Which Archimedean Copula is the right one?


 Ashlie Hancock
 2 years ago
 Views:
Transcription
1 Which Archimedean is he righ one? CPA Mario R. Melchiori Universidad Nacional del Lioral Sana Fe  Argenina Third Version Sepember 2003 Published in he YieldCurve.com ejournal (www.yieldcurve.com), Ocober 2003 BICA Coop. E.M.Lda. 25 de Mayo 774 Sano omé SANTA FE Argenina  The opinions expressed in his paper are hose of he auhor and do no necessarily reflec views shared by BICA Coop. E.M.Lda. or is saff.
2 CONTENTS ABSTRACT... 2 INTRODUCTION... 2 A NORMAL WORLD... 2 RISK MANAGEMENT APPLICATION... 4 ARCHIMEDEAN COPULAS... 4 SELECTING THE RIGHT MARGINAL DISTRIBUTION... 5 DEPENDENCE. KENDALL TAU... 8 ARCHIMEDEAN BIVARIATE COPULA... 8 Algorihm:... 9 VBA code ha Generaes random variaes from he 2dimensional Gumbel copula... 9 VBA Code ha compues Inverse Cumulaive Disribuion Funcion using numerical roofinding... 0 WHICH IS THE ARCHIMEDEAN COPULA RIGHT ONE?... VBA Code ha compues Kendall using Tau nonparameric esimaion... 3 VBA Code ha compues he pseudoobservaions T {number of ( X X ) X } 2j X2 i < / ( n ) for i,2,..., n. = < such ha j < i and i j i =... 3 X X NUMERICAL EXAMPLE... 3 The Pricing Firsadefaul Algorihm:... 4 CONCLUSIONS... 5 REFERENCES... 6 APPENDIX A OTHERS COPULA S PARAMETERS... 7 Clayon s Parameers... 7 Frank s Parameers... 7 APPENDIX B KENDALL τ REVISITED... 7 APPENDIX C  USING SIMTOOLS FEATURES IN VBA PROGRAMS... 8 APPENDIX D TAIL DEPENDENCE.... 8
3 2 Which Archimedean he righ one? Mario R. Melchiori 2 Absrac This paper presens he concep of copula from a pracical sandpoin. Given he widened use of he mulinormal disribuion, we argue is inadequacy, while advocae for using he copula as an alernaive and beer approach. We examine wha he copulas are used for wihin of risk managemen. Then we expose a guide o choose boh he margins and he Archimedean copula ha beer fi o daa. In addiion, we provide an algorihm o simulae random bivariae from Archimedean copula. In order o cover he gap beween he heory and is pracical implemenaion VBA codes are provided. They are used in a numerical example ha illusraes he use of he copula in he pricing of a firsadefaul conrac. Two spreadshees accompany o paper, by presening sep by sep all pracical applicaions covered. Inroducion Keywords:, Kendall Tau, Dependence, and Credi Derivaives Since Li (2000) firs inroduced copulas ino defaul modeling, here has been increasing ineres in his approach. Unil ha momen, he copula concep was used frequenly in survival analysis and acuaries sciences. Following o Li (2000), a copula is a funcion ha links univariae marginals o heir full mulivariae disribuion. For muniform random variables 2 U, U,..., U m he join disribuion funcion C is defined as: Cu (, u,..., u, ρ ) = Pr U u, U u,..., U u 2 m 2 2 m m where ρ 3 is a dependence parameer, can also o be called a copula funcion. can be used o link marginal disribuions wih a join disribuion. For deerminae univariae marginal disribuion funcions F ( x), F2 ( x2),..., F m ( x m ), he funcion CF ( ( x ), F ( x ),..., F ( x ) ) = F ( x, x,..., x ) 2 2 m m 2 m which is defined using a copula funcion C, resuls in a mulivariae disribuion funcion wih univariae marginal disribuions specified by F ( x), F2 ( x2),..., F m ( x m ). Sklar (959) esablished he converse. He showed ha any join disribuion funcion F can be seen as a copula funcion. He proved ha if F ( x, x2,..., xm) is a join mulivariae disribuion funcion wih univariae marginal disribuion funcions, hen here exiss a copula funcion Cu (, u2,..., u m ) such ha F ( x, x,..., x ) = CF ( ( x ), F ( x ),..., F ( x ) ). 2 m 2 2 m m If eachf i is coninuous hen C is unique. Thus, copula funcions provide an unifying and flexible way o sudy join disribuions. Anoher imporan derivaion is ha he copula allows us o model he dependence srucure independenly from he marginal disribuions. In his paper, we will focus he bivariae copula funcion Cuv (, ) for uniform variables U and V, defined over he area { ( uv, ) 0 < u,0 < v } A normal world is commonly adoped boh in marke risk models and credi risk one, eiher explicily or implicily, when he models do use of he mulinormal disribuion. The commercial credi risk models KMV and CrediMerics use ha. From he copula s poin of view he mulinormal disribuion has normal marginal disribuion and Gaussian copula dependence. 2 I am graeful o Arcady Novosyolov, Carina Srada, Glyn Holon, Luciano Alloai and Moorad Choudhry for heir generous conribuion. All remaining errors are, of course, my own. I wan o hank o Mohamoud B Dualeh for encouraging me o wrie his paper. 3 As Embrechs e. al. (200) show, he correlaion is only a limied descripion of he dependence beween random variables, excep for he mulivariae normal disribuion where he correlaion fully describes he dependence srucure.
