Dynamic Term Structure Modelling
BDT & other One-factor Models
Investments 2005
Agenda
•
•
•
•
•
•
•
•
•
Motivation and quick review of static models
The need for dynamic models
Classical dynamic models and various specifications
Drawbacks of classical models
New insight and modern models
The BDT Model in some detail
BDT solution
BDT examples
After the BDT....
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Motivation and Quick Historical Background
A simplified look at fixed income models is as follows:
Static
Models
Dynamic
Models
Equally important but different purpose.
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Static Models
•
Static models are models for the present, time zero only!
•
Concerned with fitting observed bond prices or equivalently deriving today’s term structure
of
–
–
–
–
zero-coupon yields
zero-coupon interest rates
pure discount rates
spot rates
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equivalent!
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Static Models II
•
Typically assume some functional form for the R(T)-curve, i.e. choose a model like
–
–
–
–
–
•
Nelson-Siegel, extended Nelson-Siegel (Svensson)
Polynomial (cubic) spline
exponential splines
CIR (more later)
etc.
Estimate model parameters that provide the best fit to market prices
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Zero-coupon interest rate curve estimation – The
Nelson-Siegel model
Extended Nelson-Siegel:
y  t   a  be
T
 cTe
T
1  eT

T
d 
 eT  eT  ,
2
 T

T
t
f
”Long term, initially zero, slow decay
~ T2, t  0”
Asymptotic interest rate
(t    yt  a > 0)
”Short term fast decay”
(t  0  y0 = a + b)
”Medium term, initially zero, fast
decay ~ T, t  0”
• 5 parameters
• Robust model
• Very flexible
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Static Models III
•
Use model to price other fixed income securities today, e.g.
–
–
–
–
•
bonds outside estimation sample
standard swaps
FRA’s
other with known future payments
The models used contain no dynamic element and are not used for modeling (scenarios of)
future prices or curves. On the morning of next trading day you re-fit.
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Starting to think about uncertainty
• So far a quick review of static models. But we have started to look at factor modelling and looked into the
Vasicek model in some detail.
• Why? Because interest rates are uncertain and evolving/changing through time
This insight is first step in the progression
Static
Models
Dynamic
Models
• It was a recognition of the necessity to model uncertainty and simple passage of time if you want analyze
uncertainty surrounding bond prices and interest rate derivatives.
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Modeling Uncertainty
Why is it necessary to model uncertainty?
Because there are many securities whose future payments depend on the evolution of interest rates in the
future!
E.g.
$ callable bonds
$ bond options
$ caps/floors
$ mortgage backed securities !
$ corporate bonds, etc.
$ pension liabilities
particularly ”hot” in DK right now!
$ swaptions, CMS
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Modeling Uncertainty II
These instruments cannot be priced by static yield-curve models, à la Nelson-Siegel, alone.
Some analysts add a spread to some yield, but this approach is inconsistent. Particularly common in
insurance.
”In finance we do not value interest-sensitive securities by
discounting their cash flows by a Treasury yield plus a
spread. Rather we use lattices or simulations to discount
interest-sensitive cash flows. Those are the only ways that
work.”
”So all of these methods that just add spreads to a yield are
not going to give you precision... On Wall Street, sometimes
we talk about spreads - but that is only after we have
determined price. We say, "This translates into a spread," but
we would never use the spread to come up with what the price
should be.”
David Babbel
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Modelling Uncertainty III
We must construct dynamic models that can generate future yield curve scenarios and associate
probabilities to the different scenarios.
This insight dates back to research in the mid to late 1970’es
–Merton (1973)
–Vasicek (1977)
–Cox, Ingersoll & Ross (1978, 1985)
–and others
These are the ”classical models”....
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The ideas of the Classic Models
Step 1:
We model pure discount bond prices:
P( x t , t , T )
t T
State of the world,
(vector of factors, time)
Maturity date
Model prices must have the property that
P( xT , T , T ) 1
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T
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The ideas of the Classic Models II
Step 2: Name the factors and choose stochastic process for their evolution through time
time
time

x(t )  time
time

etc.
t inflation level

t Consumer Confidence Index 


t Productivi ty Index

t " Interest Rate"


