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Skew Modeling

Bruno DupireBloomberg LP

[email protected]

Columbia University, New YorkSeptember 12, 2005
mailto:[email protected]:[email protected]


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I. Generalities


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Bruno Dupire 3

Market Skews

Dominating fact since 1987 crash: strong negative skew onEquity Markets

Not a general phenomenon

Gold: FX:

We focus on Equity Markets

K

impl

K

impl

K

impl


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Skews

Volatility Skew: slope of implied volatility as afunction of Strike

Link with Skewness (asymmetry) of the RiskNeutral density function ?

Moments Statistics Finance1 Expectation FWD price

2 Variance Level of implied vol3 Skewness Slope of implied vol

4 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 capturerisk premium b)

To arbitrage (or correctly price) exotics, find RiskNeutral dynamics calibrated to the market

K

impl Market Skew

Th. Skew

S u p p l y a n d D e m a n d


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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

imp


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2 mechanisms to produce Skews (2)

a) Negative link between prices and volatility

b) Downward jumps

1S 2S

1S 2S


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Leverage and Jumps


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Bruno Dupire 10

Dissociating Jump & Leverage effects

Variance :

Skewness :

t0 t1 t2

x = S t1-S t0 y = S t2-S t1

222 2)( y xy x y x ++=+Option prices

Hedge

FWD variance

32233 33)( y xy y x x y x +++=+Option prices

Hedge FWD skewness

Leverage


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Bruno Dupire 11


Define a time window to calculate effects from jumps and

Leverage. For example, take close prices for 3 months

Jump:

Leverage:

( )i t iS 3

( )( ) i

t t t iiS S S 2

1


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Bruno Dupire 12



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Break Even Volatilities


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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 allstrike/maturity; breaks autocorrelation and vol/spot dependency.

?

=>


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h l k


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Bruno Dupire 17

Theoretical Skew

from historical prices (3)How to get a theoretical Skew just from spot pricehistory?

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

1T S k K =

( ) PL

( ) ( )( ) 0/ =k PLk

t

S

1T 2T

K1T

S

Th i l Sk


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Bruno Dupire 18

Theoretical Skew

from historical prices (4)

Th i l Sk


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Bruno Dupire 19

Theoretical Skew


Th i l Sk


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Bruno Dupire 20

Theoretical Skew


Th ti l Sk


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Bruno Dupire 21

Theoretical Skew



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Barriers as FWD Skew trades


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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 ofEuropean options

Calibrating to the current vol surface do not imposethese dynamics


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Bruno Dupire 24

Barrier Static Hedging

If ,unwind hedge, at 0cost

If not touched, IVs are equal

Down & Out Call Strike K, Barrier L, r=0 :

With BS: K LK LK P LK C DOC 2, = LS t =

With normal model

dW dS =K LK LK PC DOC = 2,

LK L2

K

L K12L-K

L K LK K

L2


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Bruno Dupire 25

Static Hedging: Model Dominance

An assumption as the skew at L corresponds toan affine model

priced as in BS with shifted K and L givesnew hedging PF which is >0 when L is touched ifSkew assumption is conservative

LK DOC ,

LK DOC ,

LK

Back to

( )dW baS dS += (displaced LN)


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Bruno Dupire 26

Skew Adjusted Barrier Hedges

( )dW baS dS +=

( )baK

K LbaLK LK PbaLbaK

C DOC +

+++

2, 2

( ) ( )baK

K LbaL L LK LK C

baLbaK

C baL

a DigK LC UOC

+++

+

++

2, 22


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Local Volatility Model


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Bruno Dupire 28

One Single Model

We know that a model with dS = (S,t)dWwould generate smiles. Can we find (S,t) which fits market smiles? Are there several solutions?

ANSWER: One and only one way to do it.


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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

Compatiblewith Smile


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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 impliedvolatility surface,

If current implied volatility surface known, extraction oflocal volatilities,

Understanding of forward volatilities and how to lockthem.

