Review of Continuous-Time Term-Structure Models

Review of Continuous-Time

Term-Structure Mo dels

Part I: equilibrium classical mo dels

Jesp er Lund

Department of Finance

The Aarhus Scho ol of Business

DK-8210 Aarhus V, Denmark

E-mail: [email protected]

Homepage: www.hha.dk/~jel/

First draft: April, 1997

Current draft: March 2, 1998

I am grateful to Peter Honor e for helpful comments.

1 Intro duction

This pap er contains a survey of continuous-time term-structure mo dels. The general

idea is intro ducing the reader student to the sub ject, so that he or she hop efully

will b e able to read journal articles on the sub ject. The technical and mathematical

nature of this pap er re ects that ob jective.

It is now customary to divide term-structure mo dels into two groups, called

arbitrage-free and equilibrium mo dels, resp ectively. The former group contains mo d-

els which t the initial yield exactly, and prominent examples are the Heath, Jarrow

and Morton 1992 mo del and the Hull and White 1990 extended Vasicek mo del.

On the other hand, the equilibrium mo dels do not p er construction t the yield

curve exactly. Sometimes, the mo dels in equilibrium framework are called \classical"

mo dels since this approach dates back earlier than the more recent arbitrage-free

mo dels.

To a large extent, the choice between arbitrage-free and equilibrium mo dels is

dictated by the purp ose of the analysis. If we are interested in identifying b onds that

are mispriced relative to other b onds, we can only use equilibrium mo dels. On the

other hand, when pricing xed-income derivatives it is generally preferable to use an

arbitrage-free mo del, see chapter 9 in Tuckman 1995 for an excellent discussion.

For this reason, our survey pap er deals with b oth mo dels. In Part I, we discuss

equilibrium mo dels classical mo dels with either a single or multiple factors, whereas

Part II forthcoming describ es two typ es of arbitrage-free mo dels: equilibrium-typ e

mo dels with time-dep endent parameters calibrated mo dels and mo dels in Heath,

Jarrow and Morton class.

The outline of part I is as follows: section 2 contains a brief summary of the

de nition of the term structure using continuous comp ounding. Sections 3 presents

a general analysis of one-factor mo dels, whereas section 4 describ es three examples.

Finally, section 5 extends the mo deling framework to multiple factors, including a

discussion of the exp onential-ane class of mo dels. Most multi-factor mo dels with

known analytical solutions for the term structure b elong to this class.

Unfortunately, the literature is not consistent with resp ect to these de nitions. In some pap ers,

esp ecially pre-1990 pap ers, a mo del is only called an equilibrium mo del if derived from the utility

function of the representative agent. The Vasicek 1977 mo del is sometimes referred to as an

arbitrage-free or partial equilibrium mo del b ecause an \absence of arbitrage" argument is used

in the mo del derivation, cf. also section 3 in the present pap er. In summary, there is b ound to b e

some confusion, and the reader should b e careful when encountering de nitions along these lines in,

esp ecially, the \older" literature. 1

2 De nitions

There are three di erentways to represent the term structure of interest rates:

P t; T is the price, at time t, of a zero-coup on b ond maturing at time T the

maturity date . The time to maturity of this b ond is = T t. It is

imp ortant to note the distinction b etween the maturity date and the time

to maturity | they are only identical when t =0. In general, we assume

that P t; T exists for all T >t.

Rt; T The yield-to-maturity with continuous comp ounding at time t, for a zero-

coup on b ond maturing at time T .

f t; T The instantaneous forward rate at time t, for a zero-coup on b ond matur-

ing at time T .

The yield-to-maturity R t; T and forward rate f t; T are de ned as follows:

log P t; T

R t; T = 1

T t

@P t; T =@ T @ log P t; T

f t; T = = : 2

P t; T @T

Note that log denotes the natural base e logarithm. The inverse relationship ex-

presses the b ond price, P t; T , in terms of either R t; T or ft; T :

R t; T T t

P t; T = e 3

f t; sds

P t; T = e : 4

The rst formula, 3, follows simply by rearranging the de nition of R t; T in 1.

To derive 4, rst note that

Z Z

T T

@ log P t; s

log P t; T log P t; t = ds = f t; sds; 5

t t

and since P t; t=1, we get 4.

Furthermore, by equating the terms in the exp onents in 3 and 4, we get the

following relationship b etween yield-to-maturity and forward rates:

ft; sds; 6 R t; T =

T t

which may be interpreted as the average forward rate over the remaining time to

maturityof the b ond.

Unless we state otherwise in the text, all b onds are assumed to be zero-coup on b onds which

havea single paymentof one \unit of account" at time T . Bonds with more than one remaining

payment, for example bullets, are called coupon bonds. 2

3 A general one-factor mo del

In this section we carefully explain the common mathematical structure of term-

structure mo dels with a single factor, a framework that encompasses the Merton

1973, Vasicek 1977 and Cox, Ingersoll and Ross CIR 1985 mo dels. The reader

is assumed to be familiar with sto chastic di erential equations SDEs, including

Ito's lemma, at a level comparable to Hull 1997, Luenb erger 1997, or a similar

non-mathematically oriented text.

We make the following assumptions:

A{1 The b ond market is frictionless: no distorting taxes, no transactions costs,

no short-sale restrictions, and all b onds are in nitely divisible.

A{2 Investors always prefer more wealth to less, i.e., the marginal utilityofwealth

is p ositive at all levels of wealth. In e ect, this assumption rules out that

arbitrage opp ortunities can exist.

A{3 All b ond prices, i.e. P t; T for all T > t, dep end only on a single state

variable: the short rate r in addition to t and T . By implication, changes

in the yield curve at di erent maturities are p erfectly correlated.

