For Peer Review Only A Multifactor Volatility Heston Model Journal:Quantitative Finance Manuscript ID:RQUF-2006-0066.R2 Manuscript Category:Research Paper Date Submitted by the Author: 28-Jul-2007 Complete List of Authors:grasselli, martino; Universit di Padova, Matematica Pura ed Applicata Da Fonseca, Jose tebaldi, claudio; Universit di Verona, Scienze Economiche Keywords: Stochastic Volatility, Options Pricing, Options Volatility, Financial Derivatives JEL Code: G12 - Asset Pricing < G1 - General Financial Markets < G - Financial Economics, G13 - Contingent Pricing|Futures Pricing < G1 - General Financial Markets < G - Financial Economics

Note: The following files were submitted by the author for peer review, but cannot be converted to PDF. You must view these files (e.g. movies) online. QFrevisionfinal.tex figures.zip E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Financepeer-00568441, version 1 - 23 Feb 2011Author manuscript, published in "Quantitative Finance 8, 06 (2008) 591-604" DOI : 10.1080/14697680701668418For Peer Review OnlyA Multifactor Volatility Heston ModelJoseDaFonseca,ESILV andZeliadeSystems MartinoGrasselli,Universit`adegliStudidiPadova andESILVClaudioTebaldi,Universit`adegliStudidiVerona July28,2007AbstractWeconsideramodelforasingleriskyassetwhosevolatilityfollowsamultifactor (matrix) Wishart ane process, recently introduced in nancebyGourierouxandSufana(2004). AsinstandardDueandKan(1996)anemodelsthepricingproblemcanbesolvedthroughtheFastFourierTransform of Carr and Madan (1999). A numerical illustration shows thatthisspecicationprovidesaseparatetofthelongtermandshorttermimplied volatility surface and, dierently from previous diusive stochasticvolatility models, it is possible to identify a specic factor accounting for astochastic leverage eect, a well known stylized fact of FX option marketsanalyzedinCarrandWu(2004).Keywords: Wishart processes,Stochastic volatility,Matrix Riccati ODE,FFT. JEL:G12,G131 IntroductionAn accurate volatility modelling is a crucial step in order to implement realisticandecientriskminimizingstrategiesfornancialandinsurancecompanies.Forexample, pensionplansusuallyattachguaranteestotheirproductsthatare linked to equity returns. Hedging of such guarantees involves, beyond plainvanillaoptions, alsoexoticcontracts, likeforexamplecliquetoptions. TheseEcole Superieure dIngenieurs Leonard de Vinci, Departement Mathematiques etIngenierieFinanci`ere,92916ParisLaDefense,France. Email: jose.dafonseca@devinci.frZeliadeSystems,56,RueJean-JacquesRousseau75001ParisDipartimentodi MatematicaPuraedApplicata, ViaBelzoni 7, Padova, Italy. E-mail:grassell@math.unipd.itDepartment of Economics, ViaGiardinoGiusti 2, 37129Verona, Italy. E-mail: clau-dio.tebaldi@univr.it.Acknowledgements: wethankFabioMercurioaswellastheparticipantsofthe2006AsconameetingandtheSeminaireBachelierofParisforhelpful comments. Wealsothanktwoanonymousrefereesforthecareful readingof thepaper. AnearlierdraftofthispaperappearedunderthetitleWishartMulti-DimensionalStochasticVolatility.1Page 2 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlyinstruments, also called ratchet options, periodically lock in prots in a man-ner somewhat analogues to a mechanical ratchet. Exotic contracts like cliquetoptions, require an accurate modeling of the true realized variance process. Infactacliquetoptioncanbeseenasaseriesof consecutiveforwardstartop-tions whose prices depend only on realized volatility (see e.g. Hipp 1996). Aswell explainedinBergomi (2004), thereisastructural limitationwhichpre-ventsone-factorstochasticvolatilitymodelstopriceconsistentlythesetypesof options jointly with plain vanilla options. A possible reconciliation requiresthatthevolatilityprocessisdrivenbyatleast2factors, eveninasingleas-setframework, assupportedbyempirical testsliketheprincipal componentanalysis investigated in Cont and Fonseca (2002).Amongonefactorstochasticvolatilitymodels, themostpopularandeasyto implement is certainly the Heston (1993) one, in which the volatility satisesa(positive)singlefactorsquarerootprocess, wherethepricingandhedgingproblemcanbeecientlysolvedperformingaFastFourierTransform(FFThereafter, see e.g. Carr and Madan 1999).Within the Heston model an accurate modeling of the smile-skew eect fortheimpliedvolatilitysurfaceisusuallyobtainedassuminga(negative)corre-lation between the noise driving the stock return and a suitable calibration ofthe parameters driving the volatility. It is indeed a common observation that asingle factor diusive model is not exible enough to take into account the riskcomponent introduced by the variability of the skew, also known as stochasticskew(see e.g. Carr and Wu 2004). In the case of FX options this risk factor isdirectly priced in the quotes of risk reversals strategies.The aim of this paper is to extend the Heston model to a multifactor spec-ication for the volatility process in a single asset framework. While standardmultifactor modeling of stochastic volatility is based on the class of ane termstructure models introduced in Due and Kan (1996) and classied in Dai andSingleton(2000), inour model thefactor process drivingvolatilityis basedon the matrix Wishart process, mathematically developed in Bru (1991). Ourmodel takes inspiration from the multi-asset market model analyzed in Gourier-ouxandSufana(2004): intheirmodeltheWishartprocessdescribesthedy-namics of the covariance matrix and is assumed to be independent of the assetsnoises. On the contrary, we show that a symmetric matrix specication is poten-tially very useful to improve the ane factor modeling of the implied volatilitycurve. Infact, theintroductionofthematrixnotationprovidesasimpleandpowerfulparametrizationofthedependencebetweentheassetnoiseandeachvolatility factor. In particular, using a square 2 2 matrix of factors, we showthat the expression of the return-volatility covariance is linear in the o diagonalfactor, which can be directly identied as the stochastic skew risk factor: infact, suchfactorcanbespecicallyusedtogenerateastochasticleverageef-fect, which in the case of FX option can be directly calibrated on Risk Reversalquotes.Summing up the present single asset model achieves the following goals:i)the term structure of the realized volatilities is described by a (matrix) mul-2Page 3 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlytifactor model;ii)a stochastic leverage eect appears and can be used to describe stochasticskew eects