Generalized Affine Models

Generalized Affine Models Bruno Feunou Université de Montréal and CREST Nour Meddahi Imperial College London 0

Affine Models in Finance Ornstein-Uhlenbeck process (Vasicek (1977)): dx t = κ(x t θ)dt + σdw t Feller (or square-root or CIR) process (Cox, Ingersoll and Ross (1985)): dx t = κ(θ x t )dt + σ x t dw t Closed form formulas for bond yields: ( t+h )] B(t, h) = E t [exp x u du = exp(a(h)x t + b(h)) t log B(t, h) y t,h = = ã(h)x t + h b(h) Mathematical argument (Duffie and Kan (1996)) Laplace Transform: E t [exp(ux t+h )] = exp(ω(h) + α(h)x t ) Cumulant function: log(e t [exp(ux t+h )]) = ω(h) + α(h)x t Multivariate models: Duffie and Kan (1996) Volatility models: Heston (1993) and Duffie, Pan and Singleton (2000) 1

Discrete Time models L t (u) log[e[exp(ux t+1 ) x τ, τ t]] = exp(ω(u) + α(u)x t ) ψ t (u) log[e[exp(ux t+1 ) x τ, τ t]] = ω(u) + α(u)x t Closed form formulas for bond yields (affine forms). Closed form for option prices under stochastic volatility models. Macroeconomic and Finance models: x t is multivariate (inflation, GDP growth, short term interest rate): Piazzesi (2003). Mathematical argument: log E t [exp(u 1 x t+1 + u 2 x t+2 +... + u n x t+n )] = A(u 1, u 2,..., u n )x t + B(u 1, u 2,..., u n ) Characteristic function Laplace Transform. 2

Affine Models ψ t (u) log[e[exp(ux t+1 ) x τ, τ t]] ψ t (u) = ω(u) + α(u)x t. GARCH equation: It suggests r t = h t 1 u t, u t i.i.d. N (0, 1), h t = ω + αr 2 t + βh t 1, Generalized affine models. ψ t (u) = ω(u) + α(u)x t + β(u)ψ t 1 (u), 3

ARMA(1,1) with i.i.d. innovations: Hence, x t = a + bx t 1 + ε t cε t 1, ε t i.i.d., b < 1, c < 1, ψ t (u) = (ua + (1 c)ψ ε (u)) + u(b c)x t + cψ t 1 (u). 4

Advantages 1) Affine models are Markovian. Non-Markovian affine extensions: x t is multivariate and some components are latent (Macro-Finance literature). Multi-lag affine models (Monfort and Pegoraro (2007)): ψ t (u) = ω(u) + α 1 (u)x t + α 2 (u)x t 1 +... + α p (u)x t p ω(u) and α(u) are driven by a Markov chain (Bansal, Tauchen and Zhou (2004), Dai, Singleton, and Yang (2006)). 2) Persistence (like GARCH models): alpha takes care of the short time dependence while β takes care of the long time dependence. 3) Time-varying conditional distribution and time-varying higher order moments (Hansen (1994)). 5

Existence of Cumulant Functions ψ t (u) = ω(u) + α(u)x t + β(u)ψ t 1 (u) The sum of two cumulant functions is a cumulant function A positive number times a cumulant function of an infinitely divisible distribution is a cumulant function The sum of two cumulant functions of infinitely divisible distributions is a cumulant function of an infinitely divisible distribution Case of simple examples. ψ t (u) = ω(u) + α(u)x t + βψ t 1 (u), β 0 We provide in the paper an example where β( ) is a function. 6

Cumulant Structure We denote the conditional cumulant of x t+1 of order n by κ n,t, which is given by and κ n,t = ψ (n) t (0), κ n,t (κ 1,t, κ 2,t,..., κ n,t ). κ n,t = ω (n) (0) + α (n) (0)x t + n 1 j=0 n j κ n,t = ω n + α n x t + β n κ n,t 1 β (j) (0)κ n j,t 1 where ω n = ω (1) (0) ω (2) (0) : : ω (n) (0), α n = α (1) (0) α (2) (0) : : α (n) (0), 7

and β n = ( 2 1 β (0) 0 0... 0 ) β (1) (0) β (0) 0... 0 : : :... : : : :... : ( n n 1) β (n 1) ( (0) n n 2) β (n 2) ( (0)... n ) 1 β (1) (0) β (0) Consequently, the vector κ n,t is a VAR(1). Special case: β( ) is constant: κ n,t = ω (n) (0) + α (n) (0)x t + βκ n,t 1 8

