Staying at Zero with Affine Processes An Application to Term - - PowerPoint PPT Presentation

Staying at Zero with Affine Processes

An Application to Term Structure Modelling Alain Monfort1,2 Fulvio Pegoraro1,2 Jean-Paul Renne2 Guillaume Roussellet1,2,3

1CREST 2Banque de France 3Dauphine University

5th 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 ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Contents

1

Introduction

2

The ARG0 process A mixture of affine distributions Properties and extensions

3

The NATSM Short-rate specification and the affine framework Advantages of an affine framework

4

Estimation State-space formulation Estimation results

5

Assessing lift-off dates

6

Conclusion

7

Appendix

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Zero lower bound (ZLB)

Several of the major central banks now face the ZLB

1990 1995 2000 2005 2010 2015 2 4 6 8 U.S. Fed Bank of Japan Bank of England ECB Policy Rates (in %)

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Introduction The ARG0 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]

1 2 3 1995 2000 2005 2010 2015

Dates Yields (in %, annual basis)

Maturity (in years) 0.5 1 2 4 7 10

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Closed-form pricing Positivity Can stay at 0

• Gaussian Atsm
• Cir
• Qtsm
• Shadow

rate

Closed-form pricing Positivity Can stay at 0

• Gaussian Atsm
• Cir
• Qtsm
• Shadow

rate

• This Paper

Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Our ZLB model: a primer

− → We introduce a new affine process:

0.00 0.05 0.10 0.15 100 200 300 400 500

Periods

Simulation of an ARGo process

• P(R=0) = 0.6

0.00 0.25 0.50 0.75 1.00 0.00 0.05 0.10 0.15

Values Probability

Cumulative distribution function

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Introduction The ARG0 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 (ARG0 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.

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Introduction The ARG0 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)

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Contents

1

Introduction

2

The ARG0 process A mixture of affine distributions Properties and extensions

3

The NATSM Short-rate specification and the affine framework Advantages of an affine framework

4

Estimation State-space formulation Estimation results

5

Assessing lift-off dates

6

Conclusion

7

Appendix

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Introduction The ARG0 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 ∼ γZ(µ), which implies:

1

If Z = 0, X is a Dirac point mass at 0.

2

If Z > 0, X is Gamma-distributed (continuous on R+).

Definition The non-negative r.v. X ∼ γ0(λ, µ), λ > 0 and µ > 0, if X | Z ∼ γZ(µ) with Z ∼ P(λ) ⇒ P(X = 0) = P(Z = 0) = exp(−λ) .

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix A mixture of affine distributions

A mixture distribution

In other words, X ∼ γ0(λ, µ) if its (complicated) p.d.f. is:

fX(x ; λ, µ) =

+ ∞

• z=1

exp(−x/µ) xz−1 (z − 1)! µz × exp(−λ)λz z !

• 1{x>0} + exp(−λ)1{x=0}

However, simple Laplace transform: ϕX(u ; λ, µ) := E [exp(uX)] = exp

• λ

uµ (1 − uµ)

• for

u < 1 µ . = ⇒ Exponential-affine in λ.

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix A mixture of affine distributions

A mixture distribution

In other words, X ∼ γ0(λ, µ) if its (complicated) p.d.f. is:

fX(x ; λ, µ) =

+ ∞

• z=1

exp(−x/µ) xz−1 (z − 1)! µz × exp(−λ)λz z !

• 1{x>0} + exp(−λ)1{x=0}

However, simple Laplace transform: ϕX(u ; λ, µ) := E [exp(uX)] = exp

• λ

uµ (1 − uµ)

• for

u < 1 µ . = ⇒ Exponential-affine in λ.

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix A mixture of affine distributions

A mixture distribution

In other words, X ∼ γ0(λ, µ) if its (complicated) p.d.f. is:

fX(x ; λ, µ) =

+ ∞

• z=1

exp(−x/µ) xz−1 (z − 1)! µz × exp(−λ)λz z !

• 1{x>0} + exp(−λ)1{x=0}

However, simple Laplace transform: ϕX(u ; λ, µ) := E [exp(uX)] = exp

• λ

uµ (1 − uµ)

• for

u < 1 µ . = ⇒ Exponential-affine in λ.

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix A mixture of affine distributions

Introducing dynamics: the ARG0 process

Main goal: Build a dynamic affine process with zero point mass. Definition (Xt) is a ARG0(α, β, µ) if (Xt+1|Xt) is Gamma-zero distributed: (Xt+1|Xt) ∼ γ0(α + βXt, µ) for α ≥ 0, µ > 0, β > 0 . Again, simple conditional LT, exponential-affine in Xt: ϕX,t(u ; α, β, µ) := Et [exp(uXt+1)] = exp

• uµ

1 − uµ(α + β Xt)

• ,

for u < 1 µ .

