Estimating Shadow-Rate Term Structure Models with Near ... · Estimating Shadow-Rate Term Structure Models with Near-Zero Yields Jens H. E. Christensen & Glenn D. Rudebusch Federal

Estimating Shadow-Rate Term StructureModels with Near-Zero Yields

Jens H. E. Christensen&

Glenn D. Rudebusch

Federal Reserve Bank of San Francisco

Term Structure Modeling and the Lower Bound Problem

Day 2: The Lower Bound Problem

Lecture II.1

European University InstituteFlorence, September 8, 2015

The views expressed here are solely the responsibility of the authors and should not be interpreted as reflecting the views

of the Federal Reserve Bank of San Francisco or the Board of Governors of the Federal Reserve System.1 / 39

Day 2: The Lower Bound Problem

1. Lecture:Estimating Shadow-Rate Term Structure Models withNear-Zero Yields

2. Lecture:Modeling Yields at the Zero Lower Bound: Are Shadow Ratesthe Solution?

3. Lecture:How Efficient is the Extended Kalman Filter at EstimatingShadow-Rate Models?

4. Lecture:Staying at Zero with Affine Processes: An Application to TermStructure Modeling

5. Lecture:Linear-Rational Term Structure Models

2 / 39

The Lower Bound Problem

In recent years, a number of the world’s most prominentcentral banks have held their conventional policy rates ator near their effective lower bound.This makes central bank response functions asymmetric.Thanks to investors’ forward-looking behavior, this getsreflected in bond yields and returns. As a consequence,bond yields exhibit asymmetric behavior near the lowerbound.This is a problem for standard Gaussian affine models.The term structure literature offers a few solutions:

Shadow-rate modelsSquare-root modelsStay-at-zero affine modelsLinear-rational models

Today, we will study the lower bound problem and themodels above. 3 / 39

Japanese Government Bond and U.S. Treasury Yields

1995 2000 2005 2010 2015

01

23

45

Rat

e in

per

cent

Ten−year yield Four−year yield Two−year yield One−year yield

1995 2000 2005 2010 2015

02

46

8

Rat

e in

per

cent

Ten−year yield Five−year yield Two−year yield One−year yield

U.S. and Japanese yields respect the zero lower bound.Characteristics:

Systematic upward sloping yield curve near ZLB.

Asymmetric behavior of yield shocks near ZLB.

Clear compression in yield volatility near ZLB.

Today’s models should be suitable for this type of data. 4 / 39

U.K. Gilt Yields

1995 2000 2005 2010 2015

02

46

810

Rat

e in

per

cent


For U.K. gilt yields the evidence is mixed:Is there compression in yield volatility?

Is there a lower bound?

Yield curve is not always upward sloping near the lower bound. 5 / 39

German Bund and Swiss Confederation Bond Yields

1998 2002 2006 2010 2014

01

23

45

6

Rat

e in

per

cent


1998 2002 2006 2010 2014

−1

01

23

45

Rat

e in

per

cent


For German bund yields and Swiss Confederation bondyields, markets are truly testing whether there is a lowerbound.

Also, these yield curves are not always upward sloping.

In such cases, Christensen and Krogstrup (2015) arguethat a Gaussian AFNS model appears to be warranted. 6 / 39

Min-Max Range of Yield Variation

1995 2000 2005 2010 2015

01

23

Rat

e in

per

cent

US two−year yield UK two−year yield Japanese two−year yield

1998 2002 2006 2010 2014

01

23

Rat

e in

per

cent

Swiss two−year yield German two−year yield

Shown are differences between largest and smallestobserved two-year yields during 91-day windows.

U.S., U.K., and Japanese series are shown to the left,while German and Swiss series are shown to the right.

Note that German and Swiss yields have periodic burstsof high volatility since 2009. That’s a challenge!

