Stochastic Models for Interest Rates

CHAPTER 16
one-factor interest
rate modeling
The aim of this Chapter. . .
. . . is to model interest rates as random walks and bring together the instruments of
the fixed-income world and the modeling ideas of Black and Scholes. You will see
many familiar ideas and a few new ones that are not seen in the context of equity
derivatives.
In this Chapter. . .
• stochastic models for interest rates
• how to derive the bond pricing equation for many fixed-income products
• the structure of many popular interest rate models-----<~)
- '-

360 Paul Wilmott introduces quantitative finance
16.1 INTRODUCTION
Until now I have assumed that interest rates are either
constant or a known function of time. This may be a
reasonable assumption for short-dated equity con-
tracts. But for longer-dated contracts the interest
rate must be more accurately modeled. This is not
an easy task. In this chapter I introduce the ideas
behind modeling interest rates using a single source of
randomness. This is one-factor interest rate modeling. The model will allow the short-
term interest rate, the spot rate, to follow a random walk. This model leads to a parabolic
partial differential equation for the prices of bonds and other interest rate derivative
products.
The ‘spot rate’ that we will be modeling is a very loosely defined quantity, meant to
represent the yield on a bond of infinitesimal maturity. In practice one should take this
rate to be the yield on a liquid finite-maturity bond, say one of one month. Bonds with one
day to expiry do exist but their price is not necessarily a guide to other short-term rates. I
will continue to be vague about the precise definition of the spot interest rate. We could
argue that if we are pricing a complex product that is highly model dependent then the
exact definition of the independent variable will be relatively unimportant compared with
the choice of model.
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one-factor interest rate modeling Chapter 16 361
16.2 STOCHASTIC INTEREST RATES
Since we cannot realistically forecast the future course of an interest rate, it is natural to
model it as a random variable. We are going to model the behavior of r, the interest rate
received by the shortest possible deposit. From this we will see the development of a
model for all other rates. The interest rate for the shortest possible deposit is commonly
called the spot interest rate.
Figure 16.1 shows the time series of a one-month US interest rate. We will often use
the one-month rate as a proxy for the spot rate.
Earlier I proposed a model for the asset price as a stochastic differential equation, the
lognormal random walk. Now let us suppose that the interest rate r is governed by another
stochastic differential equation of the form
dr = u(r, t) dt + w(r, t) dX. (16.1)
The functional forms of u(r, t) and w(r, t) determine the behavior of the spot rate r. For the
present I will not specify any particular choices for these functions. We use this random
walk to derive a partial differential equation for the price of a bond using similar arguments
to those in the derivation of the Black–Scholes equation. Later I describe functional forms
for u and w that have become popular with practitioners.
Intuition behind stochastic interest rates
Equation (16.1) is just another recipe for generating
random numbers. Until now we’ve concentrated on
the lognormal random walk as the model for asset prices. But there’s no reason
why interest rates should behave like stock prices, there’s no reason why we
should use the same model for interest rates as for equities. In fact, such a
model would be a very poor one; interest rates certainly do not exhibit the
long-term exponential growth seen in the equity markets.
So, we need another model. But we’re going to use the same mathematical,
stochastic framework, with subtly and suitably different forms. Modeling interest
rates in this framework amounts to choosing functional forms for the dt and dX
coefficients in our random walk recipe.
From a model for the short-term interest rate r will follow a model for bonds
of all maturities and hence interest rates for all maturities. In other words, the
spot interest rate model leads to a model for the whole forward curve.
I’ll be taking the stochastic calculus and differential equation approach to the
pricing of interest rate products. But it can all be done in a binomial or trinomial
setting. Actually, trinomial is the more popular for interest rate products. The
principle is the same as in the equity tree model. I’ll give some details shortly.

