Efficient bond price approximations in non-linear equilibrium-based term structure models

Martin M. Andreasen; Pawel Zabczyk

doi:10.1515/snde-2012-0005

Published by De Gruyter February 27, 2014

Efficient bond price approximations in non-linear equilibrium-based term structure models

Martin M. Andreasen and Pawel Zabczyk

From the journal Studies in Nonlinear Dynamics & Econometrics

https://doi.org/10.1515/snde-2012-0005

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Abstract

This paper develops an efficient method to compute higher-order perturbation approximations of bond prices. At third order, our approach can significantly shorten the approximation process and its precision exceeds the log-normal method and a procedure using consol bonds. The efficiency gains greatly facilitate any estimation which is illustrated by considering a long-run risk model for the US. Allowing for an unconstrained intertemporal elasticity of substitution enhances the model’s fit, and we see further improvements when incorporating stochastic volatility and external habits.

Keywords: DSGE model; habit model; higher order perturbation method; long-run risk; stochastic volatility

JEL: C68; E0

Corresponding author: Martin M. Andreasen, Aarhus University, Fuglesangs Allé 4, 8210 Aarhus V, Denmark, e-mail: mandreasen@econ.au.dk

Acknowledgment

For helpful comments and suggestions, we thank Michel Juillard, Paul Klein, Simon Price, Juan Rubio-Ramirez, Bruce Mizrach (the Editor), and an anonymous referee. Andreasen gratefully acknowledges access to computer facilities provided by the Danish Center for Scientific Computing (DCSC) and financial support to CREATES – Center for Research in Econometric Analysis of Time Series (DNRF78), funded by the Danish National Research Foundation.

Appendix A

A general transformation of bond prices

This appendix considers the general case of an invertible transformation function R(‧)∈C^N, implying that R(p^{t, k})≡P^t^,^k. Here

Fk(x,σ)=Et[R(pt, k(x, σ))−ℳ(g(h(x, σ)+σηεt+1, σ), g(x, σ), h(x, σ)+σηεt+1, x)×R(pt+1, k−1(h(x, σ)+σηεt+1, σ))] .

First order terms

Derivative of p^k with respect to x

Consider [Fxk(xt, σ)]α1=0 which implies

Rp(pk)[pxt,k]α1−[ℳx]α1R(pt+1 ,k−1)−ℳRp(pt+1, k−1)[pxt+1, k−1]γ1[hxt]α1γ1=0

For k=1 we have Rp(p1)[px1]α1=[ℳx]α1 and M=R(p¹). Hence

Rp(pk)[pxk]α1=[px1]α1Rp(p1)R(pk−1)+R(p1)Rp(pk−1)[pxk−1]γ1[hx]α1γ1

Derivative of p^k with respect to σ

Computing Fσk(xss, 0)=0 implies

Et[Rp(pk)[pσk]−[ℳσ]R(pk−1)−ℳRp(pk−1)([pxk−1]γ1([hσ]γ1+ηt+1)+[pσk−1])]=0

For k=1 we have Et[Rp(p1)[pσ1]]=Et[ℳσ]. Similar arguments as in the text then implies pσk=0 for all values of k.

Second order terms

Derivative of p^k with respect to (x, x)

We have that [Fxxk(xss, 0)]α1, α2=0 implies

Rpp(pk)[pxk]α2[pxk]α1+Rp(pk)[pxxk]α1α2 −[ℳxx]α1α2R(pk−1)−[ℳx]α1Rp(pk−1)[pxk−1]γ2[hx]α2γ2 −[ℳx]α2Rp(pk−1)[pxk−1]γ1[hx]α1γ1 −ℳRpp(pk−1)[pxk−1]γ2[hx]α2γ2[pxk−1]γ1[hx]α1γ1 −ℳRp(pk−1)[pxxk−1]γ1γ2[hx]α2γ2[hx]α1γ1 −ℳRp(pk−1)[pxk−1]γ1[hxx]α1α2γ1 =0

The value of M equals R(p¹) and M_x is computed above. Moreover, for k=1 we have

[ℳxx]α1α2=Rp(p1)[pxx1]α1, α2+Rpp(p1)[px1]α2[px1]α1

Thus

Rp(pk)[pxxk]α1,α2=−Rpp(pk)[pxk]α2[pxk]α1 +(Rp(p1)[pxx1]α1, α2+Rpp(p1)[px1]α2[px1]α1)R(pk−1) +[px1]α1Rp(p1)Rp(pk−1)[pxk−1]γ2[hx]α2γ2 +[px1]α2Rp(p1)Rp(pk−1)[pxk−1]γ1[hx]α1γ1 +R(p1)Rpp(pk−1)[pxk−1]γ2[hx]α2γ2[pxk−1]γ1[hx]α1γ1 +R(p1)Rp(pk−1)[pxxk−1]γ1γ2[hx]α2γ2[hx]α1γ1 +R(p1)Rp(pk−1)[pxk−1]γ1[hxx]α1α2γ1

