Modeling Stock Volatility in Excel
4
1 Introduction The publication of the preference-free option pricing formula by Fischer Black and Myron Scholes in 1973 was a giant step forward in financial economics. Since then option pricing theory has developed into a standard tool for designing, pricing and hedging derivative securities of all types. The Black & Scholes formula for pricing vanilla European options in an ideal market needs six inputs: the current stock price, the strike price, the time to expiry, the riskfree interest rate, the dividends and the volatility. Of these, the first three are known from the outset and the last three must be estimated. Black & Scholes assumed a perfect world in their analysis and took the last three as constants. In the real world, however, the correct values for these parameters are only known when the option expires. This means that the future values of these quantities need to be determined if an option is to be priced correctly. The most important of the three uncertain parameters, is the volatility. Changes in interest rates (especially in a low interest rate environment) do not influence the price of an option as much as changes in volatility. Most options (and especially warrants) are short dated (expiries less than 12 months). The impact of small dividends 1 on the value of an option that is short dated is minimal. Dividend risk can be minimised through good research whereby the future dividends that will be paid during the next 12 to 18 months can be estimated quite accurately. The importance of the volatility parameter was highlighted by Black & Scholes through their model. Practitioners now needed to estimate only one parameter, the volatility, and input it into a relative simple formula to find the price of an option. Of the three uncertain parameters, changing volatility has the biggest impact on the price of an option. Volatility measures variability, or dispersion about a central tendency — it is simply a measure of the degree of price movement in a stock, futures contract or any other market. Volatility also has many subtleties that make it challenging to analyze and implement. The following question immediately comes to mind: can we estimate this seemingly complex quantity called volatility? In this short note we’ll explore 2 different ways to estimate volatility. Both of these are quite simple to implement in Microsoft Excel. 2 The Statistical Nature of Volatility Black & Scholes assumed that financial asset prices are random variables that are lognormally distributed. Therefore, returns to financial assets, the relative price changes are usually measured by taking the differences between the logarithmic prices. These differences (the so-called log-relatives) are normally distributed. A normal distribution is indicated by a bell shaped curve. This is shown in Fig. 1. -3.5 -3.1 -2.7 -2.3 -1.9 -1.5 -1.1 -0.7 -0.3 0.1 0.5 0.9 1.3 1.7 2.1 2.5 2.9 3.3 1 In SA most of the top 60 companies pay relatively small dividends. Figure 1: The bell shaped normal cumulative distribution. What does this all means in practise? Stock prices are usually observed at fixed intervals of time (daily, weekly or monthly) and we then have a time series of data. The logrelative returns are mathematically defined by the equation ( ) ( ) ¸ ¸ ¹ · ¨ ¨ © § = − = − − 1 1 ln ln ln i i i i i S S S S u (1) where i S is the stock price at the end of the i -th interval and ) ln( is the natural logarithmic function. We also assume that there are n stock prices in our sample. This equation can easily be implemented in Microsoft Excel. This is illustrated in Fig. 2 where we used a few MTN share prices. Figure 2: MTN logrelative prices. Also shown are the formulas as used in Excel. Volatility is defined as the variation or dispersion or deviation of an asset’s returns from their mean. In Fig. 1 we show two normal curves. Both have the same mean but the dotted line shows a greater dispersion than the continuous line. These two curves also illustrate that volatility indicates the range of a return’s movement. Large values of volatility mean that returns fluctuate in a wide range – large risk. The most common measure of dispersion is the standard deviation of a random variable. But, what does this all means? If we assume the mean of the logrelative returns is zero, then, a 10% volatility represents the following: in one year, returns will be within [-10%; +10%] with a probability of 68.3% (1 standard deviation from the mean); within [-20%; +20%] with a probability of 95.4% (2 standard deviations), and within [-30%; +30%] with a probability of 99.7% (3 standard deviations) — according to a normal distribution.
