Bruno Dupire
Bloomberg LP
Lecture 11
Volatility expansion
IV. Volatility Expansion
Introduction
• This talk aims at providing a better
understanding of:
– How local volatilities contribute to the value of
an option
– How P&L is impacted when volatility is
misspecified
– Link between implied and local volatility
– Smile dynamics
– Vega/gamma hedging relationship
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Framework & definitions
• In the following, we specify the dynamics
of the spot in absolute convention (as
opposed to proportional in Black-Scholes)
and assume no rates:
•

dSt   t dWt
: local (instantaneous) volatility
(possibly stochastic)
• Implied volatility will be denoted by 
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P&L of a delta hedged option
Option Value
P&L
Break-even
points
Delta
hedge
Ct  t
Ct
  t
St
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 t
S

St t
St
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P&L of a delta hedged option (2)
Correct volatility
1std
Volatility higher than expected
1std
1std
St
St t
P&L Histogram
Expected P&L = 0
1std
St
St t
P&L Histogram
Expected P&L > 0
Ito: When t  0 , spot dependency disappears
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Black-Scholes PDE
P&L is a balance between gain from G and loss from :
P& L t ,t dt 
2

  G0  0  dt
 2

From Black 02
G0
Scholes PDE:  0  
2
=> discrepancy if  different from expected:
1 2
gain over dt     02 G0dt
2
    0 : Profit 
 Magnified by G0
    0 : Loss 
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P&L over a path
Total P&L over a path
= Sum of P&L over all small time intervals
Gamma
P& L 
1 T ( 2   2 ) G dt
0
0
2 0
No assumption is made
on volatility so far
Time
Spot
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General case
• Terminal wealth on each path is:
T
1
wealthT = X   0    ( 2   02 ) G0dt
2 0
( X    is the initial price of the option)
0
• Taking the expectation, we get:


T 
1
E wealthT  X (  0 )  2   E[G0 ( 2   02 )| S ] dS dt
0 0

• The probability density φ may correspond to the density of
a NON risk-neutral process (with some drift) with volatility σ.
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Non Risk-Neutral world
• In a complete model (like Black-Scholes), the drift does
not 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 σ0 (small) to take advantage of it.
L
In summary: gamma is a volatility
collector and it can be shaped by:
• a choice of strike and maturity,
Profile of a call (L,T) for
different vol assumptions
• a choice of σ0 , our hedging volatility.
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Average P&L
• From now on, φ will designate the risk neutral density
associated with dSt  dWt .
•In this case, E[wealthT] is also X    and we have:
T 
1
X ( )  X (  0 )  2   E[G0 ( 2   02 )| S ] dS dt
0 0
• Path dependent option & deterministic vol:
1 ( 2   2 ) E [ G | S ]  dS dt
X ()  X ( 0 )  2
0
0

• European option & stochastic vol:
2
2
1
C( )  C(  0 )  2
(
E
[

|
S
]


0 ) G0  dS dt

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Quiz
• 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
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Expansion in volatility
• An important case is a European option with
deterministic vol:
C (  )  C (  0 )  21   ( 2   02 ) G0  dS dt
• The corrective term is a weighted average of the volatility
differences
• This double integral can be approximated numerically
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P&L: Stop Loss Start Gain
• Extreme case:  0  0  G0   K
T
2
1


C( )  ( S0  K )  2   K , t   K , t dt
0

• This is known as Tanaka’s formula
S
Delta = 100%
K
Delta = 0%
t
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Local / Implied volatility relationship
Differentiation
Local volatility
Implied volatility
31
Local vol
Implied vol
23
21
19
17
15
23
19
15
11
13
strike
27
maturity
spot
time
Aggregation
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Smile stripping: from implied to local
•Stripping local vols from implied vols is the
inverse operation:
C
T
2
 (S ,T )  2 2
 C
 K2
(Dupire 93)
•Involves differentiations
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From local to implied: a simple case
Let us assume that local volatility is a
deterministic function of time only:
dSt    t  dWt
In this model, we know how to combine local
vols to compute implied vol:
2

0  t dt
T
  T  
T
Question: can we get a formula with
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 S , t  ?
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From local to implied volatility
• When  0 = implied vol
1
2
2
(



