Gaussian Multifactor Interest

Rate Mo dels Theory Estimation and

Implications for Option Pricing

Frank Heitmann and Siegfried Trautmann

First draft March

Current draft June

Johannes Gutenb ergUniversity Mainz Chair of Finance Mainz

Phone Fax Email trautnbwlunimainzde

We thank Wolfgang Bhler for providing us with German b ond market data

Gaussian Multifactor Mo dels

Theory Estimation and Implications for Option Pricing

Abstract

Gaussian interest rate mo dels are attractive b ecause of their analytical tractabil

ity Examples are the onefactor mo dels of HoLee and Vasicek and twofactor

mo dels resulting from combinations of b oth We showby factor analysis that the

variation in sp ot rates of the German Governmentbondmarket in the p erio d

can essentially b e explained bytwo factors Consequentlywe restrict ourselves

to onefactor and twofactor mo dels which are estimated by nonlinear regression

We compare these mo dels with resp ect to their ability to explain observed changes

in the term structure of interest rates

On the basis of theoretical insights and our empirical ndings we conclude that

the two onefactor mo dels considered should b e discarded b ecause of the mo dels

inability to explain a twist of the term structure where short term and long term rates

move in opp osite directions Although the combination of the HoLee and Vasicek

onefactor mo del as prop osed by HeathJarrowMorton is able to explain the term

structure twist we recommend a twofactor Vasicek mo del which contains the other

mo dels as sp ecial cases for pricing interest rate options The latter mo del is the

only one which captures a ushap ed function induced by the term structure

twist This situation o ccured in two of the seven year subp erio ds analyzed A

comparison of mo del prices for calls on zero b onds which indicates that neglecting

the volatility smile might cause a substantial valuation error completes the pap er

CONTENTS i

Contents

Intro duction

Gaussian Mo dels in the HJM Framework

ArbitrageFree Term Structure Dynamics

Gaussian Mo dels

Continuous Time Version of the HoLeeMo del

Term Structure ConsistentVasicek Mo del

The HeathJarrowMorton Mo del

ATwofactor Vasicek Mo del

Volatility Estimation

Assumptions

Principal Comp onent Analysis

Nonlinear Regression

Volatility Estimates for the German Market

Data

Principal Comp onent Analysis

Nonlinear Regression

Implications for Option Pricing

Conclusions

A Pro of of Theorem

B Tables and Figures

References

ii LIST OF TABLES

List of Figures

Yield Curves in the HoLeeMo del

Yield Curves in the VasicekMo del

Yield Curves in the HJM mo del

Yield Curves in the FVMo del

Yield Curves

Yield Curves

Yield Curves

Sp ot Rate Volatility Structure

Explained Variance in the Vasicek mo del

Explained Variance in the HJM mo del

Explained Variance in the FVMo del

HoLeeCall Price versus VasicekCall Price

FVCall Price versus HJMCall Price

Probability of negative sp ot rates

List of Tables

Explained Variance of the Sp ot Rates

Parameter Estimates for the HoLeeMo del

Parameter Estimates for the VasicekMo del

Parameter Estimates for the HJMMo del

Parameter Estimates for FVMo del

Magnitude of the Smile Eect

Call Prices BS HoLee Vasicek HJM

Empirical Studies

Identication of Factors

Call Prices HJM and FV

Intro duction

The valuation of interest rate contingent claims has attracted a growing interest in

recentyears and various mo dels have b een prop osed There are essentially two ap

proaches to the mo deling of the term structure The general equilibrium approach

used for example by CoxIngersollRoss and LongstaffSchwartz

starts from a description of the economy and derives the term structure of

interest rates endogenously In contrast the arbitrage approach starts from assump

tions ab out the sto chastic evolution of one or more interest rates and derives prices

of contingent claims by imp osing the no arbitrage condition One and two factor

mo dels of only the short rate were rst suggested by Vasicek and Bren

nanSchwartz These mo dels are generally not consistent with the initial

term structure HullWhite provide a general pro cedure for short rate

mo dels whichallows to match an arbitrary initial The alternative ap

proach to mo del the complete term structure was pioneered by HoLee and

generalized by HeathJarrowMorton HJM HJM allow for a xed but

unsp ecied numb er of sto chastic factors The application of such a mo del requires

the determination of the numb er of factors driving the term structure movement

and the sp ecication of appropriate volatility co ecients

We restrict our analysis to Gaussian mo dels with deterministic volatility co e

cients b ecause of their analytical and numerical tractability Moreover we b elieve

that the assumption of normally distributed interest rates is quite reasonable if the

probability of negativeinterest rates is small We will discuss this issue based on

our parameter estimates From our p oint of view the often suggested alternative

of lognormally distributed interest rates is not appropriate since we do not exp ect

interest rates to have a drift and variance rate prop ortional to the level of interest

rates

In this pap er weshowby principal comp onent analysis of the sp ot rate movement

that in the German b ond market two factors are relevant and that these factors can

b e identied as a shift and a twist of the term structure Furthermore we develop

a sp ecic twofactor mo del within the class of HJMmo dels which can b e estimated

by nonlinear regression Our mo del includes as sp ecial cases the continuous time

version of the HoLee mo del the extended Vasicek mo del and a mo del prop osed by

HJM as an example for a simple twofactor mo del We estimate these mo dels

for seven twoyear p erio ds and identify the typ es of term structure movement that

can b e explained by these mo dels

The HoLee mo del is appropriate only when the predominantmovement of the

term structure is a parallel shift resulting in interest rate volatilities which are the

same for all maturities In con trast the Vasicek mo del is based on the often observed

phenomenon of lower volatilities for long term sp ot rates compared to short term

sp ot rates A phenomenon that can b e explained by the considered twofactor

mo dels but not by the onefactor mo dels is a twist of the term structure with short

and long rates moving in opp osite directions Moreover we nd that in subp erio ds

which are dominated byatwist of the term structure the sp ot rate variance shows

GAUSSIAN MODELS IN THE HJM FRAMEWORK

a smile pattern This smile eect is captured only by our mo del whichisessentially

atwofactor Vasicek mo del Option price simulations show that neglecting the smile

eect might cause substantial price dierences

Whereas we nd that the more elab orate mo dels are to be prefered

AminMorton obtain a slightly dierent result They compare six one

factor mo dels of the HJM typ e using implied volatility estimates In their analysis

one parameter mo dels yield more stable parameter values over time and they are

able to earn larger and more consistent abnormal returns Amin and Morton con

clude that among the one parameter mo dels the HoLee mo del seems to b e the most

preferred Flesaker in contrast rejects the HoLee mo del for Euro dollar

futures options traded at the Chicago Mercantile Exchange

The pap er is organized as follows The basic mo del and the prop erties of the

four Gaussian mo dels under consideration are analyzed in section In section we

describ e the estimation pro cedure and discuss necessary assumptions The results

of the principal comp onent analysis and the nonlinear regressions are presented in

section Implications for option pricing are discussed in section Section

concludes the pap er

Gaussian Mo dels in the HJM Framework

The HJM approach to pricing interest rate contingent claims starts with a fam

ily of forward rate pro cesses and initializes it to an arbitrary but xed initial for

ward rate curve This approachwas pioneered by HoLee for a simple

discrete onefactor mo del and extended to a continuous time multifactor setting by

