Working Paper Series. Working Paper No. 8. Affine Models. Christa Cuchiero, Damir Filipović, and Josef Teichmann. First version: April 2008

Transcription

1 Working Paper Series Working Paper No. 8 Affine Models Chrisa Cuchiero, Damir Filipović, and Josef Teichmann Firs version: April 2008 Curren version: Ocober 2008

2 AFFINE MODELS CHRISTA CUCHIERO, DAMIR FILIPOVIC, JOSEF TEICHMANN Absrac. Affine erm srucure models have gained significan aenion in he finance lieraure, mainly due o heir analyical racabiliy and saisical flexibiliy. The aim of his aricle is o presen boh heoreical foundaions as well as empirical aspecs of he affine model class. Saring from he original one-facor shorrae models of Vasiček and Cox e al, we provide an overview of he properies of regular affine processes and explain heir relaionship o affine erm srucure models. Mehods for securiies pricing and for parameer esimaion are also discussed, demonsraing how he analyical racabiliy of affine models can be exploied for pracical purposes. Key words and phrases. Affine Term Srucure, Affine Process, Characerisic Funcion, Pricing, Esimaion. The firs and hird auhor graefully acknowledge he suppor from he FWF-gran Y 328 (START prize from he Ausrian Science Fund). The second auhor graefully acknowledges he suppor from WWTF (Vienna Science and Technology Fund). 1

3 2 CHRISTA CUCHIERO, DAMIR FILIPOVIC, JOSEF TEICHMANN 1. Definiion Noaion 1. Throughou he aricle,, denoes he sandard scalar produc on R N. Definiion 1. Le r be a shor rae model specified as an affine funcion of an N-dimensional Markov process X wih sae space D R N : r = l + λ, X, (1) for some (non ime-dependen) consans l R and λ R N. This is called an Affine Term Srucure Model (ATSM) if he zero-coupon bond price has exponenial affine form, i.e. P (, T ) = E [e T r sds ] X = e G(,T )+ H(,T ),X, (2) where E denoes he expecaion under a risk neural probabiliy measure. 2. Early Examples Early well-known examples are he Vasiček [14] and he Cox, Ingersoll, Ross [5] (see eqf and eqf11-025) ime-homogeneous onefacor shor rae models. In (1), boh models are characerized by N = 1, l = 0 and λ = Vasiček Model: X follows an Ornsein-Uhlenbeck process on D = R, dx = (b + βx )d + σdw, b, β R, σ R +,

4 AFFINE MODELS 3 where W is a sandard Brownian moion. Under hese model specificaions, bond prices can be explicily calculaed and he corresponding coefficiens G and H in (2) are given by H(, T ) = G(, T ) = σ2 2 ) 1 eβ(t, β T provided ha β 0. (see also eqf11-024) T H 2 (s, T )ds + b H(s, T )ds, 2.2. Cox-Ingersoll-Ross Model: X is defined as he soluion of he following affine diffusion process on D = R +, known as Feller square roo process, dx = (b + βx )d + σ X dw, b, σ R +, β R. Like in he Vasiček model, here is a closed-form soluion for he bond price. If σ 0, G and H in (2) are hen of he form: ( ) G(, T ) = 2b (γ β)(t )/2 σ ln 2γe 2 (γ β)(e γ(t ) 1) + 2γ, H(, T ) = 2(e γ(t ) 1) (γ β)(e γ(t ) 1) + 2γ, where γ := β 2 + 2σ 2. (see also eqf11-025) Since he developmen of hese firs one-dimensional erm-srucure models, many muli-facor exensions have been considered wih he aim o provide more realisic models.

5 4 CHRISTA CUCHIERO, DAMIR FILIPOVIC, JOSEF TEICHMANN 3. Regular Affine Processes The generic mehod o consruc ATSMs is o use regular affine processes. A concise mahemaical foundaion was provided by Duffie, Filipović and Schachermayer [7]. Henceforh, we fix he sae space D = R m + R N m, for some 0 m N. Definiion 2. A Markov process X is called regular affine if is characerisic funcion has exponenial-affine dependence on he iniial sae, i.e. for R + and u ir N, here exis φ(, u) C and ψ(, u) C N, such ha for all x D E [ e u,x X 0 = x ] = e φ(,u)+ ψ(,u),x. (3) Moreover, he funcions φ and ψ are coninuous in and + φ(, u) =0 and + ψ(, u) =0 exis and are coninuous a u = 0. Regular affine processes have been defined and compleely characerized in [7]. The main resul is saed below. Theorem 3. A regular affine process is a Feller semimaringale wih infiniesimal generaor Af(x) = + N k,l=1 D\{0} A kl (x) 2 f(x) x k x l + B(x), f(x) C(x)f(x) (f(x + ξ) f(x) f(x), χ(ξ) )M(x, dξ), (4)

