Choudhry. Fixed Income Securities Derivatives Handbook

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72 Introduction to Bonds

where

a = the speed of the mean reversion

b = the mean-reversion level of r

z = the standard Weiner process with mean 0 and standard deviation 1

Note that some texts use different notation, presenting the formula as

dr r dt dz=−

()

+κθ σ

dr r dt dZ=−

()

+αµ σ

The price at time t of a zero-coupon bond that matures at time T is

given by (4.8). (For the derivation, see Vasicek (1977) and section 5.3 in

Van Deventer and Imai (1997).)

PtT AtTe

BtTrt

()

−

()

(4.8)

where

r (t) = the short rate at time t

BtT

aT t

()

−

−−

()

and

AtT

BtT T t ab

BtT

,exp

()

−+

()

−

()

−

()

⎡

⎣

⎢

⎤

⎦

⎥

In Vasicek’s model, the short rate r is normally distributed. It therefore

has a positive probability of being negative. Model-generated negative

rates are an extreme possibility. Their occurrence depends on the initial

interest rate and the parameters chosen for the model. They have been

generated, for instance, when the initial rate was very low, like those seen

in Japan for some time, and volatility was set at market levels. This pos-

sibility, which other interest rate models also allow, is inconsistent with a

no-arbitrage market: as Black (1995) states, investors will hold cash rather

than invest at a negative interest rate. For most applications, however, the

model is robust, and its tractability makes it popular with practitioners.

Hull-White Model

The well-known model described in Hull and White (1993) uses Vasicek’s

model to obtain a theoretical yield curve and ﬁ t it to the observed mar-

ket curve. It is therefore sometimes referred to as the extended Vasicek

Interest Rate Modeling 73

model, with time-dependent drift. (Haug (1998) equates Hull-White to

the Ho-Lee model, with mean reversion.) Time-dependent drift is a drift

rate whose value is dependent on the time period used to calculate it,

based on historical movement up until now. The model is popular with

practitioners because it enables them to calculate a theoretical curve that

is identical to yields observed in the market and that can be used to price

bonds and bond derivatives and to calculate hedges.

The model is expressed in equation (4.9).

dr a

rdt dz=

()

−

⎛

⎝

⎜

⎞

⎠

⎟

+ σ

(4.9)

where

a = the speed of mean reversion

()

= a time-dependent mean reversion

Given this description of the short-rate process, the price at time t of a

zero-coupon bond with maturity T may be expressed as (4.10).

PtT AtTe

BtTrt

()

−

()

(4.10)

where

r (t) = the short rate at time t

BtT

aT t

()

−

−−

()

ln , ln

AtT

Pot

BtT

vtT

()

⎡

⎣

⎢

⎤

⎦

⎥

−

()

∂

()

∂

−

()

and

vtT

eee

aT at at

()

=−

()

−

()

−−

1σ

Further One-Factor Term-Structure Models

The academic literature and market participants have proposed a large

number of alternatives to the Vasicek term-structure model and models,

such as the Hull-White model, that are based on it. Like those they seek to

replace, each of the alternatives has advantages and disadvantages.

The main advantage of Vasicek-type models is their analytic tracta-

bility. Their main weakness is that they permit negative interest rates.

Negative interest rates are not impossible in the actual market; a bond

74 Introduction to Bonds

sold as part of a repurchase transaction for which excess market demand

exists may have a negative rate. Academic researchers, however, often

prefer to work with environments where negative rates are impossible

(see Black (1995)). Since such environments are functions of more than

one variable, Vasicek-type models are considered incapable of model-

ing them. This limitation does not preclude their use; rather, the choice

of using them or not will depend on the conditions obtaining in the

economy. In Japan from 1997 to 1998, for example, money market rates

were frequently below 0.5 percent. At that level, even a volatility value

below 5 percent, when plugged into the Vasicek model, will imply a high

probability of negative interest rates. In this environment, therefore,

practitioners may wish to use an alternative model, perhaps a two-factor

or multifactor one.