HeathJarrowMorton They consider a continuous trading economy

with a trading interval T The uncertainty in the economyischaracterized by the

probability space F Q and the ltration F F t T satisfying the

t

usual conditions Furthermore they assume a continuum of discount b onds P t T

with maturities T T Let the b ond price at time t b e given by P t T where

T is the maturity of the zero b ond then the forwardrate at time t for instantaneous

and riskless b orrowing at date T is

ln P t T P t T

f t T

T P t T T

Equivalently one can dene the bond price in terms of the forward rates by

Z

T

P t T exp f t sds

t

With the assumption of a continuum of discount b onds there is a riskless investment

opp ortunityatanytimet yielding the riskless short rate

r t f t t

ArbitrageFree Term Structure Dynamics

A rolloverp osition at the short rate r t is termed money market account The

value of the money market account initialized at time with one dollar investment

is

Z

t

B t exp r sds

Bond values expressed in units of the money market account

P t T

Z t T

B t

are called relative bond prices and can b e shown to form a martingale with resp ect

to some risk neutral measure Q Analogously one can showthatforward prices

P t T

F t t T

P t t

are a martingale with resp ect to the forward risk adjusted measure Q F t t T

is the price of a zero b ond with maturity T and delivery date t at time t

ArbitrageFree Term Structure Dynamics

Let the sto chastic evolution of forward rates follow a diusion pro cess of the form

Z Z

K

t t

X

f t T f T v T dv v T dW v

k k

k 

where v T is the drift of the forward rate with maturity T while v T for

k



k K are its volatility co ecients and W v are indep endent standard

k

Brownian motions Since b ond prices dep end on forward rates the drift and volatil

ity of the b ond price pro cess

Z

t

p

P t T P T v T P v T dv

Z

K

t

X

p

v T P v T dW v

k

k

k 

must b e connected to the drift and volatility co ecients of the forward rate pro cess

HJM show using Itos lemma and a generalized version of Fubinis theorem

that

Z

T

p

t T t y dy for k K

k

k

t

Z

K

T

X

p p 

t T r t t y dy t T

k

t

k 

The drift and volatility co ecients in the general HJM mo del may also dep end on the path of

the Brownian motions Since this pap er concentrates on Gaussian mo dels with deterministic drift

and diusion co ecients we omit the dep endency of to keep the notation clear and simple

GAUSSIAN MODELS IN THE HJM FRAMEWORK

Applying Itos Lemma to the denition of the forward price yields the forward price

pro cess

Z

t

v t TF v t Tdv F t t T F t T

Z

K

t

X

v t TF v t TdW v

k

k 

with

p p

t t T t T t t

k

k k

K

X

p p p p p

t t t t t T t t t t T t T

k k k

k 

p

We refer to the volatility co ecients t T t T t t T collectively as the

k k

k

volatility structure We assume that there exist K zero b onds with maturity

S S T such that the diusion matrix t S is nonsingular

 K k k K K

This ensures the uniqueness of market prices of risk and the uniqueness of the

equivalent martingale measure if the mo del is arbitragefree

Theorem No Arbitrage Conditions

The following conditions are equivalent A Kfactor mo del satisfying these conditions

is called lo cally arbitragefree

A There exists a unique equivalent martingale measure Q such that relative b ond

prices Z t T are martingales with resp ect to this measure

A There exist market prices of risk tk K with

k

K

X

p p

t T r t t T t for all T t T

k

k

k 

A The forward rate drift is uniquely determined bythevolatility structure and

the market prices of risk

Z

K

T

X

t T t t y dy t T

k k k

t

k 

A There exists a unique equivalent martingale measure Q such that forward

rates f t t are martingales with resp ect to this measure

Pro of In app endix A

Condition A is the wellknown noarbitrage condition requiring that the

excessreturn of every b ond regardless of maturity equals the sum of K risk pre

miums Condition A relates the forward rate drift to the volatility structure and

ArbitrageFree Term Structure Dynamics

the market prices of risk The arbitragefree term structure dynamics are therefore

completely sp ecied bythevolatility structure and market prices of risk The mar

tingale measure in condition A is often called risk neutral measure since the

exp ected instantaneous return of all b onds with resp ect to this measure equals the

riskless short rate The martingale measure in condition A is the forwardrisk

adjusted measure intro duced by Jamshidian which facilitates the calculation

of closed form solutions for Europ ean style interest rate contingent claims

Using condition A one can easily show that forward prices in a lo cally

arbitragefree mo del satisfy

Z

K

t

X

p

F t t T F t T v v t v t TF v t Tdv

k k

k

k 

Z

K

t

X

dW v v t TF v t T

k k

k 

and are martingales with resp ect to Q

Z

K

t

X

F t t T F t T v t TF v t TdW v

k

k

k 

Applying Itos Lemma to the logarithm of the forward price and using the fact

that F t t T P t T yields for the sto chastic b ond price in t given the

information F

t

Z

K

t

X

P t T

p

exp P t T jF v v t v t Tdv

t k k

k

P t t

t

k 

Z Z

K K

t t

X X



v t TdW v v t T dv

k k k

t t

k  k 

Thus the b ond price at anytimet given the information F dep ends on the forward

t

price of the b ond at time tthevolatility structure and the market prices of risk For

the purp ose of contingent claim valuation we are only interested in the sto chastic

b ond price in terms of W t and W twhich are Brownian motions with resp ect

to the riskneutral measure Q and the forwardriskadjusted measure Q

Z

K

t

X

P t T

p

P t T jF v t v t Tdv exp

t k

k

P t t

t

k 

Z Z

K K

t t

X X



v t T dv v t TdW v

k k k

t t

k  k 

GAUSSIAN MODELS IN THE HJM FRAMEWORK

Z

K

t

X

P t T



P t T jF exp v t T dv

t k

P t t

t

k 

Z

K

t

X

v t TdW v

k

k

t

k 

Q and Q are identied by Girsanovs theorem The determiniation of Q and Q

requires the knowledge of the market prices of risk but the sto chastics of the b ond

price are completely sp ecied by and which are indep endentofthemarket

prices of risk This allows for a preferencefree valuation of contingent claims

Gaussian Mo dels

A Gaussian model is one in whichthevolatility structure v T is deterministic

k

Hence forward rates follow a Gaussian pro cess and are normally distributed in a



Gaussian mo del The b ond price in a Gaussian mo del is lognormally distributed



with resp ect to Q and Q

Z

K

t

X

P t T

p

P t TjF exp v t v t Tdv E



t k

k

Q

P t t

t

k 

P t T

E P t TjF

Q t

P t t

Var ln P t TjF Var ln P t TjF



t Q t

Q

Z

K

t

X



v t T dv

k

t

k 

Gaussian mo dels are often criticized b ecause of the lognormal b ond price distribu

tion since it implies normally distributed sp ot rates

ln P t T

Rt T

T t

whichmay b ecome negative with p ositive probability An imp ortant issue for the

application of Gaussian mo dels is therefore the assessment of the probabilityof

negative sp ot rates based on estimated parameters In the subsequent sections we

will analyze the prop erties of four Gaussian mo dels two onefactor and twotwo

factor mo dels



Jamshidian further restricts Gaussian mo dels to term structure mo dels with an esp e

cially simple volatility structure such that the short rate also follows a Gaussian pro cess



