Staying at Zero with Affine Processes An Application to Term Structure Modelling Alain Monfort 1 , 2 Fulvio Pegoraro 1 , 2 Jean-Paul Renne 2 Guillaume Roussellet 1 , 2 , 3 1 CREST 2 Banque de France 3 Dauphine University 5 th Conference on Fixed Income Markets Bank of Canada and SF Fed San Francisco, November 2015 All the views presented here are those of the authors and should not be associated with those of the Banque de France.
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Contents 1 Introduction 2 The ARG 0 process A mixture of affine distributions Properties and extensions 3 The NATSM Short-rate specification and the affine framework Advantages of an affine framework Estimation 4 State-space formulation Estimation results Assessing lift-off dates 5 Conclusion 6 Appendix 7 2 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Zero lower bound (ZLB) Several of the major central banks now face the ZLB Policy Rates (in %) U.S. Fed 8 Bank of Japan Bank of England ECB 6 4 2 0 1990 1995 2000 2005 2010 2015 3 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Stylized facts to match The short-term nominal rate can stay at the ZLB for several periods... and in the meantime, longer-term yields can show substantial fluctuations [JGB yields from June 1995 to May 2014] Maturity (in years) Yields (in %, annual basis) 0.5 3 1 2 4 7 10 2 1 0 1995 2000 2005 2010 2015 Dates 4 / 30
Closed-form pricing • Gaussian Atsm • Cir • Qtsm • Shadow Positivity Can stay at 0 rate
Closed-form pricing • Gaussian Atsm • Cir • Qtsm • This Paper • Shadow Positivity Can stay at 0 rate
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Our ZLB model: a primer − → We introduce a new affine process: Simulation of an ARGo process Cumulative distribution function 0.15 1.00 0.75 0.10 Probability ● P(R=0) = 0.6 0.50 0.05 0.25 0.00 0.00 0 100 200 300 400 500 0.00 0.05 0.10 0.15 Periods Values 7 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix What we do in this paper We derive Non-negative Affine processes staying at 0 (ARG 0 processes) to build a Term Structure Model which is: providing positive yields for all maturities; consistent with the ZLB (a short-rate experiencing prolonged periods at 0) WHILE long-term rates still fluctuates; affine: thus closed-form formulas for bond-pricing and lift-off probabilities are available. Empirical assessment on JGB yields (June 1995 to May 2014). Good performance of our model in terms of: fitting yield levels and conditional variances; calculating Risk-Neutral and Historical lift-off probabilities. 8 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Related literature Term structure models at the ZLB: Black (1995), Ichiue & Ueno (2007), Kim & Singleton (2012), Krippner (2012), Renne (2012), Kim & Priebsch (2013), Wu & Xia (2013), Bauer & Rudebusch (2013), Christensen & Rudebusch (2013). Conditional volatilities of yields: Almeida et al. (2011), Bikbov & Chernov (2011), Filipovic, Larsson & Trolle (2013), Creal & Wu (2014), Christensen et al. (2014). Affine and Autoregressive Gamma processes: Darolles et al. (2006), Gourieroux & Jasiak (2006), Dai, Le & Singleton (2010), Creal & Wu (2013) Lift-off probabilities: Bauer & Rudebusch (2013), Swanson & Williams (2013) 9 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Contents 1 Introduction 2 The ARG 0 process A mixture of affine distributions Properties and extensions 3 The NATSM Short-rate specification and the affine framework Advantages of an affine framework Estimation 4 State-space formulation Estimation results Assessing lift-off dates 5 Conclusion 6 Appendix 7 10 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix A mixture of affine distributions Defining the Gamma-Zero distribution We construct a new distribution in two steps: Z ∼ P ( λ ) = ⇒ Z ( ω ) ∈ { 0 , 1 , 2 , . . . } and P ( Z = 0 ) = exp ( − λ ) . We define X | Z ∼ γ Z ( µ ) , which implies: If Z = 0, X is a Dirac point mass at 0. 