Derivative of p^k with respect to (σ, σ)

Next, [Fσσ(xss, 0)]=0 implies

Et[−Rp(pk)[pσσk]+[ℳσσ]R(pk−1) +[ℳσ]Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2 +[ℳσ]Rp(pk−1)[pxk−1]γ1[η]ϕ1γ1[εt+1]ϕ1 +ℳRpp(pk−1)[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2[pxk−1]γ1[η]ϕ1γ1[εt+1]ϕ1 +ℳRp(pk−1)[pxxk−1]γ1γ2[η]ϕ2γ2[εt+1]ϕ2[η]ϕ1γ1[εt+1]ϕ1 +ℳRp(pk−1)[pxk−1]γ1[hσσ]γ1+ℳRp(pk−1)[pσσk−1]] =0

For k=1 we have Et[ℳσσ]=[pσσ1]Rp(p1). Using this and previous results, we then obtain

Rp(pk)[pσσk]=[pσσ1]Rp(p1)R(pk−1) +2[ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[I]ϕ1ϕ2 +2[ℳxt+1]γ1[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[I]ϕ1ϕ2+R(p1)Rpp(pk−1)[pxk−1]γ2[η]ϕ2γ2[pxk−1]γ1[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rp(pk−1)[pxxk−1]γ1γ2[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1+R(p1)Rp(pk−1)[pxk−1]γ1[hσσ]γ1 +R(p1)Rp(pk−1)[pσσk−1]

Third order terms

Derivative of p^k with respect to (x, x, x)

Applying the chain rule to the definition of F^k one can show that [Fxxx(xss, 0)]α1α2α3=0 equals

Rp(pk)[pxxxk]α1α2α3=−Rppp(pk)[pxk]α3[pxk]α2[pxk]α1−Rpp(pk)[pxxk]α2α3[pxk]α1−Rpp(pk)[pxk]α3[pxxk]α1α2 −Rpp(pk)[pxk]α3[pxxk]α1α2+[ℳxxx]α1α2α3R(pk−1)+(Rp(p1)[pxx1]α1,α2+Rpp(p1)[px1]α2[px1]α1)Rp(pk−1)[pxk−1]γ3[hx]α3γ3+(Rp(p1)[pxx1]α1α3+Rpp(p1)[px1]α3[px1]α1)Rp(pk−1)[pxk−1]γ2[hx]α2γ2+[px1]α1Rp(p1)/R(p0)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[hx]α2γ2+[px1]α1Rp(p1)/R(p0)Rp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[hx]α2γ2 +[px1]α1Rp(p1)/R(p0)Rp(pk−1)[pxk−1]γ2[hxx]α2α3γ2+(Rp(p1)[pxx1]α2, α3+Rpp(p1)[px1]α3[px1]α2)Rp(pk−1)[pxk−1]γ1[hx]α1γ1 +[px1]α2Rp(p1)/R(p0)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ1[hx]α1γ1 +[px1]α2Rp(p1)/R(p0)Rp(pk−1)[pxxk−1]γ1γ3[hx]α3γ3[hx]α1γ1 +[px1]α2Rp(p1)/R(p0)Rp(pk−1)[pxk−1]γ1[hxx]α1α3γ1 +[px1]α3Rp(p1)/R(p0)Rpp(pk−1)[pxk−1]γ2[hx]α2γ2[pxk−1]γ1[hx]α1γ1+R(p1)Rppp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[hx]α2γ2[pxk−1]γ1[hx]α1γ1+R(p1)Rpp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[hx]α2γ2[pxk−1]γ1[hx]α1γ1+R(p1)Rpp(pk−1)[pxk−1]γ2[hxx]α2α3γ2[pxk−1]γ1[hx]α1γ1+R(p1)Rpp(pk−1)[pxk−1]γ2[hx]α2γ2[pxxk−1]γ1γ3[hx]α3γ3[hx]α1γ1+R(p1)Rpp(pk−1)[pxk−1]γ2[hx]α2γ2[pxk−1]γ1[hxx]α1α3γ1+[px1]α3Rp(p1)/R(p0)Rp(pk−1)[pxxk−1]γ1γ2[hx]α2γ2[hx]α1γ1+R(p1)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxxk−1]γ1γ2[hx]α2γ2[hx]α1γ1+R(p1)Rp(pk−1)[pxxxk−1]γ1γ2γ3[hx]α3γ3[hx]α2γ2[hx]α1γ1+R(p1)Rp(pk−1)[pxxk−1]γ1γ2[hxx]α2α3γ2[hx]α1γ1+R(p1)Rp(pk−1)[pxxk−1]γ1γ2[hx]α2γ2[hxx]α1α3γ1+[px1]α3Rp(p1)/R(p0)Rp(pk−1)[pxk−1]γ1[hxx]α1α2γ1+R(p1)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ1[hxx]α1α2γ1+R(p1)Rp(pk−1)[pxxk−1]γ1γ3[hx]α3γ3[hxx]α1α2γ1+R(p1)Rp(pk−1)[pxk−1]γ1[hxxx]α1α2α3γ1

