A QUANTUM FIELD THEORY TERM STRUCTURE MODEL APPLIED TO HEDGING

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1 Inernaional Journal of Theoreical and Applied Finance Vol. 6, No. 5 (2003) c World Scienific Publishing Company A QUANTUM FIELD THEORY TERM STRUCTURE MODEL APPLIED TO HEDGING BELAL E. BAAQUIE and MARAKANI SRIKANT Deparmen of Physics, Naional Universiy of Singapore, Ken Ridge, Singapore phybeb@nus.edu.sg MITCH C. WARACHKA School of Business, Singapore Managemen Universiy, 469 Buki Timah Road, Singapore Received 8 November 2002 Acceped 12 February 2003 A quanum field heory generalizaion, Baaquie [1], of he Heah, Jarrow and Moron (HJM) [10] erm srucure model parsimoniously describes he evoluion of imperfecly correlaed forward raes. Field heory also offers powerful compuaional ools o compue pah inegrals which naurally arise from all forward rae models. Specifically, incorporaing field heory ino he erm srucure faciliaes hedge parameers ha reduce o heir finie facor HJM counerpars under special correlaion srucures. Alhough invesors are unable o perfecly hedge agains an infinie number of erm srucure perurbaions in a field heory model, empirical evidence using marke daa reveals he effeciveness of a low dimensional hedge porfolio. Keywords: Bond porfolio; hedging; field heory model; variance minimizaion. 1. Inroducion Applicaions of physics o finance are well known, and he applicaion of quanum mechanics o he heory of opion pricing is well known. Hence i is naural o uilize he formalism of quanum field heory o sudy he evoluion of forward raes. Quanum field heory models of he erm srucure originaed wih Baaquie [1] and he hedging properies of such models are analyzed in his paper. The inuiion behind quanum field heory models of he erm srucure sems from allowing each forward rae mauriy o boh evolve randomly and be imperfecly correlaed wih every oher mauriy. As poined ou in Cohen and Jarrow [8], his may also be accomplished by increasing he number of random facors in he original HJM owards infiniy. However, he infinie number of facors in a field heory model are linked via a single funcion ha governs he correlaion beween forward rae mauriies. Thus, insead of esimaing addiional volailiy funcions in a mulifacor HJM framework, one addiional parameer is sufficien for a field heory model o insill imperfec correlaion beween every forward rae mauriy. As he correlaion 443

2 444 B. E. Baaquie, M. Srikan & M. C. Warachka beween forward rae mauriies approaches uniy, field heory models reduce o he sandard one a facor HJM model. Therefore, he fundamenal difference beween finie facor HJM and field heory models is he minimal srucure he laer requires o insill imperfec correlaion beween forward raes. Secion 4 demonsraes he challenge o reliably esimaing volailiy funcions in a mulifacor HJM model. However, i should be sressed ha field heory models originae from he finie facor HJM framework wih he Brownian moion(s) replaced by a field. As seen in he nex secion, crucial resuls such as he forward rae drif resricion are generalized bu remain valid for an infinie facor process. The imperfec correlaion beween forward raes in a field heory model addresses he heoreical and pracical challenges posed by heir finie facor counerpars. Despie he enormous conribuion of he finie facor HJM mehodology, an imporan dilemma is inrinsic o he finie facor framework. For a finie facor model o correcly mach he movemens of N forward raes, an N facor model is required whose volailiy parameers are difficul o esimae once N increases beyond one (Amin and Moron [15] and Sec. 4). However, field heory models only require an addiional parameer o describe he correlaion beween forward rae mauriies. Consequenly, field heory models offer a parsimonious mehodology o ensure he evoluion of forward raes is consisen wih he iniial erm srucure wihou relying on a improbably large number of facors or eliminaing he possibiliy of cerain erm srucures (Björk and Chrisensen [4]). More precisely, a field heory model does no permi a finie linear combinaion of forward raes o exacly describe he innovaion in every oher forward rae. From a pracical perspecive, an N facor erm srucure model implies he exisence of an N dimensional basis for forward raes ha allows N bonds wih disinc mauriies o be arbirarily chosen when hedging. However, in he conex of a field heory model, hedging performance depends on he mauriy of he bonds in he hedge porfolio as illusraed in Sec To summarize, he primary advanage of field heory models is heir abiliy o generae a wider class of erm srucure innovaions in a parsimonious manner. Consequenly, hedging performance depends on he consiuens of he hedge porfolio. This imporan issue is addressed empirically in Sec. 4 afer he derivaion of hedge parameers in Sec. 3. As a final observaion, sochasic volailiy in a finie facor HJM model is implicily a special case of field heory. Assigning independen random variables o each forward rae mauriy has been sudied by uilizing sochasic parial differenial equaions involving infiniely many variables. Pas research ha insilled imperfec correlaion beween forward raes wih generalized coninuous random processes includes Kennedy [12], Kennedy [13], Goldsein [9], Sana Clara and Sornee [16]. The field heory approach developed by Baaquie [1] is complimenary o pas research since he expressions for all financial insrumens are formally given as a funcional inegral. However, a significan a A field heory model may also converge o a mulifacor HJM model as illusraed in Proposiion 2.1.

