Dynamic Term Structure
Modelling
BDT & other One-factor Models
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
Motivation and Quick Historical
Background
A simplified look at fixed income models is as follows:
Equally important but different purpose.
Static
Models
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
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.
Zero-coupon interest rate curve
estimation – The Nelson-Siegel
model
Extended Nelson-Siegel:
1 ,2
T
T T e T T T t
y t a be cTe d e e T
T f
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”
”Long term, initially zero, slow decay ~ T2,
t 0”
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
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
• 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.
Static
Models
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
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.”
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
The ideas of the Classic Models
Step 1:
We model pure discount bond prices:
T
t
T
t
x
P
(
t,
,
)
State of the world,
(vector of factors, time)
Maturity date
Model prices must have the property that
T
T
T
x
The ideas of the Classic Models
II
Step 2: Name the factors and choose stochastic process for their evolution through time
Process used is Itô-process/diffusion:
etc.
Rate" Interest
" time
Index ty
Productivi time
Index Confidence
Consumer time
level inflation
time
) (
t t t t
t x
t t
t
t
dt
x
t
dW
x
t
x
d
(
)
(
,
)
(
,
)
”Drift”
”Volatility”
The ideas of the Classic Models
III
Step 3: Mathematical argument (Itô’s lemma) shows bond dynamics must be (super short notation)
where P() is the price functional we are looking for.
P is also an Itô-process.
In itself this is pretty useless....
2 2
2
)
(
2
1
dx
x
P
dt
t
P
dx
x
P
dP
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:
where is market price of interest rate risk....
Solve this subject to the terminal (maturity) condition...
0
2
1
)
(
2
rP
x
x
P
x
P
t
P
T T
Alternative Representation
Alternatively the Feynman-Kac (probabilistic representation) is
where Q is risk-neutral measure.
Can these relations actually be solved for P()?
Depends on how we specified the factor process.
T
tr u xu du
Q t
e
E
T
t
x
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?
A Battle of Specifications
• Some of the more ”famous” specifications
• Merton (1973)
• Vasicek (1977)
• Dothan (1978)
• Cox, Ingersoll & Ross (1985)
Closed form solution can be found in these cases
)
(
t
dW
dt
dr
)
(
)
(
r
dt
dW
t
dr
)
(
t
rW
dr
)
(
)
(
r
dt
r
dW
t
Unrestricted model
dr=(+r)dt+rdz
Cox, Ingersoll & Ross
dr=(+r)dt+r1/2dz
Vasicek
dr=(+r)dt+dz
Brennan & Schwartz
dr=(+r)dt+rdz dr=CEVrdt+rdz
Merton
dr=dt+dz
GBM
dr=rdt+rdz
”X-model”
dr=rdz
Dothan
dr=rdz
CIR 2
dr=r3/2dz
=½ =1 =0
=0 =0
=0
=0
=3/2
=0
Example: The Vasicek model
Zero-coupon bond price
Term structure 4 ) , ( ) 2 / )( ) , ( ( exp ) , ( 1 ) , ( ) , ( ) , ( 2 2 2 2 2 ) ( ) ( ) , ( T t B t T T t B T t A e T t B e T t A T t P t T t r T t B
)
(
)
,
(
1
)
,
(
ln
1
)
,
(
B
t
T
r
t
t
T
T
t
A
t
T
T
t
R
Example: The CIR model
Zero-coupon bond price
Term structure 2 2 / 2 ) ( 2 / ) )( ( ) ( ) ( ) ( ) , ( 2 2 ) 1 )( ( 2 ) , ( 2 ) 1 ( 2 ) ( ) 1 ( 2 ) , ( ) , ( ) , ( 2 t T t T t T t T t r T t B e e T t A e e T t B e T t A T t P
)
(
)
,
(
1
)
,
(
ln
1
)
,
(
B
t
T
r
t
t
T
T
t
A
t
T
T
t
R
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.