6 AFFINE MODELS 5 for f in he se of smooh es funcions, wih m A(x) = a + x i α i, a, α i R N N, (5) i=1 N B(x) = b + x i β i, b, β i R N, (6) i=1 m C(x) = c + x i γ i, c, γ i R +, (7) i=1 M(x, dξ) = m(dξ) + m x i µ i (dξ), (8) where m, µ i are Borel measures on D\{0} and χ : R N R N some bounded coninuous runcaion funcion wih χ(ξ) = ξ in a neighborhood of 0. Furhermore, φ and ψ in (3) solve he generalized Riccai equaions, i=1 φ(, u) = F (ψ(, u)), φ(0, u) = 0, (9) ψ(, u) = R(ψ(, u)), ψ(0, u) = u, (10) wih F (u) = au, u + b, u c + D\{0} R i (u) = α i u, u + β i, u γ i + for i {1,..., m}, D\{0} R i (u) = β i, u, for i {m + 1,..., N}. ( ) e u,ξ 1 u, χ(ξ) m(dξ), ( ) e u,ξ 1 u, χ(ξ) µ i (dξ), Conversely, for any choice of admissible parameers a, α i, b, β i, c, γ i, m, µ i, here exiss a unique regular affine process wih generaor (4).

7 6 CHRISTA CUCHIERO, DAMIR FILIPOVIC, JOSEF TEICHMANN Remark 4. I is worh noing ha he infiniesimal generaor of every Feller process on R N has he form of he above inegro-differenial operaor (4) wih some funcions A, B, C and a kernel M. The specific characerisic of regular affine processes is ha hese funcions are all affine, as described in (5) - (8). Observe furhermore ha by he definiion of he infiniesimal generaor and he form of F and R, we have d d E[ e u,x X 0 = x ] =0+ = ( + φ(, u) =0 + + ψ(, u) =0 ) e u,x = (F (u) + R(u), x ) e u,x = Ae u,x. This gives he link beween he form of he operaor A and he funcions F and R in he Riccai equaions (9) and (10). Remark 5. The above parameers saisfy cerain admissibiliy condiions guaraneeing he exisence of he process in D. These parameer resricions can be found in Definiion 2.6 and equaions (2.23)-(2.24) in [7]. We noe ha admissibiliy in paricular means α i,kl = 0 for i, k, l m unless k = l = i. 4. Sysemaic analysis 4.1. Regular affine processes and ATSMs. Regular affine processes generically induce ATSMs. This relaion is explicily saed in he subsequen argumen. Under some echnical condiions which are

8 AFFINE MODELS 7 specified in [7] chaper 11, we have for r as defined in (1), E [e ] 0 rsds e u,x X0 = x = e φ(,u)+ ψ(,u),x, (11) where φ(, u) = F ( ψ(, u)), ψ(, u) = R( ψ(, u)), wih F (u) = F (u) l and R(u) = R(u) λ. Seing u = 0 in (11), one immediaely ges (2) wih G(, T ) = φ(t, 0) and H(, T ) = ψ(t, 0) Diffusion case. Conversely, for a class of diffusions dx = B(X )d + σ(x )dw (12) on D, Duffie and Kan [8] analyzed when (2) implies an affine diffusion marix A = σσt 2 and an affine drif B of form (5) and (6) respecively One dimensional nonnegaive Markov process. For D = R +, Filipović [9] showed ha (1) defines an ATSM if and only if X is a regular affine process Relaion o Heah-Jarrow-Moron framework. Filipović and Teichmann [10] esablished a relaion beween he Heah-Jarrow-Moron (HJM) framework (see eqf11-022) and ATSMs: Essenially, all generic finie dimensional realizaions 1 of a HJM erm srucure model are ime-inhomogeneous ATSMs. 1 For a precise definiion see [10].

9 8 CHRISTA CUCHIERO, DAMIR FILIPOVIC, JOSEF TEICHMANN 5. Canonical Represenaion An ATSM semming from a regular affine diffusion process X on R m + R N m can be represened in differen ways by applying nonsingular affine ransformaions o X. Indeed, for every nonsingular N N-marix K and κ R N, he ransformaion KX + κ modifies he paricular form of (12) and he shor rae process (1), while observable quaniies (e.g. he erm srucure or bond prices) remain unchanged. In order o group hose N-dimensional ATSMs generaing idenical erm srucures, Dai and Singleon [6] found N +1 subfamilies A m (N), where 0 m N is he number of sae variables acually appearing in he diffusion marix (i.e. he dimension of he posiive half space). For each class, hey specified a canonical represenaion whose diffusion marix σσ T is of diagonal form wih (σσ T (x)) kk = x k, k m 1 + m i=1 λ k,ix i k > m, where λ k,i R. For N 3 he Dai-Singleon specificaion comprises all ATSMs generaed by regular affine diffusions on R m + R N m. The general siuaion N > 3 was analyzed by Cheridio, Filipović and Kimmel [4]. 6. Empirical Aspecs 6.1. Pricing: The price of a claim wih payoff funcion f(x ) is given by he risk neural expecaion formula π(, x) = E [e ] 0 rsds f(x ) X 0 = x.