4 3 Hereafer, we will use he erm Normal for he univariae marginal disribuions and he erm Gaussian referring o he copula dependence. The advanage of using normal dependence srucure doesn arise, as should be suppose, from hisorical behavior of he financial nor credi marke, bu in is simpliciy, analyical manageabiliy and he easy esimaion he is only parameer, he correlaion marix. Empirical evidence suggess ha he use of mulinormal disribuion is inadequae 4. The nonnormaliy of univariae and mulivariae equiy reurns is hisorically unmisakable. In oher words, here is clear evidence ha equiy reurns have uncondiional fa ails, o wi, he exreme evens are more probable han anicipaed by normal disribuion, no only in marginals bu also in higher dimensions. This is imporan boh for marke risk models as credi risk one, where equiy reurns are used as a proxy for asse reurns ha follow a mulivariae normal disribuion, and, herefore, defaul imes have a mulivariae normal dependence srucure as well. As Embrechs e. al. (200) show, here many pifalls o he normaliy assumpion. For us, he main snare is he small probabiliy of exreme join evens. In credi risk case, defauls are rare evens, so ha he ail dependence has a grea impac on he defaul srucure. Tail dependence can be measure. The ail dependence for wo random variables F and Y X and Y wih marginal disribuions X F measures he probabiliy ha Y will have a realizaion in he ail of is disribuion, condiioned ha X has had a realizaion in is own ail. Tail dependence relaes he amoun of dependence in he upper righ quadran ail or lower lef one of a bivariae disribuion, so we could have upper ail dependence, lower ail dependency or boh. Upper ail dependence exiss when here is a probabiliy ha posiive ouliers happen joinly. Upper ail dependence is defined as: where λ ( limpr Y F ( u) X F ( u) ) = (.) upper Y X u F denoes he inverse cumulaive disribuion funcion and u is an uniform variable defined over ( 0, ). ( Y X ) Since Pr Y F ( u) X F ( u) can be wrien as: Pr( X FX ( u) ) Pr Y FY ( u) + Pr X FX u, Y FY u ( ) ( ( ) ( ) ) Pr( X FX ( u) ) (.2) given ha: and ( Pr X F ( ) ) Pr( X u Y FY ( u) ) u = = (.3) ( Pr X F ( ) X u, Y FY ( u) ) Cuu (, ) = (.4) an alernaive and equivalen definiion (for coninuous random variables) of (.), is he following: λ upper = 2 u + Cuu (, ) lim u u (.5) Lower ail dependence is symmerically defined: ( Y X ) Since Pr Y F ( u) X F ( u) λ ( limpr Y F ( u) X F ( u) ) = (.6) Lower Y X u 0 can be wrien as: 4 See R. Mashal, M. Naldi, and A. Zeevi. The Dependence Srucure of Asse Reurns hp://www.gsb.columbia.edu/faculy/azeevi/papers/equiy AsseRevised.pdf. Forhcoming, Risk.
5 4 Pr ( X FX ( u), Y FY ( u) ) Pr( X FX ( u) ) (.7) given (.3) and (.4) an alernaive and equivalen definiion (for coninuous random variables) of (.6), is he following: λ Lower Cuu (, ) lim u 0 u = (.8) The Gaussian copula wih correlaion ρ < does no have lower ail dependence nor upper one (, ) I is imporan o remark ha: he ail area dependency measure (, ) marginal disribuions. Lower upper λ λ. Lower upper λ λ depends on he copula and no on he NonGaussian copulas such as and Archimedean used as underlying dependence srucure wih anyone else marginal disribuion, have upper ail dependence, lower ail dependency or boh, so ha, hey could describe beer he realiy of he behavior of he financial and credi markes. See he appendix D for a nonparameric esimaion of he ail dependence 5. Risk Managemen Applicaion Unil here, we have seen wha he copula is and why he mulinormal disribuion is no an adequae assumpion. Now, we show wha he copula is used for wihin risk managemen. As i already was said, firs was used frequenly in survival analysis and acuaries sciences. In addiion, is employed in loss aggregaion, sress esing, defaul modeling and operaional risk. Hereafer, we concenrae he use of he copula in he defaul modeling scope, more concreely, in he Credi Derivaives one. Defaul risk has been exensively modeled a an individual level, bu lile is known abou defaul risk a a porfolio level where he defaul dependence is a meaningful aspec for considering. Furher, in recen years have appeared new financial insrumens, such as collaeralized deb obligaions (CDOs), n h odefaul baske, ha have coningen payoffs on he join defaul behavior of he underling securiies. In he case of n h odefaul baske, he join dependence is of vial imporance in is pricing, because he amoun of names are no large enough o ensure a correc diversificaion. Ahead, we give an example o illusrae he use of copula in he valuaion of firsodefaul conrac. The appropriae choice of he marginal disribuion is needed bu no enough o accuraely measure and price he risk exposure a a porfolio level, in addiion is criical o undersand and o model he defaul dependence o choose he fied join disribuion among he underling securiies. Archimedean copulas We will focus our aenion o one special class of copula ermed Archimedean one. An Archimedean copula can be wrien in he following way: ( ) ϕ Cu,..., u = ϕ ( u ) ϕ ( u ) (.9) n for all 0 u,..., u and where ϕ is a funcion ermed generaor, saisfying: n 0; ϕ ( ) = ' for all ( 0, ), ϕ ( ) < 0, his is o say ϕ is decreasing; for all ( 0, ), ϕ ( ) 0, his is o say ϕ is convex. n 5 For a formal calculaion of (.) see EMBRECHTS, P., A. J. MCNEIL and D. STRAUMANN (999): Correlaion and Dependence in Risk Managemen: Properies and Pifalls  hp://www.mah.ehz.ch/~srauman/preprins/pifalls.pdf 
6 5 Examples of bivariae Archimedean copulas are he following: Produc or Independen copula: ( ) ln ; C ( uv, ) uv. (.0) ϕ = = Clayon copula 6 ϕ ( ) =, > 0; ( + ) Gumbel copula 7 Frank copula 8 ϕ Cu v (.) ( lnu) + ( lnv) = = (.2) ( ) ( ln ), ; Cuv (, ) e ( ) ( ) ( ) u v e e e ( ) = ln, ; Cuv (, ) = ln + e e ϕ R (.3) The mehod described ahead is able o selec he Archimedean copula, which fis beer real daa. An Archimedean copula has he analyical represenaion given by equaion (.9). So, in order o selec he copula, i is sufficien o idenify he generaor ( ) ϕ. Selecing he righ marginal disribuion Suppose you have wo hisorical ime series compound by 000 observed daa over a period of ime, like his: 9 Series Series , Firs, i is necessary o deermine in wha manner he series are marginally disribued. For doing his, we using he char, oher daa can o reques more sophisicaed approach 0. Commercial simulaion sofware such as Crysal Ball 2 supply ools for fiing hisorical daa o deerminae probabiliy disribuion. 6 Clayon (978), CookJohnson (98), Oakes (982). 7 Gumbel (960), Hougaard (986). 8 Frank (979). 9 In he conex of his paper hese series can be considered as equiy reurns ha are used as a proxy for asse reurns. Firs, we invesigae he marginal disribuion of each series and hen we inquire which is he dependence among hem. 0 A nex paper will inroduce some of he approach such as Maximum likelihood Esimaion Decisioneering, Inc.  hp://www.decisioneering.com/  offers he Crysal Ball line of spreadshee modeling sofware for imeseries forecasing, risk analysis, and opimizaion using Mone Carlo simulaion. 2 Palisade Corporaion  hp://www.palisade.com/  develops applicaions for risk and decision analysis using Mone Carlo simulaion and opimizaion, All are addins o Excel
7 6 Which Archimedean is he righ one? Series Series 2 Average Sd Dev Sd Err Max Min Quanile 95% Series Bins Frequency Series 2 Bins Frequency
