logues to E(xxc ), etc. that were inputs to the one period problem. Therefore, a differential

or moment-to-moment characterization of the bound will tell us µC and dΛin terms of σCz

and σ Cw .

Theorem:The lower bound discount factor Λt follows

dΛ—

dΛ

(246)

= — ’ v dw

Λ Λ

314

SECTION 18.3 MULTIPLE PERIODS AND CONTINUOUS TIME

and µC , σ Cz and σCw satisfy the restriction

µ— ¶

xc 1 dΛ

(247)

σCz dz + vσ0

µC + ’ r = ’ Et Cw

—

C dt Λ

where

r

1 dΛ—2 σCw

(248)

Et —2 q

A2 ’

v=

dt Λ σ Cw σ 0 Cw

The upper bound process C t and discount factor Λt have the same representation

with v = ’v.

This theorem has the same geometric interpretation as shown in Figure 35. dΛ— /Λ— is the

combination of basis asset shocks that prices the basis assets by construction, in analogy to

x— . The term σCw dw corresponds to the error w, and σCw σ0 corresponds to E(w2 ). The

Cw

proposition looks a little different because now we choose a vector v rather than a number. We

could de¬ne a residual σCw dw and then the problem would reduce to choosing a number, the

loading of dΛ on this residual. It is not convenient to do so in this case since σCw potentially

changes over time. In the geometry of Figure 35, the w direction may change over time. The

algebraic proof just follows this logic.

Proof: Substituting equation (18.243) into the problem (18.241) in order to impose

the pricing constraint, the problem is

· ¸ µ ¶ µ —2 ¶

xc d(Λ— C) dC 1 dΛ

s.t. vv ¤ A ’ Et

0 2

0 = dt + Et ’ min vEt dw .

Λ— C Λ—2

C C dt

{v}

Using equation (18.245) for dC/C in the last term, the problem is

· ¸ µ —2 ¶

xc d(Λ— C)

1 1 dΛ

’ min vσ 0 s.t. vv0 ¤ A2 ’ Et (249)

0= + Et .

Cw

Λ— C Λ—2

C dt dt

{v}

This is a linear objective in v with a quadratic constraint. Therefore, as long as

σCw 6= 0, the constraint binds and the optimal v is given by (18.248). v = ’v gives

the maximum since σ Cw σ0 > 0. Plugging the optimal value for v in (18.249)

Cw

gives

· ¸

xc d (Λ— C)

1

’ vσ 0 .

0= + Et Cw

—C

C dt Λ

For clarity, and exploiting the fact that dΛ— does not load on dw, write the middle

term as

· ¸ µ— ¶

d (Λ— C)

1 1 dΛ

Et = µC ’ r + Et σCz dz

Λ— C Λ—

dt dt

315

CHAPTER 18 OPTION PRICING WITHOUT PERFECT REPLICATION

If σ Cw = 0, any v leads to the same price bound. In this case we can most simply

¥

take v = 0.

As in the discrete-time case, we can plug in the de¬nition of Λ— to obtain explicit, if less

intuitive, expressions for the optimal discount factor and the resulting lower bound,

q

σCw

dΛ

= ’rdt + µS ΣS σS dz ’ A2 ’ µ0 Σ’1 µS q

’1

0

(250)

˜ ˜S S ˜ dw

Λ 0

σ Cw σ Cw

q q

xc

’ r = µ0 Σ’1 σ S σCz + 2 ’ µ0 Σ’1 µ (251)

˜ S S ˜ S σCw σ0 .

µC + ˜S S A Cw

C

A partial differential equation

Now we are ready to apply the standard method; ¬nd a partial differential equation and

solve it backwards to ¬nd the price at any date. The method proceeds exactly as for the Black-

Scholes formula: Guess a solution C(S, V, t). Use Ito™s lemma to derive expressions for µC

and σ Cz , σCw in terms of the partial derivatives of C(S, V, t). Substitute these expressions

into restriction (18.251). The result is ugly, but straightforward to evaluate numerically. Just

like the Black-Scholes partial differential equation, it expresses the time derivative ‚C/‚t in

terms of derivatives with respect to state variables, and thus can be used to work back from a

terminal period.

Theorem. The lower bound C(S, V, t) is the solution to the partial differential equa-

tion

‚C

xc ’ rC + +

‚t

1 X ‚2C 1 X ‚ 2C X ‚ 2C

0 0 0

Si σSi σ0 zj =

+ Si Sj σSi σ Sj + (σ V zi σ V zj + σ V wj σV wj ) + V

2 i,j ‚Si ‚Sj 2 i,j ‚Vi ‚Vj ‚Si ‚Vj

i,j

µ ¶0 q q

¡ 0 ’1 ¢

D

(SC S ) + µS ΣS σ S σV z ’ µV C V + A2 ’ µS ΣS µS C 0 σV w σ 0 w C V

’1

0 0 0

= ’r ˜ ˜ ˜ V V

S

subject to the boundary conditions provided by the focus asset payoff xc . C V de-

T

notes the vector with typical element ‚C/‚Vj and (SC S ) denotes the vector with

typical element Si ‚C/‚Si . Replacing + with ’ before the square root gives the

partial differential equation satis¬ed by the upper bound.

The discount factor

316

SECTION 18.4 EXTENSIONS, OTHER APPROACHES, AND BIBLIOGRAPHY

In general, the Λ process (18.246) or (18.250) depends on the parameters σCw . Hence,

without solving the above partial differential equation we do not know how to spread the

loading of dΛ across the multiple sources of risk dw whose risk prices we do not observe.

Equivalently, we do not know how to optimally spread the total market price of risk across the

elements of dw. Thus, in general we cannot use the integration approach“ solve the discount

factor forward “ to ¬nd the bound by

µ ¶

ZT

Λs c ΛT c

C t = Et xs ds + Et x .

Λt T

s=t Λt

However, if there is only one shock dw, then we don™t have to worry about how the loading

of dΛ spreads across multiple sources of risk. v can be determined simply by the volatility

constraint. In this special case, dw and σCw are scalars. Hence equation (18.246) simpli¬es

as follows:

Theorem: In the special case that there is only one extra noise dw driving the V

process, we can ¬nd the lower bound discount factor Λ from directly from

q

dΛ

= ’rdt ’ µS ΣS σS dz ’ A2 ’ µ0 Σ’1 µS dw.

0 ’1

(252)

˜ ˜S S ˜

Λ

I used this characterization to solve for the case of a non-traded underlying in the last

section. In some applications, the loading of dΛ on multiple shocks dw may be constant over

time. In such cases, one can again construct the discount factor and solve for bounds by

(possibly numerical) integration, avoiding the solution of a partial differential equation.