7 / 39

Modeling Interest Rates with a Lower Bound

Problem: Standard Gaussian term structure modelsdo not restrict yields to respect a lower bound!Existing models that do respect a lower bound:

Shadow-rate modelsStochastic-volatility models with square-root processesGaussian quadratic modelsStay-at-zero affine modelsLinear-rational models

Ultimate empirical question: Which model class performswell during normal times and lower bound episodes?

In this paper, we focus on shadow-rate models. Awayfrom the lower bound, they reduce to standard Gaussianaffine models. Thus, the huge existing empirical termstructure literature remains relevant with a shadow-ratetweak in the current low-yield environment.

8 / 39

Overview of Paper

Black (1995) shadow-rate term structure models ensurenonnegative yields but have been hard to estimate.

We introduce a tractable shadow-rate arbitrage-freeNelson-Siegel model—the B-AFNS model—using anoption-based approach from Krippner (2013).

We estimate the B-AFNS model on Japanese yields.

Our option-based implementation of the B-AFNS modelprovides a very close approximation to Blackarbitrage-free bond pricing.

The B-AFNS model outperforms the standard Gaussianmodel.

The shadow short rate, which has been used to measurethe stance of monetary policy, is sensitive to modelspecification. 9 / 39

Black’s Shadow-Rate Approach

Shortly before his death in 1995, Fisher Black wrote apaper on the impossibility of negative nominal interestrates in a world with currency.

To handle this constraint, Black proposed using standardtools to model a shadow rate, st , that may be negative,while the observed short rate would be its truncatedversion due to the option to hold currency at no cost:

rt = max{st , 0}.

Numerically, this nonnegativity is straightforward tohandle—but computation can be prohibitivelyburdensome.

10 / 39

Literature Using Gaussian Shadow-rate Models

One-factor models: Gorovoi and Linetsky (2004) derivebond price formulas, but no generalization to multifactormodels. Ueno, Baba, Sakurai (2006) calibrate toJapanese data.Two-factor models for U.S.: Bomfim (2003) calibrates atwo-factor model and uses finite-difference methods tocalculate bond prices.Two-factor models for Japan: Kim and Singleton (2012)and Ichiue and Ueno (2007) do Kalman filter estimationof two-factor models. Kim and Singleton derive bondprice PDE and use finite-difference method to solve it.Macro-finance shadow-rate model: Bauer andRudebusch (2015) use macro variables to help pin downunobserved shadow short rate.

Main problem: None of these studies are easily extendedto three-factor settings due to computational challenges. 11 / 39

Krippner’s Option-Based Shadow-Rate Model (1)

Krippner (2013) provides an option-based approximationto Black’s shadow-rate concept.

Key assumptions:There exists a shadow short-rate process, st , that we onlyobserve when it is nonnegative: rt = max{st , 0}.

There exists a continuum of unobserved shadow discountbonds, P(t ,T + δ), that earn a return equal to st .

Currency exists and can be held at no cost.

Key insight: The existence of currency implies anobserved bond price that is equivalent to a portfolio oftwo assets

A long position in the corresponding shadow discount bond;

A short position in an American call option with strike 1 and theshadow discount bond as the underlying asset.

12 / 39


Shadow-rate zero-coupon bond price, P(t ,T ), may tradeabove par.

Observed zero-coupon bond price, P(t ,T ), will not riseabove par, so nonnegative yields will never be observed.

Relationship between the two bond prices:

P(t ,T ) = P(t ,T )− CA(t ,T ,T ; 1),

where CA(t , t ,T ; 1) is the value of an American call optionat time t with maturity T and strike price 1 written on theshadow bond maturing at T .

In a world with currency, bond investors effectively sell offthe possible gain from above-par bond prices.

13 / 39


0.6 0.8 1.0 1.2 1.4

0.0

0.5

1.0

1.5

Shadow bond price

Val

ue

Value of shadow bond Exercise payoff of call option Value of call option Value of constrained bond

Value of American option and constrained bond as afunction of the shadow bond price.Problem: Early exercise premium is difficult to value.Solution: The last incremental forward rate of any bond isnonnegative due to the future existence of currency andcan be isolated ... 14 / 39


Krippner introduces an auxiliary bond price equation

Paux.(t ,T + δ) = P(t ,T + δ) − CE (t ,T ,T + δ;1),

where CE (t ,T ,T + δ;1) is the value of a European call optionat time t with maturity T and strike price 1 written on theshadow discount bond maturing at T + δ.