Process used is Itô-process/diffusion:
Wiener process
d x(t )   ( xt , t )dt   ( xt , t )dWt
”Drift”
”Volatility”
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The ideas of the Classic Models III
Step 3: Mathematical argument (Itô’s lemma) shows bond dynamics must be (super short
notation)
P
P
1 2P
2
dP 
dx 
dt 
(
dx
)
x
t
2 x 2
where P() is the price functional we are looking for.
P is also an Itô-process.
In itself this is pretty useless....
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The ideas of the Classic Models IV
Step 4: Economic argument: We want no dynamic arbitrage in the model, internal consistency.
P(x,t,T) should solve the pde:
P P
1 T 2P

(  )  
 rP  0
T
t price
xof interest rate risk....2
xx
where  is market
Solve this subject to the terminal (maturity) condition...
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Alternative Representation
Alternatively the Feynman-Kac (probabilistic representation) is
  t r (u , xu ) du 
P( x, t , T )  Et e



T
Q
where Q is risk-neutral measure.
Can these relations actually be solved for P()?
Depends on how we specified the factor process.
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Solutions ??
There is a chance of finding explicit/analytic solutions if we
– limit number of factors
– choose tractable processes
The obvious first choice of ”factor” in a 1-factor model for the bond market is the ”interest rate”,
r.....but which?
Traditionally the instantaneous int. rate although a good case can be made that it is a bad choice.
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A Battle of Specifications
•
Some of the more ”famous” specifications
•
Merton (1973)
dr  dt  dW (t )
•
Vasicek (1977)
dr   (  r )dt  dW (t )
•
Dothan (1978)
dr  rW (t )
•
Cox, Ingersoll & Ross (1985)
dr   (  r )dt   r dW (t )
Closed form solution can be found in these cases
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A framework for empirical work
Unrestricted model
dr=(+r)dt+rdz
=0
Vasicek
dr=(+ r)dt+dz
=½
Cox, Ingersoll &
Ross
=0
=1
Brennan & Schwartz
dr=(+ r)dt+r1/2dz
=0
dr=(+ r)dt+rdz
=0
Merton
dr=dt+dz
GBM
dr= rdt+rdz
=0
Dothan
dr=rdz
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CEV
dr= rdt+rdz
=0
”X-model”
dr=rdz
=3/2
CIR 2
dr=r3/2dz
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Example: The Vasicek model
Zero-coupon bond price
P(t , T )  A(t , T )e  B (t ,T ) r (t )
B(t , T ) 
1  e  (T t )

 ( B(t , T )  T  t )( 2   2 / 2)  2 B(t , T ) 2 
A(t , T )  exp 


2

4


Term structure
1
1
R(t , T )  
ln A(t , T ) 
B(t , T )r (t )
T t
T t
Formulas for bond options can be derived. (Why?)
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Example: The CIR model
Zero-coupon bond price
P(t , T )  A(t , T )e  B (t ,T ) r (t )
2(e (T t )  1)
B(t , T ) 
(   )2(e (T t )  1)  2
(  )(T t ) / 2


2e
A(t , T )  

