Credit Spread Option Valuation under GARCH. Working Paper July 2000 ISSN :

Transcription

1 Credi Spread Opion Valuaion under GARCH by Nabil ahani Working Paper -7 July ISSN : Financial suppor by he Risk Managemen Chair is acknowledged. he auhor would like o hank his professors Peer Chrisoffersen Georges Dionne Geneviève Gauhier and Jean-Guy Simonao for heir suppor and heir commens. Elecronic versions pdf files of Working papers are available on our Web sie: hp://

2 Credi Spread Opion Valuaion under GARCH Nabil ahani Nabil ahani is a Ph.D. suden in Finance HEC Monreal. nabil.ahani@hec.ca Copyrigh. École des Haues Éudes Commerciales HEC Monréal. All righs reserved in all counries. Any ranslaion or reproducion in any form whasoever is forbidden. he exs published in he series Working Papers are he sole responsibiliy of heir auhors.

3 Credi Spread Opion Valuaion under GARCH Nabil ahani Absrac his paper develops closed-form soluions for opions on credi spreads wih GARCH models. We exend he mean-revering model proposed in Longsaff and Schwarz 995 and we use he Heson and Nandi's 999 GARCH specificaion raher han he radiional lognormal. Our model being more flexible capures beer he empirical properies of observed credi spreads and conains Longsaff and Schwarz 995 model as a special case. GARCH coefficiens are esimaed using spread levels for corporae bonds. Keywords : Credi spread opions GARCH models mean-reversion. Résumé Ce aricle propose une formule fermée pour l évaluaion des opions sur les écars de crédi dans le cadre des modèles GARCH. On se base sur le modèle de reour à la moyenne proposé par Longsaff e Schwarz 995 e on uilise le modèle GARCH proposé par Heson e Nandi 999 au lieu du radiionnel modèle lognormal. Ce modèle éan plus flexible s ajuse mieux aux propriéés empiriques des données observées. Il conien le modèle de Longsaff e Schwarz 995 comme cas pariculier. Les coefficiens du modèle GARCH son esimés en uilisan les niveaux des écars de crédis des obligaions corporaives. Mos clés : Écar de crédi opions modèles GARCH reour à la moyenne.

4 I Inroducion Unil recenly credi risk was considered as legiimae o do business jus as anoher uncerainy facor which was unhedgeable. Currenly credi risk can be purchased sold or resrucured wihin porfolios in he same way as radiional financial producs. his is made possible hanks o Credi Derivaives which are he mos imporan new ypes of financial producs inroduced during he las decade. hese insrumens offer invesors an imporan new ool for dynamic hedging and managing long posiions in credi risk exposures. From heir characerisics credi derivaives presen several resemblances o he radiional opions. One difficuly however comes from he fac ha he deerminan variable is he evoluion of he credi spread insead of radiional ineres raes or exchange raes. hus we need a model for credi spreads. Basing heir jusificaion on he observed empirical properies of credi spreads Longsaff and Schwarz 995 proposed a mean-revering model for he logarihm of he credi spread. hey showed ha credi spreads were meanrevering in logarihm bu hey assumed he change in logarihm o be well-approximaed by he normal disribuion. his paper provides a more general framework for he volailiy using GARCH models. I is well-known ha financial daa ses exhibi condiional heeroskedasiciy. GARCH-ype models are ofen used o model his phenomenon and show heir abiliy o explain some irregulariies e.g. in equiy reurns beer han he radiional Geomeric Brownian Moion. In he lieraure see Bollerslev and al. 99 here are many applicaions of ARCH models o ineres rae daa. All hese applicaions focused on : erm Spread Engle and al. 987 and Engle and al. 99 esimaed he relaionship beween long- and shor-erm ineres raes; Bond Yields Levels Weiss 984 esimaed ARCH models on AAA corporae bond yields and found ha ARCH effecs were significanly eviden. Alhough mos sudies involving ineres raes used linear GARCH models nonlinear dependencies could possibly exis in he condiional variance. his paper esimaes GARCH effecs on he Credi Spread. We use he Heson and Nandi's 999 GARCH specificaion and we keep he mean-revering characer showed by Longsaff and Schwarz 995. he GARCH model being more general for he volailiy fis observed credi spreads daa beer han he simple mean-revering normal model. Also