Note that we can also eliminate [ℳxxx]α1α2α3 from this expression. Again, the trick is to observe that for k=1 we have P⁰=1 for all values of (x_t, σ) and so all derivatives have to equal zero. Thus

Rp(p1)[pxxx1]α1α2α3=−Rppp(p1)[px1]α3[px1]α2[px1]α1 −Rpp(p1)[pxx1]α2α3[px1]α1 −Rpp(p1)[px1]α2[pxx1]α1α3 −Rpp(p1)[px1]α3[pxx1]α1α2 +[ℳxxx]α1α2α3R(p0)

⇕

(Rp(p1)[pxxx1]α1α2α3+Rppp(p1)[px1]α3[px1]α2[px1]α1+Rpp(p1)[pxx1]α2α3[px1]α1Rpp(p1)[px1]α2[pxx1]α1α3+Rpp(p1)[px1]α3[pxx1]α1α2)=[ℳxxx]α1α2α3

because R(p⁰)=1.

Thus we get for k>1

Rp(pk)[pxxxk]α1α2α3=−Rppp(pk)[pxk]α3[pxk]α2[pxk]α1 −Rpp(pk)[pxk]α2α3[pxk]α1 −Rpp(pk)[pxk]α2[pxk]α1α3 −Rpp(pk)[pk]α3[pxxk]α1α2 +(Rp(p1)[pxxx1]α1α2α3+Rppp(p1)[px1]α3[px1]α2[px1]α1+Rpp(p1)[pxx1]α2α3[px1]α1 +Rpp(p1)[px1]α2[pxx1]α1α3+Rpp(p1)[px1]α3[pxx1]α1α2)R(pk−1) +(Rp(p1)[pxx1]α1,α2+Rpp(p1)[px1]α2[px1]α1)Rp(pk−1)[pxk−1]γ3[hx]α3γ3+(Rp(p1)[pxx1]α1α3+Rpp(p1)[px1]α3[px1]α1)Rp(pk−1)[pxk−1]γ2[hx]α2γ2 +[px1]α1Rp(p1)/R(p0)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[hx]α2γ2 +[px1]α1Rp(p1)/R(p0)Rp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[hx]α2γ2 +[px1]α1Rp(p1)/R(p0)Rp(pk−1)[pxk−1]γ2[hxx]α2α3γ2+(Rp(p1)[pxx1]α2, α3+Rpp(p1)[px1]α3[px1]α2)Rp(pk−1)[pxk−1]γ1[hx]α1γ1 +[px1]α2Rp(p1)/R(p0)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ1[hx]α1γ1 +[px1]α2Rp(p1)/R(p0)Rp(pk−1)[pxxk−1]γ1γ3[hx]α3γ3[hx]α1γ1 +[px1]α2Rp(p1)/R(p0)Rp(pk−1)[pxk−1]γ1[hxx]α1α3γ1 +[px1]α3Rp(p1)/R(p0)Rpp(pk−1)[pxk−1]γ2[hx]α2γ2[pxk−1]γ1[hx]α1γ1 +R(p1)Rppp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[hx]α2γ2[pxk−1]γ1[hx]α1γ1 +R(p1)Rpp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[hx]α2γ2[pxk−1]γ1[hx]α1γ1 +R(p1)Rpp(pk−1)[pxk−1]γ2[hxx]α2α3γ2[pxk−1]γ1[hx]α1γ1 +R(p1)Rpp(pk−1)[pxk−1]γ2[hx]α2γ2[pxxk−1]γ1γ3[hx]α3γ3[hx]α1γ1 +R(p1)Rpp(pk−1)[pxk−1]γ2[hx]α2γ2[pxk−1]γ1[hxx]α1α3γ1 +[px1]α3Rp(p1)/R(p0)Rp(pk−1)[pxxk−1]γ1γ2[hx]α2γ2[hx]α1γ1 +R(p1)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxxk−1]γ1γ2[hx]α2γ2[hx]α1γ1 +R(p1)Rp(pk−1)[pxxxk−1]γ1γ2γ3[hx]α3γ3[hx]α2γ2[hx]α1γ1 +R(p1)Rp(pk−1)[pxxk−1]γ1γ2[hxx]α2α3γ2[hx]α1γ1 +R(p1)Rp(pk−1)[pxxk−1]γ1γ2[hx]α2γ2[hxx]α1α3γ1 +[px1]α3Rp(p1)/R(p0)Rp(pk−1)[pxk−1]γ1[hxx]α1α2γ1 +R(p1)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ1[hxx]α1α2γ1 +R(p1)Rp(pk−1)[pxxk−1]γ1γ3[hx]α3γ3[hxx]α1α2γ1 +R(p1)Rp(pk−1)[pxk−1]γ1[hxxx]α1α2α3γ1

With a log-transformation R(p^t,k)=M^k, R_p(p^t,k)=M^k, R_pp(p^t,k)=M^k, and R_ppp(p^t,k)=M^k in the deterministic steady state. Using the expressions for first and second order derivatives of bond prices derived above, we get, after simplifying, the expression stated in the body of the text.