as required in FX option markets (see Carr and Wu 2004).iii)analytic tractability,i.e. the pricing problem can be handled through theFFT methodology as in Carr and Madan (1999).WeprovideanumericalillustrationthatmotivatestheintroductionoftheWishart(multifactor)volatilityprocess: weshowthatourmodel, dierentlyfromthetraditionalHeston(singlefactor)model, cantseparatelythelong-term volatility level and the short-term volatility skew. Moreover, the correla-tionbetweenassetsreturnsandtheirvolatilityturnsouttobestochastic,sothat in our model we can deal with a stochastic skew eect as in Carr and Wu(2004).Thepaperisorganizedasfollows: insection 2weintroducethestochas-tic(Wishart)volatilitymarketmodel togetherwiththecorrelationstructure.Insection3wesolvethegeneral pricingproblembydeterminingtheexplicitexpressionoftheLaplace-Fouriertransformsoftherelevantprocesses. Inad-dition, weexplicitlycomputethepriceof theforward-startoptions, i.e. thebuildingblocksofcliquetoptions. Section4providesanumericalillustrationwhichshowstheadvantagescarriedbytheWishartspecicationwithrespectto the single factor Heston one as well as the A2(3) (in the terminology of DaiandSingleton2000)multi-Hestonmodel. InSection5weprovidesomecon-clusionsandfuturedevelopments. WegatherinAppendixAsometechnicalproofs, while in Appendix B we develop the computations in the 2-dimensionalcaseforthereadersconvenience. Finally, AppendixCdiscussesthegeneralane correlation structure in the 2-dimensional case.2 TheWishartvolatilityprocessIn an arbitrage-free frictionless nancial market we consider a risky asset whoseprice follows:dStSt= rdt +Tr__tdZt_, (1)wherer denotes the (not necessarily constant) risk-free interest rate,Tr is thetraceoperator, ZtMn(thesetof squarematrices)isamatrixBrownianmotion(i.e. composedbyn2independentBrownianmotions)undertherisk-neutral measure and tbelongs to the set of symmetricn n positive-denitematrices (as well as its square root t).From (1), it follows that the quadratic variation of the risky asset is the traceof the matrix t: that is, in this specication the volatility is multi-dimensionalsince it depends on the elements of the matrix process t, which is assumed tosatisfy the following dynamics:dt = _T+Mt + tMT_dt +_tdWtQ+QT(dWt)T _t, (2)3Page 4 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlywith , M, Q Mn, invertible, andWt Mnis a matrix Brownian motion.Equation (2) characterizes the Wishart process introduced by Bru (1991), andrepresentsthematrixanalogueofthesquarerootmean-revertingprocess. Inordertograntthestrictpositivityandthetypical meanrevertingfeatureofthevolatility, thematrixMisassumedtobenegativesemi-denite, whilesatisesT= QTQwiththerealparameter>n 1(seeBru1991p. 747). WishartprocesseshavebeenrecentlyappliedinnancebyGourierouxandSufana(2004): theyconsidered a multi-asset stochastic volatility model:dSt = diag[St]_r1dt +_tdZt_,whereSt, Zt Rn, 1=(1, ..., 1)Tandthe(Wishart)volatilitymatrixisas-sumed to be independent of Zt. In our (single-asset) specication we relax theindependency assumption: in particular, in order to take into account the skeweect of the (implied) volatility smile, we assume correlation between the noisesdriving the asset and the noises driving the volatility process.2.1 ThecorrelationstructureWecorrelatethetwomatrixBrownianmotionsWt, Ztinsuchawaythatallthe (scalar) Brownian motions belonging to the columni ofZtand the corre-sponding Brownian motions of the columnjofWthave the same correlation,sayRij. This leads to a constant matrixR Mn(identied up to a rotation)which completely describes the correlation structure, in such a way that Zt canbe written as Zt := WtRT+BtI RRT, (I represents the identity matrix andTdenotes transposition) whereBtis a (matrix) Brownian motion independentofWt.Proposition1The processZt := WtRT+BtI RRTis a matrix Brownianmotion.Proof : Itiswell knownthatZtisamatrixBrownianmotioniforany, Rn,Covt(dZt, dZt) = Et_(dZt) (dZt)T_= TIdt.HereCovt(dZt, dZt) = Et__dWtRT +dBt_I RRT__dWtRT +dBt_I RRT_T_= Covt_dWtRT, dWRT_+Covt_dBt_I RRT, dBt_I RRT_= TRRTIdt +T _I RRT_Idt= TIdt.4Page 5 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review OnlyIn line of principle one should allow for an2 n2matrix corresponding tothe(possiblydierent)correlationsbetweenWtandZt. However,inordertograntanalytical tractabilityof themodel (inparticularinordertopreservetheanity)someconstraintsshouldbeimposedonthecorrelationstructure.Itturnsoutthatsuch(nonlinear)constraintsarequitebinding: inordertogive an idea we classify in the Appendix all the possibilities in the casen = 2.Ourchoicecanbeseenasaparsimoniousway(usingonlyn2parameters)tointroduce a simple correlation structure in the model.3 ThepricingproblemInthis sectionwedeal withthepricingproblemof plainvanillacontingentclaims, in particular the European call with payo(ST K)+.Weshall seethat withintheWishart specication, analytical tractabilityispreservedexactlyasinthe(1-dimensional)Hestonmodel. Infact, itiswellknownthatinordertosolvethepricingproblemof plainvanillaoptions, itisenoughtocomputetheconditional characteristicfunction(undertherisk-neutral measure) of the underlying (see e.g. Due, Pan and Singleton 2000) orequivalently of the return processYt = ln St, which satises the following SDE:dYt =_r 12Tr [t]_dt +Tr__t_dWtRT+dBt_I RRT__. (3)We will rst compute the innitesimal generator of the relevant processes andwewill showthatthecomputationofthecharacteristicfunctioninvolvesthesolutionof aMatrixRiccati ODE. Wewill linearizesuchequations andwewill then provide the closed-form solution to the pricing problem via the FFTmethodology.3.1 TheLaplacetransformoftheassetreturnsFollowing Due, Pan and Singleton (2000), in order to solve the pricing problemfor plain vanilla options we just need the Laplace transform of the process (3).SincetheLaplacetransformof Wishartprocessesisexponentiallyane(seee.g. Bru 1991), we guess that the conditional moment generating function of theasset returns is the exponential of an ane combination ofYand the elementsof the Wishart matrix. In other terms, we look for three deterministic functionsA(t) Mn, b(t) R, c(t) R that parametrize the Laplace transform:,t() = Etexp {Yt+}= exp {Tr [A()t] +b()Yt +c()} , (4)5Page 6 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlywhere Et denotes the conditional expected value with respect to the risk-neutralmeasure and R. By applying the Feynman-Kac argument, we have,t= LY,,t(5),t(0) = exp {Yt} ,The matrix