Appendix of the Paper The Role of No-Arbitrage on Forecasting: Lessons from a Parametric Term Structure Model

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1 Appendix of the Paper The Roe of No-Arbitrage on Forecasting: Lessons from a Parametric Term Structure Mode Caio Ameida cameida@fgv.br José Vicente jose.vaentim@bcb.gov.br June Introduction In this appendix we give a suppementary materia to the paper The Roe of No- Arbitrage on Forecasting: Lessons from a Parametric Term Structure Mode. First we present a proof of Theorem 1. Next we study the agebric detais of the AFG and AFSV versions of the Legendre mode with affine dynamics. Proof of Theorem 1 Theorem 1 Assume Y t -dynamics under a probabiity measure Q equivaent to P given by: dy t = µ Q (Y t )dt + σ(y t )dw t, (1) where W is a Browian motion under Q. If µ Q (Y t ) satisfies the restriction expressed in Equation (), Q is an equivaent martingae measure and the AF conditions hod 1. N j= (j 1)L jy t,j τ j = N j=1 L jµ Q j (Y t)τ j 1 [ N ] j=1 Γ jk (Y t ) = 0 for j > [ N ] or k > [ N ] [ N ] k=1 Γ jk(y t ) τ j+k 1 k 1 In addition to the drift restriction, σ(y t ) shoud present enough reguarity to guarantee that discounted bond prices that are oca martingaes, aso become martingaes. In practica probems, a bounded or a square-affine σ(y t ) is enough to enforce the martingae condition. 1 ()

2 with Γ(Y t ) = Lσ(Y t )σ(y t )L, L j standing for the j th -ine of an upper trianguar matrix that depends ony on, and [ ] representing the integer part of a number. Proof The term structure of the Legendre poynomia mode is given by: R(τ, Y t ) = G(τ) Y t = N Y t,n P n 1 ( τ n=1 1), () that is, the oadings of the term structure are Legendre poynomias. Therefore, the τ-maturity instantaneous forward rate is ( N f(τ, Y t ) = Y t,n P n 1 ( τ N 1) + τ P n 1 ( τ Y 1) ) t,n. (4) τ n=1 In the equation above, the forward rates are expressed as inear combinations of Legendre poynomias, which can be readiy expressed as inear combinations of powers of τ: N f(τ, Y t ) = L n Y t τ n 1, (5) n=1 n=1

3 where L n is the n th row of the upper trianguar matrix L. In fact, (5) defines matrix L. If N = 6 the matrix L is L =. (7) Proposition. of Fiipovic (1999) presents conditions on f(τ, Y t ), which guarantee that discounted bond prices are martingaes under any specific interest rate mode. Using these conditions, Ameida (005) proves that if the AF restrictions () hod, then the Legendre poynomia mode is arbitrage-free. Using the first six Legendre poynomias we have f(τ, Y t ) = Y t,1 + Y t, x + Yt, (x 1) + Yt,4 (5x x)+ Y t,5 8 (5x4 0x + ) + Yt,6 8 (6x5 70x + 15x)+ τ τ [ ] Y t, + Y t, x + Yt,4 (15x ) [ ] Yt,5 8 (140x 60x) + Yt,6 8 (15x4 10x + 15). + (6) where x = τ 1. Coecting terms that are powers of τ in the expression above we obtain the upper trianguar matrix L for N = 6. Basicay, Proposition. of Fiipovic (1999) imposes a specific reationship between the partia derivatives of f(τ, Y t ).

