volatility and hedging errors - baruch...
TRANSCRIPT
Vo
lati
lity
an
d H
edg
ing
Err
ors
Jim
Gat
her
al
Sep
tem
ber
, 25 1
999
Bac
kg
rou
nd
•D
eriv
ativ
e port
foli
o b
ookru
nner
s oft
en c
om
pla
in t
hat
•h
edg
ing
at
mar
ket
-im
pli
ed v
ola
tili
ties
is
sub
-opti
mal
rel
ativ
e to
hed
gin
g a
t th
eir
bes
t gu
ess
of
futu
re v
ola
tili
ty b
ut
•th
ey a
re f
orc
ed i
nto
hed
gin
g a
t m
ark
et i
mp
lied
vola
tili
ties
to
min
imis
e m
ark-t
o-m
ark
et P
&L
vo
lati
lity
.
•A
ll p
ract
itio
ner
s re
cognis
e th
at t
he
assu
mpti
ons
beh
ind
fixed
or
swim
min
g d
elta
choic
es a
re w
rong i
n s
om
e se
nse
.
Nev
erth
eles
s, t
he
mag
nit
ude
of
the
impac
t of
this
del
ta
choic
e m
ay b
e su
rpri
sing.
•G
iven
the
P&
L i
mpac
t of
thes
e ch
oic
es, it
would
be
nic
e to
be
able
to a
void
fig
uri
ng o
ut
how
to d
elta
hed
ge.
Is
ther
e a
way
of
avoid
ing t
he
pro
ble
m?
An
Idea
lise
dM
od
el
•T
o g
et i
ntu
itio
n a
bout
hed
gin
g o
pti
ons
at t
he
wro
ng
vola
tili
ty, w
e co
nsi
der
tw
o p
arti
cula
r sa
mple
pat
hs
for
the
stock
pri
ce, both
of
whic
h h
ave
real
ised
vola
tili
ty =
20%
•a
whip
saw
pat
h w
her
e th
e st
ock
pri
ce m
oves
up a
nd
dow
n b
y 1
.25%
ever
y d
ay
•a
sine
curv
e des
igned
to m
imic
a t
rendin
g m
arket
Wh
ipsa
w a
nd
Sin
e C
urv
e S
cen
ario
s
2 P
ath
s w
ith
Vo
lati
lity
=2
0%
0
0.51
1.52
2.53
3.54
4.5
0
16
32
48
64
80
96
112
128
144
160
176
192
208
224
240
256
Day
s
Spot
Wh
ipsa
wv
sS
ine
Cu
rve:
Res
ult
s
P&
Lvs
He
dg
eV
ol.
(80,0
00,0
00
)
(60,0
00,0
00
)
(40,0
00,0
00
)
(20,0
00,0
00
)
-
20,0
00,0
00
40,0
00,0
00
60,0
00,0
00
10%
15%
20%
25%
30%
35%
40%Hed
ge
Vola
tili
ty
P&L
Whip
saw
Sin
e W
ave
Co
ncl
usi
on
s fr
om
th
is E
xp
erim
ent
•If
you k
new
the
real
ised
vola
tili
ty i
n a
dvan
ce, you w
ould
def
init
ely h
edge
at t
hat
vola
tili
ty b
ecau
se t
he
hed
gin
g e
rror
at t
hat
vola
tili
ty w
ould
be
zero
.
•In
pra
ctic
e of
cours
e, y
ou d
on’t
know
what
the
real
ised
vola
tili
ty w
ill
be.
T
he
per
form
ance
of
your
hed
ge
dep
ends
not
only
on w
het
her
the
real
ised
vola
tili
ty i
s hig
her
or
low
er t
han
your
esti
mat
e but
also
on w
het
her
the
mar
ket
is
range
bound o
r tr
endin
g.
