Functional Itô Calculus
and Volatility Risk Management
Bruno Dupire
Bloomberg L.P/NYU
AIMS Day 1
Cape Town, February 17, 2011
Outline
1)
Functional Itô Calculus
•
•
•
Functional Itô formula
Functional Feynman-Kac
PDE for path dependent options
2)
Volatility Hedge
•
•
•
•
•
Local Volatility Model
Volatility expansion
Vega decomposition
Robust hedge with Vanillas
Examples
1) Functional Itô Calculus
Why?
Most oft en,t heimpactof uncert ainty is cumulat ive. T helink
Cause  Consequence
f
Xt 

yt  f ( X t )
depends on cert ainpat t erns:
t emperat ure  wat er level
price hist ory pat hdependentpayoff
exposuret o ant igen viralreact ion
int erestrat epat h MBS prepayment
It ôcalculus deals wit h funct ionsof processes.
W e ext endit t ofunct ionsof t hecurrent pat h,
or hist ory,of t heprocess.
Review of Itô Calculus
• 1D
• nD
• infiniteD
• Malliavin Calculus
• Functional Itô Calculus
current value
possible evolutions
Functionals of running paths
Functionals defined not only on thewhole path[0,T ]but on all startingsections[0,t]
 t  {RCLL functions[0,t]  }


t
t[ 0 ,T ]
f functional: f :   , for X t   t , f ( X t )  
T he value of X t at s  t is X t ( s ) and xt  X t (t )
12.87
6.34
6.32
0
T
Examples of Functionals
- Current average
- Current drawdown
- Conditional expectation of final value
- Super - replicating price
T helast one coverstheimportantcase of pathdependentoptionprice
Finitenumber of state variables(excluding time):
- European(1)
- Asian (2)
- Optionon range (3)
Infinitenumber of state variables
- Max of rollingaverage
- First timelast valueis hit
Derivatives
 t  {RCLL functions[0,t ]  },  

t
t[ 0 ,T ]
For X t   t , f :   ,
X th ( s )  X t ( s ) for s  t X th (t )  X t (t )  h
X t ,t ( s )  X t ( s ) for s  t X t ,t ( s )  X t (t ) for s  [t , t  t ]
Space derivative
f ( X th )  f ( X t )
 x f ( X t )  lim
h 0
h
T imederivative
f ( X t ,t )  f ( X t )
 t f ( X t )  lim
t 0
t
Examples
f
xt
x f
t f
1
0

t
0
xu du
0
xt
h


f

 x
x
If f ( X t )  h( xt , t ), then
 f  h
 t
t
QVt
2 ( xt  xt  )
0
Topology and Continuity
 - distance:
For all X t , Ys in , (we can assume t  s)
d  ( X t , Ys )  X t , s t  Ys
Y
t

 s t
s
X
 - cont inuity:
A funct ionalf :    is   cont inuousat X t   if :
  0,   0 : Ys  ,
d  ( X t , Ys )    f (Ys )  f ( X t )  
Functional Itô Formula
Definit ion: a functionalf :    is sm oothif it is  - continuous,
C 2 in x and C 1 in t , wit h these derivatives themselves  - continuous.
T heorem
If x is a continuoussemi - martingaleand X t denotesits pathover[0,t],
then,for all smoothfunctionalf and for all T  0 ,
T
T
0
0
f ( X T )  f ( X 0 )    x f ( X t ) dxt  
1 T
 t f ( X t ) dt    xx f ( X t ) d x
2 0
or, in moreconcisenotation,
1
df   x f dx   t f dt   xx f d x
2
t
Fragment of proof
df  f ( )  f ( )
 ( f ( )  f ( ))
( f (
( f (
) f (
) f (
))
))
Functional Feynman-Kac Formula
Generalisat ion of FK t onon Markovdynamicsand pat hdependentpayoff.
dxt  a ( X t ) dt  b( X t ) dWt
For g suit ably int egrable, g :  T  , r :    . We define f :    by
T
f (Yt )  E[e t
 r ( Z u ) du
g ( Z T ) | Z t  Yt ]
for u  [0, t ], Z T (u )  Yt (u )
where 
for u  [t , T ], dzu  a ( Z u ) du  b( Z u ) dWu
T hen,if f is smoot h,it sat isfies
b2 ( X t )
 t f ( X t )  a( X t ) x f ( X t )  r ( X t ) f ( X t ) 
 xx f ( X t )  0
2
t
(applyfunct ionalIt ôformula t o t hemart ingale e 0
 r ( X u ) du
f ( X t ))
Delta Hedge/Clark-Ocone
If f defined by f (Yt )  E[ g ( Z T ) | Z t  Yt ] is smoot h,
t hen wehave t heexplicitMart ingaleRepresent at ion:
T
g ( X T )  E[ g ( X T ) | X 0 ]   b( X t )  x f ( X t ) dWt
0
From funct ionalIt ôand Feynman- Kac wit h r  0,
df ( X t )  b( X t )  x f ( X t ) dWt
and
T
g ( X T )  f ( X T )  f ( X 0 )   b( X t )  x f ( X t ) dWt
0
It can be comparedt o t heClark - Oconeformula
from MalliavinCalculus :
T
g ( X T )  E[ g ( X T ) | X 0 ]   b( X t ) E[ Dt g ( X T ) | X t ] dWt
0
P&L of a delta hedged Vanilla
Option Value
P&L
Break-even
points
Delta
hedge
 t
Ct t
Ct
  t
St
S