 (T  t )
 1)  2 
 (   )(e
2 /  2
   2  2 2
Term structure
1
1
R(t , T )  
ln A(t , T ) 
B(t , T )r (t )
T t
T t
Again, formulas for bond options can be derived.
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Observations and Critique
•
Note: You actually also get the time zero curve!
•
That is: You have a static model as the special case t=0!
•
At the same time nice and the major problem with these models.
•
The t=0 versions of these models rarely fit observed bond prices well! This is no surprise since
no bond price information is taken into account in the estimation. Estimation is typically
based on time series of rt.
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Some curve-examples
Vasicek Discount Function
0.06
1.2
0.05
1
Discount factor
Zero coupon Interest Rate
Vasicek Term Structure
0.04
0.03
0.02
0.8
0.6
0.4
0.01
0.2
0
0
0
5
10
15
20
0
Time to Maturity
5
10
15
20
Time to Maturity
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Vasicek estimation example
From a time series based estimation you might get
–
–
–
–
–
mean reverison rate, 
mean reversion level, 
volatility
market price of risk
initial interest rate, r0,
0.25
0.06
time consistent
0.02
0.00
0.03
Vasicek Term Structure Curve
But the Nelson-Siegel
estimation – based on
prices – is a different curve
Zero coupon interest
rate
•
0.06
0.04
0.02
0
0
10
20
30
Time
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Alternative estimation procedure
•
Estimation of a classic model such as the Vasicek can be based on prices – best fit. You are
likely to obtain a good fit!
•
Recall Nobel Laureate Richard Feynman’s opinion: ”Give me three parameters and I can fit an
•
But....estimates are likely to vary a lot from day to day
•
and estimates may make no economic sense – e.g. negative or very high mean reversion
level and volatility
elephant. Give me five and I can make it wave it’s trunk!”
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Conclusion
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The ”classic” one-factor models have a problem with the real world – which they often do
not fit very well.
•
The models are internally consistent
•
...but not externally consistent
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New Insight
•
Around early to mid 1980’es these weaknesses were realized. In particular it was realized that
– if we want to model the dynamics of the yield curve it makes no sense to ignore the
information contained in the current, observed curve
– The model for the present curve and the observed/fitted curve should coincide – they
should be externally consistent
•
Pioneers were
• Ho & Lee (1986), Heath, Jarrow & Morton (1987,1988,1992)
• Black, Derman &Toy (1990)
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The Ho & Lee model
•
Unfortunately the Ho & Lee model was quickly labelled the first no-arbitrage free model of
term structure movements.
•
This has created a lot of confusion – as if the classic models were not arbitrage-free...
•
In fact the Ho & Lee model describes price evolutions
Pi n11 (T  1)
Pi n (T )
Pi n 1 (T  1)
and is in fact not even free of arbitrage since interest rates
can become negative in the model!
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Ho & Lee Properties
•
So there are many different opinions on what ”arbitrage free” really means
•
But is is safe to say that Ho & Lee’s model was the first that obeyed the external consistency
criterion – no static arbitrage.
•
The Ho & Lee model was not really operational and very difficult to estimate.
•
But the idea was out....
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The Black, Derman & Toy model
•
The BDT quickly became a ”cult model”, especially in Denmark
•
Goldman Sachs working paper was difficult to get hold of
•
A lot of details about the model were left out in the paper. Few people knew what the model
was really about
•
ScanRate/Rio implemented the model in the systems  you had to know the model!
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A Closer Look at The BDT Model
•
BDT is a one-factor model using the short rate as the factor
•
In its original form it is a discrete time model
•
The uncertainty structure is binomial, i.e.
(i,n) (state i, time n)
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Some notation
Let us denote T-period zero-coupon price in state i and at time n as
Pi n (T )
Absence of dynamic arbitrage (internal consistency) implies


Pi n (T  1)  Pi n (1) qPi n11 (T )  (1  q) Pi n 1 (T )
where q is the risk-neutral probability. In basic version of BDT this is assumed to equal ½!
Any future state-contingent claim can be priced if all short rates are known...
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General Pricing
•
The pricing relation
Vi
n
 1 n 1 1 n 1 
 Pi (1) Vi 1  Vi 
2
2

1 Vi n11  Vi n 1 



1  ri n 
2

n
and if you have interim, state independent payments (coupons)
Vi
n
 n 1 1 n 1 1 n 1 
 Pi (1)c  Vi 1  Vi 
2
2


1  n 1 Vi n11  Vi n 1 

c 

n 
1  ri 
2

n
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Pricing Bonds
3 year 10% Bullet
Binominal tree
Bond Prices
100
98.00
12.25
97.97
11
10.50
10
100
99.55
99.59
100
101.13
9
98  99.55
1
97.97  
 10  
2

 1.11
100.92
9.00
100
Bond prices are found by discounting one period at a time, backwards, beginning at maturity.
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Implementation
•
These algorithms are very easily programmed – backwards recursion.
•
All you need is the short interest rate in every node of the lattice
r11
0
0
r
r01
r22
r12
r02
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•
But this is really the hard part... and initially we do not have these short rates. To begin with
the lattice looks as follows
?
?
r00
?
?
?
•
Before we can do anything the model must be solved
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Solving the model
•
Solving the BDT model is a complicated task since we must make sure that the lattice of short
rates is consistent with
– an observed/estimated initial term structure curve (external)
– an observed/estimated initial volatility curve
– the arbitrage pricing relation (internal)
These are required inputs – hence the BDT model
is automatically externally consistent!
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Solving the BDT model
•
The initial discount function or equivalently the term strucure curve is assumed
known/observed.
•
Note the relation (discrete compounding)
1
P (T )  P(T ) 
(1  R(T ))T
0
0
known/observed/estimated
•
Example
T
1
2
3
4
5
P(T)
0.909
0.812
0.712
0.624
0.543
R(T)
10%
11%
12%
12.5%
13%
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What volatility curve?
•
Tomorrow two TS-curves are possible:
0.12
0.1
Zero coupon interest rate
The volatility curve which must be provided as input concerns the 1-period-ahead volatilities
of zero-coupon rates as a function of time to maturity, T
0.12
0.1
0.08
0.06
0.04
0.02
0
0
0.08
1
2
3
4
5
3
4
5
Time
0.06
0.04
0.02
Zero coupon interest rate
Zero coupon interest rate
•
0
0
1
2
3
4
5
Time
0.12
0.1
0.08
0.06
0.04
0.02
0
0
1
2
Time
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Volatilities
•
For example
R11 (7)
This vol is denoted  (8)!
•
R01 (7)
Volatility is defined and calculated as

