Chp.19 Term Structure of Interest Rates (II)

>>


报告人
：陈焕华




指导老师：郑振龙

教授



厦门大学金融系

Chp.19

Term Structure of

Interest Rates (II)

>>

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:





>>

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.

>>

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
.

>>


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





>>

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.

>>


expectation
approach






•
Example
: in a riskless economy

•
With constant interest rate,


>>

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.

>>


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:




>>

Differential Equation Solution

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




•
Then
we can get:




>>


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

>>

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.

>>

Risk
-
Neutral Approach

A second approach is risk
-
neutral approach

•
Define:

•
We can then get



•
price
bonds with risk neutral probability:



>>

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.

>>

Solving the bond price
PDE

numerically

•
Now we solve the
PDE

with boundary condition

•


numerically.

•
Express the
PDE

as



•
The first step is




>>

Solving the bond price
PDE

•
At the second step








>>

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.



>>

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
don’t know how to start with the
bond price to discount factor
. Thus,
we must try a
lot of calculation to evaluate the models
.

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


>>

Vasicek

Model

•
The discount factor process is:



•
The basic bond differential equation is:



•
Method: Guess and substitute


>>

PDE

solution
:(1)

•
Guess

–
Boundary condition:

for any
r,


so

•
The result is





>>

PDE

solution
:(1)

•
To substitute back to
PDE

,we first calculate
the partial derivatives given







>>

PDE

solution
:(1)

•
Substituting these derivatives into
PDE



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

>>

PDE solution(2)

•
Solve the second ODE with

>>

PDE solution(3)

•
Solve the first ODE with

>>

PDE solution(3)


>>

PDE solution(3)


>>

PDE

solution(4)

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



•

means the term structure is
always upward sloping.


>>

Vasicek

Model by Expectation

•
The
Vasicek

model is simple enough to use
expectation approach. For other models the
algebra may get steadily worse.




•
Bond price

>>

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

>>

Vasicek

Model by Expectation





•
Take
intergral

>>

Vasicek

Model by Expectation

•
So




•
We have

>>

Vasicek

Model by Expectation

•
Next we solve the discount factor process




•
Plugging r

>>

Vasicek

Model by Expectation


>>

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

>>

Vasicek

Model by Expectation

•
And variance

>>

Vasicek

Model by Expectation

•
So



•
Plugging the mean and variance


>>

Vasicek

Model by Expectation

•
Rearrange into





•
Which is the same as in the
PDE

approach

>>

Vasicek

Model by Expectation

•
In the risk
-
neutral measure

>>

CIR Model





>>

CIR Model

•
Guess

•
Take derivatives and
substitue


•
So







>>

CIR Model

•
Solve these
ODEs





•
Where




>>

CIR Model





>>

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
rates, term spread,
(volatility),can
be added as
state variable.

>>

Multifactor Affine Model





>>

Multifactor Affine
Model

Where

>>

PDE
solution

•
Guess


•
Basically
, recall
that



•
Use Ito’s Lemma





>>

PDE
solution

>>

Multifactor Affine
Model

>>

Multifactor Affine
Model

>>

Multifactor Affine
Model

>>

Multifactor Affine
Model




Rearrange we get the ODEs for Affine Model


>>

Bibliography and Comments

•
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).

>>

Bibliography and Comments


Constantinides
(1992)

•
Nonlinear
Model based on CIR Model,

•
Analytical solution.

•
Allows for both signs of term premium.

>>

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.

>>

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).

>>

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.

>>

High
-
frequency research

•
Balduzzi,Bertola and Foresi (1996), Piazzesi(2000)
are based on diffusions with rather slow
-
moving state
variable. The one
-
day ahead densities are almost
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.


>>

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.

>>

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.

>>

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.

>>

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
.

>>

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.

>>



•

Thanks!