fixed income securities - coursework report

7/17/2019 Fixed Income Securities - Coursework Report

http://slidepdf.com/reader/full/fixed-income-securities-coursework-report 1/15

MSc Financial Mathematics

Coursework

“Fixed Income Securities”

Antonis Iaponas

London 2011

Question 1



We have downloaded data for UK government gilts from the Debt Management Oce

(DMO)1. The dates we are eamining are the following! Wednesda" #1$1#$%##&'

Wednesda" #&$1#$%##&' Wednesda" 1$1#$%##&' Wednesda" %%$1#$%##& and

Wednesda" %$1#$%##&. *or each of the above dates we etract the data for redeemable

conventional bonds ecl+ding do+ble dated bonds' indeed bonds' ,er,et+al' ann+ities

and stri,s. Then we tr" to estimate -ve dierent Mc/+lloch c+rves constrained to secondorder.

We +se Matlab to etract the 0ero co+,on "ields from the "ields of the co+,on ,a"ing

bonds' incl+ded in o+r data sam,le. Using the inb+ilt f+nction 0bt"ield.m% giving as

in,+t the time to mat+rit"' the co+,on rate' the face val+e (arbitraril" set to 1##)' the

n+mber of co+,on ,a"ments ,er "ear (set to %' i.e. semiann+all")' the "ield and settle

date the reference date of the data this can be done faster and eectivel". The o+t,+t

of this f+nction is the 0ero co+,on "ield and the corres,onding time to mat+rit". The

f+nction +ses the bootstra,,ing method in order to etract the 0ero co+,on "ields.

Then we +se M2 3cel to estimate the -tted "ield as best as we can +sing a c+bic s,line

with two 4nots5 s,litting the estimated c+rve into three e6+all" s,aced ,arts which

corres,ond to dierent time (to mat+rit") intervals. We set o+r -rst 4not to ro+ghl" 7."ears (as #7. "ears re,resent the 1$8 of o+r data) and the second one to %# "ears. 9"

e,erimenting and +sing dierent 4not ,oints we fo+nd that these two (7. and %# "ears

res,ectivel") ,rovide +s with a better and smoother c+rve -t. We +se the s+m of s6+ared

errors between the act+al and the -tted "ields as an indicator of the best -t' and o+r

ob:ective is to minimi0e this error b" estimating the o,timal coecients for the s,line

(+sing M2 3cel solver).

The c+bic s,line we +se' constrained to second order is of the form!

r0 + a1*t + b1*t2 + c1*t3 , t ≤ 10.

r(t) = r(10) + a2*(t-10) + b2*(t-10)2 + c2*(t-10)3, 10 ≤ t ≤ 20.

r(20) + a3*(t-20) + b3*(t-20)2 + c3*(t-20)3, 20 ≤ t ≤ 30.

We ta4e the -rst derivative of each e6+ation and we want them to be e6+al at 4not ,oint

t;1# for the -rst and the second e6+ation and at 4not ,oint t;%# for the second and

third e6+ation. *+rthermore' at each 4not ,oint we re6+ire the val+es of each e6+ation to

be e6+al. These constraints ens+re that the res+lts given b" o+r estimation ,rovide a

smooth c+rve with no :+m,s at the level' or the slo,e.

O+r -rst estimated c+rve for the Wednesda" #1$1#$%##& is the one shown in <ra,h 1.

The c+rve shown is +,ward slo,ing. On October 1' %##& the ocial base rate for the

9an4 of 3ngland (9o3) was =. The last time it was changed was on >,ril 1#' %##&

?9loomberg Database@. The 9o3 anno+nces the ,+rchase of A7# million of bonds with

mat+rities of to 17 "ears in the net two months. 2+ch news ca+ses the demand for

these bonds to go +,' ,+shing their ,rices +, and their "ields down. Bowever' we can

slightl" see the dierence between <ra,h 1 and <ra,h %. The act+al dierence in a bond

with aro+nd 1% "ears to mat+rit" is one basis ,oint. On the same date' there are news

con-rming the 9o3 intention to lower the interest rates b" the end of net wee4

?9loomberg Database@. Therefore' in the near f+t+re an investor wo+ld e,ect the "ield

c+rve to move +, (,arallel shift) as bond ,rices go +, d+e to the decrease in the interest

rates. This fact is not obvio+s immediatel"' between the two wee4s that ta4es ,lace' b+t

it seems more noticeable in the "ield c+rve ,rod+ced for wee4 (es,eciall" for long term