By letting δ → 0, the last incremental nonnegative forward rate,denoted f (t ,T ), is identified as

f (t ,T ) = limδ→0

[− d

dδPaux.(t ,T + δ)

].

Krippner uses this to define observed bond prices that areapproximately arbitrage-free:

Papp.(t ,T ) = e−∫ T

t f (t,s)ds.

NB: This price is only approximately arbitrage-free because theauxiliary bond price, Paux.(t ,T + δ), is not identical to the trueobserved bond price, P(t ,T ), discussed on the previous slides.15 / 39

The General Option-Based Shadow-Rate Formula

Krippner shows that the nonnegative instantaneous forward rate ofthe observed bond prices is

f (t ,T ) = f (t ,T ) + z(t ,T ),

where f (t ,T ) is the instantaneous forward rate on the shadowbond, that may go negative, while z(t ,T ) is given by

z(t ,T ) = limδ→0

[ddδ

{CE (t ,T ,T + δ;1)P(t ,T + δ)

}],

where

P(t ,T + δ) is the price of the shadow discount bond maturingat T + δ.

CE (t ,T ,T + δ;1) is the value of the European call optionmaturing at T with P(t ,T + δ) as the underlying price.

Note: These formulas have general validity independent ofmodel-dimension and assumed factor dynamics. 16 / 39

Analogy to Corporate Credit Literature

In the corporate credit literature, researchers typicallycare about the value process Vt of a firm and its volatilityσV . However, we never observe this value process.

Instead, what we actually observe are the prices of thefirm’s debt and equity, which are derivative contracts withVt as the underlying asset process, see Merton (1974).

Researchers use those derivative prices combined withMerton-like pricing formulas to draw inference about thevalue and parameters of the Vt process.

Bottom line: Provided people have few reservations aboutthis type of financial analysis, there is no obvious reasonnot to embrace shadow-rate models for similarly usefulmonetary analysis when yields are near zero.

Now, let’s make this operational ... 17 / 39

The Gaussian Option-Based Shadow-Rate Formula

For Gaussian factor dynamics, Krippner derives the nonnegativeinstantaneous forward rate:

f (t ,T ) = f (t ,T )Φ( f (t ,T )

ω(t ,T )

)+ ω(t ,T )

1√2π

exp(− 1

2

[ f (t ,T )

ω(t ,T )

]2),

where

ω(t ,T )2 =12

limδ→0

∂2v(t ,T ,T + δ)

∂δ2

and v(t ,T ,T + δ) is the conditional variance in the shadow bondoption price formula.

To make this operational, we need:

A Gaussian model;

The analytical formula for bond option prices within the model;

The limit of the double derivative of the conditional variance inthe bond option formula w.r.t. difference btw. option and bondmaturity.

Enter the AFNS model class! 18 / 39

The Arbitrage-Free Nelson-Siegel Models

Proposition: If the risk-free rate is defined by

rt = Lt + St

and the Q-dynamics of Xt = (Lt ,St ,Ct) are given by

dLt

dSt

dCt

=

0 0 00 λ −λ0 0 λ

θQ1θQ

2θQ

3

−

Lt

St

Ct

dt+ΣdW Q

t ,

where Σ is a constant matrix, then zero-coupon yields havethe Nelson-Siegel factor structure:

yt(τ) = Lt +(1 − e−λτ

λτ

)St +

(1 − e−λτ

λτ− e−λτ

)Ct −

A(τ)τ

.

This defines the AFNS model class.The constant yield-adjustment term, A(τ)/τ , ensuresabsence of arbitrage.Nice analytical pricing formulas to work with. 19 / 39

The Shadow-Rate AFNS Model Class (1)

In the shadow-rate AFNS class of models, the shadow rate is

st = Lt + St ,

while the Q-dynamics are as in the original AFNS model.