5 Heson and Nandi 999 showed ha heir condiional variance process converges weakly o Heson's 993 Sochasic Volailiy model which means ha our model has a meanrevering square-roo variance process as a coninuous-ime limi. hus our model conains coninuous-ime Longsaff and Schwarz 995 model as a special case. Deails on he convergence will be provided laer. he nex secion examines he mean-reversion characer of credi spreads in our daa se. Secion III describes he GARCH process and presens credi spread opions formulas. Secion IV esimaes GARCH coefficiens wih he maximum likelihood mehod using corporae bond spreads levels over reasuries analyses some properies of he GARCH credi spread opions and compares our resuls o hose of Longsaff and Schwarz 995. Calculaion deails are in he Appendix. Figures are presened a he end of he main ex. II Credi spread mean-reversion We examine he spread beween Moody's AAA and BAA - years bond indices and several U.S. reasury bond yields 3 and years mauriy. We use daily observaions over Summary saisics for he spreads are presened in able. We denoe he logarihm of he credi spread by. Figures and plo he ime series of for AAA and BAA indices over 3 years and years U.S. Bond. able : Summary saisics for AAA and BAA credi spreads and log-spreads over reasuries. Sepember December 99 3 years bond years bond Number of Mean Sd Dev Mean Sd Dev Observaions AAA Log AAA BAA Log BAA We also repor Skewness and Kurosis coefficiens for log-spreads in able. We can easily see ha hese coefficiens are differen from hose of a normal disribuion. his implies ha 3

6 we canno assume a normal disribuion for log-spreads. Insead we propose a differen process such as GARCH ha akes ino accoun his findings. able : Skewness and Kurosis coefficiens for AAA and BAA log-spreads over reasuries. Sepember December 99 3 years bond years bond Number of Skewness Kurosis Skewness Kurosis observaions Log AAA Log BAA I is shown ha boh ime series display mean-reversion. AAA credi spreads are apparenly more mean-revering han BAA. In order o formalize hese observaions and see how our daa evolve over ime we regressed daily changes in he value of on he value of one day before : α + β ε. he regression resuls are repored in able 3. he slope coefficien is significanly negaive in all he regressions. he AAA slope coefficiens are definiely higher han he corresponden BAA's which implies ha hey are more mean-revering. able 3 : Resuls from regressing daily changes in he logarihm of he credi spread of AAA and BAA over 3y and y US Bond α β -raio α # -raio β # R SE AAA b3y {-3.38} [5.e-8] [5.46e-8] AAA by {-.963} BAA b3y { } BAA by {-3.3} [.58e-8] [.79e-4] [.5e-4] [.56e-6] [3.e-4] [.6e-4] SE is he sandard error of he regression and R² is he deerminaion coefficien # All he coefficiens are significan a he 99% level + p-values are repored in brackes Dickey-Fuller es saisics are repored for Uni Roo es. he asympoic criical values are a % and -.86 a 5% 4

7 Noe also ha he logarihm of he AAA credi spread is more volaile han ha of he BAA's. he sandard errors of regressions ha used AAA bonds are higher han hose of regressions ha use BAA bonds. Longsaff and Schwarz 995 found he same propery wih heir daa se. he values of R² are of he same order of magniude han hose repored by Longsaff and Schwarz 995. Given hese empirical properies one should assume a mean-revering process for he logarihm of he credi spread. Bu unlike he Longsaff and Schwarz 995 model and given he skewness and kurosis analysis in able we propose a GARCH framework for he volailiy of he logarihm of he credi spread. We use Heson and Nandi's 999 GARCH specificaion which is asymmeric. We also use heir mehodology o derive closed-form soluions for credi spread opions. We assume ha he riskless ineres rae is consan and i is denoed by r. III he model and he opion valuaion formula We define as he value of he logarihm of he credi spread a he end of period and ime periods are of lengh. We assume ha follows he process given by : h + + µ + γ β + β h + λh + β z θ h h z or equivalenly o show he mean-revering feaure by : h + γ + µ + β + β h + β + z θ h + λh where h is he condiional variance of known a ime + h + z + and z {... } : is a sequence of independen sandard normal random variables. As poined ou by Heson and Nandi 999 alhough his specificaion differs from he classic GARCH models i is quie similar o he NGARCH model of Engle and Ng 993. his model has he advanage ha i provides closed-form soluions for he credi spread derivaives. When β and β are equal o zero our model is equivalen o he Longsaff and Schwarz 995 model observed a 5

