Zvi WienerContTimeFin

Zvi Wiener ContTimeFin - 8 slide 1

Financial Engineering

Term Structure Models

Zvi [email protected]

tel: 02-588-3049


Interest Rates Dynamic of IR is more complicated.

Power of central banks.

Dynamic of a curve, not a point.

Volatilities are different along the curve.

IR are used for both discounting and

defining the payoff.

Source: Hull and White seminar


Main Approaches

Black’s Model (modification of

BS).

No-Arbitrage Model.


Notations

D - face value (notional amount)

C - coupon payments (as % of par), yearly

N - maturity

NN

N

tt

t r

D

r

CP

1110

See Benninga, Wiener, MMA in Education, vol. 7, No. 2, 1998


Continuous Version

Denote by Ctdt the payment between

t and t+dt, then the bond price is given by:

N

ttr dtCeP t

0

0

Principal should be written as Dirac’s delta.


Forward IR

The Forward interest rate is a rate which investor can promise today, given the term structure.

Suppose that the interest rate for a maturity of

3 years is r3=10%, and the interest rate for 5

years is r5=11%.

No borrowing-lending restrictions.


Forward IRr3=10%, r5=11%.

Lend $1,000 for 3 years at 10%.

Borrow $1,000 for 5 years at 11%.

Year 0 -$1,000+$1,000 = $0

Year 3 $1,000(1.1)3 = $1331

Year 5 -$1,000(1.11)5 = -$1658

Is identical to a 2-year loan starting at year 3.


Forward IR

12517.11.1

11.15.0

3

5

fnt

n

tt

ntnt rr

r,

1

1

1

Forward interest rate from t to t+n.


Forward IR

fnt

t

ntr

n

tr

rnt

ee

e,

1)(

fnt

n

tt

ntnt rr

r,

1

1

1

Continuous compounding


Forward IRfnt

t

ntr

n

tr

rnt

ee

e,

1)(

tnt trrnt eenttP )(),(

fnt

t

ntnr

tr

rnt

ee

e,

)(

),(, nttPefntnr

n

nttPr f

nt

),(log,

n

nttP

),(log


Estimating TS from bond data

Idea - to take a set of simple bonds and to derive the current TS. Too many bonds. Too few zero coupons. Non simultaneous pricing. Very unstable!


Estimating TS from bond data

Assume that

r1=5.5%, r2=5.55%, r3=5.6%, r4=5.65%, r5=5.7%.

Bond prices

1 year 3% 979.766

2 years 5% 982.56

3 years 3% 918.164

4 years 7% 1030.94

5 years 0% 740.818


Estimating the TSWe can easily extract the interest rates from the prices of bonds.

However if the bond prices are rounded to a dollar, the resulting TS looks weird.

Conclusion: TS is very sensitive to small errors. Instead of solving the system of equations defining a unique TS it is recommended to fit the set of points by a reasonable curve representing TS.

Another problem - time instability.


Is flat TS possible?

Why can not IR be the same for different

times to maturity?

Arbitrage:

Zero investment.

Zero probability of a loss.

Positive probability of a gain.



Form a portfolio consisting of 3 bonds maturing in one, two, and three years and without coupons.

Choose a, b, c units of these bonds.

Zero investment:

ae-r + be-2r + ce-3r = 0

Zero duration:

-ae-r - 2be-2r - 3ce-3r = 0



Two equations, three unknowns

ae-r + be-2r + ce-3r = 0

-ae-r - 2be-2r - 3ce-3r = 0

Possible solution (r=10%):

a = 1, b = -2.21034, c=1.2214


Arbitrage in a flat TS


Arbitrage in a flat TS

However even a small costs destroy this arbitrage.

In many cases the assumption that TS is flat can be used.


YieldDenote by P(r,t,t+T) the price at time t of a pure discount bond maturing at time t+T > t.

