dynamic term structure modelling bdt & other one-factor models investments 2005

Dynamic Term Structure Modelling

BDT & other One-factor Models

Investments 2005

Prepared by Peter Løchte Jørgensen 2 of 61

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....


Motivation and Quick Historical Background

A simplified look at fixed income models is as follows:

Equally important but different purpose.

Static Models

Dynamic 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

equivalent!


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


Zero-coupon interest rate curve estimation – The Nelson-Siegel model

Extended Nelson-Siegel:

1,

2

TT T T Te T t

y t a be cTe d e e TT 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”

• 5 parameters• Robust model• Very flexible


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.


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


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!


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


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”....


The ideas of the Classic Models

Step 1:

We model pure discount bond prices:

TtTtxP t ),,(

State of the world,(vector of factors, time)

Maturity date

Model prices must have the property that

TTTxP T 1),,(


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 " timeIndexty Productivi time

Index ConfidenceConsumer time

levelinflation time

)(tt

t

t

tx

ttt dWtxdttxtxd ),(),()(

”Drift” ”Volatility”

Wiener process


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....

22

2

)(2

1dx

x

Pdtt

Pdxx

PdP


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...

02

1)(

2

rPxx

P

x

P

t

PT

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

tu duxurQ

t eETtxP),(

),,(


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.


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

)(tdWdtdr

)()( tdWdtrdr

)(trWdr

)()( tdWrdtrdr


Unrestricted modeldr=(+r)dt+rdz

Cox, Ingersoll & Ross

dr=(+r)dt+r1/2dz

Vasicekdr=(+r)dt+dz

Brennan & Schwartzdr=(+r)dt+rdz

CEVdr=rdt+rdz

Mertondr=dt+dz

GBMdr=rdt+rdz

”X-model”dr=rdz

Dothandr=rdz

CIR 2dr=r3/2dz

=½ =1=0

=0 =0

=0

=0

=3/2

=0

A framework for empirical work


Example: The Vasicek model

Zero-coupon bond price

Term structure

4

),()2/)(),((exp),(

1),(

),(),(

22

2

22

)(

)(),(

TtBtTTtBTtA

eTtB

eTtATtPtT

trTtB

)(),(1

),(ln1

),( trTtBtT

TtAtT

TtR

Formulas for bond options can be derived. (Why?)


Example: The CIR model

Zero-coupon bond price

Term structure

22

/2

)(

2/))((

)(

)(

)(),(

2

2)1)((

2),(

2)1(2)(

)1(2),(

),(),(

2

tT

tT

tT

tT

trTtB

e

eTtA

e

eTtB

eTtATtP

)(),(1

),(ln1

),( trTtBtT

TtAtT

TtR

Again, formulas for bond options can be derived.


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.


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

Zer

o c

ou

po

n I

nte

rest

Rat

e

Vasicek Discount Function

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20

Time to Maturity

Dis

cou

nt

fact

or


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, 0.03Vasicek Term Structure Curve

0

0.02

0.04

0.06

0 10 20 30

Time

Ze

ro c

ou

po

n in

tere

st

rate

But the Nelson-Siegel estimation – based onprices – is a different curve

time consistent


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


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


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)


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 TPni

)1(11 TPni

)(TPni

and is in fact not even free of arbitrage since interest rates can become negative in the model!


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....


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!


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)


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...

)(TPni

)()1()()1()1( 111 TPqTqPPTP n

ini

ni

ni


General Pricing

• The pricing relation

and if you have interim, state independent payments (coupons)

21

12

1

2

1)1(

111

111

ni

ni

ni

ni

ni

ni

ni

VV

r

VVPV

21

12

1

2

1)1(

1111

111

1

ni

nin

ni

ni

ni

nni

ni

VVc

r

VVcPV


Pricing Bonds3 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

100

100

100

11.11

102

55.999897.97

Bond prices are found by discounting one period at a time, backwards, beginning at maturity.


Implementation

• These algorithms are very easily programmed – backwards recursion.

• All you need is the short interest rate in every node of the lattice

11r

00r 1

0r

20r

21r

22r


• 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

00r

?

?

?

?

?


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!


Solving the BDT model

• The initial discount function or equivalently the term strucure curve is assumed known/observed.

• Note the relation (discrete compounding)

• Example

TTRTPTP

))(1(

1)()(0

0

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%


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

0 1 2 3 4 5

Time

Zero

cou

pon

inte

rest

rate

0

0.02

0.04

0.06

0.08

0.1

0.12

0 1 2 3 4 5

Time

Zero

cou

pon

inte

rest

rate

0

0.02

0.04

0.06

0.08

0.1

0.12

0 1 2 3 4 5

Time

Zero

cou

pon

inte

rest

rate


Volatilities

• For example

• Volatility is defined and calculated as

• Estimates are relatively easily obtained....

