Asset Pricing Under the Quadratic Class Author(S): Markus Leippold and Liuren Wu Source: the Journal of Financial and Quantitative Analysis, Vol

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Asset Pricing under the Quadratic Class Author(s): Markus Leippold and Liuren Wu Source: The Journal of Financial and Quantitative Analysis, Vol. 37, No. 2 (Jun., 2002), pp. 271 -295 Published by: University of Washington School of Business Administration Stable URL: http://www.jstor.org/stable/3595006 Accessed: 22/07/2009 10:43

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Asset Pricingunder the QuadraticClass

MarkusLeippold and LiurenWu*

Abstract Weidentify and characterize a class of termstructure models where bond yields are quadratic functionsof the statevector. We labelthis class the quadraticclass andaim to lay a solidtheoretical foundation for its futureempirical application. We consider asset pricing in generaland derivative pricing in particularunder the quadraticclass. Weprovide two generaltransform methods in pricinga widevariety of fixedincome derivatives in closed or semi-closedform. We furtherillustrate how the quadraticmodel and the transform methodscan be appliedto moregeneral settings.

I. Introduction

We identify and characterizea class of term structuremodels where bond yields are quadraticfunctions of the Markov process. We label this class the quadratic class and aim to lay a solid theoreticalfoundation for its future empirical application. We identify the necessary and sufficient conditions for the quadraticclass and considerthe generalasset pricingproblem under the quadratic framework.In particular,we proposetwo transformmethods to price a wide variety of interestrate derivativesin closed or semi-closed form. We furtherillustrate how the pricing methods can be applied to more general settings. Examples include option pricing for currenciesor stocks with quadraticstochastic volatilities. Our interests in the quadraticclass come mainly from concerns on empirical application. Recent empiricalresearch within the affine frameworkof Duffie and Kan (1996) indicates an inherenttension between i) deliveringgood empirical performancein matchingsalient featuresof the interestrate data and ii) ex- cluding positive probabilities of having negative interest rates. For example, Backus, Telmer, and Wu (1999) and Dai and Singleton (2000) find that incor- poratingGaussian state variablesin the affine frameworksignificantly increases the flexibility for model design and greatly improves its empirical performance in capturingthe conditionaldynamics of interestrates. Dai and Singleton (2001)

*Leippold, [email protected],Swiss Banking Institute, University of Zurich, Plattenstr.14, 8032 Zurich, Switzerland;and Wu, [email protected],Graduate School of Business, FordhamUni- versity, 113 West 60th Street, New York, NY 10023. We thank Marco Avellaneda, David Backus, Peter Carr,Pierre Collin, Silverio Foresi, Michael Gallmeyer,Richard Green, Massoud Heidari, Bur- ton Hollifield, Regis Van Steenkiste, Chris Telmer,Stanley Zin, and, in particular,Jon Karpoff (the editor) and Greg Duffee (the referee) for helpful comments. 271 272 Journalof Financialand QuantitativeAnalysis

and Duffee (2001) also find the need for applyingan affine marketprice of risk on Gaussianstate variablesin explainingthe dynamicbehavior of expected excess returnsto bonds and the anomaliessurrounding the variousexpectations hy- potheses. Furthermore,Backus, Foresi, Mozumdar,and Wu (2001) and Backus, Foresi,and Telmer (2001) findthat incorporating a negativesquare-root state variable also helps in explainingthe expectationanomalies in both interestrates and currencies. Yet, incorporatingeither Gaussianvariables or negative square-root variablesin the affine frameworkgenerates negative interestrates with positive probabilities.Therefore, such practicesraise concerns,among both practitioners and academics,about potential arbitrage possibilities and theirreal-time applica- bility. In contrast,the quadraticclass combines Gaussian state variableswith an affinemarket price of risk into a naturalframework and guaranteespositive interest rateby a simple parametricrestriction. Furthermore, the quadraticrelation between interestrates and the normalstate variablesadds greatflexibility for model design. The empiricalworks of Ahn, Dittmar,and Gallant(2001) and Leippold and Wu (2000) also suggest that quadraticmodels can outperformaffine models in explaininghistorical bond price behaviorin the U.S. Meanwhile,the analyticaltractability of the quadraticclass in termsof bond and option pricingis comparableto that of the affine class. We show that, under the quadraticclass, the prices of assets, whose future payoffs are exponential- quadraticin the state vector, are exponential-quadraticin the currentstate. Thus, the price of a zero-couponbond is merely a degeneratingspecial case. The co- efficientsfor the quadraticfunctions can be solved analyticallyfor the one-factor case and independentmulti-factor cases and are the solutionsto a set of ordinary differentialequations for generalmulti-factor cases. For derivativepricing, we consider the price of a general state-contingent claim and label it as the state price in its broadestmeaning. A wide variety of fixed income derivativescan be writtenas an affinefunction of such a state price. Examplesinclude European options on zero-couponbonds, interestrate caps and floors,exchange options on zero-couponbonds, and even Asian options,the payoff of which depends on the path averageof bond yields. We define two transforms on the general state price and prove that both can be regardedas an asset with exponential-quadraticpayoffs and thereforeboth can be priced analytically, up to the solutionof a series of ordinarydifferential equations. The stateprice can then be obtainedby a one-dimensionalnumerical inversion of either transform, regardlessof the dimensionof the state space. The firsttransform is similarin natureto the transformdefined in Bakshiand Madan(2000) and Duffie, Pan, and Singleton (2000). It regardsthe state price as an analogueof a cumulativedensity. The inversionis hence obtainedby an extendedversion of the Levy inversionformula for cumulativedensity functions. The second transformis inspiredby Carrand Madan(1999) and regardsthe state price as an analogueof the probabilitydensity function. For the second transform to be defined, we need to extend the transformparameter to the complex plane. The transformis hence often referredto as the generalized,or complex,transform. The choice of the imaginarydomain depends on the exact structureof the state- contingentpayoff. We identify the admissibledomain for a wide varietyof state- Leippoldand Wu 273 contingentclaims. Given the generalizedtransform, the inversioncan be cast in a way where we can apply the fast Fouriertransform (FFT). We can hence rip significantgains in computationalefficiency. The earliest example of quadraticmodels, to our knowledge, is the double square-rootmodel of Longstaff (1989) and the correctionand generalizationby Beaglehole and Tenney (1991), (1992). El Karoui, Myneni, and Viswanathan (1992) furtherdevelop this quadraticclass along the lines of Beaglehole and Ten- ney (1991). Jamshidian(1996) obtains the ordinarydifferential equations for bond pricing for the generalquadratic class and providesoption pricingformulae for a subset of the class (independentMarkov process). The SAINTS (squared- autoregressive-independent-variablenominal term structure)model of Constan- tinides (1992) is also a subset of the quadraticclass, where the pricing kernel is exogenously specified as a time-separablequadratic function of the Markov process. Rogers (1997) and Leippold and Wu (1999), starting with modeling the pricing kernel as a potential, also use examples where the pricing kernel is a time-separablequadratic function of the Markovprocess. Ahn, Dittmar,and Gallant(2001) presenta list of assumptionsthat essentially identify the complete quadraticclass. Ourpaper clarifies the identificationproblem by proving the necessity and sufficiency of the conditions. Our paper furthercontributes to the literatureby derivingasset pricing implicationsunder the quadraticframework. Most recently,Filipovic (2001) proves, undercertain regularity conditions, that if one representsthe forwardrate as a time-separablepolynomial function of the diffusion state vector, the maximal consistent order of the polynomial is two. Consistency in this context, as discussed in Bjork and Christensen(1999) and Filipovic (2000), means that the interest rate model will produce forward rate curves belonging to the parameterizedfamily. Thus, our identificationof the quadraticclass, together with the identificationof the affine class by Duffie and Kan (1996), essentially completes the search for consistent time-separable polynomialterm structuremodels. The structureof the paper is as follows. The next section identifies the quadraticclass, discusses the specificationof the pricingkernel, and analyzes the propertiesof bond yields and forwardrates underthe quadraticclass. Section III considersthe generalasset pricingproblem under the quadraticclass and presents our two transformmethods of option pricing. Section IV provides numerousex- amples and other applications.Section V concludes. Additionaltechnical details are providedin the Appendix.