Define yield to maturity R(r, t,T) as the internal rate of return at time t on a bond maturing at t+T.

TTtrReTttrP ),,(),,(

),,(log1

),,( TttrPT

TtrR


YieldThe relation between forward rates and yield:

Tt

t

dsstrFT

TtrR ),,(1

),,(

s

sttrPTtrF

),,(log

),,(

When interest are continuously compounded the average of forward rates gives the yield.


TS modelAssume that interest rates follow a diffusion process.

dZtrdttrdr ),(),(

What is the price of a pure discount bond P(r,t,T)?

dtr

Pdt

t

Pdr

r

PdP

2

22

2

Implicit one factor assumption!


TS model

Substituting dr we obtain:

Taking expectation and dividing by dt we get:

dZPdtPPPdP rrrtr

2

2

rrtr PPPdt

dPE

2

2


TS model

Using equilibrium pricing models assume that

Here is the risk premium. The basic bond pricing equation is (Merton 1971,1973):

rrtr PPPPrdt

dPE

2)1(

2

PrPPP rrtr )1(2

02


TS model

Merton has shown that in a continuous-time CAPM framework, the ration of risk premium to the standard deviation is constant (over different assets) when the utility function is logarithmic.

q

rrRE

ii

i

Sharpe ratio


TS model

For a pure discount bond we have:

dZP

Pdt

P

dP

P

dPP r

...1

Thus by Ito’s lemma

P

Ptr ri

),(


TS modelHence for the risk premium we have

P

Ptrqqr r

i

),(

The basic equation becomes

PrPPP rrtr )1(2

02

rrrtr PqrPPPP 2

02


Vasicek’s modelOrnstein-Uhlenbeck process

dZdtrdr )(

2

3

2

14

)()(11

exp

),,(

TT eTRrRe

TttrP

T

TttrPqR

T

),,(loglim

2)(

2

2


Vasicek’s modelDiscrete modeling

dZdtrdr )(

tZtrr )(

Negative interest rates.

Can be used for example for real interest rates.T

t etrTtrE ))(()(

Tt eTtrVar

12

)(2


Shapes of Vasicek’s model

All three standard shapes are possible in Vasicek’s model.

Disadvantages:

calibration, negative IR, one factor only.

There is an analytical formula for pricing

options, see Jamshidian 1989.


Extension of Vasicek

Hull, White

dZtdtrbtatdr )())(()(

dZdtrdr )(


CIR model

Precludes negative IR, but under some conditions zero can be reached.

dZrdtrdr )(

dtr

Prdt

t

Pdr

r

PdP

2

22

2


CIR model

dtr

Prdt

t

Pdr

r

PdP

2

22

2

2

22

2)(

r

Pr

t

P

r

Pr

dt

dPE


CIR model

rTtBeTtATtrP ),(),(),,(

22

)(

)(

2

)(

2)(

2

21)(

12),(

21)(

2),(

2

tT

tT

tT

tT

e

eTtB

e

eTtA


CIR modelBond prices are lognormally distributed with parameters:

dZtrdttrP

dP),(),(

rTtBtr

TtqBrtr

),(),(

),(1),(


CIR modelAs the time to maturity lengthens, the yield tends to the limit:

22 2)(

2),,(

q

qtrR

Different types of possible shapes.


One Factor TS Models

dZrtt

dtrrtrttdr

t

tttt

)()(

log)()()(

21

321


1 2 3 1 2 Cox-Ingersoll-Ross * * * 0.5Pearson-Sun * * * * 0.5Dothan * 1.0Brennan-Schwartz * * * 1.0Merton (Ho-Lee) * * 1.0Vasicek * * * 1.0Black-Karasinski * * * 1.0Constantinides-Ingersoll * 1.5

dZrtt

dtrrtrttdr

t

tttt

)()(

log)()()(

21

321


Black-Derman-Toy

The BDT model is given by

for some functions U and .