3 Quanum Field Theory Term Srucure Model 445 advanage offered by field heory is he variey of compuaional algorihms available for applicaions involving pricing and hedging fixed income insrumens. In fac, field heory was developed precisely o sudy problems involving infiniely many variables and he pah inegral described in he nex secion which serves as a generaing funcion for forward curves is a naural ool o employ in applicaions of erm srucure models. To dae, here has been lile empirical esing of previous field heory models which is undersandable given heir incomplee srucure. The purpose of his paper is o demonsrae he pracical advanages of uilizing field heory when hedging; specifically, he compuaional power offered by he pah inegral no found in previous research. Alhough field heory erm srucure models offer several improvemens over finie facor models, pas research has no fully exploied all simplifying aspecs of uilizing field heory. Specifically, implemenaion of field heory models when hedging bonds requires generalizing he concep of hedging. Since a field heory model may be viewed as an infinie facor model, perfecly hedging a bond porfolio is impossible b alhough compuing hedge parameers remains feasible. Furhermore, empirical evidence in Sec. 4 finds he correlaion beween forward rae mauriies exers a significan impac on he hedging of bonds and allows a low dimensional basis o effecively hedge erm srucure risk. I is imporan o emphasize ha field heory faciliaes raher han inhibis he implemenaion of erm srucure models as seen when he pah inegral for forward raes is inroduced. The pah inegral is a powerful heoreical ool for compuing expecaions involving forward raes required for pricing coningen claims such as fuures conracs in a sraighforward manner. As expeced, closed form soluions for fixed income coningen claims and hedge parameers reduce o sandard exbook soluions such as hose found in Jarrow and Turnbull [11] when he correlaion beween forward raes approaches uniy. In summary, his paper elaboraes on an implemenable field heory erm srucure model ha addresses he limiaions inheren in finie facor erm srucure models. This paper is organized as follows. Secion 2 briefly summarizes he field heory model wih a deailed presenaion found in he appendix. Secions 3 and 4 cover he heoreical and empirical aspecs of hedging in a field heory model. The conclusion is lef o Sec Field Theory The field heory model underlying he hedging resuls in secion hree was developed by Baaquie [1] and calibraed empirically in Baaquie and Marakani [3]. As alluded o earlier, he fundamenal difference beween he model presened in his paper and he original finie facor HJM model sems from he use of a infinie dimensional b A measure valued rading sraegy as in Björk, Kabanov, and Runggaldier [5] provides one alernaive.

4 446 B. E. Baaquie, M. Srikan & M. C. Warachka random process whose second argumen admis correlaion beween mauriies. A Lagrangian is inroduced o describe he field. The Lagrangian has he advanage over Brownian moion of being able o conrol flucuaions in he field, hence forward raes, wih respec o mauriy hrough he addiion of a mauriy dependen gradien as deailed in Definiion 2.1. The acion of he field inegraes he Lagrangian over ime and when exponeniaed and normalized serves as he probabiliy disribuion for forward rae curves. The propagaor measures he correlaion in he field and capures he effec he field a ime and mauriy x has on mauriy x a ime. In he one facor HJM model, he propagaor equals one which allows he quick recovery of one facor HJM resuls. Previous research by Kennedy [12], Kennedy [13], and Goldsein [9] has begun wih he propagaor or correlaion funcion for he field insead of deriving his quaniy from he Lagrangian. More imporanly, he Lagrangian and is associaed acion generae a pah inegral ha faciliaes he soluion of coningen claims and hedge parameers. However, previous erm srucure models have no defined he Lagrangian and are herefore unable o uilize he pah inegral in heir applicaions. The Feynman pah inegral, pah inegral in shor, is a fundamenal quaniy ha provides a generaing funcion for forward rae curves and is formally inroduced in Definiion A.1. Alhough crucial for pricing and hedging, he pah inegral has no appeared in previous erm srucure models wih generalized coninuous random processes. Remark 2.1 (Noaion). Le 0 denoe he curren ime and T he se of forward rae mauriies wih 0 T. The upper bound on he forward rae mauriies is he consan T F R which consrains he forward rae mauriies T o lie wihin he inerval [ 0, 0 + T F R ]. To illusrae he field heory approach, he original finie facor HJM model is derived using field heory principles in Appendix A. In he case of a one facor model, he derivaion does no involve he propagaor as he propagaor is idenically one when forward raes are perfecly correlaed. However, he propagaor is non rivial for field heory models as i governs he imperfec correlaion beween forward rae mauriies. Le A(, x) be a wo dimensional field driving he evoluion of forward raes f(, x) hrough ime. Following Baaquie [1], he Lagrangian of he field is defined as Definiion 2.1 (Lagrangian). The Lagrangian of he field equals { L[A] = 1 A 2 (, x) + 1 ( ) } 2 A(, x) 2T F R µ 2. (1) x Definiion 2.1 is no unique, oher Lagrangians exis and would imply differen propagaors. However, he Lagrangian in Definiion 2.1 is sufficien o explain he conribuion of field heory while quickly reducing o he one facor HJM framework. Observe he presence of a gradien wih respec o mauriy A(,x) x ha conrols

5 Quanum Field Theory Term Srucure Model 447 field flucuaions in he direcion of he forward rae mauriy. The consan c µ erm measures he srengh of he flucuaions in he mauriy direcion. The Lagrangian in Definiion 2.1 implies he field is coninuous, Gaussian, and Markovian. Forward raes involving he field are expressed below where he drif and volailiy funcions saisfy he usual regulariy condiions. f(, x) = α(, x) + σ(, x)a(, x). (2) The forward rae process in Eq. (2) incorporaes exising erm srucure research on Brownian shees, sochasic srings, ec ha have been used in previous coninuous erm srucure models. Noe ha Eq. (2) is easily generalized o he K facor case by inroducing K independen and idenical fields A i (, x). Forward raes could hen be defined as f(, x) = α(, x) + K σ i (, x)a i (, x). (3) i=1 However, a mulifacor HJM model can be reproduced wihou inroducing muliple fields. In fac, under specific correlaion funcions, he field heory model reduces o a mulifacor HJM model wihou any addiional fields o proxy for addiional Brownian moions. Proposiion 2.1 (Lagrangian of Mulifacor HJM). The Lagrangian describing he random process of a K-facor HJM model is given by where L[A] = 1 2 A(, x)g 1 (, x, x )A(, x ) f(, x) = α(, x) + A(, x) and G 1 (, x, x ) denoes he inverse of he funcion G(, x, x ) = K σ i (, x)σ i (, x ). i=1 The above proposiion is an ineresing academic exercises o illusrae he parallel beween field heory and radiional mulifacor HJM models. However, mulifacor HJM models have he disadvanages described earlier in he inroducion associaed wih a finie dimensional basis. Therefore, his approach is no pursued in laer empirical work. In addiion, i is possible for forward raes o be perfecly correlaed wihin a segmen of he forward rae curve bu imperfecly correlaed wih forward raes in oher segmens. For example, one could designae shor, medium, and long mauriies of he forward rae curve. This siuaion is no idenical o he mulifacor HJM model bu jusifies cerain marke pracices c Oher funcional forms are possible bu a consan is chosen for simpliciy.