Transcript of Modeling Stock Volatility in Excel
1 In
trod
uctio
n
The
publ
icat
ion
of th
e pr
efer
ence
-free
opt
ion
pric
ing
form
ula
by F
isch
er B
lack
and
Myr
on S
chol
es in
19
73 w
as a
gia
nt s
tep
forw
ard
in fi
nanc
ial e
cono
mic
s. S
ince
then
opt
ion
pric
ing
theo
ry h
as d
evel
oped
in
to a
sta
ndar
d to
ol fo
r des
igni
ng, p
ricin
g an
d he
dgin
g de
rivat
ive
secu
ritie
s of
all
type
s.
The
Bla
ck &
Sch
oles
for
mul
a fo
r pr
icin
g va
nilla
Eur
opea
n op
tions
in a
n id
eal m
arke
t ne
eds
six
inpu
ts:
the
curre
nt s
tock
pric
e, t
he s
trike
pric
e, t
he t
ime
to e
xpiry
, th
e ris
kfre
e in
tere
st r
ate,
the
di
vide
nds
and
the
vola
tility
. Of t
hese
, the
firs
t thr
ee a
re k
now
n fro
m th
e ou
tset
and
the
last
thre
e m
ust
be e
stim
ated
. Bla
ck &
Sch
oles
ass
umed
a p
erfe
ct w
orld
in th
eir
anal
ysis
and
took
the
last
thre
e as
co
nsta
nts.
In th
e re
al w
orld
, how
ever
, the
cor
rect
val
ues
for
thes
e pa
ram
eter
s ar
e on
ly k
now
n w
hen
the
optio
n ex
pire
s. T
his
mea
ns th
at th
e fu
ture
val
ues
of th
ese
quan
titie
s ne
ed to
be
dete
rmin
ed if
an
optio
n is
to b
e pr
iced
cor
rect
ly.
The
mos
t im
porta
nt o
f th
e th
ree
unce
rtain
par
amet
ers,
is th
e vo
latil
ity. C
hang
es in
inte
rest
rat
es
(esp
ecia
lly in
a lo
w in
tere
st r
ate
envi
ronm
ent)
do n
ot in
fluen
ce t
he p
rice
of a
n op
tion
as m
uch
as
chan
ges
in v
olat
ility.
Mos
t op
tions
(an
d es
peci
ally
war
rant
s) a
re s
hort
date
d (e
xpiri
es le
ss t
han
12
mon
ths)
. Th
e im
pact
of
smal
l div
iden
ds1
on t
he v
alue
of
an o
ptio
n th
at is
sho
rt da
ted
is m
inim
al.
Div
iden
d ris
k ca
n be
min
imis
ed th
roug
h go
od re
sear
ch w
here
by th
e fu
ture
div
iden
ds th
at w
ill be
pai
d du
ring
the
next
12
to 1
8 m
onth
s ca
n be
est
imat
ed q
uite
acc
urat
ely.
Th
e im
porta
nce
of t
he v
olat
ility
para
met
er w
as h
ighl
ight
ed b
y B
lack
& S
chol
es t
hrou
gh t
heir
mod
el.
Pra
ctiti
oner
s no
w n
eede
d to
est
imat
e on
ly o
ne p
aram
eter
, th
e vo
latil
ity,
and
inpu
t it
into
a
rela
tive
sim
ple
form
ula
to f
ind
the
pric
e of
an
optio
n. O
f th
e th
ree
unce
rtain
par
amet
ers,
cha
ngin
g vo
latil
ity h
as th
e bi
gges
t im
pact
on
the
pric
e of
an
optio
n.
Vol
atilit
y m
easu
res
varia
bilit
y, o
r dis
pers
ion
abou
t a c
entra
l ten
denc
y —
it is
sim
ply
a m
easu
re o
f th
e de
gree
of
pric
e m
ovem
ent
in a
sto
ck,
futu
res
cont
ract
or
any
othe
r m
arke
t. Vo
latil
ity a
lso
has
man
y su
btle
ties
that
m
ake
it ch
alle
ngin
g to
an
alyz
e an
d im
plem
ent.