Cox-Ingersoll-Ross (CIR) Model

Although published ofﬁ cially in 1985, the Cox-Ingersoll-Ross model

was described in academic circles in 1977, or perhaps even earlier, which

would make it the ﬁ rst interest rate model. Like Vasicek’s it is a one-

factor model that deﬁ nes interest rate movements in terms of the dynam-

ics of the short rate. It differs, however, in incorporating an additional

feature, which relates the variation of the short rate to the level of interest

rates. This feature precludes negative interest rates. It also reﬂ ects the fact

that interest rate volatility rises when rates are high and correspondingly

decreases when rates are low. The Cox-Ingersoll-Ross model is expressed

by equation (4.11).

dr k b r dt rdz=−

()

+ σ

(4.11)

k = the speed of mean reversion

Note that the CIR model is often stated with k used to denote the

speed of mean reversion as a was earlier.

Deriving the zero-coupon bond price given this model is formalized in

equation (4.12), from Ingersoll (1987), chapter 18.

Pr A e

,ττ

()

(

)

−

()

(4.12)

where

τ = the term to maturity of the bond, or (T – t)

Interest Rate Modeling 75

γλ

γτ

(

)

(

)

⎡

⎣

⎢

⎤

⎦

⎥

(

)

−−

()

(

()

−

))

(

)

=+++

()

−

()

gke

τγ λγ

γλσ

γτ

Again we use k to denote the speed of mean reversion, and λ is a user-

deﬁ ned parameter to adjust the level of this if necessary, connected with

the risk premium associated with long-dated bond yields.

Some researchers—Van Deventer and Imai (1997) cite Fleseker (1993),

for example—have stated that the difﬁ culties in determining parameters

for the CIR model have limited its use among market practitioners. Van

Deventer and Imai, however, conclude that it deserves further empirical

analysis and remains worthwhile for practical applications.

Two-Factor Interest Rate Models

This section brieﬂ y introduces a number of two-factor interest rate

models. (The References section indicates sources for further research.)

As their name suggests, these models specify the yield curve in terms of

two factors, one of which is usually the short rate. A number of factors

can be modeled when describing the dynamics of interest rates. Among

them are

❑ the short-term or instantaneous interest rate

❑ the long-term—say, 10-year—interest rate

❑ short- and long-term real inﬂ ation-adjusted rates of interest

❑ the current or expected spread between the short- and long-term

interest rates

❑ the current or expected corporate credit spread—that is, the differ-

ence in yield between the equivalent-maturity Treasury and bonds

having the same credit rating as the subject bond

❑ the current inﬂ ation rate

❑ the long-term average expected inﬂ ation rate

76 Introduction to Bonds

Which factors the model incorporates depends in part on the purpose

it is intended to serve—whether, for example, it is being used for pricing

or hedging derivative instruments or for arbitrage trading. Other consider-

ations also apply, such as the ease and readiness with which the parameters

involved can be determined.

Brennan-Schwartz Model

The model described in Brennan and Schwartz (1979) uses the short

rate and the long-term interest rate to specify the term structure. The

long-term rate is deﬁ ned as the market yield on an irredeemable, or per-

petual, bond, also known as an undated or consol bond. Both interest

rates are assumed to follow a Gaussian-Markov process. A Gaussian pro-

cess is one whose marginal distribution, where parameters are random

variables, displays normal distribution behavior; a Markov process is one

whose future behavior is conditional on its present behavior only, and

independent of its past. A later study, Longstaff and Schwartz (1992),

found that Brennan-Schwartz modeled market bond yields accurately.

In the model, the dynamics of the logarithm of the short rate are

deﬁ ned by equation (4.13).

d r a l p r dt dzln ln ln ln

(

)

[]

()

−

()

−

(

)

⎡

⎣

⎤

⎦

+ σ

(4.13)

In (4.13), p represents the relationship between the short rate, r, and

the long-term rate, l. The short rate changes in response to moves in the

level of the long rate, which follows the stochastic process described in

equation (4.14).

dl l l r dt l dz=−++

⎡

⎣

⎢

⎤

⎦

⎥

+σλσ σ

22 2 2

(4.14)

where

λ = the risk premium associated with the long-term interest rate

Extended Cox-Ingersoll-Ross Model

Chen and Scott (1992) transformed the CIR model into a two-factor

model specifying the interest rate as a function of two uncorrelated vari-

ables, both assumed to follow a stochastic process. The article demon-

strated that this modiﬁ cation of the model has a number of advantages

and useful applications.