II. QuadraticTerm Structure Models

We identify the complete non-degeneratingquadratic class of term structure models in terms of the Markovprocess and the instantaneousinterest rate function. We furtherdiscuss the specification of the pricing kernel and its impacts on bond pricing, as well as on the pricing of other assets such as currenciesand stocks. We then conclude the section by an analysis of the propertiesof bond yields and forwardrates underthe quadraticclass. 274 Journal of Financial and Quantitative Analysis

A. Necessary and Sufficient Conditions

We fix a filteredcomplete probability space {Q, , I', (Tt)0

(1) dXt = i(Xt)dt + S(Xt)dWt,

where p(Xt) is an n x 1 vector defining the drift and S(Xt) is an n x n matrix definingthe diffusionof the process. Furtherassume thatfor any time t E [0, T] and time-of-maturityT C [t, T], the marketvalue at time t of a zero-couponbond with maturityr = T - t is fully characterizedby P(Xt, r) andthat the instantaneousinterest rate, or the shortrate, r, is definedby continuity,

r (Xt) -lim In P (Xt,7) r40 T

Definition1. In the quadraticclass of termstructure models, the prices of zero- coupon bonds, P(Xt, r), are exponential-quadraticfunctions of the Markovpro- cess Xt,

- T - (2) P (Xt,r) = exp [XA () Xt b () Xt c ()], whereA(r) is a non-singularn x n matrix,b(r) is an n x 1 vector,and c(r) is a scalar.

P (Xt,0) = 1 for all Xt E D implies the boundaryconditions: A (0) = 0, b (0) = 0, and c (0) = 0. By relaxingthe non-singularityrestriction on A(r), we would have the affine class of Duffie and Kan (1996) as a subclass. The affine class is obtainedby setting A(r) - 0 for all T. A singularA(r) matrixwould imply a mixture.While we focus on the non-degeneratingcase to ease deposition and to avoid repetition,relaxing the restrictionis straightforward.2 We assumethat there exists a so-called risk-neutralmeasure, or a martingale measure, 1P*,under which the bond price can be writtenas

(3) P(Xt, T) = E* exp (- r(X,)ds) TY, whereE*[-] denotesexpectation under measure I?*.Under certain regularity conditions, the existence of such a measureis guaranteedby no-arbitrage.The measureis uniquewhen the marketis complete. Referto Duffie (1992) for details. Let t* (Xt) denote the drift functionof Xt undermeasure I?*. The diffusionfunction E(Xt) remainsthe same underthe two measuresby virtueof Girsanov'stheorem. The necessaryand sufficientconditions for the quadraticclass are identified undermeasure IP*.

'For technical details, see, for example, Jacod and Shiryaev(1987). 2Wethank Richard Green, Burton Hollifield, and Stanley Zin for pointingthis out. Leippoldand Wu 275

Proposition 1. The necessary and sufficient conditions for the quadraticclass are given by: i) The instantaneousinterest rate r(Xt) is a quadraticfunction of Xt,

(4) r(X,) = XArXt+bTXt + cr, with Ar E IRnXn,br E In and cr E IR. * ii) The drift of the Markovprocess p (Xt) = a* + b*Xt,a* E IR,b* E IRnis affine in Xt. iii) The diffusion Z(Xt) _ S E IRnXnis a constant matrix. Refer to PartA of the Appendixfor the proof. Similarconditions are listed in Ahn, Dittmar,and Gallant(2001), Beaglehole and Tenney(1991), and El Karoui, Myneni, and Viswanathan(1992). We are the first to prove its necessity and sufficiency.Filipovic (2001) furtherproves that the quadraticclass representsthe highest orderof polynomialfunctions one can apply to consistenttime-separable term structuremodels.

B. The Pricing Kernel

While the conditions are specified under the risk-neutralmeasure IP*, for most empiricalapplications, it is imperativeto identify the Markovprocess under the objective measure I. To do so, we need to furtherspecify the stochastic process for the pricing kernel ,t, which relates futurecash flows, Ks, s E (t, T], to today's price,pt, by

TsKsds Pt -E -F

where E is the expectationunder the measureIP. Given certainregularity conditions, the conditions for the existence and uniquenessof the martingalemeasure are equivalentto that for the existence and uniquenessof the pricing kernel. One can performa multiplicativedecomposition on the kernel,

t = exp (- r(Xs)ds) Mr,

where the variableMt can be interpretedas the Radon-Nikodymderivative, which takes us from the objective measure I? to the risk-neutralmeasure IP*. We can furtherdecompose it into two orthogonalparts,

dIF* (ft t = - M dItd (- (Xs)Td(_-W) Tds)

where £ denotes the Doleans exponential3 and y(Xt) is an St-adapted process satisfying the usual regularityconditions and is often referredto as the market

3See Jacod and Shiryaev (1987) for a classic reference. 276 Journal of Financial and Quantitative Analysis price of risk on the Markovprocess X. We use Yt to denote some state vector orthogonalto Xt and7y its marketprice of risk. We leave the vectorprocess Y and its marketprice of risk unspecifiedas they do not affect our bond pricingresult. Fisher and Gilles (1999) label an independentvector similar to Y as neutrino factorsand illustratehow these factorscan affect the pricingof otherassets such as currenciesand stocks, even thoughthey have no effect on bond pricing. Furthermore,direct application of the Girsanovtheorem implies thatthe drift * of the Markovprocess undermeasures P and P are linkedby

- (5) I* (Xt) = P(Xt) (Xt).