The
follo
win
g qu
estio
n im
med
iate
ly c
omes
to m
ind:
can
we
estim
ate
this
see
min
gly
com
plex
qua
ntity
cal
led
vola
tility
?
In th
is s
hort
note
we’
ll ex
plor
e 2
diffe
rent
way
s to
est
imat
e vo
latil
ity. B
oth
of th
ese
are
quite
sim
ple
to im
plem
ent i
n M
icro
soft
Exc
el.
2 Th
e St
atis
tical
Nat
ure
of V
olat
ility
Bla
ck &
Sch
oles
ass
umed
tha
t fin
anci
al a
sset
pric
es a
re r
ando
m v
aria
bles
tha
t ar
e lo
gnor
mal
ly
dist
ribut
ed. T
here
fore
, ret
urns
to fi
nanc
ial a
sset
s, th
e re
lativ
e pr
ice
chan
ges
are
usua
lly m
easu
red
by
taki
ng th
e di
ffere
nces
bet
wee
n th
e lo
garit
hmic
pric
es. T
hese
diff
eren
ces
(the
so-c
alle
d lo
g-re
lativ
es)
are
norm
ally
dis
tribu
ted.
A n
orm
al d
istri
butio
n is
indi
cate
d by
a b
ell s
hape
d cu
rve.
Thi
s is
sho
wn
in
Fig.
1.
-3.5
-3.1
-2.7
-2.3
-1.9
-1.5
-1.1
-0.7
-0.3
0.1
0.5
0.9
1.3
1.7
2.1
2.5
2.9
3.3
1 In S
A m
ost o
f the
top
60 c
ompa
nies
pay
rela
tivel
y sm
all d
ivid
ends
.
Figu
re 1
: Th
e be
ll sh
aped
nor
mal
cum
ulat
ive
dist
ribut
ion.
W
hat d
oes
this
all
mea
ns in
pra
ctis
e? S
tock
pric
es a
re u
sual
ly o
bser
ved
at fi
xed
inte
rval
s of
tim
e (d
aily
, w
eekl
y or
mon
thly
) an
d w
e th
en h
ave
a tim
e se
ries
of d
ata.
The
log
rela
tive
retu
rns
are
mat
hem
atic
ally
def
ined
by
the
equa
tion
()
() =
−=
−−
11
lnln
lnii
ii
iSS
SS
u
(1)
whe
re
iS is
the
stoc
k pr
ice
at th
e en
d of
the
i-th
inte
rval
and
)
ln(
is th
e na
tura
l log
arith
mic
func
tion.
W
e al
so a
ssum
e th
at t
here
are
n
sto
ck p
rices
in
our
sam
ple.
Thi
s eq
uatio
n ca
n ea
sily
be
impl
emen
ted
in M
icro
soft
Exc
el. T
his
is il
lust
rate
d in
Fig
. 2 w
here
we
used
a fe
w M
TN s
hare
pric
es.
Figu
re 2
: M
TN lo
grel
ativ
e pr
ices
. Als
o sh
own
are
the
form
ulas
as
used
in E
xcel
.
Vol
atilit
y is
def
ined
as
the
varia
tion
or d
ispe
rsio
n or
dev
iatio
n of
an
asse
t’s r
etur
ns f
rom
the
ir m
ean.
In F
ig. 1
we
show
two
norm
al c
urve
s. B
oth
have
the
sam
e m
ean
but t
he d
otte
d lin
e sh
ows
a gr
eate
r dis
pers
ion
than
the
cont
inuo
us li
ne. T
hese
two
curv
es a
lso
illust
rate
that
vol
atilit
y in
dica
tes
the
rang
e of
a re
turn
’s m
ovem
ent.