Derivative of p^k with respect to (σ, σ, x)

It is possible to show that [Fσσx(xss, 0)]α3=0 implies

Et{−Rpp(pk)[pxk]α3[pσσk]−Rp(pk)[pσσxk]α3 +[ℳσσx]R(pk−1)+[ℳσσ]Rp(pk−1)[pxk−1]γ3[hx]α3γ3 +2[ℳσx]α3Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[t+1]ϕ2 +2[ℳσ]Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[t+1]ϕ2 +2[ℳσ]Rp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[t+1]ϕ2 +[ℳx]α3Rpp(pk−1)[pxk−1]γ2[η]ϕ2γ2[t+1]ϕ2[pxk−1]γ1[η]ϕ1γ1[t+1]ϕ1 +ℳRppp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[ t+1]ϕ2[pxk−1]γ1[η]ϕ1γ1[t+1]ϕ1 +ℳRpp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[t+1]ϕ2[pxk−1]γ1[η]ϕ1γ1[t+1]ϕ1 +ℳRpp(pk−1)[pxk−1]γ2[η]ϕ2γ2[t+1]ϕ2[pxxk−1]γ1γ3[hx]α3γ3[η]ϕ1γ1[t+1]ϕ1 +[ℳx]α3Rp(pk−1)[pxxk−1]γ1γ2[η]ϕ2γ2[t+1]ϕ2[η]ϕ1γ1[t+1]ϕ1 +ℳRpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxxk−1]γ1γ2[η]ϕ2γ2[t+1]ϕ2[η]ϕ1γ1[t+1]ϕ1 +ℳRp(pk−1)[pxxxk−1]γ1γ2γ3[hx]α3γ3[η]ϕ2γ2[t+1]ϕ2[η]ϕ1γ1[t+1]ϕ1 +[ℳx]α3Rp(pk−1)[pxk−1]γ1[hσσ]γ1 +ℳRpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ1[hσσ]γ1 +ℳRp(pk−1)[pxxk−1]γ1γ3[hx]α3γ3[hσσ]γ1 +ℳRp(pk−1)[pxk−1]γ1[hσσx]α3γ1 +[ℳx]α3Rp(pk−1)[pσσk−1] +ℳRpp(pk−1)[pxk−1]γ3[hx]α3γ3[pσσk−1] +ℳRp(pk−1)[pσσxk−1]γ3[hx]α3γ3}=0

⇕

Rp(pk)[pσσxk]α3=−Rpp(pk)[pxk]α3[pσσk] +Et[ℳσσx]R(pk−1) +[pσσ1]Rp(p1)/R(p0)Rp(pk−1)[pxk−1]γ3[hx]α3γ3 +2Et([ℳσx]α3Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[et+1]ϕ2) +2Et([ℳσ]Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[et+1]ϕ2) +2Et([ℳσ]Rp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[et+1]ϕ2) +[px1]α3Rp(p1)/R(p0)Rpp(pk−1)[pxk−1]γ2[η]ϕ2γ2[pxk−1]γ1[η]ϕ1γ1[I]ϕ2ϕ1 +R(p1)Rppp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[pxk−1]γ1[η]ϕ1γ1[I]ϕ2ϕ1 +R(p1)Rpp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[pxk−1]γ1[η]ϕ1γ1[I]ϕ2ϕ1 +R(p1)Rpp(pk−1)[pxk−1]γ2[η]ϕ2γ2[pxxk−1]γ1γ3[hx]α3γ3[η]ϕ1γ1[I]ϕ2ϕ1 +[px1]α3Rp(p1)/R(p0)Rp(pk−1)[pxxk−1]γ1γ2[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1 +R(p1)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxxk−1]γ1γ2[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1 +R(p1)Rp(pk−1)[pxxxk−1]γ1γ2γ3[hx]α3γ3[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1 +[px1]α3Rp(p1)/R(p0)Rp(pk−1)[pxk−1]γ1[hσσ]γ1 +R(p1)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ1[hσσ]γ1 +R(p1)Rp(pk−1)[pxxk−1]γ1γ3[hx]α3γ3[hσσ]γ1 +R(p1)Rp(pk−1)[pxk−1]γ1[hσσx]α3γ1 +[px1]α3Rp(p1)/R(p0)Rp(pk−1)[pσσk−1] +R(p1)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pσσk−1] +R(p1)Rp(pk−1)[pσσxk−1]γ3[hx]α3γ3

where we have used

ℳ=R(p1)

[ℳx]α3=[px1]α3Rp(p1)/R(p0)

Et[ℳσσ]=[pσσ1]Rp(p1)/R(p0)

We now compute the terms with derivatives of σ. Here we recall that

[ℳσ]≡([ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1+[ℳxt+1]γ1[η]ϕ1γ1[εt+1]ϕ1).