1 If Z > 0, X is Gamma-distributed (continuous on R + ). 2 Definition The non-negative r.v. X ∼ γ 0 ( λ, µ ) , λ > 0 and µ > 0, if X | Z ∼ γ Z ( µ ) with Z ∼ P ( λ ) ⇒ P ( X = 0 ) = P ( Z = 0 ) = exp ( − λ ) . 11 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix A mixture of affine distributions A mixture distribution In other words, X ∼ γ 0 ( λ, µ ) if its (complicated) p.d.f. is: + ∞ � exp ( − x /µ ) x z − 1 × exp ( − λ ) λ z � � f X ( x ; λ, µ ) = 1 { x > 0 } + exp ( − λ ) 1 { x = 0 } ( z − 1 )! µ z z ! z = 1 However, simple Laplace transform: u µ u < 1 � � ϕ X ( u ; λ, µ ) := E [ exp ( uX )] = exp λ for µ . ( 1 − u µ ) = ⇒ Exponential-affine in λ . 12 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix A mixture of affine distributions A mixture distribution In other words, X ∼ γ 0 ( λ, µ ) if its (complicated) p.d.f. is: + ∞ � exp ( − x /µ ) x z − 1 × exp ( − λ ) λ z � � f X ( x ; λ, µ ) = 1 { x > 0 } + exp ( − λ ) 1 { x = 0 } ( z − 1 )! µ z z ! z = 1 However, simple Laplace transform: u µ u < 1 � � ϕ X ( u ; λ, µ ) := E [ exp ( uX )] = exp λ for µ . ( 1 − u µ ) = ⇒ Exponential-affine in λ . 12 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix A mixture of affine distributions A mixture distribution In other words, X ∼ γ 0 ( λ, µ ) if its (complicated) p.d.f. is: + ∞ � exp ( − x /µ ) x z − 1 × exp ( − λ ) λ z � � f X ( x ; λ, µ ) = 1 { x > 0 } + exp ( − λ ) 1 { x = 0 } ( z − 1 )! µ z z ! z = 1 However, simple Laplace transform: u µ u < 1 � � ϕ X ( u ; λ, µ ) := E [ exp ( uX )] = exp λ for µ . ( 1 − u µ ) = ⇒ Exponential-affine in λ . 12 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix A mixture of affine distributions Introducing dynamics: the ARG 0 process Main goal: Build a dynamic affine process with zero point mass. Definition ( X t ) is a ARG 0 ( α, β, µ ) if ( X t + 1 | X t ) is Gamma-zero distributed: ( X t + 1 | X t ) ∼ γ 0 ( α + β X t , µ ) for α ≥ 0 , µ > 0 , β > 0 . Again, simple conditional LT, exponential-affine in X t : ϕ X , t ( u ; α, β, µ ) := E t [ exp ( uX t + 1 )] u µ u < 1 � � = exp 1 − u µ ( α + β X t ) , for µ . 13 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Properties and extensions Interesting features and properties Key properties: Non-negative process. Affine process: the conditional Laplace transform is exp-affine. ϕ X , t ( u ; α, β, µ ) := E t [ exp ( uX t + 1 )] = exp [ a ( u ) X t + b ( u )] Staying at zero with probability: P ( X t + 1 = 0 | X t = 0 ) = exp ( − α ) � = 0. � α � = 0 = ⇒ zero is not absorbing. � in our multivariate yield curve model this probability will be time-varying, function of all date- t factors; Closed-form moments (affine conditional cumulants). 14 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Properties and extensions Relation to the original ARG process We extend the ARG 0 ( α, β, µ ) process to the more general ARG ν ( α, β, µ ) case: ARG ν ( α, β, µ ) process X t follows an ARG ν ( α, β, µ ) process if: X t + 1 | Z t + 1 ∼ γ ν + Z t + 1 ( µ ) with Z t + 1 | X t ∼ P ( α + β X t ) ν = 0 = ⇒ ARG 0 process. ν > 0, α = 0 = ⇒ ARG process of Gouriéroux and Jasiak (2006). ν = 0 ν > 0 Positivity Yes Yes Affine Yes Yes Zero point mass Yes No 15 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Contents 1 Introduction 2 The ARG 0 process A mixture of affine distributions Properties and extensions 3 The NATSM Short-rate specification and the affine framework Advantages of an affine framework Estimation 4 State-space formulation Estimation results Assessing lift-off dates 5 Conclusion 6 Appendix 7 16 / 30
Introduction The ARG 0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Stylized facts to match (1) short-term nominal rate at the ZLB for several periods longer-term yields showing substantial fluctuations [JGB yields from June 1995 to May 2014] Maturity (in years) Yields (in %, annual basis) 0.5 3 1 2 4 7 10 2 1 0 1995 2000 2005 2010 2015 Dates 17 / 30
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