Larg
e va
lues
of v
olat
ility
mea
n th
at re
turn
s flu
ctua
te in
a w
ide
rang
e –
larg
e ris
k. T
he m
ost c
omm
on m
easu
re o
f dis
pers
ion
is th
e st
anda
rd d
evia
tion
of a
rand
om v
aria
ble.
B
ut, w
hat d
oes
this
all
mea
ns?
If w
e as
sum
e th
e m
ean
of th
e lo
grel
ativ
e re
turn
s is
zer
o, th
en, a
10
% v
olat
ility
repr
esen
ts t
he f
ollo
win
g: i
n on
e ye
ar,
retu
rns
will
be w
ithin
[-1
0%;
+10%
] w
ith a
pr
obab
ility
of 6
8.3%
(1
stan
dard
dev
iatio
n fro
m th
e m
ean)
; with
in [-
20%
; +20
%] w
ith a
pro
babi
lity
of
95.4
% (
2 st
anda
rd d
evia
tions
), an
d w
ithin
[-3
0%;
+30%
] w
ith a
pro
babi
lity
of 9
9.7%
(3
stan
dard
de
viat
ions
) — a
ccor
ding
to a
nor
mal
dis
tribu
tion.
3 Th
e Va
rianc
e R
ate
of R
etur
n
In t
heir
pape
r in
197
3, B
lack
& S
chol
es m
entio
ned
the
para
met
er
2σ
whi
ch t
hey
said
was
the
“v
aria
nce
rate
of t
he re
turn
" on
the
stoc
k pr
ices
. Bla
ck &
Sch
oles
took
this
as
a kn
own
para
met
er th
at
is c
onst
ant t
hrou
gh th
e lif
e of
the
optio
n. D
id th
ey re
ally
kno
w w
hat t
his
para
met
er w
as?
In
a p
aper
prio
r to
thei
r sem
inal
one
, Bla
ck &
Sch
oles
gav
e m
ore
insi
ght i
nto
the
varia
nce
rate
of
retu
rn. T
here
they
sta
ted
that
they
est
imat
ed th
e in
stan
tane
ous
varia
nce
from
the
hist
oric
al s
erie
s of
da
ily s
tock
pric
es.
They
thu
s de
fined
vol
atilit
y as
the
am
ount
of
varia
bilit
y in
the
ret
urns
of
the
unde
rlyin
g as
set.
Bla
ck &
Sch
oles
det
erm
ined
wha
t is
toda
y kn
own
as th
e hi
stor
ical
vol
atilit
y an
d us
ed
that
as
a pr
oxy
for t
he e
xpec
ted
vola
tility
in th
e fu
ture
. In
that
pap
er th
ey te
sted
sev
eral
impl
icat
ions
of
thei
r mod
el e
mpi
rical
ly b
y us
ing
a sa
mpl
e of
2 0
39 c
alls
and
3 0
52 s
tradd
les
trade
d on
the
New
Yor
k st
ock
exch
ange
bet
wee
n 19
66 a
nd 1
969.
In
ana
lyzi
ng t
heir
resu
lts t
hey
note
d th
at t
he v
aria
nce
actu
ally
em
ploy
ed b
y th
e m
arke
t is
too
na
rrow
and
that
the
hist
oric
al e
stim
ates
of t
he v
aria
nce
incl
ude
an a
ttenu
atio
n bi
as, i
.e.,
the
spre
ad o
f th
e es
timat
es is
gre
ater
tha
n th
e sp
read
of
the
true
varia
nce.
Thi
s im
plie
s th
at f
or s
ecur
ities
with
a
rela
tivel
y hi
gh v
aria
nce,
the
mar
ket
pric
es u
nder
estim
ate
the
varia
nce,
whi
le u
sing
his
toric
al p
rice
serie
s w
ould
ove
rest
imat
e th
e va
rianc
e an
d th
e re
sulti
ng B
lack
& S
chol
es m
odel
pric
e w
ould
thus
be
too
high
; the
con
vers
e is
true
for
rela
tive
low
var
ianc
e se
curit
ies.
