Stefanica D. Solutions Manual: A Primer For The Mathematics Of Financial Engineering

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194

CHAPTER

LAGRANGE

MULTIPLIERS.

NEWTON'S

METHOD.

8.1.

SOLUTIONS

CHAPTER

EXERCISES

195

Problem

8: Use bootstrapping

obtain a continuously compounded zero

rate

curve given

the

prices of

the

following semiannual coupon bonds:

interest

, and with

r2(0

,t) corresponding

semi-annually compounded inter-

est.

Then

产川川

衍叫州

川呻

(ρ

川

By solving for

rc(O

,t) in (8.29),

在

that

几

，

=咐

+23

二

~r)

巧

，

t) In (

r2(~'

)"(~"))

巧

，

t) In (

十

句

:OJ))2/T2

仰

for

the

last inequality

used

the

fact

that

(1+

<民

'v'

> 0,

for x = 2/r2(0,t). In other words

the

semi-annually compounded zero rate

curve is higher

than

the

continuously compounded zero

rate

curve if

both

curves have

the

same discount factors.

While a rigorous proof is much more technical

the

same happens if

the

two curves are obtained by bootstrapping from

the

same set of bonds, i.e.,

the

zero rates corresponding to each bond

maturity

are higher if interest

compounded semi-annually

than

if interest

compounded continuously.

This

done sequentially,beginning with

the

zero rates corresponding

the

shortest bond

maturity

and moving

the

zero rates corresponding

the

longest bond

maturity

one bond

maturity

time.

口

(8.33)

(8.32)

(8.30)

γ(

咐

叫

γ(0

，

= 0.049370 = 4.9370%.

The

third

bond pays coupons in 2, 8,

, and

months, when

also

pays

the

face value of

the

bond.

The

凡

103 =

叫去(喝)

叫去(叫)

(一))叫

-ZT(072))(8

叫

马(山(

(\)

飞，

)

}\\

} J

Since

assumed

that

the

zero

rate

curve

linear on

the

intervals

0.5]

and

[0.5

the

zero rates

r(O

，

是)

and

r(O

，

击)

are known and can be obtained

by linear interpolation as follows:

命

(070)7(0705)=

0.0502

叫

4r(0,0.5) +

2γ(0

，

= 0.050214.

Let

γ(0

，

f§).

Since

r(O

,t) is linear

川

interval

f§

find

that

衍

，

(8.34)

12}

From (8.34)

follows

that

the

formula (8.31) can

written as

( 1

(一\\(

8\\

103 =

仪

(

-~r

(

，专门+

叫了(

1~2)

)

/6γ(0

，

飞

叫

-i

\ V ,

~W)

103

叫

-~x

蚓

and obtain

that

Solution: We know

that

r(O

= 0.05.

The

览

six

∞

丑叫

computed from

the

price of

the

6ι

一

囚工丑

∞

丑时

此

ths

曰

zero

∞

oupon

bOI

口

(

100\

r(0,

0.5)

2ln

(一一

) = 0.050636 = 5.0636%

飞

97.5

Using (8.30),

can solve for

the

zero

rate

r(O

from

the

formula given

the

price of

the

one year bond,i.e.,

100 _ 2.5

O.5)

+ 102.5

Assume

that

the

overnight

rate

and

that

the

zero

rate

curve

linear

the

following time intervals:

110

,0.5];

[0.5

,1]; 1

I~'

-3-1;

言，

Price

97.5

100

103

102

103

Coupon

Rate

NIaturity

6 months

1 year

months

5 years

196

CHAPTER

LAGRANGE

MULTIPLIERS.

NEWTON'S

METHOD.

8.2.

SUPPLEMENTAL

EXERCISES

197

where r

，

元)

and r

，

击)

are given by (8.32) and (8.33),

respecti

叫

Using

Newtor

内

method

solve for x in (8.35),

自 nd

时

= 0.052983, and

therefore

(

20\

、~=

I = 5.2983%.

飞

'12

Bootstrapping for

the

fourth and fifth bonds proceed similarly. For exam-

pIe

the

fourth bond makes coupon payments in 4,

, and

months.

