The arbitrage-free Multivariate Mixture Dynamics Model: Consistent single-assets and index volatility smiles

Transcription

1 The arbtrage-free Multvarate Mxture Dynamcs Model: Consstent sngle-assets and ndex volatlty smles Damano Brgo Francesco Rapsarda Abr Srd Frst verson: 1 Feb Ths verson: 23 Sept Frst posted on SSRN & arxv on Feb 2013 Abstract We ntroduce a multvarate dffuson model that s able to prce dervatve securtes featurng multple underlyng assets. Each asset volatlty smle s modeled accordng to a densty-mxture dynamcal model whle the same property holds for the multvarate process of all assets, whose densty s a mxture of multvarate basc denstes. Ths allows to reconcle sngle name and ndex/basket volatlty smles n a consstent framework. Our approach could be dubbed a multdmensonal local volatlty approach wth vector-state dependent dffuson matrx. The model s qute tractable, leadng to a complete market and not requrng Fourer technques for calbraton and dependence measures, contrary to multvarate stochastc volatlty models such as Wshart. We prove exstence and unqueness of solutons for the model stochastc dfferental equatons, provde formulas for a number of basket optons, and analyze the dependence structure of the model n detal by dervng a number of results on covarances, ts copula functon and rank correlaton measures and volatltes-assets correlatons. A comparson wth samplng smply-correlated sutably dscretzed one-dmensonal mxture dynamcal paths s made, both n terms of opton prcng and of dependence, and frst order expanson relatonshps between the two models local covarances are derved. We also show exstence of a multvarate uncertan volatlty model of whch our multvarate local volatltes model s a Markovan projecton, hghlghtng that the projected model s smoother and avods a number of drawbacks of the uncertan volatlty verson. We also show a consstency result where the Markovan projecton of a geometrc basket n the multvarate model s a unvarate mxture dynamcs model. A few numercal examples on basket and spread optons prcng conclude the paper. Key words: Mxture of denstes, Volatlty smle, Lognormal densty, Multvarate local volatlty, Complete Market, Opton on a weghted Arthmetc average of a basket, Spread opton, Opton on a weghted geometrc average of a basket, Markovan projecton, Copula functon. AMS classfcaton codes: 60H10, 60J60, 62H20, 91B28, 91B70 JEL classfcaton codes: G13. Dept. of Mathematcs, Imperal College, London. damano.brgo@mperal.ac.uk Method Investments & Advsory Ltd. Ths paper reflects solely the Author s personal opnon and does not represent the opnons of the author s employers, present and past, n any way. francesco_rapsarda@ymal.com Dept. of Mathematcs, Unversté Pars I Panthéon-Sorbonne, Pars. abr.srd@malx.unv-pars1.fr

2 CONTENTS 2 Contents 1 Introducton 3 2 The Mxture Dynamcs (MD) Model 7 3 Optons on Baskets: Motvatng multvarate models Basket opton European optons prcng Multvarate extensons of the MD model Smply Correlated Mxture Dynamcs model The Multvarate Mxture Dynamcs approach The lognormal case and the unvarate - multvarate MD connecton Dmensonalty ssues Analyss and comparson of dependence structures Instantaneous correlatons n the SCMD and MVMD models Termnal correlaton Correlaton between asset and local covarance Copula functon n MVMD Rank correlatons for normal mxtures Markovan projectons MVMD as projecton of MUVM Markovan projecton for the geometrc basket dynamcs Opton prcng Opton on an arthmetc basket Spread opton Opton on a geometrc basket Numercal Results: SCMD vs MVMD Arthmetc basket and spread optons Geometrc basket opton Conclusons and perspectves 43

3 1 INTRODUCTION 3 1 Introducton It has been known for a long tme that the Black Scholes geometrc Brownan moton model [5] does not prce all European optons quoted on a gven market n a consstent way. In fact, ths model les on the fundamental assumpton that the asset prce volatlty s a constant. In realty, the mpled volatlty, namely the volatlty parameter that, when plugged nto the Black Scholes formula, allows to reproduce the market prce of an opton, generally shows a dependence on both the opton maturty and strke. If there were no dependence on strke one could extend the model n a straghtforward fashon by allowng a determnstc dependence of the underlyng s nstantaneous volatlty on tme, so that the dynamcs could be represented by the followng stochastc dfferental equaton (SDE): ds t = µs t dt + σ t S t dw t, (1.1) σ t beng the determnstc nstantaneous volatlty referred to above. In that case, reconstructon of the tme dependence of σ t would follow by consderng that, f v(t ) denotes the mpled volatlty for optons maturng at tme T, then v(t ) 2 T = T 0 σ 2 sds. (1.2) Impled volatlty however does ndeed show a strke dependence; n the common jargon, ths behavor s descrbed wth the term smle whenever volatlty has a mnmum around the forward asset prce level, or skew when low strke mpled volatltes are hgher than hgh strke ones. In the followng we wll loosely speak of both effects as volatlty smle. In recent years, many researches have tred to ncorporate the smle effect nto a consstent theory. Several streams of nvestgaton can be dentfed n a unvarate settng. We do not am at completeness n the followng revew, but just present a few relevant examples. A frst approach s based on assumng an alternatve explct dynamcs for the asset prce process that by constructon ensures the exstence of volatlty smles or skews. Typcally, n ths dynamcs the dffuson coeffcent of the asset prce s a determnstc functon of the asset prce tself and of tme. Ths s referred to wth the term local volatlty". Examples nclude the CEV process proposed by Cox [16] and Cox and Ross [17]. A dfferent example s the dsplaced dffuson model by Rubnsten [46]. In general the alternatve explct dynamcs does not reproduce accurately enough the market volatlty structures, snce t s based on qute stylzed dynamcs, wth the mxture dynamcs excepton we wll see n a moment. A second approach s based on the assumpton of a contnuum of traded strkes [4]. Ths was extended yeldng an explct expresson for the Black Scholes mpled volatlty as a functon of strke and maturty [19, 20, 21, 22]. Ths approach however needs a smooth nterpolaton of opton prces between consecutve traded strkes and maturtes. Explct expressons for the rsk neutral stock prce dynamcs were also derved by mnmzng the