18.4 Extensions, other approaches, and bibliography

The roots of the good deal good deal idea go a long way back. Ross (1976) bounded APT

residuals by assuming that no portfolio can have more than twice the market Sharpe ratio, and

I used the corresponding idea that discount factor volatility should be bounded to generate

a robust approximate APT in Chapter 9.4. Good deal bounds apply the same idea to option

payoffs. However, the good deal bounds also impose positive discount factors, and this

constraint is important in an option pricing context. We also study dynamic models that

chain discount factors together as in the option pricing literature.

The one-period good-deal bound is the dual to the Hansen-Jagannathan (1991) bound with

positivity “ Hansen and Jagannathan study the minimum variance of positive discount factors

that correctly price a given set of assets. The good deal bound interchanges the position of

the option pricing equation and the variance of the discount factor. The techniques for solving

the bound, therefore, are exactly those of the Hansen-Jagannathan bound in this one-period

setup.

There is nothing magic about discount factor volatility. This kind of problem needs weak

317

CHAPTER 18 OPTION PRICING WITHOUT PERFECT REPLICATION

but credible discount factor restrictions that lead to tractable and usefully tight bounds. Sev-

eral other similar restrictions have been proposed in the literature.

1) Levy (1985) and Constantinides (1998) assume that the discount factor declines mono-

tonically with a state variable; marginal utility should decline with wealth.

2) The good deal bounds allow the worst case that marginal utility growth is perfectly

correlated with a portfolio of basis and focus assets. In many cases one could credibly

impose a sharper limit than ’1 ¤ ρ ¤ 1 on this correlation to obtain tighter bounds.

3) Bernardo and Ledoit (1999) use the restriction a ≥ m ≥ b to sharpen the no-arbitrage

restriction ∞ ≥ m > 0. They show that this restriction has a beautiful portfolio interpretation

“ a < m < b corresponds to limited “gain - loss ratios” just as σ(m)/E(m) corresponds to

limited Sharpe ratios. De¬ne [Re ]+ = max(Re , 0) and [Re ]’ = ’ min(’Re , 0) as the

“gains and losses” of an excess return Re . Then,

[Re ]+ sup(m)

(253)

maxe = min e

’

{m:0=E(mR )} inf(m)

{Re ∈R } [Re ]

(The sup and inf ignore measure zero states.) This is exactly analogous to

|E(Re )| σ(m)

maxe = min e

} σ(Re ) {m:0=E(mR )} E(m)

e

{R ∈R

and hints at an interesting restatement of asset pricing theory in L1 with sup norm rather than

L2 with second moment norm.

Since m ≥ a, the call option price generated by this restriction in a one-period model

is strictly greater than the lower arbitrage bound generated by m = 0; as in this case, the

gain-loss bound can improve on the good-deal bound.

4) Bernardo and Ledoit also suggest a ≥ m/y ≥ b where y is an explicit discount

factor model such as the consumption-based model or CAPM, as a way of imposing a “weak

implication” of that particular model.

These alternatives are really not competitors. Add all the discount factor restrictions that

are appropriate and useful for a given problem.

This exercise seems to me a strong case for discount factor methods as opposed to port-

folio methods. The combination of positivity and volatility constraints on the discount factor

leads to a sharper bound than the intersection of no-arbitrage and limited Sharpe ratios. I

don™t know of a simple portfolio characterization of the set of prices that are ruled out by the

good deal bound when both constraints bind. The same will be true as we add, say, gain-loss

restrictions, monotonicity restrictions, etc.

In continuous time, option pricing and terms structure problems increasingly feature as-

sumptions about the “market price of risk” of the non-traded shocks. The good-deal bounds

treat these rather formally; they choose the market prices of risks at each instant to minimize

or maximize the option price subject to a constraint that the total market price of risk is less

318

SECTION 18.5 PROBLEMS

than a reasonable value, compared to the Sharpe ratios of other trading opportunities. One

needn™t be this formal. Many empirical implementations of option pricing and term structure

models feature unbelievable sizes and time-variation in market prices of risk. Just imposing

sensible values for the market prices of risk, and trying on a range of sensible values may be

good enough for many practical situations.

The continuous-time treatment has not yet been extended to the important case of jumps

rather than diffusion processes. With jumps, both the positivity and volatility constraints will

bind.

18.5 Problems

1. Prove (18.253),

[Re ]+ sup(m)

maxe ’= min e

{m:0=E(mR )} inf(m)

{Re ∈R } [Re ]

Start with a ¬nite state space.

2. Binomial models are very popular in option pricing. This simple problem illustrates the

technique. A stock currently selling at S will either rise to ST = uS with probability

πu or decline to ST = dS with probability π d , paying no dividends in the interim. There

is a constant gross interest rate Rf .

(a) Find a discount factor that prices stock and bond. This means, ¬nd its value in each

state of nature.

(b) Use the discount factor to price a call option one step before expiration. Express

your results as an expected value using risk-neutral probabilities

(c) Do the same thing two steps before expiration.

(d) Cox, Ross and Rubinstein (1979) derive these formulas by setting up a hedge

portfolio of stocks and bonds, and ¬nding portfolio weights to exactly synthesize the

option. Rederive your result with this method.

319

Chapter 19. Term structure of interest

rates

Term structure models are particularly simple, since bond prices are just the expected value

of the discount factor. In equations, the price at time t of a zero coupon bond that comes

(j)

due at time t + j is Pt = Et (mt,t+j ). Thus, once you specify a time-series process for a

one-period discount factor mt,t+1 , you can in principle ¬nd the price of any bond by chaining

(j)

together the discount factors and ¬nding Pt = Et (mt,t+1 mt+1,t+2 ...mt+j’1,t+j ). As with

option pricing models, this chaining together can be hard to do, and much of the analytical

machinery in term structure models centers on this technical question. As with option pricing

models, there are two equivalent ways to do the chaining together: Solve the discount factor

forward and take an integral, or ¬nd a partial differential equation for prices and solve it

backwards from the maturity date.

19.1 De¬nitions and notation

A quick introduction to bonds, yields, holding period returns, forward rates, and swaps.

(N)

= log price of N period zero-coupon bond at time t.

pt

y (N) = ’ N p(N) = log yield.

1

(N) (N’1) (N)

= log holding period return.

hprt+1 = pt+1 ’ pt

dP (N,t) 1 ‚P (N,t)

= instantaneous return.

hpr = ’ ‚N dt

P P

(N’N+1) (N) (N+1)

= forward rate.

ft = pt ’ pt

‚P (N,t)

= instantaneous forward rate.