6 448 B. E. Baaquie, M. Srikan & M. C. Warachka ha disinguish beween shor, medium, and long erm duraions when hedging. However, more complicaed correlaion funcions would be required; compromising model parsimony and reinroducing he same concepual problems of finie facor models. Furhermore, here is lile economic inuiion o jusify why he correlaion beween forward raes should be disconinuous. Therefore, his approach is also no considered in laer empirical work Propagaor The propagaor is an imporan quaniy ha accouns for he correlaion beween forward raes in a parsimonious manner. The propagaor D(x, x ;, T F R ) corresponding o he Lagrangian in Definiion 2.1 is given by he following lemma where θ( ) denoes a Heavyside funcion. Lemma 2.1 (Evaluaion of Propagaor). The Propgaor equals [ D(x, x µt F R ;, T F R ) = sinh µ(t F R τ) sinh(µτ )θ(τ τ ) sinh(µt F R ) + sinh µ(t F R τ ) sinh(µτ)θ(τ τ) { ( cosh 2 2 cosh µ τ T ) ( F R cosh µ τ T ) F R ( µtf 2 R ) 2 2 }] + sinh(µτ) sinh(µτ ) + sinh µ(t F R τ) sinh µ(t F R τ ) where τ = x and τ = x boh represen ime o mauriies. Lemma 2.1 is proved by evaluaing he expecaion E[A(, x), A(, x )]. The compuaions are edious and conained in Baaquie [1] bu are well known in physics and described in common references such as Zinn-Jusin [17]. The propagaor of Goldsein [9] is seen as a special case of Lemma 2.1 defined on he infinie domain < x, x < raher han he finie domain x, x + T F R. Hence, he propagaor in Lemma 2.1 converges o he propagaor of Goldsein [9] as he ime domain expands from a compac se o he real line. The effor in solving for he propagaor on he finie domain is jusified as i allows covariances near he spo rae f(, ) o differ from hose over longer mauriies. Hence, a poenially imporan boundary condiion defined by he spo rae is no ignored. Observe ha he propagaor D(x, x ;, T F R ) in Lemma 2.1 only depends on he variables τ and τ as well as he correlaion parameer µ which implies ha he propagaor is ime invarian. This imporan propery faciliaes empirical esimaion in Sec. 4 when he propagaor is calibraed o marke daa. To undersand he significance of he propagaor, noe ha he correlaor of he field A(, x) for 0, 0 + T F R is given by E[A(, x)a(, x )] = δ( )D(x, x ;, T F R ). (4)

7 Quanum Field Theory Term Srucure Model 449 In oher words, he propagaor measures he effec he value of he field A(, x) has on A(, x ); is value a anoher mauriy x a anoher poin in ime. Alhough D(x, x ;, T F R ) is complicaed in appearance, i collapses o one when µ equals zero as flucuaions in he x direcion are consrained o be perfecly correlaed. I is imporan o emphasize ha µ does no measure he correlaion beween forward raes. Insead, he propagaor solved for in erms of µ fulfills his role. Remark 2.2 (Propagaor, Covariances, and Correlaions). The propagaor D(x, x ;, T F R ) serves as he covariance funcion for he field while σ(, x)d(x, x ;, T F R )σ(, x ) serves as he covariance funcion for forward raes innovaions. Hence, he above quaniy is repeaedly found in hedging and pricing formulae presened in he nex secion. The correlaion funcions for he field and forward rae innovaions are idenical as he volailiy funcions σ(, ) are eliminaed afer normalizaion. This common correlaion funcion is criical for esimaing he field heory model and equals Eq. (22) of Sec. 4. The following able d summarizes he imporan erms in boh he original HJM model and is exended field heory version. Quaniy Finie Facor HJM Field Theory Lagrangian 1 2 W 2 () { 1 A 2 (, x) + 1 ( ) } A(, x) 2 2T F R µ 2 x Propagaor 1 D(x, x ;, T F R ) { 1 } { 1 +TF } Pah Inegral exp dj 2 R () exp d dxdx J(, x)d(x, x ;, T F R )J(, x ) As expeced, he HJM drif resricion is generalized in he conex of a field heory erm srucure model. However, producing he drif resricion follows from he original HJM mehodology as he discouned bond price evolves as a maringale under he risk neural measure o ensure no arbirage. Under he risk neural measure, he bond price is wrien as [ P ( 0, T ) = E [0, ] e r()d 0 P (, T ) ] = DAe df(,) 0 e dxf(,x), (5) d The j() and J(, x) funcions found in he definiion of he pah inegral are source funcions used o compue he momens of forward curves (see Appendix A). They do no appear in he soluion of coningen claims or hedge parameers.

8 450 B. E. Baaquie, M. Srikan & M. C. Warachka where DA represens an inegral over all possible field pahs in he domain 0 d dx. The noaion E [0, ][S] denoes he expeced value under he risk neural measure of he sochasic variable S over he ime inerval [ 0, ]. Equaion (5) serves as he foundaion for compuing he forward rae drif resricion saed in he nex proposiion and proved in Appendix A. Proposiion 2.2 (Drif Resricion). The field heory generalizaion of o HJM drif resricion equals α(, x) = σ(, x) x dx D(x, x ;, T F R )σ(, x ). As expeced, wih µ equal o zero he resul of Proposiion 2.2 reduces o α(, x) = σ(, x) x dx σ(, x ) and he one facor HJM drif resricion is recovered. The nex secion considers he problem of hedging in he conex of a field heory model using eiher bonds or fuures conracs on bonds. 3. Pricing and Hedging in Field Theory Models Hedging a zero coupon bond denoed P (, T ) using oher zero coupon bonds is accomplished by minimizing he residual variance of he hedged porfolio. The hedged porfolio Π() is represened as Π() = P (, T ) + i P (, T i ), where i denoes he amoun of he ih bond P (, T i ) included in he hedged porfolio. Noe he bonds P (, T ) and P (, T i ) are deermined by observing heir marke values a ime. I is he insananeous change in he porfolio value ha is sochasic. Therefore, he volailiy of his change is compued o ascerain he efficacy of he hedge porfolio. For sarers, consider he variance of an individual bond in he field heory model. The definiion P (, T ) = exp( dxf(, x)) for zero coupon bond prices implies ha dp (, T ) T = f(, )d dxdf(, x) P (, T ) ( ) T = r() dxα(, x) dxσ(, x)a(, x) d and E i=1 [ ] ( ) dp (, T ) T = r() dxα(, x) d P (, )