8 7 Hisogram % Frequency (5) (4) (3) (2) () 0 Bins % 0.0% 5.0% 0.0% Series : Hisogram of he hisorical realizaion. Frequency Normal Sandard Series 2: Hisogram of he hisorical realizaion. Daa and plos show ha he Normal Sandard Probabiliy Disribuion is a fied elecion in his case. Knowing he marginal disribuion, we are able o separae marginal behavior and dependence srucure. The dependence srucure is fully described by he join disribuion of uniform variaes obained from he marginal disribuions, Normal Sandard Disribuions in our case. This poin is of fundamenal imporance and ofen cause considerably roubles. Remember, dependence srucure doesn derive from he marginal disribuions, Normal Sandard in his example, bu from he uniform variaes obained from he marginal disribuions. We jus need o know marginal disribuions so ha o recognize he cumulaive disribuion funcions (CDF) ha allows us o compue he uniform variae. For example: Series Series Φ ( ) = (.4) Φ ( ) = (.5) where Φ denoes he normal cumulaive disribuion funcion. In Excel language: = NormSDis( ) = (.6) =NormSDis( ) = (.7)
9 8 he dependence srucure refers o he relaionship beween and Now, wha if he marginal disribuion is Lognormal? Jus mus use he correc CDF (LogNormDis funcion in Excel). Dependence. Kendall Tau τ For invesigaing more deeply he dependence we need a measure for gauging i. I is known as Kendall τ (Tau). I is a rank correlaion measure, i is invarian under sricly increasing ransformaions of he underlying random variables. Linear correlaion (or Pearson s correlaion ( ρ ) ) is mos frequenly used in pracice as a measure of dependence, bu i lacks his propery. If we call c and d respecively he numbers of pairs of variables, which are concordan and discordan, hen Kendall s Tau wries : c d τ = = pc pd (.8) 3 c + d where p and p are respecively he probabiliies of concordance and discordance. c d Le V = X Y be a vecor of wo random variables a ime. In our = concordan if ( X ) ( ) > 0 Xs Y Y s. Conversely, if we have ( X ) ( ) < 0 Xs Y Y s, V, for example. Then, wo disinc observaions V and case V and V s are V are discordan s (i.e. : negaively dependen). Calculaing he lineal correlaion from boh marginal and uniform disribuion can see he propery of invariabiliy under sricly increasing ransformaions of he underlying random variables: Lineal Correlaion Uniform variables U(0,) Marginal Disribuions They are differen. Correlaion s Pearson is varian under sricly increasing ransformaions of he underlying random variables. The fundamenal reason why correlaion fails as an invarian measure of dependency is due o he fac ha he Pearson Correlaion coefficien depends no only on he copula bu also on he marginal disribuions. Thus he measure is affeced by changes of scale in he marginal variables. Now we compue he Kendall Tau dependence: τ Uniform variables U(0,) Marginal Disribuions,000 0,458 0,458,000 They are alike. Kendall Tau is invarian under sricly increasing ransformaions of he underlying random variables. Archimedean Bivariae The following algorihm generaes random variaes ( uv, ) wih generaorϕ : T whose join disribuion is an Archimedean copula C n = 2 i< j 3 The formula τn sign ( X i _ Xj) ( X2i _ X2j) (.32) can be used for esimaing τ
10 9 Algorihm:. Simulae wo independen U ( 0,) random variaes sand q. 2. Se K ( q) =, where K is he disribuion funcion Cuv (, ). 3. Se u = ϕ ( s. ϕ ( ) ) and v ϕ ( ( s) ϕ ( ) ) For each Archimedean copula we need, o wi: =. A. Kendall τ (.32) B. Thea C. Generaor ϕ ( ) D. Generaor s firs derivae ϕ '( ) E. Generaor s Inverse ϕ ( ) ϕ = = ϕ F. The disribuion funcion of Cuv (, ) K G. Disribuion funcion inverse K ( ) '( ) ( When i has no a closed form as in case of Gumbel, Frank and Clayon Archimedean copula, i can be obained hrough he equaion numerical roo finding). For doing his, we need he firs derivae regard o of For he Gumbel copula, we have: Table B. (*) C. D. E. F. G. (**) = τ (.9) (*). Only posiive dependence. ( ln) (.20) ( ln) (.2) (**) There is no a closed form for he inverse disribuion funcion So ha: e (.22) ( ln) (.23) ϕ ( ) q ϕ' ( ) ϕ ( ). ϕ' ( ) ln( ) + (.24) K Gumbel, so G. will be used for obaining i by numerical roo finding. by u ( ( ( ) ) ) s ln = (.25) e v ( ( ) ( ( ) ) ) s ln = (.26) e VBA code ha generaes random variaes from he 2dimensional Gumbel copula Funcion Gumbel(ByVal Thea As Double, Opional Random, Opional Random2) As Varian Generaes random variaes from he 2dimensional Gumbel copula Dim As Double, s As Double, q As Double, u() As Double Applicaion.Volaile ReDim u( To 2)
11 0 Simulae wo independen U ( 0,) random variaes sandq. If IsMissing(Random) Then s = Rnd Else s = Random End If If IsMissing(Random2) Then q = Rnd Else q = Random2 End If Se K ( q) =, where K is he disribuion funcion of Cuv (, ). Because Gumbel has no a closed form KCg_Inv is obained hrough by numerical roo finding = KCg_Inv(Thea, q) Se u = ϕ ( s. ϕ ( ) ) and v ϕ ( ( s) ϕ ( ) ) =. u() = Exp((s * (Log()) ^ Thea) ^ ( / Thea)) u(2) = Exp(((  s) * (Log()) ^ Thea) ^ ( / Thea)) The vecor u(2) is a pair of pseudo random numbers ha are uniformly disribued on [0,] x [0,] and ha has a Gumbel copula as a join disribuion funcion. Gumbel = u End Funcion VBA Code ha compues Inverse Cumulaive Disribuion Funcion using numerical roo finding Funcion KCg_Inv(ByVal Thea As Double, ByVal q As Double, Opional olerance As Single = ) As Double Because Gumbel has no a closed form K ( q) = is obained hrough by numerical roo finding Dim As Double, zero As Double, KCg As Double, dela As Double, diff As Double = olerancia zero = 0 Do While True The disribuion funcion of Cuv (, ) K ( ) '( ) ϕ = = ϕ.gumbel equal o ( ln) (.23) KCg =  ( * Log() / Thea) q Derivae of he disribuion funcion of Cuv (, ) K (.24) ( ) '( ) ϕ = = ϕ.gumbel equal o ln( ) + dela = (Log() / Thea)  ( / Thea) + diff = KCg  zero If Abs(diff) < olerance Then Exi Do = + (diff / dela) Loop KCg_Inv = Exi Funcion End Funcion
12 The vecor u(2) is a pair of pseudo random numbers ha are uniformly disribued on [0,] x [0,] and i has a Gumbel copula as a join disribuion funcion. Then ake he marginal disribuion funcions, in his case, normal sandard, we pu u = Φ ( r ) (.27) u=normsdis( r ) hen we have: v = Φ ( r 2 ) (.28) v=normsdis( r 2) r ( u) = Φ (.29) r = NormSInv( u) r2 ( v) = Φ (.30) r 2 = NormSInv( v) are pseudo random numbers wih disribuion funcion Φ ( Normal Sandard ) and join disribuion funcion Gumbel. Which is he Archimedean copula righ one? The disribuion funcion of an Archimedean copula, as i already had been exposed in F. is represened for he following formula: To idenifyϕ, we: ϕ = = ϕ Cuv (, ) K ( ) '( ) (.3). Esimae Kendall s correlaion coefficien using he usual nonparameric esimae: τ n n = sign ( X i _ Xj) ( X2i _ X2j) (.32) 2 i< j 2. Consruc a nonparameric esimae of, he following way: i. Firs, define he pseudoobservaions T {number of ( X X ) X j i K = < such ha i j i < X and X } 2 < X2 / ( n ) for i =,2,..., n. ii. Second, consruc he esimae of j i K as K ( ) n = proporion of i 3. Now consruc a parameric esimae of Kusing he relaionship. (.3) T s. For example, choose a generaorϕ, for his refers o Table and Appendix A, and use he esimae τ n o calculae an esimae de, say n. Use n say K n ( ). o esimae ϕ ( x), say ϕ ( ).Finally, use ϕ ( ) o esimae K ( ), In order o selec he Archimedean copula which fis beer he daa, Frees and Valdez (998) propose o use a QQ plo beween 2.ii) and 3) or by minimizing a disance such as K ( ) K ( ) dk ( ) Boh approach are presened below: x n x n 2 n n n.