Hence, for p* (Xt) to be affine in Xt, we requirethat the affine combinationof the driftjp(Xt) and the marketprice of risk 7(Xt) be affinein Xt. Obviously,there is an infinitenumber of combinationsthat result in an affine functionin Xt. In princi- ple, one can always augmentany arbitraryfunctions of Xt, sayf(Xt), in P(Xt) and at the same time add a counterpartin 7(Xt), S-~f(Xt), such thatthey cancel each other. The pricing of bonds will not be affected. In most empiricalapplications, one restrictsthat 1p(Xt) and 7(Xt) are non-triviallyaffine. By non-trivial,we mean that the functionsof Xt in tL(Xt)and 7(Xt) do not cancel each other. The canceling functionf(Xt) does not affect the term structureas the bond pricingrelation only depends on the risk-neutraldrift. It does, however,have an impact on the time-seriesproperties of interestrates. For example,if we incorporatea quadratic functioninto the drift p(X,) and cancel it out througha counterpartin the pricing kernel, interestrates are still quadraticfunctions of the state vector, yet the time-seriesproperties of the state vector are changed,as they are no longer nor- mally distributed,as impliedby an affinedrift. While not required,for tractability concerns,one often chooses 7(Xt) in such a way thatthe two functions,i(X) and p* (X) are of the same type so that the Markovprocesses are of the same type underthe two measures.In addition,the specificationof the marketprice of risk needs to satisfy certainno-arbitrage constraints. One can also exploit the indeterminacyimplied by equation(5) in practical applications.For example, one can specify the objectivedrift to matchthe time- series propertiesof interestrates while the risk-neutraldrift matches the cross- sectional property(the term structure)at each day. The differencebetween the two driftscan thenbe attributedto the marketprice of risk. The empiricalwork by Brandtand Yaron (2001) is analogousin spiritto this philosophy.The outstanding issue, then, is whetherthe implied marketprice of risk premiumis consistentor supportedby any economy.

C. Identificationof the QuadraticClass

For tractability,we adopt the "non-canceling"restriction and specify that both the drift p(Xt) and the marketprice of risk 7(Xt) are affine in Xt. In particular,under non-degenerating conditions and a possible rescaling and rotation of indices, we can take the Markovprocess in (1) to have the following simplest possible form,

(6) dXt = - Xtdt + dWt, Leippoldand Wu 277 where K E Rnxn controls the speed of mean reversion. Such a process is often referredto as a multivariateOmstein-Uhlenbeck (OU) process. We scale the process to have zero long-runmean and identityinstantaneous volatility: Z(Xt) = J.4 The affine marketprice of risk is specified as

(7) (Xt) - b,-A.Xt + with A. e IR[nxn and be E IRn. The drift of X underthe risk-neutralmeasure IP* is hence given by

P* (X) = -b.-(K+Ay) X. K*-= + A.Ycontrols the mean-reversionof the Markovprocess underthe risk- neutral measure IP*. The long run mean of the process under measure IP* is - (t +A)- by. For identificationpurposes, we furtherrestrict Ar to be symmetricwith no loss of generality as the asymmetricpart has zero contributionto the quadratic form. Furthermore,we restrict tKand AI, to be lower triangular. For the OU process X to be stationaryunder measure IP,we need all the eigenvalues of K to be positive, which amountsto a positivity constrainton the diagonalvalues of the lower triangularmatrix. Analogously, for Xt to be stationaryunder measure I?*, we need all the eigenvaluesof K* to be positive. Underthese specifications,the coefficientsof bondpricing {A(r), b(r), c(r) } 5 are determinedby the following ordinarydifferential equations (ODEs),

- (8) = Ar A () *-(*)T A (r) 2A (r)

ab(7) = br - 2A(r)bb- - (tK*)T b(r) - 2A (7) b (T), T

c(r) = Cr-b b(T)Tb. + trA (T) - b(r)Tb(T)/2, T subject to the boundaryconditions: A(0) = 0, b(0) = 0, and c(0) = 0. Finite solutions to the ordinarydifferential equations always exist for r 6 [0, T] with some fixed and finite T. We need furtherconstraint on the parametersto guarantee the existence of a stationarystate, i.e., the existence of finite solutions as T -+ oo. Closed-form solutions exist for one-factor and independentmulti-factor cases. See, for example, El Karoui, Myneni, and Viswanathan(1992) and Jamshidian (1996). Solutions for more general cases can readilybe computednumerically.

D. Properties of Bond Yields and Forward Rates

Under the non-cancelingrestriction between the drift ,u(X) and the market price of risk y(X), the Markov process has a constant diffusion matrix and an

4LettingX denote a process with a general affine drift p1(X)=a+bX and a general diffusion matrix Z(X) -= , the process in (6) is obtainedby the following linear transformation, X = S~-lX-b-la, with tK -E ~lbS. Obviously, if the short rate r is quadraticinX, it is also quadraticin its linear transformationX. The transformationis only for identificationreasons. 5Derivationsare available upon request. It follows from equation (19) in the Appendix, PartA. 278 Journalof Financialand QuantitativeAnalysis

affine drift underboth the objective measureIP and the risk-neutralmeasure IP*. * Giventhat the eigenvaluesof n and K arepositive, Xt is stationaryand distributed multivariatenormal both conditionallyand unconditionallyunder both measures,

E[Xt]=0,

V E [(Xt- I)2] = e-Se-S ds,

- Pt,T - E [XTITt] = (I e-) 0 + e-Xt

VTr E [(Xr-/it)2IFt] = e -e- ds,

where c is replacedby t* = + A^ undermeasure P* and 0 = 0 undermeasure 1 IPand- ( + Ar) b.r undermeasure P*. We dropthe subscriptt for the conditional variancesince it is independentof the currentstate Xt and is only a function of the time horizon r = T - t. Underthe quadraticclass, the yield y(Xt, r) to a zero-couponbond P(Xt, r) is given by

7 y(Xt, 7) = = (XTA(r)X + b(T) t + c(T)).

The instantaneousforward ratef(Xt, r) with maturityr is given by

r) = _0ln(Xr T-t) XT aA(r) b(r) x+ c(r) f(X, f(t-) Xt- t aT = Or 9r r+

Therefore, under the quadraticclass, both bond yields and forwardrates are quadraticforms of normalvariates, the propertiesof which are well-documented in the literature.Reviews of quadraticforms in normalvariables may be found, for example, in Holmquist(1996), Johnsonand Kotz (1970), Kathri(1980), and Mathaiand Provost(1992). In particular,the rth momentsand cumulants,as well as theirmoment-generating functions, are knownin closed form.

Property1. Let x be an n-dimensionalvector having the multivariatenormal distribution Nn(I, V), let Q(x) = xTAx, qi = xTAix, and Qk = HI=l xTAix = Hi= qi, whereAi are n x n matrices.Then i) The rth cumulantof Q(x) is

Kr = 2 r-(r- 1) ! [tr((AV)r)+rtr((AV)r-lAt,IT], r > 1.

ii) The moment-generating function of q = (q l, q2,... , qk)T is

E(e ) = In-Cl- /exp -l V- + (I - 1C) 1V where C = jk=i 2sjAjV. Leippoldand Wu 279

iii) The expectation of Qr = qlq2.. * qr is

T (r ~ r 1<2r-2i> 0). = E[Qr] (2r)! ( aj Snl 2r i!(2r- 2i)!' where ai = vecAi and 0'2 = vecV.