1 www.dmo.gov.+4

% *or more details loo4 at htt,!$$www.mathwor4s.com$access$hel,des4$hel,$toolbo$-nance$inde.html

1 C a g e

http://www.dmo.gov.uk/


http://www.mathworks.com/access/helpdesk/help/toolbox/finance/index.html?/access/helpdesk/help/toolbox/finance

http://www.mathworks.com/access/helpdesk/help/toolbox/finance/index.html?/access/helpdesk/help/toolbox/finance




mat+rities as the" are inE+enced b" the e,ectations determined b" short rates

movements).

#.% .% 1#.% 1.% %#.% %.% 8#.%

Zero Coupon Cure - !ed 01"10"200#

$aturit%

&

i

e

'

d

rap 1 Zero Coupon Curve – Wed 01/10/2008.

>s e,ected from October 1' 9o3 c+t interest rates b" #.= in a coordinated move with

other central ban4s aro+nd the world. 2o on October &' the base interest rate for the 9o3

is 7.=. This gave rise to the two "ear notes. Ft seems that the "ield c+rve from wee4 1

to wee4 is getting stee,er in the short end as mar4et ,artici,ants seem to believe that

there are solid reasons for the interest rates to be driven f+rther down ?9loomberg

Database@.

>fter the 9o3 c+t the benchmar4 rate it is antici,ated +nder the *ischer e6+ation that!

r = rr + e

meaning' that the nominal rate e6+als the real one ,l+s the inEation e,ectations. Under

this notion inEation e,ectations aro+nd the world fell shar,l" after the coordinated

interest rate c+t ?9loomberg Database@' giving an indication of a severe -nancial crisis

(for vis+al ill+stration see <ra,h &).

#.% .% 1#.% 1.% %#.% %.% 8#.%

Zero Coupon Cure - !ed 0#"10"200#

$aturit%

&

i

e

'

d


% C a g e



On October &' the ban4 resc+e ,lan is anno+nced and the Treas+r" is going to borrow A%

billion in order to save ban4s facing insolvenc". This is going to be im,lemented b"

selling the corres,onding amo+nt in bonds. These sales are going to add to the c+rrent

heavil" loaded borrowing b+dget of the 9ritish government ?9loomberg Database@. 2+ch

an action is going to ca+se the s+,,l" of bonds to go +,' ,+shing the ,rices of c+rrent

iss+es of gilts of the same mat+rit" down' and as a res+lt the "ield going +, (i.e. a,arallel +,ward movement). >ltho+gh 9o3 :+st toda" c+t its benchmar4 interest rate'

economists ,redict that is going to do so again b" %= till the end of %##&. These

e,ectations of f+t+re interest rate c+ts b" the 9o3' +nder the c+rrent -nancial climate'

are reEected in the "ield c+rve as it moves +, in a ,arallel manner from wee4 to wee4

(altho+gh the dierences are minor d+e to the small time interval ta4en into

consideration).

#.% .% 1#.% 1.% %#.% %.% 8#.%


$aturit%

&

i

e

'

d


On October %%' the UK government anno+nces that it wo+ld increase s,ending and

accommodate borrowing in a wa" to avoid the foreseeable recession. We can see that

the long term ,art of the "ield c+rve is slightl" reverting (after the %# "ears mat+rit"

,oint). 2+ch sha,es of the "ield c+rve have been often +sed as indicators of recessions.

>t the end of October %##& (October %) the ,redictions for even lower interest rates are

still vivid. 3conomists e,ect the 9o3 to c+t even more the interest rates b" the end of

the -rst wee4 of Govember' and the" even forecast an all time low interest rate

?9loomberg Database@. 2+ch e,ectations in the mar4et ma4e the "ield c+rve more

,arallel +,wards as the c+t in interest rates is going to move bond ,rices +,.