13 2 The graphical approach shows ha he Gumbel copula is he beer fi. The nonparameric approach arrives o same resul. 4 Gumbel Clayon Frank τ au au Frank K ( ) K ( ) ( ) n n dk n 2 Densiy Cumulaive Sample Gumbel Clayon Frank Gumbel Clayon Frank , Min K ( ) K ( ) ( ) n n dkn 2 = 0.00 for Gumbel, for Clayon and 0.2 for Frank. Below wo VBA code necessary for performing nonparameric esimaion ha allow us o reply he quesions Which is he copula righ one? 4 An aached Excel shee develops he nonparameric mehod horoughly. Over here we only presen is resul.
14 3 VBA Code ha compues Kendall using Tau nonparameric esimaion Funcion K_au(ByVal X As Range, ByVal X2 As Range) As Double 'Esimae Kendall's correlaion coefficien using he usual nonparameric esimae τ n n = sign ( X i _ Xj) ( X2i _ X2j) (.32) 2 i< j Dim i As Long, j As Long, s As Long, n As Long n = X.Rows.Coun For i = To n For j = i To n If j > i Then s = s + Sgn((X.Cells(i, )  X.Cells(j, )) * (X2.Cells(i, )  X2.Cells(j, ))) End If Nex Nex K_au = (Applicaion.WorksheeFuncion.Combin(n, 2) ^ ) * s End Funcion VBA Code ha compues he pseudoobservaions T {number of ( X X ) X < X and X } 2 < X2 / ( n ) for i =,2,..., n. j i j i = < such ha i j i Funcion Ts(ByVal X As Range, ByVal X2 As Range, i As Long) As Double Dim j As Long, s As Long, n As Long n = X.Rows.Coun For j = To n If X.Cells(j, ) < X.Cells(i, ) And X2.Cells(j, ) < X2.Cells(i, ) Then Ts = Ts + End If Nex Ts = Ts / (n  ) End Funcion Numerical Example This example shows how value a firsodefaul swap. For doing his, we use Li model (Li (2000)). Under his model, defauls are assumed o occur for individual asses according o Poisson process wih a deerminisic inensiy called hazard rae h. This means ha defaul imes ( T ) are exponenially disribued wih mean equal o h. Li relaes he defaul imes using a Gaussian (Normal) copula, we employ Gumbel copula, oo. We assume ha: we have a porfolio of wo credis ( n = 2). he conrac is a woyear ransacion ( = 2), which pays one dollar if he firs defaul happens during he firs wo years. h =. Each credi has a, consan for erm of de conrac, hazard rae of 0.0 A consan ineres rae of r = 0.0
15 4 The Pricing Firsadefaul Algorihm: For each Mone Carlo rial we do he following: Draw uniform bivariaes from chosen copula ( Gaussian, Gumbel ec.) Map uniform o defaul imes ( T ) using he inverse cumulaive exponenial disribuion funcion given a fixed h. Compue minimum defaul ime. If i is less han n, he presen value of he conrac is Then we average many rials and compue he expeced value of he conrac.. rt. e. We examine he impac of he asse correlaion on he value of he credi derivaive using independence, perfecly correlaed and using he following lineal correlaion marix: Lineal Correlaion Our simulaion of 30,000 rials produces he following resuls: s adefaul Swap Price Independence Perfecly Correlaed 0.65 Normal Gumbel When we assume independence or perfec correlaion below analyical soluion is possible: where, in he independence case: in he perfecly correlaed case: ht r + h T ( T ) ( r + e h ) ht ht hn. (.33) = (.34) = h (.35) The resul of he analyical soluion is following presened: Analyical Soluion Price Independence 0.30 Perfecly Correlaed 0.65
16 5 Conclusions There is clear evidence ha equiy reurns have uncondiional fa ails, o wi, he exreme evens are more probable han anicipaed by normal disribuion, no only in marginal bu also in higher dimensions. This is imporan boh for marke risk models as credi risk one, where equiy reurns are used as a proxy for asse reurns ha follow a mulivariae normal disribuion, and, herefore, defaul imes have a mulivariae normal dependence srucure as well. Oher han normal disribuion should be used boh in marginal as join disribuions. To overcome hese pifalls, he concep of copula, is basic properies and a special class of copula called Archimedean are inroduced. Then we expose a guide o choose boh he margins and he Archimedean copula ha beer fi o daa. In addiion, we provide an algorihm o simulae random bivariae from Archimedean copula. In order o cover he gap beween he heory and is pracical implemenaion VBA codes are provided. Finally we show a numerical example ha illusraes he use of he copula by pricing a firsodefaul conrac. For simpliciy s sake, and given ha he join disribuion is he major opic of his paper, when we value a firsodefaul conrac, we obviae in marginal disribuion, o use a differen disribuion o normal one, bu we employ Archimedean copula o model dependence srucure. This paper is accompanied by wo spreadshees ha presen sep by sep all pracical applicaions covered. The spreadshees are available on Coningency Analysis  hp://www.riskglossary.com/papers/.zip and hp://www.riskglossary.com/papers/firsadefaul.zip .