Was
this
the
first
obs
erva
tion
of a
vo
latil
ity s
kew
or
smile
? In
furth
er te
sts
Bla
ck &
Sch
oles
foun
d th
at th
eir
mod
el p
erfo
rmed
ver
y w
ell
whe
n th
e tru
e va
rianc
e ra
te o
f the
sto
ck w
as k
now
n.
4 E
stim
atio
n of
Vol
atili
ty
4.1
Tra
ding
or N
ontr
adin
g D
ays
To e
stim
ate
the
vola
tility
of
a st
ock
pric
e em
piric
ally
, th
e st
ock
pric
e is
usu
ally
obs
erve
d at
fix
ed
inte
rval
s of
tim
e. T
hese
inte
rval
s ca
n be
day
s, w
eeks
or
mon
ths2 .
Bef
ore
any
calc
ulat
ion
can
be
done
, ho
wev
er,
a qu
estio
n on
e ne
eds
to a
nsw
er i
s w
heth
er t
he v
olat
ility
of a
n ex
chan
ge-tr
aded
in
stru
men
t is
the
sam
e w
hen
the
exch
ange
is o
pen
as w
hen
it is
clo
sed.
S
ome
peop
le a
rgue
tha
t in
form
atio
n ar
rives
eve
n w
hen
an e
xcha
nge
is c
lose
d an
d th
is s
houl
d in
fluen
ce th
e pr
ice.
A lo
t of e
mpi
rical
stu
dies
hav
e be
en d
one
and
rese
arch
ers
foun
d th
at v
olat
ility
is
far
larg
er w
hen
the
exch
ange
is o
pen
than
whe
n it
is c
lose
d. T
he c
onse
quen
ce o
f thi
s is
that
if d
aily
da
ta a
re u
sed
to m
easu
re v
olat
ility,
the
resu
lts s
ugge
st th
at d
ays
whe
n th
e ex
chan
ge is
clo
sed
shou
ld
be ig
nore
d.
4.2
His
toric
al V
olat
ility
The
hist
oric
al v
olat
ility
is th
e vo
latil
ity o
f a s
erie
s of
sto
ck p
rices
whe
re w
e lo
ok b
ack
over
the
hist
oric
al
pric
e pa
th o
f th
e pa
rticu
lar
stoc
k. W
e pr
evio
usly
men
tione
d th
at t
he m
ost
com
mon
mea
sure
of
disp
ersi
on is
the
stan
dard
dev
iatio
n. T
he h
isto
rical
vol
atilit
y es
timat
e is
thus
giv
en b
y
=
2 )−
(1−1
=n i
iu
un
1σ
(2
)
whe
re u
is th
e m
ean
defin
ed b
y
.1
1=
=n j
ju
nu
2 One
has
to b
e co
nsis
tent
; if t
he fr
eque
ncy
of o
bser
vatio
n is
eve
ry T
hurs
day
at m
idni
ght,
the
retu
rns
all n
eed
to c
orre
spon
d to
suc
h a
perio
d
iu w
as d
efin
ed in
Equ
atio
n (1
). σ
in E
quat
ion
(2) g
ives
the
estim
ated
vol
atilit
y pe
r int
erva
l. To
ena
ble
us to
com
pare
vol
atilit
ies
for d
iffer
ent i
nter
val l
engt
hs w
e us
ually
exp
ress
vol
atilit
y in
ann
ual t
erm
s. T
o do
thi
s w
e sc
ale
this
est
imat
e w
ith a
n an
nual
izat
ion
fact
or (
norm
alis
ing
cons
tant
) h
whi
ch i
s th
e nu
mbe
r of i
nter
vals
per
ann
um s
uch
that
.*
han
σσ
=
If da
ily d
ata
is u
sed
the
inte
rval
is o
ne tr
adin
g da
y an
d w
e us
e 25
2=
h, i
f the
inte
rval
is a
wee
k,
52=
h a
nd
12=
h fo
r mon
thly
dat
a3 . E
quat
ion
(2) i
s ju
st th
e st
anda
rd d
evia
tion
of th
e sa
mpl
ed s
erie
s j
u. F
ig. 3
sho
ws
how
this
can
be
impl
emen
ted
in M
icro
soft
Exc
el w
here
we
show
the
daily
clo
sing
val
ues
for
MTN
fro
m 1
Nov
embe
r 20
04 ti
ll 25
Jan
uary
200
5. In
Fig
. 4 w
e pl
ot th
e 3
mon
th h
isto
rical
vol
atilit
y fo
r MTN
.