8 discree inervals wih a risk-premium parameer λ. his parameer was assumed o be equal o zero in Longsaff and Schwarz 995 because heir model was assumed o be riskadjused and he parameer µ incorporaed he marke price of he risk-premium. We canno use his assumpion wihin our model because he volailiy is no consan. As in Heson and Nandi's 999 model he parameer θ conrols for he skewness or he asymmery of he disribuion of he log-spreads. If θ > his implies ha negaive z raise he variance more han posiive z. he covariance of he log-spread process and he variance process is given by : h + θβ h Cov. 3 If he kurosis parameer β is posiive posiive values for θ resul in negaive correlaion beween he wo processes. As shown in Heson and Nandi 999 he discree ime variance process h converges weakly o a variance process v ha follows he square-roo process of Feller 95 Cox Ingersoll and Ross 985 and Heson 993 when he ime sep lengh ends o zero see Foser and Nelson 994. he log-spread process will also have a coninuous-ime diffusion limi. hus our wo-processes model converges weakly see Convergence in Appendix o a mean-revering square-roo variance process v : d dv η δ + λv d + ω κ v d + σ v dz v dz 4 where η is he long-run mean δ is he mean-reversion parameer λ is he risk-premium parameer ω κ and σ are he square-roo process parameers. By assuming ha κ σ and λ are all zero we ge he consan volailiy risk-adjused Longsaff and Schwarz 995 model. hus our valuaion model will conain he Longsaff and Schwarz 995 valuaion formula for consan risk-free rae as a special case. In order o value opions we mus work under a risk-neural probabiliy measure. Le us rewrie Equaion in he form : h + + µ + γ β + β h + h + β + z + z θ h 5 6

9 where z θ z + λ θ + λ Noe ha all we need is ha z : {... } h o be a sequence of risk-neural independen random variables and ha z + o be a sandard normal random variable condiional o he informaion available a ime see Derivaion of he momen generaing funcion in Appendix. his is obvious since + of independen sandard normal random variables. h is known a ime and z {... } : is a sequence A his poin we can derive closed-form soluions for European opions using he inversion of he characerisic funcion echnique following Kendall and Suar 977. We firs derive a formula for he characerisic funcion for he process given in Equaion. Le f denoe he momen generaing funcion of exp under he hisorical probabiliy measure condiional o he informaion available a ime : exp φ f φ E 6 where E denoes he ime condiional expecaion operaor under he hisorical probabiliy measure. Moivaed by he Heson and Nandi's 999 asse price model we calculae he momen generaing funcions for exp + and exp + o find ha f akes a log-linear form and is a funcion of and h +. he general form for he momen generaing funcion is given by : where f - γ φ + A + B h exp + φ 7 and A B A γ µφ + A + + β B + ln β B

10 B γ θ + λ φ θ + β B + + β B + γ φ θ Given he erminal condiions A and B can be calculaed recursively. he Appendix derives he recursion formulas for hese funcions see Derivaion of he momen generaing funcion in Appendix. Noe ha alhough his formulaion is given for he logspread process under he hisorical probabiliy measure one can ge he risk-neural condiional momen generaing funcion f by replacing θ and λ respecively by θ θ + λ and λ. he characerisic funcion of he process under he hisorical probabiliy measure condiional o ime is given by f i where i is he complex number such ha i. where E he value of a credi spread call wih mauriy and srike price K is hen given by : C + exp r e E K denoes he ime condiional expecaion operaor under he risk-neural probabiliy measure and + denoes he posiive par of a real number. We can compue he cumulaive disribuion funcion of he log-spread process by invering is characerisic funcion. Indeed Kendall and Suar 977 show ha for a random variable Y : + iφ y f i P Y y + Re dφ π iφ where Re denoes he real par of a complex number and f is he characerisic funcion. Using his resul Heson and Nandi 999 provided a formula for he expeced payoff. Evaluaed under he risk-neural probabiliy measure he call value is given by : where C f P r e KP 3 8

