simpler and more 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. For now we assume ha he saddlepoin = T ha solves κ () = K is posiive. The expecaion E(X K) + is reformulaed under an exponenially iled probabiliy measure, E (X K) + = (x K)f(x)dx = e (x K)e T (x K) f(x)dx, (7) K where κ (T ) = K and f(x) = f(x) exp(t x κ(t )). The MGF associaed wih f(x) is given by M() = M(T + )/M(T ). 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 ). riing ξ = (x K)/ κ (T ), Z = T κ (T ) and f(x)dx = g(ξ)dξ, (7) reads E (X K) + = e κ (T ) ξe Zξ g(ξ)dξ. (8) Suppose ha g(ξ) is approximaed by a normal disribuion φ( ). The inegral in (8) hen becomes 0 ξe Zξ g(ξ)dξ exp( Z ) π 0 0 K ξe (ξ+z) dξ = π Ze Z Φ(Z). (9) Insering (9) in (8) leads o he approximaion denoed by C, E (X K) + { } e κ (T )/(π) T κ (T )e Z Φ(Z) =: C. (0) Higher order erms ener if g(ξ) is approximaed by is Edgeworh expansion g(ξ) φ(ξ) + λ 3 6 (ξ 3 3ξ). Then E (X K) + C + e = C + e κ (T ) λ 3 6 = C + e Z κ (T ) λ 3 6 κ (T ) λ 3 6 e Z π 0 0 e (ξ+z) ξe Zξ φ(ξ)(ξ 3 3ξ)dξ ( ξ 4 + 3ξ ) dξ { Φ(Z)(Z 4 + 3Z ) φ(z)(z 3 + Z) } =: C. () The approximaions C and C are in agreemen wih he formulas given by Anonov e al. (005). Generally, bounds of relaive error are no available for he above approximaions. However in case ha X i, i =,..., n, are i.i.d. random variables, i is known ha g(ξ) = φ(ξ) + O(n / ), g(ξ) = φ(ξ) + λ 3 6 (ξ3 3ξ) + O(n ). So, he raes of convergence of C and C are of he order n / and n, respecively. 5

8 Negaive saddlepoin e have assumed ha he saddlepoin is posiive, when deriving C and C in (0) and (), 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 E(X µ) + (0) = π =: C 0. () 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 again be used. Noe ha κ (r) Y () = ( )r κ (r) X ( ), where he superscrip (r) denoes he r-h derivaive. Therefore, in he case of a negaive saddlepoin, E(X K) + can be approximaed by { } C = µ K + e κ (T )/(π) + T κ (T )e Z Φ(Z), (3) C = C e Z Log-reurn model revisied κ (T ) λ 3 6 { Φ(Z)(Z 4 + 3Z ) + φ(z)(z 3 + Z) }. (4) I is also possible o say wih he original probabiliy when approximaing he expeced shorfall in he log-reurn model in Suder (00). e work wih E e X {X K} which equals E e X X K P(X K). Replace x in (7) by e x and make he same change of variables, E e X {X K} = e 0 e K+ξ κ (T ) e Zξ g(ξ)dξ. Approximaing g(ξ) by he sandard normal densiy, we obain E e X {X K} = e Ż +K+ π 0 e (ξ+ż) dξ = e Ż +K+ Φ( Ż), (5) where Ż = (T ) κ (T ). Eq (5) is basically e K P, where P is given by (8), 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 Ż +K+ 5 The Lugannani-Rice ype formulas ( { Φ(Ż) λ ) 3 6 Ż3 + λ } 3 6 φ(ż)(ż ). The second ype of saddlepoin approximaions o E(X K) + can be derived in a very similar way as was done 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. To sar, we derive he following inversion formula for E X {X K}. 6

