Interest Rate Modelling One Factor

7/30/2019 Interest Rate Modelling One Factor

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One-factor Interest Rate Modeling

1 In this lecture. . .

G stochastic models for interest ratesG

how to derive the bond pricing equation for many fixed-income

products

Gthe structure of many popular interest rate models


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2 Introduction

In this lecture we see the ideas behind modeling interest rates us-

ing a single source of randomness. This is one-factor interest rate

modeling.

G

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 ma-

turity. 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.


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3 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.

GWe are going to model the behaviour ofU , 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.


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Let us suppose that the interest rate U is governed by a stochastic

differential equation of the form

G U X U W G W Z U W G ; (1)

The functional forms of X U W and Z U W determine the behaviour

of the spot rate U . For the present we will not specify any particularchoices for these functions.


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4 The bond pricing equation for the

general model

When interest rates are stochastic a bond has a price of the form

9 U W 7 .

G

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 (thebond, say) is a derivative of our independent variable U .

G

The only way to construct a hedged portfolio is by hedging one

bond with a bond of a different maturity.


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We set up a portfolio containing two bonds with different maturities

7

and 7

. The bond with maturity 7

has price 9

U W 7

and thebond with maturity 7

has price 9

U W 7

. We hold one of the

former and a number > @ of the latter. We have

D 9

> @ 9

(2)


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The change in this portfolio in a time G W is given by

G D

# 9

# W

G W

# 9

# U

G U

Z

#

9

# U

G W > @

M

# 9

# W

G W

# 9

# U

G U

Z

#

9

# U

G W

N

(3)

where we have applied Its lemma to functions of U and W . Which

of these terms are random? Once youve identified them youll see

that the choice

@

# 9

7

S

'

7

eliminates all randomness in / D . This is because it makes the coef-ficient of/ 7 zero.


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We then have

/ D

M

'

# W

Z

#

9

# U

>

M

'

7

S

'

7

N M

'

9

Z

#

9

# U

N N

/ 9

7 D / 9 7

M

'

>

M

# 9

# U

S

# 9

# U

N

9

N

G W

where we have used arbitrage arguments to set the return on the port-

folio equal to the risk-free rate. This risk-free rate is just the spot rate.

Collecting all 9

terms on the left-hand side and all 9

terms on the

right-hand side we find that '

9

Z

'

7

> 7 '

'

7

'

9

Z

'

7

> 7 '

'

7


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At this point the distinction between the equity and interest-rate

worlds starts to become apparent. This is one equation in two un-knowns. Fortunately, the left-hand side is a function of 7

but not

7

and the right-hand side is a function of 7

but not 7

. The only

way for this to be possible is for both sides to be independent of the

maturity date. Dropping the subscript from 9 , we have '

9

Z

'

7

> 7 '

'

7

D U W

for some function D U W . We shall find it convenient to writeD U W Z U W z U W > X U W

for a given X U W and non-zero Z U W this is always possible.


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The bond pricing equation is therefore

# 9

# W

Z

#

9

# U

X > z Z

# 9

# U

> U 9 (4)

The final condition corresponds to the payoff on maturity and so

for a zero-coupon bond9 U 7 7


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It is easy to incorporate coupon payments into the model. If an

amount . U W G W is received in a period G W then

# 9

# W

Z

#

9

# U

X > z Z

# 9

# U

> U 9 . U W

When this coupon is paid discretely, arbitrage considerations leadto jump condition

9 U W

>

.

7 9 U W

.

7 . U W

.

where a coupon of. U W.

is received at time W.

.


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5 What is the market price of risk?

Imagine that you hold an unhedged position in one bond with matu-

rity date 7 . In a time-step G W this bond changes in value by

G 9 Z

# 9

# U

G ;

M

# 9

# W

Z

#

9

# U

X

# 9

# U

N

G W

This may be written as

G 9 Z

# 9

# U

G ;

M

Z z

# 9

# U

U 9

N

G W

or

G 9 > U 9 G W Z

# 9

# U

G ; z G W

This expression contains a deterministic term in G W and a random

term in G ; . 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 risk the portfolio profits by z G W per unit of risk,

G ; . The function z is called the market price of risk.