( )t S ,

( )00 , t S ( )T K C ,

( )00 ,T K C


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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 financialinsight


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Bruno Dupire 32

Fokker-Planck

If Fokker-Planck Equation:

Where is the Risk Neutral density. As

Integrating twice w.r.t. x:

( )dW t xbdx ,= ( )2

22

21

x

b

t

=

2

2

K C

=

2

2

222

2

2

2

2

21

x

K C

b

t

K C

xt C

=

=

2

22

2 K

C b

t

C

=


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Bruno Dupire 33

Volatility Expansion

K,T fixed. C 0 price with LVM

Real dynamics:

Ito

Taking expectation:

Equality for all (K,T)

t t t dW dS =

t t t t dW t S dS t S ),(:),( 00 =

dt t S S

C dS

S

C S C K S t t

T T

T )),((2

1)0,()( 20

2

020

2

0

000

+

+=

+

[ ]( ) dSdt t S t S S S t S S C S C t t =+= ),(),(),(21)0,()0,(20

20000

),(202 t S S S t t ==


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Bruno Dupire 34

Summary of LVM Properties

is the initial volatility surface

compatible with local vol

compatible with = (local vol)

deterministic function of (S,t) (if no jumps)

future smile = FWD smile from local vol

( )t S , = 0

T k ,

( ) [ ]K S E T = 20

0


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Stochastic Volatility Models

H M d l


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Bruno Dupire 36

Heston Model

( )

=+=

+=

dt dZ dW dZ vdt vvdv

dW vdt S

dS

,2

Solved by Fourier transform:

( ) ( ) ( )

,,,,,,

ln

01, v xPv xPev xC

t T K

FWD x

xT K =

=

R l f


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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


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Spot dependency


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Bruno Dupire 43

Spot dependency

2 ways to generate skew in a stochastic vol model

-Mostly equivalent: similar (S t,t ) patterns, similar

futureevolutions-1) more flexible (and arbitrary!) than 2)-For short horizons: stoch vol model local vol model+ independent noise on vol.

( ) ( )( ) 0,)2

0,,,)1

==

Z W

Z W t S f xt t

0S

S T

S T

0S

SABR model


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Bruno Dupire 44

SABR model

F: Forward price

With correlation

dZ d

dW F dF t

==


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Smile Dynamics

Smile dynamics: Local Vol Model (1)


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Bruno Dupire 51


Consider, for one maturity, the smiles associatedto 3 initial spot values

Skew case

ATM short term implied follows the local vols Similar skews

+S 0S S

Local vols

S Smile0Smile S

+S Smile

K



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Bruno Dupire 52


Pure Smile case

ATM short term implied follows the local vols Skew can change sign

KS 0S +S

Local vols

S Smile

0Smile S

+S Smile

Smile dynamics: Stoch Vol Model (1)


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Bruno Dupire 53

y ( )

Skew case (r


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Bruno Dupire 54

y ( )

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 volsS Smile

0Smile S

+S Smile

S 0S +S K

Smile dynamics: Jump Model


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Bruno Dupire 55

y p

Skew case

ATM short term implied constant (does not follows thelocal vols) Constant skew Sticky Delta model

+S Smile

S Smile

0Smile S

Local vols

S 0S +S K

Smile dynamics: Jump Model


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Bruno Dupire 56

y p

Pure smile case

ATM short term implied constant (does not follows thelocal vols) Constant skew Sticky Delta model

+S Smile

S Smile

Local vols

0Smile S

S 0S +S K

Smile dynamics


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Bruno Dupire 57

y

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 S t)

StSt+dt

Smile tomorrow (Spot S t+dt)in sticky strike model

Smile tomorrow (Spot S t+dt)if ATM =constant

Smile tomorrow (Spot S t+dt)in the smile model

&

Volatility Dynamics of different models


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Bruno Dupire 58

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


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ATM volatility behavior

Forward Skews


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Bruno Dupire 60

In the absence of jump :

model fits marketThis constrains

a) the sensitivity of the ATM short term volatility wrt S;

b) the average level of the volatility conditioned to S T=K.

a) tells that the sensitivity and the hedge ratio of vanillas depend on thecalibration 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

T K K S E T K locT T ==

Sensitivity of ATM volatility / S


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Bruno Dupire 61

S t

2

At t, short term ATM implied volatility ~ t.