An
aly
sis
of
the
P&
L G
rap
h
•If
the
mar
ket
is
range
bound, hed
gin
g a
short
opti
on
posi
tion a
t a
low
ervol.
hurt
s bec
ause
you a
re g
etti
ng
conti
nuousl
y w
hip
saw
ed. O
n t
he
oth
er h
and, if
you
hed
ge
at v
ery h
igh
vol.
, an
d m
arket
is
range
bound,
your
gam
ma
is v
ery l
ow
and y
our
hed
gin
g l
oss
es a
re
min
imis
ed.
•If
the
mar
ket
is
tren
din
g, you a
re h
urt
if
you h
edge
at a
hig
her
vol.
bec
ause
your
hed
ge
reac
ts t
oo s
low
ly t
o t
he
tren
d. I
f you h
edge
at l
ow
vol.
, t
he
hed
ge
rati
o g
ets
hig
her
fas
ter
as y
ou g
o i
n t
he
money
min
imis
ing
hed
gin
g l
oss
es.
An
oth
er S
imp
le H
edg
ing
Ex
per
imen
t•
In o
rder
to s
tudy t
he
effe
ct o
f ch
angin
g h
edge
vola
tili
ty, w
e
consi
der
the
foll
ow
ing s
imple
port
foli
o:
•sh
ort
$1bn n
oti
onal
of
1 y
ear
AT
M E
uro
pea
n c
alls
•lo
ng a
one
yea
r vola
tili
ty s
wap
to c
ance
l th
e veg
a of
the
call
s at
ince
pti
on.
•T
his
is
(alm
ost
) eq
uiv
alen
t to
hav
ing s
old
a o
ne
yea
r opti
on
whose
pri
ce i
s det
erm
ined
ex-p
ost
bas
ed o
n t
he
actu
al
vola
tili
ty r
eali
sed o
ver
the
hed
gin
g p
erio
d.
•A
ny P
&L
gen
erat
ed b
y t
his
hed
gin
g s
trat
egy i
s pure
hed
gin
g e
rror.
T
hat
is,
we
elim
inat
e an
y P
&L
due
to
vola
tili
ty m
ovem
ents
.
His
tori
cal
Sam
ple
Pat
hs
•In
ord
er t
o p
reem
pt
crit
icis
m t
hat
our
sam
ple
pat
hs
are
too
unre
alis
tic,
we
take
real
his
tori
cal
FT
SE
dat
a fr
om
tw
o
dis
tinct
his
tori
cal
per
iods:
one
wher
e th
e m
arket
was
lock
ed i
n a
tre
nd a
nd o
ne
wher
e th
e m
arket
was
ran
ge
bound.
•F
or
the
ran
ge
bo
und
sce
nar
io, w
e co
nsi
der
the
per
iod f
rom
Apri
l
1991
to
Apri
l 19
92
•F
or
the
tren
din
g s
cen
ario
, w
e co
nsi
der
th
e p
erio
d f
rom
Oct
ob
er
1996
to
Oct
ober
19
97
•In
bo
th s
cenar
ios,
th
ere
alis
edvo
lati
lity
was
aro
und
12%
FT
SE
10
0 s
ince
19
85
0
1000
2000
3000
4000
5000
6000
7000 1
/1/8
55/2
2/8
610/1
0/8
72/2
7/8
97/1
8/9
012/6
/91
4/2
5/9
39/1
3/9
42/1
/96
6/2
1/9
711/9
/98
Ran
ge
Tre
nd
Ran
ge
Sce
nar
ioF
TS
E f
rom
4/1
/91 t
o 3
/31/9
2
200
0
210
0
220
0
230
0
240
0
250
0
