Transcript No Slide Title

Financial Engineering
Term Structure Models
Zvi Wiener
[email protected]
tel: 02-588-3049
Zvi Wiener
ContTimeFin - 8
slide 1
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
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slide 2
Main Approaches

Black’s Model (modification of
BS).

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No-Arbitrage Model.
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slide 3
Notations
D - face value (notional amount)
C - coupon payments (as % of par), yearly
N - maturity
N
C
D
P0  

t
N
1  rN 
t 1 1  rt 
See Benninga, Wiener, MMA in Education, vol. 7, No. 2, 1998
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slide 4
Continuous Version
Denote by Ctdt the payment between
t and t+dt, then the bond price is given by:
N
P0   e Ct dt
 rt t
0
Principal should be written as Dirac’s delta.
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slide 5
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.
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slide 6
Forward IR
r3=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
Year 5
-$1,000(1.11)5 = -$1658
= $1331
Is identical to a 2-year loan starting at year 3.
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slide 7
Forward IR
 1.11

3
 1.1
5



0.5
 1.12517
Forward interest rate from t to t+n.
 1  rt  n t  n 


 1  r t 
t


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1
n
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r
f
t ,n
slide 8
Forward IR
 1  rt  n 

 1  r t
t

t n




1
n
r
f
t ,n
Continuous compounding
 e (t  n ) rt n

trt
 e
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


1
n
e
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rt ,fn
slide 9
Forward IR
 e (t  n ) rt n
 tr
t
e




1
n
e
( t  n ) rt n
e
P(t , t  n)  e
e
e
rt ,fn
 nrt ,fn
trt
e
nrt ,fn
(t  n ) rt n trt
e
 P(t , t  n)
 log P(t , t  n)
log P(t , t  n)

r 
n
n
f
t ,n
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slide 10
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!
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slide 11
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
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slide 12
Estimating the TS
We 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.
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slide 13
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.
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slide 14
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
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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,
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b = -2.21034,
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c=1.2214
slide 16
Arbitrage in a flat TS
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slide 17
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.
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Yield
Denote 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.
P(r, t , t  T )  e
 R ( r ,t ,T )T
1
R(r , t , T )   log P(r , t , t  T )
T
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slide 19
Yield
The relation between forward rates and yield:
1
R(r , t , T ) 
T
t T
 F (r, t , s)ds
t
When interest are continuously compounded
the average of forward rates gives the yield.
 log P (r , t , t  s )
F (r , t , T )  
s
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TS model
Assume that interest rates follow a diffusion
process.
dr   (r, t )dt   (r, t )dZ
What is the price of a pure discount bond P(r,t,T)?
P
P
  P
dP 
dr 
dt 
dt
2
r
t
2 r
2
2
Implicit one factor assumption!
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slide 21
TS model
Substituting dr we obtain:
2



dP   Pr  Pt 
Prr dt  Pr dZ
2


Taking expectation and dividing by dt we get:

 dP 
E   Pr  Pt 
Prr
2
 dt 
2
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slide 22
TS model
Using equilibrium pricing models assume that

 dP 
E   r (1   ) P  Pr  Pt 
Prr
2
 dt 
2
Here  is the risk premium. The basic bond
pricing equation is (Merton 1971,1973):
0  Pr  Pt 
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
2
2
Prr  r (1   ) P
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slide 23
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.
E Ri   r
i
r

q
i
Sharpe ratio
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slide 24
TS model
For a pure discount bond we have:
P  dP
dP
Pr
 1
 ...dt 
dZ
P
P
P
Thus by Ito’s lemma
i 
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 ( r , t ) Pr
P
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slide 25
TS model
Hence for the risk premium we have
q (r , t ) Pr
r  q i 
P
The basic equation becomes
0  Pr  Pt 
0  Pr  Pt 
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
2
2

2
2
Prr  r (1   ) P
Prr  rP  qPr
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slide 26
Vasicek’s model
Ornstein-Uhlenbeck process
dr   (  r )dt  dZ
P( r , t , t  T ) 
1

T
T 2 
exp 1  e R()  r   TR()  3 1  e

4




2


q 
log P(r , t , t  T )
R()     2  lim 
 2 T 
T
2
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slide 27
Vasicek’s model
Discrete modeling
dr   (  r )dt  dZ
r   (  r)t  Z t
Negative interest rates.
Can be used for example for real interest rates.
T
Et r(t  T )    (r(t )   )e
2

T

Vart r (t  T ) 
1 e 
2
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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.
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slide 29
Extension of Vasicek
dr   (  r )dt  dZ
Hull, White
dr   (t )  a(t )(b  r )dt   (t )dZ
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slide 30
CIR model
dr   (  r)dt   r dZ
Precludes negative IR, but under some
conditions zero can be reached.
P
P
r  P
dP 
dr 
dt 
dt
2
r
t
2 r
2
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2
slide 31
CIR model
P
P
r  P
dP 
dr 
dt 
dt
2
r
t
2 r
2
2
P P r  P
 dP 
E     (  r )


