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Chp.19 Term Structure of
Interest Rates (II)
报告人：陈焕华
指导老师：郑振龙
厦门大学金融系
教授
>> 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 ur , r , 
• r starts as a state variable for the drift of
discount factor process, but it is also the short
rate process since Et d /   rft dt
• 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:
d
 rdt    dz, dr   (r  r )dt   r dz

Vasicek model is similar to AR(1) model.
• 2.CIR Model
d
 rdt     dz, dr   (r  r )   r r dz

• The 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
t  N
(N )
Pt  Et (
)
t
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
Pt
(N )
s C
 Et 
x ( s )ds
s t 
t

• We can take expectation directly or use
PDE with option payoff as boundary
conditions.
>> expectation approach
d
d2
d ln  
 1 / 2 2  (r  1 / 2 2 )dt    dz


T
T
T
2
ln
   (rs  1 / 2 s )ds    s dz
s 0
s 0
0
(N )
0
P
 E0 ( e
 sT0 ( rs 1 / 2 2 s ) ds  sT0 s dz
)
• Example: in a riskless economy
• With constant interest rate,
(N)
0
P
P0  e  rT
T
rs ds

s 0
e
,

>> 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
Et (dS / S )  rdt   Et (
dS d
)
S 
• For a bond with fixed maturity, the return is
dP( N , t ) 1 P( N , t )

dt
P
P
N
• Then we can get the basic pricing equation for
the bonds with given maturity:
Et (
dP
1 P( N , t )
dP d
)(
 r )dt   Et (
)
P
P N
P 
>> Differential Equation Solution
• Suppose there is only one state variable, r.
Apply Ito’s Lemma
P
1 P 2
P
dP  ( ur 
 r )dt   r dz
2
r
2 r
r
2
• Then we can get:
P
1  P 2 P
P
ur 
r 
 rP   r 
2
r
2 r
N
r
2
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 
P
1  2 P 2 P
P
ur 
r 
 rP 
 r
2
r
2 r
N
r
>> Implication
• 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
P
hand side as r  (.) , 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: d /   rdt , dr  (ur   r  )dt   r dzr
• We can then get
P
1  2 P 2 P
(ur   r  ) 
r 
 rP  0
2
r
2 r
N
• price bonds with risk neutral probability:
Pt
(N )
T
rs ds

s 0
 E [e
]
*
t

>> Remark
• The discount factor model carries two pieces of
information.
– The drift or conditional mean gives the short rate
– The 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
P P
1 2P 2

(ur   r  ) 
 r  rP
2
N
r
2  r
• The first step is
>> Solving the bond price PDE
• At the second step
P / r  N ,  P / r  0,
2
2
P P(2N , r )  P(N , r )

 N (ur   r  )  rP (N , r )
N
N
P(2N , r )  N 2 (ur   r  )  P(N , r ) 2
>>
5. Three Linear Term Structure Models
• Vasicek Model, CIR Model, and Affine Model
gives a linear function for log bond prices and
yields:
ln P( N , r )  A( N )  B( N )r
• 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:
d
 rdt    dz

dr   (r  r )dt   r dz
• The basic bond differential equation is:
P
1  2 P 2 P
P
 (r  r ) 
r 
 rP   r 
2
r
2 r
N
r
• Method: Guess and substitute
>> PDE solution:(1)
• Guess
P( N , r )  e A( N )  B ( N ) r
– Boundary condition: A(0)  B(0)r  0 for any r,
so A(0)  0, B(0)  0
• The result is
>> PDE solution:(1)
• To substitute back to PDE ,we first calculate
the partial derivatives given P( N , r )  e A( N )B( N ) r
>> 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
p( N , r )  A( N )  B( N )r , y ( N , r )  
•
y B( N )

0
r
N
A( N ) B( N )

r
N
N
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
mean
is normally distributed with
>> 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
,   K  K ,
y , y, b  K 1,
Con :  i  i ' y  0
 i  0  CIR.M .
i  0  V .M .
>> PDE solution
• Guess
• Basically, recall that
• Use Ito’s Lemma
dP 1 P
11
2 P

' dy 
dy '
dy
P P y
2P
yy '
>> 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 decisionmaking 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!