setting for the Wishart dynamics implies a non standard denitionof the innitesimal generator for the couple (Yt, t). The innitesimal generatorfor the Wishart process, t, has been computed by Bru (1991) p. 746 formula(5.12):L = Tr__T+M + MT_D + 2DQTQD, (6)whereD is a matrix dierential operator with elementsDi,j =_ij_.For the readers convenience, we develop the computations in the 2-dimensionalcaseinAppendixB. Endowedwiththepreviousresult, wecannowndtheinnitesimal generator of the couple (Yt, t):Proposition2The innitesimal generator of (Yt, t) is given byLY, =_r 12Tr []_y + 12Tr []2y2(7)+Tr__T+M + MT_D + 2DQTQD+ 2Tr [RQD]y.Proof : See Appendix A.Thus the explicit expression of (5) is:,t=_r 12Tr []_ ,ty+ 12Tr [] 2,ty2+Tr__T+M + MT_D,t + 2_DQTQD_,t+ 2Tr [RQD] ,ty,and by replacing the candidate (4) we obtain0 = Tr_dd A()_dd b()Y dd c() (8)+Tr__T+M + MT_A() + 2A()QTQA() + 2RQA()b()+_r 12Tr []_b() + 12Tr [] b2(),with boundary conditionsA(0) = 0 Mn,b(0) = R,c(0) = 0.6Page 7 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review OnlyBy identifying the coecients ofYwe deducedd b() = 0,henceb() = , for all.By identifying the coecients of we obtain the Matrix Riccati ODE sat-ised byA():dd A() = A()M +_MT+ 2RQ_A() + 2A()QTQA() +( 1)2In(9)A(0) = 0.Finally, as usual, the functionc() can be obtained by direct integration:dd c() = Tr_TA()+r, (10)c(0) = 0.3.2 MatrixRiccatilinearizationMatrix Riccati Equations like (9) have several nice properties (see e.g. Freiling2002): the most remarkable one is that their ow can be linearized by doublingthedimensionof theproblem, thisduetothefactthatRiccati ODEbelongto a quotient manifold (see Grasselli and Tebaldi 2004 for further details). Forsake of completeness, we now recall the linearization procedure, and provide theclosed form solution to (9) and (10). PutA() = F ()1G() (11)forF () GL(n), G() Mn, thendd[F () A()] dd[F ()] A() = F ()dd A() ,anddd G()dd[F ()] A() =( 1)2F ()+G() M+_F ()_MT+ 2RQ_+ 2G()QTQ_A() .The last ODE leads to the system of (2n) linearequations:dd G() =( 1)2F () +G() M (12)dd F () = F ()_MT+ 2RQ_2G()QTQ,which can be written as follows:dd_G() F () _ = _G() F () __M 2QTQ(1)2In_MT+ 2RQ__.7Page 8 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review OnlyIts solution is simply obtained through exponentiation:_G() F () _ = _G(0) F (0) _exp _M 2QTQ(1)2In_MT+ 2RQ__= _A(0) In_exp _M 2QTQ(1)2In_MT+ 2RQ__= _A(0)A11() +A21() A(0)A12() +A22() _,where_A11() A12()A21() A22()_ = exp _M 2QTQ(1)2In_MT+ 2RQ__(13)In conclusion, we getA() = (A(0)A12() +A22())1(A(0)A11() +A21()) ,and sinceA(0) = 0,A() = A22()1A21() , (14)which represents the closed-form solution of the Matrix Riccati (9). Let us nowturnourattentiontoequation(10). Wecanimproveitscomputationbythefollowing trick: from (12) we obtainG() = 12_dd F() +F()(MT+ 2RQ)_(QTQ)1,and plugging into (11) and using the properties of the trace we deducedd c() = 2Tr_F()1dd F() + (MT+ 2RQ)_+r.Now we can integrate the last equation and obtainc() = 2Tr_log F() + (MT+ 2RQ)_+r.Thisresultisveryinterestingbecauseitavoidsthenumerical integrationin-volved in the computation ofc().Remark3Thecomputationof theLaplaceTransformforbothassetreturnsand variance factors,t() = Etexp {Yt+ +Tr [t]}= exp_Tr_A()t_+b()Yt +c()_, (15)canbeeasilyhandledbyreplacingthecorrespondingboundaryconditionsandrepeating the above procedure.8Page 9 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Only3.3 ThecharacteristicfunctionandtheFFTmethodLet us now come back to the pricing problem of a call option, and let us brieyrecall the Fast Fourier Transform (FFT) method as in Carr and Madan (1999).For a xed > 0, let us consider the scaled call price at time 0 ascT(k) := exp {k} E_exp {rT} (ST K)+_= exp {k} E_exp {rT} (exp {YT} exp {k})+_,where k = log K. The modied call price cT() is introduced in order to obtain asquare integrable function (see Carr and Madan 1999), and its Fourier transformis given byT(v) :=_+exp {ivk} cT(k)dk= exp {rT}(v(+1)i),0(T)( +iv)( + 1 +iv),which involves the characteristic function . Recall that from the Laplace trans-form, the characteristic function is easily derived by replacing withi, wherei = 1. The inverse fast Fourier transform is an ecient method for comput-ing the following integral:Call(0) =exp {k}2_+exp {ivk} T(v)dv,whichrepresentstheinversetransformof T(v), thatisthepriceofthe(nonmodied)call option. Inconclusion, thecall optionpriceisknownoncetheparameterischosen(typically = 1.1, seeCarrandMadan1999)andthecharacteristic function is found explicitly, which is the case of the (Heston aswell as of the) Wishart volatility model.3.4 ExplicitpricingfortheForward-StartoptionIn this section we apply the methodology developed in the previous section inorder to nd out the price of a forward-start contract. This contract representsthe building block for both cliquet options and variance swaps. All these con-tracts share the common feature to be pure variance contracts. The rst stepconsists in considering a Forward-Start call option, whose payo at the maturityTis dened as follows:FSCall(T) =_STStK_+,whereStisthestockpriceataxeddatet,0 t T. Inthefollowing,wefollow the (single volatility factor) presentation of Wong (2004). By risk-neutral9Page 10 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlyvaluation, the initial price of this option is given byFSCall(0) = E_exp {rT}_STStK_+_.In particular, in the Black and Scholes framework where volatility is constant,one obtainsFSCall(0) = exp {rt} B&S(K, 1, T t, BS),where B&S(K, 1, T t, BS)denotestheBlack-Scholespriceformulaof thecorresponding call option computed with spot price (at timet)St= 1: noticethat in this way the forward start contract price is independent of the level oftheunderlyingassetanddependsonlyonthevolatility. Letusconsidertheforward log-returnYt,T= ln STSt= YT Yt,so that the price of the forward-starts call option is given byFSCall(0) = E_exp {rT} (exp {Yt,T} exp {k})+_,with as beforek = ln K. Let us denote by ,0(t, T) the characteristic functionof the log-returnYt,T, i.e. the so-called forward characteristic functiondenedby,0(t, T) := E[exp {iYt,T}] . (16)The modied option price is given byct,T(k) = exp {k} FSCall(0)and its Fourier transformt,T(v) =_+exp {ivk} ct,T(k)dk (17)= exp {rT}(v(+1)i),0(t, T)( +iv)( + 1 +iv),therefore here again we realize that in order to price a forward-starts call option,itissucienttocomputetheforwardcharacteristicfunction,0(t, T). Thiscomputationwill