4 1 INTRODUCTION 4 relatve entropy to a pror dstrbuton [1] and by assumng an analytcal functon descrbng the volatlty surface [14]. Another approach s an ncomplete market approach, and ncludes stochastc volatlty models [33, 34, 47], jump dffuson models [45] and more recently stochastc-local volatlty models [32] combnng local and stochastc volatlty. A further approach conssts of fndng the rsk neutral dstrbuton on a lattce model for the underlyng that leads to a best ft of the market opton prces subject to a smoothness crteron [13, 36]. Ths approach has the drawback of beng entrely numercal. A number of the above approaches s descrbed for the foregn exchange market n Lpton [41], see also Gatheral [26] who deals further wth volatlty surfaces parameterzaton. Recent lterature also focused on both short and long tme asymptotcs for volatlty models: we just cte [27] as a reference for small tme asymptotcs n local volatlty models, and [25] for large maturtes asymptotcs n the well known Heston stochastc volatlty model, whle pontng out that the volatlty asymptotcs lterature s much broader. In general the problem of fndng a rsk neutral dstrbuton that consstently prces all quoted optons s largely undetermned. A possble soluton s gven by assumng a partcular parametrc rsk neutral dstrbuton dependent on several, possbly tme dependent, parameters and use the latter n conjuncton wth a calbraton procedure to the market opton prces. In a number of papers, Brgo, Mercuro, Rapsarda and Sartorell [7, 9, 10, 11, 12] proposed a famly of models that carry on dynamcs leadng to a parametrc rsk neutral dstrbuton flexble enough for practcal purposes. It s relatvely straghtforward to postulate a mxture dstrbuton at a gven pont n tme, but t s less so fndng a stochastc process that s consstent wth such dstrbuton and whose stochastc dfferental equaton has a unque strong soluton. Ths s the approach adopted by the above papers. Ths famly of models s summarzed for example n Musela and Rutkowsk [43], or Fengler [24], see also Gatheral [26]. Formally, ths s part of the alternatve explct dynamcs branch of models but s typcally much rcher than the models lsted above, leadng to a practcally exact ft of the volatlty smle whle retanng analytcal tractablty. The am of ths paper s to ncorporate the effect of the volatlty smle observed on the market when prcng and hedgng multasset securtes, whle retanng sensble sngle asset volatlty structures. A whole lot of such structured securtes s nowadays offered to nsttutonal and retal nvestors, n the form of optons on baskets of stocks/fx rates and on combnatons of forward nterest rates such as e.g. European/Bermudan swaptons. In our approach we reman wthn a lognormal-mxture local volatlty model for the ndvdual assets composng the underlyng of the opton (be t a basket of stocks or a swap rate) that has proved to be qute effectve n accountng for the observed sngle assets smles, but we move one step beyond the naïve Brownan correlaton" way to connect these unvarate models when wrtng the jont mult-asset dynamcs. Indeed, gven unvarate local volatlty (one dmensonal

5 1 INTRODUCTION 5 dffuson-) models for each asset, a basc approach s ntroducng nstantaneous correlatons across the Brownan shocks of each asset, leadng to what we call the Smply Correlated Mxture Dynamcs (SCMD). For practcal mplementaton, one would then dscretse the onedmensonal sngle asset SDEs through, say, Euler or hgher order numercal schemes [39], feedng correlated nstantaneous Brownan shocks nto the scheme. In ths paper we adopt a dfferent approach and we ncorporate statstcal dependence n a new scheme that enjoys analytc multvarate denstes and a fully analytc multvarate dynamcs through a state dependent non-dagonal dffuson matrx. In so dong we are able to sample a new manfold of nstantaneous covarance structures (and a new manfold of dynamcs) whch ensures full compatblty wth the ndvdual volatlty smles and overcomes the dffcult problems created by the lack of closed form formulas for prces and senstvtes on mult-asset securtes. We call the resultng model Mult Varate Mxture Dynamcs (MVMD) and prove exstence and unqueness of the soluton for ts multvarate stochastc dfferental equaton. The tradtonal approach for prcng European style dervatves on a basket of the multdmensonal underlyng, n a SCMD type model, uses a Monte Carlo method that can be very slow as t nvolves ntensve tme dscretzaton, gven that correlaton can only be ntroduced at local shocks level. Wth ths paper we fll ths substantal gap n opton prcng and provde, wth MVMD type models, a sem-analytc soluton to the opton prcng problem where the prce can be quckly and accurately evaluated, somethng that practtoners value greatly, especally n the Rsk Management analytcs area. The level of tractablty n MVMD for both sngle assets and ndces/baskets s much hgher than wth multvarate stochastc volatlty models such as Wshart models, for whch we refer for example to [28, 18] and references theren. Ths tractablty extends to a lot of dependence measure calculatons, as we shall see shortly, whch are fundamental n a mult-asset model. Furthermore, the MVMD model leads to a complete market and hedgng s much smpler. It s practcally a tractable and flexble multvarate local volatlty model that has the potental to consstently calbrate unvarate and ndex volatlty smles through a rch but at the same tme transparent parameterzaton of the dynamcs. In mult-asset models the transparency on statstcal dependence structures and ther dynamcs s fundamental. Ths s why we study and calculate n closed form nstantaneous correlatons between assets, termnal correlatons, average correlatons, rank correlatons, squared volatlty - assets correlatons, and the whole copula functon of the MVMD model. Such explct study and formulas are not avalable n SCMD or Wshart models. We also derve an expanson of the local covarance n MVMD, showng that the frst term n the expanson concdes wth the analogous term n SCMD. As a form of comparson between MVMD and SCMD, we look at Kendall s tau rank correlaton measures across assets n detal, as mpled by the two dfferent models when the same parameters are chosen. We then ntroduce a Multvarate Uncertan Volatlty Model (MUVM). We show that the MVMD model s a Markovan projecton of the MUVM. MUVM thus gves the same

