Skew Modeling
Bruno Dupire
Bloomberg LP
bdupire@bloomberg.net
Columbia University, New York
September 12, 2005
I.
Generalities
Market Skews
Dominating fact since 1987 crash: strong negative skew on
Equity Markets
σ impl
K
Not a general phenomenon
Gold: σ impl
FX: σ impl
K
K
We focus on Equity Markets
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Skews
• Volatility Skew: slope of implied volatility as a
function of Strike
• Link with Skewness (asymmetry) of the Risk
Neutral density function ϕ ?
Moments
1
2
3
4
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Statistics
Expectation
Variance
Skewness
Kurtosis
Finance
FWD price
Level of implied vol
Slope of implied vol
Convexity of implied vol
4
Why Volatility Skews?
• Market prices governed by
– a) Anticipated dynamics (future behavior of volatility or jumps)
– b) Supply and Demand
σ impl
Sup
Market Skew
ply
and
Dem
and
Th. Skew
K
• To “ arbitrage” European options, estimate a) to capture
risk premium b)
• To “arbitrage” (or correctly price) exotics, find Risk
Neutral dynamics calibrated to the market
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Modeling Uncertainty
Main ingredients for spot modeling
• Many small shocks: Brownian Motion
(continuous prices) S
t
• A few big shocks: Poisson process (jumps)
S
t
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2 mechanisms to produce Skews (1)
• To obtain downward sloping implied volatilities
σimp
K
– a) Negative link between prices and volatility
• Deterministic dependency (Local Volatility Model)
• Or negative correlation (Stochastic volatility Model)
– b) Downward jumps
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2 mechanisms to produce Skews (2)
– a) Negative link between prices and volatility
S1
S2
– b) Downward jumps
S1
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S2
8
Leverage and Jumps
Dissociating Jump & Leverage effects
t0
t1
t2
x = St1-St0
•
Variance :
y = St2-St1
(x + y) = x + 2xy+ y
2
2
2
Option prices
FWD variance
∆ Hedge
•
Skewness :
(x + y)3 = x3 + 3x2 y + 3xy2 + y3
Leverage
Option prices
∆ Hedge
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FWD skewness
10
Dissociating Jump & Leverage effects
Define a time window to calculate effects from jumps and
Leverage. For example, take close prices for 3 months
•
Jump:
∑ (δS )
3
ti
i
•
Leverage:
∑ (S
ti
)( )
− S t1 δS ti
2
i
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Dissociating Jump & Leverage effects
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Dissociating Jump & Leverage effects
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Break Even Volatilities
Theoretical Skew from Prices
?
=>
Problem : How to compute option prices on an underlying without
options?
For instance : compute 3 month 5% OTM Call from price history only.
1) Discounted average of the historical Intrinsic Values.
Bad : depends on bull/bear, no call/put parity.
2) Generate paths by sampling 1 day return recentered histogram.
Problem : CLT => converges quickly to same volatility for all
strike/maturity; breaks autocorrelation and vol/spot dependency.
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Theoretical Skew from Prices (2)
3) Discounted average of the Intrinsic Value from recentered 3 month
histogram.
4) ∆-Hedging : compute the implied volatility which makes the ∆hedging a fair game.
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Theoretical Skew
from historical prices (3)
How to get a theoretical Skew just from spot price
history?
S
K
Example:
ST
3 month daily data
t
T1
T2
1 strike K = k ST1
– a) price and delta hedge for a given σ within Black-Scholes
1
–
–
–
–
model
b) compute the associated final Profit & Loss: PL(σ )
c) solve for σ k / PL σ k = 0
d) repeat a) b) c) for general time period and average
e) repeat a) b) c) and d) to get the “theoretical Skew”
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( )
( ( ))
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Theoretical Skew
from historical prices (4)
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Theoretical Skew
from historical prices (4)
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Theoretical Skew
from historical prices (4)
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Theoretical Skew
from historical prices (4)
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Barriers as FWD Skew trades
Beyond initial vol surface fitting
• Need to have proper dynamics of implied volatility
– Future skews determine the price of Barriers and
OTM Cliquets
– Moves of the ATM implied vol determine the ∆ of
European options
• Calibrating to the current vol surface do not impose
these dynamics
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Barrier Static Hedging
Down & Out Call Strike K, Barrier L, r=0 :
• With BS: DOCK ,L = CK − K L PL2
K
If S t = L ,unwind hedge, at 0
cost
If not touched, IV’s are equal
K
2
L
K
L
K
L2
K
L
L
K
L
K
• With normal model
DOCK , L = C K − P2 L − K
dS = σdW
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2L-K
1
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Static Hedging: Model Dominance
• Back to DOCK , L
L
K
• An assumption as the skew at L corresponds to
an affine model
dS = (aS + b )dW (displaced LN)
• DOCK ,L priced as in BS with shifted K and L gives
new hedging PF which is >0 when L is touched if
Skew assumption is conservative
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Skew Adjusted Barrier Hedges
dS = (aS + b )dW
DOC K , L
aK + b
↔ CK −
PaL2 +b (2 L − K )
aL + b
aK + b
UOC K , L
a
⎛
⎞ aK + b
C aL2 +b (2 L − K )
↔ C K − (L − K )⎜ 2 Dig L +
CL ⎟ −
aL + b ⎠ aL + b
⎝
aK + b
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Local Volatility Model
One Single Model
• We know that a model with dS = σ(S,t)dW
would generate smiles.
