Open issues in equity derivatives modelling - …euroschoolmathfi/bergomi.pdfOpen issues in equity...
15
Open issues in equity derivatives modelling Lorenzo Bergomi Equity Derivatives Quantitative Research Société Générale [email protected]
Transcript of Open issues in equity derivatives modelling - …euroschoolmathfi/bergomi.pdfOpen issues in equity...
Open
issu
es in
equi
ty d
eriva
tives
mod
ellin
g
Lore
nzo
Berg
omi
Equit
y Der
ivativ
es Q
uanti
tative
Res
earch
Socié
té Gé
néra
lelor
enzo
.berg
omi@
sgcib
.com
Talk
Outli
ne
Equi
ty d
eriva
tives
at S
G
A br
ief h
istor
y of e
quity
der
ivativ
e pro
duct
s
Preh
istor
y –
1997
Histo
ry19
97 –
2003
Mode
rn tim
es 2
003 –
Mode
lling
issue
s, alg
orith
mic
issue
s
Risk
mea
sure
men
t and
man
agem
ent
Conc
lusio
n
Lore
nzo
Berg
omi
1
Equi
ty d
eriva
tives
at S
G
SG re
gard
ed b
y ind
ustry
par
ticip
ants
as N
o 1 i
n eq
uity
der
ivativ
es
Lore
nzo
Berg
omi
2
A br
ief h
istor
y of e
quity
der
ivativ
e pro
duct
s Pr
ehist
ory
–199
7
Prod
ucts
Risk
s
Barri
er o
ptio
ns / D
igita
lsSk
ew: l
evel
/ dyn
amics
(litt
le)
Max /
Min
opt
ions
sam
e
Asian
opt
ions
Smile
Bask
et o
ptio
nsCo
rrelat
ion
(leve
l)
Volat
ility s
waps
Smile
, Vol
OfVo
l
Sim
ple c
lique
tsFo
rwar
d sm
ile
Mode
ls / a
lgos
-Bl
ack S
chol
es / l
ocal
vol
-PD
E / s
traig
ht M
onte
Car
lo
Lore
nzo
Berg
omi
+ ⎟ ⎠⎞⎜ ⎝⎛
−∑
KtS
Ni
1
+ ⎟ ⎠⎞⎜ ⎝⎛
−∑
Ki TS
N1
()+ ⎟ ⎠⎞
⎜ ⎝⎛−K
tStmax
KσkSkS
Tk
ˆ2
1ln
1−
⎟⎟ ⎠⎞
⎜⎜ ⎝⎛
−∑
+
⎟⎟ ⎠⎞
⎜⎜ ⎝⎛−
KSS
TT
12
3
A br
ief h
istor
y of e
quity
der
ivativ
e pro
duct
s Hi
stor
y -1
19
97 –
2005
Capit
al-gu
aran
teed p
rodu
cts di
stribu
ted by
retai
l netw
orks
Ever
est 1
997
5 yea
rs / 1
2 stoc
ks
Emer
ald 20
0410
year
s / 20
stoc
ksE
very
yea
r, th
e st
ock
who
se p
erfo
man
ce si
nce
t = 0
is th
e lar
gest
get
s fro
zen
and
rem
oved
from
the
bask
et, a
nd it
s lev
el is
floor
ed a
t 200
% o
t its
initi
al va
lue.
100%
+ m
axim
um p
eror
man
ce o
f yea
rly b
aske
t valu
es si
nce
t = 0
, flo
ored
at 0
.
… an
d man
y, ma
ny, m
any,
other
varia
tions
tryin
g to
find
clos
ed-fo
rm fo
rmul
as fo
r spe
cific
exot
ic pa
yoffs
now
irre
levan
t and
use
less
Lore
nzo
Berg
omi
⎟⎟⎟ ⎠⎞
⎜⎜⎜ ⎝⎛+
jS
j TS 0
min
%100
4
Lore
nzo
Berg
omi
5
A br
ief h
istor
y of e
quity
der
ivativ
e pro
duct
s Hi
stor
y -2
19
97 –
2005
Varia
nce S
waps
3 mon
ths
5 yea
rsPa
ys re
alize
d va
rianc
e –
usua
lly m
easu
red
usin
g da
ily re
turn
s
stock
s / in
dices
Napo
leon
5 yea
rs / 1
inde
xE
very
yea
r, pa
ys c
oupo
n re
duce
d by
wor
st o
f 12
mon
thly
perf
orm
ance
s of t
he in
dex.
Accu
mulat
or3 y
ears
/ 1 in
dex
At m
atur
ity p
ays t
he su
m –
if it
is po
sitiv
e –
of th
e m
onth
ly pe
rfor
man
ces,
capp
ed a
nd
and
floor
ed.