involvethecharacteristicfunctionof theWishart process,which is given in the followingProposition4Given a real symmetric matrix D, the conditional characteristicfunction of the Wishart process tis given by:D,t() = Etexp {iTr [Dt+]}= exp {Tr [B()t] +C()} , (18)10Page 11 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlywhere the deterministic complex-valued functions B() Mn(Cn), C() C aregiven byB() = (iDB12() +B22())1(iDB11() +B21()) (19)C() = Tr_T_0B(s)ds_,with_B11() B12()B21() B22()_ = exp _M 2QTQ0 MT_.Proof : See Appendix A.Now we have all the ingredients to compute the forward characteristic func-tion of the log-returns ,0(t, T):,0(t, T) = E[exp {iYt,T}]= E[Et[exp {i (YT Yt)}]]= E[exp {iYt} Et[exp {iYT}]]= E[exp {iYt} exp {Tr [A(T t)t] +iYt +c(T t)}]= exp {c(T t)} E[exp {Tr [A(T t)t]}]= exp {Tr [B(t)0] +C(t) +c(T t)} ,where the last equality comes from (18), where B(t) is given by (19) with = tand boundary conditionB(0) = A(T t).Endowed with the function ,0(t, T), it suces to plug into (17) and apply theFFT in order to nd the forward-start call price.4 NumericalillustrationIn this section we provide some examples proving that the Wishart specicationfor the volatility has greater exibility than the (single-factor as well as multi-factor) Heston one. We quote option prices using Black&Scholes volatility, whichisacommonpracticeinthemarket. LetusdenotebyVttheinstantaneousvolatility in the (single factor) Heston model, whose dynamics is given bydVt = ( Vt)dt +_VtdW2t ,where represents the long-term volatility, is the mean reversion parameter, is the volatility of volatility parameter (also called vol-of-vol), is the correlationbetween the volatility and the stock,V0is the initial spot volatility andW2tis(scalar) Brownian motion of the volatility process, which in the Heston modelisassumedtobecorrelatedwiththeBrownianmotionW1tdrivingtheassetreturns.We proceed as follows:11Page 12 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Only1. we consider the simplest modication of the previous choice which allowsto reproduce a volatility surface which cannot be generated by the (singlefactor) Heston model,2. wecompareourmodel withthemulti dimensional versionof theHes-ton model when the volatility is driven by a 2 dimensional ane processwhose state space domain is R2+ as classied in Dai and Singleton (2000):inparticularweshowthatwithinourWishartspecicationwehaveanadditionaldegreeoffreedominordertocapturethestochasticityoftheskew eect by preserving analytic tractability.4.1 WishartembeddingHestonvolatilityTheHestonmodel canbeeasilynestedintotheWishartmodel foraspecicchoiceoftheparameters. WhenallmatricesinvolvedintheWishartdynam-ics areproportional totheidentitymatrix, it is straightforwardtoseethatTr(t) follows a square root process and both models produce the same smileat dierent maturities.The original motivation for introducing multifactor models comes from theobservationthatthedynamicsoftheimpliedvolatilitysurface, aswellastherealizedvolatilityprocessaredrivenbyatleasttwostochasticfactors. Thesimplest example of implied volatility pattern that cannot be reproduced by asingle factor model is obtained by considering a diagonal model while specifyingtwo dierent mean reversion parameters in the (diagonal) matrix M. In partic-ular, if we chooseM11= 3 andM22= 0.333, then we can associate to theelement 11 the meaning of a short-term factor, while 22 has an impact on thelong-term volatility. Let us take the following values:M=_ 3 00 0.333_, R =_ 0.7 00 0.7_(20)Q =_0.25 00 0.25_, 0 =_0.01 00 0.01_,and = 3. In this case we see that in the Wishart model the long term volatilityincreases. This additional degree of freedom is interesting from a practical pointof view because on the market there are some long-term products such as forwardstart option and cliquet options whose maturity can be much higher than oneyear. It is then important to obtain prices for such contracts in closed form, inorder to investigate the properties of the long-term smile. Observe that typicallylong-termvolatilityishigherthanshort-termone. Nowwewanttogeneratethe same volatility smile with the Heston model, so in order to t the impliedvolatility at 2 years we have to set = 0.382,while the other parameters are:=6, , 0=0.15,=0.5= 0.7. However, anincreaseof thelong-termvolatilityinducesalsoanincreaseofthe3monthsvolatility,sothattheshort-term t for the implied volatility is unsatisfactory, as illustrated in Figure(1).12Page 13 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Only0.10.20.30.4ImpliedvolatilityImpliedvolatility0.8 0.9 1 1.1 1.2Moneyness MoneynessWis 3mHes 3mWis 2yHes 2yFigure1: Impliedvolatilityfor theWishart model (Wis) andHeston(Hes)model. Optionmaturitiesare3months(3m)and2years(2y). Moneynessisdened byKS0whereS0is the initial spot value.Ontheotherhand,wecantperfectlytheshort-termvolatilityproducedbytheWishartmodelbysetting=0.2952. However, inthiscasethelong-term volatility decreases and this time we arrive to an unsatisfactory t of thelong-term implied volatility level as shown in Figure (2).4.2 Wishartversus A2(3)-HestonvolatilityNoticethattheaboveobservationisnotsucienttojustifytheintroductionof the previous Wishart (matrix) ane model given by (20), whose covariancematrix can be also reproduced1using the following (vector) ane model, whichbelongs to the canonical class A2(3) of Dai and Singleton (2000):dX1t= 1(1X1t )dt +1_X1tdW1t ,dX2t= 2(2X2t )dt +2_X2tdW2t ,dYt =_r 12_X1t+X2t__dt +1_X1tdW1t+2_X2tdW2t+_(1 21) X1t+ (1 22) X2tdBt,1Wethankananonymousrefereefortheobservation13Page 14 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Only0.10.20.30.4ImpliedvolatilityImpliedvolatility0.8 0.9 1 1.1 1.2Moneyness MoneynessWis 3mHes 3mWis 2yHes 2yFigure2: Impliedvolatilityfor theWishart model (Wis) andHeston(Hes)model.with W1t , W2t , Bt are independent Brownian motions. In fact, both models leadtothesamecovariancematrixwherethestatespacedomainof thepositivefactors 11t, 22t(resp. X1t, X2tfor the A2(3) model) is R2+.Remark that in this A2(3) model once the short term and long term impliedvolatility levels are tted, there are no more free parameters in order to describethe stochasticity of the leverage eect (which leads to a stochastic skew in thespirit of Carr and Wu 2004): in fact, it turns out that the correlation betweentheassetsreturnsandtheirvolatilitiesisstochasticbutitdepends(only)onthe volatility factors:Corrt(Noise(dY ), Noise (V ol (dY ))) =Y, X1+X2_t_Y tX1+X2