0 ) G0 dSdt  0

2
•
G0
depends on  0


2

G0 dSdt

2
0 
 G0 dSdt
solve by iterations
•Implied Vol is a weighted average of Local Vols
(as a swap rate is a weighted average of FRA)
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Weighting scheme
•Weighting Scheme: proportional to G0 
At the
money
case:
Out of the
money
case:
G0 
t
S
S0=100
K=100
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S0=100
K=110
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Weighting scheme (2)
• Weighting scheme is roughly proportional to the
brownian bridge density
Brownian bridge density:

BB K ,T  x, t   P St  x ST  K
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
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Time homogeneous case
 2    ( S ) 2 ( S )dS
ATM (K=S0)
(S)
 T
G dt

 S 
 G  dSdt
0
0
OTM (K>S0)
(S)
S0
S0
K
small
S
 T
large
(S)
(S)
S0
S
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S
S0
K
S
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Link with smile
 K
1
and
 K
2
K2
are
averages of the same local
S0
vols with different weighting
schemes
K1
t
=> New approach gives us a direct expression for
the smile from the knowledge of local volatilities
But can we say something about its dynamics?
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Smile dynamics
Weighting scheme imposes
some dynamics of the smile for
a move of the spot:
For a given strike K,
S1
S0
K
S    K 
(we average lower volatilities)
Smile today (Spot St)
&
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
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t
26
25.5
25
24.5
24
23.5
St+dt
St
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Sticky strike model
A sticky strike model (  K  t    K ) is arbitrageable.
Let us consider two strikes K1 < K2
The model assumes constant vols 1 > 2 for example
 2 dt
 1 dt
 2 dt
 1 dt
By combining K1 and K2 options, we build a position with no gamma and
positive theta (sell 1 K1 call, buy G1/G2 K2 calls)
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Vega analysis
•If

& 0 are constant
1 2
C( )  C(  0 )  (   02 )  G0  dSdt
2
•
 2   02  
1
C (   )  C ( )    G0 dSdt
2

2
0
2
0
 2C
 2
C C  2 C


Vega =
2 
2  2
   
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Gamma hedging vs Vega hedging
• Hedge in Γ
insensitive to realised
historical vol
• If Γ=0 everywhere, no sensitivity to
historical vol => no need to Vega hedge
• Problem: impossible to cancel Γ now for
the future
• Need to roll option hedge
• How to lock this future cost?
• Answer: by vega hedging
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Superbuckets: local change in local vol
For any option, in the deterministic vol case:
X (  )  X (  0 )  21   ( 2   02 ) E[ G0 | S ]  dS dt
For a small shift ε in local variance around (S,t), we have:
X (  )  X (  0 )  21  E[G0 | S ] 


dX
d  (2S ,t )

 21 E[G0  S , t | S ]   S , t 
For a european option:
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dC
 
d  (2S ,t )
 21 G0  S , t   S , t 
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Superbuckets: local change in implied vol
Local change of implied volatility is obtained by combining
local changes in local volatility according a certain weighting
dC
dC d  2 

2
d  
d  2  d  2 
sensitivity in
local vol
weighting obtain
using stripping
formula
Thus:
<=>
<=>
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cancel sensitivity to any move of implied vol
cancel sensitivity to any move of local vol
cancel all future gamma in expectation
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Conclusion
• This analysis shows that option prices are
based on how they capture local volatility
• It reveals the link between local vol and
implied vol
• It sheds some light on the equivalence
between full Vega hedge (superbuckets)
and average future gamma hedge
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“Dual” Equation
2
The stripping formula   2
CT
K 2C KK
can be expressed in terms of
When
T 0
̂ :
ˆ

ˆ K
1
KX
ˆ
X
solved by ˆ 
K
du
S u u,0
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Large Deviation Interpretation
The important quantity is
dx  ax dW then
If
dy  dt  dW and
ˆ