Find conditions on 2, 3, and 2 under which the Black-Karasinski model specializes to the BDT model.

)()()( tZtt etUr


The Gaussian One-Factor Models

For 3 = 2 = 0 we get a Gaussian model, in

which the short rates r(t1), r(t2), …,r(tk) are

jointly normally distributed (under the risk-

neutral measure).

Special cases: Vasicek and Merton models.

In this case a negative 2 is mean reversion.


The Gaussian One-Factor Models

For a Gaussian model the bond-price process

is lognormal.

An undesirable feature of the Gaussian model

is that the short rate and yields on bonds are

negative with positive probability at any

future date.


The Affine One-Factor Models

The Gaussian and CIR models are special

cases of single factor models with the

property that the solution has the form:

rtTbtTatrf )()(exp),(


The Affine One-Factor Models

xtTbtTatxf )()(exp),(

The yield for all t is affine in r:

tT

txfyield

),(log

Vasicek, CIR, Merton (Ho-Lee), Pearson-Sun.


TS Derivatives

Suppose a derivative has a payoff

h(r,t) prior to maturity, and

a terminal payoff g(r,) when exercised ( <T).

Then by the definition of the equivalent martingale measure, the price at time t is defined by:

),(),(),( ,,

rgdssrhEtrF t

t

sstQtt


TS Derivatives

),(),(),( ,,

rgdssrhEtrF t

t

sstQtt

s

t

vst dvrexp,


TS Derivatives

By Feynman-Kac theorem it can be

equivalently written as a solution of PDE:

),(),(2

1),( 2 txhxFtxFtxFF xxxt

With boundary conditions:

),(),( xgxF


Bond Option

A European option on a bond is described by

setting

h(x, t) = 0,

g(x, ) = Max( f(x, ) - K, 0).


Interest Rate Swap

Can be approximated as a contract paying the

dividend rate

h(r, t) = rt - r*, where r* is the fixed leg

g(r,) = 0.


Cap

Is a loan at variable rate that is capped at

some level r*. Per unit of the principal

amount of the loan, the value of the cap is

defined when

h(rt, t) = Min(rt,r*)

g(r,) = 1 (sometimes 0)


Floor

Similar to a cap, but with maximal rate

instead of minimal:

h(rt, t) = Max(rt,r*)

g(r,) = 1 (sometimes 0)


MBS

Mortgage Backed Securities

Sinking fund bond. At origination a sinking fund bond is defined in terms of a coupon rate, a scheduled maturity date, and an initial principle.

At each time prior to maturity there is an associated scheduled principle.


MBSAssume that the coupon rate is and principal repayment is at a constant rate h.

hpdt

dpt

t

For a given initial principal p0. The schedule

is chosen so that at time T the loan is repaid.

h

eh

pp tt

0


MBSHome mortgages can be prepaid. This is typically done when interest rates decline.

Unscheduled amortization process should be defined.

It has psychological and economical factors.

Standard solution - Monte Carlo simulation.


Monte Carlo

X() - random variable

Let Y be a similar variable, which is correlated with X but for which we have an analytic formula.


Monte Carlo

Introduce a new random variable

(here Y* is the analytic value of the mean of Y() and - is a free parameter which we fix later)

*)()()( YYXX


Monte Carlo

Calculate the variance of the new variable:

*)()()( YYXX

]var[],cov[2]var[]var[ 2 YYXXX


Monte Carlo

]var[],cov[2]var[]var[ 2 YYXXX

If ]var[],cov[2 2 YYX

we can reduced variance!

The optimal value of the parameter is

]var[

],cov[*

Y

YX


Monte Carlo

This choice leads to the variance of the estimator

]var[)1(]var[ 2* XX XY

where is the correlation coefficient between X and Y.

Zvi WienerContTimeFin - 8 slide 1 Financial Engineering Term Structure Models Zvi Wiener [email protected] tel: 02-588-3049.

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