The b ond price distributions with resp ect to the risk neutral and forward risk adjusted measure

dier only in the exp ected b ond price The distribution with resp ect to the original measure Q

can not b e sp ecied without further assumptions on the market prices of risk

Continuous Time Version of the HoLeeMo del

Continuous Time Version of the HoLeeMo del

The continuous time limit of the HoLeeMo del was indep endently de

rived by Jamshidian and HeathJarrowMorton In the HJM

framework the mo del is fully sp ecied by the volatility co ecient of the forward rate

pro cess

v T



The b ond price P t T in terms of W t is



P T

exp Tt T t T t W t P t T jF

P t

which implies for the sp ot rate with time to maturity T t at time t

P T



ln

ln P t T

P t

Tt W t Rt T jF

T t T t

Future sp ot rates are therefore normally distributed with

P T



ln

P t

R

t T Tt

T t

and

R  

t T t

given the Information F

The simple volatility structure allows only for a small set of future yield curves

since the sto chastic factor W t has the same eect on all sp ot rates indep endent

of maturity Hence at anypoint of time all p ossible yield curves are parallel This

should not b e confused with a parallel shift of the term structure in time which

leads to arbitrage opp ortunities Furthermore one can easily see that starting

with a normal or at term structure inverse yield curves are not p ossible Figure

illustrates these limitations of the HoLee mo del

Term Structure ConsistentVasicek Mo del

The Vasicek mo del is based on an OrnsteinUhlenbeck pro cess for the short

rate The short rate dynamics imply an endogenous term structure whichisnot

necessarily consistent with the observable term structure HullWhite

haveshown that consistency can b e achieved byintro ducing a timedep endent drift

The functional form of the drift in terms of the initial term structure and volatility



In the discussion of the four sp ecial Gaussian mo dels we restrict ourselves to the b ond prices

and sp ot rates given the information at time



This was rst noted by Boyle In the given mo del parallel shifts are excluded by the

second drift term which dep ends on the maturity

GAUSSIAN MODELS IN THE HJM FRAMEWORK

Figure Yield Curves in the HoLeeMo del

This gure shows p ossible yield curves after one year when the initial

term structure is at at The realizations of the Brownian mo

tion are given by W t Wecho ose a large volatility

co ecientof to avoid the impression of a parallel shift in time

structure has b een derived by Jamshidian for the general Gaussian one

factor mo del In the HJMframework the term structure consistentVasicek mo del

is sp ecied by

T v

v T e

where is a real valued constant The b ond price P t T in terms of W t is

n o

P T

T t

P t T exp e xt M t T

P t

with

h i



T t  t T t t 

e e e e M t T



The HeathJarrowMorton Mo del

x t is the unique solution to the OrnsteinUhlenbeck pro cess

Z

t

t v

xt e dW v

The sp ot rate in the extended Vasicek mo del satises

P T

T t

ln

M t T e ln P t T

P t

xt Rt T

T t T t T t T t

and is normally distributed with

P T

ln

M t T

P t

R

t T

T t T t



 T t

e

R  t

t T e



T t

given the information F

Contrary to the HoLee mo del the sp ot rate volatility is a decreasing function

of time to maturitywhich reduces the probability of negative sp ot rates and is

consistent with the often encountered phenomena of lower volatilities at the long

end of the term structure Figure illustrates this dierence and shows that the

extended Vasicek mo del allows for normal at and inverse yield curves even when

the initial yield curveisat

The HeathJarrowMorton Mo del

HeathJarrowMorton prop ose a Gaussian twofactor mo del whichis

essentially a combination of the HoLee and Vasicek mo del The volatility structure

in the HJM mo del is given by

v T

 

T v

v T e

 

The b ond price P t T in terms of W t is

P T

exp fM t T M t T P t T

 

P t



T t

e xt T t W t

 





The corresp onding dierential equation is

dxtxtdt  dW t

with initial value x



The risk neutral distribution follows from the prop erties of the OrnsteinUhlenbeck pro cess cf

KaratzasShreve p



We will term this mo del in the following HJM mo del even though the HJM approachismuch more general and includes not only Gaussian mo dels

GAUSSIAN MODELS IN THE HJM FRAMEWORK

Figure Yield Curves in the VasicekMo del

This gure shows p ossible yield curves after one year when the initial

term structure is at at The realizations of the OrnsteinUhlenbeck

pro cess are given by xt The volatility and reversion

parameter are

with





M t T Tt T t



i h



 T t t T t  t



e e e e M t T





x t is the unique solution to the OrnsteinUhlenbeck pro cess like in the Vasicek

mo del The sp ot rate in the HJM mo del satises

P T

ln

M t T M t T

P t

 

Rt T

T t T t T t

T t

e

W t x t

  

T t

and is normally distributed with

P T

ln

M t T M t T

P t

 

R

t T

T t T t T t

ATwofactor Vasicek Mo del



T t 

e

t R  



e t t T





T t

given the information F From it is obvious that the twofactor HJM mo del

allows for a much larger variety of term structure movements than the onefactor

mo dels from HoLee and Vasicek More imp ortantly the HJM mo del allows for

twists of the yield curve whereas the Vasicek mo del only allows for nonparallel

shifts Figure illustrates two p ossible twists and seven nonparallel shifts of the

term structure

Figure Yield Curves in the HJM mo del

This gure shows p ossible yield curves after one year when the initial

term structure is at at The realizations of W t xt are given



by The volatility and rev ersion co ecients

are

 

ATwofactor Vasicek Mo del

Atwist of the term structure often leads to high sp ot rate volatilities for long

and short maturities but rather lowspotratevolatilities for medium maturities

This eect henceforth called smileeect cannot b e explained by the HJM mo del

R 

b ecause the sp ot rate variance t is a monotone decreasing function of time

GAUSSIAN MODELS IN THE HJM FRAMEWORK

to maturity T t Therefore we prop ose the Gaussian twofactor mo del given

by the volatility structure

T v

v T e

 

T v



v T e

 

with This mo del can b e viewed as a twofactor Vasicek mo del FV

 

mo del and is able to explain the smile eect The b ond price P t T in this mo del

is

P T



T t

P t T exp M t T M t T e x t

  

P t





T t



e x t





with

h i



T t   t T t t 



M t T e e e e







h i



T t   t T t t 



   

M t T e e e e







and

Z

t

t v

x t e dW v

 

Z

t

t v



e dW v x t

 

The sp ot rate in the FVmo del

P T

ln

M t T M t T

P t

 

Rt T

T t T t T t

T t T t



e e

x t x t

   

T t T t

 

is normally distributed with

P T

ln

M t T M t T

P t

 

R

t T

T t T t T t



 T t

e

 t R 



e t T



T t





 T t



e

 t





e



T t



ATwofactor Vasicek Mo del

Figure Yield Curves in the FVMo del

This gure shows p ossible yield curves after one year when the initial

term structure is at at The realizations of x t x t are given

 

by The volatility and reversion parameter

are and yield a strong smile

   

eect

The term structure movement is illustrated in gure Possible yield curves after one

year are similar to the ones in the HJM mo del but with a much higher variation of

sp ot rates with years time to maturity The drawback of the increasing volatility

at the long end of the term structure is the higher probability of negative sp ot rates

compared to the HJM mo del We will discuss this issue and the imp ortance of the smile eect in section

VOLATILITY ESTIMATION

Volatility Estimation

Wenow use principal comp onent analysis and nonlinear regression to estimate the



volatility structure The principal comp onent analysis is mainly used to identify

the numb er of indep endent factors It is applicable to mo dels where the volatility

co ecients are left unsp ecied apart from regularity conditions Such mo dels can

b e viewed as term structure mo dels with exogenous volatility structures For the

estimation of mo dels with endogenous volatility structures such as the four Gaussian



mo dels presented ab ove we prop ose a nonlinear regression approach

Assumptions

HJM mo del the sto chastic evolution of the complete term structure Alternatively

this can b e done by mo delling the term structure of forward rates the term structure

of sp ot rates or the discount function All three could theoretically b e used for the

purp ose of volatility estimation but the results can b e quite dierent The problem

results from the fact that in most b ond markets there is no complete set of zero

b onds Generally the term structure has to b e estimated from a set of coup on b onds

