Affine Processes. Econometric specifications. Eduardo Rossi. University of Pavia. March 17, 2009

Transcription

1 Affine Processes Econometric specifications Eduardo Rossi University of Pavia March 17, 2009 Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

2 Outline 1 Affine Processes 2 Affine Term Structure Models 3 The general case A 1 (1) canonical model A 1 (1) canonical model Measurement errors Sample log-likelihood Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

3 Introduction Affine Processes The affine processes are among the most widely studied time series processes in the empirical finance literature. Accommodation of stochastic volatility volatility jumps correlations among risk factors. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

4 Affine Processes Affine Processes Affine Process An affine process Y is one for which the conditional mean and variance are affine functions of Y. Characterization of affine processes in terms of Exponential-affine Fourier transform for continuous time Laplace transform for discrete-time. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

5 Markov Process Affine Processes Probability Space (Ω, F, P), information set F t. First-order Markov Process (MP) Y taking on values in a state space D R N. Markov Process A process is Markov if, for any measurable function, g : D R and for any fixed times t and s, s > t for some function h : D R. E t [g(y s )] = h(y t ) Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

6 Affine Processes Conditional Characteristic Fucntion CCF of a Markov Process The conditional characteristic function (CCF) of a MP, Y T, conditioned on current and lagged information about Y at date t, is given by the Fourier transform of its conditional density function: CCF t (τ, u) ( ) E t e iu Y T Y t u R N = e iu Y T f Y (Y T Y t ; γ)dy T R N where τ = (T t), i = 1, and f Y is the conditional density of Y. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

7 Affine Processes Conditional Moment-generating function CMGF of a Markov Process The conditional Moment-generating function (CMGF) of a MP, Y T, conditioned on current and lagged information about Y at date t, is given by the Laplace transform of its conditional density: CMGF t (τ, u) ( ) E t e u Y T Y t u R N = e u Y T f Y (Y T Y t ; γ)dy T R N where τ = (T t), i = 1, and f Y is the conditional density of Y. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

8 Affine Processes Affine Processes Affine Process A MP Y is said to be an affine process if either its CCF or CMGF has the exponential affine form CCF t = e φ 0t+φ Yt Yt or CMGF t = e φ 0t+φ Yt Yt where φ 0t and φ Yt are complex (real) coefficients in the case of the CCF (CMGF). They are indexed by t to allow for the possibility of time dependence of the moments of Y. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

9 Affine Processes Continuous-time Affine Processes Jump-diffusion process A jump-diffusion process is a MP solving the SDE where dy t = µ(y t, γ 0 )dt + σ(y t, γ 0 )dw t + dz t W t is an (F)-standard Brownian Motion in R N µ : D R N ; σ : D R N N ; Z pure-jump process whose jump amplitudes have a fixed probability distribution ν on R N and arrive with intensity {λ(y t ) : t 0}, for some λ : D [0, ); γ R K is the vector of unknown parameters. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

10 Jump process Affine Processes Cox process construction of jump arrivals in which, conditional on the path {Y s := 0 s t} to time t, the times of jumps arriving during the interval [0, t] are assumed to be the jump times of a Poisson process with time-varying intensity {λ(y s ) : 0 s < T } Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

11 Affine Processes Affine Processes The special case of an affine-jump diffusion is obtained by requiring that µ, σσ and λ all be affine functions on D. Y follows a jump-affine diffusion if dy t = K(Θ Y t )dt + Σ S t dw t + dz t W t N-dimensional independent standard Brownian Motion K and Σ are N N matrices, which be nondiagonal and asymmetric; S ii,t = α i + β i Y t Both drifts and the instantaneous conditional variances are affine in Y t. The jump intensity is assumed to be a positive, affine function of the state Y t λ t = l 0 + l Y Y t The jump-size distribution f J is assumed to be determined by its characteristic function J (u) = exp {ius}f J (s)ds Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

12 ATSM Affine Term Structure Models Zero-coupon price The time-t price of a zero-coupon bond maturing at time t + τ: [ ( t+τ )] P t (τ) = Et Q exp r s ds t Instantaneous short rate The instantaneous short rate r t is an affine function of a (N 1) vector of unobservable state variables x t = (x 1t,..., x Nt ) : r t = δ 0 + N δ 1i x it = δ 0 + δ 1x t i=1 Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

