Lecture 8 Some Spot Rate and Term Structure Models

8.1 Spot rate models

Different expressions of the drift µ and volatility σ in the SDE (7.14)

∆r(t + 1) = µ(t, r(t)) + σ(t, r(t)) ²t

lead to various spot rate models. In this lecture, we first consider time-homogeneous SDEs in
which µ(t, r(t)) = µ(r(t)) and σ(t, r(t)) = σ(r(t)), i.e. the drift and volatility do not involve time t
explicitly. We then study an inhomogeneous SDE in Hull-White model, along with Markov chains
represented by binomial or trinomial trees.

8.1.1 Vasicek and CIR models

Consider the following special cases in which a, b, σ are all positive constants, and β is a constant
with 0 ≤ β ≤ 1.

Example 8.1 (Vasicek model) µ(t, r(t)) = a[b − r(t)], σ(t, r(t)) = σ.

Example 8.2 [Cox-Ingersoll-Ross (CIR) model] µ(t, r(t)) = a[b − r(t)],

p
σ(t, r(t)) = σ r(t).

Example 8.3 (a more general class) µ(t, r(t)) = a[b − r(t)], σ(t, r(t)) = σ [r(t)]β .

A common feature of these models is the mean reversion property: the spot rate r tends to
decrease if r > b and increase if r < b. An advantage of CIR over Vasicek is its capability of
enforcing positive values on r. To formulate the question more precisely, assume r(1) = x > 0 and
let

τx = inf {2 ≤ t ≤ T : r(t) ≤ 0}. (8.1)

The question is: under what conditions on the parameters a, b and σ in the SDEs for Vasicek and
CIR respectively, Q(τx ≤ T ) = 0 will be satisfied? Note that Q(τx ≤ T ) = 0 and P (τx ≤ T ) = 0
are equivalent since Q is an EMM. Both imply that the values of r remain positive in the whole
process. It turns out that those conditions needed in CIR are quite mild and reasonable, but the
conditions needed in Vasicek are too strict to be realistic. Further discussion on various modeling
issues will appear in continuous-time finance.

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8.1.2 Hull-White model

In a very general form, Hull-White model can be expressed via the SDE

∆r(t + 1) = a(t) [b(t) − r(t)] + σ(t) [r(t)]β ²t , (8.2)

where β ≥ 0 is still a constant, but a(t), b(t) and σ(t) are positive-valued deterministic functions.
These functions greatly enhance modeling flexibility, e.g. the initial term structure can be incor-
porated into a(t) and b(t). The condition on {²t } is often relaxed in this model, e.g. ²t ’s are
independent with Q(²t = 1) = q(t) < 1 .

Under what conditions on functions a, b and σ, an underlying MC X can be represented by

a recombining tree such that r is defined by r(t + 1) = gt (nt ) (Q1 in Lecture 7)? A recombining
tree requires that starting from any node, a “up-down” combination (²t = 1 and ²t+1 = −1) and a
“down-up” combination (²t = −1 and ²t+1 = 1) should merge at the same node. In other words,
starting from r(t), these two combinations must arrive at the same value of r(t + 2). Standard
calculation turns this into the condition

σ(t + 1) = [1 − a(t + 1)] σ(t) ∀ t, (8.3)

which also requires a(t) < 1 ∀t (for positive volatility).

Trinomial trees are often used in this situation to provide an extra degree of freedom. In a
trinomial tree, three branches come out of each node: up, middle and down. There are 2t + 1
nodes at time t. The recombining assumption consists of “up-down = middle-middle = down-
up”, “up-middle = middle-up” and “down-middle = middle-down”. The three possible moves
up, middle and down correspond to ²t = 1, ²t = 0 and ²t = −1, with risk neutral probabilities
Q(²t = 1) = Q(²t = −1) = q(t) < 1/2 and Q(²t = 0) = 1 − 2q(t). The same argument shows that
the recombining condition is still given by (8.3).

8.1.3 Refined lattice and SDE

So far, the binomial (or trinomial tree) models and SDEs for spot rates have fixed integer 1 as the
span for both time increment and space increment. Refinement of this is needed for more realistic
models, especially for passing the limit to continuous-time finance. To carry out this technical step,
√
fix T > 0 and positive integer n, let δ = T /n be the time increment and δ the space increment in
the binomial or trinomial trees. Accordingly, for j = 0, 1, . . . , n − 1, we let

rδ (j) = r(jδ), ∆rδ (j + 1) = rδ (j + 1) − rδ (j).

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The functions µδ (j, rδ (j)), σδ (j, rδ (j)), aδ (j) and bδ (j) are defined in the same way. The SDE (7.14)
becomes

√
∆rδ (j + 1) = µδ (j, rδ (j)) δ + σδ (j, rδ (j)) δ ²j . (8.4)

Keep in mind that for a fixed δ (or n), the basic setting in Lecture 2 is still valid. However, if
we let n → ∞ (thus δ → 0), many complicated technical issues will need to be taken care of.