9 Quanum Field Theory Term Srucure Model 451 since E[A(, x)] = 0. Therefore dp (, T ) P (, T ) E [ dp (, T ) P (, T ) Squaring his expression and invoking he resul ha ] = d dxσ(, x)a(, x). (6) E[A(, x)a(, x )] = δ(0)d(x, x ;, T F R ) = D(x, x ;, T F R ) d resuls in he insananeous bond price variance Var[dP (, T )] = dp 2 (, T ) dx dx σ(, x)d(x, x ;, T F R )σ(, x ). (7) As an inermediae sep, he insananeous variance of a bond porfolio is considered. For a porfolio of bonds, ˆΠ() = N i=1 ip (, T i ), he following resuls follow direcly and dˆπ() E[d ˆΠ()] = d Var[dˆΠ()] = d i=1 j=1 i=1 i i P (, T i ) dxσ(, x)a(, x) (8) i j P (, T i )P (, T j ) i j dx dxσ(, x)d(x, x ;, T F R )σ(, x ). (9) The (residual) variance of he hedged porfolio Π() = P (, T ) + i P (, T i ) may now be compued in a sraighforward manner. For noaional simpliciy, he bonds P (, T i ) (being used o hedge he original bond) and P (, T ) are denoed P i and P respecively. Equaion (9) implies he hedged porfolio s variance equals he final resul shown below P 2 dx + 2P i=1 dx σ(, x)σ(, x )D(x, x ;, T F R ) i P i i=1 i dx dx σ(, x)σ(, x )D(x, x ;, T F R ) + i=1 j=1 i i j P i P j j dx dx σ(, x)σ(, x )D(x, x ;, T F R ). (10)

10 452 B. E. Baaquie, M. Srikan & M. C. Warachka Observe ha he residual variance depends on he correlaion beween forward raes described by he propagaor. Ulimaely, he effeciveness of he hedge porfolio is an empirical quesion since perfec hedging is no possible wihou shoring he original bond. This empirical quesion is addressed in Sec. 4 when he propagaor is calibraed o marke daa. Minimizing he residual variance in Eq. (10) wih respec o he hedge parameers i is an applicaion of sandard calculus. The following noaion is inroduced for simpliciy. Definiion 3.1. L i = P P i i dx dx σ(, x)σ(, x )D(x, x ;, T F R ), as M ij = P i P j i j dx dx σ(, x)σ(, x )D(x, x ;, T F R ). Definiion 3.1 allows he residual variance in Eq. (10) o be succincly expressed P 2 dx dx σ(, x)σ(, x )D(x, x ;, T F R ) + 2 i L i + i j M ij. (11) i=1 i=1 j=1 Hedge parameers i ha minimize he residual variance in Eq. (11) are he focus of he nex heorem. Theorem 3.1 (Hedge Parameer for Bond). Hedge parameers in he field heory model equal i = j=1 L j M 1 ij and represen he opimal amouns of P (, T i ) o include in he hedge porfolio when hedging P (, T ). Theorem 3.1 is proved by differeniaing Eq. (11) wih respec o i and subsequenly solving for i. Corollary 3.1 below is proved by subsiuing he resul of Theorem 3.1 ino Eq. (11). Corollary 3.1 (Residual Variance). The variance of he hedged porfolio equals V = P 2 dx dx σ(, x)σ(, x )D(x, x ;, T F R ) which declines monoonically as N increases. i=1 j=1 L i M 1 ij L j The residual variance in Corollary 3.1 enables he effeciveness of he hedge porfolio o be evaluaed. Therefore, Corollary 3.1 is he basis for sudying he

11 Quanum Field Theory Term Srucure Model 453 impac of including differen bonds in he hedge porfolio as illusraed in Sec For N = 1, he hedge parameer in Theorem 3.1 reduces o ( 1 = P dx ) T 1 dx σ(, x)σ(, x )D(x, x ;, T F R ) P T1 1 dx 1. (12) dx σ(, x)σ(, x )D(x, x ;, T F R ) To obain he HJM limi, consrain he propagaor o equal one. The hedge parameer in Eq. (12) hen reduces o ( 1 = P T dx 1 ) ( dx σ(, x)σ(, x T ) ) ( P T1 1 dxσ(, x) ) 2 = PP1 dxσ(, x) 1. (13) dxσ(, x) λ(t ) The popular exponenial volailiy funcion σ(, T ) = σe allows a comparison beween our field heory soluions and previous research. Under he assumpion of exponenial volailiy, Eq. (13) becomes 1 = P ( ) 1 e λ(t ). (14) P 1 1 e λ(t1 ) Equaion (14) coincides wih he raio of hedge parameers found as Eq. (16.13) of Jarrow and Turnbull [11]. In erms of heir noaion 1 = P (, T ) ( ) X(, T ). (15) P (, T 1 ) X(, T 1 ) For emphasis, he following equaion holds in a one facor HJM model [P (, T ) + 1 P (, T 1 )] r() which is verified using Eq. (15) and resuls found on pages of Jarrow and Turnbull [11], = 0 [P (, T ) + 1 P (, T 1 )] r() = P (, T )X(, T ) 1 P (, T 1 )X(, T 1 ) = P (, T )X(, T ) + P (, T )X(, T ) = 0. When T 1 = T, he hedge parameer equals minus one. Economically, his fac saes ha he bes sraegy o hedge a bond is o shor a bond of he same mauriy. This rivial approach reduces he residual variance in Eq. (11) o zero as 1 = 1 and P = P 1 implies L 1 = M 11. Empirical resuls for nonrivial hedging sraegies are found in Sec. 4.1 afer he propagaor is calibraed Fuures pricing As his paper is primarily concerned wih hedging a bond porfolio, fuures prices are derived as hey are commonly used for hedging bonds given heir liquidiy.