)7(10R

)7(11R

)!8( denoted is volThis

2

)1(R)1(R

ln

)1(~

ln)(

10

11

1

TTTRT


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:

nrstdev nn given const. is )(ln )()( Now the model can be solved!

Note: A lot of preparatory work here as opposed to the classicmodels. That is the price for external consistency.


Solving the BDT example

1st step – determining 10

11 and rr

10

11

10

11

10

11

1

1

1

1

2

)1(

)1(2

1)1(

2

1)1()2(

%192

ln

)2(

rr

P

PPPP

and

rr

We have

Two equations in two unknowns. Substitute and reduce:


Solving the BDT example II

%31.14%79.9

%79.9

1

1

1

1

)10.1(2

1

)11.1(

1

38.011

10

10

19.0210

2

er

r

rer

and we have completed the first step?

14.31%

?

?9.79%

10%


Solving the BDT example III

2nd step: Determining 22

21

20 and , rrr

210

236.010

210

211

10

11

10

11

))2(1(

1

))2(1(

1

2

)1(

))2(1(

1

))2(1(

1

2

)1(

)2(2

1)2(

2

1)1()3(

%182

)2()2(

ln

)3(

ReR

P

RR

P

PPPP

and

RR

One equation in one unknown....


Solving the BDT example IV

We find

7507.0)2(%4159.15)2()2(

8152.0)2(%7553.10)2(

11

36.010

11

10

10

PeRR

PR

Bringing back the arbitrage relation..

20

21

10

10

21

22

11

11

20

21

10

10

21

22

11

11

1

1

2

1

1

1

2

1)1()2(

1

1

2

1

1

1

2

1)1()2(

)1(2

1)1(

2

1)1()2(

)1(2

1)1(

2

1)1()2(

rrPP

rrPP

PPPP

PPPP


Solving the BDT example V

• Two equations in three unknowns.... but recall the final assumption

(2)(2)

(2)

(2)

420

221

22

220

21

20

21

21

222

20

21

21

22

)2(

ee

e

e

constant2

ln

2

ln

rrr

rr

r

r

r

r

rr

rr


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(

20

220

10

10

220

420

11

11

rerPP

ererPP

The earlier equation system is now

This is two equations in two unknowns. Solve numerically

%42.19

%77.13

%21.17

%76.9

22

21

)2(

20

r

r

r


Two year lattice

19.42%

14.31%

9.76%

13.77%9.79%

10%

Complexity does not increase as we look further out!

Alternative method of forward induction (Jamshidian 1991)


The BDT Model

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

Mean Reversion


The BDT ModelMean 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

Low

er

limit

0%

50%

100%

150%

200%

250%

300%

350%

Upper

limit


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

TSOV Local volatility

Evolution of local volatility


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%

Log normality of short rates after 15 years


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 formulationmight be warranted. A marriage of classical models and new ideas can be arranged!

)()()ln)()((ln

)()(]ln)(

)(')([ln

)())((

)()(

tdWtdtrtatr

tdWtrt

ttrd

tdWdtartdr

tdWdttdr

Ho & Lee:

Hull & White:

BDT:

Black & Karasinski:


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?


Pricing Bonds3 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

100

100

100

11.11

102

55.999897.97

Bond prices are found by discounting one period at a time, backwards, beginning at maturity.


Pricing Bond Options2 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

100

100

100

0.59

0.00

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 = 99.59 – 1.08 = 98.51


Pricing Mortgage BondsDanish 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

P P = P CALLNONDK

1.0 W , W P P = P ,ii

CALLNONDK


Pricing Danish Mortgage-Backed Securities

Zero Yields

Volatility

Short rate model, e.g.

BDT

DebtorModel

Short Rates

Price MBS

Price/Risk/Return

RentabilityCalculations


Caps/FloorsProduct 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 18

21

24 ….

Strike

Libor

Libor

Compensation from purchased floor

Time (months)3 6 9 1215 18

21

24 ….


Pricing an Interest Rate Cap3Y 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

90.011.11011

1.1

090.0 2/1

0.60

1.11

0.21

0.00

00.21225.125.2

45.01050.150.0

11.1

45.000.2 2/111.1

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

dynamic term structure modelling bdt & other one-factor models investments 2005

Documents

static models static

static models dynamic

modern models

quick review of static

static yieldcurve models

fixed income models

model uncertainty

factor models investments