The instantaneous shadow forward rates are given by

f (t ,T ) = Lt + e−λ(T−t)St + λ(T − t)e−λ(T−t)Ct + Af (t ,T ),

where Af (t ,T ) is an analytical function of model parameters.

The European call option with maturity T and strike price K writtenon the zero-coupon shadow bond maturing at T + δ is given by

C(t ,T ,T + δ;K ) = EQt

[e−

∫ Tt sudu max{P(T ,T + δ)− K ,0}

]

= P(t ,T + δ)Φ(d1)− KP(t ,T )Φ(d2),

d1 =ln(

P(t,T+δ)P(t,T )K

)+ 1

2v(t ,T ,T + δ)√

v(t ,T ,T + δ), d2 = d1 −

√v(t ,T ,T + δ),

and v(t ,T ,T + δ) is the conditional variance mentioned earlier. 20 / 39


Recall that the nonnegative instantaneous forward rate in anyGaussian model within the option-based shadow-rate framework is

f (t ,T ) = f (t ,T )Φ( f (t ,T )

ω(t ,T )

)+ ω(t ,T )

1√2π

exp(− 1

2

[ f (t ,T )

ω(t ,T )

]2).

Within the shadow-rate AFNS model, we obtain

ω(t ,T )2 =12

limδ→0

∂2v(t ,T , T + δ)

∂δ2

= σ211(T − t) + (σ2

21 + σ222)

1 − e−2λ(T−t)

2λ+(σ2

31 + σ232 + σ

233)

×

[1 − e−2λ(T−t)

4λ−

12(T − t)e−2λ(T−t)

−

12λ(T − t)2e−2λ(T−t)

]

+2σ11σ211 − e−λ(T−t)

λ+ 2σ11σ31

[

− (T − t)e−λ(T−t) +1 − e−λ(T−t)

λ

]

+(σ21σ31 + σ22σ32)[

− (T − t)e−2λ(T−t) +1 − e−2λ(T−t)

2λ

]

.

To be consistent with the notation in Kim and Singleton (2012), werefer to this as the B-AFNS(3) model class. 21 / 39


The yield-to-maturity is defined the usual way as

yt(τ) =

1τ

∫ t+τ

tf t(s)ds

=1τ

∫ t+τ

t

[ft (s)Φ

( ft(s)ω(s)

)+ ω(s)

1√2π

exp(− 1

2

[ ft (s)ω(s)

]2)]ds.

This is the measurement equation in the Kalman filter.

22 / 39

The Real-World Dynamics

We complete the model by specifying the link between therisk-neutral and real-world yield dynamics using theessentially affine risk premiums introduced by Duffee (2002):

dLt

dSt

dCt

=

κP11 κP

12 κP13

κP21 κP

22 κP23

κP31 κP

32 κP33

θP1θP

2θP

3

−

Lt

St

Ct

dt+Σ

dW P,Lt

dW P,St

dW P,Ct

,

where Σ is a lower triangular matrix, i.e. the maximally flexibleAFNS specification with 19 model parameters.

This is the transition equation in the Kalman filter.Due to the nonlinearity of the measurement equation, weuse the extended Kalman filter.Importantly, the ease and robustness of the regular AFNSmodels carries over to its shadow-rate equivalent ⇒ TheB-AFNS(3) model class is highly empirically tractable. 23 / 39

The Extended Kalman Filter Estimation (1)

In shadow-rate models, zero-coupon bond yields are notaffine functions of the state variables.

Instead, the measurement equation takes the generalform

yt = z(Xt ;ψ) + εt .

In the extended Kalman filter, this equation is linearizedthrough a first-order Taylor expansion around the bestguess of Xt in the prediction step of the Kalman filteralgorithm, Xt|t−1.

Thus, the approximation is given by

z(Xt ;ψ) ≈ z(Xt|t−1;ψ) +∂z(Xt ;ψ)

∂Xt

∣∣∣Xt=Xt|t−1

(Xt − Xt|t−1).