1

f(N, t) = ’ P ‚N

19.1.1 Bonds

The simplest ¬xed-income instrument is a zero-coupon bond. A zero-coupon bond is a

promise to pay one dollar (a nominal bond) or one unit of the consumption good (a real

(3)

bond) on a speci¬ed date. I use a superscript in parentheses to denote maturity: Pt is the

price of a three year zero-coupon bond. I will suppress the t subscript when it isn™t necessary.

(N) (N)

I denote logs by lowercase symbols, pt = ln Pt . The log price has a nice interpre-

tation. If the price of a one-year zero coupon bond is 0.95, i.e. 95/ per dollar face value, the

c

log price is ln(0.95) = ’0.051. This means that the bond sells at a 5% discount. Logs also

320

SECTION 19.1 DEFINITIONS AND NOTATION

give the continuously compounded rate. If we write erN = 1/P (N) then the continuously

compounded rate is rN = ’ ln P (N) .

Coupon bonds are common in practice. For example, a $100 face value 10 year coupon

bond may pay $5 every year for 10 years and $100 at 10 years. (Coupon bonds are often

issued with semiannual or more frequent payments, $2.50 every six months for example.)

We price coupon bonds by considering them as a portfolio of zeros.

Yield. The yield of a bond is the ¬ctional, constant, known, annual, interest rate that

justi¬es the quoted price of a bond, assuming that the bond does not default. It is not the rate

of return of the bond. From this de¬nition, the yield of a zero coupon bond is the number

Y (N) that satis¬es

1

P (N) = £ ¤N .

Y (N)

Hence

1 1

Y (N) = £ (N)

= ’ p(N) .

1; y

¤ N

P (N) N

The latter expression nicely connects yields and prices. If the price of a 4 year bond is -0.20

or a 20% discount, that is 5% discount per year, or a yield of 5%. The yield of any stream of

cash ¬‚ows is the number Y that satis¬es

N

X CFj

P= .

Yj

j=1

In general, you have to search for the value Y that solves this equation, given the cash ¬‚ows

and the price. So long as all cash ¬‚ows are positive, this is fairly easy to do.

As you can see, the yield is just a convenient way to quote the price. In using yields we

make no assumptions. We do not assume that actual interest rates are known or constant; we

do not assume the actual bond is default-free. Bonds that may default trade at lower prices or

higher yields than bonds that are less likely to default. This only means a higher return if the

bond happens not to default.

19.1.2 Holding Period Returns

If you buy an N period bond and then sell it”it has now become an N ’1 period bond”you

achieve a return of

(N’1)

P

$back

(N)

= t+1 (254)

HP Rt+1 = (N)

$paid Pt

321

CHAPTER 19 TERM STRUCTURE OF INTEREST RATES

or, of course,

(N) (N’1) (N)

hprt+1 = pt+1 ’ pt .

We date this return (from t to t + 1) as t + 1 because that is when you ¬nd out its value. If

this is confusing, take the time to write returns as HP Rt’t+1 and then you™ll never get lost.

In continuous time, we can easily ¬nd the instantaneous holding period return of bonds

with ¬xed maturity date P (T, t)

P (T, t + ∆) ’ P (T, t)

hpr =

P (T, t)

and, taking the limit,

dP (T, t)

hpr = .

P

However, it™s nicer to look for a bond pricing function P (N, t) that ¬xes the maturity rather

than the date. As in (19.254), we have to account for the fact that you sell bonds that have

shorter maturity than you buy.

P (N ’ ∆, t + ∆) ’ P (N, t)

hpr =

P (N, t)

P (N ’ ∆, t + ∆) ’ P (N, t + ∆) + P (N, t + ∆) ’ P (N, t)

=

P (N, t)

and, taking the limit

dP (N, t) 1 ‚P (N, t)

hpr = (255)

’ dt

P P ‚N

19.1.3 Forward rate

The forward rate is de¬ned as the rate at which you can contract today to borrow or lend

money starting at period N, to be paid back at period N + 1.

You can synthesize a forward contract from a spectrum of zero coupon bonds, so the

forward rate can be derived from the prices of zero-coupon bonds. Here™s how. Suppose you

buy one N period zero and simultaneously sell x N + 1 period zero coupon bonds. Let™s

track your cash ¬‚ow at every date.

322

SECTION 19.1 DEFINITIONS AND NOTATION

Buy N-Period zero Sell x N+1 Period zeros Net cash ¬‚ow

Today 0: ’P (N) +xP (N+1) xP (N+1) ’ P (N)

Time N: 1 1

Time N+1: -x -x

Now, choose x so that today™s cash ¬‚ow is zero:

P (N)

x = (N+1)

P

You pay or get nothing today, you get $1.00 at N , and you pay P (N) /P (N+1) at N + 1. You

have synthesized a contract signed today for a loan from N to N + 1”a forward rate! Thus,

(N)

Pt

(N’N+1)

= Forward rate at t for N ’ N + 1 =

Ft (N+1)

Pt

and of course

(N’N+1) (N) (N+1)

(256)

ft = pt ’ pt .

(N)

People sometimes identify forward rates by the initial date, ft , and sometimes by the

(N+1)

ending date, ft . I use the arrow notation when I want to be really clear about dating a

return.

Forward rates have the lovely property that you can always express a bond price as its

discounted present value using forward rates,

(N) (N) (N’1) (N’1) (N’2) (2) (1) (1)

pt = pt ’ pt + pt ’ pt ’ ...pt ’ pt + pt

(N’1’N) (N’2’N’1) (1’2) (1)

= ’ft ’ ft ’ ... ’ ft ’ yt

(1) (0’1)

(yt of course), so

= ft

« ’1

N’1

Y

P N’1 (j’j+1)

(N)

(N) (j’j+1)

=

= e’ ft

= ept

Pt Ft .

j=0

j=0

Intuitively, the price today must be equal to the present value of the payoff at rates you can

lock in today.

In continuous time, we can de¬ne the instantaneous forward rate

1 ‚P (N, t) ‚p(Nt )

(257)

f(N, t) = ’ =’

P ‚N ‚N

Then, forward rates have the same property that you can express today™s price as a discounted

323

CHAPTER 19 TERM STRUCTURE OF INTEREST RATES

value using the forward rate,

Z N

p(N, t) = ’ f(x, t)dx

x=0

RN

’ f (x,t)dx

P (N, t) = e .

x=0

Equations 19.256 and 19.257 express forward rates as derivatives of the price vs. maturity

curve. Since yield is related to price, we can relate forward rates to the yield curve directly.

Differentiating the de¬nition of yield y(N, t) = ’p(N, t)/N,

‚y(N, t) 1 1 ‚p(N, t) 1 1

= 2 p(N, t) ’ = ’ y(N, t) + f(N, t)

‚N N N ‚N N N

Thus,

‚y(N, t)

f(N, t) = y(N, t) + N .