– Can we find σ(S,t) which fits market smiles?
– Are there several solutions?
ANSWER: One and only one way to do it.
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The Risk-Neutral Solution
But if drift imposed (by risk-neutrality), uniqueness of the solution
Risk
Neutral
Processes
Diffusions
Compatible
with Smile
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sought diffusion
(obtained by integrating twice
Fokker-Planck equation)
29
Forward Equations (1)
• BWD Equation:
price of one option C (K 0 ,T0 ) for different (S, t )
• FWD Equation:
price of all options C (K , T ) for current (S 0 ,t0 )
• Advantage of FWD equation:
– If local volatilities known, fast computation of implied
volatility surface,
– If current implied volatility surface known, extraction of
local volatilities,
– Understanding of forward volatilities and how to lock
them.
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Forward Equations (2)
• Several ways to obtain them:
– Fokker-Planck equation:
• Integrate twice Kolmogorov Forward Equation
– Tanaka formula:
• Expectation of local time
– Replication
• Replication portfolio gives a much more financial
insight
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Fokker-Planck
• If dx = b( x, t )dW
∂ϕ 1 ∂ 2 (b 2ϕ )
• Fokker-Planck Equation:
=
∂t 2 ∂x 2
∂ 2C
• Where ϕ is the Risk Neutral density. As ϕ =
∂K 2
2 ⎛ ∂C ⎞
∂ ⎜
⎟
⎝ ∂t ⎠ =
∂x 2
2
⎛ ∂ 2C ⎞
2⎛ 2 ∂ C ⎞
⎟
∂⎜⎜ 2 ⎟⎟
∂ ⎜⎜ b
2 ⎟
⎝ ∂K ⎠ = 1 ⎝ ∂K ⎠
∂t
∂x 2
2
• Integrating twice w.r.t. x: ∂C b2 ∂2C
=
∂t 2 ∂K 2
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Volatility Expansion
•K,T fixed. C0 price with LVM σ 0 ( S t , t ) : dS t = σ 0 ( S t , t ) dWt
•Real dynamics: dSt = σt dWt
2
∂
C
∂
C0 2
1
+
2
0
dS + ∫
(
σ
−
σ
•Ito ( ST − K ) = C0 ( S 0 ,0) + ∫
0 ( S t , t )) dt
t
2
∂S
2 0 ∂S
0
T
T
• Taking expectation:
1
C ( S 0 ,0) = C0 ( S 0 ,0) + ∫∫ Γ0 ( S , t ) Ε σ t2 St = S − σ 02 ( S , t ) ϕ ( S , t )dSdt
2
2
2
•Equality for all (K,T) Ù Ε σ t S t = S = σ 0 ( S , t )
[
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([
]
]
)
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Summary of LVM Properties
Σ0 is the initial volatility surface
• σ (S,t ) compatible with Σ0 ⇔σ =
•σ
•
(ω ) compatible with Σ ⇔ E[σ
0
σ̂ k,T
⇔
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local vol
ST = K ] = (local vol)²
deterministic function of (S,t) (if no jumps)
future smile = FWD smile from local vol
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Stochastic Volatility Models
Heston Model
⎡ dS
⎢ S = µ dt + v dW
⎢
⎢⎣dv = κ v 2 ∞ − v dt + η v dZ
(
)
dW , dZ = ρ dt
Solved by Fourier transform:
FWD
x ≡ ln
τ =T −t
K
C K ,T ( x, v,τ ) = e x P1 ( x, v,τ ) − P0 ( x, v,τ )
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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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Spot dependency
2 ways to generate skew in a stochastic vol model
1) σ t = xt f (S , t ), ρ (W , Z ) = 0
2)
σρ
(W , Z ) ≠ 0
S0
σ
ST
ST
S0
-Mostly equivalent: similar (St,σt ) patterns, similar
future
evolutions
-1) more flexible (and arbitrary!) than 2)
-For short horizons: stoch vol model Ù local vol model
+ independent noise on vol.