The discount function or sp ot rates can b e estimated with sucient accuracy but

estimates of forward rates corresp onding to the slop e of the discount function are

fairly inaccurate Therefore one should b e cautious when estimating the volatility

structure directly from

K

X

t df t T t T dt t T dW

k k

k 

Rather one should use logchanges of b ond prices

K K

X X

p p  p

d ln P t T t T t T dt t T dW t

k

k k

k  k 

The estimation of the volatility structure by means of a principal comp onent analysis was rst

prop osed by HeathJarrowMorton Similar empirical tests were carried out by Kahn

and Beckers In contrast Steeley and BhlerSchulze use

the principal comp onent analysis with the primary ob jective to determine the numb er of factors

driving the term structure movement and to identify these factors

Recall that in b oth mo dell classes the term structure is exogenously given Fitting the his

torical term structures is therefore irrelevant in our context In contrast BrownDybvig

estimate the CoxIngersollRoss mo del by a crosssectional analysis of b ond prices thus calibrat

ing the mo del to t the term structure at one p oint of time GibbonsRamaswamy and

Haverkamp estimate the CoxIngersollRoss and LongstaSchwartz mo del resp ectively

by the generalized metho d of moments using rst and second order moments While rst and

second order moments are of interest for the estimation of endogenous term structure mo dels such

as the CoxIngersollRoss and LongstaSchwartz mo del only second order moments should b e used for the estimation of exogenous term structure mo dels

Assumptions



or absolute changes of sp ot rates

K K

p

X X

t T

p  p

k

t T Rt T dt t T dW t dRt T

k

k

T t T t

k  k 

For reasons which will b ecome clear later we prefer to use absolute sp ot rate changes

The term structure movement with resp ect to the risk neutral and forward ad

justed measure is completely sp ecied bythevolatility structure but the movement

with resp ect to the original probability measure dep ends on the market prices of

risk Since the risk neutral and forward risk adjusted distributions are not observ

able assumptions regarding the market prices of risk are necessary to estimate the

volatility structure To facilitate estimation we assume that all market prices of risk

are constantover time Time invariance is also assumed for the volatility co e

cients They are assumed to dep end only on the time to maturity T t but



not on the time t With these assumptions the only time dep endent comp onent

of the sp ot rate drift

K

X

p p 

t T t T Rt T

k

T t

k 

K

X

p p 

r t Rt T

k

k k

k 

is the dierence b etween the instantaneous short rate and the sp ot rate of the

corresp onding maturity divided by this maturity r t Rt T Since the in

stantaneous short rate is not directly observable wehavetoapproximate it by the

sp ot rate with the shortest maturity in our sample This approximation b ears little

risk b ecause the dierence r t Rt T tends to b e very small In contrast

when estimating the volatility structure using logchanges of b ond prices these have

to b e adjusted for the absolute value of the instantaneous short rate Therefore we

prefer to use sp ot rate changes

Since the term structure of sp ot rates can only b e estimated for discrete p oints of

time we further assume that the timecontinuous sto chastic pro cess is a reasonable

approximation of the discrete absolute sp ot rate changes

Rt Rt t t t Rt t

which are now stated in terms of the time to maturity instead of the maturity

date T Spotratechanges are observed for the time to maturities at the

 M

times t t The discrete approximation of the sp ot rate pro cess is

 N

K

X

R

Rt r t Rt t W t

n m n n m m m k n

k

k 

The sto chastic dierential equation for the sp ot rates can easily b e obtained by application of

Itos lemma to equation



They have to dep end on the time to maturity to mo del the pulltopareect of the b ond price

movement

VOLATILITY ESTIMATION

with

K

X

 p p

k

k k

k 

p

m

R

k

m

k

Principal Comp onent Analysis

Traditionally factor analysis is used to explore the p ossible underlying structure

in a set of interrelated variables with the aim to reduce the numb er of original

variables to a much smaller set of hyp othetical factors Applied to the analysis of

term structure movements we are interested in the numb er of factors driving

the evolution of sp ot rates with dierent maturities our variables and the

economic interpretation of the hyp othetical factors Apart from this traditional

use of the principal comp onent analysis it can b e applied to the estimation of the

volatility structure without restricting it by a parametric form We will demonstrate

this based on absolute sp ot rate changes but in principal the same analysis could



b e done using absolute forward rate changes or logchanges of b ond prices The

absolute sp ot rate changes have to b e corrected for the dierence of the instantaneous

short rate and the sp ot rate of the corresp onding maturity divided by the maturity

Subtracting in addition the time indep endent drift we get corrected sp ot rate

changes

R t Rt r t Rt t

n m n m n n m m



which according to must satisfy

R t R t

  N 

C B

R t R t

  N 

C B

C B

A

R t R t

 M N M

R R

W t W t

     N

 M

R R

B C B C

W t W t

     N

 M

B C B C

B C B C

A A

R R

t W t W

M  M N

M M

 M

Given this multivariate linear mo del a principal comp onent analysis can b e applied

to estimate the volatility matrix by the matrix of the factor loadings

m m MM



If forward rates are assumed to b ehave prop ortional to the level of interest rates as in

HeathJarrowMorton relative instead of absolute forward rate changes havetobe

used



Since we use a principal comp onent analysis we assume at the b eginning K  M factors and

extract only those factors with a high explanatory p ower

Nonlinear Regression



the so called factor pattern

The imp ortance of the principal comp onents can b e assessed by the p ercentof

variation explained by the factors The p ercentofvariation of the rst to last factor

is given by the ordered Eigenvalues of the covariance matrix of the standardized

sp ot rate changes divided bythenumber of variables To determine the number of

factors driving the sp ot rate changes we rely on the simple eigenvalue criterion and



the Scree test which already giveunambiguous results

Arstinterpretation of the extracted factors will b e done by examining the shap e

of the corresp onding volatility functions To identify observable variables which are

able to explain the extracted factors we estimate factor scores and test regressions

of the form

F x

in n n n n

where F denotes the factor score of the ith factor with resp ect to the nth obser

in

vation and x is an observable variable like a sp ot rate for a certain maturityora

n

spread b etween a long and a short rate The sp ot rate with a short time to maturity

as one factor would b e consistent with traditional onefactor mo dels like Vasicek

and Cox Ingersoll Ross of the term structure which are based

on the short rate development A sp ot rate with a long time to maturity as a sec

ond factor is consistent with the BrennanSchwartz mo del whereas a

spread as a second factor is consistent with SchaeferSchwartz who use

a long rate and a spread b etween a long and a short rate b ecause these are generally



uncorrelated

Nonlinear Regression

In this section wedevelop a nonlinear regression approach to the estimation of the

four Gaussian interest rate mo dels presented in section The description concen

trates on the twofactor Vasicek mo del FVmo del since this mo del includes the

others as sp ecial cases The volatility structure in the FVmo del maybestatedin

terms of the forward rates b ond prices or sp ot rates



It should b e noted that the results of the principal comp onent analysis refer to standardized

sp ot rate changes The factor loading therefore havetobemultiplied by the

of sp ot rate changes with the corresp onding maturityTo facilitate an interpretation of the esti

p

mated volatility matrix wemultiply the factor loadings furthermore by t to obtain annualized

volatilities



Statistical tests of the numb er of factors are neglected since they generally lead to a to o large

numb er of factors cf Gorsuch p and the results based on the eigenvalue criterion

and scree tests are already unambiguous



Wehave not considered any noninterest rate variables whichhave b een incorp orated by

