A JOINT STOCK AND BOND MARKET BASED ON THE HYPERBOLIC GAUSSIAN MODEL

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1 November 5, 2012 A JOINT STOCK AND BOND MARKET BASED ON THE HYPERBOLIC GAUSSIAN MODEL NICOLE BÄUERLE AND ROBIN PFEIFFER Absrac In his paper, we inroduce a join bond and sock marke model based on he sae price densiy approach as a mean o discoun fuure paymens - wheher hese are sochasic dividend paymens or secure repaymens of governmen zerobonds Based upon a recipe of Rogers 1997, we define a sae price densiy model, he so-called Hyperbolic Gaussian model which allows for closed form zerobond prices and sock prices in an arbirage-free way I is paricularly useful for insurance applicaions where large ime horizons are considered We esimae he join facor model using he exended Kalman filer The model we propose here is compuaionally much simpler han oher models which have been considered in he lieraure Key words: Ineres rae models, Poenial approach, Dividend discoun approach, Kalman filer, Mone Carlo simulaion JEL subjec classificaions: G12, C51, E43, G22 1 Inroducion In order o compare sochasic cash flows, hey have o be discouned which requires a sochasic erm srucure model Mos erm srucure models in he lieraure have been developed for banking applicaions for an overview see eg Filipovic 2009, Cairns 2008, Musiela & Rukowski 2005, Rebonao 2002 or Brigo & Mercurio 2001 For insurance companies erm srucure dynamics play a cenral role Wha is special in insurance applicaions is ha firs, he duraion of producs like life or pension insurance conracs ypically exceed he available mauriies of he currenly observable erm srucure Nex, he insurer requires a erm srucure model o discoun his conracual liabiliies which ypically requires Mone Carlo simulaion due o he complexiy Finally life and pension insurers a leas in coninenal Europe mainly inves in fixed income securiies Fuure payoffs of insurance conracs herefore depend on inermediae reurns achieved in fixed income markes and inermediae porfolio allocaion which yields pah-dependence for mos insurance applicaions In his paper we consider he so-called Hyperbolic Gaussian model which is a special case of he sae price densiy models inroduced in Rogers 1997 The sae price densiy ς is someimes called risk-neural densiy or sae-price deflaor and i can be used o price coningen claims see eg Duffie 1992 If C T is he payoff of a coningen claim a ime T, hen is price a ime can be defined by C = E[ς T C T F ] ς In he classical Black-Scholes model wih one sock and consan shorrae r, drif µ and volailiy σ, he sae price densiy is for example given by ς = L B 1 L = exp 1 2 µ r σ 2 µ r σ where B = e r and Z Karlsruhe Insiue of Technology KIT, Insiue for Sochasics, Kaisersr 89, Karlsruhe, Germany nicolebaeuerle@kiedu and robinpfeiffer@kiedu Phone: Fax: The underlying projec is funded by he Bundesminiserium für Bildung und Forschung of Germany under promoional reference 03BAPAD2 The auhors are responsible for he conen of his aricle 1 c 0000 copyrigh holder

2 2 N BÄUERLE AND R PFEIFFER The process Z is a Brownian moion and L T can be used o define he risk neural measure dq dp = L T The price C a ime of a coningen claim which pays C T a ime T is given by [ C = B E Q CT B 1 T F ] = 1 L B 1 E [ C T L T B 1 T F ] = E [ C T ς T F ] where we used he Bayes rule for he second equaion In case of a defaul free bond we have C T = 1 and an explici bond price formula essenially requires ha E [ ] ς T F can be compued One represenaive of his family is he model considered in Cairns 2004 see also Cairns 2008 There ς := φ s M, sds where φ is a deerminisic funcion and M, s for s is a family of sricly posiive diffusion maringales This model is also a special case of he Flesaker and Hughson framework Flesaker & Hughson 1996, indeed his poin of view is aken in Cairns 2004 For special choices of φ and M in paricular M is aken as a funcion of an Ornsein-Uhlenbeck sae process, Cairns shows some properies of his model and in paricular recommends i for longerm ineres rae modeling, since i provides susained periods of boh high and low ineres raes This is of paricular imporance in insurance applicaions Moreover, i guaranees posiive ineres raes The price of a bond a ime wih mauriy T in Cairns model is given by T M, sφsds P, T = M, sφsds 11 Since he price depends on he whole pah of M, s his formula is compuaionally demanding and slow in Mone Carlo simulaions The main aim of our paper is o presen he Hyperbolic Gaussian model as a model wih similar feaures bu one which is very easy o implemen and hus ineresing for pracical purposes We also show ha he Hyperbolic Gaussian model can be exended o a join bond and sock marke - a feaure which is only heoreically possible in Cairns model Indeed he advanage of an explici model for he sae price densiy, like he Hyperbolic Gaussian model, lies in avoiding he inegral over he sochasic shorrae in discouning funcions This is paricularly helpful in he pricing of complex securiies such as in life insurance, which ypically require Mone Carlo simulaion In ha case, when eg a shorrae model is used, discouning based on shorrae inegrals are necessarily pah dependen and he frequency of inermediae seps is deermined by he shorrae inegral and hence ulimaely he discouning funcion, no he securiy o be priced and is cash flow Using an explici model for he sae price densiy implies ha inermediae simulaion poins are deermined by he securiy o be priced only and hence ypically far less random variables have o be used in Mone Carlo simulaions based upon he sae price densiy Thus, we consider here he Hyperbolic Gaussian model which on he negaive side does no guaranee posiive ineres raes, however is compuaionally much more efficien han he model in Cairns 2004 and shows a similar behavior Indeed, in boh models when implemened as a wo-facor model, one sae vecor componen usually coincides wih he slope of he ineres curve whereas he oher componen coincides wih he level For a deailed comparison of boh models, including parameer esimaion resuls and properies see Pfeiffer 2010 cp also Secion 43 Moreover, wha is imporan for us, is ha he Hyperbolic Gaussian model can produce longer periods of low ineres raes and he hisorical fi using he exended Kalman filer is very good see Secion 4 Thus we hink his model migh be helpful for simulaion ools like DFA dynamic financial analysis in insurance companies Le us noe here ha also in Yao 2001 an exponenial sae price densiy has been used o derive bond prices, opion prices and foreign exchange raes Since pure erm srucure models are insufficien o simulae he asse side of an insurance company, a major ask is o exend he bond marke consisenly Wilkie 1984, 1986 argues ha an insurance or pension fund model should a leas incorporae consol yields, sock prices, ς