‚N

In the discrete case, (19.256) implies

³ ´

(N’N+1) (N) (N+1) (N+1) (N+1) (N)

ft = ’N yt + (N + 1)yt = yt + N yt ’ yt .

19.1.4 Swaps and options

Swaps are an increasingly popular ¬xed income instrument. The simplest example is a ¬xed-

for-¬‚oating swap. Party A may have issued a 10 year ¬xed coupon bond. Party B may have

issued a 10 year variable rate bond “ a bond that promises to pay the current one year rate.

(For example, if the current rate is 5%, the variable rate issuer would pay $5 for every $100

of face value. A long-term variable rate bond is the same thing as rolling over one-period

debt.) They may be unhappy with these choices. For example, the ¬xed-rate payer may not

want to be exposed to interest rate risk that the present value of his promised payments rises

if interest rates decline. The variable-rate issuer may want to take on this interest rate risk,

betting that rates will rise or to hedge other commitments. If they are unhappy with these

choices, they can swap the payments. The ¬xed rate issuer pays off the variable rate coupons,

and the variable rate issuer pays off the ¬xed rate coupons. Obviously, only the difference

between ¬xed and variable rate actually changes hands.

Swapping the payments is much safer than swapping the bonds. If one party defaults, the

other can drop out of the contract, losing the difference in price resulting from intermediate

interest rate changes, but not losing the principal. For this reason, and because they match the

patterns of cash¬‚ows that companies usually want to hedge, swaps have become very popular

tools for managing interest rate risk. Foreign exchange swaps are also popular: Party A may

swap dollar payments for party B™s yen payments. Obviously, you don™t need to have issued

324

SECTION 19.2 YIELD CURVE AND EXPECTATIONS HYPOTHESIS

the underlying bonds to enter into a swap contract “ you simply pay or receive the difference

between the variable rate and the ¬xed rate each period.

The value of a pure ¬‚oating rate bond is always exactly one. The value of a ¬xed rate bond

varies. Swaps are set up so no money changes hands initially, and the ¬xed rate is calibrated

so that the present value of the ¬xed payments is exactly one. Thus, the “swap rate” is the

same thing as a the yield on a comparable coupon bond.

Many ¬xed income securities contain options, and explicit options on ¬xed income secu-

rities are also popular. The simplest example is a call option. The issuer may have the right to

buy the bonds back at a speci¬ed price. Typically, he will do this if interest rates fall a great

deal, making a bond without this option more valuable. Home mortgages contain an interest-

ing prepayment option: if interest rates decline, the homeowner can pay off the loan at face

value, and re¬nance. Options on swaps also exist; you can buy the right to enter into a swap

contract at a future date. Pricing all of these securities is one of the tasks of term structure

modeling.

19.2 Yield curve and expectations hypothesis

The expectations hypothesis is three equivalent statements about the pattern of yields

across maturity,

1. The N period yield is the average of expected future one-period yields

2. The forward rate equals the expected future spot rate.

3. The expected holding period returns are equal on bonds of all maturities.

The expectations hypothesis is not quite the same thing as risk neutrality, since it ignores

1/2σ 2 terms that arise when you move from logs to levels.

The yield curve is a plot of yields of zero coupon bonds as a function of their maturity.

Usually, long-term bond yields are higher than short therm bond yields “ a rising yield curve

“ but sometimes short yields are higher than long yields “ an inverted yield curve. The

yield curve sometimes has humps or other shapes as well. The expectations hypothesis is the

classic theory for understanding the shape of the yield curve.

More generally, we want to think about the evolution of yields “ the expected value and

conditional variance of next period™s yields. This is obviously the central ingredients for

portfolio theory, hedging, derivative pricing, and economic explanation. The expectations

hypothesis is the traditional benchmark for thinking about the expected value of future yields.

We can state the expectations hypothesis in three mathematically equivalent forms:

325

CHAPTER 19 TERM STRUCTURE OF INTEREST RATES

1. The N period yield is the average of expected future one-period yields

³ ´

1

(N) (1) (1) (1) (1)

yt = Et yt + yt+1 + yt+2 + ... + yt+N’1 (+ risk premium) (258)

N

2. The forward rate equals the expected future spot rate

³ ´

(1)

N’N+1

= Et yt+N (+ risk premium) (259)

ft

3. The expected holding period returns are equal on bonds of all maturities

(N) (1)

(+ risk premium). (260)

Et (hprt+1 ) = yt

You can see how the expectations hypothesis explains the shape of the yield curve. If the

yield curve is upward sloping “ long term bond yields are higher than short term bond yields

“ the expectations hypothesis says this is because short term rates are expected to rise in the

future. You can view the expectations hypothesis as a response to a classic misconception.

If long term yields are 10% but short term yields are 5%, an unsophisticated investor might

think that long-term bonds are a better investment. The expectations hypothesis shows how

this may not be true. Future short rates are expected to rise: this means that you will roll

over the short term bonds at a really high rate, say 20%, giving the same long-term return.

When the short term interest rates rise in the future, long-term bond prices decline. Thus, the

long-term bonds will only give a 5% rate of return for the ¬rst year.

You can see from the third statement that the expectations hypothesis is roughly the same

as risk-neutrality. If we had said that the expected level of returns was equal across maturities,

that would be the same as risk-neutrality. The expectations hypothesis speci¬es that the

expected log return is equal across maturities. This is typically a close approximation to risk

12

neutrality, but not the same thing. If returns are log-normal, then E(R) = eE(r)+ 2 σ (r) . If

mean returns are about 10% or 0.1 and the standard deviation of returns is about 0.1, then

1/2σ 2 is about 0.005, which is very small but not zero. We could easily specify risk-neutrality

in the third expression of the expectations hypothesis, but then it would not imply the other

two “ 1 σ 2 terms would crop up.

2

The intuition of the third form is clear: risk-neutral investors will adjust positions until

the expected one-period returns are equal on all securities. Any two ways of getting money

from t to t + 1 must offer the same expected return. The second form adapts the same idea

to the choice of locking in a forward contract vs. waiting and borrowing and lending at the

spot rate. Risk-neutral investors will load up on one or the other contract until the expected

returns are the same. Any two ways of getting money from t + N to t + N + 1 must give the

same expected return.

The ¬rst form re¬‚ects a choice between two ways of getting money from t to N. You can

buy a N period bond, or roll-over N one-period bonds. Risk neutral investors will choose

one over the other strategy until the expected N ’ period return is the same.