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SABR model
• F: Forward price
β
dF = F σ t dW
dσ
σ
= α dZ
• With correlation ρ
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Smile Dynamics
Smile dynamics: Local Vol Model (1)
• Consider, for one maturity, the smiles associated
to 3 initial spot values
Skew case
Local vols
Smile S −
Smile S 0
Smile S +
S − S0 S +
K
– ATM short term implied follows the local vols
– Similar skews
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Smile dynamics: Local Vol Model (2)
• Pure Smile case
Local vols
Smile S +
Smile S −
Smile S 0
S−
S0
S+
K
– ATM short term implied follows the local vols
– Skew can change sign
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Smile dynamics: Stoch Vol Model (1)
Skew case (r<0)
Local vols
σ
Smile S −
Smile S 0
Smile S +
S+
S − S0
K
- ATM short term implied still follows the local vols
(E [σ
2
T
]
)
S T = K = σ 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 σ :
if S : S0 → S+ where is the new σ ?
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Smile dynamics: Stoch Vol Model (2)
• Pure smile case (r=0)
σ
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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Smile dynamics: Jump Model
Skew case
Local vols
Smile S −
Smile S 0
S−
Smile S +
S0 S +
K
• ATM short term implied constant (does not follows the
local vols)
• Constant skew
• Sticky Delta model
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Smile dynamics: Jump Model
Pure smile case
Local vols
Smile S 0
Smile S +
Smile S −
S−
S0
S+
K
• ATM short term implied constant (does not follows the
local vols)
• Constant skew
• Sticky Delta model
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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↑ ⇒ σ∃K ↓
(we average lower volatilities)
Smile today (Spot St)
&
Smile tomorrow (Spot St+dt)
in sticky strike model
t
26
25.5
25
Smile tomorrow (Spot St+dt)
if σATM=constant
24.5
Smile tomorrow (Spot St+dt)
in the smile model
23.5
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St+dt
St
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Volatility Dynamics of different models
• Local Volatility Model gives future short term
skews that are very flat and Call ∆ lesser than
Black-Scholes.
• More realistic future Skews with:
– Jumps
– Stochastic volatility with correlation and meanreversion
• To change the ATM vol sensitivity to Spot:
– Stochastic volatility does not help much
– Jumps are required
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ATM volatility behavior
Forward Skews
In the absence of jump :
model fits market
⇔ ∀K , T
2
E[σ T2 ST = K ] = σ 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 [σ
2
t +δt
∂σ t
∂S
2
is defined only in average:
2
∂σ loc
(S , t )
− σ Sδt = St + δS ] = σ ( St + δS , t + δt ) − σ ( St , t ) ≈
⋅ δS
∂S
2
t
2
loc
2
loc
2
2
In average, σ ATM
follows σ loc
.
Optimal hedge of vanilla under calibrated stochastic volatility corresponds to
perfect hedge ratio under LVM.
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Market Model of Implied Volatility
• Implied volatilities are directly observable
• Can we model directly their dynamics? (r = 0)
⎧dS
⎪⎪ S = σ dW1
⎨
⎪dσˆ = α dt + u dW + u dW
1
1
2
2
⎪⎩ σˆ
where σ̂ is the implied volatility of a given C K ,T
• Condition on σˆ dynamics?
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Drift Condition
C(S,σˆ , t )
•
Apply Ito’s lemma to
•
Cancel the drift term
•
Rewrite derivatives of
C(S,σˆ , t )
gives the condition that the drift
α of dσˆ
σˆ
must satisfy.