13 ATSM Affine Term Structure Models State Variable Under the risk neutral probability measure Q the state variables x t follow an affine diffusion: dx t = K Q (θ Q x t )dt + ΣS 1/2 t dwt Q (1) where Wt Q is a (N 1) vector of independent Brownian motions under Q K Q and Σ are (N N) general matrices of parameters (Σ may be asymmetric) θ Q is a (N 1) vector of parameters S t is a (N N) diagonal matrix with (i, i) element given by: [S t ] ii = α i + β ix t and for each i = 1,..., N, α i and β i respectively are a scalar and a (N 1) vector of parameters. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

14 ATSM Affine Term Structure Models The drift and the conditional variance of x t are affine functions of x t. Let us denote with: µ Q t = K Q (θ Q x t ) Σ t = ΣS 1/2 t the drift vector under Q and the diffusion matrix of x t, respectively. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

15 ATSM Affine Term Structure Models If the parametrization is admissible, Duffie and Kan (1996) have shown that: P t (τ) = exp [ a(τ) b(τ) x t ] where the coefficients a(τ) and b(τ) satisfy the following system of ordinary differential equation (ODEs): da(τ) dτ = θ Q K Q b(τ) N i=1 [ Σ b(τ) ] 2 i α i δ 0 db(τ) dτ = K Q b(τ) 1 2 N i=1 [ Σ b(τ) ] 2 i β i + δ 1 subject to the initial conditions a(0) = 0 and b(0) = 0. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

16 ATSM Affine Term Structure Models The dynamics of x t under the actual probability measure P can be derived with an application for the Girsanov s Theorem: dx t = (µ Q t + Σ t λ t )dt + ΣS 1/2 t dw t where λ t is a (N 1) vector of risk premium functions. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

17 ATSM Affine Term Structure Models Some (more or less) common choices of λ t are: Dai and Singleton (2000) λ t = S 1/2 t λ 1 where λ 1 is a(n 1) vector of parameters. Under this choice: µ Q t + Σ t λ t = K(θ x t ) K = K Q ΣΦ θ = K 1 (K Q θ Q + Σφ) where Φ is a (N N) matrix whose i-th row is given by λ 1i β i, and φ is a (N 1) vector whose i-th element is given by λ 1i α i. This choice gives rise to the so called completely affine class of term structure models. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

18 ATSM Affine Term Structure Models Duffee (2002) λ t = S 1/2 t λ 1 + (St ) 1/2 λ 2 x t where λ 1 is as above, λ 2 is a (N N) matrix of parameters, and St diagonal (N N) matrix with: is [ S t ] ii = { (αi + β i x t) 1 if inf(α i + β i x t) > 0 0 otherwise This choice gives rise to the so called essentially affine class of term structure models. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

19 ATSM Affine Term Structure Models Duffee and Stanton (2001) λ t = Σ 1 λ 0 + S t Σ 1 λ 1 where λ 0 is a (N 1) vector of parameters, and λ 1 is defined as above. Duarte (2002) with λ 0, λ 1, λ 2 and S t λ t = Σ 1 λ 0 + S 1/2 t λ 1 + (St ) 1/2 Λ 2 x t defined as above. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

20 Affine Term Structure Models The general case In a completely affine model, the relevant equations are given by: The dynamics of the state variables under P: The risk premium functions: the dynamics of x t under Q: where: dx t = K(θ x t )dt + ΣS 1/2 t dw t λ t = S 1/2 t λ dx t = K Q (θ Q x t )dt + ΣS 1/2 t dw t K Q = K + ΣΦ θ Q = K Q 1 (Kθ Σφ) where Φ is a (N N) matrix whose i-th row vis given by λ i β i, and φ is a (N 1) vector whose i-th element is given by λ i α i Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

21 The general case Affine Term Structure Models Zero coupon prices: P t (τ) = exp [ a(τ) b(τ) x t ] and zero coupon yields: y t (τ) = 1 τ lnp t(τ) = a(τ) τ where a(τ) and b(τ) satisfy the ODE: + b(τ) b t τ da(τ) dτ = θ Q K Q b(τ) N i=1 [ Σ b(τ) ] 2 i α i δ 0 db(τ) dτ = K Q b(τ) 1 2 N i=1 [ Σ b(τ) ] 2 i β i + δ 1 subject to the initial conditions a(0) = 0 and b(0) = 0. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