326

SECTION 19.3 TERM STRUCTURE MODELS “ A DISCRETE-TIME INTRODUCTION

The three forms are mathematically equivalent. If every way of getting money from t to

t + 1 gives the same expected return, then so must every way of getting money from t + 1 to

t + 2, and, chaining these together, every way of getting money from t to t + 2.

For example, let™s show that forward rate = expected future spot rate implies the yield

curve. Start by writing

(1)

ftN’1’N = Et (yt+N’1 ).

Add these up over N,

³ ´

(1) (1) (1) (1)

+ ftN’2’N’1 + ftN’1’N

ft0’1 + ft1’2 + ... = Et yt + yt+1 + yt+2 + ... + yt+N’1 .

The right hand side is already what we™re looking for. Write the left hand side in terms of the

de¬nition of forward rates, remembering P (0) = 1 so p(0) = 0,

³ ´³ ´ ³ ´

(0) (1) (1) (2) (N’1) (N)

N’2’N’1 N’1’N

0’1 1’2

ft + ft + ... + ft + ft = pt ’ pt + pt ’ pt + ... + pt ’ pt

(N) (N)

= ’pt = Nyt .

You can show all three forms are equivalent by following similar arguments. (This is a

great problem.)

It is common to add a constant risk premium and still refer to the resulting model as the

expectations hypothesis, and I include a risk premium in parentheses to remind you of this

idea. One end of each of the three statements does imply more risk than the other. A forward

rate is known while the future spot rate is not. Long-term bond returns are more volatile than

short term bond returns. Rolling over short term real bonds is a riskier long-term investment

than buying a long term real bond. If real rates are constant, and the bonds are nominal, then

the converse can hold: short term real rates can adapt to in¬‚ation, so rolling over short nominal

bonds can be a safer long-term real investment than long-term nominal bonds. These risks

will generate expected return premia if they covary with the discount factor, and our theory

should re¬‚ect this fact.

If you allow an arbitrary, time-varying risk premium, the model is a tautology, of course.

Thus, the entire content of the “expectations hypothesis” augmented with risk premia is in

the restrictions that the risk premium is constant over time. We will see that the constant risk

premium model does not do that well empirically. One of the main points of term structure

models is “ or at least ought to be “ to quantify the size and movement over time in the risk

premium.

19.3 Term structure models “ a discrete-time introduction

327

CHAPTER 19 TERM STRUCTURE OF INTEREST RATES

Term structure models specify the evolution of the short rate and potentially other state

variables, and the prices of bonds of various maturities at any given time as a function of

the short rate and other state variables. I examine a very simple example based on an AR(1)

for the short rate and the expectations hypothesis, which gives a geometric pattern for the

yield curve. A good way to generate term structure models is to write down a process for

the discount factor, and then price bonds as the conditional mean of the discount factor. This

procedure guarantees the absence of arbitrage. I give a very simple example of an AR(1)

model for the log discount factor, which also results in geometric yield curves. .

A natural place to start in modeling the term structure is to model yields statistically. You

might run regressions of changes in yields on the levels of lagged yields, and derive a model

of the mean and volatility of yield changes. You would likely start with a factor analysis

of yield changes and express the covariance matrix of yields in terms of a few large factors

that describe their common movement. The trouble with this approach is that you can quite

easily reach a statistical representation of yields that admits an arbitrage opportunity, and

you would not want to use such a statistical characterization for economic understanding of

yields, for portfolio formation, or for derivative pricing. For example, a statistical analysis

would strongly suggest that a ¬rst factor should be a “level” factor, in which all yields move

up and down together. It turns out that this assumption violates arbitrage: the long-maturity

yield must converge to a constant. (More precisely, the long-term forward rate, if it exists,

must never fall.)

How do you model yields without arbitrage? An obvious solution is to use the discount

factor existence theorem: Write a statistical model for a positive discount factor, and ¬nd

bond prices as the expectation of this discount factor. Such a model will be, by construction,

arbitrage free. Conversely, any arbitrage-free distribution of yields can be captured by some

positive discount factor, so you don™t lose any generality with this approach.

19.3.1 A term structure model based on the expectations hypothesis

We can use the expectations hypothesis to give the easiest example of a term structure model.

(This one does not start from a discount factor and so may not be arbitrage-free.) Suppose the

one-period yield follows an AR(1),

(1) (1)

yt+1 ’ δ = ρ(yt ’ δ) + µt+1 .

Now, we can use the expectations hypothesis (19.258) to calculate yields on bonds of all

328

SECTION 19.3 TERM STRUCTURE MODELS “ A DISCRETE-TIME INTRODUCTION

maturities as a function of today™s one period yield,

1 h (1) i

(2) (1)

yt = Et yt + yt+1

2

1 h (1) i

(1)

= y + δ + ρ(yt ’ δ)

2t

1 + ρ (1)

= δ+ (yt ’ δ)

2

Continuing in this way,

³ ´ 1 1 ’ ρN+1 (1)

(N)

(261)

yt ’ δ = (yt ’ δ).

N 1’ρ

You can see some features that will recur throughout the term structure models. First, the

model (19.261) can describe some movements in the yield curve over time. If the short rate is

below its mean, then there is a smoothly upward sloping yield curve. Long term bond yields

are higher as short rates are expected to increase in the future. If the short rate is above its

mean, we get a smoothly inverted yield curve. This particular model cannot produce humps

or other interesting shapes that we sometimes see in the term structure. Second, this model

(N) (1)

predicts no average slope of the term structure “ E(yt ) = E(yt ) = δ. In fact, the average

term structure seems to slope up slightly. Third, all bond yields move together in the model.

If we were to stack the yields up in a VAR representation, it would be

³ ´

(1) (1)

yt+1 ’ δ = ρ yt ’ δ + µt+1

³ ´

(2) (2) 1+ρ

yt+1 ’ δ = ρ yt ’ δ + 2 µt+1 .

... ³ ´ N +1

(N) (N)

+ N 1’ρ

1

yt+1 ’ δ = ρ yt ’ δ 1’ρ µt+1

(1)

(You can write the right hand variable in terms of yt if you want “ any one yield carries the

same information as any other.) The error terms are all the same. We can add more factors to

the short rate process, to improve on this prediction, but most tractable term structure models

maintain less factors than there are bonds, so some perfect factor structure is a common

prediction of term structure models. Finally, this model has a problem in that the short rate,

following an AR(1), can be negative. Since people can always hold cash, nominal short rates

are never negative, so we want to start with a short rate process that does not have this feature.

With this simple model in hand, you can see some obvious directions for generalization.

