ch. 15, from T. Bjork
Download
Report
Transcript ch. 15, from T. Bjork
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
slide 3
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
slide 4
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
Zvi Wiener
R (T S )
p(t , S )
p(t , T )
FinModels - 15
slide 5
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
slide 6
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
slide 7
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
slide 8
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
Zvi Wiener
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
Zvi Wiener
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
Zvi Wiener
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
- 15
T
Zvi Wiener
slide 15
Proof of Proposition 15.5
Zvi Wiener
FinModels - 15
slide 16
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
slide 17
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
Zvi Wiener
1
p(Ti 1 , Ti )
FinModels - 15
1
slide 18
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
slide 19
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
Zvi Wiener
T1
Tn-1
FinModels - 15
Tn
slide 20
Forward Swap Settled in Arrears
If you swap a fixed rate for a floating rate
(LIBOR), then at time Ti, you will receive
KL(Ti 1, Ti ) Kci
where ci is a coupon of a floater. And at Ti
you will pay the amount
K R
Net cashflow
Zvi Wiener
K L(Ti 1, Ti ) R
FinModels - 15
slide 21
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
Zvi Wiener
FinModels - 15
slide 25
Macaulay Duration
Definition of duration, assuming t=0.
n
D
Zvi Wiener
T c e
i 1
Ti y
i i
p
FinModels - 15
slide 26
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
Zvi Wiener
FinModels - 15
slide 28
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
Zvi Wiener
FinModels - 15
slide 29
Convexity
p
C 2
y
2
$
r
Zvi Wiener
FinModels - 15
slide 30
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