Financial Models 15
Zvi Wiener
02-588-3049
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
Jan-1999
T.Bjork, Arbitrage Theory in Continuous Time
Foreign Currency, Bank of Israel
Bonds and Interest Rates
Zero coupon bond = pure discount bond
T-bond, denote its price by p(t,T).
principal = face value,
coupon bond - equidistant payments as a % of
the face value, fixed and floating coupons.
Zvi Wiener
FinModels - 15
slide 2
Assumptions

There exists a frictionless market for T-
bonds for every T > 0

p(t, t) =1 for every t

for every t the price p(t, T) is differentiable
with respect to T.
Zvi Wiener
FinModels - 15
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Interest Rates
Let t < S < T, what is IR for [S, T]?

at time t sell one S-bond, get p(t, S)

buy p(t, S)/p(t,T) units of T-bond

cashflow at t is 0

cashflow at S is -$1

cashflow at T is p(t, S)/p(t,T)
the forward rate can be calculated ...
Zvi Wiener
FinModels - 15
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The simple forward rate LIBOR - L is the
solution of:
p(t , S )
1  (T  S ) L 
p(t , T )
The continuously compounded forward rate
R is the solution of:
e
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R (T  S )
p(t , S )

p(t , T )
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Definition 15.2
The simple forward rate for [S,T] contracted
at t (LIBOR forward rate) is
p(t , T )  p(t , S )
L(t; S , T )  
(T  S ) p(t , T )
The simple spot rate for [S,T] LIBOR spot
rate is
p( S , T )  1
L( S , T )  
(T  S ) p( S , T )
Zvi Wiener
FinModels - 15
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Definition 15.2
The continuously compounded forward
rate for [S,T] contracted at t is
log p(t , T )  log p(t , S )
R(t ; S , T )  
T S
The continuously compounded spot rate for
[S,T] is
log p( S , T )
R( S , T )  
T S
Zvi Wiener
FinModels - 15
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Definition 15.2
The instantaneous forward rate with
maturity T contracted at t is
 log p(t , T )
f (t , T )  
T
The instantaneous short rate at time t is
r (t )  f (t , t )
Zvi Wiener
FinModels - 15
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Definition 15.3
The money market account process is


Bt  exp  r ( s )ds 
0

t
Note that here t means some time moment in
the future. This means
dB(t )  r (t ) B(t )dt

B(0)  1
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FinModels - 15
slide 9
Lemma 15.4
For t  s  T we have


p(t , T )  p(t , s) exp   f (t , u )du 
 s

T
And in particular


p(t , T )  exp   f (t , u )du 
 t

T
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FinModels - 15
slide 10
Models of Bond Market

Specify the dynamic of short rate

Specify the dynamic of bond prices

Specify the dynamic of forward rates
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FinModels - 15
slide 11
Important Relations
Short rate dynamics
dr(t)= a(t)dt + b(t)dW(t)
Bond Price dynamics
(15.1)
(15.2)
dp(t,T)=p(t,T)m(t,T)dt+p(t,T)v(t,T)dW(t)
Forward rate dynamics
df(t,T)= (t,T)dt + (t,T)dW(t)
(15.3)
W is vector valued
Zvi Wiener
FinModels - 15
slide 12
Proposition 15.5
We do NOT assume that there is no arbitrage!
If p(t,T) satisfies (15.2), then for the forward
rate dynamics
 (t , T )  vT (t , T )v(t , T )  mT (t , T )

 (t , T )  vT (t , T )
Zvi Wiener
FinModels - 15
slide 13
Proposition 15.5
We do NOT assume that there is no arbitrage!
If f(t,T) satisfies (15.3), then the short rate
dynamics
a (t )  f T (t , t )   (t , t )

b(t )   (t , t )
Zvi Wiener
FinModels - 15
slide 14
Proposition 15.5
If f(t,T) satisfies (15.3), then the bond price dynamics
1
2

dp(t , T )  p(t , T ) r (t )  A(t , T )  S (t , T ) dt 
2


p(t , T ) S (t , T )dW (t )