A{4 The short rate instantaneous interest rate follows the general SDE:

dr = r dt + r dW ; 7

t t t t

where r and r are the drift and volatility functions, resp ectively, and

W a Brownian motion Wiener pro cess.

It is imp ortant to realize that we do not assume that the relationship b etween P t; T

and the short rate, r , is known. On the contrary, the entire purp ose of the following

is deriving that function endogenously from the ab ove assumptions, esp ecially from

the assumption ab out absence of arbitrage.

In the rst step, we determine the sto chastic pro cess SDE for the b ond price

P t; T . Note that T is xed, and t denotes calendar time. By Ito's lemma we get:

dP t; T = t; T P t; T dt + t; T P t; T dW ; 8

P P t

where

@P @P 1 @ P

t; T P t; T = r+ + r 9

@r @t 2 @r

t; T P t; T = r: 10

In equation 8, t; T is the exp ected instantaneous return of the b ond with ma-

turity date T , and t; T is the volatility standard deviation of the b ond return.

Of course, this is highly restrictive, but later we relax the assumption with the so-called multi-

factor mo dels. Rightnowwewanttokeep things simple! 3

The exp ected return and the volatility dep end on the short rate, r , but to simplify

the notation this dep endence is suppressed here.

The problem is that equilibrium exp ected returns t; T for di erent T 's are

unknown, so a general expression for the b ond price P t; T cannot be determined

at this stage. The intermediate goal in the following is developing some form of

equilibrium mo del for the exp ected returns, t; T , for all T . Concretely, we use

the principle of no-arbitrage to reduce this problem to sp ecifying a single market price

risk preference parameter.

Supp ose we construct a p ortfolio consisting of w b onds with maturity date T

1 1

and w b onds with maturity date T . We require T 6= T , but apart from that T

2 2 1 2 1

and T can b e arbitrary. The value of the resulting p ortfolio, at time t, is denoted by

= w P t; T +w Pt; T ; 11

t 1 1 2 2

and the value, , satis es the SDE:

d =[w t; T P t; T + w t; T P t; T ] dt +

t 1 P 1 1 2 P 2 2

[w t; T P t; T + w t; T P t; T ] dW : 12

1 P 1 1 2 P 2 2 t

Since there are two b onds and only one source of risk, it must b e p ossible to eliminate

the risk by cho osing w and w such that

1 2

w t; T P t; T +w t; T P t; T =0: 13

1 P 1 1 2 P 2 2

In general, this requires continuous adjustment of the p ortfolio which can be done

costlessly since wehave assumed away transactions costs. If w and w are continu-

1 2

ously readjusted according to 13, the p ortfolio SDE 12 reduces to:

d =[w t; T P t; T + w t; T P t; T ] dt; 14

t 1 P 1 1 2 P 2 2

which is lo cally deterministic riskless. To prevent arbitrage opp ortunities, the excess

return ab ove the short rate r must b e zero:

w t; T r P t; T +w t; T r P t; T = 0: 15

1 P 1 t 1 2 P 2 t 2

To summarize, we have now shown that if the 2 1 vector w = [w w ] solves the

1 2

equation

h i

t; T P t; T t; T P t; T

A w =0; 16

P 1 1 P 2 2

the same w is also a solution to the homogeneous system of equations:

" "

t; T P t; T t; T P t; T w

P 1 1 P 2 2 1

A w =0: 17

t; T r P t; T t; T r P t; T w

P 1 t 1 P 2 t 2 2

Vasicek 1977 uses the same technique. It is very similar to the metho d used to derive the

Black-Scholes sto ck option-pricing formula, see Hull 1997. 4

This is only p ossible if the rank of the matrix A is 1. To see this, note that if A

2 2

has full rank, A exists, and the solution of 17 is w = 0 | which is obviously a

contradiction. This argument proves that the rank of A must be 1.

Since A has less than full rank, it is p ossible to write the last second row as a

linear combination of the other rows | in this case a scalar function r times the

rst row. Hence, we get

t; T = r +r t; T ; j =1;2 18

P j t t P j

where r is the so-called market price of risk. Further, note that the particular

choices of T and T play no role in the ab ove derivation, so 18 must hold for all

1 2

T , and r must be indep endent of T . In summary, we have reduced the problem

of determining t; T , for all p ossible T , to that of sp ecifying a single market price

of risk parameter, r , which is at most a function of the short rate r . Of course,

this requires an additional assumption ab out market preferences, and di erent mo dels

Vasicek, Merton, CIR use di erent sp eci cations of r .

Finally, we substitute 18 into 9, and after rearranging terms and using the

de nition of t; T from equation 10, we get the following fundamental partial

di erential equation PDE for the b ond price:

1 @P @P @ P

r + [r rr] + rP =0; 19

2 @r @r @t

with b oundary condition P T; T =1. Analytical solutions to this PDE exist for sev-

eral one-factor mo dels, including the Vasicek 1977, Merton 1973 and CIR 1985

mo dels. They are describ ed in detail in section 4 b elow.

A general representation of the solution to 19 is furnished by the Feynman-Kac

formula:

r ds

; 20 P t; T = E e

where the exp ectation is taken under the probability measure probability distribu-

tion corresp onding to the drift-adjusted sto chastic pro cess:

dr = fr r r g dt + r dW ; 21

t t t t t

where W is a Brownian motion under the Q-measure, or risk-neutral distribution.

Of course we still need to calculate 20 in order to get a closed-form expression for

the b ond price P t; T , and in most cases it is actually simpler to solve the PDE

directly.