Figu
re 3
: H
isto
rical
vol
atilit
y: E
xcel
impl
emen
tatio
n.
3 Ther
e is
app
roxi
mat
ely
252
tradi
ng d
ays
per a
nnum
MTN
3 m
onth
His
toric
al V
olat
ility
20.0
0%
25.0
0%
30.0
0%
35.0
0%
40.0
0%
45.0
0%30/04/2003
30/05/2003
30/06/2003
30/07/2003
30/08/2003
30/09/2003
30/10/2003
30/11/2003
30/12/2003
30/01/2004
29/02/2004
30/03/2004
30/04/2004
30/05/2004
30/06/2004
30/07/2004
30/08/2004
30/09/2004
30/10/2004
30/11/2004
30/12/2004
Dat
e
Vol
Figu
re 4
: M
ovin
g 3
mon
th h
isto
rical
vol
atilit
y fo
r MTN
from
Feb
ruar
y 20
03.
4.3
Impl
ied
Vola
tility
A s
impl
e op
tion
pric
ing
mod
el (
like
the
Bla
ck &
Sch
oles
mod
el)
will
give
a t
heor
etic
al p
rice
for
an
optio
n as
a fu
nctio
n of
the
impl
icit
para
met
ers
— c
onst
ant v
olat
ility
bein
g on
e. H
owev
er, i
f the
opt
ion
is t
rade
d, t
he m
arke
t pr
ice
mig
ht n
ot b
e th
e sa
me
as t
he m
odel
pric
e. I
n th
at c
ase
one
mig
ht a
sk,
whi
ch v
olat
ility
estim
ate
does
one
hav
e to
use
in th
e m
odel
so
that
the
mod
el p
rice
and
the
mar
ket
pric
e ar
e th
e sa
me?
Thi
s is
the
impl
ied
vola
tility
. In
a co
nsta
nt v
olat
ility
fram
ewor
k, im
plie
d vo
latil
ity is
th
e vo
latil
ity o
f the
und
erly
ing
asse
t pric
e th
at is
impl
icit
in th
e m
arke
t pric
e of
an
optio
n ac
cord
ing
to a
pa
rticu
lar m
odel
. W
e illu
stra
te th
e ba
sic
idea
by
anal
ysin
g th
e M
TNA
BA
war
rant
. Thi
s w
arra
nt h
as a
stri
ke p
rice
of
R40
, it
expi
res
on 1
7 M
arch
200
5 an
d th
e co
ver
ratio
is 1
0.
Fig.
5 s
how
s th
e M
TN a
nd M
TNA
BA
pr
ices
fro
m F
ebru
ary
2004
. Th
e w
arra
nt p
rice
follo
ws
the
MTN
pric
e bu
t du
e to
the
gea
ring
of t
he
war
rant
the
swin
gs c
an b
e w
ilder
.
MTN
vs
MTN
ABA
2500
3000
3500
4000
4500
5000
03/02/2004
17/02/2004
02/03/2004
16/03/2004
30/03/2004
13/04/2004
27/04/2004
11/05/2004
25/05/2004
08/06/2004
22/06/2004
06/07/2004
20/07/2004
03/08/2004
17/08/2004
31/08/2004
14/09/2004
28/09/2004
12/10/2004
26/10/2004
09/11/2004
23/11/2004
07/12/2004
21/12/2004
04/01/2005
18/01/2005
Date
Price (c )
01020304050607080
MTN
ABA
Figu
re 5
: M
TN a
nd M
TNA
BA
pric
e be
havi
our.