St
St t
Functional PDE for Exotics
T
If f given by f (Yt )  E[e t
 r ( Z u ) du
g ( Z T ) | Z t  Yt ] is smooth,
thenit satisfies
1 2
 t f ( X t )  b  xx f ( X t )  r ( X t ) ( x f ( X t ) xt  f ( X t ))  0
2
T he Γ/ trade- off for Europeanoptionsalso holds for path
dependentoptions,even withan infinitenumber of st at evariables.
However,in general Γ and  will be pathdependent.
Classical PDE for Asian
T
P ayoffof Asian Call : g ( X T )  (  xu du  K ) 
0
t
Assume dxt  b( xt , t ) dWt Define I t   xu du , f ( X t )  l ( xt , I t , t )
0
l l
 t I  xt   t f  x 
I t
l
 2l
 x I  0   x f  ,  xx f  2
x
x
1 2
l
l 1 2  2l
 t f  b  xx f  0   x  b
0
2
2
t
I 2 x
l
x is a botheringconvectionterm.
I
Better Asian PDE
T
t
0
0
Define J t  Et [  xu du]   xu du  (T  t ) xt , f ( X t )  h( xt , J t , t )
h
t J  0  t f 
t
h
h


f


(
T

t
)
 x
x
J
 x J  (T  t )  
2
2
2

h

h

h
2
 f 
 2 (T  t )
 (T  t )
2
 xx
x
xJ
J 2
2
1 2
h 1 2  2 h
 2h

h
 t f  b  xx f  0   b ( 2  2 (T  t )
 (T  t ) 2 2 )  0
2
t 2
x
xJ
J
2) Robust Volatility Hedge
Local Volatility Model
• Simplest model to fit a full surface
• Forward volatilities that can be locked
dS
 (r  q) dt   ( S , t ) dW
S
C  2 K , T  2  2C
C



K

r

q
K
 q C
2
T
2
K
K
Summary of LVM
• Simplest model that fits vanillas
• In Europe, second most used model (after BlackScholes) in Equity Derivatives
• Local volatilities: fwd vols that can be locked by a
vanilla PF
2
2
• Stoch vol model calibrated  E[ t St  S ]   loc ( S , t )
• If no jumps, deterministic implied vols => LVM
S&P500 implied and local vols
S&P 500 Fit
Cumulative variance as a function of strike. One curve per maturity.
Dotted line: Heston, Red line: Heston + residuals, bubbles: market
RMS in bps
BS: 305
Heston: 47
H+residuals: 7
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
Implied and Local Volatility Bumps
P&L from Delta hedging
Assume r  q  0, dxt  v0 ( xt , t ) dWt .
For g   T , define f   by f ( X t )  E
Qv0
[g( XT ) X t ]
1
By functionalP DE, t f ( X t )  v0 ( xt , t )  xx f ( X t )  0
2
If y follows dyt  vt dWt with y0  x0 , by thefunctionalItôformula,
1 T
g (YT )  f (YT )  f (Y0 )    x f (Yt ) dyt    t f (Yt ) dt   vt  xx f (Yt ) dt
0
0
2 0
T
1 T
 f ( X 0 )    x f (Yt ) dyt   (vt  v0 ( yt , t ))  xx f (Yt ) dt
0
2 0
T
T
Model Impact
Recall
g(YT )  f (YT )  f (X 0 ) 

T
0
 x f (Yt ) dyt 
1
2

T
0
(v t  v 0 (y t ,t)) xx f (Yt ) dt
v,
Hence, with  g (v)  E Qv [g(YT ) X 0 ] and  v (x,t) t he transition density for
1 Qv T
E [  (v t  v 0 (x t ,t)) xx f (X t ) dt]
0
2
1 T
  g (v 0 )     v (x,t) E Qv [(v t  v 0 (x,t)) xx f (X t ) x t  x]dx dt
2 0
 g (v) g (v 0 ) 
Comparing calibrated models
Recall
1 T
 g (v)   g (v0 )     v ( x, t ) E Qv [(vt  v0 ( x, t ))  xx f ( X t ) xt  x] dx dt
2 0
For a knock- out Barrier opt ion,
1 T
Qv
v

