On the Quantum Field-theoretic Empirical Investigation of Forward Rates

Transcription

1 Quanum Finance 1 On he Quanum Field-heoreic Empirical Invesigaion of Forward Raes Ma Bernard 1 mab@berkeley.edu Supervisor: Seve Evans 2 evans@sa.berkeley.edu 1 Suden, Sa 251: Sochasic Analysis wih Applicaions o Mahemaical Finance, Universiy of California, Berkeley, CA 94720, USA 2 Professor, Saisics & Mahemaics (join appoinmen), Universiy of California, Berkeley, CA 94720, USA Absrac Focusing on he formal (universal) descripion of quanum field heoreic mehod for volaile classical (sochasic) pahs, we invesigae quanum sochasic model for a represenaive family of classical (sochasic) processes (F T, P); ha is, for arbirary F T -measurable processes, under he given probabiliy measures, wihin he framework of he Quanum Field Theory (QFT) mehod proposed by Baaquie, e al. (2001) for forward raes. Keywords: Sochasic analysis, quanum field heory, forwards erm srucures 1 Inroducion Inspired by he meaphysical developmens in quanum field-heoreic modeling of volailiy, quanum field-heoreic models have been applied o classical models of forward raes (ha is, ineres raes), in paricular ha of he Heah-Jarrow-Moron, HJM. Several empirical ess of he classical HJM models by researchers in he field such as: Bühler, UhrigHomburg, Waler and Weber [15], Flesker [16], Sim and Thurson [17]), however, have proved aborive: all of he ess assume a cerain form for he volailiy funcion σ. Hence, here is need for a es which is independen of he volailiy funcion. Reviewing some basic suffs: Definiion Ineres raes a any poin in ime form a usually coninuous curve (curren ineres raes for differen imes in he fuure) called he forward rae curve (FRC). A sochasic variable wih very minimal loss of generaliy, he forward rae is denoed by f(, x), which represens he ineres rae a fuure ime x for a coningen T-claim enered ino a ime < x. For example, f(1, 2) is he ineres rae one year from now for an insananeous deposi o be made 2 years ino he fuure. Definiion Bonds, he financial insrumens of deb issued by governmens and corporaions o raise money from he capials marke, have a pre-deermined (deerminisic) cash flow (i.e., a coningen T-claim). Bringing he noion of coningen claim o limeligh here: Definiion A coningen T-claim is any random variable X L 0 (F T, P) (i.e., an arbirary F T -measurable random variable). The noaion X L 0 +(F T, P) denoes se of non-negaive elemens of L 0 (F T, P), and X L 0 ++(F T, P) denoes se of elemens X of L 0 +(F, P) wih P (X > 0) > 0.

2 2 Bernard, M. Also, by definiion: Definiion A probabiliy measure Q is a maringale measure if 1. Q P, 2. The discouned price process Z is a Q-local maringale. If he discouned price process Z is Q-maringale, we say ha Q is a srong maringale measure. And, wih he concep of a maringale measure: Definiion A self-financing porfolio h such ha he corresponding value process has he properies: 1. V (0) = 0 2. V (T ) L 0 ++(F, P ) is an arbirage porfolio. And if no arbirage porfolios exiss for any T R +, hen he model is said o be free of arbirage or arbirage free. Furhermore, by a sligh modificaion of he se of admissible porfolios: Definiion A self-financing porfolio h is called Q-admissible if V Z (, h) is a Q-maringale for a given maringale measure Q Remark By definiion Z is a Q-maringale, V Z -process is he sochasic inegral of h wih respec o Z. Thus, i is clear ha every sufficienly self financing porfolio is in fac admissible. Of course, i could be annoying ha he definiion of admissibiliy is dependen upon he paricular choice of maringale measure, bu he need for he admissibiliy condiion can be seen inside he proof (given in [4]) of one of he basic resuls in he heory which saes ha: Propery A model is free of arbirage in he sense ha here exis no Q-admissible arbirage porfolio if here exis a maringale measure Q. Now, in rerospec, he inerpreaion of he coningen claim is ha a conrac which specifies ha he sochasic amoun, X, of money is o be payed ou o he holder of he conrac a ime T. As an illusraion, zero coupon bonds, also known as pure discoun bonds, wih mauriy dae T (i.e. T -bond) are a conrac which guaranees a single cash flow consising of a fixed payoff of say 1Naira a some fuure (mauriy) ime T ; he price a ime of a bond wih mauriy dae T is denoed by P (, T ). Thus, given a bond marke, a number of ineres raes (S 0,..., S k ) can be defined almos surely by a sysem of sochasic differenial equaions (SDE), driven by a finie number of Wiener processes, in a defined filered probabiliy space (Ω, F, P ) carrying he finie number of sochasic processes (S-processes) which are assumed o be all semi-maringales.