However, equation 20 o ers a lot of intuition ab out the mechanics of arbitrage-

free term-structure mo dels. The current price, P t; T , is obtained by discounting

the nal payment of one unit of account back to the present time t, and since the

future short-term interest rates are random, we take the exp ectation, conditional on

the current value of the short rate, r . Among nancial economists, the technique

is known as risk-neutral valuation, and consequently 21 is called the risk-neutral

sto chastic pro cess for the short rate. Note, however, that we are not assuming risk-

neutrality on b ehalf of the economic agents. On the contrary, investor preferences

enter the b ond-pricing formula through the drift adjustment by r r in the

t t

SDE 21. 5

3.1 Fixed-income derivatives and risk-neutral valuation

Risk-neutral valuation is an extremely powerful technique for pricing xed-income

derivatives, that is securities with uncertain sto chastic cash ows. For example,

binomial mo dels trees are based on this idea, as the tree in built with risk-neutral

probabilities, rather than the \true" probabilities, cf. Tuckman 1995, ch. 5{6.

To illustrate the scop e of risk-neutral valuation, consider a general xed income

claim, maturing expiring at time T , which o ers the following payment stream to

the holder:

For all t s T , the claim pays a continuous stream of payments prop ortional

to cr . That is, between time s and s + ds, we get cr ds from the claim,

s s

where ds is a small time interval.

At maturity, t = T ,we get a nal lump-sum payment of C r .

The value price, V r , of this claim at time t to day is given by:

R R

T s

Q Q

r ds r du

s u

t t

: 22 V r =E C r + E e cr e ds

t T s

t t

Examples of 22:

1. A zero-coup on b ond, where C r = 1 and cr =0.

T s

2. A call option, with exercise price K , on a zero-coup on b ond, maturing at time

T >T. Here, cr = 0, and the terminal payment is

1 s

C r = [Pr ;T;T K] ; 23

T T 1

where x = maxx; 0, and P r;T;T is the price at time T of a zero-coup on

b ond maturing at time T .

3. An interest rate cap on the short-rate itself , with a strikeofL. Here, C r =0

and cr =r L .

s s

If the short rate is governed by a one-factor SDE under the risk-neutral measure,

the price of the security, V r , also satis es the partial di erential equation:

1 @ V @V @V

r + [r rr] + + cr rP =0; 24

2 @r @r @t

sub ject to the b oundary condition V r = C r . Compared to the PDE 19, the

b oundary condition is mo di ed, and we have added a term cr to the left hand

side. Since the security has a continuous payout, or dividend, of cr at time t, the

exp ected risk-neutral change in the price is r P cr , instead of r P for a security

t t t

without \dividend" payments. Note that the exp ected return of the security sum of

dividends and price change is prop ortional to the short rate, r , in b oth cases.

t 6

4 Three examples of one-factor mo dels

In this section we present the b ond-pricing formula, i.e. the solution P t; T of the

fundamental PDE 19 for three di erent mo dels: Vasicek 1977, Merton 1973 and

CIR 1985. The solution technique is very much the same separation of variables,

so we only provide a detailed discussion for the Vasicek mo del.

4.1 Vasicek 1977 mo del

The short rate follows the mean-reverting Gaussian pro cess sometimes called the

Ornstein-Uhlenbeck pro cess:

dr = r dt + dW ; 25

t t t

where measures the sp eed of mean reversion the larger , the faster the sp eed of

mean reversion, is the unconditional mean, and is the instantaneous volatility

of the short rate. The Vasicek pro cess 25 is the continuous-time equivalent of a

rst-order autoregressive pro cess, or AR1 mo del. With resp ect to the market price

of risk, we assume that it is a constant r = .

This results in the following PDE:

1 @P @ P @P

+ [ r ] + rP =0; 26

2 @r @r @t

with b oundary condition P T; T = 1.

First, we guess that the solution takes the so-called exp onential-ane form:

P t; T = exp [A +Br ]; = T t: 27

Second, we di erentiate 27 with resp ect to r and t:

= B P t; T 28

@ P

= B P t; T 29

@P @P

0 0

= = [A +B r] Pt; T : 30

@t @

dB dA

0 0

, and B = . Note that A =

d d

Third, we substitute 28{30 into 26. Since all terms contain a factor P t; T ,

we move this factor outside the parenthesis braces and get:

2 2 0 0

B + B [ r ] A B r r P =0: 31

Finally,we divide by P in 31, and collect the terms containing the factor r :

2 2 0 0

B + B [ ] A fB +1+B gr =0: 32

2 7

The PDE 32 should be satis ed for all values of r , and this can only hold if b oth

expressions in braces are zero. Equating each of the two terms in braces with zero,

results in two ordinary di erential equations ODEs,

0 2 2

A = B +[ ]B 33

B = B 1: 34

If we can nd a solution to these ODEs, wehave demonstrated that 27 is indeed the

solution to 26. As with PDEs, we need b oundary conditions to solve ODEs. The

b oundary condition from the PDE, that is P T; T = 1, translates into two b oundary

initial conditions for the ODE:

A0 = 0 and B 0 = 0 : 35

Generally, a system of ODEs as we have needs to be solved simultaneously. In

the present case, however, the solution separates into two univariate ODEs with a

recursive structure since 34 only involves B . Therefore, we rst solve 34, and

after substituting the solution B into 33, we determine A .

In our e ort to solve 34, we rst rewrite it as:

B +B =1; 36

and multiply on b oth sides by exp :

B e + B e = e : 37

By the pro duct rule for di erentiation, the left hand side in 37 can also b e written

as:

n o

e B = e : 38

Since B do es not app ear on the right hand side in 38, the solution can b e obtained

by ordinary integration. By the standard relationship between di erentiation and

integration

Z Z

n o

s s

e B =B0 + e B s ds = e ds; 39

0 0

where the last equality is obtained by using the b oundary condition B 0=0,aswell

as equation 38. Finally,we arrive at the desired solution:

Z Z

s s

B = e e ds = e ds

0 0

1 e 1

= e = : 40

Note that = 0 when t = T the b ond matures. 8

Having found B , we turn to A . Again, since the function in question, i.e.