(
x
,
t
)
E
[(vt  v0 ( x, t ))  xx f ( X t ) xt  x] dx dt


0
2
1 T
    v ( x, t ) Palive ( x, t ) E Qv [(vt  v0 ( x, t ))  xx f ( X t ) xt  x, alive] dx dt
2 0
 2 f alive ( x, t ) Qv
1 T
v
   alive ( x, t )
E [(vt  v0 ( x, t )) xt  x, alive] dx dt
2 0
x 2
If v and v0 are calibrat edon t hesame vanillaopt ions,E Qv [vt xt  x]  v0 ( x, t )
 g (v)   g (v0 ) 
It amount st o evaluat et heimpactof condit ioning by " alive":
E Qv [vt xt  x, alive]  E Qv [vt xt  x]
Volatility Expansion in LVM
In general,
1 T
Qv
v

(
x
,
t
)
E
[(vt  v0 ( x, t ))  xx f ( X t ) xt  x] dx dt


0
2
In thecase where v is a LVM of theformv0  u : dxt  v0 ( xt , t )  u ( xt , t ) dWt
 g (v)   g (v0 ) 
1 T
Q
 g (v0  u )   g (v0 )     v0 u ( x, t ) u ( x, t ) E v0 u [ xx f ( X t ) xt  x] dx dt
2 0
  g (v0 )  
T
0

m( x, t ) u ( x, t ) dx dt
1 v0  u
Q
where m( x, t )   ( x, t ) E v0 u [ xx f ( X t ) xt  x]
2
Fréchet Derivative in LVM
In particular,
 g (v0  u )   g (v0 ) 

T
2 
0
 v u ( x, t ) u ( x, t ) E
0
Qv0 u
[ xx f ( X t ) xt  x] dx dt
T heFréchetderivativein thedirectionof u satisfies:
 g (v0  u )   g (v0 )
  v  g , u  lim
 0


1 T
Qv0
v0

(
x
,
t
)
u
(
x
,
t
)
E
[ xx f ( X t ) xt  x] dx dt


0
2

T
0
 m( x, t ) u ( x, t ) dx dt
where
1
Q
m( x, t )   v0 ( x, t ) E v0 [ xx f ( X t ) xt  x]
2
is thesensitivity of g to thelocal varianceat ( x, t ) (Fréchetderivative)
One Touch Option - Price
Black-Scholes model S0=100, H=110, σ=0.25, T=0.25
One Touch Option - Γ
1
O.T . : m( S , t )  . .P
2
Up-Out Call - Price
Black-Scholes model S0=100, H=110, K=90, σ=0.25, T=0.25
Up-Out Call - Γ
1
UOC : m( S , t )  . .P
2
Black-Scholes/LVM comparison
In thiscase, no volatility input of theBlack - Scholes enables to reach theLVM price.
Vanilla hedging portfolio I
1
Q
Recall m( x, t )   v0 ( x, t ) E v0 [ xx f ( X t ) xt  x] is thesensitivity of  g
2
to thelocal varianceat (x,t).
P ortfolioPF    ( K , T ) C K ,T dK dT of vanillashedges all small
volatilitymovesif and onlyif for all (x,t)
 2 PF ( x, t )
Qv0
Qv0

E
[


(
X
)
x

x
]

E
[ xx f ( X t ) xt  x]  h( x, t )
xx
PF
t
t
2
x
How can we get thefunction ( K , T )?
Vanilla hedging portfolios II
k 1  2 v0 k 
a) For k ( x, t ), we define L(k ) 

t 2 x 2
 2C K ,T
 2C K ,T
For a call C K ,T , L(
)  0 wit h boundary condition
( x, T )   K ( x )
x 2
x 2
 2 PF
b) PF    ( K , T ) C K ,T dK dT  L(
)(x, t )    ( x, t )
x 2
 2 PF
 2 PF
Qv0
Q
k ( x, t )  E [ xx f ( X t ) 
( x, t ) xt  x]  h( x, t ) 
( x, t ) wit h h( x, t )  E v0 [ xx f ( X t ) xt  x]
2
2
x
x
T hus, L(k )  L(h)   .
h 1  2 v0 h 
c) If we take   L(h)   
then L(k )  0 wit h k(x,T)  0  k  0
t 2 x 2
 2 PF
x, t , E [ xx f ( X t ) 
( x, t ) xt  x]  0 and f - PF has no sensitivity tolocal volbumps
x 2
henceno sensitivity toimplied volbumps.
Qv0
Example : Asian option

T

dxt  v 0 dWt P ay- off : g ( X t )    xt dt  K  , S0  K  100 volatility v0  20 maturityT  1
0

