Reconstructing Volatility

Finance Concepts January 08 2004

1

Reconstructing VolatilityNew techniques for understanding the implied

volatility of multi-asset options

Speaker: Marco Avellaneda

Avellaneda, Boyer-Olson, Busca and Friz:`Reconstructing Volatility’, RISK Oct 2002; `Large Deviations Methods and thePricing of Index Options in Finance’, CRAS Paris 2003

Outline• Major US indices and ETFs

• Implied volatility surfaces of single stocks and indices

• Marginalization

• `Reconstructing’ the implied volatility of index options

• Steepest-descent Approximation

• The most likely market configurations

• Multivariate stochastic volatility models

• Moment-matching technique: Lee, Wang and Karim

• Cross-currency options


2

U.S. Equities: Main Sectors & Their Indices

• Major Indices: SPX, DJX, NDXSPY, DIA, QQQ (Exchange-Traded Funds)

• Sector Indices & Index Trackers: Semiconductors: SMH, SOX

Biotech: BBH, BTKPharmaceuticals: PPH, DRG

Financials: BKX, XBD, XLF, RKHOil & Gas: XNG, XOI, OSX

High Tech, WWW, Boxes: MSH, HHH, XBD, XCIRetail: RTH

All these indices have options

Trading statistics (AMEX)

AVERAGE TRADED VOLUME IN DOLLARS LAST MONTH

0 1000 2000 3000 4000

SPY

QQQ

DIA

MM USD


3

AVERAGE TRADED VOLUME OVER THE PAST MONTH (IN USD)

0 100 200 300 400 500 600

SMH

BBH

RTH

OIH

XLF

PPH

XLK

MM USD

Sectors

COMS CMGI LGTO PSFTADPT CNET LVLT PMCSADCT CMCSK LLTC QLGCADLAC CPWR ERICY QCOMADBE CMVT LCOS QTRNALTR CEFT MXIM RNWKAMZN CNXT MCLD RFMDAPCC COST MEDI SANMAMGN DELL MFNX SDLIAPOL DLTR MCHP SEBLAAPL EBAY MSFT SIALAMAT DISH MOLX SSCCAMCC ERTS NTAP SPLSATHM FISV NETA SBUXATML GMST NXTL SUNWBBBY GENZ NXLK SNPSBGEN GBLX NWAC TLABBMET MLHR NOVL USAIBMCS ITWO NTLI VRSNBVSN IMNX ORCL VRTSCHIR INTC PCAR VTSSCIEN INTU PHSY VSTRCTAS JDSU SPOT WCOMCSCO JNPR PMTC XLNXCTXS KLAC PAYX YHOO

COMS CMGI LGTO PSFTADPT CNET LVLT PMCSADCT CMCSK LLTC QLGCADLAC CPWR ERICY QCOMADBE CMVT LCOS QTRNALTR CEFT MXIM RNWKAMZN CNXT MCLD RFMDAPCC COST MEDI SANMAMGN DELL MFNX SDLIAPOL DLTR MCHP SEBLAAPL EBAY MSFT SIALAMAT DISH MOLX SSCCAMCC ERTS NTAP SPLSATHM FISV NETA SBUXATML GMST NXTL SUNWBBBY GENZ NXLK SNPSBGEN GBLX NWAC TLABBMET MLHR NOVL USAIBMCS ITWO NTLI VRSNBVSN IMNX ORCL VRTSCHIR INTC PCAR VTSSCIEN INTU PHSY VSTRCTAS JDSU SPOT WCOMCSCO JNPR PMTC XLNXCTXS KLAC PAYX YHOO

Components of NASDAQ 100 Trust (AMEX:QQQ)

�QQQ trades as a stock

�QQQ index options are the mostheavily traded contract in AMEX

� Capitalization-weighted averageof 100 largest stocks in NASDAQ


4

Morgan Stanley 35 (MSH)

ADP JDSUAMAT JNPRAMZN LUAOL MOTBRCM MSFTCA MUCPQ NTCSCO ORCLDELL PALMEDS PMTCEMC PSFTERTS SLRFDC STMHWP SUNWIBM TLABINTC TXNINTU XLNX

YHOO

ADP JDSUAMAT JNPRAMZN LUAOL MOTBRCM MSFTCA MUCPQ NTCSCO ORCLDELL PALMEDS PMTCEMC PSFTERTS SLRFDC STMHWP SUNWIBM TLABINTC TXNINTU XLNX