The vec operatorstacks the columns of the matrixinto a vector. () denotes the Kroneckerproduct. A denotes a Kroneckerproduct of the form: A - 0)i= A =A1 ( A2 ( ... Ar with the conventionthat A <0> = 1. The symmetrizer,

Snlr =- Pnlr(7r)/r!, 7r where the summationextends over all r! permutations7r in Sr, is a projectionop- eratorof the rth tensorpower of IRnonto the rth completely symmetricspace over RI".Actually, for any matrixA, xTAx = xT ((A +AT)/2)x since the asymmetric part ((A - AT)/2) of A gives zero contribution:xT ((A - AT)/2)x = 0. In the quadraticclass specification,we hence assume Ar being symmetricwith no loss of generality. The symmetrizerbecomes an identity matrix when the weighting matrixAi is symmetric. The fact that moments and cross-momentsof all ordersfor bond yields, forward rates, and bond prices exist in analytical forms illustrates how tractable quadraticmodels are. Such results on moment conditions not only facilitate our propertyanalysis for the purpose of model design but also simplify estimation, especially when the generalizedmethods of momentsare implemented. While the analyticaltractability of the quadraticmodel is comparableto that of the affine class, the two classes often imply differentbehaviors, as discussed below.

1. Nonlinearityin InterestRate Dynamics:The Role of the QuadraticTerm Numerousstudies have documentednonlinearities in the dynamics of interest rates. Examplesinclude Ait-Sahalia(1996), Pfann, Schotman,and Tschemig (1996), Conley, Hansen, Luttmer,and Scheinkman(1997), and Stanton (1997). The quadraticterm in bond yields and forwardrates providesa directmechanism to add nonlinearityto the dynamics. As an example, the following proposition illustrateshow a one-factorquadratic model can generaterich and nonlineardy- namics in terms of the autocorrelationfunctions of bond yields.

Proposition2. A one-factorquadratic model can generatei) a more slowly de- caying autocorrelationfunction than implied by an AR(1) process, and ii) a rich (upwardor downwardsloping) term structureof persistencefor bond yields and forwardrates.

Refer to Part B of the Appendix for the proof. Intuitively, a one-factor quadraticmodel can be thought of as a two-factoraffine model with one factor being the original Gaussian factor and the other factor being the square of the Gaussianfactor. The autocorrelationfunction of bond yields is hence a weighted 280 Journal of Financial and Quantitative Analysis

averageof these two factors. Rich dynamics are generatedthrough the interac- tion between these two factors. For example, to match an autocorrelation,p, of an AR(1)-typeprocess, the autocorrelationof the Gaussianfactor needs to be set higherdue to its weightedaverage with the square.Yet, the decay of the weighted average is dominatedby the Gaussian factor and is hence slower than that of the AR(1) process. Furthermore,the relativeweight betweenthe Gaussianfactor and its squareis determinedby theirrespective coefficients A(r) and b(r) and is thereforematurity dependent. Thus, the autocorrelationcan be differentfor yields of differentmaturities due to the relativeweight change althoughthe whole yield curve is drivenby merely one Gaussianfactor. The propositionillustrates that even a one-factorquadratic model can generate a rich dynamicsfor the autocorrelationfunction and a non-trivialterm structure across maturities. Such featurescannot be obtainedfrom one-factorAR(1) type models. Withinthe affine class, one often uses multiplefactors to generate the observednonlinearities in the interestrate dynamics. In contrast,nonlinearity is intrinsicallybuilt into the quadraticmodel. Most recently,Chapman and Pearson(2000) and Duffee and Stanton(2001) find that econometricproblems make even linearmodels look nonlinearin small samples and thus cast doubt on the robustnessof the previousevidence on non- linearities. Nevertheless, the rich dynamics generatedby the quadraticmodel illustratesits flexibilityfor model design.

2. AffineMarket Price of Risk

Flexible forms for the marketprice of risk have been provento be vital in capturingthe dynamicbehavior of the expectedexcess returnsto bonds. For example, Duffee (2001) finds thatthe complete affine models of Dai and Singleton (2000) performpoorly in forecastingfuture changes in Treasuryyields in particular becausethe marketprice of risk is restrictedto be a multipleof the diffusionof the statevector. The performancecan be greatlyimproved by using statevariables thathave constantdiffusions (e.g., the Ornstein-Uhlenbeckprocess) and applying a generalaffine marketprice of risk to such variables.Dai and Singleton (2001) confirmthat such a specificationadds greatease in capturingboth the mean yield curve and the anomaliesin the expectationhypothesis. Quadraticmodels com- bine Gaussianstate variablesand marketprice of risk into a naturalframework and hence are poised to performwell in capturingthe dynamicsbehavior of expected excess returnsto both bonds and currencies. Furthermore,if one intendsto incorporatean affinemarket price of riskin the affine class, the diffusionof bond yields is forced to be constantas one is forced to apply the Ornstein-Uhlenbeckprocess as the state variable,6unless a separate square-rootfactor is incorporated.In contrast,under the quadraticclass, although one is also using the Omstein-Uhlenbeckprocess, the quadratictransformation generatesstate-dependent diffusion for bond yields and forwardrates. For example, the diffusionterm of the bond yield y(Xt, r) is given by (2A(r)Xt + b(r))/Tr,

6As discussed before, this is a must only when one intendsto exclude the canceling functions and to retainthe affine structurefor the drift underboth measures. Leippoldand Wu 281 which is affine in Xt and hence state dependent. Similarly, the diffusion of the forwardrate is also affine in Xt.

3. The Virtueof Ornstein-UhlenbeckProcesses Underthe quadraticclass, the Markovprocess follows a multivariateOrnstein- Uhlenbeck(OU) process. It has been foundunder different circumstances that the OU process rendersone with more flexibilities in matchingsalient featuresof the interest rate data than the square-rootprocesses also used in affine models. For example, Dai and Singleton (2000) and Backus, Foresi, and Telmer (2001) both observethat the unconditionalcorrelation between two square-rootstate variables can only be positive, a restrictionthat runs against evidence. The correlations between OU processes, in contrast,have no such restrictionsand can be either negativeor positive. It has been found that flexible correlationstructures between state variables significantlyimprove the model's empiricalperformance. For example, one often observes a hump shape in the mean term structureof the conditional variance of interest rates. Backus, Telmer, and Wu (1999) find that strong interactions between state variablesare necessary to generate such hump dynamics. While a multi-factorcorrelated square-root process can generatea humpshape, experience indicates that the resultinghumps are often not large enough to fit the data. The flexible correlationstructure for the multivariateOU process make it a natural choice in capturingsuch conditionaldynamics.