6 1 INTRODUCTION 6 European opton prces as MVMD and can be used nstead of MVMD to prce European optons also n the multvarate settng. MUVM features the same dependence structure as the MVMD model. The related copula s a mxture of multvarate copulas that are each a standardzed multvarate normal dstrbuton wth an approprate correlaton matrx and margnals. Despte these smlartes, the MUVM model s less smooth and convncng than the MVMD model. The fact that the uncertantly of volatlty needs to be realzed nstantly n a very near future s unrealstc and may lead to problems when hedgng wth the model and when dealng wth early exercse products, especally when exercse s consdered near the date of realzaton of the uncertan volatlty. Hence whle we show the Markovan projecton property as an nterestng mathematcal result, we recommend usage of MVMD rather than MUVM for products where the two models produce dfferent prces. We further pont out a result on correlaton between assets and ther nstantaneous varances (squared volatltes) and covarances. A drawback of local volatlty models s that they cannot decorrelate assets and volatltes, snce the latter are determnstc functons of the assets themselves. However, as ponted out n [7] for the unvarate case, n the MVMD model we have complete decorrelaton between assets and nstantaneous covarances. Whle ths s surprsng at frst sght, gven that all nstantaneous covarances are determnstc functons of the jont assets, t becomes more ntutve when thnkng about the relatonshp wth MUVM, and s the best approxmaton MVMD can attan for ts non-markovan orgnator MUVM, where nstantaneous covaratons and assets Brownan shocks are fully ndependent. We further hghlght a Markovan projecton property for the basket dynamcs mpled by MVMD. We consder the Markovan projecton of the Geometrc average basket dynamcs mpled by MVMD on one dmensonal dffusons. We fnd that the multvarate mxture dynamcs for the basket components nduces a unvarate lognormal mxture dynamcs for the basket, n a consstency result that can be used to prce European basket optons on the geometrc basket n fully closed form va a Black Scholes formula. As far as the geometrc average can be consdered as a good proxy for the arthmetc one [38], the method could be used for standard basket optons, or at the very least serve as a control varate result for the one-shot smulaton needed to prce an opton on an arthmetc basket. In the context of geometrc baskets, no other smlar consstency results are known for multvarate models. We then ntroduce opton prcng for basket optons and spread optons, dervng semclosed form formulas or one-shot smulaton schemes for MVMD aganst mult-step Monte Carlo smulaton for SCMD wth analogous parameters. In the fnal part of ths work, n order to develop a feel for the performance of our approach, we test t on a few cases, ncludng arthmetc and geometrc averages (weghted) baskets and spread optons. We compare the prces generated by MVMD to those obtaned by the SCMD model wth analogous parameters, and conclude that optons prces may not reflect the dfference n dependence structures between the two models even for payoffs, such as spread optons, that should depend heavly on the model dependence structure.

7 2 THE MIXTURE DYNAMICS (MD) MODEL 7 The paper s organsed as follows. In Secton 2, we present a bref revew of the approach to sngle asset smle modelng that has been developed n [7, 9, 10, 11]. In Secton 3, we provde examples of typcal securtes that need a multvarate settng for proper prcng. Secton 4 consders the extenson of the sngle asset model to the multvarate framework wth a thorough dscusson of the mplcatons for the dynamcs stemmng from a naïve approach (SCMD) and from ours (MVMD). In Secton 5 we provde a number of results on the dependence structure n the MVMD and SCMD models. In Secton 6, we ntroduce a new model that we call "Multvarate Uncertan Volatlty Model" so that our model s a multvarate Markovan projecton of t. We also show a consstency result for the Markovan projecton of the geometrc basket dynamcs n the MVMD model, that turns out to be a unvarate mxture dynamcs model. In Secton 7 we explan how to prce arthmetc, geometrc and spread basket optons n MVMD and how ths s much easer than wth SCMD, dervng the relevant formulas. In Secton 8, we llustrate the results of prcng European opton on a weghted arthmetc average of the underlyng assets wth postve weghts, European spread opton and European opton on weghted geometrc average n both MVMD and SCMD frameworks and we compare the results. Conclusons and suggestons for future research are gven n the fnal secton. 2 The Mxture Dynamcs (MD) Model For a maturty T > 0 denote by P (0, T ) the prce at tme 0 of the zero-coupon bond maturng at T. Let (Ω, F, P) be a probablty space wth a fltraton (F t ) t [0,T ] that s P-complete and satsfyng the usual condtons. We assume the exstence of a measure Q equvalent to P called the rsk neutral or prcng measure, ensurng arbtrage freedom n the classcal setup, for example, of Harrson, Kreps and Plska [30, 31]. At tmes, t wll be convenent to use the T forward rsk-adjusted measure Q T rather than Q. The MD model s based on the hypothess that the dynamcs of the asset underlyng a gven opton market takes the form ds(t) = µ(t)s(t)dt + ν(t, S(t))S(t)dW (t) (2.1) under Q wth ntal value S 0. Here, µ s a determnstc tme functon, W s a standard Q Brownan moton and ν (the "local volatlty") s a well behaved determnstc functon. In order to guarantee the exstence of a unque strong soluton to the above SDE, ν s assumed to be locally Lpschtz, unformly n t, and to satsfy the lnear growth condton ν 2 (t, x)x 2 L(1 + x 2 ) unformly n t (2.2) for a sutable postve constant L. Consder N purely nstrumental dffuson processes Y (t) wth dynamcs dy (t) = µ(t)y (t)dt + v (t, Y (t))y (t)dw (t) (2.3)

8 2 THE MIXTURE DYNAMICS (MD) MODEL 8 wth ntal value Y (0), margnal denstes p t and wth v satsfyng locally Lpschtz and lnear growth condtons, where each Y (0) s set to S(0). Remark 1 The reader should not nterpret the Y as real assets. They are just nstrumental processes that wll be used to defne mxtures of denstes wth desrable propertes. The margnal densty p t of S(t) s assumed to be representable as the superposton of the nstrumental processes denstes p t [9, 10, 11]: p t = λ p t wth λ 0, and λ = 1. (2.4) The problem of characterzng ν can then be cast n the followng form: s there a local volatlty ν for Eq. (2.1) such that Eq. (2.4) holds? Purely formal manpulaton of the related Kolmogorov forward equaton p t t + x (µxp t) x 2 (ν2 (t, x)x 2 p t ) = 0 (2.5) and of analogous equatons for the p t s shows that a canddate ν s N =1 ν(t, x) = λ v (t, x) 2 p t (x) N =1 λ p t (x). (2.6) We may now ntroduce the followng Defnton 2 General MD model. The general sngle-asset Mxture Dynamcs (MD) canddate model s the model gven by equatons (2.1) and (2.6). If the model equaton admts a unque soluton and f the related Kolmogorov forward equaton admts a unque soluton, then the densty of the model s a mxture accordng to Equaton (2.4), where the p terms are the denstes of the nstrumental dffuson processes (2.3). An mportant consequence of the above constructon s the followng Proposton 3 Assume that the model (2.1,2.6), wth p t from (2.3), admts a unque strong soluton and that the related Kolmogorov forward equaton admts a unque soluton. Then the prcng of European optons on S s smply a lnear-convex combnaton wth weghts λ of the opton prces under the nstrumental asset dynamcs (2.3). Smlarly for the Greeks at tme 0. In other terms, let O be the value at t = 0 of an European opton wth strke K and maturty T. O s gven by O = N =1 λ O ; where O s the European prce assocated to the hypothetcal nstrumental dynamcs (2.3). The opton prce O can be vewed as the weghted average of the European opton prces wrtten on the processes Y. Due to lnearty of dfferentaton, the same convex combnaton apples to all opton Greeks. As a consequence,

