Download - Local Vol Delta-Hedging
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IV. Volatility Expansion
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Bruno Dupire 3
Introduction
This talk aims at providing a betterunderstanding of:
How local volatilities contribute to the value ofan option
How P&L is impacted when volatility ismisspecified
Link between implied and local volatility Smile dynamics
Vega/gamma hedging relationship
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Bruno Dupire 4
In the following, we specify the dynamicsof the spot in absolute convention (as
opposed to proportional in Black-Scholes)
and assume no rates:
: local (instantaneous) volatility
(possibly stochastic) Implied volatility will be denoted by
dS dW t t t
Framework & definitions
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Bruno Dupire 5
P&L
St t
St
Break-even
points
t
t
Option Value
St
Ct
Ct t
S
Delta
hedge
P&L of a delta hedged option
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Bruno Dupire 6
P&L Histogram P&L Histogram
Expected P&L = 0 Expected P&L > 0
Correct volatility Volatility higher than expected
St tStSt t St
Ito: When , spot dependency disappearst 0
1std 1std
1std 1std
P&L of a delta hedged option (2)
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P&L is a balance between gain from Gand loss from :
P L dtt t dt & ,
2
0 02 G
=> discrepancy if different from expected:
gain over dt 1
22
0
2
0 Gdt
0
00
:
:
Profit
Loss Magnified by G
G00
2
02
From Black-Scholes PDE:
Black-Scholes PDE
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Total P&L over a path
= Sum of P&L over all small time intervals
P L
dtT
&
( )
122
0
2
00
G
Spot
Time
Gamma
No assumption is made
on volatility so far
P&L over a path
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The probability density may correspond to the density of
a NON risk-neutral process (with some drift) with volatility .
Terminal wealth on each path is:
Taking the expectation, we get:
E X E S dS dtT
wealthT
( ) [ ( )| ] G0 0 2 0200
12
( is the initial price of the option)
wealth =T X dtT
G02
0
2
00
12 ( )
X0
General case
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In a complete model (like Black-Scholes), the drift doesnot affect option prices but alternative hedging strategies
lead to different expectations
Example: mean reverting process
towards L with high volatility around L
We then want to choose K (close to L) T
and 0(small) to take advantage of it.
L
Profile of a call (L,T) for
different vol assumptionsIn summary: gamma is a volatility
collector and it can be shaped by:
a choice of strike and maturity,
a choice of 0 , our hedging volatility.
Non Risk-Neutral world
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X X E S dS dtT
( ) ( ) [ ( )| ] G
0 02 02
0012
X X E S dS dt( ) ( ) ( ) [ | ] G 0 2 02 012
C C E S dS dt ( ) ( ) ( [ | ] ) G 0 2 02 012
From now on, will designate the risk neutral densityassociated with .
In this case, E[wealthT] is also and we have:
Path dependent option & deterministic vol:
European option & stochastic vol:
dS dW t t
X
Average P&L
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Buy a European option at 20% implied vol
Realised historical vol is 25%
Have you made money ?
Not necessarily!