Now, defineAt(ψ) ≡ z(Xt|t−1;ψ)−

∂z(Xt ;ψ)

∂Xt

∣

∣

∣

Xt=Xt|t−1

Xt|t−1, Bt(ψ) ≡∂z(Xt ;ψ)

∂Xt

∣

∣

∣

Xt=Xt|t−1

.

24 / 39

The Extended Kalman Filter Estimation (2)

For the purposes of the Kalman filter, the measurementequation can be thought of as having an affine form:

yt = At(ψ) + Bt(ψ)Xt + εt .

Thus, Bt(ψ) as defined on the previous slide will be usedin the Kalman filter algorithm.

Importantly, the error term in the Kalman filter iscalculated directly (and not from the Taylor approximation)

vt = yt − E [yt |Yt−1]

= yt − z(Xt|t−1;ψ).

Note: At(ψ) is not actually calculated or used.

Beyond these two adjustments, the steps in the algorithmproceed as previously described.

25 / 39

Japanese Government Bond Yield Data

1996 2000 2004 2008 2012

01

23

45

Rat

e in

per

cent

10−year yield 4−year yield 1−year yield 6−month yield

We extend the Kim-Singleton weekly sample (1995-2008) withdata from Bloomberg until May 3, 2013, for 6 maturities.

We estimate one-, two-, and three-factor Gaussian models withand without the shadow-rate interpretation on this data.

Long sample of near-zero yields, ideal for our purposes! 26 / 39

Extended vs. Unscented Kalman Filter EstimationMaturity in months AllRMSE

6 12 24 48 84 120 yieldsMax log L

B-V(1) modelExtended KF 4.9 0.2 10.9 30.2 52.5 50.8 32.7 29,263.60Unscented KF 5.0 0.2 10.8 30.2 52.4 50.8 32.6 29,279.22B-AFNS(2) modelExtended KF 6.6 0.3 8.9 14.6 17.0 3.2 10.3 32,808.21Unscented KF 6.7 0.4 8.8 14.5 17.0 3.2 10.3 32,815.83B-AFNS(3) modelExtended KF 0.4 2.1 0.3 3.5 0.7 16.7 7.0 36,520.00Unscented KF 0.4 2.2 0.3 3.5 0.4 16.7 7.0 36,528.49

We estimated all shadow-rate models with the extended andunscented Kalman filter.In terms of RMSEs, differences are barely noticeable.We conclude that the extended Kalman filter is efficient atestimating shadow-rate models.This is also consistent with the findings in Christoffersen et al.(2014) regarding nonlinear filtering.Further evidence from a simulation study will be provided later. 27 / 39

How Good is the Option-Based Approximation? (1)

We assess the approximation provided by the option-basedB-AFNS(3) model via Monte Carlo simulations.

Input: Estimated K̂ P , Σ̂, θ̂P , λ̂, and filtered X̂t .

The continuous-time P-dynamics are given by

dXt = K̂ P(θ̂P − Xt)dt + Σ̂dW Pt .

These are approximated with the Euler approximation:

X it = X i

t−1+κ̂Pii (θ̂

Pi −X i

t−1)∆t+κ̂Pij (θ̂

Pj −X j

t−1)∆t+σ̂ii

√∆tz i

t+σ̂ij

√∆tz j

t ,

where z it , z

jt ∼ N(0,1), ∆t = 0.0001, and X0 = X̂t .

Simulated shadow and Black (1995) bond prices are given by

PSt (n) = PS

t−1(n)exp(−(X 1t + X 2

t )∆t), PS0 (n) = 1

PBt (n) = PB

t−1(n)exp(−max(X 1t + X 2

t ,0)∆t), PB0 (n) = 1.

The resulting bond prices for any maturity t > 0 are:

PSt =

1N

N∑

n=1

PSt (n) and PB

t =1N

N∑

n=1

PBt (n).