8.2 Term structure models

Sections 7.3, 7.4 and 8.1 focused on modeling the spot rate r. The corresponding term structures
can be derived based on r. This method is relatively easy to implement. A drawback is that it
lacks the capability of modeling yields (or zero-coupon bond prices) with different maturities. For
example, if one is interested in the spread between short and long term interest rates, those spot
rate models would have difficulties to model such a feature explicitly.

An alternative approach, proposed by Heath, Jarrow and Morton (HJM), is to consider an

entire yield curve as a state variable that evolves over time. In principle, one can model any one
of the three equivalent term structures: zero-coupon bond prices, yields and forward rates. It is
often convenient to work with forward rates. In this section, we start with the basic notation in
HJM set-up, followed by more detailed discussion on a special case of HJM — the Ho-Lee model,
and show how some spot rate models are derived from Ho-Lee model.

8.2.1 HJM setting

Following the basic framework in Lecture 7, we extend the idea of binomial trees to a state space
consisting of yield curves. A given value of the time t term structure {f (t, τ ), t ≤ τ ≤ T − 1}
may move “up” with a (risk neutral) probability q or “down” with probability 1 − q to one of the
two possible values of the time t + 1 term structure {f (t + 1, τ ), t + 1 ≤ τ ≤ T − 1}. In general,
q may depend on t and the value of time t state. Such a binomial tree need not be recombining.
To fix the notation for a binomial tree in this context, a typical node is denoted by (t, k), where
t = 0, 1, . . . , T − 1; k = 0, 1, . . . , 2t − 1. Let ftk be a possible value of the time t term structure,
which is a vector with components ftk (τ ), t ≤ τ ≤ T − 1. In particular, ft+1,2k and ft+1,2k+1
are the two immediate “descendants” of ftk in the binomial tree. The risk neutral (conditional)
probabilities moving from the (t, k) to the next generation (t + 1, 2k) or (t + 1, 2k + 1) are qtk and
1 − qtk respectively.

Example 8.4 We use the data in Example 7.1 to illustrate the above notation. First, Table 8.1
contains forward rates calculated from Table 7.1 by using the formula (7.4).

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 ω f (0, 0) f (0, 1) f (0, 2) f (1, 1) f (1, 2) f (2, 2) Q
ω1 0.04 0.045 0.044 0.06 0.06 0.07 0.25
ω2 ∼ ∼ ∼ ∼ ∼ 0.05 ∼
ω3 ∼ ∼ ∼ 0.03 0.03 0.04 ∼
ω4 ∼ ∼ ∼ ∼ ∼ 0.02 ∼
Table 8.1: Forward rates in Example 7.1.

Next, the values {ftk } of the term structures are given as follows:

f00 = (f (0, 0), f (0, 1), f (0, 2)) on {ω1 , ω2 , ω3 , ω4 } = (0.04, 0.045, 0.044)

f10 = (f (1, 1), f (1, 2)) on {ω1 , ω2 } = (0.06, 0.06)

f11 = (f (1, 1), f (1, 2)) on {ω3 , ω4 } = (0.03, 0.03)

f20 = f (2, 2) on {ω1 } = 0.07

f21 = f (2, 2) on {ω2 } = 0.05

f22 = f (2, 2) on {ω3 } = 0.04

f23 = f (2, 2) on {ω4 } = 0.02

There are two restrictions that a model builder must keep in mind in the specification of term
structures of forward rates: positivity of f (·, ·) and no arbitrage. It follows from (7.5), (7.8) and
(7.11) that for τ = t + 2, . . . , T ,
 ¯ 
τ
Y τ
Y ¯
¯
[1 + f (t, s − 1)]−1 = EQ  [1 + f (t + 1, s − 1)]−1 ¯¯ Ft  . (8.5)
s=t+2 s=t+2 ¯

If the left-hand side is the value at a node (t, k), say, then we have

τ
Y 1
(8.6)
s=t+2
1 + ftk (s − 1)
τ
Y τ
Y
1 1
= qtk + (1 − qtk ) .
s=t+2
1 + ft+1,2k (s − 1) s=t+2
1 + ft+1,2k+1 (s − 1)

Example 8.5 For given f (t, t), . . . , f (t, T −1), consider in each of the following three cases whether
it is possible to specify positive values of ft+1,2k+1 (t + 1), . . .,
ft+1,2k+1 (T − 1) such that (8.6) holds.

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Case 1. Positive values ft+1,2k (t + 1), . . . , ft+1,2k (T − 1) and qtk ∈ (0, 1) are fixed;

Case 2. qtk ∈ (0, 1) is fixed, but there is flexibility in choosing positive values
ft+1,2k (t + 1), . . . , ft+1,2k (T − 1).

Case 3. Positive values ft+1,2k (t + 1), . . . , ft+1,2k (T − 1) are fixed, but there is flexibility in choosing
qtk ∈ (0, 1).

8.2.2 Ho-Lee model

The paper of Ho-Lee (Journal of Finance, 1986) proposed the first yield curve model, based on
which the general HJM was developed later on. In what follows, we first present Ho-Lee model for
the term structures of zero-coupon bond prices then turn it into the equivalent version for forward
rates, which enables us to see Ho-Lee model explicitly as a special case of HJM setting in Subsection
8.2.1 and to connect Ho-Lee model to some short rate models.