A do es not app ear on the right hand side of the ODE 33, the solution can be

determined by ordinary integration:

Z Z Z

0 2 2

A =A0 + A sds = B sds +[ ] B sds: 41

0 0 0

Thus, we need to calculate the integral of B s and B s. To conserve on space, we

state the requisite results rather brie y, leaving most of the details to the reader.

= 1 e

B sds =

e 1

= = + 42

and

= + =2 2 1 e 1 e

1 1

2 2 2

B sds =

2 2

2 3

1 e 4 1 e

4 5

+ 43 =

2 2

This concludes the derivation of the Vasicek b ond-pricing formula. For convenience,

we restate the entire formula b elow after having worked a bit on the expressions:

R 1 = 44

e 1

B = 45

A = R 1 + B B : 46

This corresp onds to equation 27 on page 185 of the Vasicek 1977 pap er. Note,

though, that his notation is di erent from ours. Among other things, he uses the

opp osite sign for the market price of risk which he calls q , instead of \our" .

In 45, it straightforward to see that B < 0, so an increase in r lowers b ond

prices. The reader is encouraged to investigate how di erent parameter values for ,

, and a ect the shap e of the term structure.

4.2 The Merton 1973 mo del

The short rate is governed by the SDE:

dr = dt + dW ; 47

t t 9

and the market price of risk is a constant , as in the Vasicek mo del. It can b e shown

that the b ond price is given by 27 see section 4.1 with the following de nitions of

A and B :

B = 48

1 1

2 2 3

A = + : 49

2 6

The pro of is left to the reader. Compared to the Vasicek mo del, it is actually quite

simple and doing it is a very go o d exercise.

4.3 The CIR 1985 mo del

The famous CIR, or Cox, Ingersoll and Ross, mo del uses the so-called square-ro ot

SDE pro cess for the short rate:

dr = r dt + r dW : 50

t t t t

r= . The scaling by is CIR sp eci es the market price of risk as follows: r =

done only to simplify the subsequent derivations.

The derivations are considerably more tedious than in the Vasicek case, so we

simply state the result b elow, and refer to Ingersoll 1987, pp. 397-399 or Lund

1993, pp. 37{41 for further details. Once again, the b ond-pricing formula takes the

familiar exp onential-ane form 27, with B and A b eing de ned as follows:

2 1 e

B = 51

2 ++ 1 e

2 3

+ =2

2 2 e

4 5

A = log 52

2 + + 1 e

where

2 2

= + +2 : 53

The main advantage over the Vasicek mo del is that r is restricted to be non-

negative. However, for realistic parameter values, there is rarely much di erence

between the yield curves obtained from the Vasicek and CIR mo dels, resp ectively.

5 Multi-factor mo dels

The main advantage of one-factor mo dels is their simplicityas the entire yield curve

is a function of just one state variable. Moreover, this state variable is observable

| at least in principle in practice, we use a short-term interest rate as a proxy.

However, there are several problems with one-factor mo dels. 10

First, the mo del assumes that changes in the yield curve, and hence b ond returns,

are p erfectly correlated across maturities, and not surprisingly this assumption is

easily contradicted by the empirical evidence. Apart from that, the assumption of

p erfect correlation is highly problematic for several \practical" purp oses, for example

Value-at-Risk calculations, and pricing derivatives on interest-rate spreads. The latter

case is discussed by Canabarro 1995. Second, the shap e of the yield curve is severely

restricted. Sp eci cally, the Vasicek and CIR mo dels can only accommo date yield

curve that are monotonic increasing or decreasing and hump ed. An inversely hump ed

yield curve cannot be generated with these mo dels. Moreover, with time-invariant

parameters one-factor mo dels tend to provide a very p o or t to the actual yield curves

observed in the market.

The latter problem can be solved by calibration which is discussed in part II.

By making some parameters time-dep endent, we obtain a p erfect t to the current

initial yield curve, and the calibrated mo del can only b e used to price xed-income

derivatives. If the mo deling purp ose is identifying b onds that are mispriced, the

calibration approach cannot be used. Moreover, since the mo del is extended with

deterministic parameters, yield changes are still p erfectly correlated, so for some

securities a calibrated one-factor mo del may still b e inadequate.

For these reasons, we discuss multi-factor mo dels in the following. Sp eci cally,

the short rate is still governed by sto chastic pro cess with time-invariant parameters,

but there are now, say, m sources of innovation, and not just one as in 7 and 21.

For practical purp oses, this means that we get a b etter but not p erfect t to the

yield curve, and yield curve changes are no longer p erfectly correlated.

5.1 A general framework for multi-factor mo dels

The underlying assumptions of multi-factors mo dels are very similar to the one-factor

case, and the mo di cations relate only to the sto chastic pro cess for the short rate and

the risk premia. For convenience, however, we restate the full list of assumptions:

1. The b ond market is frictionless no taxes, transactions costs, divisibility prob-

lems, etc..

2. Investors prefers more wealth to less implies absence of arbitrage opp ortunities

in the market.

3. All b ond prices are a function of a m 1vector of state variables, denoted X .

Together with the next assumption, this implies that the market prices of risk

at time t, X , are functions of the m state variables.

4. The short rate is a known function of X , i.e. r = r X . In most case, r is the

t t t t

rst element of the vector X , that is r = w X , where w is a vector with one

t t t

as the rst element and zeros elsewhere.

5. The dynamics of the state variables are governed by:

dX = X dt + X dW ; 54

t t t t 11

where X is a m 1 drift vector, X is a m m matrix containing the

volatilities co ecients, and W an m-dimensional Brownian motion. Unless

otherwise noted, we sp ecify X as a diagonal matrix and let the m univariate

Brownian motions in the vector W be correlated. The requisite correlation

co ecients are denoted .