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Introduction The ARG0 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. ϕX,t(u ; α, β, µ) := Et [exp(uXt+1)] = exp [a(u)Xt + b(u)] Staying at zero with probability: P(Xt+1 = 0|Xt = 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).

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Properties and extensions

Relation to the original ARG process

We extend the ARG0(α, β, µ) process to the more general ARGν(α, β, µ) case: ARGν(α, β, µ) process Xt follows an ARGν(α, β, µ) process if: Xt+1 | Zt+1 ∼ γν+Zt+1(µ) with Zt+1 | Xt ∼ P(α + β Xt) ν = 0 = ⇒ ARG0 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

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Contents

1

Introduction

2

The ARG0 process A mixture of affine distributions Properties and extensions

3

The NATSM Short-rate specification and the affine framework Advantages of an affine framework

4

Estimation State-space formulation Estimation results

5

Assessing lift-off dates

6

Conclusion

7

Appendix

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Introduction The ARG0 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]

1 2 3 1995 2000 2005 2010 2015

Dates Yields (in %, annual basis)

Maturity (in years) 0.5 1 2 4 7 10

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Short-rate specification and the affine framework

Risk-neutral dynamics

The state of the economy is defined by a n-dimensional vector Xt. These factors follow a VARGν process under Q (the same under P). VARGν processes Xt follows a VARGν(α, β, µ) if, ∀t, ∀i: Zi,t+1|Xt ∼ P(αi + β′

i Xt).

Xi,t+1|Zi,t+1 ∼ γZi,t+1+νi(µi) cond. indep across i. Conditional Q-moments (same formulas under P): EQ

t (Xt+1)

= µQ ⊙ (αQ + βQ′Xt + ν) VQ

t (Xt+1)

= diag

• µQ ⊙ µQ ⊙
• ν + 2αQ + 2βQ′Xt
• Note: Conditional correlations can be allowed.

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Short-rate specification and the affine framework

Short-rate specification

The vector of factors Xt is split into two: Xt = (X (1)

t ′, X (2) t ′)′

where:

(i) All components of X (1)

t

have νj = 0 (point mass at 0). (ii) All components of X (2)

t

have νj > 0 (no point mass). (iii) µP

j = 1, βP and βQ lower-triangular (identification).

The short-term rate rt is given by: rt = δ′X (1)

t

(= rmin + δ′X (1)

t

, if LB = 0) (1) Key Property {Eq.(1) + (i)} ⇒ rt has a zero point mass.

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Short-rate specification and the affine framework

Other Properties:

{Eq.(1) + (iii)}: rt = δ′X (1)

t

 X (1)

t

X (2)

t

  = constant +  βQ

11

βQ

12

βQ

22

   X (1)

t−1

X (2)

t−1

  + ξQ

t

We have X (2) G.C. − → X (1) and thus X (2)

t

appears in the short rate conditional Q-expectations (hence in long rates). = ⇒ long-term yields can move during the ZLB.

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Short-rate specification and the affine framework

Pricing Formulas

The model belongs to the class of ATSM: Explicit closed-form bond-pricing Yields are affine in the factors for all maturities: Rt(h) = −1 h

Ah′Xt + Bh = A

′ hXt + Bh.

Recursive pricing formulas: Ah = −δ + βQ

• Ah−1 ⊙ µQ

1 − Ah−1 ⊙ µQ

• Bh

= Bh−1 + αQ′

• Ah−1 ⊙ µQ

1 − Ah−1 ⊙ µQ

• − ν′ log
• 1 − Ah−1 ⊙ µQ

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Short-rate specification and the affine framework

The historical dynamics

The SDF is exp-affine with market price of risk vector θ: dPt,t+1 dQt,t+1 = exp

• θ′Xt+1 − ψQ

t (θ)

• Change of measure property

Xt follows a VARGν(αP, βP, µP) process under the historical measure P. αP

j =

αQ

j

1 − θj µQ

j

, βP

j =

1 1 − θj µQ

j

βQ

j ,

µP

j =

µQ

j

1 − θj µQ

j

. Rk: ν is the same under both measures.

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Short-rate specification and the affine framework

Stylized facts to match (2)

Conditional volatilities: time-varying and maturity-dependent.

0.0 0.5 1.0 1.5 0.0 0.5 1.0 1.5 2−years yield 10−years yield 1995 2000 2005 2010 2015

Dates Yield volatility proxies (for yields in %, annual basis)

Model EGARCH(1,1) GARCH(1,1) Rolling window (60 days)

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Short-rate specification and the affine framework

How to treat it

Conditional variance of yields: VP

t [Rt+1(h)]

= A

′ hVP t (Xt+1)Ah

= Ah′ diag

• µP ⊙ µP ⊙
• ν + 2αP + 2βP′Xt
• Ah

Time-varying and maturity-dependent.