260
0
270
0
280
0
290
0
300
0 3/3
/91
4/2
2/9
16/1
1/9
17/3
1/9
19/1
9/9
111/8
/91
12/2
8/9
12/1
6/9
24/6
/92
5/2
6/9
2
Rea
lise
d V
ola
tili
ty 1
2.1
7%
Tre
nd S
cenar
ioF
TS
E f
rom
11/1
/96 t
o 1
0/3
1/9
7
3500
3700
3900
4100
4300
4500
4700
4900
5100
5300
5500 8/2
3/9
610/1
2/9
612/1
/96
1/2
0/9
73/1
1/9
74/3
0/9
76/1
9/9
78/8
/97
9/2
7/9
711/1
6/9
7
Rea
lise
d V
ola
tili
ty 1
2.4
5%
P&
Lvs
Hed
ge
Vola
tili
ty
(25,0
00,0
00)
(20,0
00,0
00)
(15,0
00,0
00)
(10,0
00,0
00)
(5,0
00,0
00)
-
5,0
00,0
00
10,0
00,0
00
15,0
00,0
00
20,0
00,0
00
25,0
00,0
00
5%
7%
9%
11%
13%
15%
17%
19%
21%
23%
25%
Ran
ge
Sce
nar
io
Tre
nd S
cenar
io
Dis
cuss
ion
of
P&
L S
ensi
tivit
ies
•T
he
sensi
tivit
y o
f th
e P
&L
to h
edge
vola
tili
ty d
id d
epen
d
on t
he
scen
ario
just
as
we
would
hav
e ex
pec
ted f
rom
the
idea
lise
d e
xper
imen
t.
•In
th
e ra
ng
e sc
enar
io,
the
low
er t
he
hed
ge
vola
tili
ty, th
e lo
wer
the
P&
L c
on
sist
ent
wit
h t
he
wh
ipsa
w c
ase.
•In
th
e tr
end s
cen
ario
, th
e lo
wer
th
e h
edg
e vo
lati
lity
, th
e h
igher
th
e
P&
L c
on
sist
ent
wit
h t
he
sine
curv
e ca
se.
•In
eac
h s
cenar
io, th
e se
nsi
tivit
y o
f th
e P
&L
to h
edgin
g a
t a
vola
tili
ty w
hic
h w
as w
rong b
y 1
0 v
ola
tili
ty p
oin
ts w
as
around $
20m
m f
or
a $1bn p
osi
tion.
Qu
esti
on
s?
•S
uppose
you s
ell
an o
pti
on a
t a
vola
tili
ty h
igher
than
12%
and h
edge
at s
om
e oth
er v
ola
tili
ty. I
f re
alis
ed v
ola
tili
ty i
s
12%
, do y
ou m
ake
money
?
•N
ot
nec
essa
rily
. I
t is
eas
y t
o f
ind s
cen
ario
s w
her
e you
lose
mo
ney
.
•S
uppose
you s
ell
an o
pti
on a
t so
me
impli
ed v
ola
tili
ty a
nd
hed
ge
at t
he
sam
e vola
tili
ty. I
f re
alis
ed v
ola
tili
ty i
s 12%
,
when
do y
ou m
ake
mon
ey?
•In
th
e tw
o s
cenar
ios
anal
yse
d, if
the
op
tion
is
sold
and
hed
ged
at
a
vola
tili
ty g
reat
er t
han
th
e re
alis
ed v
ola
tili
ty, th
e tr
ade
mak
es
mo
ney
. T
his
co
nfo
rms
to t
rad
ers’
intu
itio
n.
•L
ater
, w
e w
ill
sho
w t
hat
ev
en t
his
is
not
alw
ays
tru
e.
Sal
e/ H
edg
e V
ola
tili
ty C
om
bin
atio
ns
Sale
Vol.
Vola
tili
ty
P&
L
Hed
ge
Vol.