3 Quanum Finance 3 Driven by he Maringale models for he shor rae given by: dr() = µ(, r())d + σ(, r())dw () { µ = drif erm for σ = diffusion erm The erm srucure for he sysems of sochasic differenial equaions, SDE (which are some sandard models of he sough) are compleely deermined by specifying he r-dynamics under he maringale measure Q, and are in agreemen wih he Affine Term Srucure (ATS) heory which saes ha: Theorem A model is said o possess an affine erm srucure (ATS) if he erm srucure {P (, T ) 0 T, T 0} has he form P (, T ) = F (, r(), T ), where F has he form for deerminisic funcions A and B given by A(, T ) = σ2 2 Thus, we have he following derivaions: A(,T ) B(,T )r F (, r, T ) = e T B(, T ) = 1 a B 2 (, T )ds b T a(t (1 e )) B(, T )ds 1.1 The Vasicek Model Based on he SDE: dr = (b ar)d + σdw he Vasicek model has he propery of being mean revering (under he maringale measure Q) in he sense ha i will end o rever o he mean level b/a. And, wih he erm srucure compued in [5], he price propery can be saed as follows: Propery The bond prices are given by: where and 1.2 The CIR Model A(,T ) B(,T )r P (, T ) = e B(, T ) = 1 a a(t (1 e )) A(, T ) = (B(, T ) T ) ( ab 1 2 σ2) a 2 σ2 B 2 (, T ) 4a A much more difficul model o handle compared o he Vasicek model, he Cox-Ingersoll-Ross (discussed in deph in [6] and [7]), we have he following propery: Propery The erm srucure is given by: where and F T B(T r)r (, r) = A(T r)e ( B(x) = 2(e γx ) (γ + a)(e γx 1) + 2γ γ = a 2 + 2γ 2 ) 2ab σ 2

4 4 Bernard, M. 1.3 The Hull and Whie Model Deailed in [8], he Hull and Whie model has he following propery, consisen wih he Q-dynamics of he shor rae given by: dr = (φ() ar)d + σdw () where a and σ are consans, and φ is a deerminisic funcion of ime, such ha a and σ are chosen o fi he nice volailiy srucure and φ is chosen o fi he heoreical bond prices {P (0, T ) T > 0} on he evoluion curve {P (0, T ) T > 0}. Propery The bond prices are given by: A(,T ) B(,T )r P (, T ) = e where A and B solve wih he soluion given by: Thus, i is obvious ha B (, T ) ab(, T ) = 1 B(T, T ) = 0 φ()b (, T ) 1 2 σ2 B 2 (, T ) = A(, T ) A(, T ) = σ2 2 A(T, T ) = 0 a(t (1 e )) B(, T ) = 1 a T T B 2 (, T )ds b B(, T )ds (1) (2) Remark The shor rae r is he only (one-facor) explanaory variable in all of hese (classical) sandard models. Also, specifying r as he soluion of an SDE allows he use Markov process heory, so ha work can be done wihin a PDE framework. In paricular, i is ofen possible o obain analyical formulas for bond prices and derivaives. However, he drawbacks remain ha Remark hey only deal wih he spo rae (curren ineres rae for he presen ime) and he forward rae curve is reaed as a derived quaniy. And as he shor rae model becomes increasingly more realisic, he yield curve inversion described in [9] becomes increasingly more difficul. Bu from an economic poin of view, i is quie unreasonable o assume ha he enire money marke is governed by only one explanaory variable. Hence, i is hard o obain a realisic volailiy srucure for he forward raes wihou inroducing a very complicaed shor rae model. These and oher consideraion lead o he proposal of he new model - which use more han one sae variable - namely: The Maringale Modeling [4], The Musiela Parameerizaion [4]. One brigh idea however, was o presen an a priori model for he shor rae as well as for some long rae, so ha one or several inermediary ineres raes could be modeled. The mehod proposed by Healh-Jarrow-Moron, HJM is a he far end of his specrum.

