5 Interpreting the market price of risk, and risk neutrality
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Part Three ﬁxed-income modeling and derivatives
since it would predict exponentially rising or falling rates. This rules out the equity price model
as an interest rate model. So we must think more carefully about how to choose the drift and
volatility.
Modeling interest rates is far, far harder than modeling stock prices because we have no
economic clues on which to hang our model. When we set out to model equity prices we
observed that the actual level of the stock price is immaterial, only the returns matter. This
made the lognormal random walk model an obvious choice. With interest rates the level does
matter; a 5% interest rate and a 500% interest rate are clearly totally different beasts.
One thing we can do, although I don’t approve, is to choose a model that makes further
analysis easy.
Let us examine some choices for the risk-neutral drift and volatility that lead to tractable
models, that is, models for which the solution of the bond pricing equation for zero-coupon
bonds can be found analytically. We will discuss these models and see what properties we like
or dislike.
For example, assume that u − λw and w take the form
u(r, t) − λ(r, t)w(r, t) = η(t) − γ (t)r,
w(r, t) =
α(t)r + β(t).
(30.6)
(30.7)
Note that we are describing a model for the risk-neutral spot rate. I will allow the functions α, β,
γ , η and λ that appear in (30.7) and (30.6) to be functions of time. By suitably restricting these
time-dependent functions, we can ensure that the random walk (30.1) for r has the following
nice properties:
• Positive interest rates: Except for a few pathological cases, such as Switzerland in the
1960s, interest rates are positive. With the above model the spot rate can be bounded below
by a positive number if α(t) > 0 and β ≤ 0. The lower bound is −β/α. (In the special case
α(t) = 0 we must take β(t) ≥ 0.) Note that r can still go to inﬁnity, but with probability
zero.
• Mean reversion: Examining the drift term, we see that for large r the (risk-neutral) interest
rate will tend to decrease towards the mean, which may be a function of time. When the
rate is small it will move up on average.
We also want the lower bound to be non-attainable; we don’t want the spot interest rate to
get forever stuck at the lower bound or have to impose further conditions to say how fast the
spot rate moves away from this value. This requirement means that
η(t) ≥ −β(t)γ (t)/α(t) + α(t)/2,
and it is discussed further below.
With the model (30.7) and (30.6) the boundary conditions for (30.4), for a zero-coupon bond,
are, ﬁrst, that
V (r, t; T ) → 0 as r → ∞,
one-factor interest rate modeling Chapter 30
and, second, that on r = −β/α, V remains ﬁnite. When r is bounded below by −β/α, a local
analysis of the partial differential equation can be carried out near r = −β/α. Brieﬂy, balancing
the terms
1
2 (αr
+ β)
∂ 2V
∂r 2
and (η − γ r)
∂V
∂r
shows that ﬁniteness of V at r = −β/α is a sufﬁcient boundary condition only if η ≥ −βγ /α +
α/2.
I chose u and w in the stochastic differential equation for r to take the special functional
forms (30.7) and (30.6) for a very special reason. With these choices the solution of (30.4) for
the zero-coupon bond is of the simple form
Z(r, t; T ) = eA(t;T )−rB(t;T ) .
(30.8)
We are going to be looking at zero-coupon bonds speciﬁcally for a while, hence the change
of our notation from V , meaning many interest rate products, to the very speciﬁc Z for zero
coupon bonds.
The model with all of α, β, γ and η non-zero is the most general stochastic differential
equation for r which leads to a solution of (30.4) of the form (30.8). This is easily shown.
Substitute (30.8) into the bond pricing equation (30.4). This gives
∂A
∂B
−r
+ 12 w2 B 2 − (u − λw)B − r = 0.
∂t
∂t
(30.9)
Some of these terms are functions of t and T (i.e. A and B) and others are functions of r and
t (i.e. u and w). Differentiating (30.9) with respect to r gives
−
∂B
∂
∂
+ 12 B 2 (w2 ) − B (u − λw) − 1 = 0.
∂t
∂r
∂r
Differentiate again with respect to r, and after dividing through by B, you get
∂2 2
1
B
2 ∂r 2 (w )
−
∂2
(u − λw) = 0.
∂r 2
In this, only B is a function of T , therefore we must have
∂2 2
(w ) = 0
∂r 2
(30.10)
∂2
(u − λw) = 0.
∂r 2
(30.11)
and
From this follow (30.7) and (30.6).
The substitution of (30.7) and (30.6) into (30.9) yields the following equations for A and B:
∂A
= η(t)B − 12 β(t)B 2
∂t
(30.12)
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Part Three ﬁxed-income modeling and derivatives
and
∂B
= 12 α(t)B 2 + γ (t)B − 1.