9 Theorem. Le κ() = log M() be he cumulan generaing funcion of a coninuous random variable X. Then E X {X K} = κ () exp(κ() K) d (τ > 0). (6) Proof. e sar wih he case ha X has a posiive lower bound. Employing he same change of measure as in (4), we have E X {X K} = µq(x K), where Q(X K) = Plug in κ Q (), which is given by (5), we find E X {X K} = µ = exp(κ Q () K) d (τ > 0). exp κ() + log κ () log µ K d κ () exp(κ() 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 posiive 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 and E X {X K} = E (Y a){y a K} = E Y {Y a K} ap(y a K), E Y {Y a K} = κ () exp(κ() K) d + ap(y a K), we are again led o (6). Exension o variables bounded from above is sraighforward. For unbounded X, we ake X L = max(x, L), where L < /τ is a consan. Since X L is bounded from below, we have E X L {XL K} = = κ X L () exp(κ X L () K) d, M X L () exp( K) d, (7) where M X L (τ) = M (τ)+ L (LeτL xe τx )dp(x). 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 (7) as L. Due o he monoone convergence heorem, we again obain E X {X K} = = M () exp( K) d κ () exp(κ() K) d. Now, we follow Daniels (987) o approximae κ() K over an inerval conaining boh = 0 and = T by a quadraic funcion. Here, T need no o be posiive any more. Recall from secion 7

10 ha = κ(t ) T K wih aking he same sign as T. Le w be defined beween 0 and such ha (w ) = κ() K κ(t ) + T K. (8) Then we have w w = κ() κ (T ), (9) and = 0 w = 0, = T w =. Differeniae boh sides of (9) once and wice o obain w dw d dw d = κ () κ (T ), ( ) dw + (w ) d w d d = κ (). So, in he neighborhood of = T (or, equivalenly, 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, (30) dw d = κ (T ) κ (0) = K µ if T 0. Hence, in he neighborhood of = 0 we have w. Moreover, d dw w, κ () d dw µ w. (3) Based on Theorem, he inversion formula for E X {X K} can be formulaed o be E X {X K} = = = µ e w w e w w dw w + κ ()e w w d dw dw µ w + κ () d dw µ dw w +i i e w w κ () d dw µ w dw. (3) The firs inegral akes he value Φ( ). The second inegral has no singulariy because of (3). Hence here is no problem o change he inegraion conour from he imaginary axis along τ > 0 o ha along, as done in (3), no even if and T are negaive. The major conribuion o he second inegral comes from he saddlepoin. The erms in he brackes are expanded around T separaely. Only aking he leading erms ino accoun we obain κ () d dw µ w κ () d µ = K dw T w Z µ. Therefore we are led o E K X {X K} µ Φ( ) + φ( ) Z µ =: J 3. (33) Subracing KP(X K) from J 3 wih he ail probabiliy approximaed by he Lugannani-Rice formula P 3, we see immediaely ha E (X K) + (µ K) Φ( ) φ( ) =: C 3. (34) 8

11 This is a surprisingly nea formula requiring only knowledge of. A more saisical approach o derive he approximaion J 3 in (33) can be found in Appendix A. Now consider he higher order approximaion. rie U := κ (T )T κ (T ). The Taylor expansion of κ ()/ around T gives κ () κ (T ) + ( T ) U ( T ) κ (T ) + U T T T T 3. (35) By he Taylor expansion on he line = T + iy, we have G := e T +i T i T +i T i e κ() K κ () d e κ (T )( T ) + 6 κ (T )( T ) κ(4) (T )( T ) { κ 7 κ (T ) ( T ) 6 (T ) T = e + e κ (T )y π 6 κ (T )iy κ(4) (T )y 4 + { κ 7 κ (T ) y 6 (T ) + iy U T T y κ (T ) U } T T 3 dy κ ( (T ) =φ( ) Z + κ (T ) λ4 Z 8 5 ) 4 λ 3 + Uλ 3 Z λ 3 T + U Z ( 3 K =φ( ) + λ 4 Z 8 5 ) 4 λ 3 + T Z Kλ 3 Z K Z 3 =: G. + ( T ) U ( T ) κ (T ) + U } T T T 3 d Noice ha G 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 around gives H := e µe =µφ( ) +i i +i i ( 3 e (w ) µ w dw e (w ) + (w ) + ) =: H. (w ) 3 dw Finally we obain he higher order version of he Lugannani-Rice ype formulas as follows, J 4 µ Φ( ) + G H, (36) C 4 C 3 + φ( ) + (µ K) T Z 3. (37) To sudy he error convergence of C 3 and C 4 when X i, i =,..., n, are i.i.d. random variables, we look a E(X nx) + for fixed x. Combining () and (3) we obain E(X nx) + = nµ Φ( ) + G H nxp(x nx). Le κ () be he CGF of X and λ,r be he r-h sandardized cumulan of X. Since κ() = nκ (), he saddlepoin T ha solves κ(t ) = nx is also a soluion of κ () = x. Now wrie µ := EX, 9