12 454 B. E. Baaquie, M. Srikan & M. C. Warachka Proposiion 3.1 (Fuures Price). The fuures price F( 0,, T ) is given by F( 0,, T ) = E [0, ][P (, T )] = DAe dxf(,x) = F ( 0,, T ) exp{ω F ( 0,, T )} (16) where F ( 0,, T ) represens he forward price for he same conrac, F ( 0,, T ) = P ( 0,T ) P (, and 0, ) Ω F ( 0,, T ) = d dxσ(, x) dx D(x, x ;, T F R )σ(, x ). (17) 0 Deails of he proof are found in Appendix A. Observe ha for µ = 0, Eq. (17) collapses o Ω F ( 0,, T ) = d dxσ(, x) dx σ(, x ) (18) 0 which is equivalen o he one facor HJM model. For he one facor HJM model wih exponenial volailiy, Eq. (18) becomes Ω F ( 0,, T ) = σ2 2λ 3 (1 e λ(t ) )(1 e λ( 0) ) 2 which coincides wih Eq. (16.23) of Jarrow and Turnbull [11]. Observe ha he propagaor modifies he produc of he volailiy funcions wih µ serving as an addiional model parameer. Prices for call opions, pu opions, caps, and floors proceed along similar lines wih an idenical modificaion of he volailiy funcions bu heir soluions are omied for breviy. Formulae for hese coningen claims are given in Baaquie [1] where heir reducion o he closed form soluions of Jarrow and Turnbull [11] is also presened Hedging bonds wih fuures conracs The maerial in he previous subsecion allows he hedging properies of fuures conracs on bonds o be sudied. Proceeding as before, he appropriae hedge parameers for fuures conracs expiring in one year are compued. Proposiion 3.1 expresses he fuures price F(,, T ) in erms of he forward price P (,T ) P (, ) = e dxf(,x) and he deerminisic quaniy Ω F (,, T ) found in Eq. (17). The dynamics of he fuures price df(,, T ) is given by which implies df(,, T ) F(,, T ) = dω F(,, T ) df(,, T ) E[dF(,, T )] F(,, T ) dxdf(, x) (19) = d dxσ(, x)a(, x). (20)

13 Quanum Field Theory Term Srucure Model 455 Squaring boh sides leads o he insananeous variance of he fuures price Var[dF(,, T )] = df 2 (,, T ) dx dx σ(, x)d(x, x)σ(, x ). (21) The following definiion updaes Definiion 3.1 in he conex of fuures conracs. Definiion 3.2 (Fuures Conracs). Le F i denoe he fuures price F(,, T i ) of a conrac expiring a ime on a zero coupon bond mauring a ime T i. The hedged porfolio in erms of he fuures conrac is given by Π() = P + i F i where F i represen observed marke prices. For noaional simpliciy, define he following erms i L i = P F i dx dx σ(, x)d(x, x ;, T F R )σ(, x ), M ij = F i F j i i=1 j dx dx σ(, x)d(x, x ;, T F R )σ(, x ). The hedge parameers and he residual variance when fuures conracs are used as he underlying hedging insrumens have idenical expressions o hose in Theorem 3.1 and Corollary 3.1 bu are based on Definiion 3.2. Compuaions parallel hose in he beginning of his secion. Corollary 3.2 (Hedge Parameers and Residual Variance using Fuures). Hedge parameers for a fuures conrac ha expires a ime on a zero coupon bond ha maures a ime T i equals i = j=1 L j M 1 ij while he variance of he hedged porfolio equals V = P 2 dx for L i and M ij in Definiion 3.2. dx σ(, x)σ(, x )D(x, x ;, T F R ) Proof follows direcly from previous work. i=1 j=1 L i Mij 1 L j 4. Empirical Esimaion of Field Theory Models This secion illusraes he significance of correlaion beween forward rae mauriies and numerically esimaes is impac on hedging performance. The volailiy

14 456 B. E. Baaquie, M. Srikan & M. C. Warachka funcion σ(, x) and he µ parameer were previously calibraed non paramerically from marke daa in Baaquie and Srikan [3]. The daa used in he following empirical ess was generously provided by Jean-Philippe Bouchaud of Science and Finance. The daa consiss of daily closing prices for quarerly Eurodollar fuures conracs wih a maximum mauriy of 7.25 years as described in Bouchaud, Sagna, Con, and El-Karoui [6] as well as Bouchaud and Maacz [14]. As previous empirical research ino esimaing HJM models has found, Amin and Moron [15], mulifacor HJM models are difficul o esimae. This concep is revealed by considering he difference beween a one and wo facor HJM model. Le τ = x and τ = x once again represen ime o mauriy and assume he correlaion funcion is ime invarian by depending solely on τ and τ. The correlaion beween forward rae mauriies C(x, x ) may be expressed in erms of he normalized propagaor e as C(x, x ) = D(x, x ; ) D(x, x; )D(x, x ; ). (22) To clarify, he propagaor (as emphasized afer Lemma 2.1) and hence he correlaion in Eq. (22) is ime homogeneous. Once he propagaor is deermined, he correlaion beween forward raes follows immediaely and vice versa. For example, when he propagaor equals one, he correlaion beween forward raes is also idenically one. However, for he wo facor HJM model, he normalized propagaor is given by 1 + g(τ)g(τ ) (1 + g2 (τ))(1 + g 2 (τ )) for g(τ) = σ2(τ) σ 1(τ). Consequenly, he correlaion funcion above in Eq. (23) depends on he raio of volailiy funcions. Therefore, for a given correlaion srucure, obaining reliable volailiy funcion esimaes is a challenge as i is difficul o disenangle one volailiy funcion from anoher. This pracical disadvanage is overcome by field heory models. Concerning he esimaion of he field heory model, he marke volailiy funcion is used hroughou he remainder of Sec. 4. This volailiy funcion was esimaed direcly from marke daa as he variance of erm srucure innovaions for each mauriy and is graphed below. The corresponding implied correlaion parameer µ was esimaed as 0.06 (annualized). Esimaion of µ was accomplished afer esimaing he correlaion beween forward rae innovaions over he 7.25 year horizon. The µ parameer was hen found by minimizing he roo mean square difference beween he heoreical correlaion funcion in Eq. (22) which conains µ via he definiion of he propagaor in Lemma 2.1 and he empirical correlaion funcion generaed from he daa (23) e The value of T F R was se o, rescaling by and is associaed propagaor. 1 T F R preserved he Lagrangian in Definiion 2.1