2
r t
2 r
 dt 
2
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2
slide 32
CIR model


P(r, t , T ) 2e


A(t , T ) 
 (   ) e (T t )  1  2 


 (T t )
2e
1
B (t , T ) 
 (T t )
(   ) e
 1  2

T t
(  )  B (t ,T ) r
A(t, T )e2


    2
2
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2  2



2
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slide 33
CIR model
Bond prices are lognormally distributed with
parameters:
dP
  (r , t )dt   (r , t )dZ
P
 (r , t )  r 1  qB(t , T ) 
 (r , t )   B(t , T ) r
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slide 34
CIR model
As the time to maturity lengthens, the yield
tends to the limit:
2
R(r , t , ) 
  q 
  (  q )  2
2
2
Different types of possible shapes.
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slide 35
One Factor TS Models
drt  1 (t )   2 (t )rt   3 (t )rt log rt dt 
1 (t )   2 (t )rt 

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dZ
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slide 36
drt  1 (t )   2 (t )rt   3 (t )rt log rt dt 
1 (t )   2 (t )rt 

1 2
Cox-Ingersoll-Ross * *
Pearson-Sun
* *
Dothan
Brennan-Schwartz * *
Merton (Ho-Lee)
*
Vasicek
* *
Black-Karasinski
*
Constantinides-Ingersoll
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dZ
3
1
*
2
*
*
*
*
*
*
*
ContTimeFin - 8
*
*

0.5
0.5
1.0
1.0
1.0
1.0
1.0
1.5
slide 37
Black-Derman-Toy
The BDT model is given by
 (t ) Z (t )
rt  U (t )e
for some functions U and .
Find conditions on 2, 3, and 2 under which
the Black-Karasinski model specializes to the
BDT model.
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slide 38
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 riskneutral measure).
Special cases: Vasicek and Merton models.
In this case a negative 2 is mean reversion.
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slide 39
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.
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slide 40
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:
f (r, t )  expa(T  t )  b(T  t )r 
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slide 41
The Affine One-Factor Models
f ( x, t )  expa(T  t )  b(T  t ) x
The yield for all t is affine in r:
log f ( x, t )
yield  
T t
Vasicek, CIR, Merton (Ho-Lee), Pearson-Sun.
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slide 42
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:



Q
F (rt , t )  Et    t , s h(rs , s)ds   t , g (r , )
t

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slide 43
TS Derivatives



Q
F (rt , t )  Et    t , s h(rs , s)ds   t , g (r , )
t

 t ,s
Zvi Wiener


 exp   rv dv 
 t

s
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slide 44
TS Derivatives
By Feynman-Kac theorem it can be
equivalently written as a solution of PDE:
1
2
Ft  Fx  ( x, t )  Fxx ( x, t )  xF  h( x, t )
2
With boundary conditions:
F ( x, )  g ( x, )
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slide 45
Bond Option
A European option on a bond is described by
setting
h(x, t) = 0,
g(x, ) = Max( f(x, ) - K, 0).
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slide 46
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.
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slide 47
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)
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slide 48
Floor
Similar to a cap, but with maximal rate
instead of minimal:
h(rt, t) = Max(rt,r*)
g(r,) = 1 (sometimes 0)
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slide 49
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.
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slide 50
MBS
Assume that the coupon rate is  and principal
repayment is at a constant rate h.
dpt
 pt  h
dt
For a given initial principal p0. The schedule
is chosen so that at time T the loan is repaid.

h  t h
pt   p0  e 



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slide 51
MBS
Home 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.
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slide 52
Monte Carlo
X() - random variable
Let Y be a similar variable, which is
correlated with X but for which we have an
analytic formula.
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slide 53
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)
X  ( )  X ( )   Y ( )  Y *
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slide 54
Monte Carlo
X  ( )  X ( )   Y ( )  Y *
Calculate the variance of the new variable:
var[X  ]  var[X ]  2 cov[X , Y ]   var[Y ]
2
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slide 55
Monte Carlo
var[X  ]  var[X ]  2 cov[X , Y ]   var[Y ]
2
If
2 cov[X , Y ]   2 var[Y ]
we can reduced variance!
The optimal value of the parameter  is
cov[X , Y ]
 
var[Y ]
*
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slide 56
Monte Carlo
This choice leads to the variance of the estimator
var[ X  * ]  (1   XY )var[ X ]
2
where  is the correlation coefficient between X
and Y.
Zvi Wiener
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slide 57