17 6 References EMBRECHTS, P., A. J. MCNEIL and D. STRAUMANN (999): Correlaion and Dependence in Risk Managemen: Properies and Pifalls. hp://www.mah.ehz.ch/~srauman/preprins/pifalls.pdf FREES, E. W. and VALDEZ, E. A. (998): Undersanding relaionships using copulas, Norh American Acuarial Journal, 2, pp HOLTON, Glyn A. (2003): ValueaRisk: Theory and Pracice. Academic Press. hp://www.valuearisk.ne/ LI, D. (2000): On Defaul Correlaion: A Funcion Approach, working paper, RiskMerics Group, New York. hp://www.defaulrisk.com/pdf files/on Defaul Correlaion A Funcion Approach.pdF LINDSKOG, F. (2000): Modeling Dependence wih s, ETH Zurich. hp://www.mah.ehz.ch/~mcneil/fp/dependencewihs.pdf MASHAL, R. and NALDI, M. Pricing Muliname Credi Derivaives: Heavy Tailed Hybrid Approach Working Paper, Columbia Business School hp://www.columbia.edu/~rm586/pub/mashal_naldi_hybrid.pdf MENEGUZZO, D. and VECCHIATO, W. (2002): Sensiiviy in Collaeralized Deb Obligaions and Baske Defaul Swaps Pricing and Risk Monioring. MYERSON, Roger: VBA code for Simools.xla (3.3) Copyrigh hp://home.uchicago.edu/~rmyerson/addins.hm NELSON, Roger (999): An Inroducion o s. Springer Verlag. hp://www.riskbook.com/iles/nelsen_r_(999).hm ROMANO, C. (2002): Applying Funcion o Risk Managemen, Universiy of Rome, La Sapienza, Working Paper. WANG, S. S. (999): Aggregaion of Correlaed Risk Porfolios: Models & Algorihms, CAS Commiee on Theory of Risk, Working Paper. hp://www.casac.org/pubs/proceed/proceed98/ pdf
18 7 Appendix A Ohers s Parameers Clayon s Parameers B.(*) C. D. E. F. G. (**) 2τ. = ( ) + + ( ) ( + ) (.37) (.38) τ + + (.39) (.36) (*) > 0. Only posiive dependence. (**) There is no a closed form for he inverse disribuion funcion K Clayon, so G. will be used for obaining i by numerical roo finding. (.40) (.4) Frank s Parameers C. D. E. F. G. (**) e ln e (.42) (*) < e (.43) ( e e + ) ln (.44) <. Posiive and negaive dependence. (**) There is no a closed form for he inverse disribuion funcion e ln e (.45) ( e ) e e ln + ( ) e (.46) K Frank, so G. will be used for obaining i by numerical roo finding. B. Frank copula has no close form ha allows us o calculae hea parameer. We use numerical roo finding for calculaing i. Press buon in he aached Excel shee for performing he following VBA code: au au Frank Privae Sub CommandBuon_Click() Dim kau As Double Inpu waned Kendall Tau τ kau = InpuBox("Kendal Tau: ", "Inpu") In ChangingCell "G4" we use formula (.50) See Appendix B Range("G3").GoalSeek Goal:=kau, ChangingCell:=Range("G4") End Sub Appendix B Kendall τ revisied The Kendall Tau can be calculaed boh using formula (.32) or he following one: Le X and Y be random variables wih an Archimedean copula C generaed by ϕ, Kendall s Tau of X and Y is given by: τ ϕ ( ) = + 4 d (.47) ϕ' ( ) 0 when Cis Gumbel (.47) is given by:
19 8 τ ( ln) ln =+4 d = + 4 d = (.48) 0 0 ( ln) when Cis Clayon (.47) is given by: when Cis Frank (.47) is given by: + τ =+4 d = + 4 d. = + 2 (.49) 0 0 where D ( ) k x is he Debye funcion, given by: τ 4 ( D ( ) ) = (.50) x k k Dk ( x) = d k x e (.5) 0 We use (.50) for calculaing Frank copula's τ. The inegral f ( xdx ) is calculaed using Riemann sums a mehod. I approximaes he inegral by dividing he inerval ab, ino msubinervals and approximaing f wih a consan funcion on each subinerval. Riemann sum approximaes our definie inegral wih: b b a m k ( ) ( ) f xdx f x x k= 5 (.52) Appendix C  Using Simools feaures in VBA programs. For performing Mone Carlo Simulaion we use a freeware called Simools. For is righ working, i is necessary o aach Simools.xla as a reference in VBA module, by applying he Tools: References menu command in he Visual Basic Edior and checking Simools.xla as an available reference. More informaion abou Simools click hp://home.uchicago.edu/~rmyerson/addins.pdf. Appendix D Tail dependence. An example can be useful o visualize he issue. We assume: Kendall Tau τ = ρ = Gumbel copula (.2) and Gaussian one 7 5 We use a VBA code o figure ou his inegral. For is righ working, i is necessary o conain Microsof Scrip Conrol.0 as a reference in VBA module. If Microsof Scrip Conrol.0 is missing you could download from hp://www.microsof.com/downloads/deails.aspx?familyid=d7e E68C02426FEC693AC&displaylang=en π 6 We use he following relaionship: ρij = sin τij 2 2 arcsin, τ = ( ρ ) ij π ij
20 9 Use (.9) o calculae. τ Upper Tail Dependence Gumbel u CGumbel ( uu, ) λ upper (.5) Gaussian ρ u CGaussian ( uu, ) λ upper (.5) τ Lower Tail Dependence Gumbel u CGumbel ( uu, ) λ lower (.8) Gaussian ρ u CGaussian ( uu, ) λ lower (.8) In he Gumbel copula s case when u he ail upper dependence changes slighly. In Gaussian copula s case he upper ail dependence ends o zero. When u 0 he lower ail dependence ends o zero for Gaussian copula and Gumbel one. So ha, our example suggess ha Gumbel copula has upper ail dependence bu does no has lower ail one, whereas Gaussian copula does has neiher. in his case: The formulae for calculaing he upper ail dependence from Gumbel copula is: (.53) 2 2 ( ) 7 C ( uv,; ρ ) ( u), ( v) Gaussian funcion wih linear correlaion ρ, and Φ = NormSInv( u). For = Φ Φ Φ, where Φ denoes he join disribuion funcion of he bivariae sandard normal disribuion Φ denoes he inverse of he disribuion funcion of he univariae sandard normal disribuion. In Excel language Φ a VBA code is available on hp://my.dreamwiz.com/sjoo/source/bivariae_normal_disribuion.x
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationINVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS
INVESTIGATION OF THE INFLUENCE OF UNEMPLOYMENT ON ECONOMIC INDICATORS Ilona Tregub, Olga Filina, Irina Kondakova Financial Universiy under he Governmen of he Russian Federaion 1. Phillips curve In economics,
More informationMeasuring macroeconomic volatility Applications to export revenue data, 19702005
FONDATION POUR LES ETUDES ET RERS LE DEVELOPPEMENT INTERNATIONAL Measuring macroeconomic volailiy Applicaions o expor revenue daa, 1970005 by Joël Cariolle Policy brief no. 47 March 01 The FERDI is a
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationARCH 2013.1 Proceedings
Aricle from: ARCH 213.1 Proceedings Augus 14, 212 Ghislain Leveille, Emmanuel Hamel A renewal model for medical malpracice Ghislain Léveillé École d acuaria Universié Laval, Québec, Canada 47h ARC Conference
More informationFourier Series Solution of the Heat Equation
Fourier Series Soluion of he Hea Equaion Physical Applicaion; he Hea Equaion In he early nineeenh cenury Joseph Fourier, a French scienis and mahemaician who had accompanied Napoleon on his Egypian campaign,
More informationCOMPUTATION OF CENTILES AND ZSCORES FOR HEIGHTFORAGE, WEIGHTFORAGE AND BMIFORAGE