A simple feature of Ho-Lee model is that it can be represented by a recombining binomial tree (a
general HJM tree need not be recombining). Therefore, the notation in Subsection 8.2.1 is modified
to be adapted to the triangular lattice in Figure 7.2.

• On the triangle lattice {(t, k) : k = 0, 1, . . . , t; t = 0, 1, . . . , T }, the time t term structure

{B(t, τ ), t < τ ≤ T } assumes values {Btk (τ ), t < τ ≤ T }.

• For t < τ ≤ T , let s = τ − t be the time to maturity, 1 ≤ s ≤ T − t. A perturbation factor η(s)

takes either the value u(s) > 1 (an “inflation” factor or a “up” move) with probability q, or
the value 0 < d(s) < 1 (a “deflation” factor or a “down” move) with probability 1 − q, where
the (risk neutral) probability q is assumed to be a constant. The conditions

η(0) = 1 (8.7)

and

q u(s) + (1 − q) d(s) = 1 ∀1≤s≤T (8.8)

are satisfied.

• The evolution of time t term structure in (t, t + 1] follows

B(t, τ )
B(t + 1, τ ) = η(τ − t − 1), (8.9)
B(t, t + 1)

which can be spelled out as

Btk (τ )
Bt+1,k+1 (τ ) = u(s − 1), (8.10)
Btk (t + 1)

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 or

Btk (τ )
Bt+1,k (τ ) = d(s − 1). (8.11)
Btk (t + 1)

• For t + 2 ≤ τ , a “up-down” combination gives rise to

Btk (τ ) u(s − 1) d(s − 2)

Bt+2,k+1 (τ ) = ; (8.12)
Btk (t + 2) u(1)

while a “down-up” combination gives rise to

Btk (τ ) d(s − 1) u(s − 2)

Bt+2,k+1 (τ ) = . (8.13)
Btk (t + 2) d(1)

Therefore, the following “path-independent” condition that forces a recombining binomial

tree is needed:

u(s − 1) d(s − 2) d(1) = d(s − 1) u(s − 2) u(1). (8.14)

Using (8.8), we can simplify (8.14) as

1 θ
= + q(1 − θ) ∀ 2 ≤ s ≤ T, (8.15)
u(s − 1) u(s − 2)

where θ is a constant, called the interest rate spread and given by

1
q + (1 − q)θ = , (8.16)
u(1)

thus 0 < θ < 1. It follows from the induction and the condition u(0) = 1 that for 0 ≤ s ≤ T ,

1
u(s) = , (8.17)
q + (1 − q)θs

and

θs
d(s) = . (8.18)
q + (1 − q)θs

• To sum up, the dynamics of term structures in Ho-Lee model is determined by two constant
parameters q and θ, which makes Ho-Lee model computationally efficient but also too re-
strictive (underparameterized). For instance, the zero-coupon bond price may not always fall
in the range [0, 1]. Still, Ho-Lee model plays a significant role in modeling the entire yield
curve. The state at node (t, k) (the time t yield curve) (Btk (t + 1), Btk (t + 2), . . . , Btk (T ))
will change to (Bt+1,k+1 (t + 1), . . . , Bt+1,k+1 (T )) at node (t + 1, k + 1) with probability q, or
(Bt+1,k (t + 1), . . . , Bt+1,k (T )) at node (t + 1, k) with probability 1 − q.

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 • Note the relation

1 B(t, τ + 1)
= . (8.19)
1 + f (t, τ ) B(t, τ )

Following (8.10), (8.11), (8.17) and (8.18), the dynamics of forward rates is given by

1 1 q + (1 − q)θτ −t−1
= , (8.20)
1 + ft+1,k+1 (τ ) 1 + ftk (τ ) q + (1 − q)θτ −t

or

1 1 θ [q + (1 − q)θτ −t−1 ]
= . (8.21)
1 + ft+1,k (τ ) 1 + ftk (τ ) q + (1 − q)θτ −t

The state (ftk (t), ftk (t + 1), . . . , ftk (T − 1)) will turn to (ft+1,k+1 (t + 1), . . . ,
ft+1,k+1 (T − 1)) with probability q, or (ft+1,k (t + 1), . . . , ft+1,k (T − 1)) with probability 1 − q.
Compare this to Subsection 8.2.1, it can be seen clearly that Ho-Lee is a special case of HJM.
In particular, Ho-Lee corresponds to a recombining binomial tree, and assumes a constant
probability q at every node.

8.2.3 Spot rate models from Ho-Lee

In a binomial tree associated with Ho-Lee model, the recombining feature implies that the total
number of “up’s” (or “down’s) determines the dynamics, regardless of their orders. Iterating (8.20)
and (8.21) in t leads to

1 θt−nt q + (1 − q)θτ −t
= , (8.22)
1 + ft,nt (τ ) 1 + f (0, τ ) q + (1 − q)θτ

where nt represents the total number of “up’s” up to t. Set τ = t and recall (7.8), we have

r(t + 1) = [1 + f (0, t)] (qθ−t + 1 − q) θnt − 1. (8.23)

The only random variable in (8.23) is nt which follows a binomial distribution.