High vol with low gamma, low vol with high gamma
Quiz
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An important case is a European option withdeterministic vol:
C C dS dt ( ) ( ) ( ) G 02
0
2
012
The corrective term is a weighted average of the volatility
differences
This double integral can be approximated numerically
Expansion in volatility
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Bruno Dupire 14
t
Delta = 100%
KDelta = 0%
S
C S K K t K t dt T
( ) ( ) , , 02
0
12
Extreme case: 0 00 G K
This is known as Tanakas formula
P&L: Stop Loss Start Gain
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Bruno Dupire 15
13
15
17
19
21
23
Implied volatility
strike maturity
11
15
19
23
27
31
Local volatility
spot time
Aggregation
Differentiation
Local / Implied volatility relationship
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Bruno Dupire 16
2
2
2
2( , )S T
C
TC
K
Stripping local vols from implied vols is the
inverse operation:
Involves differentiations
(Dupire 93)
Smile stripping: from implied to local
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Bruno Dupire 17
Let us assume that local volatility is a
deterministic function of time only:
dS t dW t t In this model, we know how to combine local
vols to compute implied vol:
Tt dt
T
T
2
0
Question: can we get a formula with ? S t,
From local to implied: a simple case
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Bruno Dupire 18
Implied Vol is a weighted average of Local Vols
(as a swap rate is a weighted average of FRA)
When = implied vol0
solve by iterations depends onG0 0
1
202
0
2
0( ) G dSdt
0
2
2
0
0
G
G
dSdt
dSdt
From local to implied volatility
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Bruno Dupire 19
Weighting Scheme: proportional to G0
At the
moneycase:
S0=100
K=100
Out of the
moneycase:
S0=100
K=110
tS
G0
Weighting scheme
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Bruno Dupire 20
Weighting scheme is roughly proportional to thebrownian bridge density
Brownian bridge density:
BB x t P S x S KK T t T , ,
Weighting scheme (2)
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Bruno Dupire 21
( ) ( ) 2 2 S S dS
small
T
large T
S0
S
(S) S0
S
K(S)
S0
S
(S) S0
S
K(S)
ATM (K=S0) OTM (K>S0)
S dtdSdt
G
G0
0
Time homogeneous case
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Bruno Dupire 22
and are
averages of the same local
vols with different weightingschemes
K1
K2
=> New approach gives us a direct expression for
the smile from the knowledge of local volatilities
But can we say something about its dynamics?
S0
K2
K1
t
Link with smile
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Bruno Dupire 23
23.5
24
24.5
25
25.5
26
S0 K
t
S1Weighting scheme imposessome dynamics of the smile for
a move of the spot:
For a given strike K,
(we average lower volatilities)S K
Smile today (Spot St)
StSt+dt
Smile tomorrow (Spot St+dt)
in sticky strike model
Smile tomorrow (Spot St+dt)
if ATM=constant
Smile tomorrow (Spot St+dt)
in the smile model
&
Smile dynamics
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Bruno Dupire 25
C C dSdt ( ) ( ) ( ) G 0 2 02 01
2
If & are constant 0
2
0
2
G dSdtCC 02020 21
)()(
Vega =
2
2
C
C C C 2
2
2 2
Vega analysis
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Bruno Dupire 28
Local change of implied volatility is obtained by combininglocal changes in local volatility according a certain weighting
dC
d
dC
d
d
d
2 2
2
2
sensitivity in
local vol
weighting obtain
using stripping
formula
Thus:cancel sensitivity to any move of implied vol
cancel sensitivity to any move of local vol
cancel all future gamma in expectation
Superbuckets: local change in implied vol
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Bruno Dupire 29
This analysis shows that option prices arebased on how they capture local volatility
It reveals the link between local vol andimplied vol
It sheds some light on the equivalence
between full Vega hedge (superbuckets)
and average future gamma hedge
Conclusion
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Bruno Dupire 30
Dual Equation
The stripping formula
can be expressed in terms of
When
solved by
KK
T
CK
C2
2 2
:
KXK
1
0T
K
S uu
du
X
0,
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Bruno Dupire 31
Large Deviation Interpretation
The important quantity is
If then satisfies:
and
K
S uudu
0,
dWxadx
x
x ua
duxy
0
dWdtdy
KS
K
S
K
S
uu
du
SK
uu
du
u
du
0,
lnln
0,
KyWKx tt
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Bruno Dupire 32
Delta Hedging
We assume no interest rates, no dividends, and absolute(as opposed to proportional) definition of volatility
Extend f(x) to f(x,v) as the Bachelier (normal BS) price of
f for start price x and variance v:
with f(x,0) = f(x)
Then,
We explore various delta hedging strategies
dyeyfvXfEvxf v
xy
vx2
)(
,
2
)(2
1)]([),(
),(2
1),( vxfvxf xxv
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Bruno Dupire 33
Calendar Time Delta Hedging
Delta hedging with constant vol: P&L depends on thepath of the volatility and on the path of the spot price.