9 2 THE MIXTURE DYNAMICS (MD) MODEL 9 f the basc denstes p t are chosen so that the prces O are computed analytcally, one fnds an analytcally tractable model. The most natural choce for the (Y, v, p t) trplet s : Y (0) = S(0) v (t, x) = σ (t) V (t) = t 0 σ (s) 2 ds p 1 t(x) = 2πxV [ exp 1 (t) ( 2V 2(t) ln ( x S(0) ) ) ] 2 µt V (t) 2 =: l t(x) (2.7) wth σ determnstc (lognormal mxture dynamcs, LMD). Brgo and Mercuro [10] proved that, wth the above choce and addtonal nonstrngent assumptons on the σ, the correspondng dynamcs for S t ndeed admts a unque strong soluton. A greater flexblty can also be acheved by shftng the auxlary processes densty by a carefully chosen determnstc functon of tme (stll preservng rsk neutralty). Ths s the so called shfted lognormal mxture dynamcs model [11]. Theorem 4 Exstence and unqueness of solutons for the LMD model. Assume that all the real functons σ (t), defned on the real numbers t 0, are once contnuously dfferentable and bounded from above and below by two postve real constants. Assume also that n a small ntal tme nterval t [0, ɛ], ɛ > 0, the functons σ (t) have an dentcal constant value σ 0. Then the Lognormal Mxture Dynamcs model (LMD) defned by Equatons (2.1,2.7), namely ds t = µ(t)s t dt + s(t, S t )S t dw t, S 0, s(t, x) = ( N ) 1/2 k=1 λk σ k (t) 2 l k t (x) N k=1 λk l k t (x), (2.8) admts a unque strong soluton and the Kolmogorov equaton for ts densty admts a unque soluton satsfyng (2.4), whch s n ths case a mxture of lognormal denstes, leadng to opton prces that are lnear combnatons of Black-Scholes prces. In [9, 10] t s ponted out that the squared dffuson coeffcent s(t, x) 2 defned n (2.8) can be consdered as a state dependent weghted (convex combnaton) average of the basc squared volatltes (σ k ) 2 and that f the latter are unformly bounded so s s. The above descrpton gves a suffcent bass for presentng our generalzaton of the LMD to the multvarate settng, as before at frst on the bass of pure formal manpulatons, and then wth full rgor, wth the specfc am of fndng a method to nfer the mpled volatlty of a basket of securtes from the ndvdual components and/or an explct dynamcs for the

10 3 OPTIONS ON BASKETS: MOTIVATING MULTIVARIATE MODELS 10 mult-asset system. Later n the paper, formal proofs of the general consstency of the model and of the exstence and unqueness of the soluton to the multvarate verson of Eq. (2.1) wll be provded. 3 Optons on Baskets: Motvatng multvarate models A generalzaton of LMD to the multvarate settng ams to be able to compute the smle effect on the mpled volatltes for exotc optons dependng on more than one asset, such as a basket optons. Clearly, analogous technques apply to ndces. 3.1 Basket opton A basket opton s an opton whose payoff s lnked to a portfolo or "basket" of underlyng assets. We can dstngush two types of basket opton: An opton of weghted arthmetc average of the underlyngs: n B t = w k S k (t), (3.1) k=1 where B s called an arthmetc basket"; An opton of weghted geometrc average of the underlyngs: [ n B t = S k (t) w k k=1 ] 1 w wn (3.2) where B s called a geometrc basket. where S k s the k th component of the basket. Typcally the basket s consstng of several stocks, ndces or currences. Less frequently, nterest rates are also possble (S k could represent a forward rate process F k n the Lbor Market Model (LMM) and nstead of (3.1) we could have a more complcated expresson representng a swap rate). Such optons have the most vared nature: from the plan European call/put optons on the value of the basket at maturty T, to optons somewhat more complcated, such as Asan optons on the basket, Hmalaya optons, ranbow optons and so on. The weghts (w k ) k n (3.1) can be negatve. When the basket (3.1) contans short postons t s called spread and the opton known as a spread opton s wrtten on the dfference of underlyng assets. The weghts (w k ) k=1,...,n n (3.2) are postve. It s nstructve to vew a basket opton as a standard dervatve on the underlyng nstrument whose value at tme t s the basket B t so defned.

11 3 OPTIONS ON BASKETS: MOTIVATING MULTIVARIATE MODELS European optons prcng Let us assume that nterest rates are constant and equal to r > 0. We also assume the exstence and unqueness of a rsk neutral prcng measure Q that s equvalent to P under whch dscounted asset prces are martngales, mplyng the absence of arbtrage (Q s also equal to Q T as nterest rates are assumed to be determnstc). Accordng to the Black Scholes prcng paradgm [30, 31], the prce Π of an European opton at ntal tme t = 0 s gven by the rsk-neutral expectaton: Π = e rt E { [ω (B T K)] +} (3.3) where the exponental factor takes care of the dscountng and ω = ± 1 for a call/put respectvely. B T s the underlyng nstrument (can represent the value of the basket) at maturty T, K s the strke. The fundamental dffculty n prcng basket optons on a weghted arthmetc average of a basket s to determne the dstrbuton of the sum of underlyng asset prces. Let us consder the basket of securtes of Eq. (3.1). Several approxmaton methods have been proposed for optons on t when each S k follows a geometrc Brownan moton. Usually the basket value (3.1) s approxmated by the lognormal dstrbuton. Recall that here we consder baskets wth possbly negatve weghts, such as spreads. Hence, we cannot approxmate the dstrbuton of B t by a lognormal dstrbuton, snce such a basket can have negatve values or negatve skewness. However, Brgo and Masett [8] n a LIBOR market model settng and later Borovkova, Permana and Wede [6] show that a more general three-parameter famly of lognormal dstrbutons: shfted, negatve and negatve shfted lognormal, can be used to approxmate the dstrbuton of a general basket. The shfted lognormal dstrbuton s obtaned by shftng the regular lognormal densty by a fxed amount along the x-axs, and the negatve lognormal - by reflectng the lognormal densty across the y-axs. The negatve shfted lognormal dstrbuton s the combnaton of the negatve and the shfted one. Note that ths famly of dstrbutons s flexble enough to ncorporate negatve values and negatve skewness: somethng that the regular lognormal dstrbuton s unable to do. However, by usng these approxmatons we do not take nto account the nternal composton of the basket value n terms of underlyng assets havng each ts own dynamcs. Ths approach structurally cannot take nto account any smle effect on the ndvdual underlyngs volatlty, and therefore on the "basket volatlty". In the followng we wll tackle the problem n a rgorous way, through the generalzaton of the dynamcal model of Eqs. (2.1,2.7) that has proven to perform qute well on some markets [9, 10, 11] and that s under extenson to the equty markets case.