Moment Structure There is a mapping between the cumulants and the moments: Denote the conditional moments by m n,t, i.e., m n,t = E t [x n t+1], one has m 1,t = κ 1,t m 2,t = κ 2,t + κ 2 1,t m 3,t = κ 3,t + 3κ 2,t κ 1,t + κ 3 1,t m 4,t = κ 4,t + 4κ 3,t κ 1,t + 3κ 2 2,t + 6κ 2,t κ 2 1,t + κ 4 1,t m 5,t = κ 5,t + 5κ 4,t κ 1,t + 10κ 3,t κ 2,t + 10κ 3,t κ 2 1,t + 15κ 2 2,tκ 1,t + 10κ 2,t κ 3 1,t + κ 5 1,t m 6,t = κ 6,t + 6κ 5,t κ 1,t + 15κ 4,t κ 2,t + 15κ 4,t κ 2 1,t + 10κ 2 3,t + 60κ 3,t κ 2,t κ 1,t + 20κ 3,t κ 3 1,t + 15κ 3 2,t + 45κ 2 2,tκ 2 1,t + 15κ 2,t κ 4 1,t + κ 6 1,t. 9

Multi-Horizon Forecasting Important formula for the term structure of interest rates and for option pricing: ( h )] log E t [exp u i x t+i = i=1 { } h β (d k ) k 1 Ψ t (d k ) + 1 β (d k) k 1 ω (d k ) 1 β (d k ) k=1 where d k = u k + d h = u h h 1 j=k β (d j+1 ) j k α (d j+1 ) for k h 1 10

Extensions ψ t (u) = ω(u) + α(u)x t + β(u)ψ t 1 (u) = ω(u) + α(u)x t + β(u)[ω(u) + α(u)x t 1 + β(u)ψ t 2 (u)] = ω(u)(1 + β(u)) + α(u)x t + β(u)α(u)x t 1 + β(u) 2 ψ t 2 (u) : = ω(u) 1 β(u) + α(u) ( ) β(u) i x t i i=0 under the assumption u, β(u) < 1. Other models ψ t (u) = µ(u) + α i (u)x t i. i=0 ψ t (u) = ω(u) + p α i (u)x t i + i=0 q β j (u)ψ t j (u). j=1 11

Long Memory ψ t (u) = µ(u) + α i (u)x t i. i=0 Long memory in the level of x t+1 is related to the behavior of α (1) i (0) when i. Long memory in the conditional variance of x t+1 is related to the behavior of α (2) i (0) when i. Long memory in higher cumulants... 12

Term structure of interest rates The paper studies two approaches: 1) Generalized affine under the P-measure + SDF. 2) Generalized affine under the Q-measure. We will focus here on the first approach. 13

One factor: the short term interest rate r t. ψ t (u) P = log[e t [exp(ur t+1 )]] = ω(u) + α(u)r t + β(u)ψ t 1 (u) M t,t+1 = exp(γr t+1 + θ t ). Given the restriction exp( r t ) = E P t [M t,t+1 ], one gets θ t = r t ψ t (γ) and M t,t+1 = exp(γr t+1 r t ψ t (γ)). B(t, h) = E P t [ h ] M t+i 1,t+i, r t,h = i=1 log(b(t, h)). h 14

Analytical forms of the yields: r t,t+h = 1 βp (γ) h 1 β P (γ) ψ P t (γ) h + 1 h ω P (γ) 1 β P (γ) ( h 1 βp (γ) h 1 β P (γ) ) + r t h h k=1 β P (d k ) k 1 ψ P t (d k ) h h k=1 1 β P (d k ) k 1 1 β P (d k ) ω P (d k ) h with d k = u k + h 1 j=k β P (d j+1 ) j k α P (d j+1 ) for k h 1, d h = u h where u 1 = γ α P (γ) 1 βp (γ) h 1 1 β P (γ), u h = γ, and u j = γ 1 α P (γ) 1 βp (γ) h j 1 β P (γ) for 1 < j < h. ( ψ t (γ) = ω(γ) + α(γ)r t + β(γ)ψ t 1 (γ) = ω(γ) 1 β(γ) + α(γ) i=0 β(γ) i r t i ) 15