8 C a g e



#.% .% 1#.% 1.% %#.% %.% 8#.%


$aturit%

&

i

e

'

d

rap Zero Coupon Curve – Wed 22/10/2008.

*rom October 1' the ,o+nd starts to de,reciate against U2 dollar (and 3+ro) as we can

see from <ra,h H. The ca+se is the interest rate c+ts b" the 9o3' and the e,ectations

that there are going to be more and bigger of them in the near f+t+re. >s a res+ltinvestors wo+ld b+" the 9ritish ,o+nd against the 3+ro (or the U2 Dollar) as the 9o3 is

going to c+t interest rates faster than the 3+ro,ean /entral 9an45 a meas+re against

slowing down economic growth. On October %%' the ,o+nd reaches a "ear low ca+sed

b" the s,ec+lation that the central ban4s are going to c+t f+rther down interest rates.

The ,o+nd reaches a "ear low as the signs of a recession of the 9ritish econom" are

getting more and more evident.

#.% .% 1#.% 1.% %#.% %.% 8#.%


$aturit%

&

i

e

'

d

rap Zero Coupon Curve – Wed 29/10/2008.

The signs of the UK econom" entering a recession ,eriod are also evident in the 9o3

I*inancial 2tabilit" Je,ort8. >ccording to <ra,h L' we can see that the 9o3 ,redicts a

growth rate for the UK econom" e6+al to #.%= for %##. The worries for the UK

econom" are greater than ever.

8 9an4 of 3ngland' I*inancial 2tabilit" Je,ort' October %##&' Fss+e Go. %7.

7 C a g e



rap / GB Pound depreciation aainst t!e "# $o%%ar durin 01/10/08 – &1/10/08.

#ource' B%oo()er $ata)ase.

> widel" +sed a,,roach to -t all these variables movements into a form+la in order to

have an idea of their eects on the short term rates' and conse6+entl" on the "ield c+rve

is Ta"lors r+le. Using s+ch a reaction f+nction as the one below!

r0 = r*t + a1*(t *) + a2*(t *) + a3*(et - e*) + a*(st - s*)

where! r# is the target short rate' rNt is a long term e6+ilibri+m interest rate' is the

inEation' g is the <D growth or +nem,lo"ment' e is the echange rate and s is the stoc4

mar4et. The variables mar4ed with a star re,resent a target val+e for each case.

>cting as I*ed Watchers we tr" to ,redict the variables on the right hand side so as to

obtain a val+e for the short term rates. >ccording to the r+le' a1 and a% m+st be of ,ositive sign5 meaning a high interest rate when inEation rises and <D growth rates are

at their higher levels (or a negative one for a% in the case of +nem,lo"ment being +sed

instead of <D). The coecient of the echange rate sho+ld be ,ositive im,l"ing that a

high short term rate 4ee,s the c+rrenc" Istrong.

>ccording to the above anal"sis of Ta"lors r+le the e,ectations ,revailing in the

econom" at the c+rrent mar4et climate with the ,o+nd being de,reciated' the <D

growth rate being forecasted negative for the coming months (and the +nem,lo"ment

rising steadil" till P+l" %##& Q see <ra,h 1#) and the inEation going down (see the drastic

decrease in October %##& Q <ra,h ) the short term rates are e,ected to fall drasticall"'

ass+ming that the theoreticall" determined signs of the coecients are correct.

C a g e



rap Bo* +orecast o, "- G$P Grot! ,or 2009. rap # Bo* Interest ate Cuts #ource'

.)%oo()er.co(.

>s short term rates are going down bond ,rices are going +, and as a res+lt their "ields

are going down' ca+sing the "ield c+rve to be stee,er in the short term (# to "ears) and

getting a more Eat sha,e in the mid and long term ,art. Fn the reall" long mat+rities

(more than % "ears) the "ield c+rve seems to get a downward slo,ing sha,e indicating acoming recession as mentioned above.