2Et([ℳσ]Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2)=2Et([ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2)+2Et([ℳxt+1]γ1[η]ϕ1γ1[εt+1]ϕ1Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2)=2[ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[I]ϕ1ϕ2+2[ℳxt+1]γ1[η]ϕ1γ1Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[I]ϕ1ϕ2

and

2Et([ℳσ]Rp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[εt+1]ϕ2)=2Et([ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1Rp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[εt+1]ϕ2)+2Et([ℳxt+1]γ1[η]ϕ1γ1[εt+1]ϕ1Rp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[εt+1]ϕ2)=2[ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1Rp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[I]ϕ1ϕ2+2Et[ℳxt+1]γ1[η]ϕ1γ1Rp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[I]ϕ1ϕ2

To compute the [ℳσx]α3 term, we need to find an expression for [ℳσx]α3. Hence

[ℳσx]α3=([ℳyt+1yt+1]β1β3[gx]γ3β3[hx]α3γ3+[ℳyt+1yt]β1β3[gx]α3β3+[ℳyt+1xt+1]β1γ3[hx]α3γ3+[ℳyt+1xt]β1α3) ×[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1 +[ℳyt+1]β1[gxx]γ1γ3β1[hx]α3γ3[η]ϕ1γ1[εt+1]ϕ1 +([ℳxt+1yt+1]γ1β3[gx]γ3β3[hx]α3γ3+[ℳxt+1yt]γ1β3[gx]α3β3+[ℳxt+1xt+1]γ1γ3[hx]α3γ3+[ℳxt+1xt]γ1α3) ×[η]ϕ1γ1[εt+1]ϕ1

2Et([ℳσx]α3Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2)=2Et{([ℳyt+1yt+1]β1β3[gx]γ3β3[hx]α3γ3+[ℳyt+1yt]β1β3[gx]α3β3+[ℳyt+1xt+1]β1γ3[hx]α3γ3+[ℳyt+1xt]β1α3) ×[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2 +[ℳyt+1]β1[gxx]γ1γ3β1[hx]α3γ3[η]ϕ1γ1[εt+1]ϕ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2 +([ℳxt+1yt+1]γ1β3[gx]γ3β3[hx]α3γ3+[ℳxt+1yt]γ1β3[gx]α3β3+[ℳxt+1xt+1]γ1γ3[hx]α3γ3+[ℳxt+1xt]γ1α3) ×[η]ϕ1γ1[εt+1]ϕ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2}=2([ℳyt+1yt+1]β1β3[gx]α3β3[hx]α3γ3+[ℳyt+1yt]β1β3[gx]α3β3+[ℳyt+1xt+1]β1γ3[hx]α3γ3+[ℳyt+1xt]β1α3) ×[gx]γ1β1[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[I]ϕ1ϕ2 +2[ℳyt+1]β1[gxx]γ1γ3β1[hx]α3γ3[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[I]ϕ1ϕ2 +2([ℳxt+1yt+1]γ1β3[gx]α3β3[hx]α3γ3+[ℳxt+1yt]γ1β3[gx]α3β3+[ℳxt+1xt+1]γ1γ3[hx]α3γ3+[ℳxt+1xt]γ1α3) ×[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[I]ϕ1ϕ2

We finally note that Et([ℳσσx]α3) can be solved for and then substituted out by exploiting the fact that for k=1 we have P⁰=1 for all values of (x_t, σ) and so all derivatives have to equal zero. Thus

Rp(p1)[pσσx1]α3=−Rpp(p1)[px1]α3[pσσ1]+Et([ℳσσx]α3)R(p0)

⇕

(Rp(p1)[pσσx1]α3+Rpp(p1)[px1]α3[pσσ1])/R(p0)=Et([ℳσσx]α3)