Richard and Cox Ingersoll Ross since the identication of the factors is not crucial to the main purp ose of our pap er

VOLATILITY ESTIMATION

factor factor

forward rate

T t T t



t T e t T e

   

volatility

b ond price

p p

T t T t





e t T e t T

 



volatility

sp ot rate

R T t R T t





t T e e t T

 

T t T t



volatility

For the FVmo del simplies to the HoLeemo del The Vasicekmo del

 



results for and the HJMmo del is included for The volatility

 

co ecients of these mo dels satisfy the stationarity assumption of section With

the assumption of constantmarket prices of risk the drift t T dep ends only on

the time to maturity T t but not on time t The variance of sp ot rate changes

is therefore indep endentof t T

Absolute sp ot rate changes in the FVmo del are given by

Rt r t Rt t

 



e W t e W t

 

 

Adjusted for the dierence of the instantaneous short rate and sp ot rate

R

Rt r t Rt t N t v t



they are normally distributed with mean t and variance

 

 

R R



v v e e

   

 

Based on a of N sp ot rate changes for M maturities the

 M

volatility parameters can b e estimated by the nonlinear regression

   

 

R 

s

 m 

m m



m M e e

m

t

m  m 

R 

where s denotes the sample variance of the sp ot rate changes with maturity

m

We computed the least squares estimators by the GaussNewton metho d due to

m



Hartley It should b e noted that applying the metho d of least squares to

the variances in this wayisequivalent to using the generalized metho d of moments

based only on second moments with an identity matrix as weighting matrix



The sp ot rate and b ond price volatility structure have to b e derived for  starting with

the forward rate pro cess or may b e obtained as a limiting case with since  is not

dened for the ab ove expressions of the b ond price and sp ot rate volatility

See for example SeberWild p

Volatility Estimates

for the German

Data

The empirical results rep orted here are based on weekly data of the term structure

of sp ot rates from January to December The term structure of sp ot

rates are estimated from weekly prices of German TBonds Bundesanleihen and

TNotes Bundesobligationen using the p olynomial metho d prop osed by Cham

bersCarletonWaldman We use a p olynomial of order three which

results in a relatively smo oth yield curve while incurring a p otentially large mean

absolute deviation of theoretical b ond prices from market prices A spline metho d

or p olynomial of higher order allows for a smaller mean absolute deviation but leads

to strongly oscillating yield curves Since we are interested in the movementof

the whole b ond market and not the volatility caused by some outliers weprefera



smo othed yield curve for the purp ose of volatility estimation

    

Weemploy a data set containing the sp ot rates of maturities

    



years The total sample p erio d is divided into seven twoyearsubp erio ds each with

observed yield curves The mean sp ot rate standard deviation in the total

sample p erio d is for the shortest maturity and for the

longest maturity This suggest a in the short rate pro cess inducing

aspotratevolatility decreasing with the time to maturity In the long run this

may b e a go o d description of the term structure movement but if we consider the

twoyear subp erio ds other eects show up Figure illustrates the term structure

movement from to which is dominated bya reversion of the short rate

The volatility of short rates is much larger than the volatility of long rates But the

following twoyear p erio d gure shows in the rst year an almost

parallel shift and in the second year a twist of the term structure Atwist with

long and short rates moving into opp osite directions causes high volatilities of long

and short rates but lower volatilities for medium maturities A twist in the term

structure is therefore asso ciated with a smileeect in the volatility structure of sp ot

rates A subp erio d with a mo dest reversion eect is illustrated in gure

Principal Comp onent Analysis

The main results of the principal comp onent analysis are summarized in table

These indicate that the jointmovement of sp ot rates with dierent maturities can

b e explained bytwo indep endent sto chastic factors The p ercentage of the total

variance explained by the rst factor varies b etween and Adding



In contrast when valuing interest rate contingent claims we prefer an estimation metho d

which guarantees the lowest mean absolute since any estimation error of the term structure will

bias the results of the contingentclaimvaluation



Shorter maturities than half a year were not considered since the estimation pro cedure tends to yield unstable results in this region

VOLATILITY ESTIMATES FOR THE GERMAN BOND MARKET

Figure Yield Curves

This gure shows the term structure movement from to While the

rst panel shows all weekly estimated yield curves the second panel pictures

only the yield at the b eginning after one year and at the end of the twoyear

p erio d to illustrate the dominating eect From to a strong mean

reversion is observable

Principal Comp onent Analysis

Figure Yield Curves

This gure shows the term structure movement from to While the

rst panel shows all weekly estimated yield curves the second panel pictures

only the yield at the b eginning after one year and at the end of the two

year p erio d to illustrate the dominating eect From to a twist is

observable

VOLATILITY ESTIMATES FOR THE GERMAN BOND MARKET

Figure Yield Curves

This gure shows the term structure movement from to While the

rst panel shows all weekly estimated yield curves the second panel pictures

only the yield at the b eginning after one year and at the end of the twoyear

p erio d to illustrate the dominating eect From to a weak mean

reversion is observable

Principal Comp onent Analysis

the second factor which explains b etween and approximately

of the total variance is explained in all seven subp erio ds The eigenvalues of the third

factor are in all cases smaller or only slightly larger than one and the p ercentage of

variation explained by this factor is only b etween bis

Table Explained Variance of the Sp ot Rates

Perio d factor factor factor

variance explained

cumulative

variance explained

cumulative

variance explained

cumulative

variance explained

cumulative

variance explained

cumulative

variance explained

cumulative

variance explained

cumulative

The unrotated factor pattern of the rst two factors is plotted in gure for the

p erio d The other six subp erio ds show a similar result The volatility

function of the rst factor is slightly decreasing with time to maturity but p ositive

over the whole maturityrange Factor one can therefore b e regarded as a nonparallel

shift Factor two can b e viewed as a twist factor since the corresp onding volatility

function is p ositive for maturities smaller than years and negative for larger

maturities A negativevolatility in this case means that long rates decrease if short

rates increase and vice versa We will discuss the imp ortance of the twist in the



dierent subp erio ds in greater detail in section

Table in the app endix rep orts the results of the linear regressions of factors

on observable variables for the subp erio d We omit the results for the

six other subp erio ds since the results are very similar The shift is b est explained

by the sp ot rate change with four years to maturity and the twist is b est explained



by the spread b etween the sp ot rate with and year to maturity



Weomitany factor rotation since we use the principal comp onent analysis in this pap er mainly

to explore the numb er of indep endent factors driving the term structure movement