3 A JOINT STOCK AND BOND MARKET BASED ON THE HYPERBOLIC GAUSSIAN MODEL 3 dividend yield and inflaion In Wilkie 1995 he invesmen model is even expanded In order o incorporae a leas a sock we use wo approaches: A dividend discoun approach and an exension of he Black Scholes model In he dividend discoun approach, he sock price a ime is inerpreed as he value of all fuure dividends discouned a ime Such an approach easily fis ino he sae price densiy framework which is used o discoun dividends In a coninuous dividend paying seing his has been used in Graziano & Rogers 2006 In he model invesigaed in Cairns 2004 such an approach is also possible from a heoreical poin of view however pracically unfeasible due o compuaional limiaions For he Hyperbolic Gaussian model on he oher side i provides an arbirage-free, implemenable sock pricing framework The ouline of our paper is as follows: In he nex secion we review he sae price densiy approach and presen he Hyperbolic Gaussian model I is based upon mulidimensional Ornsein-Uhlenbeck sae processes The special one-dimensional case can already be found as an example in Rogers 1997 and Cairns 2008 Explici bond-price formulas are derived as well as zerobond raes and he shorrae I is also shown ha he family of bond prices is free of arbirage Secion 3 hen exends his model o an arbirage-free join bond and sock marke model by using he dividend discoun approach and an exension of he Black Scholes model Secion 4 is dedicaed o esimaing he model parameers Since we have a facor model we use he exended Kalman filer for esimaion of he model parameers in a Quasi-Maximum- Likelihood approach Only he implemenaion for he dividend discoun approach is explained The unobservable process is he sae process and he observable process is he erm srucure Since he measuremen equaion is nonlinear we use he exended Kalman filer The procedure is applied o a hree-dimensional facor process and one sock using a daase of he Federal Reserve consising of yields derived from he US Treasury securiies and he S&P 500 The esimaion resuls and possible problems are discussed A shor conclusion and an appendix conaining he proofs of some of he heorems in Secion 22 complees he paper 2 The Exended Rogers Framework and he Hyperbolic Gaussian Model 21 The Exended Rogers Framework Rogers 1997 presened a generic approach o model he sae price densiy as a funcion of an underlying Markovian sae vecor process More precisely, one has o choose a coninuous-ime Markov process X wih values in R d and a posiive funcion f wih domain R d, which, ogeher wih a parameer α R provides he sae price densiy process ς by ς = e α fx, 0 We assume ha all random variables are defined on a common probabiliy space Ω, F, P where P is he so-called pricing measure The process X is ofen called sae process In wha follows le F be he filraion generaed by X, ie F = σx s, s We will always assume ha he expecaion of fx exiss Using he definiion of ς, he curren price C of a coningen claim a ime which pays C T a ime T > under he sae price densiy approach is given by C = E [ς T C T F ] = e αt E [fx T C T X ], 21 ς fx This provides a formula which can convenienly be evaluaed by Mone Carlo mehods since only he condiional disribuion of X T X has o be simulaed This is in conras o shorrae models where usually he whole shorrae pah r has o be simulaed o obain he discoun funcion T r s ds The price of a zerobond wih mauriy a ime T is herefore given by P, T = e αt E [fx T X ] 22 fx and hence a measurable funcion of he underlying sae X The quesion now is how o choose he funcion f and he sae vecor dynamics Several imporan heoreical and pracical properies resric our choices Firs, a basic requiremen for applicabiliy is ha zerobond