15 Quanum Field Theory Term Srucure Model Sigma esimaed from daa sigma Time o mauriy/year Fig. 1. Implied volailiy funcion using marke daa. aggregaed over he sample period. The minimizaion was carried ou using he Levenberg-Marquard mehod. Sabiliy analysis performed by considering differen subsecions of he daa indicaed ha his esimae was robus wih an error of a mos The propagaor iself is graphed below. I should be emphasized ha he propagaor may also be esimaed non paramerically from he correlaion found in marke daa wihou any specified funcional form, as he volailiy funcion was esimaed. This approach preserves he closed form soluions for hedge parameers and fuures conracs illusraed in he previous secion. However, he original finie facor HJM model canno accommodae an empirically deermined propagaor since i is auomaically fixed once he HJM volailiy funcions are specified Hedging error The reducion in variance achievable by hedging a five year zero coupon bond wih oher zero coupon bonds and fuures conracs is he focus of his subsecion. The residual variances for one and wo bond hedge porfolios are shown in Figs. 3 and 4. The parabolic naure of he residual variance is because µ is consan. A more complicaed funcion would produce residual variances ha do no deviae

16 458 B. E. Baaquie, M. Srikan & M. C. Warachka Correlaion beween differen mauriies line 1 correlaion mauriy/year mauriy/year Fig. 2. Propagaor implied by µ Residual variance when hedging 5 year zero coupon bonds line Time o mauriy/year Fig. 3. Residual variance for five year bond versus bond mauriy used o hedge.

17 Quanum Field Theory Term Srucure Model 459 Residual variance when hedging 5 year zero coupon bonds line Time o mauriy/year Time o mauriy/year Fig. 4. Residual variance for five year bond versus wo bond uriies used o hedge. monoonically as he mauriy of he underlying and he hedge porfolio increases alhough he graphs appeal o our economic inuiion which suggess ha correlaion beween forward raes decreases monoonically as he disance beween hem increases as shown in Fig. 2. Observe ha he residual variance drops o zero when he same bond is used o hedge iself; eliminaing he original posiion in he process. The corresponding hedge raios are shown in Fig. 5. Numerical ess o deermine he efficiency of hedging using fuures conracs ha expire in one year are conduced by calculaing he residual variance when hedging a five year zero coupon bond wih a fuures conrac expiring in one year on various zero coupon bonds. The residual variance is shown in Fig. 6. Observe ha he zero coupon bond is bes hedged by selling fuures conracs on 4.5 year bonds which is explained by he fac ha he fuures conrac only depends on he variaion in forward rae curve from o T bu he zero coupon bond depends on he variaion of he forward rae curve from o T. Hence, a shorer underlying bond mauriy is chosen for he fuures conrac o compensae for he forward rae curve from o. A similar resul is seen when hedging wih wo fuures conracs boh expiring in one year. In his case, opimal hedging is obained when fuures conracs on he same bond as well as one on a bond wih he minimum possible mauriy (1.25 years) are shored. The use of a fuures conrac on a shor mauriy bond is consisen wih he high volailiy of shor mauriy forward raes as displayed in Fig. 1. The opimal fuures conracs o include in he hedge porfolio are shown in Table 1 when hedging a five-year bond.

18 460 B. E. Baaquie, M. Srikan & M. C. Warachka 0 Hedge raio when hedging a five year zero coupon bonds Time o mauriy/year Fig. 5. Hedge raios for five year bond Residual variance when hedging wih one fuures conrac Time o mauriy/year Fig. 6. Residual variance for five year bond hedged wih a one year fuures conrac on a T mauriy bond.

19 Quanum Field Theory Term Srucure Model 461 Table 1. Residual variance and hedge raios for a five-year bond hedged wih one year fuures conracs. Number Fuures Conracs (Hedge Raio) Residual Variance 0 none years ( 1.288) years ( ), 1.25 years ( ) The variances recorded in Table 1 correspond o daily dollar denominaed flucuaions Curren and fuure research The field heory model has been used o value call and pu opions as well as caps and floors in Baaquie [1]. Hedge parameers for hese insrumens is currenly under invesigaion. Obaining fixed income opions daa o invesigae he significance of correlaed forward raes on opion prices as well as is impac on heir hedge parameers would also be valuable. An enhanced model wih sochasic volailiy where innovaions in he erm srucure resul from he produc of wo fields has been developed by Baaquie [2]. The resuls for insananeous hedging have been exended, for he field heory model, o case of finie ime hedging in [18]. 5. Conclusion Field heory models address he heoreical limiaions of finie facor erm srucure models by allowing imperfec correlaion beween every forward rae mauriy. The field heory model offers compuaionally expedien hedge parameers for fixed income derivaives and provides a mehodology o answer crucial quesions concerning he number and mauriy of bonds o include in a hedge porfolio. Furhermore, field heory models are able o incorporae correlaion beween forward rae mauriies in a parsimonious manner ha is well suied o empirical implemenaion. Empirical evidence revealed he significan impac ha correlaion beween forward rae mauriies has on hedging performance. Despie he infinie dimensional naure of he field, i is shown ha a low dimensional hedge porfolio effecively hedges ineres rae risk by exploiing he correlaion beween forward raes. Therefore, field heory models address he heoreical dilemmas of finie facor erm srucure models and offer a pracical alernaive o finie facor models. Appendix A. Deails of Field Theory Model This appendix briefly reviews he resuls conained in Baaquie [1]. The formalism of quanum mechanics is based on convenional mahemaics of parial differenial