The purp ose of the following analysis is deriving the functional relationship be-

tween the m state variables, X , and the prices of zero-coup on b onds, P t; T , for

all T . As in section 3, we start by deriving a sto chastic pro cess for b ond prices,

including an expression for the exp ected returns on di erent b onds. Next, we use the

economic theory absence of arbitrage to imp ose an APT-like restriction on b ond re-

turns, which requires assumptions ab out the market prices of risk. Finally,we obtain

a PDE for b ond prices, as well as a risk-neutral pro cess for the short rate through

the m state variables.

By an appropriate multivariate version of Ito's lemma, b ond prices can b e shown

to evolve according to

t; T P t; T dW ; 55 dPt; T = t; T P t; T dt +

Pi it P

i=1

where drift is given by

m m m

X X X

@P @ P @P 1

t; T P t; T = X + + X X ; 56

P i i j ij

@X @t 2 @X @X

i i j

i=1 i=1 j =1

and the i'th b ond volatility is given by

t; T P t; T = X : 57

Pi i

Note that = 1 in 56. The exp ected return and the b ond volatilities dep end on the

state variables, but to keep the notation manageable, this dep endence is suppressed

here.

In order to derive the appropriate APT restriction on t; T for di erent maturity

dates T , we construct a p ortfolio consisting of K = m +1 with distinct maturities.

The number of b onds with maturity date T , i = 1;:::;K, is denoted by w . The

i i

instantaneous changes in the value of this p ortfolio, , can be written as:

w dPt; T = d =

k k t

k =1

" "

m K K

X X X

w t; T P t; T dt + w t; T P t; T dW ; 58

k P k k k Pi k k it

i=1

k =1 k =1

where we have interchanged the order of summation between i and k in the second

line of 58. Since there are more b onds than sources of risk, it must be p ossible to 12

cho ose non-zero p ortfolio weights, w , which make the p ortfolio lo cally riskless. This

means that the weights must satisfy m restrictions of the form

w t; T P t; T =0; i=1;:::;m: 59

k Pi k k

k =1

By continuously readjusting the p ortfolio weights, we can ensure that the price dy-

namics of the p ortfolio are always riskless, or deterministic. Absence of arbitrage

requires that the exp ected and realized return is equal to the short rate r | oth-

erwise there is a \free lunch" by either buying or selling the p ortfolio and taking

the opp osite p osition in the money markets b oth investments are lo cally riskless.

Stated otherwise, the exp ected excess return must b e zero,

w P t; T f t; T r g =0 60

k k P k t

k =1

We have shown that, if the vector z = [P t; T w ;:::;Pt; T w ] , with z 6= 0,

1 1 K K

solves the system of equations:

3 2 2 3

P t; T w t; T ::: t; T

1 1 P 1 1 P1 K

7 6 6 7

t; T ::: t; T P t; T w

7 6 6 7

P 2 1 P2 K 2 2

7 6 6 7 A z =0; 61

5 4 4 5

::: ::: ::: :::

t; T ::: t; T Pt; T w

Pm 1 Pm K K K

the same K 1 vector z also solves the larger system:

3 2

2 3

t; T ::: t; T

P 1 1 P1 K

P t; T w

7 6

1 1

7 6

t; T ::: t; T

P 2 1 P2 K

6 7

7 6

P t; T w

6 7

2 2

7 6

::: ::: :::

6 7 A z =0: 62

7 6

4 5

:::

7 6

t; T ::: t; T

5 4

Pm 1 Pm K

Pt; T w

K K

t; T r ::: t; T r

P 1 t P K t

Since 62 is a homogeneous system of equations and z 6= 0, this is only p ossible if the

rank of A is equal to m. If A is non-singular, the solution to 62 is z = A 0=0,

2 2

which is obviously a contradiction. Since A has m +1 rows but rank m, the last

row can be written as a linear combination of the other rows. Moreover, this result

do es not dep end on the sp eci c maturities T , so for any T we have

t; T = r + X t; T ; 63

P t t Pi

i=1

where X is the market price of risk for the i'th state variable factor. Note that

the risk premia can only dep end on X and p ossibly calendar time t, but not on the

maturity dates T or other characteristics of the K securities, for that matter.

This line of reasoning dep ends on K b eing equal to m +1. If K >m+1, we can show that

absence of arbitrage implies rankA = m by noting that A and A have the same nullspace.

2 1 2

This approach is used in pro ofs of the APT theory, see Ross 1976. Johnston 1984 contains an

intro duction to nullspaces. 13

To complete the derivation of b ond prices, we substitute 63 into 56. After

rearranging terms and using the de nition of t; T in 57, we get the following

PDE for b ond prices:

m m

X X

1 @ P

X X +

i j ij

2 @X @X

i j

i=1 j =1

@P @P

f X X X g + rP =0; 64

i i i

@X @t

i=1

with b oundary condition P T; T = 1. As in the one-factor case, cf. section 3, we

can use the Feynman-Kac theorem to represent the solution of the PDE 64 as the

risk-neutral exp ectation

rX ds

P t; T = E e ; 65

where the Q-measure corresp onds to the sto chastic pro cess SDE

dX = fX X X g dt + X dW : 66

t t t t t

Note that the drift and volatility functions of 66 are obtained from the co ecients

of the rst-order and second-order derivatives in 64, resp ectively.