11 P + π + iφ K f iφ + Re iφ f d φ P + iφ K f i + Re d π iφ φ 4 his formula looks like he Longsaff and Schwarz 995 formula. If we noe ha he f erm represens by definiion he risk-neural forward or expeced value of he spread exp f f E 5 and if we assume ha he log-spread a ime is condiionally normal wih mean υ and variance η² we have : E η exp + exp υ 6 his erm is presen in he Longsaff and Schwarz 995 formula and is equivalen o our f erm in Equaion 5. P is he risk-neural probabiliy ha he log-spread is greaer han K a mauriy. he call Dela raio is given by see Derivaion of dela in Appendix : Dela C e r e γ f P e 7 In he same way he pu value is given by see Credi spread Call-Pu pariy in Appendix : K P f P r P e 8 IV he empirical properies of credi spread opions his empirical secion sars wih he esimaion of he GARCH coefficiens using he daa described earlier. I proceeds o analyze he implied condiional disribuion. I hen presens some properies of GARCH credi spread opions and compares our model o Longsaff and Schwarz's

12 GARCH Esimaion For he esimaion we used daily daa and se he ime sep lengh equal o day. We used boh AAA and BAA spreads over y and 3y US bond over Sepember December 99. We esimaed he coefficiens using he maximum likelihood mehod. o illusrae he mean-reversion and he imporance of he skewness parameer we also esimaed resriced models by seing firs γ and hen θ. Resriced models and he unresriced model are compared o each oher using he log-likelihood raio. able 4 repors he esimaion resuls. In all cases he mean-reversion parameer γ was significanly lower han. he resriced model γ is srongly rejeced in all cases. his reinforces he resuls repored in he regression analysis of he daily changes in he logarihm of he credi spread. he parameer γ was also largely significanly differen from. We also noe ha he γ s for he BAA bonds are higher han hose of AAA. hus he AAA bonds are more meanrevering han BAA which was also repored in he regression analysis. For he skewness parameer θ he resriced model θ is easily rejeced for AAA bonds and can no be rejeced in he case of BAA over 3y bond. For BAA over y bond he -es rejecs he hypohesis θ while he log-likelihood raio es does no. When he parameer θ is significanly posiive his implies negaive correlaion beween he log-spread and he volailiy processes as shown in Equaion 3. Log-spread processes over y bond have more skweness han hose over 3y bond for boh AAA and BAA bonds θ for AAA over y bond and θ over 3y bond. he volailiy process is saionary in all cases. Saionariy coefficiens for AAA over 3y bond and AAA over y bond are quie similar. hereafer we used he esimaes in he firs line in able 4 as our GARCH parameers. Figures 3 and 4 plo he volailiy processes implied by he GARCH esimaion for AAA bonds. For AAA bonds he log-spread processes over boh 3y and y bonds have quie similar saionariy coefficiens β + β θ.9.93 which measure he degree of he volailiy mean-reversion Heson and Nandi 999. Figures 5 and 6 plo he volailiy processes implied by he GARCH esimaion for BAA bonds. Comparing Figures 3 and 5 or 4 and 6 we noe ha he volailiy of AAA bonds is more mean-revering han he BAA's. From Figures 5 and 6 and unlike AAA bonds i is clear ha he volailiy of