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6 Interpreting the market price of risk,

and risk neutrality

The bond pricing equation contains references to the functions X >

z Z and Z . 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,

G time decay,G diffusion,

Gdrift and

G

discounting.


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The equation is similar to the backward equation for a probability

density function except for the final discounting term.

G As such we can interpret the solution of the bond pricing equation

as the expected present value of all cashflows.

As with equity options, this expectation is not with respect to thereal random variable, but instead with respect to the risk-neutral

variable. There is this difference because the drift term in the equa-

tion is not the drift of the real spot rate X , but the drift of another rate,

called the risk-neutral spot rate. This rate has a drift ofX > z Z .


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When pricing interest rate derivatives (including bonds) it is im-

portant to model, and price, using the risk-neutral rate. This ratesatisfies

G U X > z Z G W Z G ;

We need the new market-price-of-risk term because our modeled

variable, U , is not traded.

Aside: If we set z to zero then any results we find are applicable

to the real world. If, for example, we want to find the distribution of

the spot interest rate at some time in the future then we would solve a

FokkerPlanck equation with the real, and not the risk-neutral, drift.


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7 Tractable models and solutions of the

bond pricing equation

We have built up the bond pricing equation for an arbitrary model.

That is, we have not specified the risk-neutral drift, X > z Z , or the

volatility, Z . How can we choose these functions to give us a good

model? First of all, a simple lognormal random walk would not be

suitable for U , since it would predict exponentially rising or falling

rates. This rules out the equity price model as an interest rate model.

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.


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For example, assume that X > z Z and Z take the form

X U W > z U W Z U W v W > r W U (5)

Z U W

5

p W U q W (6)


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By suitably restricting these time-dependent functions, we can en-

sure that the random walk for U has the following nice properties:

GPositive interest rates: Except for a few pathological cases inter-

est rates are positive. With the above model the spot rate can be

bounded below by a positive number if p W ! and q T . The

lower bound is > q p . Note that U can still go to infinity, but with

probability zero.

G Mean reversion: Examining the drift term, we see that for large

U

the (risk-neutral) interest rate will tend to decrease towards themean, which may be a function of time. When the rate is small it

will move up on average.


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We also want the lower bound to be non-attainable, we dont want

the spot interest rate to get forever stuck at the lower bound or have toimpose further conditions to say how fast the spot rate moves away

from this value. This requirement means that

v W M > q W r W p W p W


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We have chosen X and Z in the stochastic differential equation for

U to take special functional forms for a very special reason. Withthese choices the solution for the zero-coupon bond is of the simple

form

= U W % 0

9 % > 7 9 %

(7)

We are going to be looking at zero-coupon bonds specifically for

a while, hence the change of our notation from 9 , meaning many

interest rate products, to the very specific = for zero coupon bonds.

The model with all of p , q , r and L non-zero is the most general

stochastic differential equation for 7 which leads to a solution of the

form (7). Lets see how this works.


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Substitute (7) into the bond pricing equation (4). This gives

9

> 7

9

Z

%

> X > z Z % > U (8)

Some of these terms are functions of W and 7 (i.e. $ and % ) and

others are functions of U and W (i.e. X and Z ). Differentiating (8)

with respect to U gives

>

# %

# W

%

#

# U

Z

> %

#

# U

X > z Z >

Differentiate again with respect to U and divide through by % :

%

#

# U

Z

>

#

# U

X > z Z

In this, only % is a function of 7 , therefore we must have

#

# U

Z

#

# U

X > z Z


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The equations for $ and % are

# $

# W

v W % >

G 9

(9)

and

9

F 9

H 9 > (10)

In order to satisfy the final data that = U 7 7 we must have

$ 7 7 and % 7 7


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8 Solution for constant parameters

The solution for arbitrary p , q , r and v is found by integrating the

two ordinary differential equations (9) and (10). Generally speaking,

though, when these parameters are time dependent this integration

cannot be done explicitly. But in some special cases this integrationcan be done explicitly.