So for k>1 we get

Rp(pk)[pσσxk]α3=−Rpp(pk)[pxk]α3[pσσk] +(Rp(p1)[pσσx1]α3+Rpp(p1)[px1]α3[pσσ1])R(pk−1)/R(p0) +[pσσ1]Rp(p1)/R(p0)Rp(pk−1)[pxk−1]γ3[hx]α3γ3 +2([ℳyt+1yt+1]β1β3[gx]γ3β3[hx]α3γ3+[ℳyt+1yt]β1β3[gx]α3β3+[ℳyt+1xt+1]β1γ3[hx]α3γ3+[ℳyt+1xt]β1α3) ×[gx]γ1β1[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[I]φ1ϕ2 +2[ℳyt+1]β1[gxx]γ1γ3β1[hx]α3γ3[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[I]ϕ1ϕ2 +2([ℳxt+1yt+1]γ1β3[gx]γ3β3[hx]α3γ3+[ℳxt+1yt]γ1β3[gx]α3β3+[ℳxt+1xt+1]γ1γ3[hx]α3γ3+[ℳxt+1xt]γ1α3) ×[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[I]ϕ1ϕ2 +2[ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[I]ϕ1ϕ2 +2[ℳxt+1]γ1[η]ϕ1γ1Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[I]ϕ1ϕ2 +2[ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1Rp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[I]ϕ1ϕ2 +2Et[ℳxt+1]γ1[η]ϕ1γ1Rp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[I]ϕ1ϕ2 +[px1]α3Rp(p1)/R(p0)Rpp(pk−1)[pxk−1]γ2[η]ϕ2γ2[pxk−1]γ1[η]ϕ1γ1[I]ϕ2ϕ1 +R(p1)Rppp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ2[η]ϕ2γ2[pxk−1]γ1[η]ϕ1γ1[I]ϕ2ϕ1 +R(p1)Rpp(pk−1)[pxxk−1]γ2γ3[hx]α3γ3[η]ϕ2γ2[pxk−1]γ1[η]ϕ1γ1[I]ϕ2ϕ1 +R(p1)Rpp(pk−1)[pxk−1]γ2[η]ϕ2γ2[pxxk−1]γ1γ3[hx]α3γ3[η]ϕ1γ1[I]ϕ2ϕ1 +[px1]α3Rp(p1)/R(p0)Rp(pk−1)[pxxk−1]γ1γ2[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1 +R(p1)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxxk−1]γ1γ2[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1 +R(p1)Rp(pk−1)[pxxxk−1]γ1γ2γ3[hx]α3γ3[η]ϕ2γ2[η]ϕ1γ1[I]ϕ2ϕ1 +[px1]α3Rp(p1)/R(p0)Rp(pk−1)[pxk−1]γ1[hσσ]γ1 +R(p1)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pxk−1]γ1[hσσ]γ1 +R(p1)Rp(pk−1)[pxxk−1]γ1γ3[hx]α3γ3[hσσ]γ1 +R(p1)Rp(pk−1)[pxk−1]γ1[hσσx]α3γ1 +[px1]α3Rp(p1)/R(p0)Rp(pk−1)[pσσk−1] +R(p1)Rpp(pk−1)[pxk−1]γ3[hx]α3γ3[pσσk−1] +R(p1)Rp(pk−1)[pσσxk−1]γ3[hx]α3γ3

For a logarithm transformation R(p^t,k)=M^k, R_p(p^t,k)=M^k, R_pp(p^t,k)=M^k, and R_ppp(p^t,k)=M^k. Using the expressions for first and second order derivatives of bond prices derived above, we get, after simplifying,

pσσxk(1, α3)= −2ℳ−1ℳyt+1gxηη′(pxk−1)′px1(1, α3) −2ℳ−1ℳxt+1ηη′(pxk−1)′px:1(1, α3) +pσσx1(1, α3) +2ℳ−1pxk−1ηη′(gx)′ ×(ℳyt+1yt+1gxhx(:,α3)+ℳyt+1ytgx(:, α3)+ℳyt+1xt+1hx(:, α3)+ℳyt+1xt(:, α3)) +∑β1=1ny2ℳ−1ℳyt+1(1, β1)pxk−1ηη′gxx(β1, :, :)hx(:, α3) +2ℳ−1pxk−1ηη′ ×(ℳxt+1yt+1gxhx(:, α3)+ℳxt+1ytgx(:,α3)+ℳxt+1xt+1hx(:, α3)+ℳxt+1xt(:, α3)) +2ℳ−1ℳyt+1gxηη′pxxk−1hx(:, α3) +2ℳ−1ℳxt+1ηη′pxxk−1hx(:, α3) +pxk−1ηη′pxxk−1hx(:,α3) +pxk−1ηη′pxxk−1hx(:,α3) +∑γ1=1nxη′(γ1, :)η′pxxxk−1(γ1, :, :)hx(:, α3) +(hσσ)'pxxk−1hx(:, αk) +pxk−1hσσx(:, α3) +pσσxk−1hx(:, α3)

Derivative of p^k with respect to (σ, σ, σ)

It is possible to show that F_σσσ(x_ss, 0)=0 implies

[Fσσσ(xss, 0)]=Et{−Rp(pk)[pσσσk] +[ℳσσσ]R(pk−1) +3[ℳσσ]Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2 +3[ℳσ]Rpp(pk−1)[pxk−1]γ3[η]ϕ3γ3[εt+1]ϕ3[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2 +3[ℳσ]Rp(pk−1)[pxxk−1]γ2γ3[η]ϕ3γ3[εt+1]ϕ3[η]ϕ2γ2[εt+1]ϕ2 +R(p1)Rppp(pk−1)[pxk−1]γ3[η]ϕ3γ3[εt+1]ϕ3[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2[pxk−1]γ1[η]ϕ1γ1[εt+1]ϕ1 +3R(p1)Rpp(pk−1)[pxk−1]γ2[η]ϕ2γ2[εt+1]ϕ2[pxxk−1]γ1γ3[η]ϕ3γ3[εt+1]ϕ3[η]ϕ1γ1[εt+1]ϕ1 +R(p1)Rp(pk−1)[pxxxk−1]γ1γ2γ3[η]ϕ3γ3[εt+1]ϕ3[η]ϕ2γ2[εt+1]ϕ2[η]ϕ1γ1[εt+1]ϕ1 +R(p1)Rp(pk−1)[pxk−1]γ1[hσσσ]γ1 +R(p1)Rp(pk−1)[pσσσk−1]}=0

