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Skew Modeling
Bruno DupireBloomberg LP
bdupire@bloomberg.netColumbia University, New York
September 12, 2005
I. Generalities
Bruno Dupire 3
Market Skews
Dominating fact since 1987 crash: strong negative skew on Equity Markets
Not a general phenomenon
Gold: FX:
We focus on Equity Markets
K
implσ
K
implσ
K
implσ
Bruno Dupire 4
Skews
• Volatility Skew: slope of implied volatility as a function of Strike
• Link with Skewness (asymmetry) of the Risk Neutral density function ?ϕ
Moments Statistics Finance1 Expectation FWD price2 Variance Level of implied vol3 Skewness Slope of implied vol4 Kurtosis Convexity of implied vol
Bruno Dupire 5
Why Volatility Skews?
• Market prices governed by– a) Anticipated dynamics (future behavior of volatility or jumps)– b) Supply and Demand
• 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
K
implσ Market Skew
Th. Skew
Supply and Demand
Bruno Dupire 6
Modeling Uncertainty
Main ingredients for spot modeling• Many small shocks: Brownian Motion
(continuous prices)
• A few big shocks: Poisson process (jumps)
t
S
t
S
Bruno Dupire 7
2 mechanisms to produce Skews (1)
• To obtain downward sloping implied volatilities
– a) Negative link between prices and volatility• Deterministic dependency (Local Volatility Model)• Or negative correlation (Stochastic volatility Model)
– b) Downward jumps
K
impσ
Bruno Dupire 8
2 mechanisms to produce Skews (2)
– a) Negative link between prices and volatility
– b) Downward jumps1S 2S
1S 2S
Leverage and Jumps
Bruno Dupire 10
Dissociating Jump & Leverage effects
• Variance :
• Skewness :
t0 t1 t2
x = St1-St0 y = St2-St1
222 2)( yxyxyx ++=+Option prices
∆ HedgeFWD variance
32233 33)( yxyyxxyx +++=+
Option prices∆ Hedge FWD skewness
Leverage
Bruno Dupire 11
Dissociating Jump & Leverage effects
Define a time window to calculate effects from jumps andLeverage. For example, take close prices for 3 months
• Jump:
• Leverage:
( )∑i
tiS 3δ
( )( )∑ −i
ttt iiSSS 2
1δ
Bruno Dupire 12
Dissociating Jump & Leverage effects
Bruno Dupire 13
Dissociating Jump & Leverage effects
Break Even Volatilities
Bruno Dupire 15
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.
?
=>
Bruno Dupire 16
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.
Bruno Dupire 17
Theoretical Skewfrom historical prices (3)
How to get a theoretical Skew just from spot price history?Example: 3 month daily data1 strike – a) price and delta hedge for a given within Black-Scholes
model– b) compute the associated final Profit & Loss: – c) solve for– d) repeat a) b) c) for general time period and average– e) repeat a) b) c) and d) to get the “theoretical Skew”
1TSkK =σ
( )σPL( ) ( )( ) 0/ =kPLk σσ
t
S
1T2T
K1TS
Bruno Dupire 18
Theoretical Skewfrom historical prices (4)
Bruno Dupire 19
Theoretical Skewfrom historical prices (4)
Bruno Dupire 20
Theoretical Skewfrom historical prices (4)
Bruno Dupire 21
Theoretical Skewfrom historical prices (4)
Barriers as FWD Skew trades
Bruno Dupire 23
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
Bruno Dupire 24
Barrier Static Hedging
If ,unwind hedge, at 0 cost
If not touched, IV’s are equal
Down & Out Call Strike K, Barrier L, r=0 :• With BS:
KLKLK PL
KCDOC 2, −=
LSt =
• With normal model
dWdS σ=
KLKLK PCDOC −−= 2,
LKL2
K
L K1
2L-K
L KL
K
KL2
Bruno Dupire 25
Static Hedging: Model Dominance
• An assumption as the skew at L corresponds to an affine model
• 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
LKDOC ,
LKDOC ,
LK
• Back to
( )dWbaSdS += (displaced LN)
Bruno Dupire 26
Skew Adjusted Barrier Hedges
( )dWbaSdS +=
( )baK
KLbaLKLK PbaLbaKCDOC
+−++
+−↔
2, 2
( ) ( )baK
KLbaLLLKLK CbaLbaKC
baLaDigKLCUOC
+−++
+−⎟⎠⎞
⎜⎝⎛
++−−↔
2, 22
Local Volatility Model
Bruno Dupire 28
One Single Model
• We know that a model with dS = σ(S,t)dWwould 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.