For short T, we get the Short Skew Condition (SSC):
2
2
K ⎞ ⎛
K ⎞
⎛
2
σˆ = ⎜ σ + u1 ln( ) ⎟ + ⎜ u2 ln( ) ⎟
S ⎠ ⎝
S ⎠
⎝
K
2
close to the money: σˆ ~ σ + u1 ln( )
S
Î Skew determines u1
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C ( S ,σ ,̂ t)
t →T
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Optimal hedge ratio ∆H
•
C ( S , σˆ , t ) : BS Price at t of Call option with
strike K, maturity T, implied vol
• Ito: dC ( S , σˆ , t ) = 0dt + CS dS + Cσˆ dσˆ
• Optimal hedge minimizes P&L variance:
dC.dS
dσˆ .dS
H
∆ =
= CS + Cσˆ
2
2
(dS )
(dS )
Implied Vol
σ̂
BS Delta
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BS Vega
sensitivity
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Optimal hedge ratio ∆H II
dσˆ .dS
∆ = CS + Cσˆ
(dS ) 2
H
⎧ dS
⎪⎪ S = σdW1
With ⎨
⎪ dσˆ = αdt + u dW + u dW
1
1
2
2
⎪⎩ σˆ
dσˆ .dS u1σS (dW1 ) 2 u1σˆ
=
=
2
2
2
σS
(dS )
(σS ) (dW1 )
Î Skew determines u1, which determines ∆H
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Smile Arbitrage
Deterministic future smiles
It is not possible to prescribe just any future
smile
If deterministic, one must have
C K ,T (S 0 , t 0 ) = ∫ ϕ (S 0 , t 0 , S , T1 ) C K ,T (S , T1 )dS
2
2
Not satisfied in general
K
S0
ϕ
t0
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T1
T2
67
Det. Fut. smiles & no jumps
=> = FWD smile
2
2
If ∃(S , t , K , T ) / VK ,T (S , t ) ≠ σ (K , T ) ≡ lim σ imp (K , T , K + δK , T + δT )
δK → 0
δT → 0
stripped from Smile S.t
K
Then, there exists a 2 step arbitrage:
Define
∂ 2C
2
( (K , T ) − V (S , t )) ∂K (S , t , K , T )
PL t ≡ σ
(
K ,T
2
At t0 : Sell PL t ⋅ Dig S − ε ,t − Dig S + ε ,t
At t: if S ∈ [S − ε , S + ε ]
t
)
S0
S
t0
2
buy
CS K, T , sell σ
2
K
t
2
T
(K , T )δ K ,T
gives a premium = PLt at t, no loss at T
Conclusion: VK ,T (S , t ) independent of
from initial smile
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(S , t ) = VK ,T (S0 , t0 ) = σ 2 (K , T )
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Consequence of det. future smiles
• Sticky Strike assumption: Each (K,T) has a fixed σ impl ( K , T )
independent of (S,t)
• Sticky Delta assumption: σ impl ( K , T ) depends only on
moneyness and residual maturity
•
In the absence of jumps,
– Sticky Strike is arbitrageable
– Sticky ∆ is (even more) arbitrageable
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Example of arbitrage with Sticky Strike
Each CK,T lives in its Black-Scholes (σ impl ( K , T ) )world
C1 ≡ C K 1 ,T1
assume σ 1 > σ 2
C 2 ≡ C K 2 ,T2
P&L of Delta hedge position over dt:
δ PL (C 1 ) =
1
2
δ PL (C 2 ) =
1
2
((δ S ) − σ S δ t ) Γ
((δ S ) − σ S δ t ) Γ
2
2
δ PL (Γ1C 2 − Γ2 C 1 ) =
(no Γ , free Θ )
2
1
1
2
2
Γ1Γ2 2 2
S σ 1 − σ 22 δ t > 0
2
(
!
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Γ1C 2
2
)
Γ2 C 1
S t1
S t + δt
If no jump
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Arbitrage with Sticky Delta
• In the absence of jumps, Sticky-K is arbitrageable and Sticky-∆ even more so.
• However, it seems that quiet trending market (no jumps!) are Sticky-∆.
In trending markets, buy Calls, sell Puts and ∆-hedge.
Example:
K1
PF ≡ C K 2 − PK1
S
K2
σ 1 ,σ 2
VegaK > Vega K
2
S
σ 1 ,σ 2
VegaK < Vega K
2
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St
PF
1
PF
∆-hedged PF gains
from S induced
volatility moves.
1
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Conclusion
• Both leverage and asymmetric jumps may generate skew
but they generate different dynamics
• The Break Even Vols are a good guideline to identify risk
premia
• The market skew contains a wealth of information and in
the absence of jumps,
–
–
–
–
The spot correlated component of volatility
The average behavior of the ATM implied when the spot moves
The optimal hedge ratio of short dated vanilla
The price of options on RV
• If market vol dynamics differ from what current skew
implies, statistical arbitrage
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