To come to a concl+sion' October %##& (the ,eriod we +sed to ,rod+ce o+r "ield c+rves)

can be described as t+rb+lent era of the -nancial crisis ,revailing in the world econom"

the last si months. <overnments' all over the world are c+tting interest rates to avoid

economic downt+rn' b+t at the same time the" seem tremendo+sl" afraid of the Ighost

named deEation. We co+ld also go f+rther in the disc+ssion considering the case of

central ban4s (the *ed' 9o3) which have been involved either in 6+antitative' or

6+alitative easing5 disc+ssing the im,lications of the latter strateg". Bowever' the drastic

changes in interest rates and other macroeconomic ,olic" variables were im,lemented

b" the UK government d+ring Govember (Jefer to <ra,h &)' and th+s having +sed s+ch

,eriod for o+r st+d" the res+lts wo+ld have been more obvio+s in o+r gra,hs.

rap "- Ination ate #ource' .statistics.ov.u3 . rap 10 "- "ne(p%o4(ent ate.

Question 2

The Rasice4 model is essentiall" a stochastic ,rocess +sed to model the ,ath of a

variable thro+gh time. Fn the contet of this co+rsewor4' we will be +sing a Rasice4 t",e

model to tr" and moc4 the ,ath of o+r 0ero "ield Mc/+lloch c+rves obtained in S+estion

1.

The Vasicek Model

The model is de-ned b" the stochastic dierential e6+ation (2D3)'

dr ( t )=k [ θ−r ( t ) ] dt +σdW (t )where! r(#)' 4' and V # and are all constant.

The sol+tion to this' i.e. ,rice of the bonds can be fo+nd +sing the form+la'

P (t ,T )= A (t ,T )e−B (t ,T ) r (t )

where'

7 2o+rce! www.tradingeconomics.com.

H C a g e

http://www.statistics.gov.uk/

http://www.tradingeconomics.com/






A (t , T )=exp [(θ− σ 2

2k 2 ) [B (t ,T )−(T −t ) ]− σ

2

4 k B(t ,T )2]

B ( t , T )=1

k

[1−e−k (T −t )]

*rom the ,rice' we can etract the 0ero co+,on "ield'

yzc ( t ,T )=−ln ( P(t , T ))

T −t

Calibration of the Model

The idea of calibration is to -nd o,timal val+es for o+r in,+t ,arameters (r(#)' 4' and )

that will match o+r ,rices modelled b" Rasice4 to the ,rices estimated b" o+r Mc/+llochc+rve. This was done b" +sing an iterative a,,roach which sim+ltaneo+sl" assigns in,+t

,arameters' con-ned to s+itable lower and +,,er bo+nds (#= r(#) X #=' #= 4

XH##='#= X#= and #= X%##=) and meas+res the acc+rac" of o+r model b"

+sing the s+m of the least s6+ares. >s this is not an eercise in n+merical techni6+es' we

are not concerned with com,+tational e,ense (i.e. s,eed) th+s we add constraints to

o+r calibrator that reEect this. We decided to calibrate o+r model in Matlab as it was ver"

eas" to im,lement and ,rod+ce acc+rate res+lts. Fn Matlab' we made +se of the

o,timi0ation toolbo which allows +s to assign the maim+m n+mber of iterations

allowed (set to 1#'###) and the acc+rac" within which to contin+e to -nd a more o,timal

set of ,arameters (we set this to 1#H as the "ields downloaded from www.dmo.gov.+4 are

re,orted to 1#H

Q so we never sto, tr"ing to -nd a better a,,roimation +ntil we achievethe best res+lt' or eceed the maim+m n+mber of iterations).

Results

J+nning the calibrator twice for all three mat+rit" ranges' calibrating with res,ect to the

reference ,oints (Tt) ; ?#.' 1' 1.' 1.@ for mat+rities Y % "ears' (Tt) ; ?1' 1.' 1#' 1%@

for mat+rities % Y "ears V & and (Tt) ; ?1#' 1%' %#' 8#@ for mat+rities V & we get the

following res+lts.

L C a g e



rap 11 6asice3 Ca%i)ration ,or a%% t!e re,erence points used.

