Linear‐Rational Term Structure Models

• Author: ANDERS B. TROLLE,DAMIR FILIPOVIĆ,MARTIN LARSSON
• Date: 01 April 2017
• Published date: 01 April 2017
• DOI: http://doi.org/10.1111/jofi.12488

THE JOURNAL OF FINANCE •VOL. LXXII, NO. 2 •APRIL 2017

Linear-Rational Term Structure Models

DAMIR FILIPOVI ´

C, MARTIN LARSSON, and ANDERS B. TROLLE∗

ABSTRACT

We introduce the class of linear-rational term structure models in which the state

price density is modeled such that bond prices become linear-rational functions of the

factors. This class is highly tractable with several distinct advantages: (i) ensures non-

negative interest rates, (ii) easily accommodates unspanned factors affecting volatility

and risk premiums, and (iii) admits semi-analytical solutions to swaptions. A parsimo-

nious model specification within the linear-rational class has a very good fit to both

interest rate swaps and swaptions since 1997 and captures many features of term

structure, volatility, and risk premium dynamics—including when interest rates are

close to the zero lower bound.

THE CURRENT ENVIRONMENT WITH VERY low interest rates creates difficulties for

many existing term structure models, most notably Gaussian or conditionally

Gaussian models that invariably place large probabilities on negative future

interest rates. Models that respect the zero lower bound (ZLB) on interest

rates exist but often have limited ability to accommodate unspanned factors

affecting volatility and risk premiums or to price many types of interest rate

derivatives. In light of these limitations, the purpose of this paper is twofold:

∗Damir Filipovi´

c and Anders B. Trolle are both at Ecole Polytechnique F´

ed´

erale de Lausanne

and Swiss Finance Institute. Martin Larsson is at ETH Zurich. The authors wish to thank partic-

ipants at the 10th German Probability and Statistics Days in Mainz, the Stochastic Analysis and

Applications conference in Lausanne, the Seventh Bachelier Colloquium on Mathematical Finance

and Stochastic Calculus in Metabief, the Current Topics in Mathematical Finance conference in

Vienna, the Princeton-Lausanne Workshop on Quantitative Finance in Princeton, the 29th Euro-

pean Meeting of Statisticians in Budapest, the Frontiers in Stochastic Modelling for Finance con-

ference in Padua, the Term Structure Modeling at the Zero Lower Bound workshop at the Federal

Reserve Bank of San Francisco, the Symposium on Interest Rate Models in a Low Rate Environ-

ment at Claremont Graduate University,the Term Structure Modeling and the Zero Lower Bound

workshop at Banque de France, the 2014 Quant-Europe conference in London, the 2014 Global

Derivatives conference in Amsterdam, the 2015 AMaMeF and Swissquote conference in Lausanne,

the 2015 Western Finance Association conference in Seattle, and seminars at Columbia Business

School, Copenhagen Business School, the Federal Reserve Bank of New York, the London Math-

ematical Finance Seminar series, Stanford University, UCLA Anderson School of Management,

Universit´

e Catholique de Louvain, University of Cambridge, University of St. Gallen, University of

Vienna, and University of Zurich, as well as Andrew Cairns, Pierre Collin-Dufresne, Darrell Duffie,

Peter Feldhutter (discussant), Jean-Sebastien Fontaine (discussant), Ken Singleton (discussant

and Editor), and two anonymous referees for their comments. The research leading to these re-

sults has received funding from the European Research Council under the European Union’s

Seventh Framework Programme (FP/2007-2013/ERC Grant Agreement n. 307465-POLYTE). The

authors do not have any conflicts of interest as identified in the Disclosure Policy.

DOI: 10.1111/jofi.12488

655

656 The Journal of Finance R

First, we introduce a new class of term structure models, the linear-rational,

which is highly tractable and (i) ensures nonnegative interest rates, (ii) easily

accommodates unspanned factors affecting volatility and risk premiums, and

(iii) admits semi-analytical solutions to swaptions—an important class of inter-

est rate derivatives that underlie the pricing and hedging of mortgage-backed

securities, callable agency securities, life insurance products, and a wide vari-

ety of structured products. Second, we perform an extensive empirical analysis

of a set of parsimonious model specifications within the linear-rational class.