Some curve-examples
Vasicek Term Structure
0 0.01 0.02 0.03 0.04 0.05 0.06
0 5 10 15 20 Time to Maturity
Z
er
o
c
o
u
p
o
n
I
n
te
re
st
R
at
e
Vasicek Discount Function
0 0.2 0.4 0.6 0.8 1 1.2
0 5 10 15 20
Time to Maturity
D
is
co
u
n
t
fa
ct
o
Vasicek estimation example
• From a time series based estimation you might get
– mean reverison rate, 0.25 – mean reversion level, 0.06
– volatility 0.02
– market price of risk 0.00 – initial interest rate, r0, 0.03
Vasicek Term Structure Curve
0 0.02 0.04 0.06
0 10 20 30
Z
e
ro
c
o
u
p
o
n
in
te
re
s
t
ra
te
But the Nelson-Siegel estimation – based on
prices – is a different curve
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 elephant. Give me five and I can make it wave it’s trunk!”
• But....estimates are likely to vary a lot from day to day
• and estimates may make no economic sense – e.g.
Conclusion
• 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
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)
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
)
1
(
1
T
P
n i)
1
(
1
1
T
P
n i)
(
T
P
nHo & 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.
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
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.
Some notation
Let us denote T-period zero-coupon price in state i and at time
n as
Absence of dynamic arbitrage (internal consistency) implies
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...
)
(
T
P
in
(
)
(
1
)
(
)
)
1
(
)
1
(
T
P
qP
11T
q
P
1T
General Pricing
• The pricing relation
Pricing Bonds
3 year 10% Bullet
Binominal tree
Bond Prices
10
11
9
12.25
10.50
9.0 0
97.9 7
101.1 3
98.00
99.55
100.9 2
99.5 9
10 0
10 0
10 0
10 0
11 . 1
1 10 2
55 . 99 98 97 .
97
Implementation
• These algorithms are very easily programmed – backwards recursion.
• All you need is the short interest rate in every node of the lattice
1 1
r
0 0
r
10
r
2 0
r
2 1
r
2 2
• But this is really the hard part... and initially we do not have these short rates. To begin with the lattice looks as follows
• Before we can do anything the model must be solved
0 0
r
? ?
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
Solving the BDT model
• The initial discount function or equivalently the term strucure curve is assumed known/observed.
• Note the relation (discrete compounding)
• Example
T
T
R
T
P
T
P
))
(
1
(
1
)
(
)
(
0 0
known/observed/estimated
T 1 2 3 4 5
What volatility curve?
• 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
• Tomorrow two TS-curves are possible:
0 0.02 0.04 0.06 0.08 0.1 0.12 Z e ro c o u p o n in te re s t ra te 0 0.02 0.04 0.06 0.08 0.1 0.12
0 1 2 3 4 5
Time Z e ro c o u p o n in te re s t ra te 0 0.02 0.04 0.06 0.08 0.1 0.12
0 1 2 3 4 5
Volatilities
• For example
• Volatility is defined and calculated as
• Estimates are relatively easily obtained....
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
r
stdev
nn)
(ln
( ))
is
const.
given
(
Now the model can be solved!
Solving the BDT example
1st step – determining 1
0 1
1
and
r
r
1 0 1 1 1 0 1 1 1 0 1 1 1 1 1 1 2 ) 1 ( ) 1 ( 2 1 ) 1 ( 2 1 ) 1 ( ) 2 ( % 19 2 ln ) 2 ( r r P P P P P and r r We haveSolving the BDT example II
%
31
.
14
%
79
.
9
%
79
.
9
1
1
1
1
)
10
.
1
(
2
1
)
11
.
1
(
1
38 . 0 1 1 1 0 1 0 19 . 0 2 1 0 2
e
r
r
r
e
r
and we have completed the first step ? 14.31%
? ? 9.79%
Solving the BDT example III
2nd step: Determining 2
2 2
1 2
0
,
r
and
r
r
2 1 0 2 36 . 0 1 0 2 1 0 2 1 1 1 0 1 1 1 0 1 1 )) 2 ( 1 ( 1 ) ) 2 ( 1 ( 1 2 ) 1 ( )) 2 ( 1 ( 1 )) 2 ( 1 ( 1 2 ) 1 ( ) 2 ( 2 1 ) 2 ( 2 1 ) 1 ( ) 3 ( % 18 2 ) 2 ( ) 2 ( ln ) 3 ( R e R P R R P P P P P and R R Solving the BDT example IV
We find7507
.
0
)
2
(
%
4159
.
15
)
2
(
)
2
(
8152
.
0
)
2
(
%
7553
.
10
)
2
(
1 1 36 . 0 1 0 1 1 1 0 1 0
P
e
R
R
P
R
Bringing back the arbitrage relation..