BhlerSchulze obtain similar results for a principal comp onent analysis of monthly

yield curves from to

VOLATILITY ESTIMATES FOR THE GERMAN BOND MARKET

Figure Sp ot Rate Volatility Structure

This gure shows the volatility functions of the shift and twist factor estimated

by the factor pattern of the rst two principal comp onents for the subp erio d

Nonlinear Regression

The parameter estimates of the four Gaussian mo dels for the seven subp erio ds are

presented in table to The asymptotic condence intervals of the estimated



parameters are given in parentheses The R statistic is omitted for the HoLee

mo del since the restriction of the sp ot rate volatility to a constant do es not allowa

tting of a nonconstantvolatility curve The nonlinear regression reduces in this

case to a regression with only an intercept The metho d of least squares applied to

such a degenerate case gives an estimator of equal to the square ro ot of the average



sp ot rate variance across all maturities The R is always zero since no variation

is explained Hence based on an exp ostanalysis of the volatility structure the



HoLeemo del must b e rejected

The results in table reveal that the onefactor model of Vasicek leads to a vital



improvement compared to the HoLee mo del The R statistic shows that except

for the twotwist p erio ds and almost p ercent of the



But this do es not necessarily imply that the HoLeemo del is also in an exante analysis of

option prices the worst of the tested mo dels

Nonlinear Regression

Table Parameter Estimates for the HoLeeMo del

Perio d



sp ot rate variation is explained by the mo del All parameter estimates except for

the twist p erio ds are highly signicantandvary only slightly over time The Vasicek

mo del cannot t the observed volatility smile in the twotwist p erio ds illustrated

in gure The subp erio d is one the ve subp erio ds where a go o d t

of the Vasicek mo del is p ossible while the subp erio d illuminates the

limitations of the Vasicek mo del

The twofactor model of HJM comprises the two preceding mo dels The rst

factor is the parallel shift factor of HoLee and the second factor which is analogous

to the Vasicek mo del can b e viewed as a pseudo twist aecting the short rates

stronger than the long rates But the variance function

i h







e v



is monotonically decreasing

h i

v e

 

e e

z

z

so that the smile eect cannot b e mo deled The t in the two subp erio ds exhibiting

a strong smile eect is thus very p o or although b etter than for the Vasicek mo del

This is due to the fact that the variance is monotonically decreasing as in the Vasicek

mo del but b ecause of the shift factor not necessarily converging to zero Table and gure summarize the corresp onding results for the HJM mo del

VOLATILITY ESTIMATES FOR THE GERMAN BOND MARKET

Table Parameter Estimates for the VasicekMo del



Perio d R

Table Parameter Estimates for the HJMMo del



Perio d R

 

Nonlinear Regression

Figure Explained Variance in the Vasicek mo del

This gure shows the sp ot rate variance explained by the Vasicek mo del

compared to the historically sp ot rate variance for a subp erio d with only

areversion eect and a subp erio d with a strong smile eect

VOLATILITY ESTIMATES FOR THE GERMAN BOND MARKET

Figure Explained Variance in the HJM mo del

This gure shows the sp ot rate variance explained by the HJM mo del

compared to the historically sp ot rate variance for a subp erio d with only

areversion eect and a subp erio d with a strong smile eect

Nonlinear Regression

Table Parameter Estimates for FVMo del



Perio d R

   



Comparing the R of the HJM mo del in table and our FV model

in table reveals that the advantage of the FV mo del is negligible in ve of seven

subp erio ds In these p erio ds is not signicantly statistically or economically



dierent from zero except for the subp erio d The absence of a smile

eect is obvious when is not signicantly dierent from zero since the FV mo d



els reduces to the HJM mo del in this case But even with a signicantly p ositive



the smile eect may b e negligible The average slop e and convexity of the estimated

variance function serves as a measure of the smile eect When a smile eect is

presenttheaverage slop e should b e close to zero while the average convexity should

b e dierent from zero Table shows that this is the case only in the subp erio ds

and but not in the subp erio d In the twope

rio ds with a signicant smile eect the FV mo del outp erforms the HJM mo del

clearly The t of the estimated variance function is illustrated for two subp erio ds

and in gure The smile eect is largely determined by



and The reversion of the short rate to a long run mean is as in the Vasicek mo del



determined by A comparison of the estimation results for subp erio ds



and conrms the strong andweak mean

 

reversion hyp othesized in section based on the term structure plots

enow to assess the probability of negative sp ot rates in the four Gaussian W

mo dels based on historical parameter estimates The probability is plotted in gure

in the app endix for sp ot rate maturities of one to ten years based on the parameter

estimates of the subp erio d Since we observe the smile eect only in

VOLATILITY ESTIMATES FOR THE GERMAN BOND MARKET

Figure Explained Variance in the FVMo del

This gure shows the sp ot rate variance explained by the twofactor

Vasicek mo del FV compared to the historically sp ot rate variance for

a subp erio d with only a reversion eect and a subp erio d

with a strong smile eect

Nonlinear Regression

Table Magnitude of the Smile Eect

This table summarizes the average slop e and convexity of the estimated vari

ance functions as a measure of the smile eect Both are for exp ositional



purp ose multiplied by As a measure for the average slop e we use the

dierence of the variance function for the longest and shortest maturitywhich

is equivalent to the integral over the rst derivativeofthevariance function

Analogously we measure the average convexityby the dierence of the rst

derivative of the variance function for the longest and shortest maturity

Perio d average slop e average convexity

the subp erio d we use the parameter estimates of this subp erio d to

illustrate the drawback of the VF mo del Apart from the volatility parameters the

probability of negative sp ot rates is largely inuenced by the initial term structure

of forward rate The larger the initial forward rates the lower the probabilityWe

assume an unfavorable scenario of a at term structure with for all maturities

The volatility of the uncertain future sp ot rate and hence the probability of negative

sp ot rates increases with time We consider a time horizon of ten years

In all four mo dels the probability of negative sp ot rates after three years is smaller

than and thus not critical for the valuation of most interest rate futures and

options If b onds with emb edded options are to b e valued the b ehavior of the sp ot

rate dynamics after three years requires a closer lo ok In the HoLee mo del with a

sp ot rate variance prop ortional to time the probability of negative sp ot rates after

ten years is approximately The variance of the sp ot rate with maturity in the

Vasicek mo del converges to the constant





e



with time t to innity Accordingly the probability of negative sp ot rates converges

to a constant which is higher for small times to maturity Based on our parameter

estimates the probabilityiseven for sp ot rates with one year to maturity negligible

small The HJM mo del comprising the two factors of the HoLee and

Vasicek mo del yields probabilities which do not converge to a constantlevel but are

IMPLICATIONS FOR OPTION PRICING

even for a ten year horizon smaller than The examination of the probabilityof

negative sp ot rates shows a drawback of the FV mo del since the sp ot rate variance

tends to explo de with time for long maturities leading to ab out probabilityof

negative sp ot rates after veyears These results imply that the problem of negative

sp ot rates in Gaussian mo dels is negligible in the HoLee Vasicek HJM and FV

mo del when valuing interest rate contingent claims with less than three years to

maturityFor longer maturities only the Vasicek and HJM mo del are appropriate

Implications for Option Pricing

Gaussian interest rate mo dels are attractive b ecause of their analytical tractability

Closed form solutions for futures and options on zero b onds are readily available

It is well known that futures prices are martingales with resp ect to the risk neutral

measure Since the futures price equals the sp ot price at maturitytodays futures

price is simply the exp ected future sp ot price which is in a Gaussian mo del

Z

K

t

X

P t T

p

v t v t Tdv H t t T E P t TjF exp



t

Q

P t t

t

k 

Futures on coup on b onds can b e valued as p ortfolios of futures on zero b onds The

lognormal b ond price distribution allows further for a BlackScholes typ e valuation

of options on zero b onds

Theorem Calls on Zero Bonds

In a lo cally arbitragefree Gaussian Kfactor mo del the price of a call on a zero b ond

P t T with strikepriceK and maturity t is

C t K t T P t T N d KP t t N d

 

with

P t T

ln t t T d



t t T P t t K

d d t t T

 

Z Z

K K

t t

X X

 p   p

v t dv v T t t T v t T dv

k

k k

t t

k  k 

Pro of The call price may alternatively b e derived with resp ect to the risk neutral

or forward risk adjusted measure But deriving it with resp ect to the risk neutral

measure requires to compute an exp ectation of a pro duct of sto chastic variables