20 462 B. E. Baaquie, M. Srikan & M. C. Warachka equaions and funcional analysis as deailed in Zinn-Jusin [17]. The mahemaical ools underlying quanum field heory have no counerpar in radiional sochasic calculus alhough Gaussian fields are equivalen o an infinie collecion of sochasic processes. The Lagrangian has he advanage over Brownian moion of being able o conrol flucuaions in he field, hence forward raes, wih respec o mauriy hrough he addiion of a mauriy dependen gradien as deailed in Eq. (A.8). The acion inegraes he Lagrangian over ime and when exponeniaed and appropriaely normalized yields a probabiliy disribuion funcion for he forward rae curves. The propagaor measures he correlaion in he field and capures he effec he field a ime and mauriy x has on he mauriy x a ime. The Feynman pah inegral serves as he generaing funcion for forward rae curves. The pah inegral is obained by inegraing he exponenial of he acion over all possible evoluions of he forward rae curve. A.1. Resaemen of HJM Before presening he field heory of he forward raes, we briefly resae he HJM model in erms of is original formulaion bu using he noaion of pah inegrals. For simpliciy, consider a one facor HJM model of he forward rae curve whose evoluion is generaed by f(, x) = α(, x) + σ(, x)w (), (A.1) where α(, x) and σ(, x) are he drif and volailiy of forward raes. For every value of ime, he sochasic variable W () is an independen Gaussian random variable wih he propery ha E[W ()W ( )] = δ( ) (A.2) in conras o Eq. (4) involving he field. More convenionally, he HJM model is wrien as df(, x) = α(, x)d + σ(, x)dz() where Z() represens a Brownian moion process. Hence, he Gaussian process W () equals dz() d. To derive he covariance funcion implied for W (), he correlaion funcion for Z() mus be wice differeniaed. Therefore, W ()W ( ) = 2 Z()Z( ) = 2 min(, ) = θ( ) = δ( ) as seen in Eq. (A.2). The Lagrangian of he Gaussian process is defined as L[W ] = 1 2 W 2 (). (A.3) The Gaussian process may be illusraed by discreizing ime ino a discree laice of spacing ɛ, and mɛ, wih m = 1, 2 M, wih W () W (m). The

21 Quanum Field Theory Term Srucure Model 463 probabiliy measure underlying Gaussian pahs for 1 < < 2 is given by P[W ] = M e ɛ 2 W 2 (m) = m=1 dw = M m=1 ɛ 2π + M m=1 dw (m). e ɛl[w ] = e ɛ M m=1 L[W ], (A.4) The erm P[W ] represens he probabiliy of a pah raced ou by W (). For purposes of rigor, he coninuum noaion represens aking he coninuum limi of he discree muliple inegrals given above. The infinie dimensional inegraion dx() has a rigorous, measure heoreic, definiion as he inegraion over all coninuous, bu nowhere differeniable, pahs running beween poins x( 1 ) and x( 2 ). For 1 < < 2, he probabiliy disribuion funcion for he pahs of he Gaussian process equals measure given by 1<< 2 + P[W, 1, 2 ] = exp{s[w, 1, 2 ]}, (A.5) where he acion S[W, 1, 2 ] for he Gaussian process is given by he limi of he exponen in Eq. (A.4) as ɛ 0 S[W, 1, 2 ] = W 2 ()d. (A.6) The erm DW denoes pah inegraion over all he random variables W () which appear in he problem. A pah inegral approach o he HJM model has been discussed in Chiarella and El-Hassan [7] alhough he acion derived is differen han he one given above since a differen se of variables were involved. A formula for he generaing funcion of forward raes driven by a Gaussian process is given by he pah inegral Z[j, 1, 2 ] = 2 DW e 1 dj()w () e S[W,1,2] { 1 2 } = exp j 2 ()d. (A.7) 2 1 This pah inegral is crucial for applicaions involving he pricing of derivaives as demonsraed in Sec. 3. A.2. Field heory model The Lagrangian of he field equals { L[A] = 1 A 2 (, x) + 1 ( ) } 2 A(, x) 2T F R µ 2. (A.8) x

22 464 B. E. Baaquie, M. Srikan & M. C. Warachka Since he Lagrangian is quadraic, he resuling field is Gaussian and he exisence of a Hamilonian for he Lagrangian implies he field is Markovian. Forward raes are expressed as f(, x) = α(, x) + σ(, x)a(, x). (A.9) The acion of S[A] inegraes he Lagrangian over ime o yield S[A] = 0 d = 1 2T F R +TF R 0 Consider he one facor case dxl[a] { +TF R d dx A 2 (, x) + 1 ( ) } 2 A(, x) µ 2. (A.10) x W () = +TF R 0 A(, x)dx (A.11) by assuming he correlaion µ is zero. This process reduces he acion found in Eq. (A.10) o he following acion S[A] = 1 +TF R W 2 ()d dx 2T F R = W 2 ()d (A.12) which is idenical o Eq. (A.6). If one hinks of he field A( 0, x) a some insan 0 as he posiion of a sring, hen he one facor HJM model consrains he sring o be rigid. The acion S[A] given in (A.10) allows every mauriy x in A( 0, x) o flucuae as a sring wih sring rigidiy equal o 1 µ. For µ = 0, he sring is infiniely 2 rigid or perfecly correlaed. A normalizing consan Z is he resul of inegraing he pahs of he field agains he probabiliy of each pah. Z = DAe S[A] (A.13) where he noaion DA represens an inegral over all possible field pahs conained in he domain. The momen generaing funcional for he field is given by he Feynman pah inegral, Zinn-Jusin [17]. Definiion A.1 (Pah Inegral). The pah inegral over all possible pahs of he field, weighed by heir probabiliy, equals Z[J] = 1 DAe JA e S[A] Z where Z is defined in Eq. (A.13).

