Skew Modeling - Department of Industrial Engineering ...ieor.columbia.edu/files/seasdepts/industrial-engineering...Bruno Dupire 4 Skews • Volatility Skew: slope of implied volatility

Skew Modeling

Bruno DupireBloomberg LP

[email protected] University, New York

September 12, 2005

I. Generalities

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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

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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 Statistics Finance1 Expectation FWD price2 Variance Level of implied vol3 Skewness Slope of implied vol4 Kurtosis Convexity of implied vol

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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

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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

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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

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2 mechanisms to produce Skews (2)

– a) Negative link between prices and volatility

– b) Downward jumps1S 2S

1S 2S

Leverage and Jumps

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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

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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

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Dissociating Jump & Leverage effects

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Dissociating Jump & Leverage effects

Break Even Volatilities

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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 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

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Theoretical Skewfrom historical prices (4)

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Theoretical Skewfrom historical prices (4)

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Theoretical Skewfrom historical prices (4)

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Theoretical Skewfrom historical prices (4)

Barriers as FWD Skew trades

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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

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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)

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Skew Adjusted Barrier Hedges

( )dWbaSdS +=

( )baK

KLbaLKLK PbaLbaKCDOC

+−++

+−↔

2, 2

( ) ( )baK

KLbaLLLKLK CbaLbaKC

baLaDigKLCUOC

+−++

+−⎟⎠⎞

⎜⎝⎛

++−−↔

2, 22

Local Volatility Model

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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.

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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

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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

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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

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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

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Heston Model

( )⎢⎢⎢

⎣

⎡

=+−=

+=

∞ dtdZdWdZvdtvvdv

dWvdtSdS

ρηκ

µ

,2

Solved by Fourier transform:

( ) ( ) ( )τττ

τ

,,,,,,

ln

01, vxPvxPevxC

tTK

FWDx

xTK −=

−=≡

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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

-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

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SABR model

• F: Forward price

• With correlation ρ

dZddWFdF t

ασσ

σβ

=

=

Smile Dynamics

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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

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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

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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

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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

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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

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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

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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

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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 ==∀⇔

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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

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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

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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

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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

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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

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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