13 able 4 : Maximum log likelihood esimaes of he GARCH model using daily daa of he log-spread of AAA and BAA over 3y and y US bond AAA b3y AAA by BAA b3y BAA by µ γ λ β β β θ β +β θ # Long-run volailiy e-5.68e e e [.] [.] [7.e-3] [3e-79] [8. e -6] [.e-5] {6.675} [e-] [5.4e-] [4.e-3] [8.4e-5] [.] {5.96} [.] {.7693} [.] {3.8398} 4.44e e-5 [.].7e-4 8.4e-5 [.] 9.94e e-5 [.] 4.796e-5.77 [8.6e-] 5.974e [4.7e-6].54e-5.93 [.35] [e-66] [e-78] [.].836e [3.4 e -5].86e-4.44 [3. e -9].79e [.7 e -38] [5.8e-] [.3] [.9e-3] For he log-likelihood raio es he Chi-square criical values a 95% and 99% confidence levels are respecively 3.84 and raios appear in paranheses. he criical values a 95% and 99% confidence levels are respecively.645 and.36 p-values are repored in brackes. For he log-likelihood raio es he p-values are hose of he Chi-square disribuion + For γ we also repor he -raio for esing H : γ vs. H : γ < # he Saionariy coefficien β +β θ < means ha he volailiy process is saionary Long-run volailiy is he daily uncondiional volailiy given by h unc β +β / -β -β θ log L L-raio es γ [.e-] [8.4e-3] [4.3e-4] [.3e-4] L-raio es θ 4.66 [.e-] [8.5e-].7 [.43] 3.6 [.578]

14 BAA over y bond is more mean-revering han he volailiy of BAA over 3y bond which is indicaed by a higher β + β.89 >.773. θ Implied GARCH disribuion : Using Equaion we derived he implied condiional disribuion of. o see how his implied condiional disribuion varies from he normal disribuion Figure 7 plos hem for many variance raios ha is he raios beween he iniial variance and he uncondiional variance denoed by h unc. Higher raio values imply faer-ailed disribuions. We can also expec ha he condiional GARCH disribuion has a posiive skeweness which seems o be he case for a raio value of. Figure 8 plos he implied condiional disribuions for a raio value of and for wo differen ime horizons. he longer he ime horizon he higher he skeweness compared o he normal disribuion. Credi spread opion properies : We limi our discussion o call opions when we analyze he properies of our model. In order o compare our discree-ime model o he coninuous-ime model of Longsaff and Schwarz 995 we define heir coninuous-ime parameers using he coefficiens esimaes such ha he wo saionary densiies of processes and h have he same firs momens. he parameers in Equaion 6 are defined as follows : υ γ η h unc γ γ µ ln γ I is clear ha when akes high values υ and η² have finie limis because γ < : 9 υ η µ ln γ h unc µ γ he Longsaff and Schwarz 995 call price hereafer LS is evaluaed using he condiionally normal disribuion for wih mean υ and variance η². Figure 9 plos he call value as a funcion of he underlying credi spread using he GARCH parameers repored in able 4 line and heir risk-neural counerpars wih a srike K r. and for

15 differen variance raios. he LS call price and he inrinsic value are also represened. An imporan propery of GARCH credi spread calls already noiced by Longsaff and Schwarz 995 wihin heir model is ha heir value can be less han he inrinsic value which is impossible wih he Black and Scholes model. his is due o he mean-reversion characer of he credi log-spreads. Inuiively in-he-money calls are less likely o remain in he money over ime because he credi spread ends o decline owards is long-run mean. For variance raios less han he GARCH credi spread call prices are less han he LS price while variance raios greaer han give higher GARCH prices. As he underlying credi spread increases he difference beween call prices becomes small. Figure plos he difference beween LS and GARCH credi spread call prices for differen variance raios. Noe ha he higher he variance raio he greaer he difference which reinforces he resuls of Figure 9. he difference is however more imporan around he a-he-money calls. Figures and plo credi spread call prices for differen mauriies days and year. Figure shows again ha he call prices pass below he inrinsic value even when he call is only slighly in-he-money. Figure gives anoher imporan propery of GARCH credi spread calls. hey can be concave funcions of he underlying credi spread. Because of mean-reversion he dynamics of he credi spread do no saisfy he firs-degree homogeneiy propery necessary for opions o be convex funcions Meron 973. In urn his means ha he dela of a GARCH credi spread call given in Equaion 7 could be a decreasing funcion of he underlying credi spread. he dela of a GARCH credi spread call decreases o zero as he ime o mauriy increases. Alhough our model is GARCH mean-revering hese properies were somewha expeced since Longsaff and Schwarz 995 have found ha heir coninuous-ime model exhibi such characerisics. V Conclusion We have proposed a GARCH mean-revering model for credi log-spreads as an exension of he Longsaff and Schwarz 995 model which uses a consan volailiy. We used he Heson and Nandi's 999 GARCH specificaion o allow he variance process o depend on he pas levels of he credi spread. he GARCH was esimaed using he maximum likelihood mehod and he imporan coefficiens especially he mean-reversion and he skeweness parameers were found o be significan. Our model is hen more flexible 3