Hed
ge
P&
L
Tota
l
Ran
ge
Sce
nar
io13%
+$3.3
0m
m10%
-$3.8
5m
m-$
0.5
5m
m
Tre
nd
Sce
nar
io13%
+$2.2
0m
m17%
-$3.0
7m
m-$
0.8
7m
m
P&
L f
rom
Sel
lin
g a
nd
Hed
gin
g a
t
the
Sam
e V
ola
tili
ty
(60,0
00,0
00)
(40,0
00,0
00)
(20,0
00,0
00)
-
20,0
00,0
00
40,0
00,0
00
60,0
00,0
00
80,0
00,0
00
5%
7%
9%
11%
13%
15%
17%
19%
21%
23%
25%
Ran
ge
Sce
nar
io
Tre
nd S
cenar
io
Del
ta S
ensi
tivit
ies
•L
et’s
now
see
what
eff
ect
hed
gin
g a
t th
e w
rong v
ola
tili
ty
has
on t
he
del
ta.
•W
e lo
ok a
t th
e d
iffe
ren
ce b
etw
een
$-d
elta
co
mp
ute
d a
t 20%
vola
tili
ty a
nd
$-d
elta
co
mp
ute
d a
t 12
% v
ola
tili
ty a
s a
fun
ctio
n o
f
tim
e.
•In
the
range
scen
ario
, th
e dif
fere
nce
bet
wee
n t
he
del
tas
per
sist
s th
roughout
the
hed
gin
g p
erio
d b
ecau
se b
oth
gam
ma
and
veg
are
mai
n s
ignif
ican
t th
roughout.
•O
n t
he
oth
er h
and, in
the
tren
d s
cenar
io, as
gam
ma
and
veg
adec
reas
e, t
he
dif
fere
nce
bet
wee
n t
he
del
tas
also
dec
reas
es.
Ran
ge
Sce
nar
io
$ D
elta D
iffere
nce (
20%
vol. -
12%
vol.)
(150,0
00,0
00)
(100,0
00,0
00)
(50,0
00,0
00)
-
50,0
00,0
00
100,0
00,0
00
150,0
00,0
00
050
100
150
200
250
300
Tre
nd
Sce
nar
io
$ D
elta D
iffere
nce (
20%
vol. -
12%
vol.)
(120,0
00,0
00)
(100,0
00,0
00)
(80,0
00,0
00)
(60,0
00,0
00)
(40,0
00,0
00)
(20,0
00,0
00)
-
20,0
00,0
00
40,0
00,0
00
60,0
00,0
00
050
100
150
200
250
300
Fix
ed a
nd
Sw
imm
ing
Del
ta
•F
ixed
(st
icky s
trik
e) d
elta
ass
um
es t
hat
the
Bla
ck-S
chole
s
impli
ed v
ola
tili
ty f
or
a par
ticu
lar
stri
ke
and e
xpir
atio
n i
s
const
ant.
T
hen
•S
wim
min
g (
or
float
ing)
del
ta a
ssum
es t
hat
the
at-t
he-
money
Bla
ck-S
chole
s im
pli
ed v
ola
tili
ty i
s co
nst
ant.
M
ore
pre
cise
ly, w
e as
sum
e th
at i
mpli
ed v
ola
tili
ty i
s a
funct
ion o
f
rela
tive
stri
ke
o
nly
. T
henS
CB
SF
IXE
D∂∂
=δ
K
C
SK
S
C
S
C
S
CB
S
BS
BS
BS
BS
BS
BS
BS
SW
IM∂∂
∂∂−
∂∂=
∂∂
∂∂+
∂∂=
σ
σ
σ
σδ
SK
An
Asi
de:
Th
e V
ola
tili
ty S
kew
SP
X V
ola
tility
6-A
pr-
99
10.0
0%
20.0
0%
30.0
0%
40.0
0%
50.0
0%
60.0
0%
70.0
0%
80.0
0%
90.0
0%
500
700
900
1100
1300
1500
1700
4/1
5/9
9
5/2
0/9
9
6/1
7/9
9
9/1
6/9
9
12/1
6/9
9
3/1
6/0
0
6/1
5/0
0
12/1
4/0
0
Str
ike
Vola
tili
ty
Vo
lati
lity
vs
x
0.0
0%
10.0
0%
20.0
0%
30.0
0%
40.0
0%
50.0
0%
60.0
0%
70.0
0%
80.0
0%
90.0
0%
-3-2
.5-2
-1.5
-1-0
.50
0.5
11.5
4/1
5/9
9
5/2
0/9
9
6/1
7/9
9
9/1
6/9
9
12/1
6/9
9
3/1
6/0
0
6/1
5/0
0
12/1
4/0
0
()
TF
Kx
ln=
Ob
serv
atio
ns
on
th
e V
ola
tili
ty S
kew
•N
ote
how
bea
uti
ful
the
raw
dat
a lo
oks;
ther
e is
a v
ery w
ell-
def
ined
pat
tern
of
impli
ed v
ola
tili
ties
.