The

zero rates corresponding

coupon dates less

than

months,

i.e.,

the

coupon dates 4,

and

months, can be obtained from

the

part

the

zero curve

that

was already determined. By setting x

γ(0

古)

and assuming

that

the

zero rate curve is linear between

months and

months,

the

zero rates corresponding

and

months can be

written in terms of

x. Thus,

the

pricing formula for

the

fourth bond becomes

a nonlinear equation in

x which can be solved using Newton's method.

The

zero

rate

r (0,

then

determined

Using bootstrapping and Newton's method

obtain

that

。一)

4.5

帆

r(0

，

叫

(40

8.2

Supplemental

Exercises

(i)

the

盯

rent

zero

rate

curve

，

t) =

0.025

由叫古)十

find

the

yield of a four year semiannual coupon bond with coupon

rate

6%.

Assume

that

interest is compounded continuously

and

that

the

face value of

the

bond

100.

(ii)

the

zero rates have a parallel shift up by

1~0

，

and,

?O~

s.:

sis points,respectively,i.e.,

ifthe

zero rate curve changes from

rl(O

r2(0

,t) = rl

+dr

,with

γ=

{0.001,0.002,0.005,0.01,0.02},find

out

by how much does

the

yield of

the

bond increase in each case.

Note:

general, a small parallel shift in

the

zero

rate

curves results in

a shift of similar size and direction for

the

yield of most bonds (possibly

with

the

exception of bonds with

吨

maturity).

This assumption will

tested for

the

bond considered here for parallel shifts ranging from

small shifts (ten basis points)

large

曲

Summarizing,

the

zero

rate

curve obtained by bootstrapping

given by

(

2\/

民\

r(O

= 0.05;

riO

1-("\

I = 0.050212;

riO

1~("\

I = 0.050214;

飞，

}

飞，

}

= 0.052983; r

(叫

72)=om3;

叫叶=

0.0

铅

326;γ(0

，

= 0.032119,

and is linear on

the

intervals

Consider a six months

at-th

money call on an underlying asset follow-

ing a lognormal distribution with volatility 30% and paying dividends

continuously

rate

Assume

that

the

interest rates are constant

4%.

Show

that

there is a unique positive value of q such

that

.6.(

C) = 0.5,

and

find

that

val~e

using Newton's method.

How

does

this value of

q compare

十亏?

vhu

m-3

5-3

434

vhu

nunu

The

following prices of

the

easury instruments are given:

Coupon

Rate

Price

6 -

lonth

T-bill 0 99.4565

- Month T-bill 0 98.6196

2 - Year

T-bond

1011

号

3 - Year

T-bond

4.5

107~I

5 - Year T-bond 3.125

102

立

- Year T-bond 4

103~';

The

Treasury bonds pay semiannual coupons. Assume

that

interest is

continuously compounded.

(i) Use

bootstrappi

吨

obtain a zero

rate

curve from the prices of

the

6-months and

12-monthsτ

easury

bills, and of

the

2-year, 5-year and

10-year Treasury bonds;

198

CHAPTER

LAGRANGE

IULTIPLIERS.

NEWTON'S

METHOD.

8.3.

SOLUTIONS

SUPPLEMENTAL

EXERCISES

199

(ii) Find

the

relative

prici

吨

error

correspondi

吨

the

3-year Treasury

bond if

the

zero

rate

curve obtained

part

(i) is used. In

other

words,

price a 3-year semiannual coupon bond with 4.5 coupon

rate

and find

its relative error

the

price

107

去

the

3-year

easury bond

8.3

Solutions

Supplemental

Exercises

Problem

1: (i)

the

current zero rate curve

，

= 0.025

+忐叫古

)+τ:71

且

the

yield of a four year semiannual coupon bond with coupon

rate

6%.

Assume

that

interest

compounded continuously and

that

the

face value of

the

bond is 100.

