functional ito calculus and pde for path-dependent options bruno dupire bloomberg l.p. pde and...
TRANSCRIPT
![Page 1: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/1.jpg)
Functional Ito Calculus
and PDE for Path-Dependent Options
Bruno DupireBloomberg L.P.
PDE and Mathematical Finance
KTH, Stockholm, August 19, 2009
![Page 2: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/2.jpg)
Outline1) Functional Ito Calculus
• Functional Ito formula• Functional Feynman-Kac• PDE for path dependent options
2) Volatility Hedge
• Local Volatility Model• Volatility expansion• Vega decomposition• Robust hedge with Vanillas• Examples
![Page 3: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/3.jpg)
1) Functional Ito Calculus
![Page 4: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/4.jpg)
Why?
process. theof history,or
path,current theof functions it to extend We
processes. of functions with deals calculus Ito
prepayment MBS path rateinterest
reaction viralantigen toexposure
payoffdependent path history price
crop theofquality etemperatur
:patternscertain on depends
)(
eConsequencCause
link The .cumulative isy uncertaint ofimpact theoften,Most
ttf
t XfyX
![Page 5: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/5.jpg)
Review of Ito Calculus
• 1D
• nD
• infiniteD
• Malliavin Calculus
• Functional Ito Calculus
current value
possible evolutions
![Page 6: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/6.jpg)
Functionals of running paths
)( and )( is at of valueThe
)(,for ,: :functional
}t][0, functions RCLL bounded{
t][0, sections starting allon but T][0,path wholeon theonly not defined sFunctional
],0[
tXxsXtsX
XfXff
tttt
ttt
Ttt
t
0 T
12.87
6.32
6.34
![Page 7: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/7.jpg)
Examples of Functionals
hit is last value First time-
average rolling ofMax -
variablesstate ofnumber Infinite
(3) rangeon Option -
(2)Asian -
(1)European -
: time)(excluding variablesstate ofnumber Finite
priceoption dependent path of caseimportant thecovers onelast The
valuefinal ofn expectatio lConditiona-
variationQuadratic-
drawdownCurrent -
averageCurrent -
![Page 8: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/8.jpg)
Derivatives
t
XfXfXf
h
XfXfXf
tttstXsXtssXsX
htXtXtssXsX
fX
ttt
ttt
tht
htx
tttttt
thtt
ht
tt
Tttt
)()(lim)(
derivative Time
)()(lim)(
derivative Space
],[for )()( for )()(
)()( for )()(
,:,For
},t][0, functions RCLL bounded{
,
0
0
,,
],0[
![Page 9: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/9.jpg)
Examples
00
)(2010
tt
ttx
t
t
ut
xf
xxf
QVduxxf
t
hf
x
hf
txhXf tt
t
x
then ),,()( If
![Page 10: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/10.jpg)
Topology and Continuity
tsYXYXd
stYX
ststst
st
,),(
) assumecan (we ,in , allFor
:distance -
)()(),(
,:0,0
: if at continuous is : functionalA
:continuity -
tsst
s
t
XfYfYXd
Y
Xf
t s
X
Y
![Page 11: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/11.jpg)
Functional Ito Formula
xdfdtfdxfdf
xdXfdtXfdxXfXfXf
T
tCxCf
Xx
xxtx
T
ttxx
T
tt
T
ttxT
t
2
1
notation, concise morein or,
)(2
1)()()()(
, 0 allfor then,,continuous- themselves
sderivative e with thes,in and in ,continuous- is :
t],[0,over path its denotes and martingale-semi continuous a is If
Theorem
0000
12
![Page 12: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/12.jpg)
Fragment of proof
))()((
))()((
))()((
)()(
ff
ff
ff
ffdf
![Page 13: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/13.jpg)
Functional Feynman-Kac Formula
€
dx t = a(X t ) dt + b(X t ) dwt
For g and r suitably integrable, g : ΛT →ℜ, r : Λ →ℜ we define f :Λ →ℜ by
f (Yt ) ≡ E[e− r(Z u )
t
T
∫ dug(ZT ) | Zt = Yt ]
where for u∈ [0, t],ZT (u) = Yt (u)
for u∈ [t,T],dzu = a(Zu) du + b(Zu) dwu
⎧ ⎨ ⎩
Then f satisfies
Δ t f (X t ) + a(X t )Δx f (X t ) − r(X t ) f (X t ) +b2(X t )
2Δxx f (X t ) = 0
(From functional Ito formula, using that e− r(Z u )
0
t
∫ duf (X t ) is a martingale)
![Page 14: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/14.jpg)
Delta Hedge/Clark-Ocone
ttTtt
T
TT
ttxt
T
TT
ttxtt
dWXXgDEXbXXgEXg
dWXfXbXfXfXg
dWXfXbXdf
]|)([)(]|)([)(
:CalculusMalliavin from
formula Ocone-Clark the tocompared becan which
)()()()()(
:tionRepresenta Martingaleexplicit thehave weand
)()()(
Kac,-Feynman and Ito functional From
00
00
![Page 15: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/15.jpg)
P&L
St t
St
Break-even points
t
t
Option Value
St
CtCt t
S
Delta hedge
P&L of a delta hedged Vanilla
![Page 16: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/16.jpg)
Functional PDE for Exotics
€
Recallthat
df (X t ) = Δx f (X t ) dx + Δ t f (X t ) dt +1
2b2Δxx f (X t ) dt
The portfolio PF of option f with a short position of Δx f stocksgives :
dPF(X t ) = Δ t f (X t ) dt +1
2b2Δxx f (X t ) dt
In the absence of arbitrage,
Δ t f (X t ) +1
2b2Δxx f (X t ) + r(X t )(Δx f (X t )x t − f (X t )) = 0
The ?/Θ trade - off for European options also holds for
path dependent options, even with an infinite number of state variables.