J+nning the calibrator twice' we can clearl" see that there is a large degree of

randomness in the ,aths of the Rasice4 -tted c+rves' which are not within their

res,ective mat+rit" ranges!

• /alibration for Mat+rities Y % "ears ( ) estimates the Mc/+lloch c+rve ver" acc+ratel"

+, +ntil Mat+rit" ; % and then deviates randoml". Fn the case of the calibration for % th

October in <ra,h 1%' it estimates ver" acc+ratel" +, +ntil the 1%th "ear' however as this

does not occ+r in all the other r+ns (refer to <ra,hs 11' 1% Z 18) we can sa" that this is

d+e to l+c4.• /alibration for Mat+rities V& "ears ( ) estimates the Mc/+lloch c+rve ver" acc+ratel"

after the &th "ear' however it is a fairl" ,oor estimator for the ,rior mat+rities.

& C a g e



rap 12 6asice3 Ca%i)ration ,or a%% t!e re,erence points used 7to i%%ustrate rando(ness.

• /alibration for Mat+rities % Y "ears V & ( ) estimates the Mc/+lloch c+rve ver"

acc+ratel" +, +ntil mat+rit" 1% [ 1 and then deviates. Goticing a trend develo, for o+r

var"ing mat+rities the reason for this a,,arent fail+re in s,eci-c ,arts (mat+rities) of

o+r model has to do with o+r reference ,oints. *or this s,eci-c calibration' as o+r

reference ,oints ((Tt) ; ?1 1. 1# 1%@) onl" go +, to 1%' the Rasice4 calibration is onl"

calibrating acc+ratel" within this range./learl"' to get a more acc+rate Rasice4 model' we need to calibrate it +sing reference

,oints from across the whole range of mat+rities (#8#)' th+s we ta4e (Tt) ; ?#. 1 1.

1. 1# 1% 1 %# % 8#@

rap 13 Ris+al >cc+rac" of the calibrated Rasice4 Model.

C a g e



This gra,h shows 6+ite clearl" how acc+rate the Rasice4 model can be calibrated to o+r

Mc/+lloch c+rve. Fn fact the estimates are so close to the Mc/+lloch c+rves that the"

almost overla, each other the whole wa" (the arrows ,oint towards the ,art of the

Rasice4 c+rve that can be seen).

The Rasice4 model calibrates well in o+r case ,rimaril" d+e to the fairl" smooth and+niform nat+re of o+r Mc/+lloch c+rve. This means that the fo+r ,arameters that Rasice4

+ses is s+cient to ca,t+re the nat+re of it. Ff the c+rve had another 4in4 or s+dden

change(s) in direction' the Rasice4 model wo+ld have been ins+cient and fail miserabl".

The val+es of the ,arameters estimated for the c+rves can be seen in the tables below!

$aturities 2

%ears

$aturities 4 #

1st October %th October 1st October %th October

5e6erence

7oints

(8-t) = 90., 1, 1.,

1.:

(8-t) = 910 12 20

30:

r(0) #.#%HH&8 #.#%H#&#H #.#81&H1 #.#%%#L%#L; #.#H#8718L #.#7#&7HL1 #.18#&#%8L #.17H7#888

< #.%#H%7L7 #.H8L&8& #.#%7#%%& #.#&###8

#.#1#H %.#3#7 #.#%H7#8 #.#8%878L

8ab'e 1 6asice3 ode% coe:cients ,or aturities %ess t!an 2 and (ore t!an 8 4ears

$aturities 2 %ears 4 #

1st October %th

October

1st

October

%th

October

1st

October

%th

October5e6ere

nce

7oints

(8-t) = 91, 1., 10, 12:(8-t) = 91, 1., 10,

20:

(8-t) = 90. 1 1. 1. 10

12 1 20 2 30:

r(0) #.#%H#%%#H

#.#%78718

#.#%L#HL8

#.#%H7&11&

#.#%H8H87#%

#.#%H%L%#7%

;#.1#18#

1

#.#H187L8

&

#.#&H

#L

#.1HH7L

%#.1787L%L

#.17%7H7

7

< #.1#H&1#

81

#.8%8&&%%

%

#.%1&#

7

#.#7LL

18#.#H%LH887

#.#H7&711

11

#.#888

H

#.#7#&1#

8

#.##77

%L

#.#8##H

%#.#81H1&%

#.#8%7#%8

L8ab'e 2 6asice3 ode% coe:cients ,or aturities )eteen 2 and 8 4ears 72 and 8 e;c%uded.