t=Et___X1tdZ1t+_X2tdZ2t__

1_X1tdW1t+2_X2tdW2t___X1t+X2t_

21X1t+22X2t=1

1X1t+2

2X2t_X1t+X2t_

21X1t+22X2t. (21)4.2.1 StochasticleverageeectintheWishartmodelIn order to compute the analogue of (21) in the general Wishart model, let usnow consider the correlation between the stock noise and the noise driving itsscalarvolatility, represented byTr(t): this is computed in the followingProposition5The stochastic correlation between the stock noise and the volatil-14Page 15 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlyity noise in the Wishart model is given byt =Tr [RQt]_Tr [t]_Tr [QTQt]. (22)Proof : See Appendix A.The previous proposition highlights the analytical tractability of the Wishartspecication: infact, withintheWishartmodel itispossibletohandlethe(stochastic) correlation (and in turn the stochastic skew eect) by means of theproductRQ.Whentheproduct RQisamultipleof theidentitymatrix, werecoverthe usual constant correlation parameter as in the (single factor) Hestonmodel as well as in the multi-Heston model with1 = 2and1 = 2;When the product RQ is diagonal then the Wishart model is qualitativelyequivalent to a A2(3) multi-Heston model, in the sense that the stochasticcorrelationdependsonlyonthevolatilityfactors11t, 22t(whiletheodiagonal factor 12tdoes not appear): in fact, in this case (22) readsA2(3)t=R11Q1111t+R22Q2222t_11t+ 22t_Q21111t+Q22222t,which is exactly the analogue of (21);When the productRQ is not diagonal (i.e. whenR orQ is not diagonal)from (22) it turns out thattdepends also on the o diagonal volatilityterm 12t:Wist= A2(3)t+Q22R12_11t+ 22t_Q21111t+Q22222t12tthat is, in the Wishart specication, the o diagonal elements of the vol-of vol matrixQandthecorrelationmatrixRareadditional degreesoffreedom w.r.t. the A2(3) multi-Heston model in order to control the sto-chasticity of (the correlation and in turn of) the leverage eect once theshort-termandlong-termimpliedvolatilitylevelsaretted. Thisrep-resentsasuitablefeatureof astochasticvolatilitymodel whichcanbecalibrated on Risk Reversal quotes in the spirit of Carr and Wu (2004).This model cannot be nested into a A2(3) since the admissible domains ofA2(3) and the Wishart model are crucially dierent: while the former hasthe linear structure of Rm+R(nm), the Wishart domain is the symmetricconeof positivedenitematrices(seealsoGrasselli andTebaldi 2004),which is non linear in the factors (the domain of 12tis given by the set11t22t_12t_2> 0). This non linearity allows the Wishart specicationto reproduce new eects w.r.t. the classic (vector) ane models.15Page 16 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Only 0.18 0.19 0.2 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28Smile: implied BlackScholes volatility - 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2Moneyness 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time to maturity (in years) 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34 0.225 0.23 0.235 0.24 0.245 0.25 0.255 0.26Smile: implied BlackScholes volatility - 0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2Moneyness 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Time to maturity (in years) 0.16 0.18 0.2 0.22 0.24 0.26 0.28 0.3 0.32 0.34Figure 3: Wishart implied volatility forR1(left) andR2(right).4.2.2 TheimpactofRonthestochasticleverageeectIn the following examples we compare the Wishart specication with diagonalmatrixparameters(equivalenttotheA2(3)multi-Hestonmodel)withanondiagonal one, inordertohighlighttheadditional exibilityintroducedbyodiagonal terms.It is well known that in the Heston model the skew is related to a (negative)correlation between the volatility and the stock price. Taking the matricesM=_ 5 00 3_, Q =_0.35 00 0.25_, 0 =_0.02 00 0.02_(23)R1 =_ 0.7 00 0.5_, R2 =_0 00 0_and = 3 in the Wishart model, we get forR1(resp.R2) the left (resp. right)hand side of Figure (3) , which conrms that R is strictly related to the leverageeectinboththeWishartand A2(3)multi-Hestonmodels. Inparticular,theshort and long term implied volatility levels can be tted by using the diagonalterms in the matrix R1 as well as the parameters 1, 2 in the A2(3) multi-Hestonmodel.Now let us consider the Wishart model with a non-diagonal correlation ma-trixR3given byR3 =_ 0.7 R120 0.5_.16Page 17 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Only-0,99-0,92-0,85-0,79-0,72-0,65-0,58-0,52-0,45-0,38-0,31-0,25-0,18-0,11R12=0R12=0,25R12=0,500,020,040,060,080,10,120,140,16CorrelationCorrelation distribution (3 years)R12=0R12=0,25R12=0,5Figure 4: Distribution functions of the correlation process in the Wishart modelwith non diagonal matrix R.From (22) we obtain:Wist= 0.7 (0.35) 11t+ 0.25_R1212t0.522t__11t+ 22t_(0.35)211t+ (0.25)222t.The presence of the o diagonal parameter R12 introduces the new factor 12tin the correlation, which is described by an additional source of uncertainty. InFigure (4) we considered the distribution of the correlation process for dierentvalues of R12:notice that the distribution becomes more sparse as R12 increases,a new eect which cannot be reproduced by the A2(3) multi-Heston model.5 ConclusionWe showed that the multifactor volatility extension of the Heston model consid-ered in this paper is exible enough to take into account correlations with theunderlying asset returns. In the meanwhile it preserves analytical tractability,i.e. a closed form for the conditional characteristic function, and a linear factorstructure which can be potentially very useful in the calibration procedure. Fi-nally, our numerical results show that the exibility induced by the additionalfactors allow a better t of the smile-skew eect at both long and short matu-rities. In particular, contrarily to the Heston model, the Wishart specicationdoes permit a separate t of both long-term and short-term skew (or volatility17Page 18 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlylevel), sothatwecanallowformorecomplextermstructuresfortheimpliedvolatility surface. Future work will be devoted to the calibration of this modeltooptionpricesandfurtherstudiesareneededinordertoillustratetheim-provementsoncalibrationwithrespecttothe(scalarand A2(3)multi-factor)Hestonmodel. Fromanancial econometricperspective, ontheotherhand,thismodelseemstobeanaturalcandidatetoanalyzeanddescribevolatilityand stochastic correlations eects on the risk premia valued by the market.References[1] Bergomi, L. (2004) Smile dynamics, Risk September, 117-123.[2] Bergomi, L. (2005) Smile dynamics II, Risk October.[3] Bru, M. F. (1991) Wishart Processes. Journal of Theoretical Probability,4, 725-743.[4] Cont, R. andJ. daFonseca(2002)Dynamicsof impliedvolatilitysur-faces. Quantitative Finance, 2, No 1, 45-60.[5] Carr, P. and D. B. Madan (1999) Option valuation using the fast Fouriertransform. Journal of Computational Finance, 2 No 4.[6] Carr, P. and L. Wu.(2004) Stochastic Skewin Currency Options,preprint.[7] Dai, Q. andK. Singleton(2000)SpecicationAnalysisof AneTermStructure Models. Journal of Finance, 55, 1943-1978.[8] Due, D. andR. Kan(1996)AYield-FactorModel ofInterestRates.Mathematical Finance, 6 (4), 379-406.[9] Due, D., J. Pan and K. Singleton (2000) Transform analysis and assetpricing for ane jump-diusions. Econometrica, 68, 1343-1376.[10] Freiling, G. (2002):A Survey of Nonsymmetric Riccati Equations. LinearAlgebra and Its Applications, 243-270.[11] Gourieroux, C. and R. Sufana (2003) Wishart Quadratic Term StructureModels. CREF 03-10, HEC Montreal.[12] Gourieroux, C. and R. Sufana (2004) Derivative Pricing with MultivariateStochastic Volatility: Application to Credit Risk. Working paper CREST.[13] Grasselli M. and C. Tebaldi (2004) Solvable Ane Term Structure Mod-els. Mathematical Finance, to appear.[14] Heston, S. L. (1993) A Closed-Form Solution for Option with StochasticVolatilitywithApplicationstoBondandCurrencyOptions.ReviewofFinancial Studies, 6, (2), 327-343.18Page 19 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Only[15] C. Hipp(1996) Options for GuaranteedIndex-linkedLife Insurance.AFIR 1996 Proceedings, Vol. II, S. 1463-1483.[16] Wong, G. (2004) Forward Smile and Derivative Pricing . Working paperUBS.6 AppendixA:ProofsProof of Proposition2: Theonlynontrivial termin(7)comesfromthecovariationd < ij, Y>t, fori, j = 1, ..., n.It will be useful to introduce the square root matrixt := t, so thatijt=n

l=1iltljt=n

l=1iltjlt,where the last equality follows from the symmetry oft. Now we identify thecovariation terms with the coecients of_2xijy_, thus obtainingd < ij, Y>t= Et____n

l,k=1iltdWlkQkj +n

l,k=1jltdWlkQki____n

l,k,h=1lktdWkhRlh____=n

l,k,h=1_iltQkj +jltQki_hltRhkdt=n

k,h=1__n

l=1ilthlt_Qkj +_n

l=1jlthlt_Qki_Rhkdt=n

k,h=1_ihtQkj + jhtQki_Rhkdt,which corresponds to the coecient of the term_2xijy_, since2Tr [RQD]y= 2n