To c
alcu
late
the
impl
ied
vola
tility
we
ask
ours
elve
s: o
n 1
Dec
embe
r 20
04, t
he w
arra
nt p
rice
was
R
0.44
and
MTN
’s p
rice
was
R40
(the
sam
e as
the
strik
e pr
ice)
, we
now
wan
t to
know
, if w
e su
bstit
ute
this
pric
e (4
4 ce
nts)
, int
o th
e B
lack
& S
chol
es e
quat
ion,
wha
t vol
atilit
y w
ill po
p ou
t!
Bef
ore
we
can
do a
nyth
ing,
we
need
to
know
the
par
amet
ers
men
tione
d in
the
Int
rodu
ctio
n.
Cur
rent
inte
rest
rate
s ar
e at
8.5
% a
nd M
TN’s
div
iden
d yi
eld
is 1
%.
If yo
u ha
ve a
n op
tion
pric
ing
spre
adsh
eet,
you
can
subs
titut
e al
l the
par
amet
ers
into
that
and
use
E
xcel
’s G
oals
eek
to s
earc
h fo
r the
vol
atilit
y4 . If
you
do n
ot h
ave
such
a s
prea
dshe
et y
ou c
an u
se th
e fo
rmul
a du
e to
Cor
rado
and
Mille
r. Th
ey re
fer t
o it
as th
e im
prov
ed q
uadr
atic
form
ula
whe
re
.)
(2
22
22
−′−
−′−
+−′
−+′
=π
πσ
XS
XS
VX
SV
XS
T
Her
e rT
eK
X−
= w
hich
is th
e di
scou
nted
stri
ke p
rice,
dT
eS
S−
=′ w
here
S is
the
stoc
k pr
ice,
Kth
e st
rike
pric
e, V
is th
e w
arra
nt p
rice
mul
tiplie
d by
the
cove
r rat
io,
r is
the
risk-
free
inte
rest
rate
, d
is t
he d
ivid
end
yiel
d,
1415
9265
.3=
π (
Arc
him
edes
’ co
nsta
nt)
and
T i
s th
e tim
e to
exp
iry.
It is
ac
cura
te o
ver a
wid
e ra
nge
of s
trike
pric
es.
Fig.
6 s
how
s an
impl
emen
tatio
n in
Exc
el (
ensu
re t
hat t
he s
heet
is s
et u
p as
sho
wn
with
all
the
para
met
ers
in t
he c
ells
as
show
n).
We
show
the
impl
ied
vola
tiliti
es c
alcu
late
d fo
r a
few
MTN
AB
A w
arra
nt p
rices
. In
Fig
. 7 w
e pl
ot th
e M
TNA
BA
war
rant
pric
e an
d th
e im
plie
d vo
latil
ity ti
me
serie
s.
4 Rem
embe
r to
mul
tiply
the
war
rant
pric
e by
the
cove
r rat
io.
Figu
re 6
: Im
plie
d vo
latil
ity: i
mpl
emen
tatio
n in
Exc
el.
If w
e ha
d m
any
war
rant
s, w
hich
var
y in
stri
ke p
rice
and
time
to e
xpira
tion,
that
wer
e w
ritte
n on
the
sam
e un
derly
ing
like
MTN
, we
wou
ld o
bser
ve a
term
stru
ctur
e of
vol
atilit
ies
and
a vo
latil
ity “s
mile
" or
“s
kew
". Th
is is
due
to s
yste
mat
ic d
evia
tions
from
the
pred
ictio
ns o
f th
e B
lack
& S
chol
es m
odel
and
w
arra
nts
anot
her b
road
er d
iscu
ssio
n.