∂t
In order to satisfy the ﬁnal data that Z(r, T ; T ) = 1 we must have
(30.13)
A(T ; T ) = 0 and B(T ; T ) = 0.
30.7 SOLUTION FOR CONSTANT PARAMETERS
The solution for arbitrary α, β, γ and η is found by integrating the two ordinary differential
equations (30.12) and (30.13). Generally speaking, though, when these parameters are timedependent this integration cannot be done explicitly. But in some special cases this integration
can be done explicitly.
The simplest case is when α, β, γ and η are all constant. In this case we have
dB
= 12 αB 2 + γ B − 1.
dt
This can be integrated when written in the form
B
0
1
dB = 12 α
(B − a)(B + b)
t
dt,
T
where
±γ +
γ 2 + 2α
.
α
This incorporates the ﬁnal condition at t = T . The result is that
b, a =
B(t; T ) =
2(eψ 1 (T −t) − 1)
,
(γ + ψ 1 )(eψ 1 (T −t) − 1) + 2ψ 1
(30.14)
where
ψ1 =
γ 2 + 2α and ψ 2 =
η − aβ/2
.
a+b
The equation for A is
dA
= ηB − 12 βB 2 .
dt
Dividing this by the ordinary differential equation for B gives
ηB − 12 βB 2
dA
.
= 1
2
dB
2 αB + γ B − 1
This can be integrated to give
α
A = aψ 2 log(a − B) + (ψ 2 + 12 β)b log((B + b)/b) − 12 Bβ − aψ 2 log a,
2
which has incorporated the ﬁnal condition.
(30.15)
one-factor interest rate modeling Chapter 30
When all four of the parameters are constant it is obvious that both A and B are functions
of only the one variable τ = T − t, and not t and T individually; this would not necessarily
be the case if any of the parameters were time-dependent.
A wide variety of yield curves can be predicted by the model. As τ → ∞,
B→
2
γ + ψ1
and the yield curve Y has long-term behavior given by
Y →
2
η(γ + ψ 1 ) − β .
(γ + ψ 1 )2
Thus for constant and ﬁxed parameters the model leads to a ﬁxed long-term interest rate,
independent of the spot rate.
The probability density function, P (r, t), for the risk-neutral spot rate satisﬁes
∂P
∂
∂2
= 12 2 (w2 P ) − ((u − λw)P ).
∂t
∂r
∂r
In the long term this settles down to a distribution, P∞ (r), that is independent of the initial
value of the rate. This distribution satisﬁes the ordinary differential equation
1
2
d
d2
((u − λw)P∞ ).
(w2 P∞ ) =
dr 2
dr
The solution of this for the general afﬁne model with constant parameters is
P∞ (r) =
2γ k
α
(k)
r+
β
α
k−1
2γ
e− α
(r+ βα )
(30.16)
where
k=
2η 2βγ
+ 2
α
α
and (·) is the gamma function. The boundary r = −β/α is non-attainable if k > 1. The mean
of the steady-state distribution is
αk
β
− .
2γ
α
30.8
NAMED MODELS
There are many interest rate models, which are associated
with the names of their inventors. The stochastic differential
equation (30.1) for the risk-neutral interest rate process, with
risk-neutral drift and volatility given by (30.6) and (30.7), incorporates the models of Vasicek, Cox, Ingersoll & Ross, Ho &
Lee, and Hull & White.
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Part Three ﬁxed-income modeling and derivatives
30.8.1
Vasicek
The Vasicek model takes the form of (30.6) and (30.7) but with α = 0, β > 0 and
with all other parameters independent of time:
dr = (η − γ r) dt + β 1/2 dX.
This model is so ‘tractable’ that there are explicit formulae for many interest rate
derivatives. The value of a zero-coupon bond is given by
eA(t;T )−rB(t;T )
where
B=
1
(1 − e−γ (T −t) )
γ
and
A=
1
βB(t; T )2
1
(B(t;
T
)
−
T
+
t)(ηγ
−
β)
−
.
2
γ2
4γ
The model is mean reverting to a constant level, which is a good property, but interest rates
can easily go negative, which is a very bad property.
In Figure 30.2 are shown three types of yield curves predicted by the Vasicek model, each
using different parameters. (It is quite difﬁcult to get the humped yield curve with reasonable
numbers.)
The steady-state probability density function for the Vasicek model is a degenerate case of
(30.16), since α = 0. We ﬁnd that
P∞ (r) =
γ − γβ
e
βπ
r− γη
2
.