4 4 N BÄUERLE AND R PFEIFFER prices and hence ineres raes are available in closed form Given 22, his implies ha E [fx T X ] mus be available in closed form Ye anoher imporan requiremen is ha ineres raes are mean revering in he sense ha we wan hisorically observed ineres raes o occur repeaedly in simulaions Following Lierman & Scheinkman 1991, we know ha o realisically model erm srucure dynamics, muli-dimensional sae processes X are required Typical choices for X herefore are processes whose componens are Ornsein-Uhlenbeck or Cox-Ingersoll-Ross processes The following heorems derive furher imporan condiions The firs saemen can be found in Rukowski 1997, Secion 3 Theorem 21 If he sae price densiy is a supermaringale, hen he erm srucure model is free of arbirage The nex heorem links non-negaiviy of ineres raes wih he sae price densiy cf also Cairns 2008, Theorem 82 Theorem 22 Ineres raes are non-negaive if and only if he sae price densiy is a supermaringale Proof Ineres raes are non-negaive if and only if 1 P, T holds for all [0, T ], hence 1 P, T = E [ς T X ] ς This is equivalen o ς E [ς T X ] and hence o ς being an F -supermaringale Noe, hough, ha in Theorem 21 he supermaringale propery of he sae price densiy is only a sufficien condiion for obaining an arbirage-free marke, he condiion in Theorem 22 is necessary If we give up he supermaringale propery of ς, all ha is required for a erm srucure model is a muli-dimensional mean revering process and a posiive funcion f which guaranee ha E [fx T X ] is available in closed form If no-arbirage can be shown, we have a viable erm srucure model, albei one which allows for negaive ineres raes as many currenly used erm srucure models 22 The Hyperbolic Gaussian model In his secion, we will presen an arbirage-free erm srucure model which we call Hyperbolic Gaussian model in which he sae price densiy is no a supermaringale and hence no-arbirage has o be shown separaely The Hyperbolic Gaussian model is he fourh example of Rogers 1997, specified by he choice of he funcion fx = coshγ x + c where γ R d, c R and he sae process X which is given by he follow dynamics dx = κ µ X d + CdZ 23 where κ = diagκ i R d d is a diagonal marix wih κ 1,, κ d on he diagonal, µ R d, C is a d n marix and Z = Z 1,, Z n is an n-dimensional Brownian moion under he pricing measure wih insananeous correlaion marix CC =: ρ = ρ ij The sae process herefore is an Ornsein-Uhlenbeck process under he pricing measure Hence he sae price densiy formula yields ς := e α coshγ X + c = 1 2 e α expγ X + c + exp γ X c 24 To simplify noaion, we inroduce he real-valued process V := γ X From 23 i follows ha dv = γ κ µ X d + ĈdZ 25 where Ĉ := γ C Hence we can express he sae price densiy as ς := e α coshv + c = 1 2 e α expv + c + exp V c 26

5 A JOINT STOCK AND BOND MARKET BASED ON THE HYPERBOLIC GAUSSIAN MODEL 5 An imporan propery of Ornsein-Uhlenbeck processes is ha he condiional disribuion of X T given X is mulivariae normal wih E [X T X ] = e κt X + 1 e κt µ 27 whereby e κt := diage κ it R d d, 1 e κt := diag1 e κ it R d d and condiional covariance marix Σ, T := Cov [X T X ] = ρlk 1 e κ k+κ l T 28 κ k + κ l l,k=1,,d Theorem 23 For he Hyperbolic Gaussian model as defined above, he price of a zerobond a ime wih mauriy T is P, T = e αt cosh E [V T X ] + c e 1 2 γ Σ,T γ coshv + c Proof Using 21 wih CT = 1, he bond price is given by P, T = e αt E [fx T X ] fx wih fx = coshγ X + c = coshv + c Using he fac ha X T X has a condiional mulivariae normal disribuion, we ge E [expv T + c X ] = exp E [V T + c X ] Cov [V T + c X ] = exp γ e κt X + 1 e κt µ + c γ Σ, T γ Thus, we arrive a E [fx T X ] = E [coshv T + c X ] = 1 [ exp γ e κt X + 1 e κt µ + c 2 + exp γ e κt X + 1 e κt ] 1 µ c exp 2 γ Σ, T γ 1 = cosh E [V T X ] + c exp 2 γ Σ, T γ which yields he resul From he zerobond price formula we can direcly derive he nominal zerobond raes y, T := 1 T log P, T which are given in he nex corollary: Corollary 24 For he Hyperbolic Gaussian model, nominal zerobond raes y, T a ime wih expiry dae T are given by y, T = α log cosh E [V T X ] + c T + log cosh V + c T γ Σ, T γ 2T Insananeous forward raes f, T = T log P, T and shorraes r = f, can be derived as well In wha follows denoes he Euclidean norm Corollary 25 For he Hyperbolic Gaussian model, he shorrae is given by r = α γ κ µ X anhv + c 1 2 Ĉ 2 and insananeous forward raes f, T are given by f, T = α anhe[v T X ] + cγ κe κt X µ + γ e κt ρe κt γ