Bruno Dupire 29
The Risk-Neutral Solution
But if drift imposed (by risk-neutrality), uniqueness of the solution
sought diffusion(obtained by integrating twice
Fokker-Planck equation)
DiffusionsRisk
NeutralProcesses
Compatible with Smile
Bruno Dupire 30
Forward Equations (1)
• BWD Equation: price of one option for different
• FWD Equation: price of all options for current
• 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.
( )tS ,
( )00 , tS( )TKC ,
( )00 ,TKC
Bruno Dupire 31
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
Bruno Dupire 32
Fokker-Planck
• If• Fokker-Planck Equation:
• Where is the Risk Neutral density. As
• Integrating twice w.r.t. x:
( )dWtxbdx ,= ( )2
22
21
xb
t ∂∂
=∂∂ ϕϕ
2
2
KC
∂∂
=ϕϕ
2
2
222
2
2
2
2
21
xKCb
tKC
xtC
∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂=
∂
⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
∂=
∂
⎟⎠⎞
⎜⎝⎛∂∂
∂
2
22
2 KCb
tC
∂∂
=∂∂
Bruno Dupire 33
Volatility Expansion
•K,T fixed. C0 price with LVM
•Real dynamics:
•Ito
• Taking expectation:
•Equality for all (K,T)
ttt dWdS σ=tttt dWtSdStS ),(:),( 00 σσ =
dttSSCdS
SCSCKS tt
TT
T )),((21)0,()( 2
02
020
2
0
000 σσ −
∂∂
+∂∂
+=− ∫∫+
[ ]( ) dSdttStSSStSSCSC tt∫∫ −=ΕΓ+= ),(),(),(21)0,()0,( 2
02
0000 ϕσσ
[ ] ),(20
2 tSSStt σσ ==Ε
Bruno Dupire 34
Summary of LVM Properties
is the initial volatility surface
• compatible with local vol
• compatible with = (local vol)²
• deterministic function of (S,t) (if no jumps)
future smile = FWD smile from local vol
( )tS,σ =⇔Σ σ0
Tk,σ̂
( )ωσ [ ]KSE T=⇔Σ 20 σ
⇔
0Σ
Stochastic Volatility Models
Bruno Dupire 36
Heston Model
( )⎢⎢⎢
⎣
⎡
=+−=
+=
∞ dtdZdWdZvdtvvdv
dWvdtSdS
ρηκ
µ
,2
Solved by Fourier transform:
( ) ( ) ( )τττ
τ
,,,,,,
ln
01, vxPvxPevxC
tTK
FWDx
xTK −=
−=≡
Bruno Dupire 37
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
Bruno Dupire 43
Spot dependency
2 ways to generate skew in a stochastic vol model
-Mostly equivalent: similar (St,σt ) patterns, similar future evolutions-1) more flexible (and arbitrary!) than 2)-For short horizons: stoch vol model local vol model + independent noise on vol.