COMPUTATION OF CENTILES AND ZSCORES FOR HEIGHTFORAGE, WEIGHTFORAGE AND BMIFORAGE The mehod used o consruc he 2007 WHO references relied on GAMLSS wih he BoxCox power exponenial disribuion (Rigby
More informationThe Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationRisk Modelling of Collateralised Lending
Risk Modelling of Collaeralised Lending Dae: 4112008 Number: 8/18 Inroducion This noe explains how i is possible o handle collaeralised lending wihin Risk Conroller. The approach draws on he faciliies
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationGraphing the Von Bertalanffy Growth Equation
file: d:\b1732013\von_beralanffy.wpd dae: Sepember 23, 2013 Inroducion Graphing he Von Beralanffy Growh Equaion Previously, we calculaed regressions of TL on SL for fish size daa and ploed he daa and
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationEconomics 140A Hypothesis Testing in Regression Models
Economics 140A Hypohesis Tesing in Regression Models While i is algebraically simple o work wih a populaion model wih a single varying regressor, mos populaion models have muliple varying regressors 1
More informationA Brief Introduction to the Consumption Based Asset Pricing Model (CCAPM)
A Brief Inroducion o he Consumpion Based Asse Pricing Model (CCAPM We have seen ha CAPM idenifies he risk of any securiy as he covariance beween he securiy's rae of reurn and he rae of reurn on he marke
More informationYTM is positively related to default risk. YTM is positively related to liquidity risk. YTM is negatively related to special tax treatment.
. Two quesions for oday. A. Why do bonds wih he same ime o mauriy have differen YTM s? B. Why do bonds wih differen imes o mauriy have differen YTM s? 2. To answer he firs quesion les look a he risk srucure
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationPrincipal components of stock market dynamics. Methodology and applications in brief (to be updated ) Andrei Bouzaev, bouzaev@ya.
Principal componens of sock marke dynamics Mehodology and applicaions in brief o be updaed Andrei Bouzaev, bouzaev@ya.ru Why principal componens are needed Objecives undersand he evidence of more han one
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationSPEC model selection algorithm for ARCH models: an options pricing evaluation framework
Applied Financial Economics Leers, 2008, 4, 419 423 SEC model selecion algorihm for ARCH models: an opions pricing evaluaion framework Savros Degiannakis a, * and Evdokia Xekalaki a,b a Deparmen of Saisics,
More informationA Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation
A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011 1. The risk free ineres rae in corporae valuaion
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More informationLIFE INSURANCE WITH STOCHASTIC INTEREST RATE. L. Noviyanti a, M. Syamsuddin b
LIFE ISURACE WITH STOCHASTIC ITEREST RATE L. oviyani a, M. Syamsuddin b a Deparmen of Saisics, Universias Padjadjaran, Bandung, Indonesia b Deparmen of Mahemaics, Insiu Teknologi Bandung, Indonesia Absrac.
More informationMorningstar Investor Return
Morningsar Invesor Reurn Morningsar Mehodology Paper Augus 31, 2010 2010 Morningsar, Inc. All righs reserved. The informaion in his documen is he propery of Morningsar, Inc. Reproducion or ranscripion
More informationAn empirical analysis about forecasting Tmall airconditioning sales using time series model Yan Xia
An empirical analysis abou forecasing Tmall aircondiioning sales using ime series model Yan Xia Deparmen of Mahemaics, Ocean Universiy of China, China Absrac Time series model is a hospo in he research
More informationHierarchical Mixtures of AR Models for Financial Time Series Analysis
Hierarchical Mixures of AR Models for Financial Time Series Analysis Carmen Vidal () & Albero Suárez (,) () Compuer Science Dp., Escuela Poliécnica Superior () Risklab Madrid Universidad Auónoma de Madrid
More informationComplex Fourier Series. Adding these identities, and then dividing by 2, or subtracting them, and then dividing by 2i, will show that
Mah 344 May 4, Complex Fourier Series Par I: Inroducion The Fourier series represenaion for a funcion f of period P, f) = a + a k coskω) + b k sinkω), ω = π/p, ) can be expressed more simply using complex
More informationMath 201 Lecture 12: CauchyEuler Equations
Mah 20 Lecure 2: CauchyEuler Equaions Feb., 202 Many examples here are aken from he exbook. The firs number in () refers o he problem number in he UA Cusom ediion, he second number in () refers o he problem
More informationVector Autoregressions (VARs): Operational Perspectives
Vecor Auoregressions (VARs): Operaional Perspecives Primary Source: Sock, James H., and Mark W. Wason, Vecor Auoregressions, Journal of Economic Perspecives, Vol. 15 No. 4 (Fall 2001), 101115. Macroeconomericians
More informationTable of contents Chapter 1 Interest rates and factors Chapter 2 Level annuities Chapter 3 Varying annuities
Table of conens Chaper 1 Ineres raes and facors 1 1.1 Ineres 2 1.2 Simple ineres 4 1.3 Compound ineres 6 1.4 Accumulaed value 10 1.5 Presen value 11 1.6 Rae of discoun 13 1.7 Consan force of ineres 17
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationON THURSTONE'S MODEL FOR PAIRED COMPARISONS AND RANKING DATA
ON THUSTONE'S MODEL FO PAIED COMPAISONS AND ANKING DATA Alber MaydeuOlivares Dep. of Psychology. Universiy of Barcelona. Paseo Valle de Hebrón, 171. 08035 Barcelona (Spain). Summary. We invesigae by means
More informationChapter 6: Business Valuation (Income Approach)
Chaper 6: Business Valuaion (Income Approach) Cash flow deerminaion is one of he mos criical elemens o a business valuaion. Everyhing may be secondary. If cash flow is high, hen he value is high; if he
More informationHedging with Forwards and Futures
Hedging wih orwards and uures Hedging in mos cases is sraighforward. You plan o buy 10,000 barrels of oil in six monhs and you wish o eliminae he price risk. If you ake he buyside of a forward/fuures
More informationDYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus University Toruń 2006. Ryszard Doman Adam Mickiewicz University in Poznań
DYNAMIC ECONOMETRIC MODELS Vol. 7 Nicolaus Copernicus Universiy Toruń 26 1. Inroducion Adam Mickiewicz Universiy in Poznań Measuring Condiional Dependence of Polish Financial Reurns Idenificaion of condiional
More informationChabot College Physics Lab RC Circuits Scott Hildreth
Chabo College Physics Lab Circuis Sco Hildreh Goals: Coninue o advance your undersanding of circuis, measuring resisances, currens, and volages across muliple componens. Exend your skills in making breadboard
More informationWhy Did the Demand for Cash Decrease Recently in Korea?