6 6 N BÄUERLE AND R PFEIFFER Proof By definiion we obain and f, T = log P, T T = T log = T e αt coshe[v T X ] + c e 1 2 γ Σ,T γ coshv + c αt + log coshe[v T X ] + c γ Σ, T γ = α anhe[v T X ] + cγ κe κt X µ γ e κt ρe κt γ r = f, = α anhv + cγ κx µ 1 2 Ĉ 2 which yields he resul We can see from he shorrae ha he Hyperbolic Gaussian model allows for negaive ineres raes Neverheless, he probabiliy of negaive ineres raes is small due o non-lineariy of he funcion f in he sae process and he sae process iself being condiionally normal The higher α, he lower he condiional probabiliy of negaive ineres raes, so ha we would prefer model esimaes wih high α However, α has a direc economic inerpreaion which migh bound α and hence shape he probabiliy of negaive ineres raes Theorem 26 The parameer α equals he asympoic long rae lim T y, T Proof Since lim T E [X T X ] = µ and since log and cosh are coninuous and cosh 1 we obain direcly lim T y, T = α The asympoic long rae is very influenial for long-erm applicaions as i deermines long-erm discouning funcions As can be seen in Yao 1999, i is consan for many currenly used erm srucure models, paricularly also he affine model framework see Duffie & Kan 1996 and Dai & Singleon 2000 Noe ha in mos models he consan asympoic long rae is a funcion of several model parameers, which makes sensiiviy analysis wih respec o he asympoic long rae impossible, whereas he Hyperbolic Gaussian model allows for such sensiiviy analysis As he Hyperbolic Gaussian model also allows for an expansion o sock marke dynamics, i is paricularly ineresing for long-erm usage in pension or insurance applicaions which require long-erm discouning Since he sae price densiy in he Hyperbolic Gaussian model is in general no a supermaringale, i does no follow from sandard lieraure ha i is free of arbirage The proof of he following heorem can be found in he appendix Theorem 27 For he Hyperbolic Gaussian model as defined above, he bond marke is free of arbirage Remark 28 Noe ha he proof of Theorem 27 implies ha he marke price of risk is given by ΛV = anhv + cĉ and hus bounded I can be used o define an equivalen maringale measure Q which does no depend on he ime o mauriy and o define a corresponding Q- Brownian moion Z Q Finally noe here ha he Bond price formula in Theorem 23 is very easy o implemen since i only requires he simulaion of X which is normally disribued This is in conras o formula 11 used by Cairns which is compuaionally demanding Moreover, i has been shown in Pfeiffer 2010 ha boh models have similar properies hough Cairns model guaranees posiive ineres raes Hence for pracical purposes he Hyperbolic Gaussian model seems o be superior

7 A JOINT STOCK AND BOND MARKET BASED ON THE HYPERBOLIC GAUSSIAN MODEL 7 3 Join Bond and Sock Marke We exend now he Hyperbolic Gaussian bond model o include a sock This is done by wo differen approaches: The dividend discoun approach and a simple exension of he Black Scholes model 31 The Dividend Discoun Approach In he dividend discoun approach one assumes ha he sock pays random dividends a deerminisic, discree ime poins 0 τ 1 < τ 2 < wih τ n for n More precisely we denoe he dividend paid a ime τ n by D τn 0 The sock can hen be inerpreed as a porfolio of infiniely many securiies which pay D τn a imes τ n Hence S = τ n E [ς τn D τn F ] ς 31 This approach can be raced back o Gordon 1959 A coninuous version in he sae price densiy approach can be found in Graziano & Rogers 2006 Remark 31 I is imporan o noe ha due o dividend paymens, we have o disinguish beween he sock price and he wealh of a sock holder, which includes dividends already paid For example he wealh of an invesor a ime T who bough he sock a ime is given by W T = S T + ς τn D τn ς T τ n T For implemenaion of he join sock and bond marke model using he dividend discoun approach, we have o specify he dividend paymen process D depending on he sae process X in such a way ha he expecaion E [ς τn D τn F ] in 31 can be calculaed in closed form Furhermore, we require dividends o be non-negaive and we implemen an expeced dividend growh over ime o compensae for inflaion, as company income from which dividends are o be paid should be inflaion proeced An inuiive proposal defines D = D, X := exp c + µ + γ D X, 32 for consans c R, µ R and γ D R d This specificaion guaranees posiive dividends, varying around an exponenial rend defined by µ This dividend growh rend is inroduced o capure he assumpion of company income being adaped o rising prices Unlike nominal bonds, sock holders herefore hold some proecion agains inflaion implemened by he growh rend µ The facor c is a muliplicaive scaling facor for sock prices and dividends, which proved o be necessary in implemenaions bu which is omied in furher heoreical consideraions From an implemenaion poin of view, i is imporan o noe ha our choice of D allows for closed form soluions for he prices of dividend-paying securiies, as he produc of he wo lognormally disribued variables D and ς is again lognormally disribued The curren price S n of he sochasic dividend D τn o be paid a ime τ n herefore is available in closed form by 1 ς S n = E [ς τn D τn X ] = exp ατ n + µτ n + γ D E [X τn X ] + 1 γ Σ, τ n γ + γ D Σ, τ n γ D 2 cosh c + γ E [X τn X ] + γ Σ, τ n γ D 33 which defines a sock price process by equaion 31 We assume ha he sum in 31 converges This is eg he case when κ i 0, α µ and τ n = n which are realisic assumpions I remains now o prove ha he join Hyperbolic Gaussian bond and sock marke model is free of arbirage The proof of he nex heorem can be found in he appendix 1 Noe ha he same definiion of he dividend paymen process may also be used o expand Rogers 1997 firs and second examples which specify fx := exp x and fx := exp c + x Qx, respecively

