# Chp.19 Term Structure of Interest Rates (II)

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28 Οκτ 2013 (πριν από 4 χρόνια και 8 μήνες)

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：陈焕华

Chp.19

Term Structure of

Interest Rates (II)

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Continuous
time
models

Term structure models are usually more
convenient in continuous time.

Specifying a discount factor process and then
find bond prices.

A wide and popular class models
for the discount
factor:

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Implications

Different term structure models give different
specification of the function for

r starts as a state variable for the drift of
discount factor process, but it is also the short
rate process since

Dots(.) means that the terms can be function
of state variables
.(And so are
time
-
varying
)

Some orthogonal components can be added
to the discount factor with on effect on bond
price.

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Some famous term structure models

1.Vasicek

Model:

Vasicek

model is similar to AR(1) model
.

2.CIR

Model

T
he
square root
terms captures
the fact that
higher interest rate seem to be more volatile,
and keeps the interest rate from zero
.

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Continuous
time
models

Having specified a discount factor process, it is
simple matter to find bond prices

Two way to solve

1. Solve
the discount factor model forward and
take the expectation

2. Construct a
PDE

for prices, and solve that
backward

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Implication

Both
methods naturally adapt to pricing term
structure
derivatives : call options on bonds,
interest rate floors or caps,
swaptions

and
so forth, whose payoff is

We can take expectation directly or use
PDE

with option payoff as boundary
conditions.

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expectation
approach

Example
: in a riskless economy

With constant interest rate,

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Remark

In more situations, the expectation approach is
analytically not easy.

But in numerical way, it is a good way. We can
just stimulate the interest rate process
thousands of times and take the average.

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Differential Equation Approach

Similar to the basic pricing equation for a
security price S with no dividend

For a bond with fixed maturity, the return is

Then we can get the basic pricing equation for
the bonds with given maturity:

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Differential Equation Solution

Suppose there is only one state variable, r
.
Apply Ito’s Lemma

Then
we can get:

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Market
Price of Risk and

Risk
-
neutral Dynamic Approach

The above mentioned
PDE

is derived with
discount factors.

Conventionally the
PDE

is derived without
discount factors.

One approach is write short
-
rate process and
set market price of risk to

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I
mplication

If the discount factor and shocks are imperfectly
correlated,

Different authors use market price of risk in
different ways.

CIR(1985) warned against modeling the right
hand side as , it will lead to positive
expected return when the shock is zero, thus
make the Sharpe ration infinite.

The covariance method can avoid this.

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Risk
-
Neutral Approach

A second approach is risk
-
neutral approach

Define:

We can then get

price
bonds with risk neutral probability:

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Remark

The discount factor model carries two pieces of
information.

The
drift or conditional mean gives the short
rate

T
he
covariance generates
market
price of risk.

It is useful to keep the term structure model
with asset pricing, to remind where the market
price of risk comes from
.

This beauty is in the eye of the beholder, as
the result is the same.

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Solving the bond price
PDE

numerically

Now we solve the
PDE

with boundary condition

numerically.

Express the
PDE

as

The first step is

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Solving the bond price
PDE

At the second step

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5. Three Linear Term Structure Models

Vasicek

Model, CIR Model, and Affine Model
gives a
linear function
for log bond prices and
yields
:

Term structure models are easy in principle
and numerically. Just specify a discount factor
process and find its conditional expectation or
solve the differential equation.

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Overview

Analytical
solution is important since the term
structure model can not be reverse
-
engineered.
We can only start from discount factor process to
bond price, but
bond price to discount factor
. Thus,
we must try a
lot of calculation to evaluate the models
.

-
hoc time series models of discount factor
should be connected with macroeconomics, for
example, consumption, inflation, etc.

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Vasicek

Model

The discount factor process is:

The basic bond differential equation is:

Method: Guess and substitute

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PDE

solution
:(1)

Guess

Boundary condition:

for any
r,

so

The result is

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PDE

solution
:(1)

To substitute back to
PDE

,we first calculate
the partial derivatives given

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PDE

solution
:(1)

Substituting these derivatives into
PDE

This equation has to hold
for every r
, so we get
ODEs

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PDE solution(2)

Solve the second ODE with

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PDE solution(3)

Solve the first ODE with

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PDE solution(3)

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PDE solution(3)

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PDE

solution(4)

Remark
: the log prices and log yields are
linear function of interest rates

means the term structure is
always upward sloping.

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Vasicek

Model by Expectation

The
Vasicek

model is simple enough to use
expectation approach. For other models the

Bond price

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Vasicek

Model by Expectation

First we solve r from

The main idea is to find a function of r, and by
applying Ito’s Lemma we get a
SDE

whose
drift is only a function of t. Thus we can just
take
intergral

directly.