i,j,k,h=1DijjhRhkQkiyand since by denitionD is symmetric.Proof of Proposition4: Werepeatthereasoningasin(4)wherethistime there is no dependence on Yt, so that the (complex-valued non symmetric)Matrix Riccati ODE satised byB() becomesdd B() = B()M +MTB() + 2B()QTQB()B(0) = iD,19Page 20 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review OnlywhileC() = Tr_T_0B(s)ds_.Applying the linearization procedure, we arrive to the explicit solutionB() =F ()1G() , with_G() F () _ = _G(0) F (0) _exp _M 2QTQ0 MT_= _B(0) In_exp _M 2QTQ0 MT_= _iDB11() +B21() iDB12() +B22() _,which gives the statement.Proof of Proposition 5: The rst step consists in nding the stock noise:dStSt= rdt +Tr__tdZt_= rdt +_Tr [t]Tr_tdZt_Tr [t]= rdt +_Tr [t]dzt,where zt is a standard Brownian Motion. We now compute the (scalar) standardBrownian motionwtdriving the processTr [t]:dTr [t] = _Tr_T+ 2Tr [Mt]_dt + 2Tr__tdWtQ_= ...dt + 2_Tr [tQTQ]Tr_tdWtQ_Tr [tQTQ]= ...dt + 2_Tr [tQTQ]dwt,20Page 21 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlywhere we used the fact thatd Tr [.]t =

ijCovt_eTidtei, eTj dtej_= 4

ijCovt_eTi_tdWtQei, eTj_tdWtQej_= 4

ijEt_eTi_tdWtQeieTj QTdWTt_tej_= 4

ijeTi_tTr_QeieTj QT_tejdt= 4

ijTr_QTQeieTjeTi tejdt= 4

ijeTj QTQeieTi tejdt= 4

jeTj QTQtejdt= 4Tr_tQTQdt,where we used thatEt_dWtQeieTj QTdWTt = Tr_QeieTj QTdtsince from Proposition 1:Et_dWtTdWTt = TIdt= Tr_TIdt.Inconclusion, thecorrelationbetweenthestocknoiseandthevolatilitynoisein the Wishart model is stochasticand corresponds to the correlation betweenthe Brownian motionsztandwt, whose covariation is given by:Covt(dzt, dwt) = Covt_Tr_tdZt_Tr [t],Tr_tdWtQ_Tr [tQTQ]_= Et_Tr_tdWtRT_Tr [t]Tr_tdWtQ_Tr [tQTQ]_=

ij Covt_eTitdWRTei, eTjtdWtQej__Tr [t]_Tr [tQTQ]=1_Tr [t]_Tr [tQTQ]