12 Z := T κ (T ) and := sgn(t ) xt κ (T ). I is obvious ha µ = nµ, Z = nz, = n, λ 3 = λ,3 / n and λ 4 = λ,4 /n. Apparenly, he remainder erm H H is of he order O(n 3/ ). Less obvious bu by ermby-erm muliplicaion and inegraion we find ha he remainder erm G G is also O(n 3/ ). More precisely, we have, G =φ( ) ( n x + xλ,4 5xλ,3 + xλ,3 Z n 8Z 4Z T Z n H =µ φ( ) n 3 + O ( n 3/). Z x Z 3 In addiion, from Daniels (987), we know ( ) P(X nx) = Φ( ) + φ( ) {n / Z + ( ) n 3/ λ,4 Z 8 5λ,3 λ,3 4 Z Z ) ( + O n 3/), ( + O n 5/)}. Subsiuion of above hree formulas gives { E(X nx) + =(µ x) n Φ( ) + n φ( ) } { + φ( ) n + µ x ( T Z 3 + O n 3/)}. (38) I follows ha he raes of convergence of C 3 and C 4 in (34) and (37) are of order O(n / ) and O(n 3/ ), respecively. Noice ha in he problem considered here, he raes of convergence of C and C, in (0) and (), are of order O() and O(n / ), respecively, since in eq. (8) he erm κ (T ) in fron of he inegral should be wrien as nκ (T ), which gives rise o an addiional n erm. Remark. Ineresingly, Marin (006) gives an approximaion formula for E(X K) +, decomposing he expecaion o one erm involving he ail probabiliy and anoher erm involving he probabiliy densiy, E (X K) + (µ K)P(X K) + K µ f X (K). T Marin (006) suggess o approximae P(X K) by he Lugannani-Rice formula P 3 in (0) and f X (K) by he Daniels formula f D in (7). In he i.i.d. case, his leads o an approximaion C M := (µ K)P 3 + (K 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/. e propose o replace P 3 by is higher order version, P 4 in (). This gives he following formula, E (X K) + C 3 + (µ K)φ( ) ( 3 λ 3 Z Z 3 ). (39) No only eq. (39) is simpler han C M as λ 4 is no involved, bu also i has a higher rae of convergence of order n 3/. However compared o C 4 eq. (39) 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 (34). For hese reasons, C 4 is o be preferred. 0

13 Zero saddlepoin I is menioned in Daniels (987) ha in case ha he saddlepoin T = 0, or in oher words, µ = K, he approximaions o ail probabiliy P o P 4 all reduce o P(X K) = λ 3(0) 6 π. e would like o show ha, under he same circumsances, C 3 and C 4 also reduce o he formula C 0 in (). To show ha C 3 = C 0 when T = 0, we poin ou ha lim C κ (0) κ (T ) 3 = lim T ( Φ( )) φ( ) T. T 0 T 0 T Noe ha when T 0, κ (0) κ (T ) T κ (0), T ( Φ( )) 0 and T κ (0) (see (30)). 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 only considered approximaions o coninuous variables. Le us now urn o he laice case. This is largely ignored in he lieraure, even in applicaions where laice variables are much more 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 p(k). The inversion formula of E( ˆX K) + can hen be formulaed as E( ˆX K) + = = = k=k+ (k K)p(k) = exp(κ() K) exp(κ() K) k=k+ me m d m= e ( e ) d. (k K) exp(κ() k)d Expanding he wo erms in he inegrand separaely, we find, for laice variables, he following formulas corresponding o C and C in (0) and (), respecively, T e T Ĉ = C ( e T ), (40) T e T Ĉ = C ( e T ) + e + T e ( T T e T T e ) T Z {φ(z) Z( Φ(Z)}. (4) κ (T )( e T ) 3 For he approximaions o E ˆX ˆX K, we also need he laice version for he ail probabiliy P( ˆX K) = e + Z Φ(Z) T e T =: ˆP (4)