As

t is random, the sensitivity is defined only in average:

S S

t S t S t t S S S S S E loct loct loct t t t t t

++=+=+),(

),(),(][2

2222

2 ATM In average, follows .

Optimal hedge of vanilla under calibrated stochastic volatility corresponds to

perfect hedge ratio under LVM.

2loc

Market Model of Implied Volatility


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Bruno Dupire 62

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

dW udW udt d

dW S

dS

T K C ,

Drift Condition


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Bruno Dupire 63

Apply Itos 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 u 1

( )t S C ,,

( )t S C ,,

d

2

2

2

12 )ln()ln(

+

+=

S K

uS K

u

)ln(~ 12

S K

u+

( ) t S C ,,

T t

Optimal hedge ratio H


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Bruno Dupire 64

),,( t S C

22 )(.

)(.

dS dS d C C

dS dS dC

S H

+==

: BS Price at t of Call option withstrike K, maturity T, implied vol

Ito: Optimal hedge minimizes P&L variance:

0),,( d C dS C dt t S dC S ++=

Implied Vol

sensitivityBS VegaBS Delta

Optimal hedge ratio H II


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Bruno Dupire 65

With

Skew determines u 1, which determines H

2 )(.

dS dS d

C C S H

+=

++=

=

2211

1

dW udW udt d

dW S

dS

S u

dW dW

S S u

dS dS d

)()(

)()(. 1

21

21

21

2 ==


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Smile Arbitrage

Deterministic future smiles


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Bruno Dupire 67

S0

K

T1 T2t0

It is not possible to prescribe just any futuresmile

If deterministic, one must have

Not satisfied in general

( ) ( ) ( )dS T S C T S t S t S C T K T K 1,10000, ,,,,, 22 =

Det. Fut. smiles & no jumps=> = FWD smile


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Bruno Dupire 68

FWD smileIf

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 offrom initial smile

( ) ( ) ( ) ( )T T K K T K T K t S V T K t S impT K T K

++

,,,lim,,/,,, 2

00

2,

( ) ( )( ) ( )T K t S K

C t S V T K PL T K t ,,,,, 2

2

,2

t S t S t Dig DigPL ,, +

S

t 0 t T

S 0K

[ ] ( ) T K t T K K S S S ,2

TK,2 ,sell,CS2

buy,if +

( ) ( ) ( )T K t S V t S T K ,,, 200, ==( )t S V T K ,,

Consequence of det. future smiles


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Bruno Dupire 69

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

),( T K impl

),( T K impl

Example of arbitrage with Sticky Strike


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Bruno Dupire 70

21C 12C

1t S

t t S +

Each CK,T lives in its Black-Scholes ( )world

P&L of Delta hedge position over dt:

If no jump

21,2,1 assume2211 > T K T K C C C C

!

( ) ( )

( ) ( )( )( ) ( )

( )

>==

=

free,no

02

22

21

2211221

22

22

212

12

12

21

1

t S C C PL

t S S C PL

t S S C PL

),( T K impl

Arbitrage with Sticky Delta


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Bruno Dupire 71

1K

2K

t S

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 K K PC PF

21 , S

Vega > Vega2K 1K

PF

21 ,

Vega < Vega2K 1K

S PF

-hedged PF gainsfrom S inducedvolatility moves.

Conclusion


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Bruno Dupire 72

Both leverage and asymmetric jumps may generate skewbut they generate different dynamics

The Break Even Vols are a good guideline to identify riskpremia

The market skew contains a wealth of information and inthe 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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