12 4 MULTIVARIATE EXTENSIONS OF THE MD MODEL 12 4 Multvarate extensons of the MD model To fx deas, suppose we are faced wth the followng problem: we want to prce an opton maturng at T on the basket of n securtes gven by Eq. (3.1) or Eq. (3.2). Each of these n securtes wll have a smley volatlty structure, and we expect the basket to show a smle n ts mpled volatlty, too. Through Eqs. ( ) we now have a pece of machnery that allows us to calbrate an LMD to each mpled volatlty smle structure of the ndvdual component S k of the basket. Suppose we have already calbrated the ndvdual LMDs to such smle surfaces, thus fndng the LMD local volatltes governng the dynamcs of each S k. We denote by Yk 1,..., Y k N the nstrumental processes for asset S k. Namely, for each asset n the basket we have a famly of nstrumental processes lke (2.3) that refer to that specfc asset mxture dstrbuton, each (2.3) beng specalzed accordng to Equaton (2.7). To gude the reader through notaton, we recall as a smple conventon that for us upper ndces n general denote a component n the mxture, whereas lower ndces denote dfferent assets. So for example σk h wll refer to asset S k and to the h-th component of the mxture, whereas the densty of Y k at tme t wll be denoted by l k,t. We are now nterested n connectng these unvarate LMD models S 1,..., S n nto a multvarate model that embeds statstcal dependence among the dfferent asset. The most mmedate way to do ths s to ntroduce a non-zero quadratc covaraton between the Brownan motons drvng the LMD models for S and S j respectvely. 4.1 Smply Correlated Mxture Dynamcs model Defnton 5 SCMD Model. We defne the Smply Correlated multvarate Mxture Dynamcs (SCMD) model for S = [S 1,..., S n ] as a vector of unvarate LMD models, each satsfyng Theorem 4 wth dffuson coeffcents s 1,..., s n gven by Formula (2.8) and denstes l 1,..., l n appled to each asset, and connected smply through quadratc covaraton ρ j,j between the Brownan motons drvng assets and j. Ths s equvalent to the followng n-dmensonal dffuson process where we keep the W s ndependent and where we embedded Brownan covaraton nto the dffuson matrx C, whose -th row we denote by C : ds(t) = dag(µ)s(t)dt + dag(s(t)) C(t, S(t))dW (t), ã,j (t, S) := C CT j (4.1) ( N k=1 ã,j (t, S) = s (t, S )s j (t, S j )ρ j = λk σk (t)2 l k,t (S ) N k=1 λk lk,t (S ) where T represents the transposton operator. N k=1 λk j σk j (t)2 l k j,t (S ) 1/2 j) N k=1 λk j lk j,t (S ρ j. j) (4.2) Assumpton. Throughout the paper we assume ρ to be postve defnte.

13 4 MULTIVARIATE EXTENSIONS OF THE MD MODEL 13 Remark 6 SCMD: no multvarate mxture. It s mportant to pont out n SCMD that whle sngle assets probablty denstes are mxtures by constructon, the multvarate densty s not a mxture of multvarate basc denstes. The mxture property does not extend from the mono-dmensonal dynamcs to the multdmensonal one. The practcal use of the SCMD model s related to the followng consderaton. Most often, one realstc way to prce a plan European opton dependng on more than one asset, especally n large dmenson, s to use a Monte Carlo smulaton that samples sutably dscretzed paths accordng to the drft rate of each component (rsk free mnus dvdend yeld) and to the dffuson matrx gven by the local volatlty functon n the mxture of denstes model. Therefore, assumng to have an exogenously computed structure of nstantaneous correlatons ρ j (computed e.g. through hstorcal analyss or mpled by market nstruments and supposed constant over tme) among the assets returns, we could apply a naïve Euler Monte Carlo scheme and smulate the jont evoluton of the assets through a sutably dscretzed tme grd τ 1 = 0 τ N = T wth a covarance matrx whose (, j) component over the (τ l, τ l+1 ) propagaton nterval s gven by (4.2) computed at t = τ l. It s mmedate by constructon that the SCMD approach s consstent wth both the ndvdual dynamcs nduced by a LMD model for each underlyng asset and wth the mposed "nstantaneous correlaton" (Brownan quadratc covaraton) structure ρ j. However, besdes the practcal possblty of controllng the nstantaneous correlaton, and that the number of base unvarate denstes to mx does not ncrease wth the number of underlyng assets, one must be aware of the SCMD man lmtatons, especally the followng one. For European type basket optons we do not really need the full dynamcs when t comes to actually computng the prce, even wth several maturtes n the pcture. Indeed, for each maturty T the payout depends only upon the values of the assets at tme T,.e., upon the values S k (T ), k, regardless of the hstory of prces. So n order to compute the rsk neutral expectaton n (3.3) gvng the prce Π, the only nformaton we need s the jont densty of the process (S 1 (T ), S 2 (T ),..., S n (T )) of random varables under that partcular rsk neutral measure. Ths densty s usually called the state prce densty. In SCMD we do not know ths densty, so we have to generate samples from the entre path B t for 0 t T. The dscretzaton tme steps τ l+1 τ l should be chosen carefully to be sure that the numercal scheme used to generate the dscrete samples produces reasonable approxmatons. Notce that when the maturty T ncreases, more tme steps are needed. Ths s partcularly relevant n calbratng the model for rsk management applcatons, for example, where the nverse problem can become dauntng f the dmenson s large and the dscretzaton step small. 4.2 The Multvarate Mxture Dynamcs approach One could try to do somethng dfferent and approach the problem so that, under sutable assumptons, the ndvdual LMD models (one for each underlyng asset, separately calbrated