Solving the BDT example V
Solving the BDT example VI
) 1 ( 1 2 1 ) 1 ( 1 2 1 ) 1 ( ) 2 ( ) 1 ( 1 2 1 ) 1 ( 1 2 1 ) 1 ( ) 2 ( 2 0 2 2 0 1 0 1 0 2 2 0 4 2 0 1 1 1 1 r e r P P e r e r P P The earlier equation system is now
This is two equations in two unknowns. Solve numerically
Two year lattice
19.42% 14.31%
9.76% 13.77% 9.79%
10%
Complexity does not increase as we look further out!
The BDT Model
TSOI: 3->5%, TSOV: 25->11%
0% 5% 10% 15% 20% 25% 30% 35% 40%
The BDT Model
Mean fleeing
TSOI: 3->5%, TSOV: 10% flat
0,0% 0,5% 1,0% 1,5% 2,0% 2,5% 3,0% 3,5%
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
Lo
w
e
r
lim
it
0% 50% 100% 150% 200% 250% 300% 350%
U
p
p
e
r
lim
The BDT Model
TSOI: 3->5%
0% 5% 10% 15% 20% 25% 30%
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
The BDT Model
TSOI: 3->5%; TSOV: 25->11%;0-30Y
0% 5% 10% 15% 20% 25%
0% 5% 10% 15% 20% 25% 30% 35%
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!
)
(
)
(
)
ln
)
(
)
(
(
ln
)
(
)
(
]
ln
)
(
)
('
)
(
[
ln
)
(
)
)
(
(
)
(
)
(
t
dW
t
dt
r
t
a
t
r
t
dW
t
r
t
t
t
r
d
t
dW
dt
ar
t
dr
t
dW
dt
t
dr
Ho & Lee:
Hull & White:
BDT:
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
BDT Model
Applications
© SimCorp Financial Training A/S
Pricing Bonds
3 year 10% Bullet
Binominal tree
Bond Prices
10
11
9
12.25
10.50
9.0 0
97.9 7
101.1 3
98.00
99.55
100.9 2
99.5 9
10 0
10 0
10 0
10 0
11 . 1
1 10 2
55 . 99 98 97 .
97
Pricing Bond Options
2 year American call on 3 year 10% bullet, strike 99
10
11
9
12.25
10.50
9.0 0
97.9 7
101.1 3
98.00
99.55
100.9 2 99.5
9
10 0
10 0
10 0
10 0
0.59
0.0 0
0.00
0.55
1.9 2
Binominal
tree Bond Prices American Call
1.08
0.2 5 2.1 3*
1.1 3
* The option is exercised immediately
Using the BDT model the price of the American call option can be found to be 1.08.
Pricing Mortgage Bonds
Danish Mortgage-Backed Securities
Bond Pool of underlying Loans
• Callable Bond Model Prepayment Risk of Call Option
• Debtors are not homogenous: Several Call options
• Other Features:
– Cost of Prepaying – Premium required
– Prepayment behaviour (first, optimal) – Prepayment Model
– Tax
– DK Cash flows
Debtor Model
P P
=
PDK NON CALL
1.0 W
, W P
P =
Pricing Danish Mortgage-Backed
Securities
Zero Yields
Volatilit y
Short rate model, e.g.
BDT
Debtor Model
Short Rates
Price MBS
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.
Time (months) Strike
Libor
Libor Compensation from purchased cap
3 6 9 12 15 1
8 21 24 ….
Strike Libor
Libor Compensation from
purchased floor
Time (months)
3 6 9 12 15 1
Pricing an Interest Rate Cap
3Y Cap on 1Y rate, strike 10%
3Y Cap (1Y) = 1Y Call IRG (1Y) + 2Y Call IRG (1Y)
10 11 9 12.25 10.50 9 Binomial tree
1Y Call IRG 2Y Call IRG
0.41 0.00 90 . 0 11 . 1 10 11
1 . 1 0 90 . 0 2 / 1 0.60 1.11 0.21 0.00 00 . 2 1225 . 1 25 . 2 45 . 0 1050 . 1 50 . 0 11 . 1 45 . 0 00 . 2 2 / 1 11 .
1
Value 3Y Cap = 0.41 + 0.60 = 1.01
Tree is in Bond yields, strike is Money Market Rate (here is no difference)