Option Pricing Models The paper studies two approaches: 1) Generalized affine under the P-measure + SDF. 2) Generalized affine under the Q-measure. Heston and Nandi (2000): r t = µ + ε t = µ + h t 1 z t, z t i.i.d. N (0, 1), h t = ω + α(z t γ h t 1 ) 2 + βh t 1. ψ t (u, v) P log[e P [exp(ur t+1 + vh t+1 ) r τ, τ t]] = ω(u, v) + α(u, v)h t We extend the model to ψ t (u, v) P log[e P [exp(ur t+1 +vh t+1 ) r τ, τ t]] = ω P (u, v)+α P (u, v)h t +β P (u, v)ψ P t 1(u, v) which implies 2 ω P u 2 (0, 0) = 0, 2 α P u 2 (0, 0) = 1, 2 β P (0, 0) = 0 u2 2 ψ t u 2 (0, 0) = V arp t [r t+1 ] = h t. 16

SDF: M t,t+1 = exp(γr t+1 + λh t+1 + θ t ). The price at t of a European call option with pay off (S t+h X) + at t + h is given by where and C 1,t = ) exp (ψ Q t,t+h (1) + 2 C 2,t = 1 2 + + C t = exp( rh)s t C 1,t exp( rh)xc 2,t + 0 [ 1 2πu Im exp 1 πu Im ( iu ln [ ( ))] exp (Ψ Qt,t+h (1 + iu) iu ln du XSt ( ) )] X + Ψ Qt,t+h (iu) du [ ] Ψ Q t,t+h (u) = Ψ 1 β (γ, λ)h ω (γ, λ) 1 β (γ, λ)h t (γ, λ) h 1 β (γ, λ) 1 β (γ, λ) 1 β (γ, λ) { } h + β (d k ) k 1 Ψ t (d k ) + 1 β (d k) k 1 α (d k ) 1 β (d k ) k=1 S t 17

with d k = (u + u k, v k ) + d h = (u + u h, v h ) h 1 j=k β (d j+1 ) j k (0, α (d j+1 )) for k h 1 and u h = γ, v h = λ u j = γ α (γ, λ) v j = λ α (γ, λ) 1 β (γ, λ)h j 1 β (γ, λ) 1 β (γ, λ)h j 1 β (γ, λ) for 1 j < h for 1 j < h 18

Non-linear mean with MA(1) structure: x t = f(x t 1 ) + ε t cε t 1, ε t i.i.d., c < 1. ψ t (u) = (1 c)ψ ε (u) + u(f(x t ) cx t ) + cψ t 1 (u) = ω(u) + α(u, x t ) + β(u)ψ t 1 (u) i.e., α(u, x t ) non linear function of x t No term structure of interest rates and options prices formulas. 19

Affine models: ψ t (u) log[e[exp(ux t+1 ) x τ, τ t]] ψ t (u) = ω(u) + α(u)x t. L t (u) E[exp(ux t+1 ) x τ, τ t] L t (u) = exp(ω(u) + α(u)x t ) L t (u) = γ(u) + exp(ω(u) + α(u)x t ) + β(u)l t 1 (u). Sufficient conditions for the existence of Laplace transforms. We have term structure of interest rates and options prices formulas. 20

Summary of Models Generalized affine models: Generalized non-affine models: ψ t (u) = ω(u) + α(u)x t + β(u)ψ t 1 (u) Generalized Laplace models: ψ t (u) = ω(u) + α(u, x t ) + β(u)ψ t 1 (u) L t (u) = γ(u) + exp(ω(u) + α(u)x t ) + β(u)l t 1 (u). 21

Estimation MLE. QMLE. Method of Moments. Characteristic function (Singleton (2001), Carrasco and Florens (2005)). 22

Application Affine and generalized affine models allow to compute the term structure of Value-at-Risk (Duffie and Pan (2001)) and the expected shortfall. We use daily realized volatility. Discrete time model for (r t+1, RV t+1 ) ψ t (v, u) = log[e t [exp(vr t+1 + yrv t+1 )] = ω(v, u) + α(v, u)rv t + βψ t 1 (v, u), We characterize the conditional cumulant function of r t+1,t+h = 1 p r t+1. h Term structure of Value-at-Risk i=1 23

Specification of the model: Motivation from continuous time: d log p u = (a + bσu)du 2 + σ u dw u When no leverage effect, daily log-return r t+1 = log(p t+1 ) log(p t ) has the following distribution: r t σ(p τ, σ s, τ t, s t + 1) N (a + biv t+1, IV t+1 ), which suggests the following model: r t+1 σ(r τ, RV τ, RV t+1, τ t) N (a + brv t+1, c + drv t+1 ). 24

