>>
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!
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