<<

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( 17)



>>

σCz and σ Cw capture the stochastic evolution of the bound over the next instant “ the ana-
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—

(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

= ’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

= ’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

(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 ,

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


= ’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


(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


Vasicek (19.264)
= ’rdt ’ σΛ dz
Λ
dr = φ(¯ ’ r)dt + σ r dz
r




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,

(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


= ’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.

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