4 Technica Detais about the Sub-Cass of Arbitrage-Free Legendre Modes with Affine Dynamics. The affine cass of dynamic term structure modes is composed by processes whose state vector Y is an affine diffusion 4, and whose impied short term rate is affine in Y. Dai and Singeton (000) proposed the foowing notation to describe the dynamics of canonica affine modes under the risk neutra measure Q: dy t = µ Q (Y t )dt + Σ S t (Y t )dw t = κ Q (θ Q Y t )dt + Σ S t (Y t )dw t (8) where κ Q and Σ are N N matrices, θ Q is a R N -vector, and S t is diagona matrix with eements St ii = α i + β i Y t for some scaar α i and some R N -vector β i. Now suppose we want to equip the affine cass of modes with a oadings structure composed by Legendre poynomias 5. To this end, we have to impose the AF restrictions of Theorem 1. Consider the auxiiary state space vector Ỹt defined by Ỹ t = LY t, (9) where L is the upper trianguar matrix of Theorem 1. This auxiiary process characterizes term structure movements when the oadings come from a power series in the maturity variabe τ. It appears as an intermediate step in cacuations. The dynamics of Ỹt under probabiity measure Q is given by dỹt = µ Q (Ỹt)dt + Σ S t (Ỹt)dWt, (10) where the parameters of this stochastic differentia equations system are defined in simiar way to (8) (i.e., S ii t = α i + β i Ỹ t for some scaar α i and some R N -vector β i and so on) and are reated through (9) with the corresponding parameters in (8). It shoud be cear that Ỹt is affine if, and ony if, Y t is affine, because L is invertibe. Under this particuar sub-cass (affine pus poynomia oadings), the first requirement of AF restrictions becomes N (j 1)Ỹt,jτ j = j= [ N N ] [ µ Q j (Ỹt)τ N ] j 1 ( H 0,jk + H 1,jk Ỹ t ) τ j+k 1, (11) k j=1 j=1 k=1 4 This means that the drift and the squared diffusion terms of Y are affine functions of Y. 5 Note that it is not possibe to make use of Duffie and Kan (1996) separation arguments that ead to their pair of Ricatti equations since the Legendre poynomias do not satisfy one of the agebraic conditions stated in their main theorem. 4

5 where ( Σ St Σ ) = H 0ij + H 1ij Ỹ t, with H 0ij R and H 1ij R N. ij In particuar, by matching coefficients on the maturity variabe τ in (11), we obtain an expicit expression for the drift of the auxiiary process: µ Q ( Ỹ t )i = iỹt,i+1 + Min{i 1,[ N ]} j=max{1,i [ N ]} H 0,j(i j) + H 1,j(i j) Ỹ t. (1) i j This expression can be readiy transated to a simiar expression for the drift of the origina state vector Y with the use of (9). In the empirica section of our paper, we compare a three factor CS version with corresponding AF versions that present three stochastic factors with nonnu diffusions. By Theorem 1, a natura way to impement this appication, is to work with AF versions driven by six factors (three stochastic, three conditionay deterministic). In the next ines, we provide the restrictions that shoud be impemented to generate affine modes with poynomia oadings, and how to transate those restrictions to generate affine modes with Legendre poynomia oadings. After that, we expain in detais the two AF versions chosen to be impemented in this work: the Arbitrage-Free Gaussian (AFG) version, in which the voatiity of Y is deterministic and time independent, and the Arbitrage-Free Stochastic Voatiity (AFSV) version, in which ony one stochastic factor determines the voatiity of Y. 5

6 When N = 6 the dynamics of Ỹt has the foowing form: { S t ii αi + (Ỹt) = β iỹt if i 0 if i >, Σ i,j = 0 i, j >, µ Q (Ỹt) 1 = Ỹt,, µ Q (Ỹt) = Ỹt, + H 0,11 + H 1,11 Ỹ t, µ Q (Ỹt) = Ỹt,4 + H 0,1 µ Q (Ỹt) 4 = 4Ỹt,5 + H 0,1 µ Q (Ỹt) 5 = 5Ỹt,6 + H 0, + H 1,1 Ỹ t + H 0,1 + H 1,1 Ỹ t, + H 1,1 Ỹ t + H 0, + H 1, Ỹ t + H 0, + H 1, Ỹ t + H 0,1 + H 1,1 Ỹ t, + H 1, Ỹ t, (1) µ Q (Ỹt) 6 = H 0, + H 1, Ỹ t. The dynamics of the term structure movements Y under the origina Legendre poynomia parameterization can then be obtained by soving (9). To that end, et us rewrite the drift µ Q in matrix notation, as an affine transformation of Ỹ : µ Q (Ỹt) = M + UỸt, (14) where U = U 1 + U, and U 1, U and M are given by: 0 H 0,11 H 0,1 + H 0,1 M = H 0,1 + H 0, + H, (15) 0,1 H 0, + H 0, 6 H 0,