5.2 The Brennan and Schwartz mo del

Thus far, our discussion of multi-factor mo dels may app ear somewhat abstract. For

example, we have not made any attempts to interpret the state vector, X , except

for b eing the driving force of changes in the yield curve. However, once we have

sp eci ed a sto chastic pro cess for the state variables and made assumptions ab out the

risk premia, we can solve the b ond-pricing equation and determine the functional

relationship between P t; T and X , for any maturity date T . Given m di erent

b ond prices i.e., p oints on the yield curve, we can invert the b ond-pricing equation

and express the m state variables in terms of m zero-coup on yields. This approachis

useful in practical implementations of the mo dels, but the interpretation of the state

variables may not b e straightforward.

The last problem suggests that we should use observable state variables when

building term-structure mo dels. Using macro economic variables is an interesting

idea, and in ation an obvious candidate for any nominal term-structure mo del, but

there are several problems. In particular, macro economic variables are observed rela-

tively infrequently monthly, at most, and the data quality is often rather poor due

to measurement problems. Instead, we concentrate on mo deling approaches that

directly identify the state variable with b ond yields for sp eci c maturities. The rst

mo del in this vein is the Brennan and Schwartz 1979 mo del which is a two-factor

We are not saying that macro economic variables, such as in ation, playno role in the deter-

mination of the term structure. However, the relevantvariables, e.g, exp ected future in ation, are

not directly observable, and the b est way to estimate measure these variables is probably to use

information in the yield curve itself. That makes the sp eci cation problem circular. 14

mo del with the short rate, r , and the consol yield, l , as state variables. Under the

t t

true original probability measure, the state variables X = r ;l are governed by

t t t

the general sto chastic pro cess

dr = r ;l dt + r ;l dW 67

t 1 t t 1 t t 1t

dl = r ;l dt + r ;l dW ; 68

t 2 t t 2 t t 2t

and the price of a zero-coup on b ond, P t; T , satis es the PDE:

2 2 2

1 @ P @ P 1 @ P

2 2

r;l + r;l + r;l r;l+

1 2

2 2

2 @r 2 @l @r@l

f r;l r;l r;lg +

1 1 1

@P @P

f r;l r;l r;lg + rP =0; 69

2 2 2

@l @t

sub ject to the b oundary condition P T; T = 1. This PDE involves two risk premia,

r;l and r;l. However, as we show in the following, it is p ossible to eliminate

1 2

r;l and r;l from the PDE, so only one market price of risk parameter must

2 2

be sp eci ed.

If we normalize the continuous coup on of the consol to one, the price V of the

consol b ond is given by:

1 1

l s l s

t t

V = e ds = e = 70

l l

t t

Of course, the consol price, V , must also satisfy the PDE 69. Contrary to the

normal case, the functional relationship between X = r;l and V is known, so we

can simply substitute the requisite partial derivatives into 69. Sp eci cally, since

all partial derivatives with resp ect to r and t vanish, and since @ V =@ l = l and

2 2 3

@ V=@l =2l , we have that

3 2 2 1

l r;l l f r;l r;l r;lg +1rl =0: 71

2 2 2

Note that wehave added one to the left hand side of the PDE in 71 b ecause of the

\dividend" payments, cf. equation 24 in section 3.1. Finally, it follows from 71

that the risk-neutral drift for the consol yield is given by:

1 2 2

r;l r;l r;l = l r;l+l rl: 72

2 2 2

This eliminates and from the PDE, and the resulting b ond-pricing formula only

2 2

dep ends on the parametric sp eci cations of , , , , and .

1 1 2 1

A consol is an annuity or bullet which never matures, so there is no repayment of principal,

only interest rate payments coup ons. Concretely, Brennan and Schwartz 1979 assume that the

b ond makes continuous payments at the annual rate c, and let l b e the continuously comp ounded

consol yield. In practice, this b ond do es not trade in the market, but go o d proxies can usually b e

found, for example consol yields with discrete payments, or p erhaps the yield-to-maturityofa very

long bullet. 15

In fact, there are di erentversions of the BS mo del, but in the following we fo cus

on the version in Brennan and Schwartz 1979, where

d log r = [ log l log r log p] dt + dW 73

t t t 1 1t

dl = r;ldt + l dW ; 74

t 2 2 t 2t

where , p, and are time-invariant parameters. Note that r;l delib erately

1 2 2

is left unsp eci ed. If the market price of risk is sp eci ed as a constant, = ,

1 1

the PDE b ecomes use Ito's lemma to determine the SDE for r in 73

2 2 2

@ P @ P 1 1 @ P

2 2 2 2

r + l + rl +

1 2

2 2

2 @r 2 @l @r@l

@P 1

r logl=r logp+ r +

1 1

@r 2

n o

@P @P

1 2 2

l r;l+l rl + rP =0: 75

@l @t

Unfortunately, there is no closed-form solution for b ond prices, and the PDE can only

solved numerically, e.g., by using nite-di erence approximations. Alternatively, b ond

prices can computed with Monte Carlo simulations of the risk-neutral exp ectation

in 65. Numerical PDE solutions are used in the original pap er by Brennan and

Schwartz 1979, whereas Schwartz and Torous 1989 rely on Monte Carlo simulation

when pricing mortgage-backed securities under the Brennan-Schwartz mo del. Both

techniques are very time-consuming, even with mo dern computing equipment.

5.3 The exp onential-ane class of mo dels

The lack of an analytical solution for b ond prices is the main drawback of the Brennan

and Schwartz mo del. Moreover, the mo del requires the consol yield as input, and this

b ond may not trade in all b ond markets.

In this section we consider a general class of mo dels, called exponential-ane mo d-

els, where a general analytical solution is available [Due and Kan 1996]. For some

parametric sp eci cations, we can obtain a closed-form expression like in the Vasicek

mo del, but in the worst case we will have to solve a system of ordinary di erential

equations numerically, and this can be done very eciently with the Runge-Kutta

metho d. This is esp ecially true for multi-factor mo dels since the complexity of the

numerical solution only increases linearly in the number of state variables, whereas

the complexity of nite-di erence PDE solutions generally increases exp onentially in

the number of state variables.