First, we will want more complex driving processes than an AR(1). For example, a hump-

shape in the conditionally expected short rate will result in a hump-shaped yield curve. If

there are multiple state variables driving the short rate, then we will have multiple factors

driving the yield curve which will also result in more interesting shapes. We also want pro-

cesses that keep the short rate positive in all states of nature. Second, we will want to add

329

CHAPTER 19 TERM STRUCTURE OF INTEREST RATES

some “market prices of risk” “ some risk premia. This will allow us to get average yield

curves to not be ¬‚at, and time-varying risk premia seem to be part of the yield data.

The yield curve literature proceeds in exactly this way: specify a short rate process and

the risk premia, and ¬nd the prices of long term bonds. The trick is to specify suf¬ciently

complex assumptions to be interesting, but preserve our ability to solve the models.

19.3.2 The simplest discrete time model

The simplest nontrivial model I can think of is to let the log of the discount factor follow an

AR(1) with normally distributed shocks. I write the AR(1) for the log rather than the level in

order to make sure the discount factor is positive, precluding arbitrage. Log discount factors

are typically slightly negative, so I denote the unconditional mean E (ln m) = ’δ

(ln mt+1 + δ) = ρ (ln mt + δ) + µt+1 .

In turn, you can think of this discount factor model as arising from a consumption based

power utility model with normal errors.

µ ¶γ

Ct+1

= e’δ

mt+1

Ct

ct+1 ’ ct = ρ(ct ’ ct’1 ) + δ t+1 .

The term structure literature has only started to think whether the empirically successful

discount factor processes can be connected empirically back to macroeconomic events in this

way.

From this discount factor, we can ¬nd bond prices and yields. This is easy because the

conditional mean and variance of an AR(1) are easy to ¬nd. (I am following the strategy of

solving the discount factor forward rather than solving the price backward.) We need

(1) (1)

= ’pt = ’ ln Et (eln mt+1 )

yt

1 (2) 1

(2)

= ’ pt = ’ ln Et (eln mt+1 +ln mt+2 )

yt

2 2

and so on. Iterating the AR(1) forward,

(ln mt+2 + δ) = ρ2 (ln mt + δ) + ρµt+1 + µt+2

(ln mt+3 + δ) = ρ3 (ln mt + δ) + ρ2 µt+1 + ρµt+2 + µt+3

so

(ln mt+1 + δ) + (ln mt+2 + δ) = (ρ + ρ2 ) (ln mt + δ) + (1 + ρ)µt+1 + µt+2

330

SECTION 19.3 TERM STRUCTURE MODELS “ A DISCRETE-TIME INTRODUCTION

(ln mt+1 + δ) + (ln mt+2 + δ) + (ln mt+3 + δ)

= (ρ + ρ2 + ρ3 ) (ln mt + δ) + (1 + ρ + ρ2 )µt+1 + (1 + ρ)µt+2 + µt+3 .

2

1

Using the rule for a lognormal E(ex ) = eE(x)+ 2 σx , we have ¬nally

1

(1)

= δ ’ ρ(ln mt + δ) ’ σ2

yt

2µ

(ρ + ρ2 ) 1 + (1 + ρ)2 2

(2)

yt = δ’ (ln mt + δ) ’ σµ

2 4

(ρ + ρ2 + ρ3 ) 1 + (1 + ρ)2 + (1 + ρ + ρ2 )2 2

(3)

yt = δ’ (ln mt + δ) ’ σµ .

3 6

Notice all yields move as linear functions of a single state variable, ln mt + δ. Therefore,

we can substitute out the discount factor and express the yields on bonds of any maturity as

functions of the yields on bonds of one maturity. Which one we choose is arbitrary, but it™s

conventional to use the shortest interest rate as the state variable. With E(y (1) ) = δ ’ 1 σ2 ,

2µ

we can write our term structure model as

h i

(1) (1)

yt ’ E(y (1) ) = ρ yt’1 ’ E(y (1) ) + µt (19.262)

(ρ + ρ2 ) ³ (1) ´ 1 + (1 + ρ)2

(2) (1)

σ2

yt = δ’ yt ’ E(y ) ’ µ

2 4

(ρ + ρ2 + ρ3 ) ³ (1) ´ 1 + (1 + ρ)2 + (1 + ρ + ρ2 )2

(3) (1)

σ2 .

yt = δ’ yt ’ E(y ) ’ µ

3 6

ρ 1 ’ ρN ³ (1) ´ X X (1 + ρk )2

∞j

(N) (1)

σ2

yt = δ’ yt ’ E(y ) + µ

N 1’ρ 2j

j=1 k=1

This is the form in which term structure models are usually written “ an evolution equation

for the short rate process (together, in general, with other factors or other yields used to

identify those factors), and then longer rates written as functions of the short rate, or the other

factors.

This is not a very realistic term structure model. In the data, the average yield curve “

(N)

the plot of {E[yt ]} versus N “ is slightly upward sloping. The average yield curve from

this model is slightly downward sloping as the σ2 terms pile up. The effect is not large; with

µ

(2) (1) (3) (1)

ρ = 0.9 and σ µ = 0.02, I ¬nd E(yt ) = E(yt ) ’ 0.02% and E(yt ) = E(yt ) ’ 0.06%.

Still, it does not slope up. More importantly, this model only produces smoothly upward

sloping or downward sloping term structures. For example, with ρ = 0.9, the ¬rst three

terms multiplying the one period rate in (19.262) are 0. 86, 0. 81, 0.78. Two, three and four

period bonds move exactly with one-period bonds using these coef¬cients. This model shows

no conditional heteroskedasticity “ the conditional variance of yield changes is always the

same. The term structure data show times of high and low volatility, and times of high yields

and high yield spreads seem to track these changes in volatility Finally, this model shows a

331

CHAPTER 19 TERM STRUCTURE OF INTEREST RATES

weakness of almost all term structure models “ all the yields move together; they follow an

exact one-factor structure, they are perfectly conditionally correlated.

The solution, of course, is to specify more complex discount rate processes that give rise

to more interesting term structures.

19.4 Continuous time term structure models

The basic steps.

1. Write a time-series model for the discount factor, typically in the form

dΛ

= ’rdt ’ σ Λ (·)dz

Λ

dr = µr (·)dt + σ r (·)dz

2. Solve the discount factor model forward and take expectations, to ¬nd bond prices

µ ¶

Λt+N

(N)

Pt = Et .

Λt

3. Alternatively, from the basic pricing equation 0 = E [d(ΛP )] we can ¬nd a differential

equation that the price must follow,

‚P 1 ‚P 2 ‚P ‚P

µr + σr ’ ’ rP = σr σΛ.

2 ‚r2

‚r ‚N ‚r

(0)

You can solve this back from PN = 1.

I contrast the discount factor approach to the market price of risk and arbitrage pricing

approaches.