YHOO

�35 Underlying Stocks

� Equal-dollar weighted index, adjustedannually

�Each stock has typically O(30) options over a 1yr horizon

SOX, XNG, XOI

XNG

APAAPCBRBRREEXENEEOGEPGKMINBLNFGOEIPPPSTRWMB

… & many others

SOX

ALTRAMATAMDINTCKLACLLTCLSCCLSIMOTMUNSMNVLSRMBSTERTXNXLNX

XOI

AHCBPCHVCOC.BXOMKMGOXYPREPRDSUNTXTOTUCLMRO


5

Example: Basket of 20 Biotechnology Stocks ( Components of BBH)

32-BBH5614GENZ

476.8271SHPGY813ENZN

846SEPR53.544DNA

645QLTI554CRA

9212MLNM3716CHIR

8215MEDI4113BGEN

7212IDPH4046AMGN

644ICOS1064ALKS

848HGSI644AFFX

468GILD5518ABI

ATM ImVolSharesTickerATM ImVolSharesTicker

75

80

85

90

95

100

ImpliedVol

VarSwap

ImpliedVol 96.6191 94.5071 88.4581 83.9929 81.7033 82.5468 81.4319 80.1212 78.6667 80.7064 78.8035

VarSw ap 87.1215 87.1215 87.1215 87.1215 87.1215 87.1215 87.1215 87.1215 87.1215 87.1215 87.1215

31.3 32.5 33.8 35 37.5 40 41.3 42.5 43.8 45 46.3

Strike

Vol.

AOL Jan 2001 Options:Implied volatility curve on Dec 20,2000

Market close

Stock probability is not lognormal


6

60

62

64

66

68

70

72

74

76

78

80

30.00 32.50 35.00 37.50 40.00 42.50 45.00 47.50 50.00

ImpliedVol

VarSwap

54

56

58

60

62

64

66

68

70

30.00

32.50

35.00

37.50

40.00

42.50

45.00

47.50

50.00

55.00

60.00

ImpliedVol

VarSwap

46

48

50

52

54

56

58

60

62

30.00

35.00

40.00

45.00

50.00

60.00

70.00

ImpliedVol

VarSwap40

42

44

46

48

50

52

54

56

58

27.50

35.00

42.50

50.00

57.50

65.00

72.50

80.00

87.50

100.0

0

ImpliedVol

VarSwap

Expiration2/17/01

Expiration4/21/01

Expiration7/21/01

Expiration1/19/02

The AOL ``volatility skew’’ for several expiration dates

30

35

40

45

50

55

75 80 85 90 95 100 105

strike

vol bid vol

ask vol

BBH March 2003 Implied VolsPricing Date: Jan 22 03 10:42 AM


7

DJX Mar 03 Pricing Date: 10/25/02

20

22

24

26

28

30

88 85 77 67 62 56 44 38 32 21 14

Delta

Vo

l BidVol

AskVol

Implied Volatility Curve for Options on Dow Jones Average

Stylized facts about equity volatility curves

� Implied volatility curves are typically downward sloping

� Counterexamples: precious metal stocks are upward sloping

� There is little curvature (or smile). Skew is important.


8

( )

( ) ( ) ( )( ) Fit Empirical/ln1,,

Volatility Stochastic

Volatility Local sDupire',

impliedimplied SKaTSTK

dZd

tS

dWS

dS

tt

t

t

t

+⋅=

=

=

=

σσ

κσσ

σσ

σ

Modeling single-stock volatility curves

Price dynamicsnot lognormal

Jumps …

What is the relation between index options and options on the

components?

Standard (log-normal) Volatility Formula for Index Options

ijjiji

jij

N

jjI ppp ρσσσσ ∑∑

≠=

+= 2

1

22 (*)

Does not apply when volatilities are strike-dependent

How can we incorporate volatility skew information into (*)?