23 Quanum Field Theory Term Srucure Model 465 In he conex of he presen erm srucure model, he pah inegral equals Z[J] = 1 { } +TF R DA exp d dxj(, x)a(, x) e S[A]. (A.14) Z 0 The above formula is inerpreed as follows, he Z erm operaes as a normalizing consan while es[a] Z represens he probabiliy disribuion funcion for each random pah generaed by he field A(, x). The inegrand { } +TF R exp d dxj(, x)a(, x) 0 conains an exernal source f funcion J(, x) coupled o he field A(, x) ha is operaed on o produce momens of he disribuion. The pah inegral Z[J] is evaluaed explicily in Baaquie [1] and simplifies Eq. (A.14) o { 1 } +TF R Z[J] = exp d dxdx J(, x)d(x, x ;, T F R )J(, x ). (A.15) 2 0 The pah inegral is he basis for all applicaions of erm srucure modeling such as he pricing and hedging of fixed income coningen claims since i represens he generaing funcion for forward rae curves. A.3. Proof of Proposiion 2.2 Saring from Eq. (5) P ( 0, T ) = DAe df(,) 0 e dxf(,x) and using he ideniy P ( 0, T ) = e dxf( 0,x) 0 implies e dxf( 0,x) 0 = DAe d 0 d α(,)+σ(,)a(,) 0 e 0 d dxα(,x)+σ(,x)a(,x), 1 = DAe d 0 dxα(,x)+σ(,x)a(,x), 1 = e d dxα(,x) and isolaes he drif in he exponen as 0 d dxα(, x) d dx dx σ(,x)d(x,x ;,T 0 F R)σ(,x ) = d dx dx σ(, x)d(x, x ;, T F R )σ(, x ), (A.16) f To make an analogy wih univariae generaing funcions, he funcion J(, x) serves a similar role o in he momen generaing funcion e µ+ 1 2 σ2 2 for normal random variables N (µ, σ 2 ).

24 466 B. E. Baaquie, M. Srikan & M. C. Warachka and consequenly dxα(, x) = 1 2 dxdx σ(, x)d(x, x ;, T F R )σ(, x ). (A.17) This no arbirage condiion mus hold for any Treasury bond mauring a any ime x = T. Hence, differeniaing he above expression wih respec o T produces he final version of he drif resricion α(, x) = σ(, x) x A.4. Proof of Proposiion 3.1 F( 0,, T ) = DAe dxf(,x) dx D(x, x ;, T F R )σ(, x ). (A.18) = e dxf(0,x) DAe d 0 dxα(,x)+σ(,x)a(,x) = F ( 0,, T )e d 0 dxα(,x) Z[σ(, x)θ(x )θ(t x)] where Z[J] is defined in Eq. (A.14). Proceeding furher leads o F( 0,, T ) = F ( 0,, T )e d 0 dx x dx σ(,x)d(x,x ;,T F R)σ(,x ) e 1 2 where Ω F ( 0,, T ) is given by Eq. (17). d 0 dxdx σ(,x)d(x,x ;,T F R)σ(,x ) = F ( 0,, T ) exp{ω F ( 0,, T )} (A.19) References [1] B. E. Baaquie, Quanum field heory of reasury bonds, Physical Review E 64 (2001) [2] B. E. Baaquie, Quanum field heory of forward raes wih sochasic volailiy, Physical Review E 65 (2002) [3] B. E. Baaquie and M. Sririkan, Empirical invesigaion of a quanum field of forward raes, o appear in Physical Review E (2003). hp://xxx.lanl.gov/abs/condma/ [4] T. Bjork and B. J. Chrisensen, Ineres rae dynamics and consisen forward rae curves, Mahemaical Finance 9(4) (1999) [5] T. Bjork, Y. Kabanov and W. Runggaldier, Bond marke in he presence of marked poin rocesses, Mahemaical Finance 7(2) (1997) [6] J.-P. Bouchaud, N. Sagna, R. Con, N. El-Karoui and M. Poers, Phenomenology of he ineres rae curve, Applied Financial Mahemaics 6 (1999) 209. [7] C. Chiarella and N. El-Hassan, Evaluaion of derivaive securiy prices in he Heah Jarrow Moron framework as pah inegrals using fas fourier ransform echniques, Journal of Financial Engineering 6(2) (1997)

25 Quanum Field Theory Term Srucure Model 467 [8] J. Cohen and R. Jarrow, Markov modeling in he Heah, Jarrow, and Heah erm srucure framework, Cornell Universiy (2000). [9] R. Goldsein, The erm srucure of ineres rae curve as a random field, Review of Financial Sudies 13(2) (2000) [10] R. Jarrow, D. Heah and A. Moron, Bond pricing and he erm srucure of ineres raes: A new mehodology for coningen claims, Economerica 60(1) (1992) [11] R. Jarrow and S. Turnbull, Derivaive Securiies, Second Ediion (Souh Wesern College Publishing, 2000). [12] D. P. Kennedy, The erm srucure of ineres raes as a gaussian field, Mahemaical Finance 4(3) (1994) [13] D. P. Kennedy, Characerizing Gaussian models of he erm srucure of ineres raes, Mahemaical Finance 7(2) (1997) [14] A. Maacz and J.-P. Bouchaud, An empirical invesigaion of he forward ineres rae erm srucure, Inernaional Journal of Theoreical and Applied Finance 3(4) (2000) [15] A. Moron and K. Amin, Implied volailiy funcions in arbirage free erm srucure models, Journal of Financial Economics 35 (1994) [16] P. Sana-Clara and D. Sornee, The dynamics of he forward ineres rae curve wih sochasic sring shocks, Review of Financial Sudies 14(1) (2001) [17] J. Zinn-Jusin, Quanum Field Theory and Criical Phenomenon (Cambridge Universiy Press, 1992.) [18] B. E. Baaquie and M. Srikan, Finie hedging in field heory of ineres rae, condma/ , o appear in Physical Review E (2003).

26