Lore
nzo
Berg
omi
+ ⎟⎟⎟ ⎠⎞
⎜⎜⎜ ⎝⎛
⎟⎟ ⎠⎞
⎜⎜ ⎝⎛
−+
1min
kSkS
kC
+ ⎟⎟⎟ ⎠⎞
⎜⎜⎜ ⎝⎛
⎟⎟⎟ ⎠⎞
⎜⎜⎜ ⎝⎛−
⎟⎟ ⎠⎞
⎜⎜ ⎝⎛−
−∑ k
kSkS
%1,
%1,11
min
max
Tσ
kSkS
k
2
2
ˆ1
ln−
⎟⎟ ⎠⎞
⎜⎜ ⎝⎛
−∑
6
Mode
rn ti
mes
Corri
dor v
arian
ce sw
aps
Dail
y va
rianc
e on
ly co
unte
d w
hen
unde
rlyin
g is
insid
e gi
ven
inte
rval
Corre
lation
swap
sPa
ys re
alize
d co
rrela
tion
over
3 y
ears
by
stoc
ks o
f an
inde
x
Gap n
otes
Mat
urity
= 1
yea
r, a
serie
s of d
aily
puts
on
daily
retu
rns o
f an
inde
x
with
strik
es 8
5%, 9
0%
Optio
ns on
reali
zed v
arian
ceO
n in
dice
s, m
atur
ities
: 3 m
onth
s to
2 ye
ars
Timer
optio
nsV
anill
a pa
yoff
, paid
whe
n re
alize
d va
rianc
e Q
tre
ache
s set
leve
l:
Hybr
idsE
quiti
es /
Rat
es /
For
ex /
Com
mod
ities
Arb
itrar
y pa
yoffs
Lore
nzo
Berg
omi
[]
∑⎟⎟⎟ ⎠⎞
⎜⎜⎜ ⎝⎛∆
−⎟⎟ ⎠⎞
⎜⎜ ⎝⎛
−∈
kH
LkS
tσ
kSkS
2
2
,ˆ
1ln
1[
][
][
]H
LH
L,0
,,
,,
∞+
+ ⎟⎟ ⎠⎞
⎜⎜ ⎝⎛−
⎟⎟ ⎠⎞⎜⎜ ⎝⎛
∑−
22
1
ˆln
1K
τττ
σSS
T
7
∑ =
+⎟⎟ ⎠⎞
⎜⎜ ⎝⎛=
τ iii
τSS
Q1
2
1ln
Mode
lling
issue
s –1
Why
not
just
delt
a-he
dge ?
Varia
nce o
f res
idua
l P&L
too
large
use o
ther
opt
ions
Optio
ns ar
e hed
ged
with
opt
ions
Once
we s
tart
usin
g op
tions
as h
edgi
ng in
stru
men
tsLe
ss se
nsiti
vity t
o hi
stor
ical p
aram
eter
s, m
ore s
ensis
tivity
to im
plied
par
amet
ers
Mode
l the
dyn
amics
of i
mpl
ied p
aram
eter
s
Exam
ple o
f sim
ple c
lique
t
Lore
nzo
Berg
omi
8
t1T
2T
12σ̂
()
L,,
ˆ 12r
σP
+ ⎟⎟ ⎠⎞⎜⎜ ⎝⎛
−1
12 TT SS
2008
2007
2006
2005
2004
2003
2002
60 55 50 45 40 35 30 25 20 15 10
Smile
3 m
ois
K =
95
- K=
105
01234567
2/5/2001
2/5/2002
2/5/2003
2/5/2004
2/4/2005
2/4/2006
2/4/2007
2/4/2008
Mode
lling
issue
s –2
How
shou
ld ca
libra
tion
be d
one ?
Do
we re
ally n
eed
to ca
libra
te ?
•N
ot c
omp
uls
ory:
char
ge a
hed
gin
g co
st.
We
hed
ge p
aram
eter
pby
tra
din
g in
stru
men
t O
so t
hat
sen
siti
vity
to
pva
nis
hes
:
•M
odel
pri
ce P
is a
dju
sted
so
as t
o in
clu
de
hed
gin
g co
st:
•Th
en w
hat i
s the
poi
nt in
calib
ratin
g ?
•E
nsu
res
pri
ce fa
ctor
s in
hed
gin
g co
sts
incu
rred
at
t =
0–
no
t fu
ture
co
sts
!