8 8 N BÄUERLE AND R PFEIFFER Theorem 32 For he Hyperbolic Gaussian model wih he sock price process S defined as he infinie sum of discouned dividends wih payoffs given by32, he join bond and sock marke is free of arbirage 32 Exension of Black Scholes Model Anoher simple possibiliy o obain an arbiragefree join bond and sock marke is o use he classical Black Scholes model for he non-dividend paying sock and assume ha under he risk neural measure Q cf Remark 28, he drif of he sock is given by he shorrae of he Hyperbolic Gaussian model in Corollary 25, ie ds = S r d + σ T CdZ Q By consrucion his yields an arbirage-free marke An example for his approach may be found in Albrech 2007, where a one-facor Vasicek model is used for he shorrae and a Black-Scholes model for he sock dynamics Since he sock price is a sochasic exponenial, i is also possible o derive an explici formula for S 4 Esimaion For esimaion and risk-managemen, he physical measure is required Whereas we are essenially free o specify he marke price of risk and hence he physical measure, a sandard approach derives he physical measure in such a way ha he sae process follows similar dynamics under boh he physical measure and he measure ypically used for pricing For he Hyperbolic Gaussian model, his implies ha he sae process under he physical measure P should follow Ornsein-Uhlenbeck dynamics as under he pricing measure Using Girsanov s heorem wih d P d P = E Λ P,P X s dz s where E is he sochasic exponenial we know ha 0 dz P = dz + Λ P,P X d is a Brownian moion wr P Thus, implies dx = κ µ X d + CdZ = CΛ P,P X + κ µ κx d + CdZ P, κµ! = CΛ P,P X + κ µ so Λ P,P X := C 1 κ µ µ As his drif correcion erm is consan, he Novikov condiion is fulfilled and boh measures are equivalen Noe ha alernaive specificaions migh provide beer hisorical fi and in paricular superior erm premiums, see for example Duffee Esimaion by Exended Kalman Filer We esimaed boh models wih he exended Kalman filer Since he approach is similar for he Black Scholes model, we resric our presenaion o he firs model A deailed descripion for he second model can be found in Pfeiffer 2010 The Hyperbolic Gaussian model allows for implemenaion of he exended Kalman filer for esimaion of he model parameers in a Quasi-Maximum-Likelihood approach The sae process X is he hidden process and he nominal zero bond raes Y are observed for differen mauriies as well as he sock For a general inroducion o he Kalman filer, see eg Harvey 1991 or Kellerhals 2001 Using a discree ime grid and he sandard noaion

9 A JOINT STOCK AND BOND MARKET BASED ON THE HYPERBOLIC GAUSSIAN MODEL 9 F Y = σ {Y 0, Y 1,, Y }, N and X := E[X F Y ] X 1 := E[X F Y 1] he ransiion equaions are given by and Σ := E [ X X 1 X X 1 F Y Σ 1 := E [ X X 1 X X 1 F Y 1], X 1 = e κ X e κ µ, 41 Σ 1 = e κ Σ 1 1 e κ + Qθ 42 where θ denoes he vecor of model parameers and he covariance marix Qθ is given by compare equaion 28 Qθ = ρlk The measuremen equaion is defined by y M, + σ 1 Y = y M, + σ n = S M 1 e κ k+κ l κ k + κ l l,k=1,,d g 1 X ; θ g n X ; θ + g n+1 X ; θ ] ɛ 1 θ ɛ n θ θ ɛ n+1 where y M, + σ i is he ineres rae observed in he marke wih ime o mauriy σ i, i = 1,, n and S M is he observed sock price in he marke a ime The funcion g i, i = 1,, n is given by g i X ; θ = α log cosh γ E[X +σi X ] σ i + log cosh γ X + c σ i γ Σ, + σ i γ σ i The funcion g n+1 describes he model-implied sock price and is given by g n+1 X ; θ = 1 [ exp ατ n + µτ n exp c + γ D + γ E[X τn X ] 2ς τ n> γd + γ Σ, τ n γ D + γ + exp c + γ D γ E[X τn X ] + 1 ] 2 γd γ Σ, τ n γ D γ Furhermore ɛ θ = ɛ 1 θ,, ɛ n+1 θ R n+1 is a mulivariae normal error erm wih covariance marix Cov ɛ := H = diag ν,, ν, ν S R n+1 n+1 where ν is used for he measuremen errors of yields and ν S for he measuremen errors of he sock price This specificaion is due o he scaling difference beween ineres raes, roughly varying beween 0 and 02, and sock prices roughly varying beween 100 and 1500 Individual measuremen errors for each mauriy may allow for furher insighs, ye his benefi is made up for by he subsanially higher number of model parameers one had o esimae in his case For he updaing sep, we require he Kalman gain marix K := Σ 1 B 1 F 1 1