362 Paul Wilmott introduces quantitative finance
16.3 THE BOND PRICING
EQUATION FOR THE
GENERAL MODEL
When interest rates are stochastic a bond has a price
of the form V(r, t; T ). The reader should think for the
moment in terms of simple bonds, but the governing
equation will be far more general and may be used to
price many other contracts. That’s why I’m using the
notation V rather than our earlier Z for zero-coupon bonds.
Pricing a bond presents new technical problems, and is in a sense harder than pricing
an option since there is no underlying asset with which to hedge. We are therefore not
modeling a traded asset; the traded asset (the bond, say) is a derivative of our independent
variable r. The only way to construct a hedged portfolio is by hedging one bond with a
bond of a different maturity. We set up a portfolio containing two bonds with different
maturities T1 and T2. The bond with maturity T1 has price V1(r, t; T1) and the bond with
maturity T2 has price V2(r, t; T2). We hold one of the former and a number −
of the latter.
We have
 = V1 −
V2. (16.2)
The change in this portfolio in a time dt is given by
d = ∂V1
∂t dt + ∂V1
∂r dr + 1
2 w2 ∂2V1
∂r2 dt −

( ∂V2
∂t dt + ∂V2
∂r dr + 1
2 w2 ∂2V2
∂r2 dt
)
, (16.3)
where we have applied It ˆo’s lemma to functions of r and t. Which of these terms are
random? Once you’ve identified them you’ll see that the choice

= ∂V1
∂r
/ ∂V2
∂r
eliminates all randomness in d. This is because it makes the coefficient of dr zero. We
then have
d =
( ∂V1
∂t + 1
2 w2 ∂2V1
∂r2 −
( ∂V1
∂r
/ ∂V2
∂r
) ( ∂V2
∂t + 1
2 w2 ∂2V2
∂r2
))
dt
= r dt = r
(
V1 −
( ∂V1
∂r
/ ∂V2
∂r
)
V2
)
dt,
where we have used arbitrage arguments to set the return on the portfolio equal to the
risk-free rate. This risk-free rate is just the spot rate.
Collecting all V1 terms on the left-hand side and all V2 terms on the right-hand side we
find that
∂V1
∂t + 1
2 w2 ∂2V1
∂r2 − rV1
∂V1
∂r
=
∂V2
∂t + 1
2 w2 ∂2V2
∂r2 − rV2
∂V2
∂r
.
At this point the distinction between the equity and interest rate worlds starts to become
apparent. This is one equation in two unknowns. Fortunately, the left-hand side is aS"UlsTL YDIFFE./2£NT r::'RDM

one-factor interest rate modeling Chapter 16 363
function of T1 but not T2 and the right-hand side is a function of T2 but not T1. The
only way for this to be possible is for both sides to be independent of the maturity date.
Dropping the subscript from V, we have
∂V
∂t + 1
2 w2 ∂2V
∂r2 − rV
∂V
∂r
= a(r, t)
for some function a(r, t). I shall find it convenient to write
a(r, t) = w(r, t)λ(r, t) − u(r, t);
for a given u(r, t) and non-zero w(r, t) this is always possible. The function λ(r, t) is as
yet unspecified. λ is ‘universal’ in that through this function all interest rate products are
linked.
The bond pricing equation is therefore
∂V
∂t + 1
2 w2 ∂2V
∂r2 + (u − λw) ∂V
∂r − rV = 0. (16.4)
Is this like Black–Scholes?
Pretty much, yes. Mathematically, it’s of the same form as
the Black–Scholes equation, but with different coefficients
in front of two of the partial derivative terms. That’s why I like to teach people
about BS before interest rates . . . the math is almost identical but there are no
problems with one equation for two unknowns.
The downside of this kind of modeling for interest rates is rather severe.
Finding the best (correct?) form for w and u − λw is not easy. And it’s not even
possible to determine u − λw from observing time series for r, since that time
series depends on u and w not on λ.
To find a unique solution of (16.4) we must impose one final and two boundary
conditions. The final condition corresponds to the payoff on maturity and so for a
zero-coupon bond
V(r, T; T ) = 1.
Boundary conditions depend on the form of u(r, t) and w(r, t) and are discussed later for a
special model.