28 / 39


0 2 4 6 8 10

−1

01

2

Time to maturity in years

Rat

e in

per

cent

0 2 4 6 8 10

−1

01

2


Rat

e in

per

cent

0 2 4 6 8 10

−1

01

2


Rat

e in

per

cent

Yield curve, B−AFNS(3) model Shadow curve, B−AFNS(3) model Yield curve, Black (1995) simulation Shadow curve, Black (1995) simulation Observed yields

We simulate N = 25,000 ten-year-long factor paths and calculateshadow and Black zero-coupon bond yields.

The difference between simulated and analytical shadow bond yieldsprovides a measure of the simulation accuracy.

The difference between simulated Black and option-based yieldsmeasures the accuracy of the option-based approximation. 29 / 39


Average Absolute Simulation Error in Shadow Yields andAverage Absolute Approximation Error in Observed Yields:

Averaged over 19 dates—first observation in each year in sample.

Maturity in months12 36 60 84 120

Shadow yields 0.14 0.29 0.44 0.55 0.72Yields 0.09 0.22 0.43 0.82 2.25

Differences (in bps) between simulated and option-impliednonnegative yields are only marginally larger than differencesbetween simulated and analytical shadow yields.

We conclude that the option-based B-AFNS(3) model providesa very close approximation to the Black model for our data.

Note: If greater accuracy is required, Priebsch (2013) providesa second-order approximation, but it is very time consuming.

30 / 39

Fit of Standard and Shadow-Rate Models (RMSE)

Maturity in months All6 12 24 48 84 120 yields

Max log L

One-factor modelsV(1) 5.8 0.0 12.1 32.8 55.6 52.4 34.4 28,362.97B-V(1) 4.9 0.2 10.9 30.2 52.5 50.8 32.7 29,263.60Two-factor modelsAFNS(2) 5.9 0.0 9.0 17.6 21.6 0.0 12.2 32,186.23B-AFNS(2) 6.6 0.3 8.9 14.6 17.0 3.2 10.3 32,808.21Three-factor modelsAFNS(3) 0.0 2.4 0.2 4.2 0.0 23.3 9.7 35,469.67B-AFNS(3) 0.4 2.1 0.3 3.5 0.7 16.7 7.0 36,520.00

Standard one-, two-, and three-factor models, denotedV(1), AFNS(2), and AFNS(3), respectively.

Equivalent shadow-rate models, denoted B-V(1),B-AFNS(2), and B-AFNS(3), respectively.

Shadow-rate models have closer fit, higher log L.31 / 39

Fitted Yield Curves: Two- and Three-Factor Models

0 2 4 6 8 10

0.0

0.5

1.0

1.5

2.0


Rat

e in

per

cent

AFNS(2) model B−AFNS(2) model Observed yields

0 2 4 6 8 10

0.0

0.5

1.0

1.5

2.0


Rat

e in

per

cent

AFNS(3) model B−AFNS(3) model Observed yields

Fitted yield curves from standard and shadow-rate two-and three-factor AFNS models on July 1, 2005.

For two-factor models, there is a clear visible gain fromadopting a shadow-rate implementation.

For three-factor models, the improvement is less clear.32 / 39

Estimated Shadow Short Rates

1995 2000 2005 2010

−3

−2

−1

01

23

Rat

e in

per

cent

1995 2000 2005 2010

−3

−2

−1

01

23

Rat

e in

per

cent

B−V(1) model B−AFNS(2) model B−AFNS(3) model B−AG2 model

The models tend to agree about the location of theshadow short rate when it is positive.On the other hand, there is much disagreement about theshadow short rate when it is negative. 33 / 39

Probability of Negative Future Short Rates

In the AFNS models, we assess the probability of negativefuture short rates at any horizon t + τ > t .




The conditional mean of Xt under the P-measure is

EPt [Xt+τ ] = (I − exp(−K̂ Pτ))θP + exp(−K̂ Pτ)X̂t .