ijeTi_tTr_RTeieTj QT_tejdt=1_Tr [t]_Tr [tQTQ]Tr_tQTRTdt=Tr [tRQ]_Tr [t]_Tr [tQTQ]dt21Page 22 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Only7 AppendixB:The2-dimensionalcaseIn this Appendix we develop the computations in (6) in the casen = 2. Thismeans that the Wishart process tsatises the following SDE:dt = d_11t12t12t22t_=__11122122__11211222_+_M11M12M21M22__11t12t12t22t_+_11t12t12t22t__M11M21M12M22__dt+_11t12t12t22t_1/2_dW11tdW12tdW21tdW22t__Q11Q12Q21Q22_+_Q11Q21Q12Q22__dW11tdW21tdW12tdW22t__11t12t12t22t_1/2.Let be_11t12t12t22t_ :=_11t12t12t22t_1/2,so that2t= t =_ _11t_2+_12t_211t12t+12t22t11t12t+12t22t_12t_2+_22t_2_. (24)We obtaind11t= (.)dt + 211t_Q11dW11t+Q21dW12t_+ 212t_Q11dW21t+Q21dW22t_,d12t= (.)dt +11t_Q12dW11t+Q22dW12t_+12t_Q12dW21t+Q22dW22t_+12t_Q11dW11t+Q21dW12t_+22t_Q11dW21t+Q21dW22t_,d22t= (.)dt + 212t_Q12dW11t+Q22dW12t_+ 222t_Q12dW21t+Q22dW22t_,22Page 23 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlyand using (24):d < 11, 11>t= 411t(Q211 +Q221)dt,d < 12, 12>t= _11t_Q212 +Q222_+ 212t(Q11Q12 +Q21Q22) + 22t(Q211 +Q221)_dt,d < 22, 22>t= 422t(Q212 +Q222)dt,d < 11, 12>t= _211t(Q11Q12 +Q21Q22) + 212t_Q211 +Q221__dt,d < 11, 22>t= 412t(Q11Q12 +Q21Q22) dt,d < 12, 22>t= 2_12t_Q212 +Q222_+ 22t(Q11Q12 +Q21Q22)_dt.On the other hand, from (6) we can identify the coecient of_2ijlk_ in thetrace of the matrix 2tDQTQD, that is2_11t12t12t22t__11121222__Q11Q21Q12Q22__Q11Q12Q21Q22__11121222_.After some computations, we obtain:Tr_2tDQTQD = 2Tr_tDQTQD= 211t_Q211 +Q221_2(11)2+ 2_11t_Q212 +Q222_+ 212t(Q11Q12 +Q21Q22) + 22t(Q211 +Q221)_2(12)2+ 222t_Q212 +Q222_2(22)2+ 4_11t(Q11Q12 +Q21Q22) + 12t_Q211 +Q221__21112+ 412t(Q11Q12 +Q21Q22)21122+ 4_12t_Q212 +Q222_+ 22t(Q11Q12 +Q21Q22)_21222,thus proving the equality in (6), sinceL = Tr__T+M + MT_D+ 12_< 11, 11>t2(11)2+ 4 < 12, 12>t2(12)2+ < 22, 22>t2(22)2+ 4 < 11, 12>t21112+2 < 11, 22>t21122+ 4 < 12, 22>t21222_,23Page 24 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlywhere we recall that2 < 12, 12>t2(12)2=< 12, 12>t2(12)2+ < 21, 21>t2(21)2;4 < 11, 12>t21112= 2 < 11, 12>t21112+ 2 < 11, 21>t21121.8 AppendixC: Theanityconstraints onthecorrelationstructureIn this Appendix we study the general correlation structure in the casen = 2.We introduce 4 matrices R11, R12, R21, R22 M2 representing the correlationsamong the matrix Brownian motions (in total 16 =n2n2correlations: Rabijdenotes the correlation betweenZabtandWijt). In this way we can writeZ11t= Tr_WtR11T+_1 Tr [R11R11T]B11t(25)Z12t= Tr_WtR12T+_1 Tr [R12R12T]B12t(26)Z21t= Tr_WtR21T+_1 Tr [R21R21T]B21t(27)Z22t= Tr_WtR22T+_1 Tr [R22R22T]B22t(28)First of all we notice that there are some constraints on the parameters in orderto grant thatZtis indeed a matrix Brownian motion.Proposition6Ztis a matrix Brownian motion iTr_RijRlmT = 0 for (i, j) = (l, m), i, j, l, m {1, 2} . (29)Proof : Let us consider the rst element of the matrixCovt(dZt, dZt) :Covt(dZt, dZt)11 = Et_Tr_dWtR11T1 +_1 Tr [R11R11T]dB11t1+Tr_dWtR12T2 +_1 Tr [R12R12T]dB12t2_._Tr_dWtR11T1 +_1 Tr [R11R11T]dB11t1+Tr_dWtR12T2 +_1 Tr [R12R12T]dB12t2_= 11dt +22dt+ (12 +21) (R1111R1211 +R1112R1212 +R1121R1221 +R1122R1222) dt.Since we have to prove thatCovt(dZt, dZt) = TIdt for all vectors, , itmust be thatR1111R1211 +R1112R1212 +R1121R1221 +R1122R1222 = 0,24Page 25 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlythat is Tr_R11R12T = 0. Similar computations for the other components leadto the conclusion.Now we look for the additional constraints on the matricesRijin