•W
hen
im
pli
ed v
ola
tili
ty i
s plo
tted
agai
nst
, a
ll o
f
the
skew
curv
es h
ave
roughly
the
sam
e sh
ape.
T
FK
)/
ln(
Ho
w B
ig a
re t
he
Del
ta D
iffe
ren
ces?
•W
e as
sum
e a
skew
of
the
form
•F
rom
the
foll
ow
ing t
wo g
raphs,
we
see
that
the
typic
al
dif
fere
nce
in d
elta
bet
wee
n f
ixed
and s
wim
min
g
assu
mpti
ons
is a
round $
100m
m. T
he
erro
r in
hed
ge
vola
tili
ty w
ould
nee
d t
o b
e ar
ound 8
poin
ts t
o g
ive
rise
to a
sim
ilar
dif
fere
nce
.
•In
the
range
scen
ario
, th
e dif
fere
nce
bet
wee
n t
he
del
tas
per
sist
s th
roughout
the
hed
gin
g p
erio
d b
ecau
se b
oth
gam
ma
and v
ega
rem
ain s
ignif
ican
t th
roughout.
•O
n t
he
oth
er h
and, in
the
tren
d s
cenar
io, as
gam
ma
and
veg
a dec
reas
e, t
he
dif
fere
nce
bet
wee
n t
he
del
tas
also
dec
reas
es.
TxB
S3.
0−
=∂
∂σ
Ran
ge
Sce
nar
io
Sw
imm
ing
De
lta
-Fix
ed
De
lta
(20,0
00,0
00)
-
20,0
00,0
00
40,0
00,0
00
60,0
00,0
00
80,0
00,0
00
100,0
00,0
00
120,0
00,0
00
140,0
00,0
00
050
100
150
200
250
300
Tre
nd
Sce
nar
io
Sw
imm
ing
De
lta
-Fix
ed
De
lta
(20,0
00,0
00)
-
20,0
00,0
00
40,0
00,0
00
60,0
00,0
00
80,0
00,0
00
100,0
00,0
00
120,0
00,0
00
140,0
00,0
00
050
100
150
200
250
300
Su
mm
ary
of
Em
pir
ical
Res
ult
s•
Del
ta h
edgin
g a
lway
s giv
es r
ise
to h
edgin
g e
rrors
bec
ause
we
cannot
pre
dic
tre
alis
edvola
tili
ty.
•T
he
resu
lt o
f hed
gin
g a
t to
o h
igh o
r to
o l
ow
a v
ola
tili
ty
dep
ends
on t
he
pre
cise
pat
h f
oll
ow
ed b
y t
he
under
lyin
g
pri
ce.
•T
he
effe
ct o
f hed
gin
g a
t th
e w
rong v
ola
tili
ty i
s of
the
sam
e ord
er o
f m
agnit
ude
as t
he
effe
ct o
f hed
gin
g u
sing
swim
min
g r
ather
than
fix
ed d
elta
.
•F
iguri
ng o
ut
whic
h d
elta
to u
se a
t le
ast
as i
mport
ant
than
gues
sing f
utu
re v
ola
tili
ty c
orr
ectl
y a
nd
pro
bab
ly m
ore
im
port
ant!