We next use the expression for [M_σ] found previously. We also have from differentiation of ℳ that

[ℳσσ]=([ℳyt+1yt+1]β1β3[gx]γ3β3[η]ϕ3γ3[εt+1]ϕ3+[ℳyt+1xt+1]β1γ3[η]ϕ3γ3[εt+1]ϕ3)[gx]γ1β1[η]ϕ1γ1[εt+1]ϕ1 +[ℳyt+1]β1[gxx]γ1γ3β1[η]ϕ3γ3[εt+1]ϕ3[η]ϕ1γ1[εt+1]ϕ1 +[ℳyt+1]β1[gx]γ1β1[hσσ]γ1 +[ℳyt+1]β1[gσσ]β1 +[ℳyt]β1[gσσ]β1 +([ℳxt+1yt+1]γ1β3[gx]γ3β3[η]ϕ3γ3[εt+1]ϕ3+[ℳxt+1xt+1]γ1γ3[η]ϕ3γ3[εt+1]ϕ3)[η]ϕ1γ1[εt+1]ϕ1 +[ℳxt+1]γ1[hσσ]γ1

For [M_σσσ], we exploit the fact P⁰=1 for all values of (x_t, σ) and so all derivatives have to equal zero. Thus Rp(p1)[pσσσ1]=Et{[ℳσσσ]}.

To evaluate the expectations in the term for [F_σσσ(x_ss, 0)], we define

[m3(εt+1)]ϕ2ϕ3ϕ1={m3(εt+1(ϕ1))if ϕ1=ϕ2=ϕ30otherwise

where m³(ϵ_t₊₁) denotes the third moment of ϵ_t₊₁(φ₁) for φ₁=1, 2, …, n_ε. Notice that m³(ϵ_t₊₁) is a n_ε×n_ε×n_ε matrix. Following some simplifications we finally get

Rp(pk)[pσσσk]= +Rp(p1)[pσσσ1]R(pk−1) +3[ℳyt+1yt+1]β1β3[gx]γ3β3[η]ϕ3γ3[gx]γ1β1[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1 +6[ℳyt+1xt+1]β1γ3[η]ϕ3γ3[gx]γ1β1[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1 +3[ℳyt+1]β1[gxx]γ1γ3β1[η]ϕ3γ3[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1 +3[ℳxt+1xt+1]γ1γ3[η]ϕ3γ3[η]ϕ1γ1Rp(pk−1)[pxk−1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1 +3([ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1+[ℳxt+1]γ1[η]ϕ1γ1)[m3(εt+1)]ϕ2ϕ3ϕ1Rpp(pk−1)[pxk−1]γ3[η]ϕ3γ3[pxk−1]γ2[η]ϕ2γ2 +3([ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1+[ℳxt+1]γ1[η]ϕ1γ1)[m3(εt+1)]ϕ2ϕ3ϕ1Rp(pk−1)[pxxk−1]γ2γ3[η]ϕ3γ3[η]ϕ2γ2 +R(p1)Rppp(pk−1)[pxk−1]γ3[η]ϕ3γ3[pxk−1]γ2[η]ϕ2γ2[pxk−1]γ1[η]ϕ1γ1[m3(et+1)]ϕ2ϕ3ϕ1 +3R(p1)Rpp(pk−1)[pxk−1]γ2[η]ϕ2γ2[pxxk−1]γ1γ3[η]ϕ3γ3[η]ϕ1γ1[m3(εt+1)]ϕ2ϕ3ϕ1 +R(p1)Rp(pk−1)[pxxxk−1]γ1γ2γ3[η]ϕ3γ3[η]ϕ2γ2[η]ϕ1γ1[m3(εt+1)]ϕ2ϕ3ϕ1 +R(p1)Rp(pk−1)[pxk−1]γ1[hσσσ]γ1 +R(p1)Rp(pk−1)pσσσk−1