( ) ( )( ) 0,)2
0,,,)1≠
==ZW
ZWtSfxtt
ρρσ
0S
ST
σ
ST
σ
0S
Bruno Dupire 44
SABR model
• F: Forward price
• With correlation ρ
dZddWFdF t
ασσ
σβ
=
=
Smile Dynamics
Bruno Dupire 51
Smile dynamics: Local Vol Model (1)
• Consider, for one maturity, the smiles associated to 3 initial spot values
Skew case
– ATM short term implied follows the local vols– Similar skews
+S0S−S
Local vols
−S Smile
0 Smile S+S Smile
K
Bruno Dupire 52
Smile dynamics: Local Vol Model (2)
• Pure Smile case
– ATM short term implied follows the local vols– Skew can change sign
K−S 0S +S
Local vols
−S Smile
0 Smile S
+S Smile
Bruno Dupire 53
Smile dynamics: Stoch Vol Model (1)
Skew case (r<0)
- ATM short term implied still follows the local vols
- Similar skews as local vol model for short horizons- Common mistake when computing the smile for anotherspot: just change S0 forgetting the conditioning on σ :if S : S0 → S+ where is the new σ ?
[ ] ( )( )TKKSE TT ,22 σσ ==
Local vols
+S0S−S
−S Smile
0 Smile S+S Smile
K
σ
Bruno Dupire 54
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 vol
model• Again, do not forget conditioning of vol by S
Local vols−S Smile
0 Smile S
+S Smile
−S 0S +S K
σ
Bruno Dupire 55
Smile dynamics: Jump Model
Skew case
• ATM short term implied constant (does not follows the local vols)
• Constant skew• Sticky Delta model
+S Smile
−S Smile
0 Smile S
Local vols
−S 0S +S K
Bruno Dupire 56
Smile dynamics: Jump Model
Pure smile case
• ATM short term implied constant (does not follows the local vols)
• Constant skew• Sticky Delta model
+S Smile−S Smile
Local vols
0 Smile S
−S 0S +S K
Bruno Dupire 57
Smile dynamics
23.5
24
24.5
25
25.5
26
S0 K
t
S1
Weighting scheme imposes some dynamics of the smile for a move of the spot:For a given strike K,
(we average lower volatilities)S K↑⇒ ↓∃σ
Smile today (Spot St)
StSt+dt
Smile tomorrow (Spot St+dt)in sticky strike model
Smile tomorrow (Spot St+dt)if σATM=constant
Smile tomorrow (Spot St+dt)in the smile model
&
Bruno Dupire 58
Volatility Dynamics of different models
• Local Volatility Model gives future short termskews that are very flat and Call lesser thanBlack-Scholes.
• More realistic future Skews with:– Jumps– Stochastic volatility with correlation and mean-
reversion• To change the ATM vol sensitivity to Spot:
– Stochastic volatility does not help much– Jumps are required
∆
ATM volatility behavior
Bruno Dupire 60
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 σσ ==∀⇔
Bruno Dupire 61
Sensitivity of ATM volatility / S
St
∂∂ 2σ
At t, short term ATM implied volatility ~ σt.
As σt is random, the sensitivity is defined only in average:
SS
tStSttSSSSSE loctloctloctttttt δσσδδσδσσ δδ ⋅
∂∂
≈−++=+=−+),(),(),(][
22222
2ATMσIn average, follows .
Optimal hedge of vanilla under calibrated stochastic volatility corresponds to perfect hedge ratio under LVM.
2locσ
Bruno Dupire 62
Market Model of Implied Volatility
• Implied volatilities are directly observable• Can we model directly their dynamics?
where is the implied volatility of a given• Condition on dynamics?
( )0=r
⎪⎪⎩
⎪⎪⎨
⎧
++=
=
2211
1
ˆˆ
dWudWudtd
dWSdS
ασσ
σ
σ̂ TKC ,
σ̂
Bruno Dupire 63
Drift Condition
• Apply Ito’s lemma to
• Cancel the drift term
• Rewrite derivatives of
gives the condition that the drift of must satisfy.