9

“Marginalization”

( ) ( )dttYdWtYYd ,, κκ +=

( )

( ) ( )

( ) ( )

+=

=+=

∈

∑

∑

=

=

+

dttYXdWtYXdY

nidttYXdWtYXdX

RYX

m

j

jj

im

j

jij

i

n

,,,,

,,...,1 ,,,,

,

1

1

1

νκ

µσ

( )000 ,, tYX

Consider an n+1 dimensional diffusion:

Problem: given a starting point, , find a one-dimensional diffusion

such that and have the same probability distributions for all( )tY ( )tY 0tt >

( ) ( )

( ) ( )( ) ( )

( ) ( )( ) ( )

...,,...

...,,2

1,,

,,;,,,,

2

22

2

2

000

∑∑

∑

∂∂−

∂∂−

∂∂∂

+∂∂

∂+∂∂=

∂∂

≡

ii

ijji

ii

xtyx

yxx

xytyx

yt

tyx

tyxtYXtyx

νπ

πκπ

ππ transition probability function

…

Fokker-Planck Equation


10

( ) ( )

( ) ( )( ) ( )( )

( ) ( )( ) ( ) ( )

( ) ( )

⋅∂∂−

⋅∂∂=

∂∂

∴

⋅∂∂−⋅

∂∂=

∂∂

≡ ∫

tyty

ty

yty

ty

ty

yt

ty

tyy

tyyt

ty

xdtyxty n

,,

,,,

,

,,

2

1,

,,,,2

1,

,,,

2

2

2

22

2

ππ

νππππκπ

νππκπ

ππ

Computing the marginal distributions…

integration with respectto x eliminates terms withx-derivatives

new 1D diffusion diffusion equation

Equivalent 1-D diffusion

( ) ( ) ( )( )

( ) ( )( )

( ) ( ) ( )( )

( ) ( )( )

( ) ( ) ( )( ) ( ){ }

( ) ( ) ( )( ) ( ){ }ytYttYtXEty

ytYttYtXEty

xdtyx

xdtyxtyx

ty

tytyty

xdtyx

xdtyxtyx

ty

tytyty

n

n

n

n

==

==

=⋅⋅≡

=⋅⋅≡

∫∫

∫∫

,,,

,,,

,,

,,,,

,

,,,,,

,,

,,,,

,

,,,,,

22

222

νν

κκ

π

πνπ

πνν

π

πκπ

πκκ

Equivalent parameters = conditional expectations of local parameters


11

Application to Index Options

( )

( )

( ) ( )

( )( ) ( )

( )I

SwtS

I

SwSwtStS

tS

dttSdZtSI

dI

dtdWdWE

drdtdWtSS

dS

SwI

iiii

ijijjjiijjii

ijji

iiiiiii

i

i

n

ii

∑∑

∑

==

+=

=

−=+=

==

µµ

ρσσσ

µσ

ρ

µµσ

,

,,

,

,,

, ,,

loc2loc2

locloc

1

Index = weighted sum of stock prices (constant weights)

Diffusion eq.for each stockreflects vol skew

Characterization of the equivalent volatility for the index

( )( )( ) ( )( ) ( ) ( )

( )

( )( ) ( )( ) ( )( ) ( )( ) ( )

( ) .,...,1 ,

,,

,,

,2

2

niSw

SwSp

ItSwttSttStSptSpE

ItSwI

wwtStSttSttS

EtI

jjj

iii

iii

ijijjjiiji

iii

ijijjijijjii

==

==

==

∑

∑∑

∑∑

ρσσ

ρσσσ

• sigma_loc can be seen asa ‘stochastic vol’ driving the index

• sigma_bar is then the ``averaged vol”


12

Varadhan’s Formula

( ) { }( )

( ) ( ){ } ( ) ( ) 1 , 2

,0Prob. log

0

,

22

1

<<−≈==

=

==∑=

tt

yxdxXytX

xX

dtdWdWEdWtXdX

ii

n

jjkkjj

jii

σ

ρσ

( )( ) ( )

( )( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ijjiij

jin

ijij

yx

xxxaxaxg

dssssgyxd

ρσσ

γγγγγ

0,0,

inf,

1

1

0 11,0

2

==

=

−

⋅⋅

=== ∫∑ Riemannian metric

In practice: dimensionlesstime ~ 0.02

Dupire localvolatility model for each stock

Steepest-descent approximation

( ) ( ) .,...,2,1 log 0

log nitF

S

eS

Sx

i

it

i

ii i

=

=

≡ µ

( ) ( ) ( )

( ) ( )