16 and capures he empirical properies of credi spreads in a beer way han he radiional lognormal model. We also derived closed-form soluions for European opions on credi spreads. Call prices exhibi he same unusual properies found by Longsaff and Schwarz 995. he call value can be less han is inrinsic value and can be a concave funcion of he underlying credi spread. Comparing our model o Longsaff and Schwarz 995 model we have found ha he difference beween hem is more imporan for a-he-money calls. Alhough he closed-form soluions derived here are only for simple calls and pus on credi spreads wihin a GARCH framework valuaion expressions for oher credi spreads European exoic derivaives such as barrier opions could be derived in he same way. 4

17 -3 Figure : Moody's AAA and BAA log-spread over 3y US Bond BAA AAA Figure : Moody's AAA and BAA log-spread over y US Bond -3.5 BAA AAA

18 .6 Figure 3 : Moody's AAA log-spread over 3y US bond - Volailiies Figure 4 : Moody's AAA log-spread over y US bond - Volailiies

19 .8 Figure 5 : Moody's BAA log-spread over 3y US bond - Volailiies Figure 6 : Moody's BAA log-spread over y US bond - Volailiies

20 8 Figure 7 : Implied condiional log-spread disribuion for differen variance raios he normal disribuion Figure 8 : Implied condiional log-spread disribuion for differen mauriies GARCH days Normal days GARCH day Normal day

21 x Figure 9 : Credi spread call prices for differen variance raios Longsaff & Schwarz Inrinsic value Credi Spread -4 x Figure : Credi spread call price difference beween GARCH and Longsaff and Schwarz model for differen variance raios Longsaff and Schwarz a-he-money price is e Credi Spread 9

22 3.5-3 x 4 Garch days Longsaff & Schwarz Inrinsic value Figure : Credi spread call prices for a differen mauriy Credi Spread -4 x.54 Garch 5 days Inrinsic value Figure : Credi spread call prices for long erm mauriy

23 Appendix Convergence : Our model converges weakly o a coninuous-ime model. Firs given he mehodology in Heson and Nandi 999 when he ime sep lengh ends o zero he variance process in our GARCH model converges weakly o a square-roo process of Feller 95 Cox Ingersoll and Ross 985 and Heson 993. Define v as he limi process of variance h. Now we have o show ha he log-spread process also converges weakly o a coninuous-ime process. he dynamics of can be wrien as : + η δ + λv+ + v + z+ where h v + + and δ -γ. In he same way as in Heson and Nandi 999 who worked wih he special case δ no mean-reversion his process converges weakly o : d dv η δ + λv d + ω κ v d + σ v dz where Z is a Wiener process and ω σ and κ are defined as in Heson and Nandi 999. v dz Derivaion of he momen generaing funcion : Recall ha he momen generaing funcion of he spread wih a ime horizon condiional o he ime informaion is given by : exp φ f φ E. Firs we calculae his funcion for + in order o ge he log-linear form. Wihou any loss of generaliy we assume ha he ime sep lengh is equal o. We hen have : f + E E exp exp φ + exp φµ + φ γ + φλ h + + φ h + z + φµ + φ γ + φλ h E exp φ h z We obain a log-linear form for f + by noing ha z + is condiionally normally disribued and ha h + is known a ime. Hence we can wrie :