14 4 MULTIVARIATE EXTENSIONS OF THE MD MODEL 14 each on ts volatlty surface) could be merged so as to provde a coherent mult-asset model that allows for a degree of (sem)analytc tractablty comparable to the one typcal of the unvarate case. Ths wll lead to a model where the mxture property s lfted to the multvarate densty, contrary to the SCMD case (Remark 6 above). Consder an n dmensonal stochastc process S(t) = [S 1 (t),, S n (t)] T whose generc th component follows the SDE ds (t) = µ S (t)dt + S (t)c (t, S)dW (t) (4.3) where µ s a constant, W = [W 1,, W n ] T s a standard n dmensonal Brownan moton and C (t, S) s a row vector whose components are determnstc functons of tme and of the state of the process S. Denote a j (t, S) = C (t, S) Cj T (t, S). The assocated Kolmogorov forward PDE to be satsfed by the probablty densty p S(t) of the stochastc process S s p S(t) t + n =1 x [µ x p S(t) ] 1 2 n,j=1 2 x x j [a j x x j p S(t) ] = 0 (4.4) where all functons are evaluated at (t, x) for all t 0, x R n. Wth ths notaton S s gven by the SDE ds(t) = dag(µ)s(t)dt + dag(s(t))c(t, S(t))dW (t) (4.5) where C s the n n matrx whose th row s C. C must be chosen so as to grant a unque strong soluton to the SDE (4.5). In partcular, C s assumed to lead to a locally Lpschtz a(t, x) and to satsfy, for a sutable postve constant K, the generalzed lnear growth condtons The symbol denotes here vector and matrx norms. trace(a(t, x)) x 2 K(1 + x 2 ). (4.6) Consder an n dmensonal stochastc process X (k) whose generc th component follows the dynamcs dx (k) (t) = µ X (k) (t)dt + X (k) (t)σ (k) (t, X (k) )dw (t) (4.7) wth σ (k) (t, X (k) ) an 1 n matrx satsfyng partcular condtons ensurng that the resultng SDE gvng the dynamc of X (k) has a unque strong soluton. Denote a (k) j (t, X(k) ) = σ (k) (t, X (k) ) σ (k) j (t, X (k) ) T and p (k) t the probablty densty functon of X (k). The assocated Kolmogorov equaton to be satsfed by p (k) t s p (k) t (x) + t n =1 x [µ x p (k) t (x)] 1 2 n,j=1 2 x x j [a (k) j (t, x)x x j p (k) t (x)] = 0. (4.8)

15 4 MULTIVARIATE EXTENSIONS OF THE MD MODEL 15 Inspred by the unvarate approach whch gave rse to the LMD model, let us postulate that the densty at any tme t of the multvarate process S be equal to a weghted average of the p (k) t p S(t) (x) = N k=1 λ k p (k) t (x), λ k 0 k, The condton that p S(t) satsfy Eq.(4.4) and that each p (k) t through standard algebra to the PDE 1 2 n,j=1 2 x x j [( a j (t, x) N k=1 λ k p (k) t (x) N k=1 λ k a (k) j (t, x)p(k) t Proposton 7 The unque canddate soluton of the PDE (4.10) s N λ k = 1 (4.9) k=1 satsfy the equaton (4.8) leads (x) ) x x j ] = 0. (4.10) a j (t, x) = N k=1 λk a (k) j (t, x)p(k) t n k=1 λk p (k) t (x) (x), a (k) j (t, x) = σ(k) (t, x) σ (k) j (t, x) T. (4.11) Proof. It can be easly proven that the most general soluton of the equaton 2 j x x j f j (x) = 0 has a Fourer transform satsfyng (q, f(q)q) = 0. The only matrx functon f(q) satsfyng t and nfntely dfferentable wth respect to q s constant. Ths constant must be zero n order to have fnte frst and second moments of the multvarate densty p S(t). Ths leads to the followng defnton. Defnton 8 The general Multvarate Mxture Dynamcs (MVMD) canddate Model for the vector of asset prces S s defned as gven by Equatons (4.5) and (4.11). If a unque soluton for the model equatons exsts and t admts a multvarate probablty densty, ths s a mxture of basc multvarate denstes accordng to Eq (4.9), where each p k s a multvarate basc densty assocated wth an nstrumental multvarate dffuson process (4.7). Remark 9 MVMD: multvarate mxture. MVMD has been desgned so as to have a mxture law for the multvarate model, contrary to SCMD, see Remark 6 above. Of course to show that ths s ndeed a model we need to prove that the equaton has a unque soluton. We thus specalze our framework to a fully tractable case. 4.3 The lognormal case and the unvarate - multvarate MD connecton We now specalze our framework by assumng that the volatlty coeffcent matrx for the k th "base" densty p (k) t of Eq. (4.8) s a determnstc functon of tme, ndependent of the state, and of the partcular form a (k) j (t, x) = σ(k) (t) σ (k) j (t) T. Under ths hypothess we already know the dynamcs correspondng to Eq. (4.8), snce we are dealng wth multvarate geometrc Brownan motons for (4.7), and we can explctly

16 4 MULTIVARIATE EXTENSIONS OF THE MD MODEL 16 wrte ther denstes p (k) t p (k) t (x) = [ 1 (2π) n 2 det Ξ (k) (t)π n =1 x exp xt (Ξ (k) (t)) 1 x 2 ], (4.12) where Ξ (k) (t) s the n n ntegrated covarance matrx of returns for the many components of the process X (k) : Ξ (k) j (t) = t 0 σ (k) (s)σ (k) j (s) T ds (4.13) (Ξ (k) s assumed to be nvertble at all tmes and nstantaneous correlaton s ncluded nto the vector components) and t x (k) = ln x ln x (0) µ t + 0 σ (k)2 (s) ds. (4.14) 2 Calculatons are smpler under the further assumpton that nstantaneous correlaton s constant n tme, namely σ (k) (t)σ (k) or n other terms, assumng that j (t) T = σ (k) (t) σ (k) j (t) ρ j =: σ k (t)σj k (t)ρ,j, ρ = BB T, σ (k) (t) := σ (k) (t)b (4.15) va dagonalzaton or Cholesky decomposton and for postve and regular scalar tme functons σ (k) (t), where B s the -th row of B. The fact that the denstes wll get mxed up through Eq. (4.9) wll have mportant consequences on the actual structure of correlatons, both nstantaneous and average. But frst, let us prove that under a further assumpton we can be fully consstent wth the dynamcs specfed by the LMD model for the ndvdual assets. Let s assume that we have calbrated an LMD model for each S (t): f p S (t) s the densty of S, we wrte p S (t)(x) = N k=1 λ k l k,t(x), wth λ k 0, k and k λ k = 1 (4.16) where Y 1,..., Y N are nstrumental processes for S evolvng lognormally accordng to the stochastc dfferental equaton: wth densty l k,t. dy k (t) = µ Y k (t)dt + σ k (t)y k (t)dz (t), d Z, Z j t = ρ j dt (4.17) For notatonal smplcty we wll assume that the number of base denstes N wll be the same, N, for all assets. The exogenous correlaton structure ρ j s gven by the symmetrc, postve defnte matrx ρ.