==

=≅

∑i

xtii

xtii

IeeSwxdx

eeSStStI

ii

ii

µ

µσσ

0,0minarg

0 ,,

2*

**loc

22 *

Change to log-scale:

( ) ( )( ){ }( )( ){ }ItIE

ItIEtI

−−=

δδσσ

2loc

2,Formally,

Applying Varadhan’s Formula,

where


13

Steepest Descent=Most Likely Stock Price Configuration

IeF ix

ii =∑

Replace conditional distribution by “Dirac function” at most likely configuration

x*

Characterization of MLC

( ) ( ) ( ) ( ) ( ) ijjjiij

n

iji xxxpxptI ρσσσ ***

1

*2, ∑

=

=

( ) ( ) ( )

( )

=

==

∑

∫ ∑

=

+

=

n

i

txii

x

ijjj

n

jj

i

IeSw

nixxpu

du

ii

i

1

0

*

1

*

*

*

0

,...,1

µ

ρσλσ

Euler-Lagrange equations: find such that( )λ,*x


14

Solution of linearized systemin terms of the stock betas

( ) ( ) ( ) ( ) ( )

( )

( ) ( ) ( ) ( ) ( ) ( )

∆

∆∆==

=≅

≡

≡

∑

∑

=

=

I

IVar

I

I

S

SCovxx

xxCovx

px

x

eI

Ix

pp

ii

iI

jij

n

jij

Ii

t

ijjij

n

ijiI

/,

,0

0000

0ln

00000

^^*

21

2*

1

2

ββ

σσσρ

σ

ρσσσ

µ

Most likely config. : described by the risk-neutral regression coefficients of stock returns with the index return (``micro” CAPM)

Expression in terms of Implied Volatilities and Black- Scholes Deltas

( ) ( )

( ) ( ) ( )( )xx

u

du

xx

x

σσσ

σσ

+≈

≈

−

∫

02

1

1

impl.impl.

1

0

impl.

� Seek direct relation between implied volatilities of single-stock options and implied volatility of index options

� Tool: Berestycki-Busca-Florent large-deviations result for single-stock(“1/2 slope rule”)

303234363840424446

-2 -1.5 -1 -0

.5 00.

5 11.

5 2

log(K/S(0))

vol (

%)

local vol

implied vol


15

The Formula In Terms of BlackScholes Deltas

( )( )

( )( )

( )( )

( )( )

( ) ( )( )

×∆=∆

=

+

≡∆

∆∆⋅=

×≈

∑

∫

∑

=

−

∞−

−

=

f

jjn

jjijIi

y z

i

if

f

jjn

jjij

f

ii

ii

I

FpNN

dzeyNtS

F

tN

I

I

S

SCorr

I

II

I

Fp

I

II

S

FS

implI

impl.

1

1

2*

implI

implI

impl.

1implI

*impl.

*

2

2

1)( ,

2

1ln

1

,)/ln)/ln/ln

σσ

ρ

πσ

σ

σσσ

ρσσ

From Euler-Lagrange equations:

Translate into Black-Scholes deltas

Delta of eachstock is a function ofdelta of index option

Reconstruction Formula for the Index Volatility

( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )( )

5.025.02

2

5.0

1

impl.impl.impl.impl.

impl.impl.

=

−∆−∆=∆

∆+≅∆

∑=

i

n

ijjjjiiijiijI

IIII

p

pp σσσσρσ

σσσ

capitalization of stock in index (%)


16

Correlation & Optimal Configurations

( ) ( )( )

( )( )

( ) iI

II

S

FS

IFp

f

ii

ii

IifI

n

jjjj

∀≈

∆≡∆∴=∑=

)/ln/ln

implI

*impl.

*

impl

1

impl

σσ

σσ�If the stocks are perfectly correlated,

“ Equal-delta approximation “

�If the stocks are uncorrelated,

( )( )

( )( )

( )( )

( ) ( )( )

××∆=∆

××≈

−

fI

iiiIi

fI

jii

f

ii

ii

I

FpNN

I

Fp

I

II

S

FS

impl

impl.1

impl

impl.

implI

*impl.