Term structure models are usually more convenient in continuous time. As with the last

model, I specify a discount factor process and then ¬nd bond prices. A wide and popular

class of term structure models are based on discount factor process of the form

dΛ

(19.263)

= ’rdt ’ σΛ (·)dz

Λ

dr = µr (·)dt + σ r (·)dz

332

SECTION 19.4 CONTINUOUS TIME TERM STRUCTURE MODELS

This speci¬cation is analogous to a discrete time model of the form

mt+1 = xt + σµt+1

xt+1 = ρxt + µt+1 .

This is a convenient representation since the state variable x carries the mean discount factor

information.

The r variable starts out as a state variable for the drift of the discount factor. However,

f

you can see quickly that it will become the short rate process since Et (dΛ/Λ) = ’rt dt. The

dots (·) remind you that these terms can be functions of state variables. Of course, we can

add orthogonal components to the discount factor with no effect on the bond prices. Thus,

the perfect correlation between interest rate and discount factor shocks is not essential.

Term structure models differ in the speci¬cation of the functional forms for µr , σ r , σ Λ .

We will study three famous examples, the Vasicek model, the Cox-Ingersoll-Ross model and

the general af¬ne speci¬cation. The ¬rst two are

dΛ

Vasicek (19.264)

= ’rdt ’ σΛ dz

Λ

dr = φ(¯ ’ r)dt + σ r dz

r

√

dΛ

CIR (19.265)

= ’rdt ’ σΛ Λdz

Λ √

dr = φ(¯ ’ r)dt + σr rdz

r

The Vasicek model is quite similar to the AR(1) we studied in the last section. The CIR model

adds the square root terms in the volatility. This speci¬cation captures the fact in US data

that higher interest rates seem to be more volatile. In the other direction, it keeps the level of

the interest rate from falling below zero. (We need σr ¤ 2φ¯ guarantee that the square root

r

process does not get stuck at zero.)

Having speci¬ed a discount factor process, it™s a simple matter to ¬nd bond prices once

again,

µ ¶

Λt+N

(N)

.

Pt = Et

Λt

We can simply solve the discount factor forward and take the expectation. We can also use

the instantaneous pricing condition 0 = E(d(ΛP )) to ¬nd a partial differential equation for

prices, and solve that backward.

Both methods naturally adapt to pricing term structure derivatives “ call options on bonds,

interest rate ¬‚oors or caps, “swaptions” that give you the right to enter a swap, and so forth.

We simply put any payoff xC that depends on interest rates or interest rate state variables

333

CHAPTER 19 TERM STRUCTURE OF INTEREST RATES

inside the expectation

Z ∞

Λs C

(N)

Pt = Et x (s)ds.

Λt

s=t

Alternatively, the price of such options will also be a function of the state variables that drive

the term structure, so we can solve the bond pricing differential equation backwards using

the option payoff rather than one as the boundary condition.

19.4.1 Expectation approach

As with the Black-Scholes option pricing model, we can think of solving the discount factor

forward, and then taking the expectation. We can write the solution8 to (19.263)

RT RT

ΛT ’ s=0 (rs + 1 σ 2 )ds ’ s=0 σ Λs dz

=e 2 Λs

Λ0

and thus,

³ RT ´

RT

12

e’ s=0 (rs + 2 σΛs )ds ’ s=0 σΛs dz .

(N)

(266)

P0 = E0

For example, in a riskless economy σΛ = 0, we obtain the continuous-time present value

formula,

RT

(N)

= e’ rs ds.

P0 s=0

With a constant interest rate r,

P0 = e’rT .

In more interesting situations, solving the Λ equation forward and taking the expectation

analytically is not so easy. Conceptually and numerically, it is easy, of course. Just simulate

the system (19.263) forward a few thousand times, and take the average.

If this is mysterious, write ¬rst

8

µ ¶

1 dΛ2

dΛ 12

d ln Λ = ’ = ’ r + σ Λ dt ’ σ Λ dz

2 Λ2

Λ 2

and then integrate both sides from zero to T .

334

SECTION 19.4 CONTINUOUS TIME TERM STRUCTURE MODELS

19.4.2 Differential equation approach

Recall the basic pricing equation for a security with price S and no dividends is

µ ¶ µ ¶

dS dS dΛ

(267)

Et ’ rdt = ’Et .

S SΛ

The left hand side is the expected excess return. As we guessed an option price C(S, t) and

used (19.267) to derive a differential equation for the call option price, so we will guess a

bond price P (N, t) and use this equation to derive a differential equation for the bond price.

If we speci¬ed bonds by their maturity date T , P (t, T ), we could apply (19.267) directly.

However, it™s nicer to look for a bond pricing function P (N, t) that ¬xes the maturity rather

than the date. Equation (19.255) gives the holding period return for this case, which adds an

extra term to correct for the fact that you sell younger bonds than you buy,

dP (N, t) 1 ‚P (N, t)

return = ’ dt

P P ‚N

Thus, the fundamental pricing equation, applied to the price of bonds of given maturity

P (N, t) is

µ ¶ µ ¶ µ ¶

dP 1 ‚P (N, t) dP dΛ

(268)

Et ’ + r dt = ’Et .

P P ‚N PΛ

Now, we™re ready to ¬nd a differential equation for the bond price, just as we did for the

option price to derive the Black-Scholes formula. Guess that all the time dependence comes

through the state variable r, so P (N, r). Using Ito™s lemma

µ ¶

1 ‚ 2P 2

‚P ‚P

dP = µr + σ dt + σ r dz.

2r

‚r 2 ‚r ‚r

(There is no ‚P/‚N because we de¬ned the P function as the price of bonds with constant

maturity N . If we had de¬ned P (T, r) as the price of bonds that come due on date T , then

there would be a ’‚P/‚T term. This would take the place of the ‚P/‚N term in (19.268).)

Plugging in to (19.268) and canceling dt, we obtain the fundamental differential equation for

bonds,

‚P 1 ‚P 2 ‚P ‚P

(269)

µr + σr ’ ’ rP = σr σΛ.

2 ‚r2

‚r ‚N ‚r

All you have to do is specify the functions µr (·), σr (·), σ Λ (·) and solve the differential

equation.

335

CHAPTER 19 TERM STRUCTURE OF INTEREST RATES

19.4.3 Market price of risk and risk-neutral dynamic approaches

The bond pricing differential equation (19.269) is conventionally derived without discount

factors.

One conventional approach is to write the short rate process dr = µr (·)dt + σ r (·)dz, and

then specify that any asset whose payoffs have shocks σr dz must offer a Sharpe ratio of »(·).

We would then write

‚P 1 ‚P 2 ‚P ‚P

µr + σr ’ ’ rP = σr ».

