zvi wienerconttimefin - 8 slide 1 financial engineering term structure models zvi wiener...

Zvi Wiener ContTimeFin - 8 slide 1

Financial Engineering

Term Structure Models

Zvi Wienermswiener@mscc.huji.ac.il

tel: 02-588-3049

Zvi Wiener ContTimeFin - 8 slide 2

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

Zvi Wiener ContTimeFin - 8 slide 3

Main Approaches

Black’s Model (modification of

BS).

No-Arbitrage Model.

Zvi Wiener ContTimeFin - 8 slide 4

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

Zvi Wiener ContTimeFin - 8 slide 5

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.

Zvi Wiener ContTimeFin - 8 slide 6

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.

Zvi Wiener ContTimeFin - 8 slide 7

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.

Zvi Wiener ContTimeFin - 8 slide 8

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.

Zvi Wiener ContTimeFin - 8 slide 9

Forward IR

fnt

t

ntr

n

tr

rnt

ee

e,

1)(

fnt

n

tt

ntnt rr

r,

1

1

1

Continuous compounding

Zvi Wiener ContTimeFin - 8 slide 10

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

Zvi Wiener ContTimeFin - 8 slide 11

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!

Zvi Wiener ContTimeFin - 8 slide 12

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

Zvi Wiener ContTimeFin - 8 slide 13

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.

Zvi Wiener ContTimeFin - 8 slide 14

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.

Zvi Wiener ContTimeFin - 8 slide 15

Is flat TS possible?

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

Zvi Wiener ContTimeFin - 8 slide 16

Is flat TS possible?

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

Zvi Wiener ContTimeFin - 8 slide 17

Arbitrage in a flat TS

Zvi Wiener ContTimeFin - 8 slide 18

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.

Zvi Wiener ContTimeFin - 8 slide 19

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

Zvi Wiener ContTimeFin - 8 slide 20

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.

Zvi Wiener ContTimeFin - 8 slide 21

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!

Zvi Wiener ContTimeFin - 8 slide 22

TS model

Substituting dr we obtain:

Taking expectation and dividing by dt we get:

dZPdtPPPdP rrrtr

2

2

rrtr PPPdt

dPE

2

2

Zvi Wiener ContTimeFin - 8 slide 23

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

Zvi Wiener ContTimeFin - 8 slide 24

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

Zvi Wiener ContTimeFin - 8 slide 25

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

),(

Zvi Wiener ContTimeFin - 8 slide 26

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

Zvi Wiener ContTimeFin - 8 slide 27

Vasicek’s modelOrnstein-Uhlenbeck process

dZdtrdr )(

2

3

2

14

)()(11

exp

),,(

TT eTRrRe

TttrP

T

TttrPqR

T

),,(loglim

2)(

2

2

Zvi Wiener ContTimeFin - 8 slide 28

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

Zvi Wiener ContTimeFin - 8 slide 29

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.

Zvi Wiener ContTimeFin - 8 slide 30

Extension of Vasicek

Hull, White

dZtdtrbtatdr )())(()(

dZdtrdr )(

Zvi Wiener ContTimeFin - 8 slide 31

CIR model

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

dZrdtrdr )(

dtr

Prdt

t

Pdr

r

PdP

2

22

2

Zvi Wiener ContTimeFin - 8 slide 32

CIR model

dtr

Prdt

t

Pdr

r

PdP

2

22

2

2

22

2)(

r

Pr

t

P

r

Pr

dt

dPE

Zvi Wiener ContTimeFin - 8 slide 33

CIR model

rTtBeTtATtrP ),(),(),,(

22

)(

)(

2

)(

2)(

2

21)(

12),(

21)(

2),(

2

tT

tT

tT

tT

e

eTtB

e

eTtA

Zvi Wiener ContTimeFin - 8 slide 34

CIR modelBond prices are lognormally distributed with parameters:

dZtrdttrP

dP),(),(

rTtBtr

TtqBrtr

),(),(

),(1),(

Zvi Wiener ContTimeFin - 8 slide 35

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

22 2)(

2),,(

q

qtrR

Different types of possible shapes.

Zvi Wiener ContTimeFin - 8 slide 36

One Factor TS Models

dZrtt

dtrrtrttdr

t

tttt

)()(

log)()()(

21

321

Zvi Wiener ContTimeFin - 8 slide 37

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

Zvi Wiener ContTimeFin - 8 slide 38

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

Zvi Wiener ContTimeFin - 8 slide 39

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.

Zvi Wiener ContTimeFin - 8 slide 40

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.

Zvi Wiener ContTimeFin - 8 slide 41

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),(

Zvi Wiener ContTimeFin - 8 slide 42

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.

Zvi Wiener ContTimeFin - 8 slide 43

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

Zvi Wiener ContTimeFin - 8 slide 44

TS Derivatives

),(),(),( ,,

rgdssrhEtrF t

t

sstQtt

s

t

vst dvrexp,

Zvi Wiener ContTimeFin - 8 slide 45

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

Zvi Wiener ContTimeFin - 8 slide 46

Bond Option

A European option on a bond is described by

setting

h(x, t) = 0,

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

Zvi Wiener ContTimeFin - 8 slide 47

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.

Zvi Wiener ContTimeFin - 8 slide 48

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)

Zvi Wiener ContTimeFin - 8 slide 49

Floor

Similar to a cap, but with maximal rate

instead of minimal:

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

g(r,) = 1 (sometimes 0)

Zvi Wiener ContTimeFin - 8 slide 50

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.

Zvi Wiener ContTimeFin - 8 slide 51

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

Zvi Wiener ContTimeFin - 8 slide 52

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.

Zvi Wiener ContTimeFin - 8 slide 53

Monte Carlo

X() - random variable

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

Zvi Wiener ContTimeFin - 8 slide 54

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

Zvi Wiener ContTimeFin - 8 slide 55

Monte Carlo

Calculate the variance of the new variable:

*)()()( YYXX

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

Zvi Wiener ContTimeFin - 8 slide 56

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

Zvi Wiener ContTimeFin - 8 slide 57

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.