364 Paul Wilmott introduces quantitative finance
It is easy to incorporate coupon payments into the model. If an amount K(r, t) dt is
received in a period dt then
∂V
∂t + 1
2 w2 ∂2V
∂r2 + (u − λw) ∂V
∂r − rV + K(r, t) = 0.
When this coupon is paid discretely, arbitrage considerations lead to jump condition
V(r, t−
c ; T ) = V(r, t+
c ; T ) + K(r, tc),
where a coupon of K(r, tc) is received at time tc.
Pricing by binomial and trinomial trees
Remember how we built up the binomial tree in Chapter 3
for equities? The process is the same for interest rate
products, after all, the pricing differential equation is mathematically very similar
to the Black–Scholes equation.
Here’s how the binomial model works. There are several stages.
Stage 1: Build your tree There are several possibilities for this, just as there
were when building up the equity tree. The simplest is to put all the diffusion
into the up and down moves. For example, the interest rate r goes to
r + w δt1/2
on an up move, or
r − w δt1/2
on a down move. See the figure below.
r
r + w dt
r − w dt
1/2
1/2
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

one-factor interest rate modeling Chapter 16 365
Stage 2: Define the risk-neutral probabilities Simple. The probability of an
up move is
1
2 + u δt1/2
2w .
But the risk-neutral probability is
1
2 + (u − λw)δt1/2
2w .
It’s the risk-neutral probability you will use when working out expected values.
Stage 3: Discounting Discount at the rate r at the base of the two branches.
Now you just follow the same procedure as in Chapter 3 to work out contract
values. You could even modify the VB code in that chapter for interest rate
products.
Often trinomial models are used because of the extra degree of freedom they
allow in choosing parameters—you are still only going to fit the volatility and
risk-neutral drift.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.4 WHAT IS THE MARKET PRICE
OF RISK?
I now give an interpretation of the function λ(r, t).
Imagine that you hold an unhedged position in one
bond with maturity date T. In a time-step dt this bond
changes in value by
dV = w ∂V
∂r dX +
( ∂V
∂t + 1
2 w2 ∂2V
∂r2 + u ∂V
∂r
)
dt.
From (16.4) this may be written as
dV = w ∂V
∂r dX +
(
wλ ∂V
∂r + rV
)
dt,
or
dV − rV dt = w ∂V
∂r (dX + λ dt). (16.5)
The right-hand side of this expression contains two terms: a deterministic term in dt and a
random term in dX. The presence of dX in (16.5) shows that this is not a riskless portfolio.
The deterministic term may be interpreted as the excess return above the risk-free rate
for accepting a certain level of risk. In return for taking the extra risk the portfolio profits
by an extra λ dt per unit of extra risk, dX. The function λ is therefore called the market
price of risk.

366 Paul Wilmott introduces quantitative finance
16.5 INTERPRETING THE MARKET PRICE OF RISK,
AND RISK NEUTRALITY
The bond pricing equation (16.4) contains references to the functions u − λw and w. The
former is the coefficient of the first-order derivative with respect to the spot rate, and the
latter appears in the coefficient of the diffusive, second-order derivative. The four terms
in the equation represent, in order as written, time decay, diffusion, drift and discounting.
We can interpret the solution of this bond pricing equation as the expected present value
of all cashflows, just like we could with equity derivatives.
Suppose that we get a ‘Payoff’ at time T then the value of that contract today would be
E
[
e− ∫ T
t r(τ ) dτ Payoff
]
.
Notice that the present value (exponential) term goes inside the expectation since it is
also random when interest rates are random.
We exploit this relationship in Chapter 29, and see exactly how to price via simulations.
As with equity options, this expectation is not with respect to the real random variable,
but instead with respect to the risk-neutral variable. There is this difference because the
drift term in the equation is not the drift of the real spot rate u, but the drift of another rate,
called the risk-neutral spot rate. This rate has a drift of u − λw. When pricing interest
rate derivatives (including bonds of finite maturity) it is important to model, and price,
using the risk-neutral rate. This rate satisfies
dr = (u − λw) dt + w dX.
We need the new market-price-of-risk term because our modeled variable, r, is not traded.
Because we can’t observe the function λ, except possibly via the whole yield curve (see
Chapter 17), I tend to think of it as a great big carpet under which we can brush all kinds
of nasty, inconvenient things.
16.6 NAMED MODELS
There are many interest rate models associated with
the names of their inventors. The stochastic differ-
ential equation (16.1) for the risk-neutral interest rate
process incorporates the models of Vasicek, Cox,
Ingersoll & Ross, Ho & Lee, and Hull & White.
16.6.1 Vasicek
The Vasicek model takes the form
dr = (η − γ r)dt + β1/2dX.
This model is so ‘tractable’ that there are explicit formulæ for many interest rate derivatives.
The value of a zero-coupon bond is given by
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