Question 3

Ho!ee !attice Method Calibration

Fn order to ,rice an o,tion on a bond we are re6+ired to +se the Bo\ee lattice method. To

do this we m+st set +, a short rate lattice. This can be done b" +sing the e6+ation rs(<)

= a(<) + b(<)*s for # s 4. Where a(4) is the initial val+e (g+essed) of the "ield at

time ; 4' +sed to constr+ct the lattice and b(4) is the volatilit" which is set constant

thro+gho+t the lattice.

$cCu''oc &ie'd >a'ues and uessed >a'ues 6or a(<)

$aturit% 0 1 2 3 /

%?c2.<<<8

22.9822

9&.2<59

0&.519&

2&.=2

5&.92&

5.115&

1.2<=

9

P+sti-cation of the volatilit" estimate choice is ,rovided later on.

1# C a g e



uessed -a(<)

2.1809

2.85<&9

&.&25<9

&.<589=

&.=&98<

&.95<2&

.158=<

.25<&1

8ab'e 3 cCu%%oc! >ie%d Curve 6a%ues and Initia% Guessed 6a%ues.

Construction o, t!e #!ort – ate ?attice

@ort - 5ate Lattice

@tate "8iAe

0 1 2 3 /

1<.20&

/1.&99

1<1.9<

=1

12.89

90

12.<92

&

12.=89

98

10.5<<

=910.=8&

1<10.985

<911.08&

2

38.==9

1=8.8<00

<9.0=<

&9.2=89

<9.&=<5

1

2<.=&9

1<=.0=2

=.15&&

&=.&<9=

0=.5=22

&=.<<9=

8

1.5<&

125.0&2

25.&<5

=05.<5

95.<<29

<5.8<5

95.9<&0

02.18

092.85<

&9&.&25

<9&.<58

9=&.=&98

<&.95<2

&.158=

<.25<&

18ab'e Constructed #!ort – ate ?attice.

Get we constr+ct the Bo\ee lattice which calc+lates ris4 ne+tral ,rices for o+r bonds

based on o+r initial g+esses of a(4). *rom these ,rices we get a set of im,lied "ields and

com,are them with o+r act+al "ields from o+r Mc/+lloch /+rve for the %th October %##&

at t ; ?# 1 % 8 7 H L@.

Using the addin M2 3cel solver' we tr" to minimise the dierences between these two'

i.e.

∑t =0

7

(implied yield (t ) – actual yield (t ))2 )

s+b:ect to changing onl" the g+essed "ields a(4). Once the model is calibrated correctl"'

we can now ta4e o+r -nal short rate lattice (which will have changed now as o+r a(4)sare dierent) and calc+late the bond ,rice.

11 C a g e



The "ond#s $rice

To calc+late the bond ,rice' we +se a ris4 ne+tral i.e. ,robabilit" of an +, movement (+)

; ,robabilit" of a down movement (d) ; ]) binomial tree t",e method to disco+nt the

,rices bac4 from mat+rit". We 4now that at mat+rit" o+r bond ,rice is worth 1## (i.e.

(T'T) ; *ace Ral+e ; 1## ' T ; H)' in all states (nodes). Disco+nting this bac4 +sing thecorres,onding short rates from o+r short rate lattice (see Table 7) as a disco+nting

factor' we can -nd the val+e of the bond at each dierent node' as

7(t-1,8) =

u × P u (t , T )+d × Pd( t , T )

1+r ( t −1 )100

Bond 7rice Lattice

@tate "8iAe

0 1 2 3 /

/100.0

0

88.89<8=

100.00

81.020&

990.2<<

2100.0

0

3=5.<52=

28&.5<8

291.<=88

&100.0

0

2=2.5825

9=9.295&

88<.2&85

<9&.1&<1

5100.0

0

1=1.<915

===.&&&

08&.1=5=

089.0&8=

<9.<05

100.0

0

0=2.9&&

8==.=025

882.5008

8=.&1&

191.9==<

09<.19&

&100.0

08ab'e Constructed Bond@s Price ?attice.