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Advantages of an affine framework

Advantages of an affine framework

NATSM properties Yields Rt(h) are non-negative; Long-term yields can move while rt = 0 for several periods; Unconditional first two moments are available in closed-form; Conditional first two moments of yields are affine in Xt (available in closed-form); Yields forecasts are explicitly affine in Xt;

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Contents

1

Introduction

2

The ARG0 process A mixture of affine distributions Properties and extensions

3

The NATSM Short-rate specification and the affine framework Advantages of an affine framework

4

Estimation State-space formulation Estimation results

5

Assessing lift-off dates

6

Conclusion

7

Appendix

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix State-space formulation

Estimation technique

State vector Yt = (R′

t, V ′ t, S′ t)′ affine in Xt:

Rt = yield levels (6 maturities); Vt = 2- and 10-y yield conditional (Egarch) variance; St = SPF for 3-m and 1-y ahead 10-y yield;

• prelim. estimations have suggested dim(X (1)

t

) = 1, dim(X (2)

t

) = 3 and ν = 0; Estimation technique Linear Kalman-filter-based QML:

  

Xt+1 = m + MXt + Σ1/2

t

εt+1 Yt = Γ0 + Γ1Xt + Ω ηt ,

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Estimation results

Filtered factors

2 4 1 2 3 4 0.0 2.5 5.0 7.5 1 2 3 4 5

Factor n°1 Factor n°2 Factor n°3 Factor n°4

1995 2000 2005 2010 2015

Dates

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Estimation results

Factor loadings of yields and conditional variances

• 2

4 6 8 10 0.0 0.2 0.4 0.6

(a) Factor loadings of yields

Loadings Maturities (in years)

• First factor

Second factor Third factor Fourth factor

• 2

4 6 8 10 0.000 0.004 0.008

(b) Factor loadings of conditional variances

Loadings Maturities (in years)

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Estimation results

Fit of Conditional Variances and SPFs

2−year yield 10−year yield 0.00 0.02 0.04 0.06 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015

Dates Conditional variance proxy

• bserved

fitted 3−month ahead 10−year yield 12−month ahead 10−year yield

• 1

2 3 4 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015

Dates Forecast Surveys (in %, annual basis)

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix Estimation results

Fit of Yields

0.00 0.25 0.50 0.75 0.0 0.5 1.0 1.5 1 2 3

6−month yield 2−year yield 7−year yield

1995 2000 2005 2010 2015

Dates Yields (in %, annual basis)

0.00 0.25 0.50 0.75 1.00 0.0 0.5 1.0 1.5 2.0 1 2 3

1−year yield 4−year yield 10−year yield

1995 2000 2005 2010 2015

Dates

• bserved

fitted

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Contents

1

Introduction

2

The ARG0 process A mixture of affine distributions Properties and extensions

3

The NATSM Short-rate specification and the affine framework Advantages of an affine framework

4

Estimation State-space formulation Estimation results

5

Assessing lift-off dates

6

Conclusion

7

Appendix

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Lift-off probability dates under P and Q

We calculate the following probabilities: P(rt+k = 0 | Xt) and Q(rt+k = 0 | Xt); P(rt+k < 25 bps. | Xt) and Q(rt+k < 25 bps. | Xt). Useful formula If z ∈ R+ and ϕz(u) its Laplace transform. P(z = 0) = lim

u→−∞ϕz(u) .

Next two plots (Q is the black solid line): Time-series dimension: t varies (k = 2yrs and 5yrs). Horizon dimension: k varies (t = 11/30/07 and 05/30/14).

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

2−years ahead 5−years ahead 0.0 0.2 0.4 0.6 0.8 0.00 0.25 0.50 0.75 1.00 lambda = 0 lambda = 25 bps 1995 2000 2005 2010 2015 1995 2000 2005 2010 2015

Dates Probabilities

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Horizon dimension of probabilities

lambda = 0 lambda = 25 bps 0.00 0.25 0.50 0.75 1.00 1 2 3 4 5 1 2 3 4 5

Forecast horizon Probabilities

2007−11−30 2014−05−30 Q probability P probability

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Contents

1

Introduction

2

The ARG0 process A mixture of affine distributions Properties and extensions

3

The NATSM Short-rate specification and the affine framework Advantages of an affine framework

4

Estimation State-space formulation Estimation results

5

Assessing lift-off dates

6

Conclusion

7

Appendix

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Summary and further research

We have derived affine non-negative processes staying at 0 and built an affine term-structure model (NATSM) gathering: a short-rate consistent with the ZLB experiencing periods at 0 while long-run rates still fluctuates; closed-form formulas for bond-pricing and lift-off probabilities. An empirical assessment showed performance of our model for: fitting yield levels and conditional variances; calculating risk-neutral and historical lift-off probabilities. Further research: Empirical comparison of NATSMs, derivatives pricing.