So
me
Th
eory
•C
onsi
der
a E
uro
pea
n c
all
opti
on s
truck
at
K e
xpir
ing a
t
tim
e T
and d
enote
the
val
ue
of
this
opti
on a
t ti
me
t
acco
rdin
g t
o t
he
Bla
ck-S
chole
s fo
rmula
by
. I
n
par
ticu
lar,
.
•W
e as
sum
e th
at t
he
stock
pri
ce S
sat
isfi
es a
SD
E o
f th
e
form
wher
e m
ay i
tsel
f be
stoch
asti
c.
•P
ath-b
y-p
ath, w
e hav
e
Ct
BSb g
CT
SK
BS
Tbgbg
=−
+
dS S
dt
dZ
t t
tt
St
=+
µσ
,
σt
St
,
CT
CdC
C Sd
SC
td
tS
C Sdt
C Sd
SS
C Sd
t
BS
BS
BS
T
BS t
tB
SS
tB
S
t
T
BS t
tS
tt
BS
t
T
t
t
bgb g
−= =
∂ ∂+
∂
∂+
∂ ∂
R S| T|U V| W|
=∂ ∂
+−
∂ ∂
R S TU V W
z z z
0
2
0
22
2
20
1 2
22
22
20
σ
σσ
(ɵ
)
wher
e th
e fo
rwar
d v
aria
nce
.
So, if
we
del
ta h
edge
usi
ng t
he
Bla
ck-S
chole
s (f
ixed
) del
ta,
the
outc
om
e of
the
hed
gin
g p
roce
ss i
s:
ɵ.
σσ
tB
St
tt
22
=∂ ∂b g
ns
CT
CS
C Sdt
BS
BS
St
tB
S
t
T
tbgb g
−=
−∂ ∂
z0
1 2
22
22
20
(ɵ
)σ
σ
•In
the
Bla
ck-S
chole
s li
mit
, w
ith d
eter
min
isti
c vola
tili
ty,
del
ta-h
edgin
g w
ork
s pat
h-b
y-p
ath b
ecau
se
•In
rea
lity
, w
e se
e th
at t
he
outc
om
e dep
ends
both
on g
amm
a
and t
he
dif
fere
nce
bet
wee
nre
alis
edan
d h
edge
vola
tili
ties
.
•If
gam
ma
is h
igh w
hen
vo
lati
lity
is
low
and
/or
gam
ma
is l
ow
wh
en
vola
tili
ty i
s h
igh
you
wil
l m
ake
mo
ney
and
vic
e ve
rsa
.
•N
ow
, w
e ar
e in
a p
osi
tion t
o p
rovid
e a
counte
rexam
ple
to
trad
er i
ntu
itio
n:
•C
on
sid
er t
he
par
ticu
lar
pat
h s
how
n i
n t
he
foll
ow
ing
sli
de:
•T
he
real
ised
vo
lati
lity
is
12
.45
% b
ut
vo
lati
lity
is
clo
se t
o z
ero
wh
en g
amm
a is
lo
w a
nd h
igh
wh
en g
amm
a is
hig
h.
•T
he
hig
her
th
e h
edg
e vo
lati
lity
, th
e lo
wer
the
hed
ge
P&
L.
•In
th
is c
ase,
if
yo
u p
rice
and
hed
ge
a s
hort
opti
on
po
siti
on a
ta
vola
tili
ty l
ow
er t
han
18
%, you l
ose
mo
ney
.