The %&tion#s $rice

The o,tion ,rice can now be calc+lated +sing a similar binomial tree. We ta4e the stri4e

,rice e6+al to the s+m of the bond ,rices in all states for T ; 7 (K ; H.8H&L) and -nd

the ,a"o at mat+rit" +sing the e6+ation below!

payoff ( T , T )=max ( P ( i ,T ,T )− K ,0) ,T =4∧i=node

We then disco+nt this bac4 in the same fashion (as that of the bond ,rice binomial tree)

to obtain the ,rice of the 3+ro,ean /all ; #.8&#7 .

C(t-1,8) = u × C u (t , T )+d ×C d (t , T )

1+r ( t −1 )100

ption 7rice - @tri<e 7rice #/.3/#

@tate "8iAe

0 1 2 3

0.0000

0

30.0000

00.0000

0

2

0.0000

0

0.0000

0

0.0000

0

1% C a g e



10.288

20.<0&1

<1.2<=0

22.<=00

2

0'()))

'*1.5&0<

22.555

2&.99&&

25.<088

58ab'e Constructed ption@s Price ?attice.

Volatilit+ ,stimateFt m+st be noted that the ,rice of the o,tion on the bond de,ends considerabl" on the

short rate series that were chosen as a volatilit" estimate. O+r choice is the one "ear

\ibor rate (dail" fre6+enc") from Pan+ar" to December %##&H. We avoided +sing a

government short rate. The reason is that short rates are fre6+entl" mani,+lated b" the

government in order to aect the long term rates and the economic activit" level. Fn this

wa" the" do not freel" reEect the mar4et conditions and e,ectations' es,eciall" d+ring

t+rb+lent economic ,eriods. Fnstead' we +se \ibor which is a mar4et determined short

rate and ,ossibl" reEects more acc+ratel" the conditions mentioned above. Moreover' it

is fre6+entl" +sed in ,ractice for ,ricing b" traders etc. Using a longer sam,le wo+ld not

have ,rovided a more acc+rate estimate. *+rthermore' it wo+ld not reEect better the

mar4et e,ectations in the f+t+re.Using this set of data of %L observations' we obtain an estimate of the volatilit" b"

+sing the standard inb+ilt M2 3cel f+nction (2T3DR). 2ince o+r data fre6+enc" is dail"'

b+t o+t lattice time interval is going to be "earl" we ad:+st it b" m+lti,l"ing it b" the

s6+are root of %%. Th+s o+r volatilit" estimate for the constr+ction of the lattice is

#.&88L. Ft is necessar" to mention that the a,,lication of the s6+are root Tr+le relies on

the ass+m,tion that the observations are inde,endentl" distrib+ted. This ass+m,tion is

,ossibl" more valid for a mar4et determined interest rate' than for one that occasionall"

inE+enced b" the government for economic and ,olitical reasons.

H The data were downloaded from the website of the 9ritish 9an4s >ssociation (99>) Q www.bba.gov.+4

18 C a g e



5e6erences

!iterature

T+c4man' 9' I*ied Fncome 2ec+rities' Wille" *inance' 2econd 3dition Q %##%.

-atabases9loomberg Database

Internet

www.ft.com$home$+4

www.bloomberg.com

www.ban4ofengland.co.+4

www.dmo.gov.+4

www.statistics.gov.+4

www.bba.gov.+4

17 C a g e

http://www.ft.com/home/uk

http://www.bloomberg.com/

http://www.bankofengland.co.uk/



http://www.bba.gov.uk/

http://www.ft.com/home/uk

http://www.bloomberg.com/

http://www.bankofengland.co.uk/



http://www.bba.gov.uk/

fixed income securities - coursework report

Documents