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Thank you for your attention.

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Contents

1

Introduction

2

The ARG0 process A mixture of affine distributions Properties and extensions

3

The NATSM Short-rate specification and the affine framework Advantages of an affine framework

4

Estimation State-space formulation Estimation results

5

Assessing lift-off dates

6

Conclusion

7

Appendix

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Table : Parameter estimates

P-parameters Q-parameters Estimates Std. Estimates Std. α4 3.2455 0.1118 3.2347 0.1113 β1,1 0.9663 0.0078 0.9794 0.0042 β2,2 0.9978 0.0005 0.9957 0.0006 β3,3 0.9486 0.0044 0.9705 0.0023 β4,4 0.9967 0.0005 0.9933 0.0003 β2,1 0.0308 0.0041 0.0308 0.0041 β3,2 0.1094 0.0059 0.1120 0.0061 β4,3 3.88·10−4 2.28·10−5 3.87·10−4 2.27·10−5 µ1 1 – 1.0135 0.0040 µ2 1 – 0.9980 0.0005 µ3 1 – 1.0231 0.0023 µ4 1 – 0.9967 0.0003 Other Parameters δ1 0.0030 0.0003 θ1

• 0.0133

0.0039 θ2 0.0020 0.0005 θ3

• 0.0226

0.0022 θ4 0.0033 0.0003 σR 0.0407 0.0003 σV 3 · 10−3 − σS 0.15 − 40 / 30

Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

ARG0 Summary

time t Xt realized α + βXt Zt+1|Xt ∼ P(α + βXt) time t + 1 Xt+1|Zt+1 ∼ γZt+1(µ)

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Univariate case: lift-offs formulas

Z ∈ R+ and ϕZ(u) its Laplace transform. PZ{0} = lim

u→−∞ϕZ(u) .

Lift-off probabilities: (Xt) ∼ ARG0(α, β, µ) and ϕt,h(u1, . . . , uh) its multi-horizon conditional Laplace transform.

P(Xt+h = 0 | Xt) = lim

u→−∞ϕt,h(0, . . . , 0, u)

P

• Xt+1 = 0, . . . , Xt+h = 0
• Xt
• =

lim

u→−∞ϕt,h(u, . . . , u)

= exp(−α h − β Xt) , P

• Xt+1 = 0, . . . , Xt+h = 0, Xt+h+1 > 0
• Xt)
• = exp [−α h − β Xt] [1 − exp(−α)] ,

h > 1.

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Multivariate Case

Z ∈ Rn

+ and ϕZ(u1, . . . , un) its Laplace transform.

PZ{0, . . . , 0} = lim

u→−∞ϕZ(u, . . . , u) .

Notations: Multi-horizon conditional LT. ϕP

t,k(u1, . . . , uk)

= EP

• exp
• u

′

1Xt+1 + . . . + u

′

kXt+k

• Xt
• =

exp

• A

′

k Xt + Bk

• ϕ(h)P

R,t,k(v1, . . . , vk)

= E [exp (v1 Rt+1(h) + . . . + vk Rt+k(h)) | Xt]

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Lift-offs

P [rt+k = 0 | Xt] = lim

u→−∞ϕ(1)P R,t,k(0, . . . , 0, u)

P [rt+1 = 0, . . . , rt+k = 0 | Xt] = lim

u→−∞ϕ(1)P R,t,k(u, . . . , u) = pr,t,k

(say) P [rt+1 = 0, . . . , rt+k−1 = 0, rt+k > 0 | Xt] = pr,t,k−1 − pr,t,k P

• v′R(t+k)

t+1 (h) > λ | Xt

• = 1

2 + 1 π

+∞

Im

• ϕ(h)P

R,t,k(i v x) exp(−i λ x)

• x

dx

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Introduction The ARG0 process The NATSM Estimation Assessing lift-off dates Conclusion Appendix

Useful remarks

Remark 1 Stationarity conditions are easily imposed: Xt stationary ⇐ ⇒ ∀j ∈ {1, . . . , n}, ρj := µjβj,j < 1 . Remark 2 The assumption of conditional independence can be relaxed keeping the affine structure of the multivariate process Xt. = ⇒ Recursive discrete-time affine process (mimeo).

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