Calendar time delta hedge: replication cost of
In particular, for sigma = 0, replication cost of
t
uxx dudQVfTXf0
2
,0
2
0 )(
2
1).,(
)).(,(2
tTXf t
t
uxxdQVfXf0
,002
1)(
)( tXf
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Bruno Dupire 34
Business Time Delta Hedging
Delta hedging according to the quadratic variation:
P&L that depends only on quadratic variation and
spot price
Hence, for
And the replicating cost of is
finances exactly the replication of f until
txtxxtvtxtt dXfdQVfdQVfdXfQVLXdf ,0,0,021),(
,,0 LQV T
t
t
uuxtt dXQVLXfLXfQVLXf 0 ,00,0 ),(),(),(
),( ,0 tt QVLXf ),( 0 LXf
),( 0 LXf LQV ,0:
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Bruno Dupire 35
Daily P&L Variation
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Bruno Dupire 36
Tracking Error Comparison
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Bruno Dupire 38
Stochastic volatility model Hull&White (87)
P
tt
P
tt
t
t
dZdtd
dWrdtS
dS
Incomplete model, depends on risk premium
Does not fit market smile
Strike price
Impliedvola
tility
Strike price
Impliedvolatility
Z W,
0Z W, 0
Hull & White
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Bruno Dupire 39
Role of parameters
Correlation gives the short term skew Mean reversion level determines the long term
value of volatility
Mean reversion strength Determine the term structure of volatility
Dampens the skew for longer maturities
Volvol gives convexity to implied vol
Functional dependency on S has a similar effect
to correlation
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Bruno Dupire 40
Heston Model
dtdZdWdZvdtvvdv
dWvdtS
dS
,
Solved by Fourier transform:
,,,,,,
ln
01, vxPvxPevxC
tTK
FWDx
x
TK
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Bruno Dupire 46
2 ways to generate skew in a stochastic vol model
-Mostly equivalent: similar (St,t) patterns, similar
futureevolutions
-1) more flexible (and arbitrary!) than 2)
-For short horizons: stoch vol modellocal vol model
+ independent noise on vol.
Spot dependency
0,)2
0,,,)1
ZW
ZWtSfxtt
0S
STST0S
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Bruno Dupire 47
Convexity Bias
0,
?| 2022
ZW
KSEdZd
dWdS
ttt
t
2
0
2
onlyNO! tE
tlikely to be high if
00or SSSS tt
KSE tt |2
0S
2
0
K
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Bruno Dupire 48
Risk Neutral drift for instantaneous forwardvariance
Markov Model:
fits initial smile with local vols
Impact on Models
dWtSfSdS
t, tS,
]|[
,,
2
2
SSE
tStSf
tt
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Bruno Dupire 49
Skew case (r
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Bruno Dupire 50
Smile dynamics: Stoch Vol Model (2)
Pure smile case (r=0)
ATM short term implied follows the local vols
Future skews quite flat, different from local volmodel
Again, do not forget conditioning of vol by S
Local vols
SSmile
0Smile S
SSmile
S 0S S
K
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Forward Skew
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Bruno Dupire 52
Forward Skews
In the absence of jump :
model fits market
This constrains
a) the sensitivity of the ATM short term volatility wrt S;
b) the average level of the volatility conditioned to ST=K.
a) tells that the sensitivity and the hedge ratio of vanillas depend on the
calibration to the vanilla, not on local volatility/ stochastic volatility.To change them, jumps are needed.
But b) does not say anything on the conditional forward skews.
),(][, 22 TKKSETK locTT
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Sensitivity of ATM volatility / S
S 2
dSS
tS
tSttSSSSSE loc
tloctloctttttt
),(
),(),(][
22222
2
ATM
At t, short term ATM implied volatility ~ t.
As tis random, the sensitivity is defined only in average:
In average, follows .
Optimal hedge of vanilla under calibrated stochastic volatility corresponds to
perfect hedge ratio under LVM.
2
loc