The model speciﬁ es the relationship expressed in equation (4.15).

dy k y dt y dz

iiii iii

=−

()

+θσ (4.15)

Interest Rate Modeling 77

where

= the independent variables y

and y

k and θ = the parameters that describe the drift rate and the dz or

stochastic factor

In this model, the formula for deriving the price of a zero-coupon

bond is (4.16).

Py y tT AAe

By By

12 12

11 22

,,,

()

−−

(4.16)

where A and B are deﬁ ned as before

Heath-Jarrow-Morton (HJM) Model

The approach described in Heath-Jarrow-Morton (1992) represents a

radical departure from earlier interest rate models.

The previous models

take the short rate as the single or (in two- and multifactor models)

key state variable in describing interest rate dynamics. The speciﬁ cation

of the state variables is the fundamental issue in applying multifactor

models. In the HJM model, the entire term structure and not just the

short rate is taken to be the state variable. Chapter 3 explained that the

term structure can be deﬁ ned in terms of default-free zero-coupon bond

prices or yields, spot rates, or forward rates. The HJM approach uses

forward rates.

The single-factor HJM model captures the change in forward rates at

time t, with a maturity at time T, using

❑ a volatility function

❑ a drift function

❑ a geometric Brownian or Weiner process, which describes the

shocks, or noise, experienced by the term structure

Consider a forward-rate term structure f (0, T ) that is T-integrable—

for which the integral can be computed in terms of the variable T. The

dynamics of this structure may be described by the stochastic differential

equation (4.17).

df t T a t T dt dz,,

()

+ σ (4.17)

where

a = the drift rate, denoted in some texts by α

In some texts the term k is used for a.

σ = the constant volatility level

78 Introduction to Bonds

dz = the geometric Brownian motion or Weiner process, denoted in

some texts by W

By applying Ito calculus, i.e., Ito’s lemma, (4.17) can be transformed

to solve for the price of an asset. Taking the integral of expression (4.17)

results in equation (4.18), which derives the forward rate,

f t T f T a s T ds dz

,, ,

()

∫

(4.18)

where

s = an incremental move forward in time, so that

0 ≤≤ ≤≤tT tsT and

Equation (4.18) assumes that the forward rate is normally distributed.

Crucially, the forward rates with maturities f (0,1), f (0,2) .… f (0, T ) are

assumed to be perfectly correlated. The random element is the Brownian

motion dz. The impact of this process is felt over time, rather than over

different maturities.

The single-factor HJM model states that, given an initial forward-rate

term structure f (t, T ) at time t, the forward rate for each maturity T is

given by (4.20), which is the integral of (4.19).

dftT atTdt tTdz,, ,

()

(4.19)

f tT f T a sT ds sT dzs

,, , ,()

()

∫∫

(4.20)

Note that the expressions at (4.19) and (4.20) are simpliﬁ ed versions

of the formal model being discussed.

Equations (4.19) and (4.20) state that the development of the for-

ward rate for any maturity period T can be described in terms of the drift

and volatility parameters a(t, T ) and σ(t, T ). The HJM model’s primary

assumption is that, for each T, the drift and volatility processes are de-

pendent only on the histories up to the current time t of the Brownian

motion process and of the forward rates themselves.

The Multifactor HJM Model

In the single-factor HJM model, forward rates of all maturities move in

perfect correlation. For actual market applications—pricing an interest

rate instrument that is dependent on the spread between two points on

Interest Rate Modeling 79

the yield curve, for instance—this assumption can be too restrictive. In

the multifactor model, each of the state variables is described by its own

Brownian motion process. An m-factor model, for example, would include

Brownian motions dz

, dz

, ..... , dz

. This allows each T-maturity forward

rate to be described by its own volatility level σ

,T ) and Brownian mo-

tion process dz

. Under this approach, the forward rates derived from the

bonds of differing maturities that deﬁ ne the current term structure evolve

under more appropriate random processes, and different correlations

among forward rates of differing maturities can be accommodated.