7 and U 1 = U = H 1,11 H 1,1 + H 1,1 H 1,1 + H 1, + H 1,1 H 1, + H 1, H 1, Finay, the drift and diffusion of process Y are given by:, (16). (17) µ Q (Y t ) = L 1 µ Q (Ỹt) = L 1 µ Q (LY t ) = L 1 M + L 1 ULY t (18) σ(y t ) = L 1 Σ S t (LY t ). (19) In our empirica appication, the maximum maturity is equa to = 10 years. Then, matrix L is given by: L = (0) Now we are ready to speciaize the drift restriction (1) to each particuar AF version impemented in this paper (AFG and AFSV), and aso to obtain the corresponding restrictions for the process of interest Y, the one that drives term structure movements within the Legendre poynomia mode. 7

8 .1 The AFG Version Noting that in this version the matrix controing the diffusion structure of vector Ỹ, i.e. S(.), is the identity matrix, we directy obtain Σ Σ = H 0, and from (19) we obtain the reation between H 0 and Σ: H 0 = LΣ ((L 1 ) ) 1 = LΣ L. (1) If we adopt a diagona matrix representation for Σ 6, with Σ ii as the i th -diagona term, then, in order to match the second requirement of AF restrictions we must have Σ ii = 0 for i 4. Therefore, using transformation L between Y and Ỹ, H0 can be expicity reated to the non-nu diagona terms in Σ: H 0 = Σ 11 + Σ + Σ 0.4Σ 1.Σ 0.18Σ Σ 1.Σ 0.16Σ Σ 0.16Σ Σ 0.16Σ 0.04Σ () Since U is nu under the Gaussian version, we earn from (18) that the two matrices (L 1 M and L 1 U 1 L) necessary to obtain an expicit expression for the 6 This representation for Σ provides exacty the same identification structure of Dai and Singeton (00). 8

9 drift µ Q (Y t ) are given by: and L 1 U 1 L = L 1 M = 5 Σ Σ + 1 Σ 5 Σ 11 + Σ Σ 5 Σ Σ Σ + Σ 9 7 Σ 5 7 Σ () (4) Note that Y 4, Y 5, and Y 6 are deterministic factors under the Gaussian case. This is a consequence of two facts: (i) their dynamics do not depend on the Brownian motion vector, and (ii) their drifts do not depend on the first three components of the state vector. With matrices L 1 M and L 1 U 1 L in hands, we obtain the drift of vector Y, and in particuar, the drifts of the deterministic factors Y 4, Y 5, and Y 6 : µ Q (Y t ) 4 = Σ + Σ Y t,5 0.6Y t,6, µ Q (Y t ) 5 = 9 7 Σ +.16Y t,6, (5) µ Q (Y t ) 6 = 5 7 Σ. By expicity soving the ordinary differentia equations impied for these factors, we have Y t,4 = Y 0,4 + (Σ + Σ Y 0,5 0.6Y 0,6 )t + (0.9Σ Y 0,6 )t Σ t, (6) 9