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The simplest case is when p , q , r and v are all constant, in which

casep

$ D

O R J , > U

G - O R J - - >

G > , U

O R J ,

and

% W 7

H

U

% > 9

>

H U

0

U

% > 9

>

where

E D

D H

5

H

p

p

and

5

H

F and U

L > , G

D E


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When all four of the parameters are constant it is obvious that both

$ and % are functions of only the one variable 7 > W , and not W

and 7 individually; this would not necessarily be the case if any of

the parameters were time dependent.

26


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A wide variety of yield curves can be predicted by the model. As

,

%

r

and the yield curve < has long term behaviour given by

<

r

v r

> q

Thus for constant and fixed parameters the model leads to a fixed

long-term interest rate, independent of the spot rate.

27


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The probability density function, 3 U W , for the risk-neutral spot

rate satisfies# 3

# W

#

# U

Z

3 >

#

# U

X > z Z 3

In the long term this settles down to a distribution, 3

U , that is

independent of the initial value of the rate. This distribution satisfiesthe ordinary differential equation

G

G U

Z

3

G

G U

X > z Z 3

28


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The solution of this for the general affine model with constant pa-

rameters is

3

U

?

r

p

@

N

c N

M

U

q

p

N

N >

0

>

H

F

7

G

F

where

N

v

p

q r

p

and c ? is the gamma function. The boundary U > q p is non-

attainable if N ! . The mean of the steady-state distribution is

p N

r

>

q

p

29


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9 Named models

There are many interest rate models, associated with the names of

their inventors.

9.1 Vasicek

The Vasicek model takes the form

G U v > r U G W q

G ;

The value of a zero-coupon bond is given by

H

9 % > 7 9 %

where

H

> 0

> H % > 9

and

H

9 % > % 9 L H >

G >

G 9 %

H

30


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0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 2 4 6 8 10

1.Three types of yield curve given by the Vasicek model.

31


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The steady-state probability density function for the Vasicek model

is

3

U

7

H

q ~

H

>

H

G

7 >

L

H

Thus, in the long run, the spot rate is Normally distributed.

0

2

4

6

8

10

12

14

-0.05 0 0.05 0.1 0.15

2.The steady-state probability density function for the risk-neutral

spot rate in the Vasicek model.

32


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9.2 Cox, Ingersoll & Ross

The CIR model takes the form

G U v > r U G W

5

p U G ;

The spot rate is mean reverting and ifv ! p the spot rate stays pos-

itive. 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

H

9 % > 7 9 %

where and are as given above with G . The resulting ex-

pression is not much simpler than in the non-zeroG

case.

33


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0

5

10

15

20

25

30

0 0.05 0.1 0.15

3.The steady-state probability density function for the risk-neutral

spot rate in the CIR model.

34


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0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 0.5 1 1.5 2 2.5

Vasicek

CIR

4.A simulation of the Vasicek and CIR models using the same ran-dom numbers.

35


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9.3 Ho & Lee

Ho & Lee have

/ 7 L 9 / 9 G

G ;

The value of zero-coupon bonds is given by

H

9 % > 7 9 %

where

% > 9

and

>

=

%

9

L 8 % > 8 / 8

q 7 > W

36


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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 func-tion v W 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

v W >

#

# W

O R J =

9

A

9 G 9 > W

A

where today is time W W A . In this =

9

A

% is the market price

today of zero-coupon bonds with maturity % . Clearly this assumes

that there are bonds of all maturities and that the prices are twicedifferentiable with respect to the maturity.

37


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9.4 Hull & White

Hull & White have extended both the Vasicek and the CIR models

to incorporate time-dependent parameters:

/ 7 L 9 > H 9 7 / 9 G 9

G ;

G U v W > H 9 7 / 9

5

p W U G ;

This time dependence again allows the yield curve (and even a volatil-

ity structure) to be fitted.

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Interest Rate Modelling One Factor

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