Specification of RV t+1 given r τ, RV τ, τ t: Two possible affine Affine models (given that RV t is a positive variable): ψ t (u) = log E t [exp(urv t+1 )] = ω(u) + α(u)rv t. Inverse Gaussian : ω(u) = ν(1 1 2uµ), α(u) = ρ µ (exp(1 1 2uµ) 1) Gamma : ω(u) = ν log(1 uµ), α(u) = ρu 1 uµ When extend our analysis to the generalized affine case, i.e., ψ t (u) = log E t [exp(urv t+1 )] = ω(u) + α(u)rv t + βψ t 1 (u), 25

Table 2: Joint Estimation Panel A: R-RV- DM/USD: inverse gaussian 30 min 5 min Affine G-Affine Affine G-Affine par Est STD Est STD Est STD Est STD β 0.6111 0.0396 0.5449 0.041 ρ 0.3255 0.0203 0.1754 0.0179 0.3444 0.0193 0.2150 0.019 µ 0.2341 0.0114 0.1834 0.0087 0.1642 0.0071 0.1328 0.005 ν 1.2565 0.0398 0.5045 0.0545 2.0818 0.0647 0.9380 0.096 a 0.0063 0.0139 0.0063 0.0140 0.0064 0.0180 0.0064 0.018 c -0.0214 0.0448-0.0214 0.0449-0.0180 0.0433-0.0180 0.043 b 1.74E-08 5.709E-06 1.44E-08 6.024E-06 4.98E-08 1.093E-05 1.54E-08 5.777 d 0.9282 0.0290 0.9282 0.0290 0.7551 0.0236 0.7551 0.023 LIK -1600.0932-1547.4719-1838.6743-1790.0531 BIC 0.8069 0.7850 0.9234 0.9034 26

Panel B:R-RV- DM/USD: gamma 30 min 5 min Affine G-Affine Affine G-Affine par Est STD Est STD Est STD Est STD β 0.6172 0.0381 0.5696 0.0383 ρ 0.3043 0.0211 0.1744 0.0174 0.3599 0.0203 0.2206 0.0190 µ 0.2033 0.0075 0.1668 0.0065 0.1503 0.0054 0.1239 0.0046 ν 1.5165 0.0447 0.5542 0.0610 2.2383 0.0698 0.8907 0.0935 a 0.0063 0.0139 0.0063 0.0139 0.0064 0.0180 0.0064 0.0179 c -0.0214 0.0447-0.0214 0.0448-0.0180 0.0433-0.0180 0.0432 b 7.0E-09 2.701E-06 3.36E-08 7.974E-06 1.75E-08 7.241E-06 5.9E-09 3.153E d 0.9282 0.0290 0.9282 0.0290 0.7551 0.0236 0.7551 0.0236 LIK -1782.7432-1726.3807-1975.3616-1915.0602 BIC 0.8961 0.8723 0.9901 0.9644 27

Our approach outperforms the Heston and Nandi (2000) model (based on daily data).

Figure 1 Term structure of Value-at-Risk generated by Affine inverse gaussian 2.5 Term structure of Value at Risk / SQRT (Maturity): Affine VaR / SQRT(Maturity) 2 1.5 Low Median High 1 10 20 30 40 50 60 70 80 90 100 Maturity Figure 2 Term structure of Value-at-Risk generated by G-Affine inverse gaussian Term structure of Value at Risk / SQRT(Maturity) : G Affine 1.8 VaR / SQRT(Maturity) 1.6 1.4 1.2 Low Median High 1 0.8 10 20 30 40 50 60 70 80 90 100 Maturity in days 7

Figure 3 Term structure of Value-at-Risk: low volatility 1.06 Term structure of Value at Risk / SQRT(Maturity) : Low Variance 1.04 1.02 VaR / SQRT(Maturity) 1 0.98 0.96 0.94 0.92 G Affine Affine 0.9 0.88 0.86 10 20 30 40 50 60 70 80 90 100 Maturity in days Figure 4 Term structure of Value-at-Risk: median volatility Term structure of Value at Risk / SQRT(Maturity) : Median Variance 1.6 VaR / SQRT(Maturity) 1.5 1.4 1.3 1.2 G Affine Affine 1.1 10 20 30 40 50 60 70 80 90 100 Maturity in days 8

Figure 5 Term structure of Value-at-Risk: high volatility 2.1 Term structure of Value at Risk / SQRT(Maturity) : High Variance 2 1.9 VaR / SQRT(Maturity) 1.8 1.7 1.6 1.5 1.4 G Affine Affine 1.3 1.2 1.1 10 20 30 40 50 60 70 80 90 100 Maturity in days 9