10 Y t,5 = Y 0,5 + ( 9 7 Σ +.16Y 0,6 )t Σ t, (7) Y t,6 = Y 0, Σ t. (8) Note that, under this Gaussian version, the dynamics of the state variabes Y t,4, Y t,5 and Y t,6, in addition to being deterministic, are competey determined by parameters Σ, Σ, and the initia conditions Y 0,4, Y 0,5 and Y 0,6.. The AFSV Version The AFSV version, presents one stochastic factor driving the stochastic voatiity of the three stochastic factors (Y 1, Y and Y ). In order to keep the risk-neutra dynamics of both Y t and Ỹt within the sub-cass of affine modes with ony one factor determining the voatiity, we choose factor Y to drive the stochastic voatiity 7. Specificay we set β i = [0 0 β i 0 0 0], what gives: H 1,ij = [0 0 h ij 0 0 0] 1 i, j 6; where (ΣS t Σ ) ij = H 0ij + H 1ij Y t with H 1ij R N. This specifications impy that H 1 Y t = Y t, H, where in the right-hand side we have a tensor product, with H being a 6 6-matrix with eements h ij (see Duffie (001) for the tensoria notation). From (19) and the reation Ỹt = LY t we obtain H 0 = Σ (diag [α 1,..., α 6 ]) Σ, H 0 = LH 0 L, H = Σ (diag [b 1,..., b 6 ]) Σ. Since H 1 Ỹt = L (H 1 Y t ) L = Y t, LHL we have [ 70 H 1,ij = 0 0 z ij 11.5z ij 7 z ij with z ij = (LHL ) ij. ] z ij, (9) 7 Choices of any of the two remaining factors to capture stochastic voatiity coud have been impemented but with higher computationa costs. 10

11 Hence from (15), (16) and (17) we can express M and U as we as the drift of Y t as functions of α i, β i (i = 1,..., 6) and Σ. Finay, for identification purposes, in the empirica impementation of this version, we fix Σ to be a diagona matrix (with Σ ii = 0 for i 4 in order to match the second requirement of AF restrictions) and α = [ ].. Estimation Procedures How does one estimate CS and AF versions of the Legendre poynomia mode? For the CS version, we run cross-sectiona independent regressions for each point t in time, within the sampe period. In a market of zero coupon bonds, assuming that we observe yieds R obs with measurement error, the mode is estimated with the use of the foowing inear regression: Ŷ t = (F F ) 1 F R t obs, (0) where Robs t is a vector containing observed yieds, at time t, for different maturities (τ 1,...τ k ), and F is the foowing matrix: F = P 0 ( τ 1 1) P 1 ( τ 1 1)... P N 1 ( τ 1 1) P 0 ( τ 1) P 1 ( τ 1)... P N 1 ( τ 1).. P 0 ( τ k 1) P 1 ( τ k 1)... P N 1 ( τ k 1) For both AF versions we use the Quasi-Maximum Likeihood (QML) procedure, adopting the methodoogy proposed by Chen and Scott (199), with -,5-, and 10-year maturity zero-coupon bonds priced exacty and -, and 7-year maturity zero-coupon bonds priced with i.i.d zero-mean errors. The conditiona transition densities are obtained with the use of cosed-form formuas for the first and the second moments of Y within the affine framework (see for instance, Duffee, 00, and Jacobs and Karoui, 006). Observe that for the AFG version the QML is, in fact, a pure maximum ikeihood procedure since the transitions densities come exacty from a norma distribution... 11

12 References [1] Ameida, C.R. (005). Affine Processes, Arbitrage-Free Term Structures of Legendre Poynomias, and Option Pricing, Internationa Journa of Theoretica and Appied Finance, 8,, 1-. [] Chen, R.R. and Scott, L. (199). Maximum Likeihood Estimation for a Mutifactor Equiibrium Mode of the Term Structure of Interest Rates. Journa of Fixed Income,, [] Dai, Q. and Singeton, K. (000). Specification Anaysis of Affine Term Structure Modes. Journa of Finance, LV, 5, [4] Dai, Q. and Singeton, K. (00). Expectation Puzzes, Time-Varying Risk Premia, and Affine Modes of the Term Structure. Journa of Financia Economics, 6, [5] Duffee, G. R. (00). Term Premia and Interest Rates Forecasts in Affine Modes. Journa of Finance, 57, [6] Duffie, D. (001). Dynamic Asset Pricing Theory. Princeton University Press. [7] Duffie, D. and Kan, R. (1996). A Yied Factor Mode of Interest Rates. Mathematica Finance, 6, 4, [8] Fiipovic, D. (1999). A Note on the Neson and Siege Famiy. Mathematica Finance, 9, 4, [9] Jacobs, K. and Karoui, L. (006). Affine Term Structure Modes, Voatiity and the Segmentation Hypothesis, Working Paper, McGi University. 1