4. PositiveInterest Rates In regardto limits aboutthe square-rootprocess, the OU processhas regained its popularityin empiricalapplications within the affine framework.See the most recent applicationsin Backus, Telmer, and Wu (1999), Duffee (2001), and Dai and Singleton (2001). In addition, Backus, Foresi, Mozumdar,and Wu (2001) find that some of the limitations of the square-rootprocess can be mitigatedby using a negative square-rootprocess as a state variable. However,affine models with either OU processes or negativesquare-root processes imply positive probabilities of having negativeinterest rates. While it may be a worthwhilesacrifice if the empiricalperformance of the model can be significantlyimproved and if the real probabilityof havingnegative interest rates, albeit positive, is small given ap- propriatechoices of parameterestimates, some practitionersand academics alike hold strong opinions against term structuremodels that imply negative interest rates. Similar concerns also arise when one uses the term structureanalogue to model stochastic volatility in currenciesand stocks (see Section IV). These concerns can be partiallyrelieved under the quadraticclass by restrictingA r to be positive definite and by setting Cr= 4AbrTArlbr. Under such a restriction,the lower bound for the instantaneousinterest rate is zero. Pan (1998) guarantees that the lower bound of all interest rates is zero by furtherrestricting b(r) = 0 and c(r) = 0 for all r. Such restrictions,however, lead to a degenerationof the quadraticstructure such that it is equivalentto a parameterizedone-factor affine model. 282 Journalof Financialand QuantitativeAnalysis III. Asset Pricing

In this section, we first extend the bond pricingresult to assets with general exponential-quadratictype payoffsand then apply the resultto the pricingof state- contingentpayoffs.

A. Assets with Exponential-Quadratic Payoffs

Consideran asset which has the following exponential-quadraticpayoff struc- tureat time T,

(9) exp (-ql(XT) - f q2(X)ds),

whereqj(X) denotesa quadraticfunction of X, namelyqj(X) = XTAjX+bfTX+ cj. The quadraticform qj(Xt) can eitherbe regardedas interestrates (bond yield or forwardrate) or ratesof returnon otherassets. The integralcan be regardedeither as an averagerate in Asian style payoffsor as a cumulationof continuouspayoffs. Forexample, if we let q (XT)=TY(XT, r) andq2 =0, the payoff would be a r-year zero-couponbond; if we set q I (Xt) = -Ty(XT, T) and q2 = 0, the payoff would be equivalentto a gross returnon a r-year simple rate,

1 + TRT = eTy(XTT).

Due to the additivityof quadraticforms, we can also regardq 1 and/orq2 as linear combinationsof manydifferent interest rates (quadraticforms). We show that assets with such generalpayoff structurescan be priced analytically underthe quadraticframework.

Proposition3. Underthe quadraticclass, the time-t price of an asset with a payoff functionas in (9) is exponential-quadraticin Xt,

(10) (ql + q2,r) E* exp (- r(Xs)ds)

- j x exp -ql (XT) q2(X,)ds) St

= - exp (-XTA(r)Xt b(r)TXt - c(r)).

The coefficientsA(r), b(r), and c(r) satisfy the ordinarydifferential equations in (8) with boundaryconditions A(0) = A 1, b(0) = bl, and c(0) = cl and with {Ar, br, r} being replacedby {Ar + A2, br + b2, Cr + C2}.

The proof is given in PartC of the Appendix. Note that the price of a zero- couponbond is just a degeneratingspecial case of the generalpayoff structurein (9) with q -= q2 = 0. Leippoldand Wu 283

B. State-ContingentClaims

Now consider the time-t price of a contingentclaim that pays exp(-q i(XT) at time T in case qj(XT) < y is true for some fixed numbery,

(11) Gq,qj(y,r) E* exp (-j r(Xs)ds) e-q(XT)Iq(x)

1. FourierTransform Method Let Xq,,q,(z) denote the Fouriertransform of Gqi,qj(y) defined as

f+00 (12) Xqe,qi(Z) ]- elZYdGqi,q{(y), z E R, -00

where we omit the second argumentin r in the state prices and their transforms in case no confusion occurs. The following propositionderives a closed-form solution for this transform. Proposition4. Underthe quadraticclass, the Fouriertransform of the state price Gqi,qj(y), defined in (12), is equivalentto the price of an asset with exponential- quadraticterminal payoffs,

Xqi,q (z) = (qi - izqj).

Proof 1. The result is obtainedby applying Fubini's theoremand applying the result on the Fouriertransform of a Dirac density. o

The Fouriertransform of the state price Gq,qjq(y) can be regardedas an asset price characterizedin (10). Here, of course, the term asset price has to be used with caution since the asset has a complex valued payoff function. But 284 Journalof Financialand QuantitativeAnalysis

more importantly,Proposition 4 implies that the Fouriertransform of the state- contingentclaim retainsthe exponential-quadraticform and hence the tractability of the quadraticclass. Given the Fourier transform Xqi,qj(z) = (qi - izqj), the state price Gq1,qj(y) can be obtainedby an extendedversion of the Levy inversionformula, which we prove in the Appendix,Part D.

Proposition5. The state price Gqi,qj(y) is given by the following inversionformula,

Gqi,qj(y) =- Xq,qj(0)+ 1 o eiZyxqiq,(-z) -e-iZXqi,qj (z) ^ W 2 2 o iz

The above inversionformula involves a numericalintegration, similar to the numericalvaluation of the normalcumulative densities in the Black-Scholesfor- mula. The prices of many existing fixed income derivativescan be expressedas functionsof the generalstate price Gqi,qj(y). We can thereforeprice them through the inversionformula given in Proposition5. Duffie, Pan, and Singleton (2000) apply a similarapproach to asset pricingunder an affinejump-diffusion environ- ment. The key advantageof such a transformmethod for derivativepricing is its great computationalefficiency. In particular,regardless of the dimensionof the state space, we only need one numericalintegration for the inversion.In contrast, methods based on Arrow-Debreuprices7 in, for example, Beaglehole and Ten- ney (1991) and Foresi and Steenkiste(1999), requireat least as many numerical integrations(and in general more than) as there are state space dimensions. In these methods, one first prices Arrow-Debreusecurities, which are claims with a Dirac functiontype payoff. Prices of general state-contingentclaims are then calculatedby integratingthe Arrow-Debreuprice-weighted cash flows over con- tingentstates. In a generalaffine or quadraticframework, the Arrow-Debreuprice is obtainedin a way analogousto our transformmethod. Furthernumerical inte- grationsover the statespace, and sometimesthe time space,need to be performed, finally yielding the state-contingentclaim prices. Obviously,for state-contingentclaims of the type in (11), such a procedure is deliberatelyinefficient. A more direct approachlike ours is called for. On the otherhand, for claims with more complex payoffs that cannotbe representedas a simple functionof Gqi,qj(y), one may need to resortto the Arrow-Debreuprice approachor othernumerical procedures.