22 Affine Term Structure Models The general case Let us denote with ψ the vector of parameters: ψ = (K, θ, Σ, B, α, δ 0, δ 1, λ) where B = (β 1,..., β N ) is a (N N) matrix. To be admissible, a parametrization must guarantee that [S t ] ii is strictly positive, for all i. Following Dai and Singleton (2000), it turns out that the condition to impose on ψ depend on the rank of B, which we shall denote by m. Formally, for any given N, there are N + 1 subfamilies A m (N) of admissible N factor models, corresponding to m = 0, 1,..., N. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

23 Affine Term Structure Models The general case For each m, the canonical representation of A m (N) is defined as follows. To begin with, partition x t as: x t N 1 = x B t m 1 x D t (N m) 1 Accordingly, if m > 0, partition K as follows: K BB 0 K N N = m m m (N m) K DB K DD (N m) m (N m) (N m) Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

24 Affine Term Structure Models The general case If m = 0, K is either upper or lower triangular. Furthermore: [K] ij 0 for 1 j m, i j To insure stationarity, the (real part of the) eigenvalues of K must be positive. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

25 Affine Term Structure Models The general case On θ: Furthermore: [Kθ] i = θ N 1 = θ B m 1 0 (N m) 1 m [K] ij [θ] j > 0 for 1 i m, j=1 This condition must be strengthened to: m [Kθ] i = [K] ij [θ] j > 1 2 for 1 i m j=1 to insure that 0 is not an absorbing state for x B t. Finally: [θ] i 0 for 1 i m Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

26 Affine Term Structure Models The general case On Σ: On α: On B: B N N = [ α = Σ (N N) = I N 0 m 1 ι (N m) 1 I m ] (N 1) B BD m m m (N m) 0 0 (N m) m (N m) (N m) Furthermore: [B] ij 0 for 1 j m, m + 1 j N Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

27 Affine Term Structure Models The general case On δ 1 δ 1i 0, for m + 1 i N Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

28 Affine Term Structure Models The general case To complete the definition, the subfamilies A m (N) consist of all the models which are nested special cases of the canonical representation, or of any equivalent model obtained by an invariant transformation of it. Invariant transformations preserve admissibility and identification. These are sufficient (but not necessary) conditions for admissibility; Aït-Sahalia and Kimmel (2002)provide some examples of admissible two- and three-factors affine models which are not invariant transformations of the corresponding canonical representation. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

29 A 1 (1) A 1 (1) canonical model Dynamics under P: dx t = k(θ x t )dt + x t dw t Short rate: r t = δ 0 + δ 1 x t Risk premium: λ t = λ x t Dynamics under Q: where: and dx t = k Q (θ Q x t )dt + x t d W t k Q = k + λ θ Q = kθ k + λ Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

30 A 1 (1) A 1 (1) canonical model Yields: y t (τ) = a(τ) τ where a(τ) and b(τ) solve: Parameters: Restrictions on ψ: + b(τ) x t τ da(τ) = θ Q k Q b(τ) δ 0 dτ db(τ) = k Q b(τ) 1 dτ 2 b(τ)2 i β i + δ 1 ψ = (k, θ, δ 0, δ 1, λ) k > 0 θ 0 kθ > 1 2 Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

31 A 1 (1) Measurement errors Assume that at each date t we observe P yields: y t = [y t (τ 1 ),..., y t (τ P )] The yield of maturity τ x is observed without error: Hence: y t (τ x ) = a(τ x) + b(τ x) τ x x t = τ x x t τ [ x y t (τ x ) + a(τ ] x) b(τ x ) τ x Notice: x t must be strictly positive at each date. We take into account these constraints by introducing a huge penalization in the log-likelihood function. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

32 A 1 (1) Measurement errors Using the Jacobian formula and Aït-Sahalia (2001) analytic approximation, it is easy to compute the log-likelihood of the yields observed without error(see the following). The other P 1 yields are observed with error: where: In vector terms: y t (τ i ) = ŷ t (τ i ) + e i for i = 1,..., P 1 ŷ t (τ i ) = a(τ i) + b(τ i) τ i y xt = ŷ xt + e t where e t is a [(P 1) 1] vector of measurement errors. If we make an assumption on the stochastic structure of e t, than it is straightforward to compute the log-likelihood of the yields observed with error. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40 τ i x t