Nece
ssar
y to
calib
rate
mod
el on
relev
ant s
et o
f hed
ging
inst
rum
ents
Usele
ss if
one
is u
nabl
e to
spec
ify h
ow to
hed
ge th
e exo
tic w
ith th
e hed
ge in
stru
men
ts
Lore
nzo
Berg
omi
9
dpdO λdpdP
=
()
()
()
()
()
Market
Market
Price
pp
Pp
Op
Oλ
pP
=≈
−+
=ˆ
ˆ
Mode
lling
issue
s –3
Volat
ility r
isk –
mod
els
•«O
ld m
odels
»•
Loc
al v
olat
ilit
y
•H
esto
n
•SA
BR
•M
odel
s ba
sed
on
pro
cess
of i
nst
anta
neo
us
vari
ance
:
•Ju
mp
/ L
évy
•Ch
allen
ge: B
uild
mod
els th
at g
ive co
ntro
l on
join
t dyn
amics
of i
mpl
ied vo
latilit
ies an
d sp
ot:
•Firs
t ste
p: m
odel
dyna
mics
of c
urve
of f
orwa
rd va
rianc
es
•Nex
t ste
p: m
odel
dyna
mics
of t
he im
plied
volat
ility s
urfa
ce
•D
irec
t m
odel
lin
g of
dyn
amic
s of
imp
lied
vol
atil
itie
s is
a d
ead
en
d
•L
ow-d
imen
sion
al M
arko
v re
pre
sen
tati
on d
esir
able
•H
ow m
uch
free
dom
are
we
allo
wed
?
Lore
nzo
Berg
omi
VS
dWdt
dVdW
SV
dtdS
)(
+=
+=
KK
10
Euro
stox
x 50
- m
at 1
an
10152025
8090
100
110
120
Smile
orig
inal
Smile
S =
105
%Sm
ile S
= 9
5%
Mode
lling
issue
s –4
Hybr
ids
•Equ
itie
s
•In
tere
st r
ates
•F
orex
•C
omm
odit
ies
•Hyb
rid m
odels
are n
ot b
uilt
by si
mpl
y glu
eing
toge
ther
mod
els fo
r eac
h as
set c
lass
•Pa
ssive
hyb
rids:
pay
off i
nvol
ves o
ne as
set c
lass o
nly
•L
ong-
dat
ed e
quit
y, F
orex
op
tion
s
•C
red
it /
Equ
ity:
con
vert
ible
bon
ds
•Ac
tive h
ybrid
s:pa
yoff
invo
lves a
ll ass
et cl
asse
s•
Req
uir
e st
ate-
of-t
he
art
mod
els
for
each
ass
et c
lass
•E
ven
loca
l vol
cal
ibra
tion
for
equ
ity
smil
es n
ot e
asy
wh
en in
tere
st r
ates
are
sto
chas
tic
Lore
nzo
Berg
omi
11
Mode
lling
issue
s –5
Corre
latio
n –h
ow d
e we p
ut to
geth
er co
rrelat
ion
mat
rices
?•H
ow d
o w
e bu
ild
th
e la
rge
corr
elat
ion
mat
rice
s n
eed
ed in
hyb
rid
mod
elli
ng
?
•Si
mp
ler
ques
tion
: im
agin
e a
1-fa
ctor
sto
ch. v
ol m
odel
an
d a
pay
off
invo
lvin
g 2
sec
uri
ties
•H
ow d
o w
e se
t th
e cr
oss-
corr
elat
ion
s ?
•E
ven
sim
ple
r qu
esti
on –
how
do
we
mea
sure
cor
rela
tion
s ?
•E
xam
ple
of
Eu
rop
ean
/ J
apan
ese
stoc
ks –
no
over
lap
Corre
latio
n –h
ow d
e we m
easu
re co
rrelat
ion
risk ?
Corrr
elatio
n –h
ow to
mod
el co
rrelat
ion
smile
?
Lore
nzo
Berg
omi
12
oC
oC
oC
oC
oC
oC
oC
oC
Europe
Japan
?
?
Algo
rithm
ic iss
ues
Mont
e Car
lo •Ho
w ca
n we
spee
d up
pric
ing
?
•Qu
asi-r
ando
m n
umbe
rs
•Di
scre
tizat
ion
of S
DEs ?
•Call
able
/ put
able
optio
ns
•Com
putin
g se
nsiti
vies t
o •In
itial
cond
ition
s
•Par
amet
ers o
f dyn
amics
(vol
atilit
ies / c
orre
latio
ns, e
tc..)
Lore
nzo
Berg
omi
13
Conc
lusio
n
Thes
e are
excit
ing
times
for d
oing
qua
ntita
tive f
inan
ce
Lot
s of
new
inst
rum
ents
/ p
rod
uct
/ a
lgor
ith
mic
issu
es
Ric
h m
ath
emat
ical
too
lbox
fro
m w
hic
h t
o p
ick
Lore
nzo
Berg
omi
14
