financial models 15

Jan-1999T.Bjork, Arbitrage Theory in

Continuous TimeForeign Currency,

Bank of Israel

Zvi Wiener

02-588-3049http://pluto.mscc.huji.ac.il/~mswiener/zvi.html

Financial Models 15

Zvi Wiener FinModels - 15 slide 2

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.


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.


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 ...


The simple forward rate LIBOR - L is the solution of:

),(

),()(1

Ttp

StpLST

The continuously compounded forward rate R is the solution of:

),(

),()(

Ttp

Stpe STR


Definition 15.2

The simple forward rate for [S,T] contracted at t (LIBOR forward rate) is

),()(

),(),(),;(

TtpST

StpTtpTStL

The simple spot rate for [S,T] LIBOR spot rate is

),()(

1),(),(

TSpST

TSpTSL


Definition 15.2

The continuously compounded forward rate for [S,T] contracted at t is

ST

StpTtpTStR

),(log),(log

),;(

The continuously compounded spot rate for [S,T] is

ST

TSpTSR

),(log),(


Definition 15.2

The instantaneous forward rate with maturity T contracted at t is

T

TtpTtf

),(log

),(

The instantaneous short rate at time t is

),()( ttftr


Definition 15.3

The money market account process is

t

t dssrB0

)(exp

Note that here t means some time moment in the future. This means

1)0(

)()()(

B

dttBtrtdB


Lemma 15.4

For t s T we have

T

s

duutfstpTtp ),(exp),(),(

And in particular

T

t

duutfTtp ),(exp),(


Models of Bond Market

Specify the dynamic of short rate

Specify the dynamic of bond prices

Specify the dynamic of forward rates


Important Relations

Short rate dynamics

dr(t)= a(t)dt + b(t)dW(t) (15.1)

Bond Price dynamics (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


Proposition 15.5We do NOT assume that there is no arbitrage!

),(),(

),(),(),(),(

TtvTt

TtmTtvTtvTt

T

TT

If p(t,T) satisfies (15.2), then for the forward

rate dynamics


Proposition 15.5We do NOT assume that there is no arbitrage!

),()(

),(),()(

tttb

ttttfta T

If f(t,T) satisfies (15.3), then the short rate

dynamics


Proposition 15.5

)(),(),(

),(2

1),()(),(),(

2

tdWTtSTtp

dtTtSTtAtrTtpTtdp

If f(t,T) satisfies (15.3), then the bond price dynamics

T

t

T

t

dsstTtS

dsstTtA

),(),(

),(),(


Proof of Proposition 15.5


Fixed Coupon Bonds

n

iiin TtpcTtpKtp

1

),(),()(

n

iin TtprTtpKtp

1

),(),()(

KTTrciTT iiiii 10


Floating Rate Bonds

1),(

1

1

ii

i TTpc

KTTLTTc iiiii ),( 11

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.


1),(

1

1

ii

i TTpc

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

iiin TtpTtpTtptp

11 ),(),(),()(

),()( 0Ttptp


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 T1 Tn-1 Tn



If you swap a fixed rate for a floating rate (LIBOR), then at time Ti, you will receive

iii KcTTLK ),( 1where ci is a coupon of a floater. And at Ti you

will pay the amount

RK

Net cashflow RTTLK ii ),( 1



At t < T0 the value of this payment is

),()1(),( 1 ii TtpRKTtKp

The total value of the swap at time t is then

n

iii TtpRTtpKt

11 ),()1(),()(


Proposition 15.7

At time t=0, the swap rate is given by

n

ii

n

Tp

TpTpR

1

0

),0(

),0(),0(


Zero Coupon Yield

The continuously compounded zero coupon yield y(t,T) is given by

tT

TtpTty

),(log),(

),()(),( TtytTeTtp

For a fixed t the function y(t,T) is called the zero coupon yield curve.


The Yield to Maturity

The yield to maturity of a fixed coupon bond y is given by

n

i

ytTi

iectp1

)()(


Macaulay Duration

Definition of duration, assuming t=0.

p

ecTD

n

i

yTii

i

1


Macaulay Duration

What is the duration of a zero coupon bond?

T

tt

tT

tt y

CFt

iceBondwtD

11 )1(Pr

1

A weighted sum of times to maturities of each coupon.


Meaning of Duration

Dpecdy

d

dy

dp n

i

yTi

i

1

r

$


Proposition 15.12 TS of IR

With a term structure of IR (note yi), the

duration can be expressed as:

Dpecds

d

s

n

i

syTi

ii

01

)(

p

ecTD

n

i

yTii

ii

1


Convexity

r

$

2

2

y

pC


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=0hint: it is equal to forward rate


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.

financial models 15

Documents