Define

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Vasicek

Model by Expectation

Take
intergral

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Vasicek

Model by Expectation

So

We have

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Vasicek

Model by Expectation

Next we solve the discount factor process

Plugging r

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Vasicek

Model by Expectation

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Vasicek

Model by Expectation

The first integral includes a deterministic
function, so gives rise to a normally distributed
r.v
. for

Thus is normally distributed with

mean

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Vasicek

Model by Expectation

And variance

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Vasicek

Model by Expectation

So

Plugging the mean and variance

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Vasicek

Model by Expectation

Rearrange into

Which is the same as in the
PDE

approach

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Vasicek

Model by Expectation

In the risk
-
neutral measure

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CIR Model

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CIR Model

Guess

Take derivatives and
substitue

So

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CIR Model

Solve these
ODEs

Where

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CIR Model

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Multifactor Affine
Models

Vasicek

Model and CIR model are special
cases of affine models (
Duffie

and
Kan

1996,
Dai and Singleton 1999).

Affine Models maintain the convenient form
that the log bond prices are linear functions of
state
variables(The short rate and conditional
variance be linear functions of state variables).

More state variables, such as long interest
(volatility),can
state variable.

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Multifactor Affine Model

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Multifactor Affine
Model

Where

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PDE
solution

Guess

Basically
, recall
that

Use Ito’s Lemma

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PDE
solution

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Multifactor Affine
Model

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Multifactor Affine
Model

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Multifactor Affine
Model

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Multifactor Affine
Model

Rearrange we get the ODEs for Affine Model

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The choice between discrete and continuous time is
just for convenience.

Campbell
, Lo and
MacKinlay
(1997) give a discrete time treatment,
showing that the bond prices are also linear in discrete
time two parameters square root model.

In addition to affine, there are many other kinds of term
structure models, such as Jump, regime shift model,
nonlinear stochastic volatility model, etc. For the
details,

refer to Lin(2002).

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Constantinides
(1992)

Nonlinear
Model based on CIR Model,

Analytical solution.

Allows for both signs of term premium.

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Risk
-
neutral method

The
risk
-
neutral probability method rarely make
reference to the separation between drifts and
market price of risk. This was not a serious
problem for the option pricing, since volatility is
more important.

However
, it is not suitable for the portfolio
analysis and other uses. Many models imply
high and time
-
varying market price of risk and
conditional Sharpe ratio.

Duffee
(1999
) and Duarte(2000) started to fit
the model to the empirical facts about the
expected returns in term structure models.

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Term Structure and Macroeconomics

In finance, term structure models are often based
on AR process.

In macroeconomics, the interest rates are
regressed on a wide variety of variables, including
lagged interest rate, lagged inflation, output,
unemployment, etc.

This equation is interpreted as the decision
-
making rule for the short rate.

Taylor rule(Taylor,1999), monetary VAR literature
(
Eichenbaum

and Evans(1999).

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The criticism of finance model

The criticism of term structure model in finance
is hard when we only use one factor model.

Multifactor models are more subtle.

But if any variable forecasts future interest
rate, it becomes a state variable, and should
be revealed by bond yields.

Bond yields should completely drive out other
macroeconomic state variables as interest rate
forecasters.

But in fact, it is not.

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High
-
frequency research

Balduzzi,Bertola and Foresi (1996), Piazzesi(2000)
are based on diffusions with rather slow
-
moving state
variable. The one
-
exactly normal.

Johannes(2000) points out the one day ahead
densities have much fatter tails than normal
distribution. This can be modeled by fast
-
moving state
variables. Or, it is more natural to think of a jump
process.

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Other Development

All the above mentioned models describe the
bond yields as a function of state variables.

Knez, Litterman and Scheinkman(1994) make
a main factor analysis on the term structure
and find that most of the variance of yields can
be explained by three main factors, level,
slope, hump. It is done by a simple eigenvalue
decomposition method.

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Remark

Remark: This method is mainly used in portfolio
management, for example, to realize the asset
immunation

of insurance fund.

It is a good approximation, but just an
approximation. The remaining
eigenvalues

are
not zero. Then the maximum likelihood method
is not suitable, maybe GMM is better.

The importance of approximation depends on
how you use the model, if you want to find some
arbitrage opportunity, it has risk. The deviation
from the model is at best a good Sharpe ratio
but K factor model can not tell you how good.

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Possible Solution

Different parameters at each point in time (Ho
and Lee 1986). It is useful, but not satisfactory.

The whole yield curve as a state variable,
Kennedy(1994), SantaClara and
Sornette(1999) may be the potential way.

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Market Price of Risk

The
market price of interest rate risk reflects bond the market
price of real interest rate change and the market price of
inflation.

The relative contribution is very important for the nature of
risk.

If the real interest rate is constant and nominal rates change
with inflation, the short term bonds are safest long term
investment
.

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Market Price of Risk

If
the inflation is constant and nominal rates change with the
real rate, the long term bonds are safest long term
investment
.

Little work is done on the separation of interest rate
premia

between real and inflation premium components.
Buraschi

and
Jiltsov
(1999) is one recent effort.

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Thanks!