~
 (T )   ln R 1 (T  1)

•

 R 11 (T  1) 

ln  1
 R 0 (T  1) 
2
Estimates are relatively easily obtained....
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BDT’s example
T
1
2
3
4
5
 (T)
20%
19%
18%
17%
16%
(meaningless)
Otherwise the decreasing pattern is typical – we often estimate
smaller volatilities for longer maturities.
One additional asumption:

(n)
 stdev(ln r ) is const. given n
(n)
Now the model can be solved!
Note: A lot of preparatory work here as opposed to the classic
models. That is the price for external consistency.
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Solving the BDT example
1st step – determining
We have
r11 and r01
r11
ln 1
r0
 (2) 
 19%
2
and
1
1

P(2)  P(1) P11 (1)  P01 (1)
2
2

P(1)  1
1 




2 1  r11 1  r01 
Two equations in two unknowns. Substitute and reduce:
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Solving the BDT example II
1
1 
1
1 




2
1 20.19
(1.11)
2(1.10) 1  r0 e
1  r01 

r01  9.79%
r11  9.79%  e 0.38  14.31%
and we have completed the first step
10%
?
14.31%
9.79%
?
?
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Solving the BDT example III
2nd step: Determining
r02 , r12 and r22
R11 (2)
ln 1
R0 (2)
 (3) 
 18%
2
and
P(3) 


1
1

P (1) P11 (2)  P01 (2)
2
2


P (1) 
1
1



2  (1  R11 (2)) 2 (1  R01 (2)) 2 

P (1) 
1
1



2  (1  R01 (2)e 0.36 ) 2 (1  R01 (2)) 2 
One equation in one unknown....
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Solving the BDT example IV
We find
R01 (2)  10.7553%  P01 (2)  0.8152
R11 (2)  R01 (2)e 0.36  15.4159%  P11 (2)  0.7507
Bringing back the arbitrage relation..
1
1

P11 (2)  P11 (1)  P22 (1)  P12 (1)
2
2

1
1

P01 (2)  P01 (1)  P12 (1)  P02 (1)
2
2


1 1
1 1 
P11 (2)  P11 (1) 

2
2
2
1

r
2
1

r
2
1 

1 1
1 1 
P01 (2)  P01 (1) 

2
2
 2 1  r1 2 1  r0 
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Solving the BDT example V
•
Two equations in three unknowns.... but recall the final assumption
 ( 2)
r12
r22
ln 2 ln 2
r0
r1


 constant
2
2

e
2 (2)
r22 r12
 2  2
r1
r0

r12  r02 e 2
(2)
r22  r12 e 2
(2)
 r02 e 4
(2)
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Solving the BDT example VI
The earlier equation system is now
1

1
1
1
P11 (2)  P11 (1) 

2 4
2 2 
2
(
1

r
e
)
2
(
1

r
)
0
0e

1
1
1 1 
P01 (2) 
P01 (1) 

2 2
2 
2
(
1

r
e
)
2
(
1

r
0
0 )