MTN
AB
A a
nd it
s Im
plie
d Vo
latil
ity
515253545556575
03/02/2004
17/02/2004
02/03/2004
16/03/2004
30/03/2004
13/04/2004
27/04/2004
11/05/2004
25/05/2004
08/06/2004
22/06/2004
06/07/2004
20/07/2004
03/08/2004
17/08/2004
31/08/2004
14/09/2004
28/09/2004
12/10/2004
26/10/2004
09/11/2004
23/11/2004
07/12/2004
21/12/2004
04/01/2005
18/01/2005
30%
35%
40%
45%
50%
55%
60%
MTN
ABA
Impl
ied
Vol
Figu
re 7
: M
TNA
BA
war
rant
pric
e an
d im
plie
d vo
latil
ity.
5 D
iffer
ence
bet
wee
n Im
plie
d an
d St
atis
tical
Vol
atili
ties
Impl
ied
vola
tiliti
es s
houl
d be
vie
wed
diff
eren
tly f
rom
sta
tistic
al v
olat
ilitie
s ev
en t
houg
h th
ey b
oth
fore
cast
the
vol
atilit
y of
the
und
erly
ing
asse
t ov
er t
he l
ife o
f th
e op
tion.
The
tw
o fo
reca
sts
diffe
r be
caus
e th
ey u
se d
iffer
ent
data
and
diff
eren
t m
odel
s. Im
plie
d m
etho
ds u
se c
urre
nt d
ata
on m
arke
t pr
ices
of o
ptio
ns, s
o th
e im
plie
d vo
latil
ity c
onta
ins
all t
he fo
rwar
d ex
pect
atio
ns o
f inv
esto
rs a
bout
the
likel
y fu
ture
pric
e pa
th o
f the
und
erly
ing.
Als
o, d
ue to
the
Bla
ck &
Sch
oles
ass
umpt
ions
this
met
hod
assu
mes
that
the
unde
rlyin
g’s
pric
e pa
th is
con
tinuo
us.
Con
trast
this
with
sta
tistic
al m
etho
ds w
hich
use
his
toric
dat
a on
the
unde
rlyin
g as
set r
etur
ns in
a
disc
rete
tim
e m
odel
for t
he v
aria
nce
of a
tim
e se
ries.
6 R
ealiz
ed/A
ctua
l Vol
atili
ty
This
is
the
hist
oric
al v
olat
ility
calc
ulat
ed l
ooki
ng “
back
war
d" w
hen
an o
ptio
n ha
s ex
pire
d. A
s an
ex
ampl
e, le
t’s s
ay a
trad
er w
ants
to w
rite
an o
ptio
n to
day
that
exp
ires
in 3
mon
ths
time.
To
estim
ate
the
vola
tility
he/
she
mig
ht c
alcu
late
the
hist
oric
al v
olat
ility
of th
e pa
st 3
mon
ths.
If s
imila
r op
tions
are
tra
ding
in th
e m
arke
t he/
she
mig
ht c
alcu
late
the
impl
ied
vola
tility
. Th
e ac
tual
vol
atilit
y w
ill, h
owev
er,
only
be
know
n at
exp
iry.
Onc
e th
e 3
mon
ths
have
pas
sed,
one
can
cal
cula
te t
he r
ealiz
ed v
olat
ility
(act
ual v
aria
nce)
bet
wee
n th
e or
igin
al t
rade
dat
e an
d ex
piry
bec
ause
the
act
ual p
rice
path
is t
hen
know
n.
This
arti
cle
is p
ublis
hed
for
gene
ral i
nfor
mat
ion
and
is n
ot in
tend
ed a
s ad
vice
of
any
natu
re. T
he v
iew
poin
ts e
xpre
ssed
are
not
nec
essa
rily
that
of
Fina
ncia
l Cha
os T
heor
y P
ty.
Ltd.
A
s ev
ery
situ
atio
n de
pend
s on
its
own
fact
s an
d ci
rcum
stan
ces,
onl
y sp
ecifi
c ad
vice
sho
uld
be
relie
d up
on.