2. GeneralizedFourier Transform and FFT Traditionalnumerical integration methods for the inversionin Proposition 5, such as the quadraturemethod used in Singleton (1999) and Duffie, Pan, and Singleton (2000), can be inefficient due to the oscillating natureof the Fourier transform.Instead of workingwith the inversionformula in Proposition5, we can also cast the problemin a way such that we can applythe fast Fouriertransform

7They are also referredto as Green'sfunctions of the state process. Leippoldand Wu 285

(FFT) and thus take full advantageof its considerableincrease in computational efficiency. For this purpose,let Pqi,',(z) denoteyet anotherFourier transform of Gqi,qj(y) defined as

(13) oqji,q(z) J eZYGqi,,(y)dy, z E C C C. J-00 Comparingthe two Fourier transformsdefined in (12) and (13), we see that Gqi,qj(y)is treated as an analogue of a cumulative density function in Xqi,qj(Z) while it is treatedas a probabilitydensity in oq,,qj(z). But more importantly,the transformparameter z in (13) is extendedto the complex plane with C being the complex domain of z where goqi,qj(z)is well-defined. A Fouriertransform that extends to the complex plane is often referredto as the generalized,or complex, Fouriertransform. Refer to Titchmarsh(1975) for a comprehensivetreatment. As it turns out later, such an extension and the choice of the complex domain are critical for the applicationof the FFT algorithm. Proposition6. Under the quadraticclass, the generalizedFourier transform of the state price Gqi,qj(y)defined in (12), when well-defined,is given by

= -i(qi- izqj). poqj,qj(z) Z The result is obtainedvia integrationby partsand Proposition4, +00 r . i eiZY 1 - qi,qj(Z) = Gqi (y) _ elZYdGqi,qJ(y)= ). iz 00 iz O0 Z Since Gqi, (oo) = V/(qi) > 0, the limit term is well-defined and vanishes only when Im z > 0. In general, the admissible domain C of z depends on the exact payoff structureof the contingentclaim. Table 1 presentsthe generalizedFourier transformsof various contingent claims and their respective admissible domain for the value of z. Similarly, they are derived via integrationby parts and by checking the boundaryconditions as y -> ±oo. Carrand Madan(1999) consider the special case of pricing a call option on stocks and refer to the imaginarypart of z as the dampeningfactor as the call option price needs to be dampenedfor its transformto be finite. Let z = Zr+ izi, where Zrand zi denote, respectively,the real and imaginary part of z. Let gp(z)denote the generalizedFourier transform of some state price functionG(y), which can be in any of the forms presentedin Table 1. Then, given that yo(z)is well-defined,the correspondingstate price function G(y) is obtained via the inversionformula, 1 fizi+00 G(y) = 2 ee'z (z)dz. iZi-00 This is an integralalong a straightline in the complex z-plane parallelto the real axis. zi can be chosen to be any real numbersatisfying the restrictionin Table 1 for the correspondingstate price function. The integralcan also be writtenas GZiy r°° - G(y) - eZ iZ'Yp(zr + izi) dzr, 7 J 286 Journalof Financialand QuantitativeAnalysis

TABLE1 GeneralizedFourier Transforms of VariousContingent Claims (a, f3,a, b are real constants witha < 3)

Contingent GeneralizedTransform Restrictions Claim -izp(z) on Imz - Gqi,qj(Y) i(q izqj) (0, 0o) Gqi,qj(-y) (qi + izqj) (-00, O) - eaYGqi,qj(y) b(qi (a + iz)qj) (a, oo) e3y Gqi,j (-y) /(q, + (1/ + iz)qj) (-oo, 3) ae'YGqi,j(y) a¢(qi - (a + iz)qj) +be3yGq,,qm(-y) +b((q, + (3 + iz)qm) (a,/3) which can be approximatedon a finite intervalby

ziyN-I + (14) G(y) E e-r(k)Y(Zr(k) ii) AZr, k=0 where zr(k) are the nodes of Zrand AZrthe grid of the nodes. Recall thatthe FFT is an efficientalgorithm for computingthe discreteFourier coefficients. The discreteFourier transform is a mappingof f = (fo,... ,fN-)T on the vector of Fourier coefficients d = (do,... , dN- 1)T, such that

N-1 (15) dj = Efke jkNi j=O,1,... N-1. k=O

FFT allows the efficient calculationof d if N is an even number,say N = 2m, m E N. The algorithmreduces the numberof multiplicationsin the requiredN summationsfrom an orderof 22mto thatof m2m-1,a very considerablereduction. By a suitable choice of AZr and a discretizationscheme for y, we can cast the approximationin the form of (15) to take advantageof the computingefficiency of FFT. For instance,if we set zr(k)= rk andyj= -b+ Aj, andrequire rA = 27r/N, we can cast the stateprice approximationin (14) into the form of FFT summation in (15),

N-1 G(yj) N Efke-jk Ni j = o, i, N- , k=O with

fk = Nezy+ibzr(k) r((Zr(k) + iZi). 71

Undersuch a discretizationscheme, the effective upperlimit for the integrationis a = Nr7,the rangeof strikelevel y is from -b to NA - b, with a regularspacing of size A. The restrictionthat r]A= 27/N indicatesthe trade-offbetween a fine grid in strikeand a fine gridin summation.Thus, if we choose r/ small to obtaina fine gridfor the integration,then we can only computestate prices G at strikespacings Leippoldand Wu 287 that are relativelylarge. To increase the accuracyof integrationwith relative fine grids on strikey, one can also incorporateSimpson's rule into the summation,

Z 1-k(-1)'--j, G(yj) Efke-'kN-i 3 k= where An is the Kroneckerdelta functionthat is unity for n =0 and zero otherwise. See Carrand Madan (1999) for an applicationto pricing call options on stocks. Obviously,such an algorithmcan also be appliedto the affine economy. Applica- tion of the FFT algorithmdrastically increases the computationalefficiency as one can obtain option prices on the whole spectrumof strikes with merely one FFT transformation.In particular,in some computinglanguages (such as MATLAB) that allow for vectorizingthe FFT algorithm,option prices on the whole surface of strike and maturityare obtainedat one stroke.

IV. Examples and Applications A. Examples of Fixed Income Derivatives 1. EuropeanOptions on Bonds Let Ct denotea Europeancall optionat time t on a zero-couponbond P (X t, Tp) of maturityrp. Let Tc denote the maturityof the call option, and K the strike price. Since the bond price P(XT, Tp) has a quadraticform, we write P(XT, Tp) - e-q(T,)(XT). Then, from equation (11) we obtain

Ct = Gq(rp),q(rp)(-ln K, Tc) -KGo,q(rp) (- In K, Tc). Similarly,the price of a put option on the same bond with the same maturityTc and strikeprice K can be writtenas

Pt = KGo,-q(p) (In K, Tc)-Gq(rp),-q(rp) (n K, rc). The pricing formulaefor caps and floors take a similar structureas they can be writtenas options on bonds.

2. Exchange Options The payoff of an exchange option on zero-couponbonds can be writtenas - (mlP(XT, r1) m2P(XT, T2))+,

which denotes the rightto exchangem2 bonds with maturityr2 for ml bonds with maturityTr at time T = t + r. Again, straightforwardapplication of the state price definitionyields the price of such an exchange option,

= l M(t,rTi, T2) mlGq(r,),q(r')-q(r2) (In ~) -m2Gq(T2),q(,)-q(r2) (In m

Note that given the exponential-quadraticform for the bond prices, the exercise condition mlP(XT, Ti) > m2P(XT, T2) implies

[q(rl) - q(T2)] (XT) < In m. m2 288 Journalof Financialand QuantitativeAnalysis

3. Asian Options Consideran Asian-stylecall option whose payoff dependson the path average of the fixed maturity (rp) bond yields over the maturity (rc = T - t) of the option,

(exp (-- /y(X, p)ds) -K).