The forward risk adjusted measure in contrast eectively decouples that pro duct



cf Jamshidian

The forward price of the call is a martingale with resp ect to Q

C t t TjF C t t T E

t Q

C t T

C t TjF E

t Q

P t t

The forward price given the information F maythus b e calculated by

t

C t TjF P t t C t T E

t Q



P t T K jF P t t E

t Q

E P t T jF P t t

Q t

fP t T K g

KP t t Prob P t T KjF

Q t

P t T N d KP t t N d

 

The last equation follows by substituting P t T with equation and a standard

calculation of the exp ectation

This option pricing formula resembles the BlackScholes formulainanobvious

r t t

way The price of a zero b ond P t t with maturity t replaces e in the



traditional BlackScholes mo del Moreover the variance t t is replaced by

 

t t T whichisgiven in our Gaussian mo dels by

  

t t T T t t t

HL





 T t t t

t t T e e

Vas







T t t t   



e e t t T T t t t

HJM









  t t T t



t t T e e

FV









T t  t t



 

e e





Despite the similarity of the option pricing formulas in the BlackScholes mo del and

the Gaussian term structure mo dels there is a crucial dierence Even the Black

mo dication which is based on forward prices to eliminate the changing

variance of the b ond prices as time elapses is not able to price options on b onds with

dierent maturities in a consistent manner The volatility has to b e estimated for

every option separatelyIncontrast the four estimated Gaussian mo dels can price

all interest rate contingent claims with one set of volatility parameters For example

all four mo dels incorp orate the pulltopareect b ecause t t T converges to zero



It should b e noted that the option pricing formula for the Vasicek mo del is equivalentto

the one derived by Jamshidian for the original Vasicek mo del which is not necessarily

consistent with the initial term structure This is not surprising b ecause the adjustment to the

initial term structure can b e achieved by correcting the drift rate whichdoesnotenter the option

pricing formula

IMPLICATIONS FOR OPTION PRICING

with t T In the following wethus analyze the ability of the four mo dels to price

dierentcontingent claims We use atthemoney calls on zero b onds with maturities

ranging from to years The option maturityvaries b etween and years

First we compare the onefactor mo dels of HoLee and Vasicek The key dif

ference b etween the two mo dels is the sp ot rate variance decreasing with time to

maturityintheVasicek mo del and b eing constant in the HoLee mo del We take

as volatility parameters for the Vasicek mo del the parameter estimates for the sub

p erio d and calculate the implied volatility for the HoLee mo del which

gives the same option value for the twoyear option on the veyear b ond The call

price dierence b etween the HoLee and Vasicek mo del for the previously describ ed

set of options is shown in gure Naturally the dierence is zero if the option

Figure HoLeeCall Price versus VasicekCall Price

This gure shows call price dierences C C for atthemoney options

HL Vas

with a maturity of years The maturity of the underlying b onds with a face

value of varies b etween and years The initial term structure is at at

The volatility parameters are

HL Vas

expires immediately and for the threeyear option on a three year b ond b ecause

prices of atthemoney calls have to b e zero in these cases The price dierence is

also zero or close to zero for all options on the veyear b ond whichwas used for

the calculation of the implied volatility in the HoLee mo del But options on b onds

with maturities smaller larger than veyears are undervalued overvalued and

the absolute relative price dierence for the twoyear option ranges from

to Weinterpret this large price dierentials as a valua

tion error of the HoLee mo del b ecause the observed reversion in is not

captured by the HoLee mo del Prices for a twoyear atthemoney call according

Table Call Prices BS HoLee Vasicek HJM

This table presents call prices of atthemoney options with a maturityof

years for the BlackScholes HoLee Vasicek and HJM mo del The volatil

ity parameters are

BS HL Vas Vas

The maturity

H J M H J M HJM

of the underlying b onds varies b etween and years and the initial term

structure is at at

Bond Maturity C C C C

BS HL Vas HJM

to the BlackScholes HoLee Vasicek and HJM mo del are summarized in table

The volatility parameters for the BlackScholes and HJM mo del are again implicitly

calculated to t the twoyear option on the veyear b ond The call prices for the

BlackScholes mo del show clearly that the pulltopareect is not captured b ecause

options on b onds with a maturity close to the maturity of the option are strongly

overvalued and options on b onds with a maturity far larger than the option maturity

are undervalued The HJM mo del with three volatility parameters is tted not only

to the veyear b ond but also to the tenyear b ond Price dierentials are fairly small for all options

IMPLICATIONS FOR OPTION PRICING

Finally we examine pricing dierences b etween our twofactor mo dels A compar

ison of the HJM and FV mo del makes sense only for a subp erio d with a signicant

smile eect b ecause otherwise the FV mo del reduces to the HJM mo del Hence

we use the parameters estimates for the subp erio d and compare the call

prices for the same set of options as b efore The call price dierences are plotted in

gure Obviously the HJM mo del misprices calls on b onds with large maturities

Figure FVCall Price versus HJMCall Price

This gure shows call price dierences C C for atthemoney options

FV HJM

with a maturity of years The maturity of the underlying b onds with a face

value of varies b etween and years The initial term structure is at at

The volatility parameters are

HJM HJM HJM

bzw

FV FV FV FV

when a smile eect is present The absolute relative price dierence for the two

year option on the tenyear b ond is Besides the undervaluation

of calls on long term b onds the HJM mo del also overvalues calls on medium term

bonds years maturity This mispricing is caused by our estimation pro ce

dure which tries to t the variance function Since the ushap ed historical variance

function cannot b e tted in the HJM mo del the estimated variance is to o large for



maturities b etween and years and to o small thereafter see gure

 Call prices for the HJM and FV mo del are summarized in table in the app endix

Conclusions

Gaussian mo dels of term structure movements are attractive b ecause of their sim

plicity and analytical tractability The contribution of this pap er is threefold

First we examine the prop erties of four Gaussian mo dels the p opular onefactor

mo dels of HoLee and Vasicek resp ectivelyandatwofactor mo del suggested by

HeathJarrowMorton The fourth mo del is a twofactor Vasicek mo del whichwe

prop ose to capture the smile eect in the volatility structure

Second we examine the mo dels ability to explain the historical volatility struc

ture for the German b ond market We nd three basic patterns in the historical

yield curvemovement a parallel shiftareversion and a twist The parallel shift

implies a sp ot rate variance which is constant across all maturities as in the HoLee

mo del The reversion of the short rate to a long run average implies higher sp ot

rate volatilities for short time to maturities This monotonically decreasing variance

function is mo deled byVasicek and HeathJarrowMorton But only the twofactor

mo dels are able to incorp orate twists of the term structure Despite this similar

ity the HJM and FV mo del dier in their ability to explain a smile eect in the

volatility structure which is generally asso ciated with a twist of the term structure