Due and Kan 1996 prop ose a general class of term-structure mo dels that in-

clude Gaussian mo dels as a sp ecial case. Under the original true probability mea-

sure, the m state variables in the vector X are governed the pro cess

dX = K X dt + CX dW ; 76

t t t t

Actually, there are ab out 4{5 consol b onds in the Danish b ond market, but they trade very

infrequently, i.e., the liquidity is extremely low. 16

where X is a m m diagonal matrix whose i'th diagonal element given by

X = + X : 77

ii t i t

In this setup, the m univariate Brownian motions are indep endent, and the dep en-

dence structure between the innovations to X is captured by the m m matrix C .

Lo osely sp eaking, this means that the m m variance-covariance matrix for changes

in X is given by

2 0

CovdX =C X C dt; 78

t t

with representative elementi; j

[CovdX ] =CovdX ;dX = C C X dt: 79

t ij it jt ik jk t

k=1

We refer to Due and Kan 1996 for a thorough discussion of conditions ensuring

that 76 is a well-de ned sto chastic pro cess. The short rate, or instantaneous interest

rate, r , is sp eci ed as an ane function of X :

t t

r = r X = w + w X = w + w X : 80

t t 0 i it 0 t

i=1

Generally, the vector w consists of either zeros or ones. Finally, to complete the mo del

sp eci cation, we make the following assumptions ab out the market prices of risk:

X =X 81

t t

With these assumptions, the fundamental PDE can b e written as

m m m

X X X

@ P 1

C C X +

ik jk t

2 @X @X

i j

i=1 j =1

k=1

m m m

X X X

@P @P

K X C X + rX P = 0; 82

ik k k ik k

@X @t

i=1

k=1 k =1

or more compactly using matrix algebra and the trace op erator,

h i

1 @ P @P

2 0 2

Tr C X C + K X C X

0 0

2 @X @X @X

+ [w + w X ] P = 0: 83

If A and B are symmetric square m m matrices,

m m m m m

X X X X X

TrAB = fAB g = A B = A B ;

ii ij ji ij ij

i=1 i=1 j =1 i=1 j =1

that is TrAB is a convenientway to write the sum of the pro duct of all m elements in A and B . 17

Following the general idea of section 4, we guess that the solution takes the fol-

lowing form

P ; X = exp [A +B X]; 84

where A is a scalar function, and B anm1vector. In order to verify whether

the solution is of the form 84, we compute the requisite partial derivatives of 84

and substitute these expressions into the PDE 83. If we can obtain an ODE system

de ning A and B , we have demonstrated that the solution is of the form 84.

Moreover, by solving the ODE system, either analytically or by numerical metho ds

Runge-Kutta, we obtain the b ond-pricing formula.

First, we have after straightforward calculations

= B P t; T ; i =1;:::;m 85

@ P

= B B P t; T ; i =1;:::;m j =1;:::;m 86

i j

@X @X

i j

@P @P dB dA

= = + X P t; T : 87

@t @ d d

If we substitute these expressions into the PDE, and collect terms involving X ,

for i = 1;:::;m and the constant one, we obtain the following ODEs | after

rearranging several terms and using the prop erty that TrAB C =TrBC A:

m m

X X

dB 1

0 2 0 0

= [C B ] K B [C B ] w 88

i i i i

d 2

i=1 i=1

m m

X X

1 dA

0 2 0 0

= [C B ] + B K [C B ] w : 89

i i i i 0

d 2

i=1 i=1

0 0

Here, [C B ] refers to the i'th element of the m 1 vector C B . Finding a

general closed-form solution to this ODE system do es not seem to be p ossible, but

many sp ecial cases mo dels can be solved in closed form.

Having derived the ODEs, it is worth emphasizing the sp eci c restrictions which

result in the exp onential-ane b ond price 84. Due and Kan 1996 show that we

obtain 84 if the following conditions hold:

The short rate is an ane function of the state variables, that is r = w + w X .

t 0 t

The risk-neutral drift function vector is ane in X .

The covariances b etween dX and dX for all i; j are ane functions of X . For

i j t

the SDE 76 this holds if X is ane in X , cf. equation 79.

t t

The sto chastic pro cess 76 combined with the risk premia 81 is the most general

sp eci cation satisfying the sucient conditions in Due and Kan 1996. In appli-

cations, further restrictions are often needed to obtain a tractable mo del. 18

In the ab ove setup, nothing is assumed ab out the state variables, and accordingly

they are taken as unobserved variables. As mentioned during the intro ductory re-

marks of section 5.2, we can always invert the b ond-pricing formula and express the

m state variables in terms of m distinct zero-coup on yields. Still, the starting p oint

of the mo deling e ort is an unobserved sto chastic pro cess, and the identi cation with

m b ond yields is only made indirectly. Alternatively, Due and Kan 1996 prop ose

taking m \reference" yields as the state variables, that is sp ecifying the sto chastic pro-

cess under the Q-measure directly for these m yields. This approach | called yield

factor mo dels | facilitates direct identi cation with observable quantities p oints on

the yield curve, but as the state variables are now traded assets, we need to imp ose

parameter restrictions on 76, such that the m b onds are priced correctly, see Due

and Kan 1996. Unfortunately, the requisite parameter restrictions are often quite

complex and therefore dicult to imp ose. In essence, there is a tradeo between

direct interpretation of the state variables and the time-invariant mo del parameters.

5.4 Examples of ane multi-factor mo dels

The three one-factor mo dels in section 4 b elong to the exp onential-ane class. For

mo dels with multiple factors, there are a lot of di erent sp eci cations, and we cannot

provide an exhaustive list. Instead, we o er a few examples from the multi-factor

literature.