 A(t , T )     (t , s)ds

t

T
S (t , T )    (t , s)ds


t

FinModels
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T
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slide 15
Proof of Proposition 15.5
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FinModels - 15
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Fixed Coupon Bonds
n
p(t )  K p(t , Tn )   ci p(t , Ti )
i 1
Ti  T0   i
ci  ri Ti  Ti 1 K


p(t )  K  p(t , Tn )  r  p(t , Ti ) 
i 1


n
Zvi Wiener
FinModels - 15
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Floating Rate Bonds
ci  Ti  Ti 1 L(Ti 1, Ti ) K
L(Ti-1,Ti) is known at Ti-1 but the coupon is
delivered at time Ti. Assume that K =1 and
payment dates are equally spaced.
ci 
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1
p(Ti 1 , Ti )
FinModels - 15
1
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ci 
1
p(Ti 1 , Ti )
1
This coupon will be paid at Ti. The value of -1
at time t is -p(t, Ti). The value of the first term
is p(t, Ti-1).
n
p(t )  p(t , Tn )    p(t , Ti 1 )  p(t , Ti )
i 1
p(t )  p(t , T0 )
Zvi Wiener
FinModels - 15
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Forward Swap Settled in Arrears
K - principal, R - swap rate,
rates are set at dates T0, T1, … Tn-1 and paid at
dates T1, … Tn.
T0
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T1
Tn-1
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Tn
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Forward Swap Settled in Arrears
If you swap a fixed rate for a floating rate
(LIBOR), then at time Ti, you will receive
KL(Ti 1, Ti )  Kci
where ci is a coupon of a floater. And at Ti
you will pay the amount
K R
Net cashflow
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K L(Ti 1, Ti )  R
FinModels - 15
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Forward Swap Settled in Arrears
At t < T0 the value of this payment is
Kp(t, Ti 1 )  K (1  R) p(t, Ti )
The total value of the swap at time t is then
n
(t )  K   p(t , Ti 1 )  (1  R) p(t , Ti )
i 1
Zvi Wiener
FinModels - 15
slide 22
Proposition 15.7
At time t=0, the swap rate is given by
R
p(0, T0 )  p(0, Tn )
n
  p (0, Ti )
i 1
Zvi Wiener
FinModels - 15
slide 23
Zero Coupon Yield
The continuously compounded zero coupon
yield y(t,T) is given by
log p(t , T )
y (t , T )  
T t
p(t , T )  e
(T t ) y ( t ,T )
For a fixed t the function y(t,T) is called
the zero coupon yield curve.
Zvi Wiener
FinModels - 15
slide 24
The Yield to Maturity
The yield to maturity of a fixed coupon
bond y is given by
n
p(t )   ci e
(Ti t ) y
i 1
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FinModels - 15
slide 25
Macaulay Duration
Definition of duration, assuming t=0.
n
D
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T c e
i 1
Ti y
i i
p
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Macaulay Duration
T
T
CFt
1
D   t wt 
t

t
Bond Price t 1 (1  y)
t 1
A weighted sum of times to maturities of each
coupon.
What is the duration of a zero coupon bond?
Zvi Wiener
FinModels - 15
slide 27
Meaning of Duration
dp d 
Ti y 
  ci e    Dp
dy dy  i 1

n
$
r
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FinModels - 15
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Proposition 15.12 TS of IR
With a term structure of IR (note yi), the
duration can be expressed as:
n
D
T c e
i 1
Ti yi
i i
p
d 
Ti ( yi  s ) 
 ci e
   Dp
ds  i 1
 s 0
n
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FinModels - 15
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Convexity
 p
C 2
y
2
$
r
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FinModels - 15
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FRA Forward Rate Agreement
A contract entered at t=0, where the parties (a
lender and a borrower) agree to let a certain
interest rate R*, act on a prespecified principal,
K, over some future time period [S,T].
Assuming continuous compounding we have
at time S: -K
at time T: KeR*(T-S)
Calculate the FRA rate R* which makes PV=0
hint: it is equal to forward rate
Zvi Wiener
FinModels - 15
slide 31
Exercise 15.7
Consider a consol bond, i.e. a bond which
will forever pay one unit of cash at t=1,2,…
Suppose that the market yield is y - flat.
Calculate the price of consol.
Find its duration.
Find an analytical formula for duration.
Compute the convexity of the consol.
Zvi Wiener
FinModels - 15
slide 32