The price of such an option has the representation,

CA = - Gf q(rp),f q(rp)(ln K, Tc) KGo,f q(rp)(ln K,r c). An Asian put option is pricedin very much the same way,

= ln - Pt KGo,- f q(rp)(- K, Tc) Gf q(rp), f q(rq)(-ln K, Tc) Given the path dependenceof the Asian payoff, the tractabilityof the pricing relationis remarkable.

B. Option Pricing under Quadratic Stochastic VolatilityModels

As the affinejump-diffusion framework has been widely appliedto stochastic volatility modeling,8 we illustratehow the quadraticframework can also be appliedto model stochasticvolatility. Let S denotethe price of an asset (stock or exchangerate), which is assumed to satisfy the following stochasticdifferential equation under measure P *,

dSt/St = (r - 6)dt + /vd(t)Zt, where Zt denotes a standardscalar Brownian motion, r the instantaneousinterest rate,and 6 the continuouslycompounded dividend yield for stocks andthe foreign interestrate for currencies. To be consistent with the quadraticframework, we assumethat both r and 6 are quadraticfunctions of Markovprocess X. The instantaneousvariance rate of the process is denotedby v(t). We allow it to be stochasticand model it by a quadraticfunction of the Markovprocess Xt: v(t) - v(X,),

v(Xt) = XtAvXt + b Xt + cv. Similarto instantaneousinterest rate, positivity of variancerate can be guaranteed easily by a simple parametricconstraint. Furtherassume that Zt is independentof the Brownianmotion vector Wt in Xt. The generalizedFourier transform of the log returns T =ln ST/St overmaturity T = T - t is given by = (16) s (u) E* [e'uS-It], u c C

= E* exp (J (iur(X,) - iu6(X,) - \v(Xs))ds) ,Yt

8Prominentexamples include, Bakshi, Cao, and Chen (1997), Bates (1996), (2000), and Heston (1993). Leippoldand Wu 289 where A = (iu + u2) /2. The last line is obtainedby the principleof conditional expectation. Note thatby redefiningan interestrate ?(X,)= -iur(Xt)+iu6(Xt)+Av(Xt), the transform , (u) has the same form as the bond pricingformula in (3) and with the same boundarycondition: Pb(u) = 1 at r = 0. We can thus solve the transforman- alytically as an exponential-quadraticfunction of Xt, with coefficients determined by the series of ordinarydifferential equations in (8). Under such a setup, interestrates, dividendyields (or foreign interestrates), and stochasticvolatility are tightly linkedtogether by the Markovprocess Xt. Such a tight link, however, can be broken easily, if necessary,by expandingthe state vector and assuming that r, 6, and v are each a quadraticform of a subsect of the state vector. The subsect can be orthogonalor overlapping,depending on the requiredcorrelation structure. Alternatively, as is common practicefor option pricingon stocks and currencies,we can simply assumeconstant interest rates and dividendyields and factorout the termexp(-(r - 6)7r).The residualexpectation still has the bond pricing form with interest rate redefined as 7(Xt) = Av(Xt). An analytical solution is readily obtained. Empirically,Bakshi, Cao, and Chen (1997) have found thatincorporating stochastic interest rate does not significantly improvethe model's performancein pricing S&P 500 index options. Given the transform Ps(u) on asset returnsand analogous to the previous section, we can derive transformson many state-contingentclaims with the asset as the underlying. In particular,consider the time-t price of a contingent claim that pays exp(-bs,) at time T in case cs, < y is truefor some fixed numbery, * (17) Gb,c (y) e-e [e- bsTIcsT

(b,c(Z) j eizyGbc(y)dy = _e-r`s(zc+bi), z E C C,

where s (-) is the Fouriertransform of the asset returnsr defined in (16). The proofs are analogousto those for Propositions4 and 6. If we keep the quadratic interest rate assumption,the interest rate term will be absorbedinto a modified transformOs(-). In solving the coefficients for the modified transform s(.), we need to modify the interestrate termyet again: r(Xt) = r(Xt) + I(Xt). Given the two transforms,state price Gb,c(y) can be solved numericallyby either of the two inversionmethods proposedin the previous section. Many Eu- ropeanstyle options can be writtenin terms of Gb,c(y). For example, the price of a call option on the asset with strikeK can be writtenas

Ct = StG-1,-i (ln(St/K))- KGo,_1(ln(St/K)). The price of a put option with the same strikeK is given by

Pt = KGo,, (-In(St/K))-StG_l,1 (-In(St/K)). 290 Journal of Financial and Quantitative Analysis

C. Estimation of Quadratic Models

Regardingts(u) in (16) as the characteristicfunction of the returns r, we can invertit to obtainthe conditionaldensity of the asset return.An analyticalform for /s(u) is hence also useful for maximumlikelihood calibration of the model. Char- acteristicfunctions for bond yields and forwardrates can also be obtainedfrom = Proposition 10 by setting r(Xs) = q2(Xs) 0 and letting ql (XT) = -iuy(XT, Tp).9 For example, Singleton (1999) exploits the knowledgeof / to derive maximum likelihoodestimators for affinemodels. He obtainsthe conditionaldensity via in- vertingthe characteristicfunction. Chacko (1999) andChacko and Viceira (2000) also propose a spectral generalizedmethods of moments estimationtechnique based on the characteristicfunction. However,due to the nonlinearrelation between yields and the state vector underthe quadraticframework, identifying the state variablesfrom the yields be- comes a more challengingtask. That also limits the applicationof the maximum likelihood calibrationas the conditionaldensities derivedabove are conditional on the state vector.

V. Conclusion

We identify and characterizea class of term structuremodels and price assets with generalpayoff structuresunder such a class. In particular,we propose two transformmethods for efficiently pricing a wide varietyof state-contingent claims. The transformmethods can also be appliedto econometricestimation and to optionpricing on othersecurities, such as currenciesand stocks, with quadratic stochasticvolatilities. These resultslay a solid foundationfor futureempirical ap- plicationsof the quadraticclass to term structuremodeling, fixed income derivatives pricing,and asset pricingin general.

Appendix

A. Proof of Proposition 1

Assume that the bond pricingformula under the risk-neutralmeasure yields finitebond prices,

- P(X,, r) = E* exp r(Xs)ds) t .