1
Q
m( x, t )   v0 ( x, t ) E v0 [ xx f ( X t ) xt  x] is thesensitivity of  g to thelocal varianceat (x,t).
2
T
T
K
K
Asian Option Hedge
Robust volatilityhedge with PF    ( K , T ) CK ,T dK dT
 h( K , T ) 1  2 v0 ( K , T )h( K , T )  

 ( K , T )  

2
t
2
x


T
T
K
K
Forward Parabola - Γ
Payoff:
g( XT )  (ST2  ST1 )2
T1 = 1, T2 = 2, S0 = 100, σ = 10
Forward Parabola Hedge
Forward Cube - Γ
Payoff:
g( XT )  (ST2  ST1 )3
T1 = 1, T2 = 2, S0 = 100, σ = 10
Forward Cube Hedge
Forward Start - Γ
Payoff
g( XT )  (ST2  ST1 )
T1 = 1, T2 = 2, S0 = 100, σ = 10
Forward Start Hedge
One Touch - Γ
Payoff
g ( X T )  1{maxST H 104}
T1 = 1, H = 104 ,S0 = 100, σ = 5
One Touch Hedge
Link Γ/Vega
Robustness
• Superbucket hedge protects today from first order moves
in implied volatilities; what about robustness in the
future?
• After delta hedge, the tracking error is
1
TE 
2

T
0
(v t  v 0 (y t ,t)) xx f (Yt ) dt
• In the case of discrete hedging in the model or fast mean
reversion of vol, variance of TE is proportional to
E[  0 ( xx f (Yt )) dt] 
T

2
 
T
0
E[( xx f (Yt ))2 y t  y] (y,t)dydt
Hedgeability
• The optimal vanilla hedge PF minimizes
 
T
E[( xx f (Yt )   xx PF(y,t))2 y t  y] (y,t) dy dt
0
and thus coincides with the superbucket hedge as
 xx PF(y,t)  E[ xx f (Yt ) y t  y]
• The residual risk is


 
T
0
Var[ xx f (Yt )) y t  y]  (y,t) dy dt
• The hedgeability of an option is linked to the dependency
of the functional gamma on the path
The Geometry of Hedging
Bruno Dupire
53
• In practice, determining the best hedge is not sufficient
 Need to measure of residual risk
M
o
hedging
easure
f performan
e

2
H

2
H
X X

X
remaining
risk
after
hedging
H

1

 2
1
 2
initial
risk
before
hedging
X
X
2



2
Y
H
for
0


2
and

1


1
:
X


X
2
If



and



1
:H

0
.
X
Y
2
Y
If


1
or

0
:H

1
.
X
&
Y
highly
correlated
,
Basket
options
hedgeable.

with
similar
latility
Spread
vo
options
not
hedgeable.
H-Squared Parabola vs Cube
Skewness Products
• Negative Skewness: fat tail to the left
• Linked to UP Dispersion < Down Dispersion
i
n
/2 j
1n/2S
S
T
T
U
Dispersion
p
 


i
j
n
/2
S
i
1S
j

1
0
0
i
j
n
1 n S
S
T
T
Down
Dispersion




i
j
n
/2
S
i
n
/2

1S
j

n
/
2

1
0
0
Some products are based on
Up Dispersion – Down Dispersion
Hedge with Components
• Decorrelated Case
• Good hedge with risk reversals on the
components
Hedge with Components
• Correlated Case
• No hedge with components
BMW
Sample Mean of Best third
+
Sample Mean of Worst third
-2 x
Sample Mean of Middle third
Perfect Hedge for BMW, Correlation = 0%
Short 6 Shares, Short 9 Puts at –K*, Long 9 Calls at K*
Component Hedge
6
Norm Cdf = 2/3
4
Payoff
2
0
-2
Norm Cdf = 1/3
-4
-6
-3
-2
-1
0
Level
K* = 0.4307
1
2
3
BMW with 30 stocks , Correlation = 0
Hedge with Puts and Calls using 120 strikes
R-squared = 80.7%, MC runs = 10,000
1.5
Target
1
0.5
0
-0.5
-1
-1.5
-1
-0.5
Hedge
0
0.5
1
BMW with 30 stocks , Correlation = 40%
Hedge with Puts and Calls using 120 skrikes
R-squared = 5.8%, MC runs = 10,000
1
0.8
0.6
Target
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-0.4
-0.3
-0.2
-0.1
0
Hedge
0.1
0.2
0.3
Analysis
• Correlation shifts the returns, which affects the
components hedge but not the target
• In the absence of hedge, it is a mistake to price the
product with a calibrated model
• Implied individual skewness is irrelevant !
• It is a wrong way to play implied versus realized
skewness
Conclusion
• Itô calculus can be extended to functionals of price paths
• Price difference between 2 models can be computed
• We get a variational calculus on volatility surfaces
• It leads to a strike/maturity decomposition of the volatility
risk of the full portfolio