Why Did he Demand for Cash Decrease Recenly in Korea? Byoung Hark Yoo Bank of Korea 26. 5 Absrac We explores why cash demand have decreased recenly in Korea. The raio of cash o consumpion fell o 4.7% in
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationImpact of Debt on Primary Deficit and GSDP Gap in Odisha: Empirical Evidences
S.R. No. 002 10/2015/CEFT Impac of Deb on Primary Defici and GSDP Gap in Odisha: Empirical Evidences 1. Inroducion The excessive pressure of public expendiure over is revenue receip is financed hrough
More informationMarkit Excess Return Credit Indices Guide for price based indices
Marki Excess Reurn Credi Indices Guide for price based indices Sepember 2011 Marki Excess Reurn Credi Indices Guide for price based indices Conens Inroducion...3 Index Calculaion Mehodology...4 Semiannual
More informationCointegration Analysis of Exchange Rate in Foreign Exchange Market
Coinegraion Analysis of Exchange Rae in Foreign Exchange Marke Wang Jian, Wang Shuli School of Economics, Wuhan Universiy of Technology, P.R.China, 430074 Absrac: This paper educed ha he series of exchange
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationRepresenting Periodic Functions by Fourier Series. (a n cos nt + b n sin nt) n=1
Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals
More informationUSE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES
USE OF EDUCATION TECHNOLOGY IN ENGLISH CLASSES Mehme Nuri GÖMLEKSİZ Absrac Using educaion echnology in classes helps eachers realize a beer and more effecive learning. In his sudy 150 English eachers were
More informationChapter 6 Interest Rates and Bond Valuation
Chaper 6 Ineres Raes and Bond Valuaion Definiion and Descripion of Bonds Longerm debloosely, bonds wih a mauriy of one year or more Shorerm debless han a year o mauriy, also called unfunded deb Bondsricly
More informationDoes Option Trading Have a Pervasive Impact on Underlying Stock Prices? *
Does Opion Trading Have a Pervasive Impac on Underlying Sock Prices? * Neil D. Pearson Universiy of Illinois a UrbanaChampaign Allen M. Poeshman Universiy of Illinois a UrbanaChampaign Joshua Whie Universiy
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationCredit Index Options: the noarmageddon pricing measure and the role of correlation after the subprime crisis
Second Conference on The Mahemaics of Credi Risk, Princeon May 2324, 2008 Credi Index Opions: he noarmageddon pricing measure and he role of correlaion afer he subprime crisis Damiano Brigo  Join work
More informationA Reexamination of the Joint Mortality Functions
Norh merican cuarial Journal Volume 6, Number 1, p.166170 (2002) Reeaminaion of he Join Morali Funcions bsrac. Heekung Youn, rkad Shemakin, Edwin Herman Universi of S. Thomas, Sain Paul, MN, US Morali
More informationPricing Guaranteed Minimum Withdrawal Benefits under Stochastic Interest Rates
Pricing Guaraneed Minimum Wihdrawal Benefis under Sochasic Ineres Raes Jingjiang Peng 1, Kwai Sun Leung 2 and Yue Kuen Kwok 3 Deparmen of Mahemaics, Hong Kong Universiy of Science and echnology, Clear
More informationThe Interaction of Guarantees, Surplus Distribution, and Asset Allocation in With Profit Life Insurance Policies
1 The Ineracion of Guaranees, Surplus Disribuion, and Asse Allocaion in Wih Profi Life Insurance Policies Alexander Kling * Insiu für Finanz und Akuarwissenschafen, Helmholzsr. 22, 89081 Ulm, Germany
More informationTEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS
TEMPORAL PATTERN IDENTIFICATION OF TIME SERIES DATA USING PATTERN WAVELETS AND GENETIC ALGORITHMS RICHARD J. POVINELLI AND XIN FENG Deparmen of Elecrical and Compuer Engineering Marquee Universiy, P.O.
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationIndividual Health Insurance April 30, 2008 Pages 167170
Individual Healh Insurance April 30, 2008 Pages 167170 We have received feedback ha his secion of he e is confusing because some of he defined noaion is inconsisen wih comparable life insurance reserve
More informationBALANCE OF PAYMENTS. First quarter 2008. Balance of payments
BALANCE OF PAYMENTS DATE: 20080530 PUBLISHER: Balance of Paymens and Financial Markes (BFM) Lena Finn + 46 8 506 944 09, lena.finn@scb.se Camilla Bergeling +46 8 506 942 06, camilla.bergeling@scb.se
More informationThe Generalized Extreme Value (GEV) Distribution, Implied Tail Index and Option Pricing
he Generalized Exreme Value (GEV) Disribuion, Implied ail Index and Opion Pricing Sheri Markose and Amadeo Alenorn his version: 6 December 200 Forhcoming Spring 20 in he Journal of Derivaives Absrac Crisis
More informationIssues Using OLS with Time Series Data. Time series data NOT randomly sampled in same way as cross sectional each obs not i.i.d
These noes largely concern auocorrelaion Issues Using OLS wih Time Series Daa Recall main poins from Chaper 10: Time series daa NOT randomly sampled in same way as cross secional each obs no i.i.d Why?