2 ‚r2

‚r ‚N ‚r

With » = σΛ , this is just (19.269) of course. (If the discount factor and shock are imper-

fectly correlated, then » = σ Λ ρ.) Different authors use “market price of risk” in different

ways. Cox Ingersoll and Ross (1985, p.398) warn against modeling the right hand side as

‚P/‚r ψ (·) directly; this speci¬cation could lead to a positive expected return when σr = 0

and hence an in¬nite Sharpe ratio or arbitrage opportunity. By generating expected returns as

the covariance of payoff shocks and discount factor shocks, we naturally avoid this mistake

and other subtle ways to introduce arbitrage opportunities without realizing that you have

done so.

A second conventional approach is to use an alternative interest rate and discount factor

process,

dΛ

(19.270)

= ’rdt

Λ

dr = (µr ’ σ r »)dt + σr dzr .

If we use this alternative process, we obtain

‚P 1 ‚P 2 ‚P

(µr ’ σr ») + σ’ ’ rP = 0

2 ‚r2 r ‚N

‚r

which is of course the same thing. This is the “risk neutral probability” approach, since the

drift term in (19.270) is not the true drift. Since (19.270) gives the same prices, we can ¬nd

and represent the bond price via the integral

h RT i

(N) — ’ s=0 rs ds

Pt = Et e

where E — represents expectation with respect to the risk-neutral process de¬ned in (19.270)

rather than the true probabilities de¬ned by the process (19.263).

When we derive the model from a discount factor, the single discount factor carries two

pieces of information. The drift or conditional mean of the discount factor gives the short

rate of interest, while the covariance of the discount factor shocks with asset payoff shocks

generates expected returns or “market prices of risk.” I ¬nd it useful to write the discount

factor model to keep the term structure connected with the rest of asset pricing, and to remind

336

SECTION 19.5 THREE LINEAR TERM STRUCTURE MODELS

myself where “market prices of risk” come from, and reasonable values for their magnitude.

Of course, the result is the same no matter which method you follow.

The fact that there are fewer factors than bonds means that once you have one bond price

you can derive all the others by “no arbitrage” arguments and make this look like option

pricing. Some derivations of term structure models follow this approach, setting up arbitrage

portfolios.

19.4.4 Solving the bond price differential equation

Now we have to solve the partial differential equation (19.269) subject to the boundary con-

dition P (N = 0, r) = 1. Solving this equation is straightforward conceptually and numeri-

cally. Express (19.269) as

1 ‚ 2P 2

‚P ‚P

= (µr ’ σ r σ Λ ) + σ ’ rP.

2 ‚r2 r

‚N ‚r

We can start at N = 0 on a grid of r, and P (0, r) = 1. For ¬xed N , we can work to one step

larger N by evaluating the derivatives on the right hand side. The ¬rst step is

‚P

P (∆N, r) = ∆N = ’r∆N.

‚N

At the second step, ‚P/‚r = ∆N , ‚ 2 P/‚r2 = 0, so

P (2∆N, r) = ∆N 2 (µr ’ σr σ Λ ) ’ r2 ∆N 2 .

Now the derivatives of µr and σr with respect to r will start to enter, and we let the computer

take it from here. Analytic solutions only exist in special cases, which we study next.

19.5 Three linear term structure models

I solve the Vasicek, Cox Ingersoll Ross, and Af¬ne model. Each model gives a linear

function for log bond prices and yields, for example,

ln P (N, r) = A(N ) ’ B(N )r

As we have seen, term structure models are easy in principle and numerically: specify a

discount factor process and ¬nd its conditional expectation or solve the bond pricing partial

differential equation back from maturity. In practice, the computations are hard. I present

337

CHAPTER 19 TERM STRUCTURE OF INTEREST RATES

next three famous special cases of term structure models “ speci¬cations for the discount

factor process “ that allow analytical or quickly calculable solutions.

Analytical or close to analytical solutions are still important, because we have not yet

found good techniques for reverse-engineering the term structure. We know how to start with

a discount factor process and ¬nd bond prices. We don™t know how to start with the charac-

teristics of bond prices that we want to model and construct an appropriate discount factor

process. (At least one whose parameters do not change every day, as in the “arbitrage free”

literature.) Thus, in evaluating term structure models, we will have to do lots of the “forward”

calculations “ from assumed discount factor model to bond prices “ and it is important that

we should be able to do them quickly.

Obviously, this ¬eld would be dramatically changed if we could ¬nd a way to reverse-

engineer the term structure to directly estimate the discount factor process. Also, the ad-hoc

time-series models of the discount factor obviously need to be connected to macroeconomics;

if not to consumption, then at least to in¬‚ation, marginal products of capital, and macroeco-

nomic variables used by the Federal Reserve in its interventions in the interest rate markets.

19.5.1 Vasicek model via by pde

The Vasicek (1977) model is a special case that allows a fairly easy analytic solution. The

method is the same as the more complex analytic solution in the CIR and af¬ne classes, but

the algebra is easier, so this is a good place to start.

The Vasicek discount factor process is

dΛ

= ’rdt ’ σΛ dz

Λ

dr = φ(¯ ’ r)dt + σr dz

r

Using this process in the basic bond differential equation (19.269), we obtain

1 ‚ 2 P 2 ‚P

‚P ‚P

(271)

φ(¯ ’ r) +

r σr ’ ’ rP = σr σΛ .

2 dr2

‚r ‚N ‚r

I™ll solve this equation with the usual unsatisfying non-constructive technique “ guess the

functional form of the answer and show it™s right. I guess that log yields and hence log prices

are a linear function of the short rate,

(272)

P (N, r) = eA(N)’B(N)r .

I take the partial derivatives required in (19.271) and see if I can ¬nd A(N ) and B(N ) to

make (19.271) work. The result is a set of ordinary differential equations for A(N) and

B(N ), and these are of a particularly simple form that can be solved by integration. I solve

338

SECTION 19.5 THREE LINEAR TERM STRUCTURE MODELS

them, subject to the boundary condition imposed by P (0, r) = 1. The result is

1¡ ¢

1 ’ e’φN (19.273)

B(N ) =

φ

µ2 ¶

σ2

1 σr σ r σΛ

’ r (N ’ B(N)) ’ r B(N )2 . (19.274)

A(N ) = 2+ ¯

2φ φ 4φ

The exponential form of (19.272) means that log prices and log yields are linear functions

of the interest rate,

p(N, r) = A(N ) ’ B(N )r

A(N ) B(N )

y(N, r) = ’ + r.

N N

Solving the pde: details.

The boundary condition P (0, r) = 1 will be satis¬ed if

A(0) ’ B(0)r = 0.