* )/ln/ln

σσ

σσ

σσ Less volatile & smallstocks have smallerdeltas closer to 50%(their skew does notaffect the index as much)

DJX: Dow Jones Industrial Average

DJX Nov 02 Pricing Date: 10/25/02

20

25

30

35

40

89 86 81 78 72 67 61 54 47 40 34 27 20 15

Delta

Vo

l

BidVol

AskVol

SDA

T=1 month


17

DJX Dec 02 Pricing Date: 10/25/02

2022

2426

2830

3234

87 82 80 74 71 67 59 54 50 46 41 37 32 28 21 14

Delta

Vo

l

BidVol

AskVol

SDA

T= 2 months

DJX Jan 03 Pricing Date: 10/25/02

202224262830323436

88 84 79 75 69 62 55 48 40 33 26 15

Delta

Vo

l

BidVol

AskVol

SDA

T=3 months


18

DJX Mar 03 Pricing Date: 10/25/02

20

22

24

26

28

30

32

88 85 77 67 62 56 44 38 32 21 14

Delta

Vo

l

BidVol

AskVol

SDA

T= 5 months

DJX June 03 Pricing Date: 10/25/02

20

22

24

26

28

89 87 83 75 66 57 48 43 38 29 21 15

Delta

Vo

l

BidVol

AskVol

SDA

T=7 months


19

BBH Nov 02 Date: Oct 25 02

30

35

40

45

50

55

60

89 79 62 41 22

Delta

Vol

BidVol

AskVol

WKB

BBH: Biotechnology HLDRT = 1 month

BBH Dec 02 Date: Oct 25 02

30

35

40

45

50

55

60

88 81 72 60 47 34 23

Delta

Vo

l

BidVol

AskVol

SDA

T = 2 months


20

BBH Apr 03 Date: Oct 25 02

30

35

40

45

50

55

90 86 81 75 68 61 53 45 37

Delta

Vo

l BidVol

AskVol

SDA

T = 6 months

Is dimensionless time is too long? (Error bars: Juyoung Lim)

Is correlation causing the discrepancy?

Expiration: Sep 02

15

20

25

30

35

40

440

445

450

455

460

465

470

475

480

485

490

495

500

505

strike

impl

ied

vol

BidVol

AskVol

WMC vol

Steepest Desc

S&P 100 Index Options(Quote date: Aug 20, 2002)


21

Expiration: Oct 02

15

20

25

30

35

40

430

440

450

460

470

480

490

500

510

520

strike

imp

lied

vol BidVol

AskVol

WMC vol

Steepest Desc



Expiration: Nov 02

15

17

19

21

23

25

27

29

31

33

430 440 450 460 470 480 490 500 510 520

strike

imp

lied

vo

l

BidVol

AskVol

WMC vol

Steepest Desc


22


Expiration: Dec 02

15

17

19

21

23

25

27

29

31

33

420 440 460 470 480 490 500 510 520 530 540

strike

imp

lied

vol BidVol

AskVol

WMC vol

Steepest Desc

General Stochastic Volatility Systems

( ) ( )

i

ii

i

ii

ijjiijji

iii

iii

i

i

dy

S

dSx

I

dIx

dtrdZdWEdtdWdWE

dZd

dWS

dS

σσ

ρ

κσσσ

===

==

==

,

( )nn yyxx ,...,,,..., 11

Look for most likely configuration of stocks and volscorresponding to a given index

displacement x


23

Most likely configuration forStochastic Volatility Systems

ijxx

ji

n

ijjiI

I

iIiiii

I

iIiiii

ji eettpptx

rxy

xx

ρσσσ

σκγγ

σρσββ

γγ),0(),0(),(1

2

*

*

∑=

≅

==

==Most likely configurationfor stocks moves and volatility moves, giventhe index move

SDA

Numerical Example

%40

5.0 %,301 %,20

5.02

2211

21

=−==−==

===

ργσγσ

ppn

10%

15%

20%

25%

30%

35%

40%

45%

50%

-1.0

-0.8

-0.6

-0.4

-0.2 0.

00.2 0.

40.

60.8 1.

0

dI/ILocal Index volImplied Index vol


24

Method I: Dupire & Most LikelyConfiguration for Stock Moves

N-dimensionalEquity market

( )tx ,11σ

( )txii ,σ

( )txnn ,σ

( )txI ,σ

• Step 1: Local volatilityfor each stock

•Step 2: Find most likelyconfiguration for stocks

Method II: Stochastic Volatility System and joint MLC for Stocks and Volatilities

N-dimensionalEquity market ( )txI ,σ

•Only one step: compute themost likely configuration ofstocks and volatilities at thesame time


25

Are methods I and II ``equivalent’’?