Since the FV mo del includes the HJM mo del as a sp ecial case and a smile eect is

presentintwoofseven analyzed subp erio ds we prefer the FV mo del The applica

tion of the FV mo del is only limited by the probability of negative sp ot rates which

increases sharply with the time to maturity of the contingent claim and should b e

analyzed carefully when the maturity of the con tingent claim is larger than three

years

Finally the relevance of these ndings is judged by their implications for option

pricing The purp ose of a term structure mo del like the four under consideration

is the valuation of a broad varietyofinterest rate contingent claims in a consistent

manner We use options on zero b onds for which analytical solutions in all these

mo dels exist and vary only the maturity of the option and the underlying b ond

Even this simple set of interest rate contingent claims yields signicant price dif

ferences which can b e attributed to the mo del characteristics An op en question

remaining for future research is the predictabilityofthetwist and reversion eect

Moreover it remains to show that the prop osed twofactor Vasicek mo del enables its

user to earn abnormal prots from apparent mispricings

A PROOF OF THEOREM

A Pro of of Theorem

The equivalent martingale measures Q and Q are dened through

Z Z

K K

t t

X X



v dv dQdQt exp v dW v

k k k

k  k 

and

Z

K

t

X

p

v v t dW v dQ dQt exp

k k

k

k 

Z

K

t

X

p



v v t dv

k

k

k 



The corresp onding Brownian motions are given by Girsanovs theorem

Z

t

W t W t v dv

k k k

Z

t

p

W t W t v v t dv

k k

k

k

A  A Application of Itos Lemma gives the relative b ond prices pro cess

p p

Z t T dt t T Z t T dW t dZ t T t T r t

In terms of the Brownian motion W t with resp ect to the risk neutral measure Q

the relative b ond price pro cess is

p p

dZ t T t T r tZ t T dt t T Z t T tdt

p

t T Z t T dW t

p p

t T tZ t T t T r t

p

t T Z t T dW t

Relative b ond price are thus martingales with resp ect to Q if and only if

p p

t T r t t T t

A  A The equivalence of A and A follows immediately from the relation

of the forward rate and b ond price pro cess Substituting the b ond price drift and

diusion rate in condition A with and gives

Z Z Z



T T T

t y dy t r t t y dy t y dy r t

t t t



To apply Girsanovs theorem we need to assume that the market prices of risk t satisfy the

k

Novikov condition or condition C of HeathJarrowMorton With this assumption

and certain regularity conditions regarding the drift and diusion co ecients the existence of an

equivalent martingale measure is ensured see HeathJarrowMorton Prop osition

Taking the partial derivative with resp ect to T yields

Z

T

t y dy t T t t T t T

t

Z

T

t y dy  t T t T t

t

Analogously condition A follows byintegrating A with resp ect to T and

substitution of the forward rate drift and diusion rate according to and

A  A The forward rate pro cess

df t t t t dt t t dW t

in terms of the Brownian motion W t with resp ect to the forw ard risk adjusted

measure Q is

p

df t t t t dt t t t t t dt t t dW t

Forward rates are thus martingales with resp ect to Q if and only if

p

t t t t t t t

B TABLES AND FIGURES

B Tables and Figures

Table Empirical Studies

Author Market Perio d Results

HJM US TBonds May Two factors which can b e identi

ed as shift and twist

Kahn US TBonds s Two factors which explain

Shift and Twist of the

total variance

Steeley UK Three factors which explain

government and of the total

gilts variance The rst factor is a par

allel shift the second factor deter

mines the slop e and the third the

convexity of the yield curve

Bhler German Two factors which explain

Schulze b onds issued and of the total variance

byBund The rst factor is explained b est

Bahn and by a medium term sp ot rate the

Post second factor by a spread b etween

a long and a short rate

Beckers Dierent Two factors Shift und Twist

Countries in Australia Belgium Denmark

France Germany Italy Nether

lands Spain Three factors

Shift Twist und Buttery in Canada Japan UK USA

Figure Probability of negative sp ot rates

This gure shows the probability of negative sp ot rates with maturities of

years in the four Gaussian mo dels as time elapses The probabilities are

computed based on an initially at term structure at and the parameters

HoLee

Vasicek

HJM

 

FV

   

The parameters for the HoLee Vasicek and HJM mo del are the historical

estimates for the subp erio d which do es not reveal a smile eect

Since the FV mo del reduces in this case to the HJM mo del we use for the

FV mo del the historical parameter estimates for the subp erio d

exhibiting a strong smile eect

HoLee Vasicek

HJM FV

B TABLES AND FIGURES

Table Identication of Factors

This table rep orts the results of the regressions

F R

in n n n n

where F denotes the factor score of the ith factor with resp ect to the nth

in

observation As indep endentvariables the sp ot rate changes for the maturities



years and the changes of two spreads b etween a long and a short



rate are chosen

explanatory factor factor

 

variable R tWert R tWert



R



R

R

R

R

R

R

R

 

R R

 



R R 

Table Call Prices HJM and FV

This table shows call prices of atthemoney options with a maturityof

years for the HJM and FV mo del The volatility parameters are

HJM

H J M HJM FV FV

The maturity of the underlying

FV FV

b onds varies b etween and years and the initial term structure is at at

Bond Maturity C C

HJM FV

REFERENCES

REFERENCES

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Free Term Structure Mo dels Working Pap er May Scho ol of Business

Administration The UniversityofMichigan

Beckers S Interest Rate VolatilityandFactor Mo dels Working Pap er

BARRA International

Black F The Pricing of Commo dity Contracts The Journal of Financial

Economics pp

Boyle P Immunization Under Sto chastic Mo dels of the Term Structure

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Brennan MJ E Schwartz A Continuous Time Approach to the Pric

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Brown SJ PH Dybvig The Empirical Implications of the Cox Inger

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pp

Bhler W M Schulze Ein ZweiFaktoren Mo dell fr das b er

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Mannheim

Chamb ers DR WT Carleton DW Waldman A New Ap

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Financial and Quantitative Ananlysis pp

Cox JC JE Ingersoll SA Ross A Theory of the Term Structure

of Interest Rates Econometrica pp

Flesaker B Testing the HeatJarrowMortonHoLee Mo del of Interest

Rate Contingent Claim Pricing Journal of Financial and Quantitative Analy

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Gibb ons MR K Ramaswamy ATest of the Cox Ingersoll and

Ross Mo del of the Term Structure Review of Financial Studies pp

Gorsuch RL Factor Analysis Hillsdale NJ L Erlbaum Asso ciates

Heath D RA Jarrow A Morton Contingent Claim Valuation

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pp

Heath D RA Jarrow A Morton Bond Pricing and the Term

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REFERENCES

Ho T S Lee The Term Structure Movements and Pricing Interest Rate

Contingent Claims Journal of Finance pp

Hull J A White Pricing InterestRate Derivative Securities Review

of Financial Studies pp

Hull J A White OneFactor InterestRate Mo dels and the Valuation

of InterestRate Derivative Securities Journal of Financial and Quantitative

Analysis pp

Jamshidian F An Exact Formula Journal of Finance

pp

Jamshidian F Bond and Option Evaluation in the Gaussian Interest

Rate Mo del Research in Finance pp

Kahn R Risk Mo delling in F Fab ozzi ed The Hand

book of FixedIncome Securities edpp

Karatzas I SE Shreve Brownian Motion and

New York et al SpringerVerlag

Longsta FA ES Schwartz Interest Rate Volatility and the Term

Structures A TwoFactor General Equilibrium Mo del Journal of Finance

S

Pedersen H E Shiu A Thorlacius ArbitrageFree Pricing of Inter

est Rate Contingent Claims Transactions of the Society of Actuaries

pp

Richard SF An Arbitrage Mo del of the Term Structure of Interest

Rates Journal of pp

Rindel K P Sandas An Empirical Examination of the Pricing of Eu

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Schaefer SM ES Schwartz ATwoFactor Mo del of the Term Struc

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Seb er G C Wild Nonlinear RegressionNewYork John Wiley

Steeley JM Mo delling the Dynamics of the Term Structure of Interest

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Steeley JM Some Developments in Measuring and Mo deling the Term

Structure of Interest Rates in Ho dges S ed Options Recent Advances

in Theory and PracticeVol pp

REFERENCES

Vasicek O An Equilibrium Characterization of the Term Structure

Journal of Financial Economics pp