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5 Interpreting the market price of risk, and risk neutrality

# 5 Interpreting the market price of risk, and risk neutrality

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514

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

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

αk

β

− .

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

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.

521

522

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

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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Part Three ﬁxed-income modeling and derivatives

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.

• 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.

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