5 Quanum Finance 5 2 The Classical Heah-Jarrow-Meron Model Where he enire forward rae curve is he (infinie dimensional) sae variable, he assumpion here is ha Assumpion For every fixed T > 0, he forward rae f(, T ) has a sochasic differenial which under he objecive measure P is given by df(, T ) = α(, T )d + σ(, T )dw () (3) f(0, T ) = f (0, T ) where W is a (d-dimensional) P-Wiener process where as α(, T ) and σ(, T ) are adaped processes. Thus concepually from (1), he HJM model is one sochasic differenial in he -variable for each fixed choice of (mauriy) T, wih he observed forward rae curve {f (0, T ) T 0} as he iniial condiion, auomaically giving a good fi beween observed and heoreical bond prices a = 0, hence ranquilizing he ask of invering he yield curve. However, observe ha Remark The HJM approach o ineres raes is no a proposal of a specific model. Raher, i is a framework o be used for analyzing ineres rae models. Hence, every shor rae model can be equivalenly formulaed in forward rae erms, and for every forward rae model, he arbirage free price of a coningen T -claim X is sill given by he pricing formula [ ( T ) ] Π(0, X ) = E exp r(s)ds X (4) 0 where he shor rae as usual is given by r(s) = f(s, s). In addiion, Remark Specifying α, σ, wih {f (0, T ) T 0}, is essenially he same as specifying he enire forward rae srucure. In fac, by he relaion ( T ) P (, T ) = exp f(, s)ds 0 he enire erm srucure {P (, T ) T > 0, 0 T } is specified. (5) Thus, based on he HJM drif heory (proved in [4]) we have he following heorem: Theorem There exiss a d-dimensional column-vecor process wih he propery T 0 & T (where [ ] T denoes ranspose). λ() = [λ 1 (),..., λ d ()] T (6) T α(, T ) = σ(, T ) σ(, s) T ds σ(, T )λ() (7)

6 6 Bernard, M. Tha is, i is possible o observe how he processes α and σ should be relaed so ha he induced sysem of bond prices admis no arbirage possibiliies wihin he d sources of randomness (one for every Wiener process) and infinie number of raded asses (one bond for each mauriy T ). Remark Hence, i is obvious ha he Brownian moions on which he HJM model depends are independen of x; ha is, he HJM is limied. 3 Quanum-heoreic Model of he One-facor Forward Raes Thus far, he classical models underlying empirical ess done by erswhile researchers such as Bühler, Uhrig-Homburg, Waler and Weber [15], Flesker [16], Sim and Thurson [17], e ceeras, have been disseced, and he anaomy characerizing heir defecs have been annoaed. Of he classical models, wha seems closer o realiy is he HJM; provided he reasoning becomes plausible, given ha i is represenaive, if i is logical and if here is empirical suppor (proof-es) for is represenaion from he quanum-heoreical model of he classical HJM. Following he Baaquie, e al. model in [1], he forward rae curve as a quanum-heoreical version of he HJM model which originaes in [2] is realized as follows: Assumpion Assuming he heory is ime ranslaion invarian. The saed one-facor quanum field-heoreic sring of he forward raes is given by: f(, x) = α(, x)σ(, x)a(, x) (8) where A(, x) is a quanum field wih he acion given by +TF R S[A] = d dxl[a] (9) 0 ( L[A] = 1 A 2 (, x) + 1 ( ) ) A(, x) 2 2 µ 2 (10) x and T F R (inroduced o ensure he acion is well defined and does no affec final resuls as he limi o infiniy is aken) is he larges ime-o-mauriy for which he forward raes are defined (in he domain of semi-infinie parallelogram given by > 0, < x < + T F R ) such ha: σ(, x) is assumed o be dependen only upon he variable θ = x based on Assumpion he iniial forward rae curve f( 0, x) is fixed he field values of A(, x) resing on he boundary poins of he domain are arbirary and are inegraion variables he second erm in he acion given in (7) valid from [12], hence no abrogaed by any arbirage. Remark As µ 0, Baaquie, e al. in [5], showed ha he model reduces o he HJM model up o a re-scaling. Recalling from [2], Baaquie, e al. gave he momen generaing funcion of he quanum field heory by he Feynman pah inegral: Z[J] = 1 DAe 0 d +T F R dxj(,x)a(,x)e S[A] (11) Z