14 or is higher order version P( ˆX K) =e + Z { ( Φ(Z) T e T λ3 + φ(z) 6 (Z ) + Z T Z(e T ) λ 3 6 Z3 T ) e T } =: ˆP. (43) Recall ha he Lugannani-Rice formula for laice variables reads P( ˆX Ẑ K) = Φ( ) + φ( ) =: ˆP 3, (44) where Ẑ = ( e T ) κ (T ). A similar laice formula can also be obained for J 3, which we will denoe by Ĵ3. e firs wrie down he inversion formula of he ail probabiliy of a laice variable, Q( ˆX K) = k=k Combining (45) wih Theorem, we obain E ˆX{ ˆX K} Q( ˆX = k) = exp(κ Q () K) e d. (45) = By he same change of variables as in secion 5, we have E ˆX{ ˆX K} = = κ exp(κ() K) () e d. κ ()e w w d e dw dw e w w µ w + κ () d e dw µ w dw. Now we can proceed exacly as in secion 5 as lim 0 e =. This leads o KẐ Ĵ 3 = µ Φ( ) + φ( ) µ, (46) Ĉ 3 = (µ K) Φ( ) φ( ) C 3. (47) Including higher order erms we obain e T Ĉ 4 = Ĉ3 + φ( ) Ẑ( e T ) + (µ K) 3. (48) A higher order version of ˆP 3 can be derived similarly, ( P( ˆX Ẑ K) = Φ( ) + φ( ) + λ ) 4 8 5λ 3 4 e T λ 3 e T ( + e T ) Ẑ Ẑ3 + 3 =: ˆP 4. (49) 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 we shall no elaborae furher.

15 7 Numerical resuls By wo numerical experimens we evaluae he qualiy of he various approximaions ha are derived in he earlier secions. In our firs example X = X i where X i are i.i.d. exponenially disribued wih densiy p(x) = 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. Their exac values are available as X Gamma(n, ). The ail probabiliy is hen given by and P(X K) = EX {X K} = n γ(n, K) Γ(n), γ(n +, K), Γ(n + ) where Γ and γ are he gamma funcion and he incomplee gamma funcion, respecively. In he second example we se X = X i where X i are i.i.d. Bernoulli variables wih P(X i = ) = P(X 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 ) = K( p) (n K)p K(n K), λ 3 = n n K, λ 4 = n 6nK + 6K. nk(n K) nk(n K) In his specific case, X is binomially disribued wih ( ) n P(X = k) = p k ( p) n k, k which means ha C and S as defined in (4) can also be calculaed exacly. e repor in Tables and on he approximaions obained in he exponenial case and in Tables 3 and 4 approximaions in he Bernoulli case. For he approximaions o S we ake S r = C r /P r + K for r =,, 3, 4. The saddlepoin approximaions in he Bernoulli case are based on he formulas for laice variables derived in secion 6. In general we see ha all approximaions work remarkably well in our experimens. The higher order Lugannani-Rice ype formula, S 4, C 4 and heir laice versions, produce almos exac approximaions. Paricularly worh menioning is he qualiy of approximaions C 4 and Ĉ4, ha use he same informaion as C and Ĉ, bu show errors ha are significanly smaller han C and Ĉ. 8 Conclusions e have derived wo ypes of saddlepoin approximaions o E(X K) + and EX X 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 version and a higher order version. e 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 show ha all hese approximaions work remarkably well. The Lugannani-Rice ype formulas o E(X K) + are paricularly aracive because of heir simpliciy. 3