The multifactor HJM model is represented by equation (4.21).

f tT f T a sT ds sT dz s

,, , ,

()

(

)

∫

∑

∫

(4.21)

Equation (4.21) states that the dynamics of the forward-rate process,

beginning with the initial rate f (0, T ), are speciﬁ ed by the set of Brownian

motion processes and the drift parameter. For practical applications, the

evolution of the forward-rate term structure is usually derived in a bino-

mial-type path-dependent process. Path-independent processes, however,

have also been used, as has simulation modeling based on Monte Carlo

techniques (see Jarrow (1996)). The HJM approach has become popular

in the market, both for yield-curve modeling and for pricing derivative

instruments, because it matches yield-curve maturities to different volatil-

ity levels realistically and is reasonably tractable when applied using the

binomial-tree approach.

Choosing a Term-Structure Model

Selecting the appropriate term-structure model is more of an art than a

science, depending on the particular application involved and the user’s

individual requirements. The Ho-Lee and BDT versions, for example, are

arbitrage, or arbitrage-free, models, which means that they are designed to

match the current term structure. With such models—assuming, of course,

that they specify the evolution of the short rate correctly—the law of no-

arbitrage can be used to determine the price of interest rate derivatives.

Equilibrium interest rate models also exist. These make the same as-

sumptions about the dynamics of the short rate as arbitrage models do,

but they are not designed to match the current term structure. The prices

of zero-coupon bonds derived using such models, therefore, do not match

prices seen in the market. This means that the prices of bonds and inter-

est rate derivatives are not given purely by the short-rate process. In brief,

arbitrage models take as a given the current yield curve described by the

80 Introduction to Bonds

market prices of default-free bonds; equilibrium models do not.

Among the considerations that should be taken into account when

deciding which term-structure model to use are the following:

❑ Ease of application. In this respect, arbitrage models have the

advantage. Their key input is the current spot-rate term structure. This,

unlike the input to equilibrium models, can be determined in a straight-

forward process from the market price of bonds currently trading in the

market.

❑ Desirability of capturing market imperfections. The term

structure generated by an arbitrage model will reﬂ ect the current mar-

ket term structure, which may include pricing irregularities arising from

liquidity and other considerations. Equilibrium models do not reﬂ ect such

irregularities. Selection of a model will depend on whether or not model-

ing market imperfections is desirable.

❑ Application in pricing bonds or interest rate derivatives. Tra-

ditional seat-of-the-pants bond pricing often employs a combination of

good sense, prices observed in the market (often from interdealer-broker

screens), and gut feeling. A more scientiﬁ c approach may require a yield-

curve model, as will relative value trading—trading bonds of different ma-

turities against each other, for example, in order to bet on changes in yield

spreads. In such cases, equilibrium models are clearly preferable, since

traders will want to compare the theoretical prices given by the model

with the prices observed in the market. Arbitrage models, in contrast,

assume that the market bond prices are correct. So, an arbitrage model

would always suggest that there was no gain to be made from a relative

value trade.

Pricing derivative instruments such as interest rate options (or swap-

tions) requires a different emphasis. The primary consideration of the de-

rivative market maker is the technique and price of hedging the derivative.

When writing derivative contracts, market makers simultaneously hedge

their exposure using either the underlying assets or a combination of

these and other derivatives, such as exchange-traded futures. They proﬁ t

from the premium they extract and from the difference in price over time

between the derivative and the hedge. In this enterprise, only arbitrage

models are appropriate, because they price derivatives relative to the actual

market. An equilibrium model, in contrast, prices derivatives relative to a

theoretical market, which is not appropriate, since those used in the hedge

are market instruments.

❑ Use of models over time. The parameters in an interest rate model

—most notably the drift, volatility, and, if applicable, mean reversion

rate—reﬂ ect the current state of the economy. This state is not constant;

the drift rate used today, for example, may well differ from the value used

Interest Rate Modeling 81

tomorrow. Over time, any model must be recalibrated. For arbitrage mod-

els, however, this is a constant process, since their parameters change con-

tinuously. Equilibrium model parameters, in contrast, are calculated from

historical data or using logic, and so may not change as frequently. On the

other hand, the accuracy of these models may suffer over time, as current

rates diverge from historic average rates. Users must decide whether the

greater accuracy of the arbitrage model is worth the constant tweaking

that makes it possible.

This list is just a sample. Users must consider a wide range of issues

when selecting an interest rate model. It has been observed, for instance,

that models that incorporate mean reversion are more accurate than those

that do not. Another crucial consideration is the computer processing

power available to the user. Single-factor models are often preferred pre-

cisely because their processing is more straightforward.