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K
S
du

uˆ

K
S

K
S
du
u u,0
du
yx   
x0 a u 
x
satisfies:
xt  K  Wt  yK 
du
ln K  ln S
ˆ
  K
du
u u ,0
S u u,0
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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:
( y x)
f ( x, v)  E x ,v [ f ( X )] 
1
2v

f ( y )e

2
2v
dy
with f(x,0) = f(x)
1
f
(
x
,
v
)

f xx ( x, v)
• Then,
v
2
• We explore various delta hedging strategies
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Calendar Time Delta Hedging
• Delta hedging with constant vol: P&L depends on the
path of the volatility and on the path of the spot price.
• Calendar time delta hedge: replication cost of f ( X t ,  2 .(T  t ))
1 t
f ( X 0 ,  .T )   f xx (dQV0,u   2 du )
2 0
2
• In particular, for sigma = 0, replication cost of f ( X t )
f (X0) 
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1 t
f xx dQV0,u

0
2
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Business Time Delta Hedging
• Delta hedging according to the quadratic variation:
P&L that depends only on quadratic variation and
spot price
df ( X t , L  QV0,t )  f x dX t  f v dQV0,t 
1
f xx dQV0,t  f x dX t
2
• Hence, for QV0,T  L,
t
f ( X t , L  QV0,t )  f ( X 0 , L)   f x ( X u , L  QV0,u ) dX t
0
And the replicating cost of f ( X t , L  QV0,t )is f ( X 0 , L)
f ( X 0 , L) finances exactly the replication of f until  : QV0,  L
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Daily P&L Variation
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Tracking Error Comparison
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V.
Stochastic Volatility Models
Hull & White
•Stochastic volatility model Hull&White (87)
dS t
 rdt   t dWt P
St
d t  dt   dZ tP
Implied volatility
Implied volatility
•Incomplete model, depends on risk premium
•Does not fit market smile
Strike price
 Z ,W  0
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Strike price
 Z ,W  0
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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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Heston Model
 dS
 S   dt  v dW

dv   v  v dt   v dZ
dW , dZ   dt
Solved by Fourier transform:
FWD
x  ln
 T t
K
C K ,T x, v,   e x P1 x, v,   P0 x, v, 
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Spot dependency
2 ways to generate skew in a stochastic vol model
1)  t  xt f S , t ,  W , Z   0
2)

W , Z   0
S0

ST
ST
S0
-Mostly equivalent: similar (St,t ) patterns, similar
future
evolutions
-1) more flexible (and arbitrary!) than 2)
-For short horizons: stoch vol model  local vol model
+ independent noise on vol.
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Convexity Bias
dS   t dW
 2
d t  dZ
  W , Z   0



E  t2 | S t  K   02 ?

 
NO!only E  t2   02

E  t2 | St  K

 02
S0
K
 t likely to be high if St  S0 or St  S0
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Impact on Models
• Risk Neutral drift for instantaneous forward
variance
• Markov Model:
dS
fits initial smile with local vols  S, t 
 f S , t  t dW
S
 2 S , t 
 f S , t  
E[ t2 | St  S ]
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Smile dynamics: Stoch Vol Model (1)
Local vols
Skew case (r<0)

Smile S 
Smile S 0
Smile S 
S
S  S0
K
- ATM short term implied still follows the local vols
E
2
T


ST  K   2  K , T 
- Similar skews as local vol model for short horizons
- Common mistake when computing the smile for another
spot: just change S0 forgetting the conditioning on  :
if S : S0  S+ where is the new  ?
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Smile dynamics: Stoch Vol Model (2)
• Pure smile case (r=0)

Local vols
Smile S 
Smile S

Smile S 0
S
S0
S
K
• ATM short term implied follows the local vols
• Future skews quite flat, different from local vol
model
• Again, do not forget conditioning of vol by S
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Forward Skew
Forward Skews
In the absence of jump :
model fits market
 K , T
2
E[ T2 ST  K ]   loc
(K ,T )
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.
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Sensitivity of ATM volatility / S
At t, short term ATM implied volatility ~ σt.
As σt is random, the sensitivity
Et [
2
t t
 2
S
is defined only in average:
2
 loc
(S , t )
  St  St  S ]   ( St  S , t  t )   ( St , t ) 
 dS
S
2
t
2
loc
2
loc
2
2
In average,  ATM
follows  loc
.
Optimal hedge of vanilla under calibrated stochastic volatility corresponds to
perfect hedge ratio under LVM.
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