10 Suppose f can be expressed by AFFINE MODELS 9 f(x) = e C+iλ,x f(λ)dλ, R N λ R N, (13) for some inegrable funcion f and some consan C R N. If, moreover E [e ] 0 rsds e C,X X0 = x <, hen (11) implies [ ( ) ] π(, x) = E e 0 R rsds e C+iλ,X f(λ)dλ X 0 = x N = = R N E [e 0 rsds e C+iλ,X X0 = x] f(λ)dλ R N e φ(,c+iλ)+ ψ(,c+iλ),x f(λ)dλ. Hence, he price π(, x) can be compued via numerical inegraion, since he inegrands are in principle known. For insance, in he case N = 1, he payoff funcion of a European call (e x e k ) +, where x corresponds o he log price of he underlying and k o he log srike price, saisfies (13). In paricular, we have he following inegral represenaion (see [11]) (e x e k ) + = 1 2π R e (C+iλ)x e k(1 C iλ) (C + iλ)(c + iλ 1) dλ. Therefore, he previous formula o compue he price of he call π(, x) is applicable. An alernaive approach leading o he same resul can be found in Carr and Madan [3].

11 10 CHRISTA CUCHIERO, DAMIR FILIPOVIC, JOSEF TEICHMANN 6.2. Esimaion: Saisical mehods o esimae he parameers of ATSMs have been based on maximum likelihood and generalized mehod of momens. Concerning maximum likelihood echniques, he condiional log densiies enering ino he log likelihood funcion can in general be obained by inverse Fourier ransformaion. Since his procedure is compuaionally cosly, several approximaions and limied-informaion esimaion sraegies have been considered (e.g. [13]). Anoher possibiliy is o use closed form expansions of he log likelihood funcion which are available for general diffusions [1] and which have been applied o ATSMs. In he case of Gaussian and Cox-Ingersoll-Ross models, one can forgo such echniques, since he log densiies are known in closed form (e.g [12]). As condiional momens of he form E[X m X s] n for m, n 0 can be compued from he derivaives of he condiional characerisic funcion and are in general explicily known up o he soluion of he Riccai ODEs (9) and (10), he generalized mehod of momens is an alernaive o maximum likelihood esimaion (e.g. [2]). 7. Relaed aricles eqf eqf eqf eqf eqf eqf13-009

12 AFFINE MODELS 11 References [1] Y. Ai-Sahalia. Closed-form likelihood expansions for mulivariae diffusions. Annals of Saisics, 36(2): , [2] T. G. Andersen and B. E. Sørensen. GMM esimaion of a sochasic volailiy model: A Mone Carlo sudy. Journal of Business & Economic Saisics, 14(3): , [3] P. Carr and D. Madan. Opion valuaion using he fas Fourier ransform. Journal of Compuaional Finance, 2(4):61 73, [4] P. Cheridio, D. Filipović, and R. L. Kimmel. A noe on he Dai-Singleon canonical represenaion of affine erm srucure models. forhcoming in Mahemaical Finance. [5] J. C. Cox, J. E. Ingersoll, and S. A. Ross. A heory of he erm srucure of ineres raes. Economerica, 53(2): , [6] Q. Dai and K. J. Singleon. Specificaion analysis of affine erm srucure models. Journal of Finance, 55: , [7] D. Duffie, D. Filipović, and W. Schachermayer. Affine processes and applicaions in finance. The Annals of Applied Probabiliy, 13(3): , [8] D. Duffie and R. Kan. A yield-facor model of ineres raes. Mahemaical Finance, 6(4): , [9] D. Filipović. A general characerizaion of one facor affine erm srucure models. Finance and Sochasics, 5(3): , [10] D. Filipović and J. Teichmann. On he geomery of he erm srucure of ineres raes. Proceedings of The Royal Sociey of London. Series A. Mahemaical, Physical and Engineering Sciences, 460(2041): , [11] F. Hubalek, J. Kallsen, and L. Krawczyk. Variance-opimal hedging for processes wih saionary independen incremens. Annals of Applied Probabiliy, 16(2): , 2006.

13 12 CHRISTA CUCHIERO, DAMIR FILIPOVIC, JOSEF TEICHMANN [12] N. D. Pearson and T.-S. Sun. Exploiing he condiional densiy in esimaing he erm srucure: An applicaion o he Cox, Ingersoll, and Ross model. Journal of Finance, 49(4): , [13] K. J. Singleon. Esimaion of affine asse pricing models using he empirical characerisic funcion. Journal of Economerics, 102: , [14] O. Vasiček. An equilibrium characerizaion of he erm srucure. Journal of Financial Economics, 5: , Vienna Insiue of Finance, Universiy of Vienna, and Vienna Universiy of Economics and Business Adminisraion, Heiligensäder Srasse 46-48, A-1190 Wien, Ausria; Vienna Universiy of Technology, Deparmen of Mahemaical Mehods in Economics, Wiedner Haupsrasse 8 10, A-1040 Wien, Ausria address: cuchiero@fam.uwien.ac.a, damir.filipovic@vif.ac.a, jeichma@fam.uwien.ac.a