(ii)

the

zero rates have a parallel shift up by

, 100, and

200

basis points,

respecti

叫

，

, if

the

zero

rate

curve

cha

吨

from

'r1

,t)

r2(0,t)

η(OJ)+d71with

dT={0.00170.00270.00570.0170.02}7and

out

how much does

the

yield of

the

bond increase in each case.

SoJUUonJ(i)TEle bond provides coupOBpaymmts equal

to3every

sixmo

毗

until 3.5 years from now, and a final cash

flow

of 103

且

four

years. By

disc

，?

u~t

，

i?g

t~is

ca~h

flo

;V~

the.

resent using

the

zero

rate

cur~e

直

that

the

value of

the

bond

乌

工

兰

倪吨叫叶

寸(-一仆俨

η1

= 106.1995. (8.36)

The

yield of

the

bOIId

is found by solving

the

formuia for

the

price of

the

bond in terms of its yield, i.e, by solving

3 exp (

-y

D+ 103

exp(

一句)

(8.37)

for

U7WIleEe

BIis

given by

(836)7i.e-7BI=1061995.Using

NewtOI

内

method,

obtain

that

the

yield of

the

bond

(ii)

the

zero rates

increa

毗

the

value of

the

bond decreases, and therefore

the

yield of

the

bond will increase. Our goal here is

investigate whether a

parallel shift of

the

zero curve up by dr results in an

increωe

the

yield of

the

bond also equal

When

the

zero rates increase from

,t) to

俨

2(0

，

t) =

r1(0

,t) + dr ,

the

value of

the

bond decreases from B

given by (8.36)

了

叫

(

-r2

( 0,

)斗+

103

吨

(-4

州，

4))

仨

飞飞，

2) 2)

The

new yield of

the

bond,denoted by

,will be larger

than

the

initial yield

到，

and

obtained by solving

B2 =

￡

仪叫叫叶

寸(-一叶

Uω2

for

,where B

is given by (8.38).

For parallel shifts equal

dr = {0.001,0.002,0.005,0.01,0.02},

obtain

the

following bond prices and yields:

Zero

rate

shift New bond price New yield

Yield increase

的

-Y

10bp = 0.001 105.8150

0.043511

0.00099979

20bp = 0.002 105

319

0.044510

0.00199957

50bp = 0.005 104.2915

0.047510

0.00499893

100bp = 0.01 102

199

0.052509

0.00999784

200bp = 0.02 98.7829

0.062506

0.01999562

As expected

the

increase of

the

yield of

the

bond

slightly

smalle

飞

but

very close

the

parallel shift of

the

zero

rate

curve,

i.e.

，的

-Y

自

dr.

口

Problem

2: Consider a six months

at-th

money call on an underlying asset

following a lognormal distribution with volatility 30%

and

paying dividends

continuously

rate

Assume

that

the

interest rates are constant

4%.

Show

that

there

a unique positive value of q such

that

.6.(

C) =

0.5

, an

白

口

that

value

岳

口

Newtωon

旷

l'S

method. How does this value of q compare

号?

Solution: Recall

that

the

Delta of a

plai

口

vanilla

call option on

underlying

asset paying dividends continuously

rate

Y = 0.042511 = 4.2511%.

.6.(

C) =

N(d

)

(8.39)

200

CHAPTER

LAGRANGE

MULTIPLIERS.

NEWTON'S

METHOD.

8.3.

SOLUTIONS

SUPPLEMENTAL

EXERCISES

201

川♂

σf

一-

From (8.39) and (8.40),

find

that

叮

σd

飞

~(C)

)

Iσ;

)

other words, if

the

interest rates are flat

the

Delta

of a six

months

at-th

money call option on

口

日

derlying

asset with volatility 30%

is equal

0.5 if

the

underlying asset pays 6.69% dividends continuously.

兰

直缸

from

(何

8.ι40

创)

and

.4钊钊

tha

创

出

l=Oa

卫

~(C)

巳

一叫

\V(

仰

0.5e

一

矿

< 0.5.