0.1
0.09
0.08
0.07
0.06
Yield
518
0.05
0.04
0.03
0.02
0.01
0
0
2
4
6
8
T
Figure 30.2 Three types of yield curve given by the Vasicek model.
10
one-factor interest rate modeling Chapter 30
14
12
10
PDF
8
6
4
2
−0.05
0
0
0.05
r
0.1
0.15
Figure 30.3 The steady-state probability density function for the risk-neutral spot rate in the
Vasicek model.
This is plotted in Figure 30.3. Thus, in the long run, the spot rate is Normally distributed in
the Vasicek model. The mean of this distribution is
η
.
γ
(The parameters in the ﬁgure have been deliberately chosen to give an alarming probability of
a negative interest rate. For reasonable parameters the probability of negative rates is not that
worrying, but then with reasonable parameters it’s hard to get realistic looking yield curves.)
30.8.2
Cox, Ingersoll & Ross
The CIR model takes (30.6) and (30.7) as the interest rate model but with β = 0, and
again no time dependence in the parameters:
√
dr = (η − γ r) dt + αr dX.
The spot rate is mean reverting and if η > α/2 the spot rate stays positive. There
are some explicit solutions for interest rate derivatives, although typically involving integrals of the non-central chi-squared distribution. The value of a zero-coupon
bond is
eA(t;T )−rB(t;T )
where A and B are given by (30.15) and (30.14) with β = 0. The resulting expression is not
much simpler than in the non-zero β case.
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Part Three ﬁxed-income modeling and derivatives
30
25
PDF
20
15
10
5
0
0
0.05
0.1
0.15
r
Figure 30.4 The steady-state probability density function for the risk-neutral spot rate in the
CIR model.
The steady-state probability density function for the spot rate is a special case of (30.16). A
plot of this function is shown in Figure 30.4. The mean of the steady-state distribution is again
η
.
γ
In Figure 30.5 are simulations of the Vasicek and CIR models using the same random numbers. The parameters have been chosen to give similar mean and standard deviations for the
two processes.
0.08
0.07
0.06
0.05
r
520
0.04
0.03
Vasicek
CIR
0.02
0.01
0
0
0.5
1
1.5
2
2.5
t
Figure 30.5 A simulation of the Vasicek and CIR models using the same random numbers.
one-factor interest rate modeling Chapter 30
30.8.3
Ho & Lee
Ho & Lee have α = γ = 0, β > 0 and constant but η can be a function of time:
dr = η(t) dt + β 1/2 dX.
The value of zero-coupon bonds is given by
eA(t;T )−rB(t;T )
where
B =T −t
and
T
A=−
t
η(s)(T − s)ds + 16 β(T − t)3 .
This model was the ﬁrst ‘no-arbitrage model’ of the term structure of interest rates. By this
is meant that the careful choice of the function η(t) will result in theoretical zero-coupon bonds
prices, output by the model, which are the same as market prices. This technique is also called
yield curve ﬁtting. This careful choice is
∂2
log ZM (t ∗ ; t) + β(t − t ∗ )
∂t 2
where today is time t = t ∗ . In this ZM (t ∗ ; T ) is the market price today of zero-coupon bonds
with maturity T . Clearly this assumes that there are bonds of all maturities and that the prices
are twice differentiable with respect to the maturity. We will see why this should give the
‘correct’ prices later. This analytically tractable model also yields simple explicit formulae for
bond options. The business of ‘yield curve ﬁtting’ is the subject of Chapter 31.
η(t) = −
30.8.4
Hull & White
Hull & White have extended both the Vasicek and the CIR models to incorporate time-dependent
parameters. This time dependence again allows the yield curve (and even a volatility structure)
to be ﬁtted. We will explore this model in greater depth later.
30.9
EQUITY AND FX FORWARDS AND FUTURES WHEN
RATES ARE STOCHASTIC
Recall from Chapter 5 that forward prices and futures prices are the same if rates are constant?
How does this change, if at all, when rates are stochastic? We must repeat the analysis of that
chapter but now with
dS = µS dt + σ S dX1
and
dr = u(r, t) dt + w(r, t) dX2 .
We are in the world of correlated random walks, as described in Chapter 11. The correlation
coefﬁcient is ρ.
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Part Three ﬁxed-income modeling and derivatives
30.9.1
Forward Contracts
V (S, r, t) will be the value of the forward contract at any time during its life on the underlying
asset S, and maturing at time T . As in Chapter 5, I’ll assume that the delivery price is known
and then ﬁnd the forward contract’s value.