5.4.1 Gaussian central tendency mo del

This mo del has b een prop oses by, among others, Beaglehole and Tenney 1991 and

Jegadeesh and Pennacchi 1996. The short rate is governed by the two-factor Gaus-

sian pro cess

dr = r dt + dW 90

t 1 t t 1 1t

d = dt + dW ; 91

t 2 t 2 2t

and the two Brownian motions maybe correlated with correlation co ecient . The

market prices of risks are sp eci ed as constants, and . The central tendency

1 2

mo dels generalized the Vasicek mo del by letting the short rate revert towards a time-

varying sto chastic mean which is governed by a separate pro cess. Sometimes this

feature is referred to as a \double decay" mo del.

The PDE is given by:

2 2 2

1 @ P @ P @P 1 @ P

2 2

+ [ r ] + +

1 2 1 1 1

1 2

2 2

2 @r 2 @ @r@ @r

@P @P

+ [ ]+ rP =0; 92

2 2 2

@ @t

We note, in passing that the same problem applies to the Brennan and Schwartz 1979 mo del.

In section 5.2, we assumed that the price formula for the consol was V = l , but we did not

provide a pro of, and, in general, the result requires further restrictions on the parameters in 73{

74. Apparently, this problem seems to b e ignored by Brennan and Schwartz 1979. 19

sub ject to the b oundary condition P T; T = 1: It is straightforward to verify that

this mo del is exp onential-ane since the mo del is Gaussian. Therefore,

P t; t + = exp [A +B r +B ]: 93

1 t 2 t

If we substitute the requisite partial derivatives into 92 and divide by P on b oth

sides of the equation, get

1 1

2 2 2 2

B + B + B B + B [ r ]

1 2 1 2 1 1 1 1

1 1 2 2

2 2

0 0 0

+B [ ] A B r B r =0: 94

2 2 2 2

1 2

Since 94 must hold for all values of r and , we obtain the following ODE systems

after collecting terms:

B = B 1 95

1 1

B = B B 96

1 1 2 2

1 1

0 2 2 2 2

A = B + B + B B

1 2 1 2

1 1 1 2 2

2 2

B + B ; 97

1 1 1 2 2 2 2

with b oundary initial conditions B 0 = 0, B 0 = 0, and A0 = 0 as P T; T =1

1 2

for all r and . It is p ossible to solve the entire ODE system in closed form, but

t T

for reasons of space we concentrate on B and B . First, note that the ODE

1 2

de ning B is exactly the same as in the Vasicek mo del. This means that

e 1

B = 98

If we substitute 98 into 96, we get another linear ODE, which can be solved by

the same technique as we used in section 4.1:

2 1 2

e 1 e e

B = : 99

2 1 2

Finally, we can substitute 98 and 99 into 97, and A can be calculated by

ordinary integration. Since the expression for A is rather length, it is omitted

here.

5.4.2 Fong-Vasicek sto chastic volatility mo del

Fong and Vasicek 1991 prop ose another extension of the Vasicek mo del, where the

Ornstein-Uhlenbeck pro cess is augmented with a sto chastic volatility factor:

V dW 100 dr = r dt +

t 1t t 1 t

dV = V dt + V dW 101

t 2 t t 2t 20

The correlation co ecient between two Brownian motions is denoted . Fong and

Vasicek 1991 sp ecify the market prices of risk as

V; i =1;2 102 =

i i

since this is the only sp eci cation which preserves the ane prop erty under the Q-

measure. With these assumptions, the PDE b ecomes

2 2 2

@ P @ P 1 1 @ P @P

V + V + V + [ r V ]

1 1

2 2

2 @r 2 @V @r@V @V

@P @P

+ [ V V ]+ rP =0: 103

2 2

@V @t

The solution is of the form

P t; t + = exp [A +B r +B V]: 104

1 2

After substituting the requisite partial derivatives of 104 into the PDE and collecting

terms, we get the following ODE system de ning the functions A , B and B :

1 2

B = B 1 105

1 1

0 2 2 2

B = B + B + B B

1 2

2 1 2

2 2

B + B 106

1 1 2 2 2

A = B + B : 107

1 1 2 2

The solution for B isthe same as in the Vasicek mo del. Closed-form expressions

for B and A are presented in Selby and Strickland 1995. The expressions

are quite complicated | involving in nite-order series expansions | so it might be

worthwhile to consider solving the ODEs numerically instead.

5.4.3 Multi-factor CIR mo dels

Neither the Gaussian central-tendency mo del nor the Fong-Vasicek sto chastic volatil-

ity mo del restrict the short rate to be non-negative. A p opular multi-factor mo del

with this prop erty is the m-factor CIR mo del which is obtained by adding m inde-

p endent square ro ot pro cesses,

dr = y 108

t it

i=1

dy = y dt + y dW ; 109

it i i it i it it

and the market price of risk for the i'th factor is sp eci ed as in the one-factor CIR

mo del, that is

y : 110 = =

it i i i 21

The easiest way to derive an expression for b ond prices is using risk-neutral exp ecta-

tions

y ds

i=1

P t; T = E e ; 111

where y evolves according to

dy = f y y g dt + y dW 112

it i i it i it i it

under the Q-measure. By interchanging the order of integration and summation,

equation 111 maybe rewritten as

y ds

i=1

P t; T = E e

y ds

= E e

i=1

y ds

E e 113 =

i=1

= P t; T ; 114

i=1

where P t; T is the price formula for a one-factor CIR mo del with parameters ,

i i

, and , as well as \short rate" y . Note that the third line follows b ecause of

i i i it

indep endence between the m square-ro ot pro cesses. 22

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