Applyingthe Feynman-Kacformula gives

) (18) r(Xt)P(Xt, ) = (X, + A*P(Xt, T),

9The unconditionaldensity can also be obtained similarly by letting T -+ oo, given that a stationarystate exists. Characteristicfunctions under the objective measurescan also be obtainedanalo- gously by replacing/* (Xt) with IL(Xt)in the partialdifferential equation in (19) or by settingA? and b, to zero in the ordinarydifferential equations in (8). Leippoldand Wu 291 where A* denotes the infinitesimalgenerator on Xt undermeasure IP*,

A* P _t (Xt, 7T) - (xt) A* (Xt)

i1 j= 1 - ii where the subscriptij denotes the (i,j)th element of the matrix in the bracket. Assume thatindeed P(Xt, r) has an exponential-quadraticform as in (2), since the instantaneousinterest rate r(Xt) is assumed to be well-definedby continuity,the exponential-quadraticform for the bondprice implies thatr(Xt) is also a quadratic function of Xt. Evaluatethe partialderivatives of the bond price P(Xt, r) in (2), plug them into the partialdifferential equation in (18), and rearrange,we have

'A ( (19) r(Xt) = X Xt or

+ 0( X) Xt + ( -[2A (T) Xt ()]bb * (Xt)

- - b + E E [2A (T) [2A (T) Xt+ (T)][2A (T)Xt b ()T] i=l j=1

X [a (Xt) a (Xt) ] ij for all T <T and for all X E P. Under mild non-degeneracyconditions (e.g., A(r) being nonsingular),equation (19) and the principleof matchingimply that i) a(Xt)ur(Xt)Tis a constantmatrix, independent of Xt. ii) p* (Xt) is affine in Xt. This providesthe necessity part. Conversely,suppose that p,* (Xt) is affinein Xt and a(Xt) is a constantmatrix. Considerthe candidateexponential-quadratic function for the bond price given in (2) for some A(r), b(r), and c(r). If we can choose A(r), b(r), and c(T) so that (19) is satisfied, then the bond price will indeed be exponential-quadraticin Xt. Given that we have a finite solution to the ordinarydifferential equations in (8), there is indeed a solution for A(r), b(r), and c(T) satisfying (19), implying that the bond price is exponential-quadraticin Xt, as in (2). This provesthe sufficiency part. o

B. Proof of Proposition 2 Under the specificationof a one-factorquadratic term structuremodel, the varianceand auto-covarianceof bond yields y t with maturityr are given by

var(yt) 2 (A ) + (b(T)) V

cov(yt,yt) = o2n2 (A()v) + n (() 292 Journalof Financialand QuantitativeAnalysis

where0 = e -h denotesthe autocorrelationof the Markovprocess X with discrete intervalh. The nth orderautocorrelation function is definedas = p(n) V(yt+nh yt) var(y)t Straightforwardmanipulation yields equation - a(-T)2n + b(T)n = p(n), with the weights given by

a2(r) = 2(A(r)V)2 b(rr) b(T)2V 2(A(r)V)2 +b(T)2V' 2(A(r)V)2+b(T)2V Note that the weights a(r) and b(T) are positive and sum to one. In case of the shortrate, we replaceA(r)/r and b((T)/T with Ar and br. i) For any AR(1) type process,the nth-orderautocorrelation, PAR (n), is equal to the nth-powerof its first-orderautocorrelation, PAR(1 )n: PAR(n) = PAR(l)- Lettingp(n) denotethe nthorder autocorrelation of the one-factorquadratic model, we claim that,for any ordern > 1, given thatPAR(n) = p(n), we have PAR(2n) < p(2n). To see this, we computethe differencebetween the two,

p(2n) - PAR(2n) = p(2n) - PAR(n)2

= a(r)4 + b(T)q2 - (a(T)2 + b(T)r )

= a(T)b(T) (02n - 0n)2 > 0, which is always greaterthan zero since the weights a(r) and b(T) are positive. Since this result holds for any n > 1, it implies that the autocorrelationfunction of an AR(1) specificationdecays faster than implied by a quadraticone-factor model. ii) The nth orderautocorrelation is determinedby the autocorrelationof the statevariable X andthe relativeweight a(r) andb(r) = 1-a(r), which dependson the maturityof the bondyield. The termstructure for the nthorder autocorrelation is upward(downward) sloping if a(r) decreases(increases) with r. r

C. Proof of Proposition 3

First,due to the additivityof quadraticforms, we can rewritethe expectation in (10) as

- - (ql + q2, r) E* exp W(Xs)ds exp(-qI (XT)) St , wherei(Xs)= r(Xs)+q2 (Xx) retains the quadraticform of the instantaneousinterest rate. Applyingthe Feynman-Kacformula gives

7(Xt) (.T)- = at(, T)+ ( Leippoldand Wu 293 an equation analogous to (18). Assume that indeed tb(-, r) has an exponential- quadraticform as in (10), the partialdifferential equation is reducedto

T 'OA (r) - ' (r)-clb 9C (T) (20) 7(X) = X, A (r X + ) ab X, +

+ [2A(r)Xt + b (r)]T [n*Xt+ b,]

-2 nE -E b) [2A ()t+b )][2A (r)Xt + b (r)]T] i=1 j=1 which has the same form as (19), with r(Xt) replacedby i(Xt) and with z*(Xt) - -b~ + c*Xtand cr(X) = I replacedby their respectiveparametric specifications. Collecting terms, we obtain the same ordinarydifferential equations as in (8), with only a substitutionof {Ar, br, Cr} by {Ar + A2, br + b2, Cr + c2} to reflect the instantaneousinterest rate adjustment.The boundaryconditions also differ to reflect the terminalpayoff difference:A(0) = A i, b(0) = bl, and c(0) = cl.

D. Proof of Proposition 5

To prove the inversion formula for state prices, we follow the proof of the inversionformula for cumulativedensity functions. See, for example, Chapter4 of Alan and Ord (1987). The only differenceis that the limit of the state prices is given by limu-+oo Gq ,qj(u) = b (qi) while the limit of a cumulativedensity goes to unity,limu_+oo F (u) = 1. We also need the following results,

Z eiZ1l e- -e-izy -2 in szygny, dz = = sgny, 7Ir J Iz Jo z lim = 0, Gqi,qj} (u) U->-00

/oo ry f0o - sgn (u y) dGqi,qj(u) = - dGqi,qj(u) + j dGqi,qj(u) 00 - = (qi) 2Gqi,q (y).

For a positive numberc, the uniformlyconvergent integral

Ic -qq Iz d 27r Iz c - =1 eiY f00 e dGqi,qj (u) e eidGqi,q (u) dz

00 - 1 J (u-y) ez(uy) ,q d j ( = C_ e-iz(- )-eiz Y)dGqi,qj (u) dz

1 .c 2sinz(u-y) (u z 27r n dJ-ino 294 Journal of Financial and Quantitative Analysis

Because the integralis uniformlyconvergent, we may change the orderof inte- grationto obtain

Ic = C-2sinz(u Y)dzdGq, (u).

The integralwith respectto z is continuousand bounded.We may thereforelet c tend to infinityto obtain

1 r00 - lim Ic = - -2 sgn (u y) dGqi,qj(y)

=- [0 (qi)- 2Gq,,qj(y)].

We thereforehave the resultin Proposition5. Furthermore,note that Xqi,q,(z)and Xqi,qj(-z) are conjugatequantities and hence, if Z(z)and I(z) are the real and imaginaryparts of Xqi,, (z), we have - cos = Xq,q (0) 1 f0 R(z) sinyz 1(z) yz d Gqiqjy 2qij (0 - _

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