For a logarithm transformation, it is straightforward to show that

pσσσk=pσσσ1+3ℳ−1[ℳyt+1yt+1]β1β3[gx]γ3β3[η]ϕ3γ3[gx]γ1β1[η]ϕ1γ1[pxk−1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1 +6ℳ−1[ℳyt+1xt+1]β1γ3[η]ϕ3γ3[gx]γ1β1[η]ϕ1γ1[pxk−1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1 +3ℳ−1[ℳyt+1]β1[gxx]γ1γ3β1[η]ϕ3γ3[η]ϕ1γ1[pxk−1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1 +3ℳ−1[ℳxt+1xt+1]γ1γ3[η]ϕ3γ3[η]ϕ1γ1[pxk−1]γ2[η]ϕ2γ2[m3(εt+1)]ϕ2ϕ3ϕ1 +3ℳ−1([ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1+[ℳxt+1]γ1[η]ϕ1γ1)[m3(εt+1)]ϕ2ϕ3ϕ1[pxk−1]γ3[η]ϕ3γ3[pxk−1]γ2[η]ϕ2γ2 +3ℳ−1([ℳyt+1]β1[gx]γ1β1[η]ϕ1γ1+[ℳxt+1]γ1[η]ϕ1γ1)[m3(εt+1)]ϕ2ϕ3ϕ1[pxxk−1]γ2γ3[η]ϕ3γ3[η]ϕ2γ2 +[pxk−1]γ3[η]ϕ3γ3[pxk−1]γ2[η]ϕ2γ2[pxk−1]γ1[η]ϕ1γ1[m3(εt+1)]ϕ2ϕ3ϕ1 +3[pxk−1]γ2[η]ϕ2γ2[pxxk−1]γ1γ3[η]ϕ3γ3[η]ϕ1γ1[m3(εt+1)]ϕ2ϕ3ϕ1 +[pxxxk−1]γ1γ2γ3[η]ϕ3γ3[η]ϕ2γ2[η]ϕ1γ1[m3(εt+1)]ϕ2ϕ3ϕ1 +[pxk−1]γ1[hσσσ]γ1+pσσσk−1

Appendix B

Matlab implementation of the POP method

The approximation method presented in the body of the text is implemented in Matlab. For the first perturbation step, we apply the codes accompanying Schmitt-Grohé and Uribe (2004) to compute first and second order derivatives, while the routines underlying Andreasen (2012b) are used for all third-order terms. For the second perturbation step, the user only needs to specify the stochastic discount factor in Anal_PricingKernel_derivatives.m and the position of the one-period bond price in y_t. Analytical derivatives of the pricing kernel are then computed based on symbolic differentiation, and these derivatives are evaluated in the steady state by num_eval_PricingKernel.m. Bond prices are then computed in Get_Bond_Prices_3rd.m up to third order, either for the level of bond prices or for a log-transformation.

Appendix C

Closed-form solution to the endowment model with habits

For the considered habit model, we have

ℳt+1:=β(Ct+1−hZt+1)−γ(Ct−hZt)−γ=β(1−h exp{zt+1})−γ(1−h exp{zt})−γexp{−γxt+1}.

A closed-form solution for zero-coupon bond prices is given by (see Zabczyk 2014)

Ptk=(1−hexp{zt})γβkexp−γkμ+xt−μρ(1−ρk)(1−ρ)×∑n=0+∞−γn(−h)exp(1−φ)kzt−μ1−(1−φ)kφ−(xt−μ)ρρk−(1−φ)kρ−(1−φ)n×∏j=1k𝕃ξγ(1−ρj)(1−ρ)+nρj−(1−φ)jρ−(1−φ).

Here, L_ξ is the Laplace transform of ξ, and (αn) denotes a generalized binomial coefficient, i.e.,

(αn):=∏k=1n(α−k+1)/k, for n>0 and (α0):=1,

where αℝ and n∈𝒩. The condition for convergence of this solution is β exp{Var(ξ)2(γ1−ρ)2}<1.

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Article note

A previous version of the paper was entitled: “An Efficient Method of Computing Higher Order Bond Price Perturbation Approximations.”

Supplemental Material

The online version of this article (DOI: 10.1515/snde-2012-0005) offers supplementary material, available to authorized users.

Published Online: 2014-2-27

Published in Print: 2015-2-1

Efficient bond price approximations in non-linear equilibrium-based term structure models

Abstract

Acknowledgment

Appendix A

A general transformation of bond prices

First order terms

Derivative of p^k with respect to x

Derivative of p^k with respect to σ

Second order terms

Derivative of p^k with respect to (x, x)

Derivative of p^k with respect to (σ, σ)

Third order terms

Derivative of p^k with respect to (x, x, x)

Derivative of p^k with respect to (σ, σ, x)

Derivative of p^k with respect to (σ, σ, σ)

Appendix B

Matlab implementation of the POP method

Appendix C

Closed-form solution to the endowment model with habits

References

Article note

Supplemental Material

Journal and Issue

Articles in the same Issue

Efficient bond price approximations in non-linear equilibrium-based term structure models

Abstract

Acknowledgment

Appendix A

A general transformation of bond prices

First order terms

Derivative of pk with respect to x

Derivative of pk with respect to σ

Second order terms

Derivative of pk with respect to (x, x)

Derivative of pk with respect to (σ, σ)

Third order terms

Derivative of pk with respect to (x, x, x)

Derivative of pk with respect to (σ, σ, x)

Derivative of pk with respect to (σ, σ, σ)

Appendix B

Matlab implementation of the POP method

Appendix C

Closed-form solution to the endowment model with habits

References

Article note

Supplemental Material

Journal and Issue

Articles in the same Issue

Derivative of p^k with respect to x

Derivative of p^k with respect to σ

Derivative of p^k with respect to (x, x)

Derivative of p^k with respect to (σ, σ)

Derivative of p^k with respect to (x, x, x)

Derivative of p^k with respect to (σ, σ, x)

Derivative of p^k with respect to (σ, σ, σ)