7 Quanum Finance 7 Thus, wih some changes of variables and subsequen calculaions given in [2], Z[J] = e d T F R dθθ J(,θ)D(θ,θ ; T F R )J(,θ ) (12) for θ = x, θ = x, and he propagaor D(θ, θ ;, T F R ) given by D(θ, θ, T F R ) = µ sinh 3 (µt F R ) sinhµ(t F R θ)sinhµθ {1 + sinh 2 (µt F R )Θ(θ θ)}+ sinhµ(t F R θ)sinhµθ {1 + sinh 2 (µt F R )Θ(θ θ)}+ cosh(µt F R ){1 + sinhµ(t F R Θ)sinhµ(T F R Θ )} (13) ha is, he unconsrained boundary condiions, as discovered by Baaquie, e al. And, by he following: Assumpion Assuming he field a boundary = x (ha is, A(, )) is disribued normally wih he variance a. Since i is well known ha shor erm ineres raes are heavily influenced by cenral banks, he propagaor becomes: D 1 (θ, θ ) = D(θ, θ ) D(0, θ)d(0, θ ) D(0, θ (14) ) + a Remark Thus, i is obvious ha any of he resuls due o he no arbirage condiion is no affeced by he mean of he field a he boundary. In addiion, from he propagaor D(θ, θ ;, T F R ), also noed Baaquie, e al., he correlaor of he field A(, θ), is given by E(A(, θ)a(, θ )) = δ( )D(θ, θ ;, T F R ) (15) Thus, i can be readily shown ha he no arbirage condiion is saisfied only when α(, x) = σ(, x) x dx D(s, x ;, T F R )σ(, x ) (16) And again, in he limi µ 0, D 1, he one-facor HJM model is obained as: α(, x) = σ(, x) 4 Empirical Experimenaion x dx σ(, x ) (17) The empirical experimenaion wih he quanum-heoreic model of HJM wih Baaquie, e al. Uses daily closing prices for eurodollar fuures prices as a measure of he forward raes. Linearly inerpolaes he eurodollar fuures prices, covering 846 days over he 1990s, o calculae forward raes a 3 monh inervals (aken o be a good approximaion o he insananeous forward rae) following from [12], such ha he daase spanned 846 rading days covering he 1990s wih forward raes 7 years ino he fuure available. Parameerizes he forward raes as f(, θ), raher han f(, x) o simplify analysis considerably (since he domain shape in he (, σ) variables is recangular) and he focus is on he main quaniies: V (θ) = < δf 2 (, θ) > (18)