10 10 N BÄUERLE AND R PFEIFFER where B 1 is he Jacobi marix of he non-linear measuremen funcion g = g 1,, g n+1 of he Hyperbolic Gaussian model given by x 1 g 1 x, θ x d g 1 x; θ B 1 = x 1 g n x; θ x d g n x; θ R n+1 d x 1 g n+1 x; θ x d g n+1 x; θ x=x 1 Noe ha θ = γ, γ D, µ, µ, κ, ρ, c, ν, ν S The required derivaives for he yield and sock price measuremens can be derived using and x i P, T = γ i P, T [ anhγ E [X T X ] + ce κ i anhγ X + c ] 43 S = S γi D e κ it + anhh, X γ i e κ it γ i anhγ X + c x i respecively wih h, X = E[V T X ] + γ Σ, T γ D + c Wih F 1 = B 1 Σ 1 B 1 + H, he Kalman gain marix can be calculaed The predicion error yields he updaing seps v = y M, + σ 1,, y M, + σ n, S M gx 1, θ 44 and which concludes he filer X = X 1 + K v Σ = Σ 1 K B 1 Σ 1, For implemenaion, we have o cu off he infinie sum of dividend paying securiies for compuaional reasons As he dividends are non-negaive, even for lower ineres raes an addiional n + 1-h dividend likely increases he sock price Noe ha a posiive dividend growh rend µ may compensae he discouning of fuure dividends, hus he curren value of expeced fuure dividends will no necessarily decrease wih higher imes o mauriy In general, we expec ha cuing off dividend paymens of laer daes implies ha early dividends will be overesimaed o make up for omied dividends No surprisingly, we found a conflicing role of he asympoic long rae α deermining longerm discouning, and he dividend growh rae µ In he join model, esimaes of boh hese parameers were very unsable Resricing α = 0042 in our esimaes resuled in beer sabiliy of boh parameers The economic jusificaion of he specificaion α = 0042 is based on a simplifying applicaion of he Fisher hypohesis Fisher 1930, which pariions nominal ineres raes ino he expeced inflaion rae and a real rae Now he long-erm average of he real rae is aken as 22% and he expeced asympoic inflaion rae is inerpreed as he inflaion arge of he cenral bank, currenly 2% The Fisher hypohesis implies ha he sum of boh equals he nominal ineres rae Using hisorical dividend daa should solve hese problems, ye due o raher small dividend paymens in absolue erms he cu-off levels are necessarily high, implying a higher compuaional effor for esimaion Furhermore, a leas in case of US daa, each ime sep would require implemenaion of an individual dividend paymen schedule as he imes beween dividend paymens are no necessarily consan Finally, we wan o use sock index daa, moivaed by our proposiion of he Hyperbolic Gaussian model for long-erm insurance applicaions, and he problems of irregular dividend paymens are even worse for an index or generally for a porfolio of socks Therefore, we omied he inclusion of hisorical dividend daa Effecively, we model a heoreical finie cash flow whose curren value equals he sock price Omiing hisorical dividend paymens allows for simplifying assumpions regarding

11 A JOINT STOCK AND BOND MARKET BASED ON THE HYPERBOLIC GAUSSIAN MODEL y 05y 1y 2y 3y 5y 7y 10y S&P Table 1 Hisorical mean absolue errors in basis poins for he yields and in icks for he index frequency and regulariy of dividend paymen daes We deem hese simplifying assumpions no less viable han he frequen assumpion of a coninuous dividend yield on socks The componens of he sae process are assumed o be Ornsein-Uhlenbeck processes These are ypically easy o be esimaed, unless mean reversion is low In ha case, he exended Kalman filer may be unable o derive he rue value of long-erm mean µ, as µ eners he filer equaions only hrough he ransiion equaion 41 and he impac of µ in his equaion decreases wih he mean reversion facor κ Consequenly, one has o examine he filered sae process o assess he fi of µ In he following esimaes, we resriced he long-erm mean of he sae process under he pricing measure o be zero, µ = 0 The exended Kalman filer is able o esimae non-resriced µ in case of he pure bond marke model, alhough improvemen of he more general model is negligible In he join bond and sock marke model, resricing µ improved esimaion speed and sabiliy subsanially 42 Esimaion-Resuls For esimaion of ineres raes, we use a hree-dimensional sae process and a daase of he Federal Reserve consising of 025, 05, 1, 2, 3, 5, 7 and 10-year yields derived from US Treasury securiies from January 1984 o January 2008 The daase is obained from he Federal Reserve download poral For esimaion of he sock marke we use S&P 500 price index daa This reflecs well he assumpion ha he insurance company invess in a well diversified sock fund raher han a single sock Furhermore, here is no dividend jump in he daa 43 Resuls for he Dividend Discoun Approach For simpliciy, we did no use dividend daa direcly, bu assume ha he dividend paymen process is semiannually and unobservable In pracice, we are replicaing a dividend paymen process which implies he same price daa as he S&P 500 index We found ha he exended Kalman filer very quickly provides parameer esimaes which closely fi hisorical ineres raes and he dynamics of he sock price, bu only a few of hose provide a close fi of he absolue sock price as well Typically, he model-implied sock price is highly correlaed o he observed sock price, ye differs in absolue value To overcome his drawback, a second esimaion sep was implemened which uses hese firs esimaes and fis only he parameers describing he sock price dynamics, ha is γ D, ρ ij, µ and c, o he sock price observaions The final resuls differ from he iniial values only slighly in Loglikelihood, ye guaranee in mos cases a good hisorical fi of he sock price see Figure 1 As he exended Kalman filer provides only Quasi-ML esimaes, we deem such a correcing approach as viable Table 2 provides hree esimaed parameer ses We see ha he parameer esimaes sill show considerable variabiliy, which may be due o overparamerizaion Noe ha muliple local maxima of he Loglikelihood funcion are a ypical finding in complex applicaions However noe ha he measuremen errors ν and ν S for he sock are very small Table 1 provides absolue hisorical errors of he model We see ha for all esimaes hisorical errors are exremely small, in paricular regarding he sock, and very similar across differen esimaed parameer ses We examine he filered underlying sae vecor in figure 2 We find clear correlaion beween one of he sae facor componens and he slope of he ineres curve and a second sae facor componen and he sock price The hird sae facor is correlaed o he 10-year rae Boh