The conditional covariance matrix of Xt under the P-measure is

V Pt [Xt+τ ] =

∫ τ

0e−K̂ PsΣ̂Σ̂′e−(K̂ P)′sds.

Now, the short rate is normally distributed:

EPt [rt+τ ] =

(

1 1 0)

EPt [Xt+τ ], V P

t [rt+τ ] =(

1 1 0)

V Pt [Xt+τ ]

110

.

PP(rt+τ ≤ 0) = Φ( −EP

t [rt+τ ]√V P

t [rt+τ ]

).

34 / 39

Probability of Negative Future Short Rates

1995 2000 2005 2010

0.0

0.2

0.4

0.6

0.8

1.0

Pro

babi

lity

AFNS(3) model

Illustration of the probability that the short rate will benegative at a three-month forecast horizon according tothe estimated AFNS(3) model.

35 / 39

Yield Volatility Simulations in Shadow-Rate Models (1)

In standard affine models, where zero-coupon yields areaffine functions of the state variables, model-impliedconditional predicted yield volatilities are given by thesquare root of

V Pt [yT (τ)] =

1τ2

B(τ)′V Pt [XT ]B(τ),

whereT − t is the prediction period;τ is the yield maturity;B(τ) contains the yield factor loadings;V P

t [XT ] is the conditional covariance matrix.

In shadow-rate models, zero-coupon bond yields arenonlinear functions of the state variables. Thus,model-implied conditional yield volatilities have to begenerated via Monte Carlo Simulations.

36 / 39

Yield Volatility Simulations in Shadow-Rate Models (2)




The conditional mean of XT under the P-measure:

EPt [XT ] = (I − exp(−K̂ P(T − t)))θP + exp(−K̂ P(T − t))X̂t .

The conditional covariance matrix of XT under the P-measure:

V Pt [XT ] =

∫ T−t

0e−K̂ PsΣ̂Σ̂′e−(K̂ P)′sds.

We simulate N = 1,000 draws directly at the desired horizon:

X nT ∼ N

(EP

t [XT ],VPt [XT ]

), n = 1, . . . ,1000.

The resulting conditional bond volatility is:

√

V Pt [yT (τ )] =

√

√

√

√

1N − 1

N∑

n=1

(y(X nT , τ )− y(XT , τ ))2, y(XT , τ ) =

1N

N∑

n=1

y(X nT , τ ).

37 / 39

Volatility Compression of Medium-Term Yields

1995 2000 2005 2010

020

4060

8010

0

Rat

e in

bas

is p

oint

s

1995 2000 2005 2010

020

4060

8010

0

Rat

e in

bas

is p

oint

s

Correlation = 72.4%

AFNS(3) model B−AFNS(3) model Three−month realized volatility of two−year yield

Standard Gaussian models assume constant yield volatility.

However, near the ZLB, there is a clear positive correlationbetween yield levels and volatilities.

Gaussian shadow-rate models can replicate this phenomenonfor short- and medium-term yield maturities. 38 / 39

Conclusion

We combine Krippner’s general option-based shadow-rateframework with the arbitrage-free Nelson-Siegel model class.

This delivers highly tractable models that are approximatelyarbitrage-free and respect the zero lower bound for yields.

We estimate one-, two-, and three-factor versions of theoption-based shadow-rate AFNS model and matching standardmodels on a long sample of near-zero Japanese yields.

We find that introducing the shadow-rate interpretationimproves model fit and allows the models to capture otherimportant aspects of yields near the ZLB, in particular thepositive correlation between yield levels and volatilities.

We show that the B-AFNS(3) model provides a very closeapproximation to the corresponding Black shadow-rate model.

Finally, results for the shadow short rate are sensitive to factordimension and dynamics. Thus, it is not likely to be a usefultool for assessing the stance of monetary policy near the ZLB. 39 / 39

Documents

Estimating Shadow-Rate Term Structure Models with Near ... · Estimating Shadow-Rate Term Structure Models with Near-Zero Yields Jens H. E. Christensen & Glenn D. Rudebusch Federal