8 8 Bernard, M. C(θ) = < δf(, θ min)(δf(, θ) δf(, θ min )) > < δf 2 (, θ min ) > r(θ) = V (θ) C(θ) + 1 (again in line wih [17] such ha differences aken over one rading day (ɛ), δf(, θ) = f(+ɛ, θ) f(, θ) for θ min hree monhs assuming here are 250 rading days in a year o obain he discreizaion θ(0) = 1 ɛ ). In addiion, By he one-facor HJM model, expressions for he above quaniies, accurae o zeroh order in ɛ, are derived as follows: (19) (20) V HJM (θ) = σ(θ) ɛ (21) C HJM (θ) = σ(θ) σ(θ min ) 1 (22) r HJM (θ) = σ(θ min ) ɛ (23) Thus, discreizing he Brownian moion process W as W () = x (where x is a random number wih he sandard normal disribuion), noing in paricular ha he raio r HJM (θ) is independen of σ(θ) and is in fac consan. However, as saed by Baaquie, e al. he raio calculaed from he daa was far from consan, confirming ha he ime ranslaion invarian one-facpr HJM model remains inconsisen wih he real evoluion of he FRC (Forward Rae Curve) for any choice of funcion σ(θ) as expeced. Again deriving expressions for he above quaniy o zeroh-order accuracy in ɛ using he unconsrained quanum field-heoreic model, we have he following: 1 ɛ C QF T (θ) = V QF T (θ) = σ(θ) D(θ, θ;, T F R )ɛ (24) σ(θ)d(θ, θ min ;, T T R ) σ(θ min )D(θ min, θ min ;, T F R ) 1 (25) Thus, giving he raion r(θ) in his model o be: r QF T (θ) = σ(θ min) ɛd(θ, θ;, T F R )D(θ min, θ min ;, T F R ) D(θ, θ min ;, T T R ) which Baaquie, e al. noed o be no longer consan. (26) However sill independen of σ(θ), his suddenly no-longer-consan raio has he possibiliy of fiing he raio in order o find he µ and σ(θ min ).

9 Quanum Finance 9 Thus, Baaquie, e al. aking he limi of T F R (as required) and using he Levenberg-Marquard mehod [13] obained he non-linear leas squares fi (as shown in able I of his paper [1]), wih confidence inervals obained hrough he boosrap mehod [14] and an alernaive confidence inerval by dividing he daa ino series of 500 days saring from he firs day, second day,... calculaing funcion r(σ), and fiing parameers (for he resuling 346 daa ses as shown in figure 1 of [1]). And, similar o esimaes of σ(θ) (ploed in figure 3 of [1]) for he one-facor HJM model, wo differen esimaes (ploed in figure 2 of [1]) of he funcion σ(θ) are obained using equaions 22 and 23. Remark Thus, he HJM model was shown o be inconsisen wih he daa, while on he conrary, he quanum field-heoreic model was consisen wih daa. Besides, consan or exponenial forms, commonly used in he lieraure, was very far from he volailiy funcion for he HJM model derived from he daa. Also, Baaquie, e al. repeaed he same procedure for he consrained quanum field-heoreic model, and showed ha he agreemen beween he wo funcions is beer han in he case of he unconsrained model as may be expeced due o he addiional parameer involve based on he obained resuls depiced in able II, he fied raio shown in figure 4 of [1], and he wo esimaes of σ(θ) presened in figure 5 of [1], alhough he model may be over-specified since differen values of he parameers give rise o very similar values for r(θ) as refleced by he large confidence inervals. Furhermore, based on he assumpion Assumpion ha σ is only a funcion of σ, and α is also only a funcion of θ, and ha he iniial FRC is fla or ha he effec of he iniial FRC becomes negligible afer a long ime he mean spread beween he forward raes and he spo rae, given by: s(θ) =< f(, θ) f(, θ min ) > (27) which is essenially a linear sum of wo pars in he model: he marke price of risk and he no arbirage condiion) is derived hus as: θ S QF T = (θ θ min ) lim α() α()d θ min (28) (for he no arbirage condiion in he quanum field-heoreical model) where α QF T () = σ() 0 σ(θ)d(, θ;, T F R )dθ (29) Baaquie, e al. calculaed he spread (due o he no arbirage condiion), by applying numerical inegraion by rapezoidal mehod (chosen due o he relaive inaccuracy in he esimaion of σ(θ) in he firs place) o one of he esimaes of σ(θ) eiher one giving similar resuls and observed ha he calculaed spread was significanly smaller han he acual spread, even for when he procedure is repeaed for he consrained quanum field-heoreic model. Remark Thus, giving showing consisency wih he exisence of he spread due o risk aversion (alhough a significan porion of he spread could be derived from he way he forward rae curve evolves).