Since

~(C)

is a decreasing function of q,we

obtain

that

the

value of q such

that

(C)

= 0.5 must

lower

than

十号

Indeed

，

the

value previously

obtained for

q satisfies this condition,i.e.,

where

(去)

(r-q

十号

\(/

‘

飞

For

at-th

money option,i.e.,for S = K ,

find

that

q = 0.066906 <

附

十二口

is easy

see

that

~(C)

is a decreasing function of

the

dividend

rate

q, smce

Problem

The

following prices of

the

Treasury instruments are given:

θA

θq

8d1

-Te

句

)

十

俨卢

叩咐川

川/气(刷

出

ω1)

‘~

中

飞

一

卡

口

一庐句川

川(刷

仇创

十+

俨一吁

飞

TτE

乒

一亏叮{

一工立土

'l:7r

\σ/

6 - Month T-bill

- Month T-bill

2 - Year

T-bond

3 - Year

T-bond

5 - Year

T-bond

- Year

T-bond

Coupon

Rate

4.5

3.125

Price

565

98.6196

1011

旦

107

里

102

立

~?b

103~'~

When

q = 0,

find

that

(0) = N

(手+手)

…5

J!!

炉

(C)

豆豆

(e-

句

))

= 0

The

Treasury bonds pay semiannual coupons. Assume

that

interest is con-

tinuously compounded.

(i) Use bootstrapping

obtain

a zero

rate

curve from

the

prices of

the

一

months

and

12-叮-

￥

easur)γy

bonds;

(ii)

Find

the

relative pricing error corresponding

the

3-year

easury bond

the

zero

rate

curve obtained

part

(i) is used. In other words, price a

3-year semiannual coupon bond with 4.5 coupon

rate

and

五日

its relative

error

the

price

107~

the

3-year

eas

町

bond

Solution: (i)

The

6-mo

叫

and

12-months zero rates can be obtained di-

rectly from

the

prices of

the

easury

bills, i.e.,

Also, since 0 <

N(d

)

< 1,

follows

that

We conclude

that

~(C)

is a decreasing function of q

and

that

,for

主

，

the

values of

~(C)

decrease from

N(O)

> 0.5

Therefore, there exists

a unique value

q > 0 such

that

~(C)

= 0.5. From

follows

that

this value can be obtained by solving for x

the

nonlinear equation f

(x)

= 0,

where

叮

σn\

叫

' N I

+ v

~.L

I - 0.5.

1σ

二~

Using Newton's

method

obtain

that

q = x = 0.066906.

21n

(品

= 1.09%;

叫品)

究

r(0,

0.5)

202

CHAPTER

LAGRANGE

MULTIPLIERS.

NEWTON'S

IETHOD.

Using bootstrapping,

obtain

the

following 2-year, 5-year and 10-year

zero rates:

γ(0

，

10)

2099%;

2.6824%;

3.7371%.

(ii)

The

zero rates computed above correspond

the

followi

吨

zero

rate

curve which

piecewise linear between consecutive bond maturities:

(到一

1)γ(

+ 2

- t)

,0.5), if

0.5

三

-1)

，

十

- t) 1'

, if

三

)平

γ(0

，

+于

γ(0

，

if 2:::;t:::;5;

孚

γ(0

，

10)

+半响的，

if 5 < t <

With

this zero rate curve,

the

value of

the

3-year semiannual coupon bond

with 4.5 coupon rate

vhu

\liI/

-z-2

\1111/

t-2

/It--\

/III-\

The

price of

the

3-year Treasury bond was given to be 107.5625. There-

fore

the

relative pricing error given by

the

bootstrapped zero

rate

curve

which does not include

the

3-year bond

1107.5625

108.19301

0.005862 -

0.59%.

口

107.5625

Bibliography

[1]

Milton Abramowitz

时

Irene

Stegun.

Handbook

of Mathematical Func-

tions. National Bureau of Standards

Gaithersburg

，扣

1aryland

，

10th cor-

rected printing edition

, 1970.

[2]

Dan

Stefa

旧

ca.

lathematical Primer with Numerical

lethods for Fi-

nancial Engineering.

Press, New York, 2007.

203