12 12 N BÄUERLE AND R PFEIFFER Figure 1 Model implied blue and hisorical S&P 500 prices in he dividend discoun model γ 1 γ 2 γ 3 γ1 D γ2 D γ3 D µ 1 µ 2 µ 3 µ κ 1 κ 2 κ ρ 12 ρ 13 ρ 23 c ν ν S LogL Table 2 Esimaes by he exended Kalman filer for he Hyperbolic Gaussian model using US Treasury erm srucure daa and S&P500 sock marke daa from January 1984 o January 2008 he dynamics of he level and he sock price facor provide very small mean reversion, as could be seen in our esimaes of µ 2 and µ 3 44 Resuls for Black Scholes model We implemened again a hree-facor model One main difference beween he Black-Scholes based approach and he dividend discoun model is compuaional speed Whereas he dividend discoun model is compuaionally slow, he Black-Scholes based approach is very fas in esimaion and simulaion alike, since calculaion of he curren sock reurn is compuaionally equivalen o compuaion of an ineres rae wih given mauriy Firs noe ha sock reurns differ subsanially from ineres raes in auocorrelaion and variance To accoun for hese differences, we implemened wo approaches, based on resricions of he parameer vecor γ The framework presened by Albrech 2007 implies a pure sock marke facor, which in our model would be equivalen o γ 3 = 0 We consider wo model frameworks, one wihou resricions on γ, which implies ha he sae vecor drives boh sock and bond markes, and one wih γ 3 = 0, which implies wo sae vecor

13 A JOINT STOCK AND BOND MARKET BASED ON THE HYPERBOLIC GAUSSIAN MODEL 13 Figure 2 Filered sae vecor componens lef and empirical proxies of he firs principal componens of he erm srucure componens driving bond and sock markes, and one sae vecor componen driving only he sock marke Noe ha implemenaion of hese resricions is very easy, and pricing formulae sill hold Table 3 provides MAEs of implied yield curves We see ha resricing γ implies a poorer erm srucure fi The reason is ha all hree vecor componens conain erm srucure daa for general γ, whereas γ 3 = 0 guaranees ha he hird sae vecor componen drives sock reurns only and herefore improves sock reurn fi Considering hisorical fi of he sock price, we have MAEs in icks of more han 1700 for general γ and 18 basis poins for γ 3 = 0 The reason is ha for γ 3 = 0, he Exended Kalman filer fis Z 3 o he observed sock price, whereas wih general γ a rade-off exiss beween fiing he sock price and he erm srucure Given he lower Loglikelihood values of he resriced approach, we can expec ha he disribuion of

14 14 N BÄUERLE AND R PFEIFFER Assuming γ y 05y 1y 2y 3y 5y 7y 10y Assuming γ 3 = Table 3 Hisorical mean absolue errors in basis poins for he yields Z 3 according o he filering in he resriced case deviaes from he heoreical model-implied disribuion of Z 3 By definiion of he model, sock reurns are normally disribued, whereas i is well known ha his is no he case in realiy Therefore he good hisorical fi of he model assuming γ 3 = 0 does no reflec he basic problems his approach akes from he Black-Scholes model We expec ha non-normaliy of sock reurns is responsible for he lower Loglikelihood values of he resriced model A sraighforward improvemen of he join model would allow for sochasic volailiy of he sae vecors, hereby inroducing sochasic volailiy in he sock marke as well as he bond marke 45 Comparison In general, sock marke models may be implemened using reurn-based or price-based approaches Boh approaches have heir meris: banking applicaions ypically consider sock derivaives, which are based on sock prices raher han reurns Therefore, pricebased approaches are superior for banking applicaions Once dividend paymens are inroduced, however, he siuaion changes To realisically implemen discree dividend paymens, we require pah-dependen approaches and consider reinvesmen of dividends payed In insurance applicaions, reinvesmen of dividends is an imporan ask since, over he long run, dividend reurns make up a sizeable par of overall sock reurns and furhermore inermediae dividend paymens provide free cash flows wihou he need o liquidae asses under managemen The Black Scholes model is compuaionally superior o he dividend discoun model, ye i does no fi hisorical daa o he same exen as he dividend discoun model 5 Conclusion The Hyperbolic Gaussian model is based on he sae price densiy approach and allows for an easy esimaion of parameers and fas Mone Carlo simulaion I can also be exended o a join bond and sock marke Based upon is good long-erm fi of boh sock prices and ineres raes and he increased imporance of changes in ineres raes, he Hyperbolic Gaussian model migh provide a useful ool for long-erm risk managemen like DFA due o he simple and fas implemenaion of Mone Carlo simulaions I herefore could be useful for insurance applicaions The proof of Theorem 27 is as follows: 6 Appendix Proof Using he Iô-Doeblin formula i is easy o see ha he sae price densiy saisfies he SDE ς = 1 2 e α e V+c + e V c dς = ας d + anhv + cς dv ς d < V >