( ) ( )i

iiii

xiii

rettx ii

σκϖσσ ϖ =≈ ,0,

( ) ( ) ( ) xxijjij

ijiI

jjii eettpptxβϖβϖρσσσ ,0,0,2 ∑=

Answer is NO, in general.

( ) ( ) ( ) xxijjij

ijiI

ji eettpptx γγρσσσ ,0,0,2 ∑=

Dupire local vol. forsingle names

Index vol.,Method I

Index vol.,Method II

Equivalence holds only under additional assumptions on stock-volatility correlations

ijiiij

iIiiiI

I

iIii

I

iIiii

I

iIi

i

iiiii

rr

rr

r

rr

ρ

ρ

σκγ

σρκ

σρσ

σκβϖ

=

=

=

== Method I

Method II

Conditions underwhich both methods give equivalent valuations


26

The corresponding stock-volatility correlation structure is `diagonal’

( )

iiiiiiijiiij

kjk

ikij

lkiikk

iki

ddWrdzrr

mr

ddWEddWmdz

ςαρ

ρ

ςςα

+=⇒=

=∴

=+=

∑

∑ 0,

Examples: Dupire, CEV, uncoupled SV mode

%50 ,7.0

5.0

6.0

7.0

%40 %,30 %,20

21

21

==

−−

−−

=

===

κκ

ρσσ

r

10%

15%

20%

25%

30%

35%

40%

-0.5 -0.4 -0.3 -0.2 -0.1 -0 0.1 0.2 0.3 0.4 0.5

X

IVO

L(%

)

Method II

Method I

Numerical Example


27

Lee, Wang and KarimRISK, Dec 2003Gram-Charlier expansion

( ) ( )

( ) SmilexSkewxx

k

TSmile

Tks

TSkew

Tk

Ts

xEk

xEsxVar

F

Kx

⋅+⋅+=

=

+=

−++=

−===

=

2atm

2atm

4

4

3

32

!41

,!4

2!3

1

147

!4!31

3 , ),( ,log

σσ

σσ

σσσσ

σσσ

Calculate Index Moments

( )( ) ( )

( ) ( )lkjilkjilkjijk

i

kjikjikjijk

i

ijjijij

i

zzzzEppppxE

zzzEpppxE

ppxE

σσσσ

σσσ

ρσσ

∑

∑

∑

=

=

=

4

3

2

``Gaussian’’ closure for third and 4th moments


28

3rd Moments

( )

( ) ( )

( ) ( )

iikijkjikjijk

iI

iiiii

iikikikii

iikijikikikji

sppps

szEzzzE

szEzzCorrzzzE

kjisSymszzCorrzzCorrSymzzzE

ρρσσσσ

ρ

ρρ

∑=

==

==

≠≠==

3

3

3

1

),(

if ][]),(),([

Consistent with the SDA approximation formula!

Stochastic Correlation Model

• Lee, Wang and Karim: stochastic correlation

• Linear fit

34.0:

66.0 :

ln

−=−=

++=

ββ

εβαρ

BKX

OEX

I Rho_bar is the`average’ correlation

This model can be used to improve the accuracy of the SDA.


29

Cross-Currency OptionsMajor currency triad: USD/JPY, USD/EUR, JPY/EUR

( )

( )

( )

222112

21

2atm,

12

)(

2

222

1

111

2

0

ln/

0ln/

0ln/

12

2

1

σσσρσσ

µµ

µ

µ

+−=

−=

==

==

==

−

I

t

t

t

xxx

eI

IxJPYEURI

eS

SxUSDEURS

eS

SxUSDJPYS

-1

-0.8

-0.6

-0.4

-0.2

-1.3

878E

-16

0.2

0.4

0.6

0.8 1

-1-0.9-0.8-0.7-0.6-0.5-0.4-0.3-0.2-0.1-1.38778E-160.10.20.30.40.50.60.70.80.91

X-bar=-0.5

Sig(USD/JPY)=20 pctSig(USD=11.2 pctRho=50 pct

X-bar=+0.6

Formally, the answer is the sameas in the index case. The ``geometry’’is different.

Reconstructing Volatility

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