For short T, we get the Short Skew Condition (SSC):
close to the money:
Skew determines u1
( )tSC ,ˆ,σ
( )tSC ,ˆ,σα
σσˆˆd
2
2
2
12 )ln()ln(ˆ ⎟
⎠⎞
⎜⎝⎛+⎟
⎠⎞
⎜⎝⎛ +=
SKu
SKuσσ
)ln(~ˆ 12
SKu+σσ
( )tSC ,̂,σ
Tt →
Bruno Dupire 64
Optimal hedge ratio ∆H
),ˆ,( tSC σ
σ̂
2ˆ2 )(.ˆ
)(.
dSdSdCC
dSdSdC
SH σ
σ+==∆
• : BS Price at t of Call option with strike K, maturity T, implied vol
• Ito: • Optimal hedge minimizes P&L variance:
σσ σ ˆ0),ˆ,( ˆ dCdSCdttSdC S ++=
Implied VolsensitivityBS Vega
BS Delta
Bruno Dupire 65
Optimal hedge ratio ∆H II
With
Skew determines u1, which determines ∆H
2ˆ )(.ˆ
dSdSdCCS
H σσ+=∆
⎪⎪⎩
⎪⎪⎨
⎧
++=
=
2211
1
ˆˆ
dWudWudtd
dWSdS
ασσ
σ
Su
dWdW
SSu
dSdSd
σσ
σσσ ˆ
)()(
)()(.ˆ 1
21
21
21
2 ==
Smile Arbitrage
Bruno Dupire 67
Deterministic future smiles
S0
K
ϕ
T1 T2t0
It is not possible to prescribe just any future smileIf deterministic, one must have
Not satisfied in general( ) ( ) ( )dSTSCTStStSC TKTK 1,10000, ,,,,,
22 ∫= ϕ
Bruno Dupire 68
Det. Fut. smiles & no jumps => = FWD smile
If
stripped from Smile S.t
Then, there exists a 2 step arbitrage:Define
At t0 : Sell
At t:
gives a premium = PLt at t, no loss at T
Conclusion: independent of from initial smile
( ) ( ) ( ) ( )TTKKTKTKtSVTKtS impTKTK δδσσδδ
++≡≠∃→→
,,,lim,,/,,, 2
00
2,
( ) ( )( ) ( )TKtSKCtSVTKPL TKt ,,,,, 2
2
,2
∂∂
−≡ σ
( )tStSt DigDigPL ,, εε +− −⋅S
t0 t T
S0
K
[ ] ( ) TKt TKK
SSS ,2
TK,2 , sell , CS2buy , if δσεε +−∈
( ) ( ) ( )TKtSVtS TK ,,, 200, σ==( )tSV TK ,,
Bruno Dupire 69
Consequence of det. future smiles
• Sticky Strike assumption: Each (K,T) has a fixed independent of (S,t)
• Sticky Delta assumption: depends only on moneyness and residual maturity
• In the absence of jumps,– Sticky Strike is arbitrageable– Sticky ∆ is (even more) arbitrageable
),( TKimplσ
),( TKimplσ
Bruno Dupire 70
Example of arbitrage with Sticky Strike
21CΓ
12CΓ
1tS
ttS δ+
Each CK,T lives in its Black-Scholes ( )world
P&L of Delta hedge position over dt:
If no jump
21,2,1 assume2211
σσ >≡≡ TKTK CCCC
!
( ) ( )( )( ) ( )( )( ) ( )
( )ΘΓ
>−ΓΓ
=Γ−Γ
Γ−=
Γ−=
free , no
02
22
21
2211221
22
22
21
2
12
12
21
1
tSCCPL
tSSCPL
tSSCPL
δσσδ
δσδδ
δσδδ
),( TKimplσ
Bruno Dupire 71
Arbitrage with Sticky Delta
1K
2K
tS
• 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:
12 KK PCPF −≡
21,σσS
Vega > Vega2K 1K
PF
21,σσ
Vega < Vega2K 1K
S PF
∆-hedged PF gains from S induced volatility moves.
Bruno Dupire 72
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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