The first contribution of the paper is to introduce the class of linear-rational

term structure models. A sufficient condition for the absence of arbitrage op-

portunities in a model of a financial market is the existence of a state price

density; that is, a positive adapted process ζtsuch that the price (t,T)at

time tof any time-Tcash flow CTis given by1

(t,T)=1

ζt

Et[ζTCT].(1)

Following Constantinides (1992), our approach to modeling the term structure

is to directly specify the state price density.Specifically, we assume a multivari-

ate factor process Zt, which has a linear drift, and a state price density, which

is a linear function of Zt. In this case, bond prices and the short rate become

linear-rational functions—that is, ratios of linear functions—of Zt,whichis

why we refer to the framework as linear-rational. We show that one can easily

ensure that the short rate stays nonnegative.2

We distinguish between factors that are spanned by the term structure and

those that are unspanned, and we provide conditions such that all of the factors

in Ztare spanned. A key feature of the framework is that the term structure

depends only on the drift of Zt. This leaves freedom to specify exogenous fac-

tors feeding into the martingale part of Zt. Such factors give rise to unspanned

stochastic volatility (USV) and can be recovered from bond derivatives prices.

We further distinguish between USV factors that directly affect the instan-

taneous bond return covariances and those that affect expected future bond

return covariances.

Within the linear-rational framework we show how to construct a model in

which Ztis m-dimensional and there are n≤mUSV factors. The joint factor

process is affine, and swaptions can be priced semi-analytically. This model

is termed the linear-rational square-root (LRSQ) model. We also discuss an

extension of the state price density specification that allows for much richer risk

premium dynamics than the baseline model. It also allows for the introduction

1Throughout, we assume there is a filtered probability space (,F,Ft,P) on which all random

quantities are defined, and Et[·] denotes Ft-conditional expectation.

2While zero is a natural lower bound on nominal interest rates, any lower bound is accommo-

dated by the framework. In the United States, the Federal Reserve kept the federal funds rate in

a range between 0 and 25 basis points from December 2008 to December 2015, and other money

market rates mostly remained nonnegative during this period. However, in the Eurozone (as well

as in Denmark, Sweden, and Switzerland), the ZLB assumption has recently been challenged as

both policy rates and money market rates have moved into negative territory.

Linear-Rational Term Structure Models 657

of unspanned risk premium factors, although this is not a focus of our empirical

analysis.

The second contribution of the paper is an extensive empirical analysis of the

LRSQ model. We use a panel data set consisting of both swaps and swaptions

from January 1997 to August 2013. At a weekly frequency, we observe a term

structure of swap rates with maturities from 1 year to 10 years as well as a

surface of at-the-money implied volatilities of swaptions with swap maturities

from 1 year to 10 years and option expiries from 3 months to 5 years. The esti-

mation approach is quasi-maximum likelihood in conjunction with the Kalman

filter. The term structure is assumed to be driven by three factors, and we vary

the number of USV factors between one and three. A robust feature across

all specifications is that parameters align such that under the risk-neutral

measure and after a normalization of the factor process, the short rate mean-

reverts to a factor that affects the intermediate part of the term structure (a

curvature factor), which in turn mean-reverts toward a factor that affects the

long end of the term structure (a slope factor). The preferred specification has

three USV factors and simultaneously fits both swaps and swaptions well. This

result continues to hold for the part of the sample period in which short-term

rates are very close to the ZLB.

Using long samples of simulated data, we investigate the ability of the model

to capture the dynamics of the term structure, volatility, and swap risk premi-

ums. First, the model captures important features of term structure dynamics

near the ZLB. Consistent with the data, the model generates extended periods

of very low short rates. Furthermore, when the short rate is close to zero, the

model generates highly asymmetric distributions of future short rates, with

the most likely values of future short rates being significantly lower than the

mean values. Related to this, the model also replicates how the first principal

component of the term structure changes from a level factor during normal

times to more of a slope factor during times of near-zero short rates.

Second, the model captures important features of volatility dynamics near

the ZLB. Previous research shows that a large fraction of variation in volatility

is effectively unrelated to variation in the term structure. We provide an impor-

tant qualification to this result: volatility becomes compressed and gradually

more level-dependent as interest rates approach the ZLB. This is illustrated by

Figure 1, which shows the 3-month implied volatility of the 1-year swap rate

plotted against the level of the 1-year swap rate. More formally, for each swap

maturity, we regress weekly changes in the 3-month implied volatility of the

swap rate on weekly changes in the level of the swap rate. Conditional on swap

rates being close to zero, the regression coefficients are positive, large in mag-

nitude, and very highly statistically significant, and the R2s are around 0.50.

However, as the level of swap rates increases, the relation between volatility

and swap rate changes becomes progressively weaker, and volatility exhibits

very little level-dependence at moderate levels of swap rates. Capturing these

dynamics—strong level-dependence of volatility near the ZLB and predomi-

nantly USV at higher interest rate levels—poses a significant challenge for

existing dynamic term structure models. Our model successfully meets this