This is two equations in two unknowns. Solve numerically
r02  9.76%
 ( 2 )  17.21%
r12  13.77%
r22  19.42%
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Two year lattice
19.42%
14.31%
10%
9.79%
13.77%
9.76%
Complexity does not increase as we look further out!
Alternative method of forward induction (Jamshidian 1991)
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The BDT Model
Mean Reversion
TSOI: 3->5%, TSOV: 25->11%
40%
35%
30%
25%
20%
15%
10%
5%
0%
1
3
5
7
9
11
13
15
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17
19
21
23
25
27
29
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The BDT Model
Mean fleeing
3,5%
350%
3,0%
300%
2,5%
250%
2,0%
200%
1,5%
150%
1,0%
100%
0,5%
50%
0,0%
0%
1
3
5
7
Upper limit
Lower limit
TSOI: 3->5%, TSOV: 10% flat
9 11 13 15 17 19 21 23 25 27 29
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The BDT Model
Evolution of local volatility
TSOI: 3->5%
30%
25%
20%
15%
10%
5%
0%
1
3
5
7
9
11
13
15
17
TSOV
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19
21
23
25
27
29
Local volatility
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The BDT Model
Log normality of short rates after 15 years
TSOI: 3->5%; TSOV: 25->11%;0-30Y
25%
20%
15%
10%
5%
0%
0%
5%
10%
15%
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20%
25%
30%
35%
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Further developments of BDT ideas
Depending on the particular application the BDT model can be too
hard to implement in practice and a nicer/more flexible formulation
might be warranted. A marriage of classical models and new ideas
can be arranged!
Ho & Lee:
Hull & White:
BDT:
Black & Karasinski:
dr   (t )dt  dW (t )
dr  ( (t )  ar )dt  dW (t )
 ' (t )
d ln r  [ (t ) 
ln r ]   (t )dW (t )
 (t )
ln r  ( (t )  a(t ) ln r )dt   (t ) dW (t )
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Exercise
•
•
•
•
In the first spread sheet lattice: Check for the first three years
that the model is calibrated, ie. determine P(1), P(2), P(3) and
find volatilities (2) and (3).
The initial term structure is calibrated on Aug 15, 2004. Check
the pricing of st.lån 5%2005 in BOTH LATTICES.
Determine the first zero-coupon rates (e.g. five years out) of the
two possible curves and show the curves in the same graph.
The two lattices assign identical prices to fixed income
securities at time 0 because the models are calibrated to the
same initial curve, but what about interest rate derivatives?
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BDT Model Applications
© SimCorp Financial Training A/S
www.simcorp.com
Pricing Bonds
3 year 10% Bullet
Binominal tree
Bond Prices
100
98.00
12.25
97.97
11
10.50
10
100
99.55
99.59
100
101.13
9
98  99.55
1
97.97  
 10  
2

 1.11
100.92
9.00
100
Bond prices are found by discounting one period at a time, backwards, beginning at maturity.
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Pricing Bond Options
2 year American call on 3 year 10% bullet, strike 99
Binominal tree
Bond Prices
American Call
100
98.00
12.25
97.97
11
10.50
10
9.00
0.59
1.08
99.55
99.59
0.00
0.25
100
100
101.13
9
0.00
0.55
2.13*
1.13
100.92
1.92
100
* The option is exercised immediately
 Using the BDT model the price of the American call option can be found to be 1.08.
• Value of Callable Bond is: NonCall-CallOption = 99.59 – 1.08 = 98.51
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Pricing Mortgage Bonds
Danish Mortgage-Backed Securities
 Bond  Pool of underlying Loans
• Callable Bond  Model Prepayment Risk of Call Option
PDK = PNON  PCALL
• Debtors are not homogenous: Several Call options
PDK = PNON   PCALL  W
i
• Other Features:
– Cost of Prepaying
–
–
–
–
i
,  W ,  1.0
Premium required
Prepayment behaviour (first, optimal)
Prepayment Model
Tax
Debtor Model
– DK Cash flows
• Path Dependency?
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Pricing Danish Mortgage-Backed Securities
Zero Yields
Short Rates
Debtor
Model
Volatility
Short rate model, e.g.
BDT
Price MBS
Rentability
Calculations
Price/Risk/Return
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Caps/Floors
Product description
Long term options based on a money market rate at future dates (often 3M or 6M LIBOR). Caps ensure a
maximum funding rate compared to floors which ensure a minimum deposit rate. A purchased collar is a
combination of a long cap and a short floor.
Libor
Libor
Compensation from purchased
cap
Strike
Strike
Libor
Libor
Compensation from purchased floor
3
6
9 12 15 18 21 24 ….
Time (months)
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3
6
9 12 15 18 21 24 ….
Time (months)
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Pricing an Interest Rate Cap
3Y Cap on 1Y rate, strike 10%
3Y Cap (1Y) = 1Y Call IRG (1Y) + 2Y Call IRG (1Y)
1Y Call IRG
Binomial tree
2Y Call IRG
2.25
 2.00
1.1225
12.25
11  10
 0.90
1.11
11
10.50
10
0.41
9
1.11
0.50
 0.45
1.1050
0.60
0.00
0.21
9
0.00
1/ 2 0.90  0
1.1
1.11 
1/ 2 2.00  0.45
1.11
 Value 3Y Cap = 0.41 + 0.60 = 1.01
 Tree is in Bond yields, strike is Money Market Rate (here is no difference)
 Also beware of Day Counts
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