10 10 Bernard, M. 5 Conclusion Classical (sochasic) models of forward raes have been shown o be inadequae o fi for real phenomena happening wih jusifiable dyamical quanum causes. However, he quanum field heory (QFT) mehod, which is sill undergoing developmen has been shown o hold much nicer feaures, even for raher fundamenal models such he binomial model for forward raes. In addiion, he QFT: Presens a new way o es models such as he one-facor, ime ranslaion invarian Heah-Jarrow-Moron, Baaquies one-facor, and he ime ranslaion invarian quanum field-heoreic model Shows relaively higher consisency wih daa (even when he boundary condiions are consrained so ha i may reflec he special naure of he spo rae, and he parameers can no be sufficienly and accuraely derived using he mehod), and Explains a significan porion of he spread beween he forward raes and spo rae beer han he classical models. References [1] Baaquie, B.E., Srikan, M An Empirical Invesigaion of a Quanum Field Theory of Forward Raes, Physical Review E, o , [2] Baaquie, B.E. Quanum Field Theory of Treasury Bonds, Inl. Workshop on Nonlinear Physics, Theory and Experimen II, Gallipoli, Lecce, [3] Baaquie, B.E., Coriano, C., and Srikan, M. Quanum Mechanics, Pah Inegrals and Opion Pricing: Reducing he Complexiy of Finance, Inl. Workshop on Nonlinear Physics, Theory and Experimen II, Gallipoli, Lecce, [4] Björk, T. Arbirage heory in coninuous ime, Oxford Universiy Press, [5] Vasicek, O. An equilibrium characerizaion of he erm srucure, Journal of Financial Economics, 5, [6] Cox, J., Ingersoll, J., and Ross, S. An iner-emporal general equilibrium model of asse prices, Economerica, 53, 2, 1985a. [7] Cox, J., Ingersoll, J., Ross, S. An iner-emporal general equilibrium model of asse prices, Economerica, 53, 2, 1985b. [8] Hull, J., and Whie, A. Pricing ineres rae derivaive securiies, The Review of Financial Sudies, 3, 1, [9] Bjork, T., and Hyll, M. On he inversion of he yield curve, Working paper, Sockholm School of Economics, [10] Heah, D., Jarrow, R., and Moron, A. Bond pricing and he erm srucure of ineres raes: a new mehodology, Economerica, 60, 1, [11] Brown, R.G., and Schaefer, S.M. Ineres rae volailiy and he shape of he erm srucure, Phil. Trans. R. Soc., Lond., [12] Bouchaud, J.P., Con R., El-Karoui N., Poers M., and Sagna N. Phenomenology of he Ineres Rae Curve, Applied Mahemaical Finance, 6, 1997.

11 Quanum Finance 11 [13] Marquard, D.W. An algorihm for leas-squares esimaion of nonlinear parameers, Journal of he Sociey for Indusrial and Applied Mahemaics, 11, [14] Efron, B. The jacknife, he boosrap, and oher resampling plans, SIAM, Philadelphia, [15] Buhler W., Uhrig-Homburg M., Waler U., and Weber T. An Empirical Comparison of Forward-Rae and Spo-Rae Models for Valuing Ineres-Rae Opions, Journal of Finance, 54, [16] Flesker B., Tesing of he Heah-Jarrow-Moron / Ho-Lee Model of Ineres Rae Coningen Claims Pricing, Journal of Financial and Quaniaive Analysis, 38, [17] Sim, A.B., and Thurson, D.C. An empirical sudy of a new class of no-arbirage-based discree models of he erm srucure, The Journal of Financial Research, 19, 1996.