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 23.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationForecasting Malaysian Gold Using. GARCH Model
Applied Mahemaical Sciences, Vol. 7, 2013, no. 58, 28792884 HIKARI Ld, www.mhikari.com Forecasing Malaysian Gold Using GARCH Model Pung Yean Ping 1, Nor Hamizah Miswan 2 and Maizah Hura Ahmad 3 Deparmen
More informationUsing Monte Carlo Method to Compare CUSUM and. EWMA Statistics
Using Mone Carlo Mehod o Compare CUSUM and EWMA Saisics Xiaoyu Shen Zhen Zhang Absrac: Since ordinary daases usually conain change poins of variance, CUSUM and EWMA saisics can be used o deec hese change
More informationµ r of the ferrite amounts to 1000...4000. It should be noted that the magnetic length of the + δ
Page 9 Design of Inducors and High Frequency Transformers Inducors sore energy, ransformers ransfer energy. This is he prime difference. The magneic cores are significanly differen for inducors and high
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationRelationships between Stock Prices and Accounting Information: A Review of the Residual Income and Ohlson Models. Scott Pirie* and Malcolm Smith**
Relaionships beween Sock Prices and Accouning Informaion: A Review of he Residual Income and Ohlson Models Sco Pirie* and Malcolm Smih** * Inernaional Graduae School of Managemen, Universiy of Souh Ausralia
More information23.3. Even and Odd Functions. Introduction. Prerequisites. Learning Outcomes
Even and Odd Funcions 3.3 Inroducion In his Secion we examine how o obain Fourier series of periodic funcions which are eiher even or odd. We show ha he Fourier series for such funcions is considerabl
More informationDIFFERENTIAL EQUATIONS with TI89 ABDUL HASSEN and JAY SCHIFFMAN. A. Direction Fields and Graphs of Differential Equations
DIFFERENTIAL EQUATIONS wih TI89 ABDUL HASSEN and JAY SCHIFFMAN We will assume ha he reader is familiar wih he calculaor s keyboard and he basic operaions. In paricular we have assumed ha he reader knows
More informationThe naive method discussed in Lecture 1 uses the most recent observations to forecast future values. That is, Y ˆ t + 1
Business Condiions & Forecasing Exponenial Smoohing LECTURE 2 MOVING AVERAGES AND EXPONENTIAL SMOOTHING OVERVIEW This lecure inroduces imeseries smoohing forecasing mehods. Various models are discussed,
More informationChapter 4: Exponential and Logarithmic Functions
Chaper 4: Eponenial and Logarihmic Funcions Secion 4.1 Eponenial Funcions... 15 Secion 4. Graphs of Eponenial Funcions... 3 Secion 4.3 Logarihmic Funcions... 4 Secion 4.4 Logarihmic Properies... 53 Secion
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationINTRODUCTION TO FORECASTING
INTRODUCTION TO FORECASTING INTRODUCTION: Wha is a forecas? Why do managers need o forecas? A forecas is an esimae of uncerain fuure evens (lierally, o "cas forward" by exrapolaing from pas and curren
More informationExponentially Weighted Moving Average Control Charts with TimeVarying Control Limits and Fast Initial Response
Exponenially Weighed Moving Average Conrol Chars wih TimeVarying Conrol Limis and Fas Iniial Response Sefan H. Seiner Dep. of Saisics and Acuarial Sciences Universiy of Waerloo Waerloo, Onario N2L 3G1
More information4. International Parity Conditions
4. Inernaional ariy ondiions 4.1 urchasing ower ariy he urchasing ower ariy ( heory is one of he early heories of exchange rae deerminaion. his heory is based on he concep ha he demand for a counry's currency
More informationTime Series Analysis Using SAS R Part I The Augmented DickeyFuller (ADF) Test
ABSTRACT Time Series Analysis Using SAS R Par I The Augmened DickeyFuller (ADF) Tes By Ismail E. Mohamed The purpose of his series of aricles is o discuss SAS programming echniques specifically designed
More informationResearch Question Is the average body temperature of healthy adults 98.6 F? Introduction to Hypothesis Testing. Statistical Hypothesis
Inroducion o Hypohesis Tesing Research Quesion Is he average body emperaure of healhy aduls 98.6 F? HT  1 HT  2 Scienific Mehod 1. Sae research hypoheses or quesions. µ = 98.6? 2. Gaher daa or evidence
More informationChapter 4. Properties of the Least Squares Estimators. Assumptions of the Simple Linear Regression Model. SR3. var(e t ) = σ 2 = var(y t )
Chaper 4 Properies of he Leas Squares Esimaors Assumpions of he Simple Linear Regression Model SR1. SR. y = β 1 + β x + e E(e ) = 0 E[y ] = β 1 + β x SR3. var(e ) = σ = var(y ) SR4. cov(e i, e j ) = cov(y
More informationMortality Variance of the Present Value (PV) of Future Annuity Payments
Morali Variance of he Presen Value (PV) of Fuure Annui Pamens Frank Y. Kang, Ph.D. Research Anals a Frank Russell Compan Absrac The variance of he presen value of fuure annui pamens plas an imporan role
More informationWorking Paper No. 482. Net Intergenerational Transfers from an Increase in Social Security Benefits
Working Paper No. 482 Ne Inergeneraional Transfers from an Increase in Social Securiy Benefis By Li Gan Texas A&M and NBER Guan Gong Shanghai Universiy of Finance and Economics Michael Hurd RAND Corporaion
More informationThis document is downloaded from DRNTU, Nanyang Technological University Library, Singapore.
This documen is downloaded from DRNTU, Nanyang Technological Universiy Library, Singapore. Tile A Bayesian mulivariae riskneural mehod for pricing reverse morgages Auhor(s) Kogure, Asuyuki; Li, Jackie;
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationDETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUNSHAN WU
Yugoslav Journal of Operaions Research 2 (22), Number, 67 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUNSHAN WU Deparmen of Bussines Adminisraion
More informationAP Calculus AB 2007 Scoring Guidelines
AP Calculus AB 7 Scoring Guidelines The College Board: Connecing Sudens o College Success The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and
More informationStochastic Calculus and Option Pricing
Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 BlackScholes
More informationINVESTMENT GUARANTEES IN UNITLINKED LIFE INSURANCE PRODUCTS: COMPARING COST AND PERFORMANCE
INVESMEN UARANEES IN UNILINKED LIFE INSURANCE PRODUCS: COMPARIN COS AND PERFORMANCE NADINE AZER HAO SCHMEISER WORKIN PAPERS ON RISK MANAEMEN AND INSURANCE NO. 4 EDIED BY HAO SCHMEISER CHAIR FOR RISK MANAEMEN
More informationModeling a distribution of mortgage credit losses Petr Gapko 1, Martin Šmíd 2
Modeling a disribuion of morgage credi losses Per Gapko 1, Marin Šmíd 2 1 Inroducion Absrac. One of he bigges risks arising from financial operaions is he risk of counerpary defaul, commonly known as a
More informationUnderstanding Sequential Circuit Timing
ENGIN112: Inroducion o Elecrical and Compuer Engineering Fall 2003 Prof. Russell Tessier Undersanding Sequenial Circui Timing Perhaps he wo mos disinguishing characerisics of a compuer are is processor
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationThe Application of Multi Shifts and Break Windows in Employees Scheduling
The Applicaion of Muli Shifs and Brea Windows in Employees Scheduling Evy Herowai Indusrial Engineering Deparmen, Universiy of Surabaya, Indonesia Absrac. One mehod for increasing company s performance
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationRelative velocity in one dimension
Connexions module: m13618 1 Relaive velociy in one dimension Sunil Kumar Singh This work is produced by The Connexions Projec and licensed under he Creaive Commons Aribuion License Absrac All quaniies
More informationTHE PERFORMANCE OF OPTION PRICING MODELS ON HEDGING EXOTIC OPTIONS
HE PERFORMANE OF OPION PRIING MODEL ON HEDGING EXOI OPION Firs Draf: May 5 003 his Version Oc. 30 003 ommens are welcome Absrac his paper examines he empirical performance of various opion pricing models
More information