Set up the portfolio of one long forward contract and short
of the underlying asset, and
1 of a risk-free bond:
= V (S, t) −
S−
1
Z.
I won’t go through all the details because the conclusion is the obvious one:
∂V
∂ 2V
∂V
∂ 2V
∂ 2V
∂V
+ 12 σ 2 S 2 2 + ρσ Sw
+ 12 w2 2 + rS
+ (u − λw)
− rV = 0.
∂t
∂S
∂S∂r
∂r
∂S
∂r
The ﬁnal condition for the equation is simply the difference between the asset price S and
the ﬁxed delivery price S. So
V (S, r, T ) = S − S.
The solution of the equation with this ﬁnal condition is
V (S, r, t) = S − SZ.
At this point Z is not just any old risk-free bond, it is a zero-coupon bond having the same
maturity as the forward contract. This is the forward contract’s value during its life.
Remember that the delivery price is set initially to t = t0 as the price that gives the forward
contract zero value. If the underlying asset is S0 at t0 then
0 = S0 − SZ
or
S=
S0
.
Z
The quoted forward price is therefore
Forward price =
S
.
Z
Remember that Z satisﬁes
∂Z
∂ 2Z
∂Z
+ 12 w2 2 + (u − λw)
− rZ = 0
∂t
∂r
∂r
with
Z(r, T ) = 1.
one-factor interest rate modeling Chapter 30
30.9.2
Futures contracts
Use F (S, r, t) to denote the futures price.
Set up a portfolio of one long futures contract and short
risk-free bond:
=−
S−
1
of the underlying, and
1
of a
Z.
(Remember that the futures contract has no value.)
d
= dF −
dS −
1 dZ.
Following the usual routine we get
∂ 2F
∂F
∂ 2F
∂ 2F
∂F
∂F
+ 12 σ 2 S 2 2 + ρσ Sw
+ 12 w2 2 + rS
+ (u − λw)
= 0.
∂t
∂S
∂S∂r
∂r
∂S
∂r
The ﬁnal condition is
F (S, r, T ) = S.
Let’s write the solution of this as
F (S, r, t) =
S
.
p(r, t)
Why? Two reasons. First, a similarity solution is to be expected; the price should be proportional to the asset price. Second, I want to make a comparison between the futures price and
the forward price. The latter is
S
.
Z
So it’s natural to ask, how similar are Z and p?
It turns out that p satisﬁes
2
∂p 1 2 ∂ p
∂p
+ (u − λw)
+ 2w
− rp−w2
2
∂t
∂r
∂r
∂p
∂r
q
2
+ ρσ β
∂p
= 0.
∂r
(30.17)
(Just plug the similarity form into the equation to see this.)
The ﬁnal condition is
p(r, T ) = 1.
The differences between the p and Z equations are in the underlined terms in
Equation (30.17).
30.9.3
The Convexity Adjustment
There is clearly a difference between the prices of forwards and futures when interest rates are
stochastic. From Equation (30.17) you can see that the difference depends on the volatility of the
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spot interest rate, the volatility of the underlying and the correlation between them. Provided
that ρ ≥ 0 the futures price is always greater than the equivalent forward price. Should the
correlation be zero then the volatility of the stock is irrelevant. If the interest rate volatility is
zero then rates are deterministic and forward and futures prices are the same.
Since the difference in price between forwards and futures depends on the spot rate volatility,
market practitioners tend to think in terms of convexity adjustments to get from one to the
other. Clearly, the convexity adjustment will depend on the precise nature of the model. For
the popular models described above, the p Equation (30.17) still has simple solutions.
30.10 SUMMARY
In this chapter I introduced the idea of a random interest rate. The interest rate that we modeled
was the ‘spot rate,’ a short-term interest rate. Several popular spot rate models were described.
These models were chosen because simple forms of the coefﬁcients make the solution of the
basic bond pricing equation straightforward analytically.
From a model for this spot rate we can derive the whole yield curve. This is certainly
unrealistic and in later chapters we will see how to make the model of more practical use.
FURTHER READING
• See the original interest rate models by Vasicek (1977), Dothan (1978), Cox, Ingersoll &
Ross (1985), Ho & Lee (1986) and Black, Derman & Toy (1990).
• For details of the general afﬁne model see the papers by Pearson & Sun (1989), Dufﬁe
(1992), Klugman (1992) and Klugman & Wilmott (1994).
• The comprehensive book by Rebonato (1996) describes all of the popular interest rate
models in detail.