σσ
St
tt
S2
2=
∀ɵ
A C
oo
ked
Sce
nar
io
3500
4000
4500
5000
5500
6000
050
100
150
200
250
Co
ok
ed S
cen
ario
: P
&L
fro
m S
elli
ng
and
Hed
gin
g a
t th
e S
ame
Vo
lati
lity
(15,0
00,0
00)
(10,0
00,0
00)
(5,0
00,0
00)
-
5,0
00,0
00
10,0
00,0
00
15,0
00,0
00
20,0
00,0
00
5%
7%
9%
11%
13%
15%
17%
19%
21%
23%
25%
Co
ncl
usi
on
s
•D
elta
-hed
gin
g i
s so
unce
rtai
n t
hat
we
must
del
ta-h
edge
as
litt
le a
s poss
ible
and w
hat
del
ta-h
edgin
g w
e do m
ust
be
opti
mis
ed.
•T
o m
inim
ise
the
nee
d t
o d
elta
-hed
ge,
we
must
fin
d a
sta
tic
hed
ge
that
min
imis
esgam
ma
pat
h-b
y-p
ath. F
or
exam
ple
,
Avel
laned
aet
al.
hav
e der
ived
such
sta
tic
hed
ges
by
pen
alis
ing g
amm
a pat
h-b
y-p
ath.
•T
he
ques
tion o
f w
hat
del
ta i
s opti
mal
to u
se i
s st
ill
open
.
Tra
der
s li
ke
fixed
and s
wim
min
g d
elta
. Q
uan
ts p
refe
r
mar
ket
-im
pli
ed d
elta
-th
e del
ta o
bta
ined
by a
ssum
ing t
hat
the
loca
l vola
tili
ty s
urf
ace
is f
ixed
.
An
oth
er D
igre
ssio
n:
Lo
cal
Vo
lati
lity
•W
e as
sum
e a
pro
cess
of
the
form
:
wit
h a
det
erm
inis
tic
funct
ion o
f st
ock
pri
ce a
nd t
ime.
•L
oca
l vola
tili
ties
can b
e co
mpute
d f
rom
mar
ket
pri
ces
of
opti
ons
u
sing
•M
arket
-im
pli
ed d
elta
ass
um
es t
hat
the
loca
l vola
tili
ty
surf
ace
stay
s fi
xed
thro
ugh t
ime.
dS S
dt
dZ
t t
tt
St
=+
µσɵ
,
ɵ,
σt
St
2
2
,2 2
ˆ
KCTC
KT
∂∂∂∂
=σK
T,
σ̂ C
CT
CdC
C SdS
C td
tS
C Sdt
C SdS
SC S
dt
LL
L
T
L t
tL
tS
tL
t
T
L t
tt
St
St
L
t
T
t
tt
bgb g
−= =
∂ ∂+
∂ ∂+
∂ ∂
R S| T|U V| W|
=∂ ∂
+−
∂ ∂
R S TU V W
z z z
0
2
0
22
2
20
1 2
22
22
20
σ
σσ
,
,,
(ɵ
)
We
can e
xte
nd t
he
pre
vio
us
anal
ysi
s to
loca
l vola
tili
ty.
So, if
we
del
ta h
edge
usi
ng t
he
mar
ket
-im
pli
ed d
elta
,th
e
outc
om
e of
the
hed
gin
g p
roce
ss i
s:
CT
CS
C Sdt
LL
tS
tS
tL
t
T
tt
bgb g
−=
−∂ ∂
z0
1 2
22
22
20
(ɵ
),
,σ
σ
∂ ∂C S
L t
ΓL
tt
L
t
SS
C Sbg≡
∂ ∂
22
2D
efin
e
Then
If t
he
clai
m b
eing h
edged
is
pat
h-d
epen
den
t th
en i
s al
so
pat
h-d
epen
den
t. O
ther
wis
e al
l th
e ca
n b
e det
erm
ined
at
ince
pti
on.