33 Sample log-lik A 1 (1) Sample log-likelihood Let us denote with ν the vector of parameters entering the distribution of the measurement errors e t, and define: ( ) ψ γ = ν the vector of all (structural and auxiliary) parameters in the model. We also assume that the measurement errors e t are independent through time, but may be correlated in cross section. The conditional density of the yields at date t given previous observations can be factorized as: f Y (y t Y t 1 1 ; γ) = f Y (y t y t 1 ; γ) = f Y (y t (τ x ) y t 1 ; γ)f Y (y xt y t (τ x ); γ) = f X ( x t x t 1 ; ψ) τ x b(τ x ) f E (e t x t ; γ) Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

34 Sample log-lik A 1 (1) Sample log-likelihood The factor component of the total sample log-likelihood is given by: l Y [ψ; y (τx )] = T lnf X ( x t x t 1 ; ψ) + (T 1)[ln(τ x ) lnb(τ x )] t=2 The measurement error component of the total sample log-likelihood depend on the assumed stochastic structure of e t. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

35 Sample log-lik A 1 (1) Sample log-likelihood If we assume: e t IIDN(0, ω 2 I P 1 ) then ν reduces to the scalar ω, and: T (P 1) l Y [γ; Y ( x) ] = log (2π) T (P 1) log (ω) 1 2 2ω 2 T e te t t=1 Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

36 Sample log-lik A 1 (1) Sample log-likelihood If we assume: e t IIDN(0, Ω) with Ω = diag(ω 2 1,..., ω2 P 1 ), then ν = (ω2 1,..., ω2 P 1 ), and: P 1 T (P 1) l Y [γ; Y ( x) ] = log (2π) T log (ω i ) 2 i=1 1 T P 1 ( e ) 2 it 2 ω where: t=1 i=1 P 1 T (P 1) = log (2π) T log (ω i ) v t = i=1 ( eit,..., e ) P 1t ω 1 ω P 1 T v tv t Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40 t=1

37 Sample log-lik A 1 (1) Sample log-likelihood If we assume: e t IIDN(0, Ω) with general symmetric positive definite Ω = C C, where C is the upper triangular matrix provided by the Cholesky decomposition of Ω, then ν consist of the P(P 1)/2 free elements of C, and: T (P 1) l Y [γ; Y ( x) ] = log (2π) T 2 2 log Ω 1 2 T (P 1) = log (2π) T 2 2 log C C 1 T e 2 t(c C) 1 e t t=1 T e tω 1 e t t=1 Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

38 Sample log-lik A 1 (1) Sample log-likelihood T (P 1) l Y [γ; Y ( x) ] = log (2π) T log C T e tc 1 (C ) 1 e t t=1 T (P 1) log (2π) T (P 1) log [C] ii 2 i=1 1 2 where v t = (C ) 1 e t. T [(C ) 1 e t ] [(C ) 1 e t ] t=1 T (P 1) = log (2π) T (P 1) log [C] ii i=1 T v tv t t=1 Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

39 Optimization issues Sample log-likelihood Things are complicated by the existence of T non linear constraints of the form x t > 0.To solve the problem, we follow Duffee (2002) strategy: 1 Pick a random initial value of γ 2 Check if it is admissible, i.e. if x t > 0 for all t. If the answer is positive, go on; otherwise, go back to step (1). 3 the optimization method; continue iterations until a stable convergence is achieved. 4 In the point thus attained, start a derivative-based optimization routine. Procedure (1)-(4) is repeated until a given number (say 1000) of estimates are obtained. The final estimate is the one associated to the maximum value of the log-likelihood. Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40

40 References Sample log-likelihood 1 Aït-Sahalia, Y. (2001), Closed-Form Likelihood Expansions for Multivariate Diffusions, working paper, Princeton University 2 Aït-Sahalia, Y. and R. Kimmel (2002), Estimating Affine Multifactor Term Structure Models Using Closed-Form Likelihood Expansions, working paper, Princeton University 3 Dai, Q. and K.Singleton (2002), Specification Analysis of Affine Term Structure Models, Journal of Finance, vol. LV, Duarte, J (2002), Evaluating Alternative Risk Preferences in Affine Term Structure Models, working paper, University of Washington. 5 Duffee, G.(2002), Term Premia and Interest Rate Forecasts in Affine Models, Journal of Finance, vol.lvii, Duffee, G. and R. Stanton (2001), Estimation of Dynamic Term Structure Models, working paper, University of California, Berkeley. 7 Duffie, D. and R. Kan (1996), A Yield-Factor Model of Interest Rates, Mathematical Finance, vol. 6, n.4, Eduardo Rossi (University of Pavia) Affine Processes March 17, / 40