However, in general ? and Θ will be path dependent.
![Page 17: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/17.jpg)
Classical PDE for Asian
term.convection bothering a is
02
10
2
1
,0
),,()(, Define ),( Assume
)()( :CallAsian of Payoff
2
222
2
2
0
0
I
lx
x
lb
I
lx
t
lfbf
x
lf
x
lfI
t
l
I
lxfxI
tIxlXfduxIdWtxbdx
KduxXg
xxt
xxxx
ttt
t
tttutttt
T
uT
![Page 18: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/18.jpg)
Better Asian PDE
0))()(2(2
10
2
1
)()(2
)()(
0
),,()(,)(][ Define
2
22
2
2
222
2
22
2
2
2
00
J
htT
Jx
htT
x
hb
t
hfbf
J
htT
Jx
htT
x
hf
J
htT
x
hf
tTJ
t
hfJ
tJxhXfxtTduxduxEJ
xxt
xx
x
x
tt
tttt
t
u
T
utt
![Page 19: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/19.jpg)
2) Robust Volatility Hedge
![Page 20: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/20.jpg)
Local Volatility Model• Simplest model to fit a full surface• Forward volatilities that can be locked
CqK
CKqr
K
CK
TK
T
C
dWtSdtqrS
dS
2
22
2
2
,
),()(
![Page 21: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/21.jpg)
Summary of LVM
• Simplest model that fits vanillas
• In Europe, second most used model (after Black-Scholes) in Equity Derivatives
• Local volatilities: fwd vols that can be locked by a vanilla PF
• Stoch vol model calibrated
• If no jumps, deterministic implied vols => LVM
),(][ 22 tSSSE loctt
![Page 22: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/22.jpg)
S&P500 implied and local vols
![Page 23: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/23.jpg)
S&P 500 FitCumulative variance as a function of strike. One curve per maturity.Dotted line: Heston, Red line: Heston + residuals, bubbles: market
RMS in bpsBS: 305Heston: 47H+residuals: 7
![Page 24: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/24.jpg)
Hedge within/outside LVM
• 1 Brownian driver => complete model
• Within the model, perfect replication by Delta hedge
• Hedge outside of (or against) the model: hedge against volatility perturbations
• Leads to a decomposition of Vega across strikes and maturities
![Page 25: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/25.jpg)
Implied and Local Volatility Bumps
implied to
local volatility
![Page 26: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/26.jpg)
P&L from Delta hedging
T
txxtt
T
ttx
T
txxt
T
tt
T
ttxTT
ttt
txxttt
tT
Q
tT
ttt
dtYftyvvdyYfXf
dtYfvdtYfdyYfYfYfYg
xydWvdyy
XftxvXf
XXgEXffg
dWtxvdxqr
v
0 000
0000
00
0
0
)()),((2
1)()(
)(2
1)()()()()(
formula, Ito functional by the , with follows If
0)(),(2
1)( PDE, functionalBy
])([)(by define,For
.),(,0 Assume
0
![Page 27: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/27.jpg)
Model Impact
T
ttxxtQv
g
T
txxttQ
gg
vT
Qg
T
txxtt
T
ttxTT
dtdxxxXftxvvEtxv
dtXftxvvEvv
vtxXYgEv
dtYftyvvdyYfXfYfYg
v
v
v
0 00
0 00
0
0 000
])()),([(),(2
1)(
])()),(([2
1)()(
,for density n transitio the),( and ])([)( with Hence,
)()),((2
1)()()()(
Recall
![Page 28: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/28.jpg)
Comparing calibrated models
][],[
:alive""by ngconditioni ofimpact theevaluate toamountsIt
),(][ options, vanillasame on the calibrated are and If
],)),([(),(
),(2
1
],)()),([(),(),(2
1
])()),([(),(2
1)()(
option,Barrier out -knock aFor
])()),([(),(2
1)()(
Recall
00
0 02
2
0 0
0 00
0 00
xxvEalivexxvE
txvxxvEvv
dtdxalivexxtxvvEx
txftx
dtdxalivexxXftxvvEtxPtx
dtdxxxXftxvvEtxvv
dtdxxxXftxvvEtxvv
ttQ
ttQ
ttQ
T
ttQalivev
alive
T
ttxxtQ
alivev
T
ttxxtQv
gg
T
ttxxtQv
gg
vv
v
v
v
v
v
![Page 29: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/29.jpg)
Volatility Expansion in LVM
])([),(2