17 4 MULTIVARIATE EXTENSIONS OF THE MD MODEL 17 The most natural tentatve choce for the base denstes of Eq. (4.9) s p S(t) (x) = or more explctly N k 1,k 2, k n=1 λ k 1 1 λkn n l k 1,...,k n 1,...,n;t (x), l k 1,...,k n 1,...,n;t (x) = 1 (2π) n 2 det Ξ (k 1 k n) (t)π n =1 x lk 1,...,k n 1,...,n;t (x) := p [Y k 1 1 exp [ ] kn T (x), (4.18) (t),...,yn (t) x(k 1 k n)t Ξ (k 1 k n) (t) 1 x (k 1 k n) 2 ]. Here, Ξ (k 1 k n) (t) s the ntegrated covarance matrx whose (, j) element s and, generalzng Eq. (4.14) Ξ (k 1 k n) j (t) = t 0 σ k (s)σk j j (s)ρ jds (4.19) t x (k 1 k n) = ln x ln x (0) µ t + 0 Then, specalzng (4.11), we have the followng σ k2 (s) ds. (4.20) 2 Defnton 10 The multvarate extenson of the LMD model that we call Lognormal Mult Varate Mxture Dynamcs (LMVMD) model s gven by Eqs. (4.5) and (4.11) under specfcaton (4.15), leadng to ds(t) = dag(µ) S(t) dt + dag(s(t)) C(t, S(t))B dw (t), (4.21) C (t, x) := N k 1,...,k n=1 λk λkn n N k 1,...,k n=1 λk λkn n σ k (t)b l k 1,...,k n 1,...,n;t (x) l k 1,...,k n 1,...,n;t (x) and therefore, defnng consstently wth earler notaton a = CB(CB) T, where a(t, x) = N k 1,...,k n=1 λk 1 V k 1,...,k n (t) = 1...λkn n V k 1,...,k n (t) l k 1,...,k n 1,...,n;t (x) N k 1,...,k n=1 λk λkn n l k 1,...,k n 1,...,n;t (x) (4.22) [ σ k (t) ρ,j σ k j j (t) ],j=1,...,n. (4.23) To avod lengthy acronyms and wth a slght abuse of notaton, we wll refer to the LMVMD model smply as MVMD, assumng mplctly from now on that we are dealng wth the lognormal case. Puttng notatonal complexty asde, what we ultmately dd s to mx n all possble ways the component denstes for the ndvdual assets, stll ensurng consstency wth the startng models for the components assets, and mposng the nstantaneous correlaton structure ρ at the level of the consttuent denstes. To confrm that wth MVMD we have a full model and not just a canddate model, we need the followng theorem.

18 4 MULTIVARIATE EXTENSIONS OF THE MD MODEL 18 Theorem 11 Under the assumpton that the volatltes σ k (t) for all are once contnuously dfferentable and unformly bounded from below and above by two postve real numbers σ and ˆσ respectvely, and that they take a common constant value σ 0 for t [0, ɛ] for a small postve real number ɛ, namely ( σ = nf t 0 ( ˆσ = sup t 0 mn =1 n,k =1, N max =1 n,k =1 N (σk (t)) ) (σk (t)) ) σ k (t) = σ 0 > 0 for all t [0, ɛ], and assumng the matrx ρ to be postve defnte, the MVMD n-dmensonal stochastc dfferental equaton (4.21) admts a unque strong soluton. The dffuson matrx a(t, x) n (4.22) s postve defnte for all t and x. Proof. Exstence and unqueness of a strong soluton follows analogously to the unvarate case [9]. Indeed, from (4.22) we see that a s a weghted average of V s n (4.23), wth postve (and state-dependent) weghts. Snce we are assumng σ s to be unformly bounded and ρ s to be postve defnte, all V matrces are postve defnte and such s a. Moreover, all a s entres are mmedately seen to be bounded above and and below, the dagonal terms beng bounded below by postve quanttes. Standard algebra yelds n n σ 2 a 2 = a j (t, S) 2 n 2ˆσ 2,,j=1 so that we have a unformly bounded contnuously-dfferentable functon (hence locallly Lpschtz), and the usual lnear growth condton holds. The common value σ 0 n an ntal transent nterval s needed to smplfy the analyss of the component denstes lmt for t 0 when the ntal condtons for the sngle asset denstes are taken as Drac delta functons. We now check that MVMD s ndeed consstent wth the mxture of denstes models LMD through whch we have specfed the dynamcs of the sngle components of S n the begnnng. Proposton 12 For any smooth test functon f : R R and any t 0, the expectaton of f(s (t)) s the same under the SCMD model (4.1), (4.2) and the MVMD model (4.5), (4.22). Proof. The proof s trval: let us start from the MVMD model. It s enough to compute the multple ntegral E 0 {f(s (t))} = dx 1 dx dx n f(x )p S(t) (x) = N k 1,k 2, k n=1 λ k 1 1 λkn n dx1 dx (4.24) dx n f(x )l k 1,...,k n 1,...,n;t (x) Integratng out all varables but x n each of the ntegrals n the rght hand sde we have E 0 {f(s (t))} = N k 1,k 2, k n=1 λk 1 1 λkn n dx f(x )l k,t (x ) = N k =1 λk dx f(x )l k,t (x ) = dx f(x )p S (t)(x ) (4.25)

19 4 MULTIVARIATE EXTENSIONS OF THE MD MODEL 19 where the last ntegral s the same under SCMD, snce by the condton that probablty ntegrate up to one we know that N k=1 λk = 1 for all. 4.4 Dmensonalty ssues The computatonal scheme shown above ensures full consstency between the sngle asset and the mult asset formulatons of the mxture of lognormal denstes model. It must be borne n mnd, however, that the number of base multvarate denstes of the formulaton of Eqs. (4.18) (4.20) explodes as N n f we have N base unvarate denstes for each of the n underlyng assets (more generally, f asset reles on a sngle asset mxture theory based on N denstes, the number of multvarate denstes enterng the superposton amounts to n =1 N ). Ths combnatoral exploson seems to lmt the applcablty of the theory to baskets made of very few assets. However, as already observed elsewhere [11] two emprcal facts appear n the unvarate mxture of denstes model, that encourage the applcaton of the model to real world multvarate settngs. They are brefly summed up here: the number of base denstes N needed to reproduce accurately enough the mpled volatlty surface for a sngle asset s typcally 2 to 3; there appears to be a clear herarchy between denstes composng the mxture, dctated by the weghts λ k borne by each densty n the superposton (2.4): n fact, typcally one densty takes up most of the weght, the second takes up most of the remanng weght (remember that N k=1 λ k = 1) and the last weghs lttle compared to the frst two. The consequences of the frst ssue are evdent: the base n the power law N n s of the order of two/three. Ths s not enough to completely solve the exploson problem: takng N = 3 and n = 8 stll mples that n order to compute the prce of an European opton on the basket, we should compute 6561 multdmensonal ntegrals. However, the second pont ensures that most of the multvarate coeffcents λ k 1 1 λkn n result from the product of the smallest λ, thus renderng the correspondng terms n the expanson of Eq. (4.18) neglgble. Gven any 0 κ 1, a possble soluton can therefore be to approxmate Eq. (4.18) through p S(t) (x) p S(t) (x, κ) = I(λ) := {(k 1, k 2,..., k n ) : (k 1,...,k n) I(λ) n j=1 λ k j j k N [1, N], l k 1,...,k n 1,...,n;t (x), (4.26) n j=1 λ k j j > κ} κ therefore plays the role of a "cutoff parameter" that ensures that only sgnfcant contrbutons to the multvarate expanson are retaned; n order to preserve normalzaton of the