order togrant the anity of the model, that is such that LY,is ane on the elementsof t. Let us consider the rst element:d < 11, Y>t= Et__11tdZ11t+12tdZ12t+12tdZ21t+22tdZ22t_d11t= 2__11t_2Q11R1111 +_11t_2Q21R1112 +11t12tQ11R1121+11t12tQ21R1122 +11t12tQ11R1211 +11t12tQ21R1212+_12t_2Q11R1221 +_12t_2Q21R1222 +11t12tQ11R2111+11t12tQ21R2112 +_12t_2Q11R2121 +_12t_2Q21R2122+11t22tQ11R2211 +11t22tQ21R2212 +12t22tQ11R2221 +12t22tQ21R2222_dtIt follows thatR2211 = 0R2212 = 0R1111 = R1221 +R2121R1112 = R1222 +R2122R2221 = R1121 +R1211 +R2111R2222 = R1122 +R1212 +R2112From the expression ofd < 22, Y>twe obtainR1121 = 0R1122 = 0and it turns out that the other conditions are redundant, as well as those comingfromd < 12, Y>t. In conclusion, the anity constraints lead to the followingspecication for the 4 correlation matrices:R12 =_a bc d_R21 =_e fg h_R11 =_c +g d +h0 0_R22 =_0 0a +e b +f_.25Page 26 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review OnlyNow we impose (29) and obtain:R11 R21 e(c +g) +f(d +h) = 0 (30)R11 R12 a(c +g) +b(d +h) = 0 (31)R22 R21 g(a +e) +h(b +f) = 0 (32)R22 R12 c(a +e) +d(b +f) = 0 (33)R12 R21 ae +bf +cg +dh = 0. (34)After some manipulations we arrive toae(a +e)2+cg(b +f)2= 0. (35)Here we see that there are 8 parameters but subject to 5 (nonlinear) constraints,allowing only a few compatible choices for the parameters. Now we are readytowritedowntheinnitesimal generatorassociatedtothegeneral (ane)2-dimensional case:Proposition7The innitesimal generator of (Yt, t) is given byLY, =_r 12Tr []_y + 12Tr []2y2(36)+Tr__T+M + MT_D + 2DQTQD+ 2Tr [(R11 +R22) QD]y.Proof : We focus on the covariation termsd < ij, Y>t, fori, j = 1, ..., 2 :d < 11, Y>t= 2Q11_(c +g) 11+ (a +e) 12_dt+ 2Q21_(d +h) 11+ (b +f) 12_dtd < 22, Y>t= 2Q12_(a +e) 22+ (c +g) 12_dt+ 2Q22_(d +h) 12+ (b +f) 22_dtd < 12, Y>t= Q12_(c +g) 11+ (a +e) 12_dt+Q22_(d +h) 11+ (b +f) 12_dt+Q11_(c +g) 12+ (a +e) 22_dt+Q21_(d +h) 12+ (b +f) 22_dtand we obtain the statement, sinced < ij, Y>t corresponds to the coecientof the term_2xijy_, andTr [(R11 +R22) QD]y= Tr__11t12t12t22t__c +g d +ha +e b +f__Q11Q12Q21Q22__11121222__yand by denitionD is symmetric.26Page 27 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review OnlyBy applying the Feynman-Kac argument to the Laplace transform,t() = Etexp {Yt+} (37)= exp {Tr [A()t] +b()Yt +c()} , (38)we obtainb() anddd A() = A()M +_MT+ 2 (R11 +R22) Q_A() + 2A()QTQA() +( 1)2In(39)A(0) = 0.We have proved the followingProposition8TheRiccati equationssatisedbythematrixcoecient A()associated to the Laplace transform (37) are given by (39), whereR11 =_c +g d +h0 0_R22 =_0 0a +e b +f_,where the parametersa, b, c, d, e, f, g, h satisfy the (non-linear) constraints (30),(31), (32), (33), (34), (35).Remark9Ourmodelcorrespondstochoosingc =d =e =f= 0(orequiva-lentlya = b = g = h = 0): we obtainR12 =_21220 0_R21 =_0 01112_R11 =_11120 0_R22 =_0 02122_,andZ11t= W11t11 +W12t12 +_1 211212B11t(40)Z12t= W11t21 +W12t22 +_1 221222B12t(41)Z21t= W21t11 +W22t12 +_1 211212B21t(42)Z22t= W21t21 +W22t22 +_1 221222B22t(43)27Page 28 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011For Peer Review Onlywe can then introduce a matrixR =_11122122_,in such a way thatZt := WtRT+ BtI RRT, where Bt is a matrix Brownianmotion which can be deduced fromBt.28Page 29 of 29E-mail: quant@tandf.co.ukURL://http.manuscriptcentral.com/tandf/rqufQuantitative Finance123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960peer-00568441, version 1 - 23 Feb 2011