1. BDT & other One-factor Models
Dynamic Term Structure Modelling
BDT & other One-factor Models
Investments 2005

2. Prepared by Peter Løchte Jørgensen
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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3. Motivation and Quick Historical Background
A simplified look at fixed income models is as follows:
Equally important but different purpose.
Static Models
Dynamic Models
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4. Prepared by Peter Løchte Jørgensen
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
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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6. Zero-coupon interest rate curve estimation – The Nelson-Siegel model
Extended Nelson-Siegel:
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”
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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8. 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
Dynamic Models
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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
swaptions, CMS
particularly ”hot” in DK right now!
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10. 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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11. 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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12. The ideas of the Classic Models
Step 1:
We model pure discount bond prices:
State of the world,
(vector of factors, time)
Maturity date
Model prices must have the property that
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13. 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:
Wiener process
”Drift”
”Volatility”
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14. 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....
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15. 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...
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16. 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.
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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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18. 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
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19. Prepared by Peter Løchte Jørgensen
A framework for empirical work
Unrestricted model
dr=(+r)dt+rdz
=0
=0
=½
=1
Vasicek
dr=(+r)dt+dz
Cox, Ingersoll & Ross
dr=(+r)dt+r1/2dz
Brennan & Schwartz
dr=(+r)dt+rdz
CEV
dr=rdt+rdz
=0
=0
=0
Merton
dr=dt+dz
GBM
dr=rdt+rdz
”X-model”
dr=rdz
=0
=3/2
Dothan
dr=rdz
CIR 2
dr=r3/2dz
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20. Example: The Vasicek model
Zero-coupon bond price
Term structure
Formulas for bond options can be derived. (Why?)
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21. Prepared by Peter Løchte Jørgensen
Example: The CIR model
Zero-coupon bond price
Term structure
Again, formulas for bond options can be derived.
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22. 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
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24. Vasicek estimation example
From a time series based estimation you might get
mean reverison rate,  0.25
mean reversion level,  0.06
volatility
market price of risk 0.00
initial interest rate, r0, 0.03
time consistent
But the Nelson-Siegel
estimation – based on
prices – is a different curve
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25. 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. negative or very high mean reversion level and volatility
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26. Prepared by Peter Løchte Jørgensen
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
...but not externally consistent
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27. Prepared by Peter Løchte Jørgensen
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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28. Prepared by Peter Løchte Jørgensen
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
and is in fact not even free of arbitrage since interest rates
can become negative in the model!
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29. Prepared by Peter Løchte Jørgensen
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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30. 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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31. 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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32. Prepared by Peter Løchte Jørgensen
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...
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33. Prepared by Peter Løchte Jørgensen
General Pricing
The pricing relation
and if you have interim, state independent payments (coupons)
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34. Prepared by Peter Løchte Jørgensen
Pricing Bonds
3 year 10% Bullet
Binominal tree
Bond Prices 10 11 9
12.25
10.50
9.00
97.97
101.13
98.00
99.55
100.92
99.59
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
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36. Prepared by Peter Løchte Jørgensen
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 ? ? ? ? ?
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37. Prepared by Peter Løchte Jørgensen
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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38. Prepared by Peter Løchte Jørgensen
Solving the BDT model
The initial discount function or equivalently the term strucure curve is assumed known/observed.
Note the relation (discrete compounding)
Example
known/observed/estimated 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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39. Prepared by Peter Løchte Jørgensen
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:
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40. Prepared by Peter Løchte Jørgensen
Volatilities
For example
Volatility is defined and calculated as
Estimates are relatively easily obtained....
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41. Prepared by Peter Løchte Jørgensen
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:
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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42. Solving the BDT example
1st step – determining
We have
Two equations in two unknowns. Substitute and reduce:
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43. Solving the BDT example II?
and we have completed the first step
14.31%
10% ?
9.79% ?
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44. Solving the BDT example III
2nd step: Determining
One equation in one unknown....
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45. Solving the BDT example IV
We find
Bringing back the arbitrage relation..
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46. Solving the BDT example V
Two equations in three unknowns.... but recall the final assumption
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47. Solving the BDT example VI
The earlier equation system is now
This is two equations in two unknowns. Solve numerically
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48. Prepared by Peter Løchte Jørgensen
Two year lattice
19.42%
14.31%
10%
13.77%
9.79%
9.76%
Complexity does not increase as we look further out!
Alternative method of forward induction (Jamshidian 1991)
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49. Prepared by Peter Løchte Jørgensen
The BDT Model
Mean Reversion
TSOI: 3->5%, TSOV: 25->11% 0% 5%
10%
15%
20%
25%
30%
35%
40% 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29
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50. Prepared by Peter Løchte Jørgensen
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
Lower limit 0%
50%
100%
150%
200%
250%
300%
350%
Upper limit
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51. Prepared by Peter Løchte Jørgensen
The BDT Model
Evolution of local volatility
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
TSOV
Local volatility
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52. Prepared by Peter Løchte Jørgensen
The BDT Model
Log normality of short rates after 15 years
TSOI: 3->5%; TSOV: 25->11%;0-30Y 0% 5%
10%
15%
20%
25%
30%
35%
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53. 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:
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54. Prepared by Peter Løchte Jørgensen
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 s(2) and s(3).
The initial term structure is calibrated on Aug 15, 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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55. BDT Model Applications
© SimCorp Financial Training A/S

56. Prepared by Peter Løchte Jørgensen
Pricing Bonds
3 year 10% Bullet
Binominal tree
Bond Prices 10 11 9
12.25
10.50
9.00
97.97
101.13
98.00
99.55
100.92
99.59
100
Bond prices are found by discounting one period at a time, backwards, beginning at maturity.
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57. Prepared by Peter Løchte Jørgensen
Pricing Bond Options
2 year American call on 3 year 10% bullet, strike 99 10 11 9
12.25
10.50
9.00
97.97
101.13
98.00
99.55
100.92
99.59
100
0.59
0.00
0.55
1.92
Binominal tree
Bond Prices
American Call
1.08
0.25
2.13*
1.13
* 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 = – 1.08 = 98.51
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58. 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
Path Dependency?
Debtor Model
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59. Pricing Danish Mortgage-Backed Securities
Zero Yields
Volatility
Short rate model, e.g. BDT
Debtor
Model
Short Rates
Price MBS
Price/Risk/Return
Rentability
Calculations
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60. Prepared by Peter Løchte Jørgensen
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
Compensation from purchased cap 3 6 9 12 15 18 21 24 ….
Compensation from purchased floor
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61. Pricing an Interest Rate Cap
3Y Cap on 1Y rate, strike 10%
3Y Cap (1Y) = 1Y Call IRG (1Y) + 2Y Call IRG (1Y)
Binomial tree
1Y Call IRG
2Y Call IRG
12.25 11
1.11 10
10.50
0.41
0.60 9
0.00
0.21 9
0.00
Value 3Y Cap = = 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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