20 5 ANALYSIS AND COMPARISON OF DEPENDENCE STRUCTURES 20 resultng densty, n j=1 λ k j j = n j=1 λk j j (k 1,...,k n) I(λ) n j=1 λk j j. (4.27) Note that p S(t) (x) = p S(t) (x, κ = 0), whereas ncreasng κ decreases the number of base multvarate denstes n the approxmate expanson of Eq. (4.27); κ therefore controls the tradeoff between the accuracy n the approxmaton and the computatonal effcency. In order to have an estmate of the computatonal gan due to a choce of κ 0, we can compute how the volume n n dmensonal space of the regon κ < n =1 x 1 scales for fxed cutoff as a functon of n 1: the recursve law s V n (κ) = 1 κ dx 1 1 κ x 1 dx 2 settng V 0 = 1 conventonally. 1 κ n 1 =1 x dx n = V n 1 (κ) + ( 1)n (n 1)! κ (ln κ)n 1 (4.28) Now, let us neglect the strkng feature that there exsts a strong herarchy between components n the unvarate mxtures of denstes (pont two above). Suppose nstead that we are n a less favorable case, namely that the densty of coeffcents λ of the mxture model for each asset s unform and equal to ρ (.e. the dstance on the [0, 1] nterval between consecutve λ s equal to 1 ρ ); then, the densty of coeffcents n the multvarate theory s ρn. An estmate of the number of multvarate denstes nvolved n the expanson of Eq. (4.27) s N n (κ) = V n (κ)ρ n. To gve an example, f κ = 5% and ρ = 3, the number of denstes has a maxmum at N n (5%) 80 for n 8: neglectng the denstes that contrbute to 5% of the normalzaton already yelds much less than the full 6581 set of eght varate denstes. 5 Analyss and comparson of dependence structures 5.1 Instantaneous correlatons n the SCMD and MVMD models From (4.2) and (4.22)-(4.23) we can compare the expresson for the nstantaneous local covarance, or quadratc covaraton, between asset returns n the two models, SCMD and MVMD. Wthout loss of generalty consder a two dmensonal process, namely take n = 2. To lghten notaton we omt the tme argument n volatltes σ k (t). Recall that n a SCMD scheme the nstantaneous varance for the log S 1 asset, say, at tme t would be (see Eq. (4.2)) N k=1 C 11 (x 1, t) = λ 1 k σ (k)2 1 l (1k) t (x 1 ) N k=1 λ 1 k l (1k) t (x 1 ) (5.1) and the nstantaneous covarance of returns, or quadratc covaraton, between the two assets would be N C 12 (x 1, x 2, t) = k=1 λ 1 k σ (k) 1 l(1k) t (x 1 ) N k=1 λ 2 k σ (k ) 2 l (2k) t (x 2 ) N k=1 λ 1 k l (1k) t (x 1 ) N k=1 λ 2 k l (2k) ρ (5.2) t (x 2 )

21 5 ANALYSIS AND COMPARISON OF DEPENDENCE STRUCTURES 21 to be compared wth the expressons C 11 (x, x 2, t) = N k,k =1 λ 1 k k λ 2 σ (k)2 1 l (kk ) t (x 1, x 2 ) N k,k =1 λ 1 k k λ 2 l (kk ) t (x 1, x 2 ) (5.3) and C 12 (x, t) = ν k,k =1 λ 1 k k λ 2 σ (k) ) 1 σ(k 2 ρ l (kk ) t (x 1, x 2 ) ν k,k =1 λ 1 k k λ 2 l (kk ) t (x 1, x 2 ) (5.4) of MVMD. An evdent dfference s that, whle n Eq. (5.1) the nstantaneous covarance of log S 1 depends only on x 1 tself, and not on x 2, the opposte s true of Eq. (5.3). In other words, the dffuson matrx s now fully dependent on the components of the multdmensonal process. Moreover, the two equatons (5.2) and (5.4) are structurally dfferent. However, there must be a lnk between the two: we know that n the lmt when the correlaton ρ between varables ln(s 1 ) and ln(s 2 ) vanshes, they wll n fact evolve gnorng one another n both models. By the choce we made at the begnnng, l (kk ) t s a bvarate lognormal densty,.e. t has the expresson l (kk ) 1 t (x 1, x 2 ) = [ 1 2π α 11 α 22 ρ 2 α 2 x 1 x 2 exp 1 2 x 1 2 α 22 α 2 x α 11 α 22 ρ 2 α 2 1 x 12 ρ α 11 α 22 ρ 2 α 2 12 ] 2 α + x 11 2 α 11 α 22 ρ 2 α 2 12 (5.5) wth x 1 and x 2 defned as n Eq. (4.14) and α 11 = t 0 σ(k)2 1 (s)ds α 22 = t 0 σ(k ) 2 2 (s)ds (5.6) α 12 = t 0 σ(k) ) 1 (s)σ(k 2 (s)ds. The tetrachorc expanson for the bvarate normal densty wth correlaton ρ reads [48] n(x 1, x 2, ρ) = n(x 1 )n(x 2 ) k=0 (H k s the k th Hermte polynomal); ths, appled to l (kk ) t [ ] l (kk ) 1 t (x 1, x 2 ) 1 2πα11 x 1 exp 1 2 2α 11 x πα22 x 2 exp ρ k k! H k(x 1 )H k (x 2 ) (5.7) yelds [ ] 1 2 2α 22 x 2 [ ] [ ] πα11 x 1 exp α 11 x 1 1 2πα22 x 2 exp 1 2 2α 22 x 2 and smlarly expandng Eqs. (5.3) and (5.4) we get the followng x 1 x 2 α 12 α 11 α 22 ρ + O(ρ 2 ) (5.8)