24 f φ exp φµ + φ γ + λφ + φ h. + + In he same way one can calculae f + and find ha we sill obain a log-linear form for he generaing funcion. Le us assume ha a ime + we have : f φ γ + A + + B + h + exp + + where A and B are deerminisic funcions. A ime he generaing funcion can be wrien using he ieraed expecaions law : f E E E hen we have by definiion of f + : f exp φ E+ exp φ f + { exp φ γ + A + + B + h } E + + φ. Replacing + and h + by heir expressions as funcions of and h + see Equaion we ge : f E exp exp + + exp β + + B - φ γ + φµγ + A + + β B + β B + h + φλγ h z + θ h + φγ h + z+ he wo firs erms in he expecaion operaor are known a ime hence he only erm ha we need o compue is he las one. his is where he Heson and Nandi's 999 GARCH specificaion akes he advanage over oher GARCH models. he las erm can be compued as a funcion of h + using he fac ha for a sandard normal variable z and consans a and b we have : E ab exp a z + b exp ln a exp a Under he risk-neural probabiliy measure we only need ha z + mus be a sandard normal variable condiional o ime and his is he case since h + is known and is consan

25 3 condiional o ime. hus he las erm can be developed and rearranged o obain a "perfec" square of z + and h + added o a remaining erm ha depends on h + and he GARCH parameers. Rearranging he erms in a log-linear form we ge Equaions 9 and in he main ex. Derivaion of dela : Noe ha he dela of a credi spread call is by definiion equal o : e C e C Dela. Using Equaion 3 ha gives he call valuaion expression we can wrie : P K P f C e r. Noe ha φ φγ φ f f. We hen have using Equaion 4 : + d φ φ π γ φ i f K Re P i A and d φ φ π γ γ φ i f K Re P f P f i. A In order o obain Equaion 7 we use a change of variable ϕ φ + i in Equaion A o see ha : d + + φ φ π γ φ i f K Re P K i he only remaining erm he firs on he righ-hand side of Equaion A muliplied by he discoun facor and by exp gives he dela expression as in Equaion 7.

26 4 Credi spread Call-Pu pariy : As for Black and Scholes model we proceed by noing ha he payoff of a porfolio ha is long in a call and shor in a pu is given by : K e e K K e + + hus he ime porfolio's value is simply he discouned forward value ha is : K e E e K e E e r r which is equal o : K f e r. his means ha : K f e P C r.

27 References Bollerslev. "Generalized Auoregressive Condiional Heeroskedasiciy" Journal of Economerics Vol pp Bollerslev. R.Y. Chou and K.F. Kroner "ARCH Modelling in Finance : A review of he heory and empirical evidence" Journal of Economerics Vol 5 Apr/May 99 pp Cox J. J. Ingersoll and S. Ross "A heory of he erm Srucure of Ineres Raes" Economerica Vol pp Engle R.F. D.M. Lilien and R.P. Robins "Esimaing ime Varying Risk Premia in he erm Srucure : he ARCH-M Model" Economerica Vol 55 March 987 pp Engle R.F. and V.K. Ng "Measuring and esing he Impac of News on Volailiy" Journal of Finance Vol pp Engle R.F. V.K. Ng and M. Roschild "Asse Pricing wih a Facor-ARCH Covariance Srucure : Empirical Esimaes for reasury Bills" Journal of Economerics Vol 45 July 99 pp Feller W. An Inroducion o Probabiliy heory and Is Applicaions Vol 966 Wiley & Sons New York. Foser D. and D. Nelson "Asympoic Filering heory for Univariae ARCH Models" Economerica Vol pp. -4. Heson S.L. "A Closed-Form Soluion for Opions wih Sochasic Volailiy wih Applicaions o Bond and Currency Opions" he Review of Financial Sudies Vol pp Heson S.L. and S. Nandi "A Closed-Form GARCH Opion Pricing Model" Forhcoming in he Review of Financial Sudies Kendall M. and A. Suar he Advanced heory of Saisics Vol 977 Macmillan Publishing Co. Inc. New York. Longsaff F.A. and E.S. Schwarz "Valuing Credi Derivaives" he Journal of Fixed Income June 995 pp. 6-. Weiss A.A. "ARMA Models wih ARCH Errors" Journal of ime Series Analysis Vol pp