15 A JOINT STOCK AND BOND MARKET BASED ON THE HYPERBOLIC GAUSSIAN MODEL 15 Since < V > = Ĉ 2 we obain dς = ς α + γ κ µ X anhv + c + 1 2Ĉ Ĉ In view of Corollary 25 we obain dς = ς r d + ς anhv + cĉdz d + ς anhv + cĉdz Le us denoe by EX he sochasic exponenial of he process X Since 0 r sds is of bounded variaion we obain ς = E r s ds E anhv s + cĉdz s 61 = exp r s ds E 0 anhv s + cĉdz s If we define B = exp 0 r sds and L := E 0 anhv s + cĉdz s we obain ς = B 1 L Noe ha since anh is bounded, L is an F -maringale wih expecaion 1 Hence we can define he probabiliy measure Q by dq d P F = L Noe ha Q does no depend on T From he Bayes formula we obain P, T = E[ς T F ] ς = E[B 1 T L T F ] B 1 = B E Q [B 1 T L Hence he discouned bond price is a maringale under Q which shows ha he marke is free of arbirage The proof of Theorem 32 is as follows: Proof I is sufficien o show ha ς 1 B 1 E[ς τn D τn F ] is a Q-maringale for he same Q as in he proof of Theorem 32 However, he Bayes formula again implies which implies he saemen E[ς τn D τn F ] = E[B 1 τ n L τn D τn F ] ς B 1 L = B E Q [B 1 τ n D τn F ] References Albrech, P 2007 Einige Überlegungen zur simulanen Modellierung von Akienindex und Zinssrukur Mannheimer Manuskripe zu Risikoheorie, Porfolio Managemen und Versicherungswirschaf Brigo, D & Mercurio, F 2001 Ineres Rae Models: Theory and Pracice Springer Cairns, A J G 2004 A family of erm-srucure models for long-erm risk managemen and derivaive pricing Mahemaical Finance 14, Cairns, A J G 2008 Ineres Rae Models Princeon Universiy Press, Princeon, NJ Dai, Q & Singleon, K J 2000 Specificaion analysis of affine erm srucure models Journal of Finance 55, Duffee, G R 2002 Term premia and ineres rae forecass in affine models Journal of Finance 57, Duffie, D 1992 Dynamic Asse Pricing Princeon Universiy Press, Princeon, NJ Duffie, D & Kan, R 1996 A yield-facor model of ineres raes Mahemaical Finance 6, Filipovic, D 2009 Term-Srucure Models Springer Fisher, I 1930 The Theory of Ineres The Macmillan Co, New York Flesaker, B & Hughson, L 1996 Posiive ineres Risk Gordon, M 1959 Dividends, earnings, and sock prices The Review of Economics and Saisics 41, F ]

16 16 N BÄUERLE AND R PFEIFFER Graziano, G D & Rogers, L 2006 Hybrid derivaives pricing under he poenial approach Working paper, Universiy of Cambridge Harvey, A C 1991 Forecasing, Srucural Time Series Models and he Kalman Filer Cambridge Books, Cambridge Universiy Press Kellerhals, B P 2001 Financial Pricing Models in Coninuous Time and Kalman Filering Lecure noes in economics and mahemaical sysems, Springer, Berlin Lierman, R & Scheinkman, J 1991 Common facors affecing bond reurns Journal of Fixed Income Musiela, M & Rukowski, M 2005 Maringale Mehods in Financial Modeling, vol 36 of Sochasic Modelling and Applied Probabiliy 2 ed, Springer Pfeiffer, R 2010 Sae price densiy models for he erm srucure of ineres raes applicaions o insurance and expansions o he sock marke macroeconomic variables hp: //wwwdigbibubkauni-karlsruhede/vollexe/ PhD hesis, KIT Rebonao, R 2002 Modern Pricing of Ineres-Rae Derivaives: The Libor Marke Model and Beyond Princeon Universiy Press Rogers, L 1997 The poenial approach o he erm srucure of ineres raes and foreign exchange raes Journal of Mahemaical Finance 7, Rukowski, M 1997 A noe on he Flesaker-Hughson model of he erm srucure of ineres raes Applied Mahemaical Finance Wilkie, A 1984 Seps owards a comprehensive sochasic model Research discussion paper, The Insiue of Acuaries Wilkie, A 1986 A sochasic invesmen model for acuarial use Transacions of he Faculy of Acuaries 39, Wilkie, A 1995 More on a sochasic asse model for acuarial use Briish Acuarial Journal 1, Yao, Y 1999 Term srucure modeling and asympoic long rae Insurance: Mahemaics and Economics 25, Yao, Y 2001 Sae price densiy, Esscher ransforms, and pricing opions on socks, bonds, and foreign exchange raes Norh American Acuarial Journal 5,