16 K Exac C C C 3 C e- 3.79e e e e e e e e e e e e-3.388e e e e e e e-5 Table : Exac values of E(X K) + and heir saddlepoin approximaions. X = n i= X i where X i is exponenially disribued wih densiy f(x) = e x (x 0) and n = 00. K Exac S S S 3 S Table : Exac values of EX X K and heir saddlepoin approximaions. X = n i= X i where X i is exponenially disribued wih densiy f(x) = e x (x 0) and n = 00. K Exac Ĉ Ĉ Ĉ 3 Ĉ e e e e e- 0.50e-.5757e-.57e-.5397e-.509e e-.433e-.359e-.4075e-.3353e e e e e e e e e e e-4 Table 3: Exac values of E(X K) + and heir saddlepoin approximaions. X = n i= X i where X i is X i is Bernoulli disribued wih p(x i = ) = 0.5 and n = 00. K Exac Ŝ Ŝ Ŝ 3 Ŝ Table 4: Exac values of EX X K and heir saddlepoin approximaions. X = n i= X i where X i is Bernoulli disribued wih p(x i = ) = 0.5 and n = 00. 4

17 A Alernaive derivaion of J 3 The approximaion J 3 in (33) can also be derived by a more saisical approach. Le us replace he densiy of X by is saddlepoin approximaion (7), we hen obain E X {X K} x eκ() x + λ 4() 5λ 3() dx (50) π κ () 8 4 T K where x = κ (). Le again K = κ (T ). A change of variables from x o gives E X {X K} κ () κ ()e κ() κ () + λ 4() 5λ 3() d π 8 4 Le w / = κ () κ() and / = κ (T )T κ(t ) so ha wdw = κ ()d, = 0 w = 0, = T w =. A second change of variables from o w gives E X {X K} e w wκ () π + λ 4() 5λ 3() dw, κ () 8 4 which is precisely in he form of eq. (3..) in Jensen (995). According o Theorem 3.. herein, one finds E X {X K} = Φ( ) {q(0) + λ 4(0) 5λ 3(0) } + q (0) 8 4 q( ) q(0) + φ( ), (5) where q(w) = Lemma. wκ () κ (). Le q(w) = w κ (), hen q(w) = κ () q(w). q(w) = 5 6 λ 3(0) + 4 λ 3(0) 8 λ 4(0) w + O( w 3 ), q(0) =, q λ4 (0) (0) = 5λ 3(0), 8 4 = w κ (0) 3 λ 3(0)w + O( w 3 ). Proof. See Jensen(995) Lemma According o Lemma, we have q(0) = µ, q( ) = κ (T ) T κ (T ), (5) q (w) = q (w)κ () + q (w)κ () d dw + q(w) κ () where d dw = κ (0) 3 λ 3(0)w d, dw = λ 3(0). hen w = 0 we find 3 κ (0) q λ4 (0) (0) = 8 5λ 3(0) µ + 4 λ4 (0) = 5λ 3(0) 8 4 λ 3(0) 6 ( ) d + κ () d dw dw κ (0) κ (0) + κ (0) κ (0) + κ (0) λ 3(0) 3 κ (0) µ. (53), 5

18 Plugging (5) and (53) in (5) we again ge E κ (T ) X {X K} = µ Φ( ) + φ( ) T κ (T ) µ J 3. (54) References Anonov, A., Mechkov, S. & Misirpashaev, T. (005), Analyical echniques for synheic CDOs and credi defaul risk measures, Technical repor, Numerix. Daniels, H. E. (954), Saddlepoin approximaions in saisics, The Annals of Mahemaical Saisics 5(4), 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, Jensen, J. (995), Saddlepoin Approximaions, Oxford Universiy Press. 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), 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