Wri
ting t
he
last
equat
ion o
ut
in f
ull
, fo
r tw
o l
oca
l vola
tili
ty
surf
aces
w
e get
Then
, th
e fu
nct
ional
der
ivat
ive
CT
CS
dt
LL
tS
tS
Lt
T
tt
bgb g
bg
−=
−z
01 2
22
0(
ɵ)
,,
σσ
Γ
ΓL
tS bg
ΓL
tS bg
CC
dt
dS
ES
SS
Sdt
LL
tt
St
SL
tt
t
T
tt
~ɵ
(~ɵ
),
,Σ
ΣΓ
didi
bgch
−=
−∞ zz
1 2
22
00
0σ
σϕ
ΣΣ
ˆ an
d
~
δ δσϕ
CE
SS
SS
L
tS
Lt
tt
t
ɵ,
21 2
0=
Γbgch
In p
ract
ice,
we
can s
et by b
uck
et h
edgin
g.
Note
in p
arti
cula
r th
at E
uro
pea
n o
pti
ons
hav
e al
l th
eir
sensi
tivit
y
to l
oca
l vola
tili
ty i
n o
ne
buck
et -
at s
trik
e an
d e
xpir
atio
n. T
hen
by b
uyin
g a
nd s
elli
ng E
uro
pea
n o
pti
ons,
we
can c
ance
l th
e ri
sk-
neu
tral
expec
tati
on o
f gam
ma
over
the
life
of
the
opti
on b
eing
hed
ged
-a
stat
ic h
edge.
This
is
not
the
sam
e as
can
cell
ing g
amm
a
pat
h-b
y-p
ath.
If y
ou d
o t
his
, you s
till
nee
d t
o c
hoose
a d
elta
to h
edge
the
rem
ainin
g r
isk. I
n p
ract
ice,
whet
her
fix
ed, sw
imm
ing o
r m
arket
-
impli
ed d
elta
is
chose
n, th
e par
amet
ers
use
d t
o c
om
pute
thes
e ar
e
re-e
stim
ated
dai
ly f
rom
a n
ew i
mpli
ed v
ola
tili
ty s
urf
ace.
Du
mas
, F
lem
ing a
nd W
hal
ey p
oin
t out
that
the
loca
l vola
tili
ty
surf
ace
is v
ery u
nst
able
over
tim
e so
agai
n, it
’s n
ot
obvio
us
whic
h d
elta
is
opti
mal
.
δ δσ
CL
tS
t,
20
=
Ou
tsta
nd
ing
Res
earc
h Q
ues
tio
ns
•Is
ther
e an
opti
mal
choic
e of
del
ta w
hic
h d
epen
ds
only
on
obse
rvab
le a
sset
pri
ces?
•H
ow
should
we
pri
ce
•P
ath-d
epen
den
t op
tion
s?
•F
orw
ard s
tart
ing
opti
on
s?
•C
om
po
un
d o
pti
on
s?
•V
ola
tili
ty s
wap
s?
So
me
Ref
eren
ces
•A
vel
lan
eda,
M.,
an
d A
. P
arás
. M
anag
ing
th
e vo
lati
lity
ris
k o
f
port
foli
os
of
der
ivat
ive
secu
riti
es:
the
Lag
ran
gia
nU
nce
rtai
n
Vo
lati
lity
Mo
del
. A
ppli
ed M
ath
emati
cal
Fin
an
ce, 3, 2
1-5
2 (
199
6)
•B
lach
er,
G. A
new
app
roac
h f
or
und
erst
andin
g t
he
imp
act
of
vola
tili
ty o
n o
pti
on
pri
ces.
R
ISK
98
Con
fere
nce
Han
dou
t.
•D
erm
an,
E. R
egim
es o
f vo
lati
lity
. R
ISK
Apri
l, 5
5-5
9 (
1999
)
•D
um
as,
B.,
J. F
lem
ing
, an
d R
.E. W
hal
ey. I
mp
lied
vo
lati
lity
fun
ctio
ns:
em
pir
ical
tes
ts. T
he
Jou
rna
l o
f F
inan
ce V
ol.
LII
I, N
o. 6,
Dec
emb
er 1
998
.
•G
up
ta, A
. O
n n
eutr
al g
rou
nd
. R
ISK
Ju
ly, 37
-41 (
19
97)