1),( where
),(),()(
])([),(),(2
1)()(
),(),( : form theof LVM a is wherecase In the
])()),([(),(2
1)()(
general,In
00
00
00
000
00
0 00
xxXfEtxtxm
dtdxtxutxmv
dtdxxxXfEtxutxvuv
dWtxutxvdxuvv
dtdxxxXftxvvEtxvv
ttxx
Quv
T
g
T
ttxx
Quvgg
tttt
T
ttxxtQv
gg
uv
uv
v
![Page 30: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/30.jpg)
Frechet Derivative in LVM
)derivative(Frechet ),(at variancelocal the to ofy sensitivit theis
])([),(2
1),(
where
),(),(
])([),(),(2
1
)()(lim,
:satisfies ofdirection in the derivativeFrechet The
])([),(),(2
)()(
,particularIn
00
00
00
0
0
00
0v
000
txg
xxXfEtxtxm
dtdxtxutxm
dtdxxxXfEtxutx
vuvu
u
dtdxxxXfEtxutxvuv
ttxx
Qv
T
T
ttxx
Qv
ggg
T
ttxx
Quvgg
v
v
uv
![Page 31: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/31.jpg)
One Touch Option - Price
Black-Scholes model S0=100, H=110, σ=0.25, T=0.25
![Page 32: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/32.jpg)
One Touch Option - Γ
![Page 33: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/33.jpg)
PtSmTO ..2
1),(:..
![Page 34: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/34.jpg)
Up-Out Call - Price
Black-Scholes model S0=100, H=110, K=90, σ=0.25, T=0.25
![Page 35: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/35.jpg)
Up-Out Call - Γ
![Page 36: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/36.jpg)
PtSmUOC ..2
1),(:
![Page 37: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/37.jpg)
Black-Scholes/LVM comparison
price. LVM reach the toenables Scholes-Black theofinput y volatilitno case, In this
![Page 38: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/38.jpg)
Vanilla hedging portfolio I
?),(function get the can we How
),(])([])([),(
t)(x, allfor ifonly and if moves volatility
small all hedges vanillasof ),( Portfolio
.at variancelocal theto
ofy sensitivit theis ])([),(2
1),( Recall
00
00
2
2
,
g
TK
txhxxXfExxXEx
txPF
dTdKCTKPF
(x,t)
xxXfEtxtxm
ttxx
Q
ttPFxx
Q
TK
ttxx
Qv
vv
v
![Page 39: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/39.jpg)
Vanilla hedging portfolios II
bumps. volimplied y tosensitivit no hence
bumps vollocal y tosensitivit no has and 0]),()([,,
00 with 0)(then 2
1)( take weIf c)
.)()( Thus,
])([),( with ),(),(]),()([),(
),(),)((),( b)
)(),(condition boundary with 0)( , call aFor
2
1)( define we),,(For a)
2
2
20
2
2
2
2
2
2
2
,
2,
2
2,
2
,
20
2
0
00
PF-fxxtxx
PFXfEtx
kk(x,T)kLx
hv
t
hhL
hLkL
xxXfEtxhtxx
PFtxhxxtx
x
PFXfEtxk
txtxx
PFLdTdKCTKPF
xTxx
C
x
CLC
x
kv
t
kkLtxk
ttxx
Q
ttxx
Q
ttxx
Q
TK
KTKTK
TK
v
vv
![Page 40: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/40.jpg)
Example : Asian option
.at variancelocal theto ofy sensitivit theis ])([),(2
1),(
1maturity20 volatility100S, )( : off-Pay v
g
00
0
0
00 (x,t) xxXfEtxtxm
TvKKdtxXgdWdx
ttxx
Qv
T
tttt
v
K
T
KT
![Page 41: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/41.jpg)
Asian Option Superbuckets
KT
20
2
,
),(),(
2
1),(),(
),( with hedgeatility Robust vol
x
TKhTKv
t
TKhTK
dTdKCTKPF TK
K
T
![Page 42: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/42.jpg)
Γ/VegaLink
![Page 43: Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009](https://reader036.vdocument.in/reader036/viewer/2022062304/56649f225503460f94c3aeb3/html5/thumbnails/43.jpg)
Conclusion
• Ito calculus can be extended to functionals of price paths
• Local volatilities are forward values that can be locked
• LVM crudely states these volatilities will be realised
• It is possible to hedge against this assumption
• It leads to a strike/maturity decomposition of the volatility risk of the full portfolio