modeling the implied volatility surface - baruch...

49
Modeling the Implied Volatility Surface Jim Gatheral Global Derivatives and Risk Management 2003 Barcelona May 22, 2003

Upload: trandan

Post on 23-Mar-2018

241 views

Category:

Documents


5 download

TRANSCRIPT

Page 1: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Modeling the Implied Volatility Surface

Jim GatheralGlobal Derivatives and Risk Management 2003BarcelonaMay 22, 2003

Page 2: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

This presentation represents only the personal opinions of the author andnot those of Merrill Lynch, its subsidiaries or affiliates.

Page 3: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Outline of this talk

n A compound Poisson model of stock tradingn Empirical verification of modeling assumptionsn Stochastic volatilityn Empirical dynamics of SPX and VIXn Dynamics of the implied volatility skewn Which stochastic volatility model?n Do stochastic volatility models fit option prices?n Jumpsn The impact of large option trades

Page 4: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Stock trading as a compound Poisson process

n Consider a random time change from conventional calendar time totrading time such that the rate of arrival of stock trades in a given(transformed) time interval is a constant .

n Intuitively, relative to real time, trading time flows faster when there ismore activity in the stock and more slowly when there is less activity.

n Suppose that the (random) size of a trade is independent of the level ofactivity in the stock.

n Assume further that each trade impacts the mid-log-price of the stock byan amount proportional to .

• This is a standard assumption in the market microstructure literature

n Then the change in log-mid-price over some time interval is given by

n Note that both the number of trades and the size of each trade in agiven time interval are random.

λ

n

n

1

sgn( )N

i ii

x n n=

∆ = ∑

N int∆

Page 5: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Volatility and volume: a relationship

n The variance of this random sum of random variables is given by

n Rewriting this in terms of volatility, we obtain

n But is just the expectation of the volume over the timeinterval . Let where is the trading rate.

n The factor cancels and transforming back to real time, we see thatvariance is directly proportional to volume in this simple model.

n Moreover, the distribution of returns in trading time is approximatelyGaussian for large .

[ ] [ ] [ ][ ] [ ]

2

2

i i

i

Var x E N Var n Var N E n

E N E n

α α

α

∆ = + =

[ ] [ ] [ ]2iVar x t E N E nσ α2∆ = ∆ =

[ ] [ ]iE N E nt∆

t∆

t∆

[ ]E N tλ= ∆ λ

Page 6: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

The relationship

n A key assumption in our simple model is that market impact isproportional to the square root of the trade size . The followingargument shows why this is plausible:

• A market maker requires an excess return proportional to the risk of holdinginventory.

• Risk is proportional to where is the holding period.• The holding period should be proportional to the size of the position.• So the required excess return must be proportional to .

n

n

Tσ T

n

Page 7: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Independence of trade size and trade frequency*

* Excluding trades over 10,000 shares

INTC: 30 minute buckets from 10/1/2002 to 4/25/2003

y = -0.0011x + 910.03R2 = 0.0003

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

Frequency

Ave

rage

Siz

e

Page 8: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Independence of trade size and trade frequency*

* Excluding trades over 10,000 shares

MER: 30 minute buckets from 10/1/2002 to 4/25/2003

y = -0.0221x + 910.37R2 = 9E-05

0

500

1000

1500

2000

2500

0 100 200 300 400 500 600 700 800 900

Frequency

Ave

rag

e S

ize

Page 9: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Empirical variance vs volume

INTC from 4/30/2001 to 4/25/2003

y = 5E-11xR2 = 0.2183

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0 40,000,000 80,000,000 120,000,000 160,000,000 200,000,000

Volume (Shares)

ln^2

(hi/l

o)

Page 10: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Empirical variance vs volume

MER from 2/26/2001 to 4/25/2003

y = 3E-10xR2 = 0.2854

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

0.018

0.02

0 5,000,000 10,000,000 15,000,000 20,000,000 25,000,000

Volume (Shares)

ln^2

(hi/l

o)

Page 11: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Implications of our simple model

n Our simple but realistic model has the following modeling implications• Log returns are roughly Gaussian with constant variance in trading time

(sometimes call intrinsic time) defined in terms of transaction volume• Trading time is the inverse of variance• When we transform from trading time to real time, variance appears to be

random

n It is natural to model the log stock price as a diffusion processsubordinated to another random process which is really trading volumein our model - stochastic volatility

n What form should the volatility process take?

Page 12: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Empirical observations

n In the underlying index return data we observe• Clustering: large moves follow large moves, small moves follow small

moves• Mean reversion of volatility• Negative correlation between implied volatility moves and index returns

n In the implied volatility data we observe• The volatility envelope: short-dated volatilities move more than long-dated

ones• A pronounced skew with downside implied volatilities significantly higher

than upside implied volatilities.• Implied volatilities move more when implied volatility is high.

Page 13: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

A generic stochastic volatility model

n We are now in a position to write down a generic stochastic volatilitymodel consistent with our observations. Let denote the log stock priceand denote its variance. Then

with .

n is a mean-reversion term, is the correlation between volatilitymoves and stock price moves and is called “volatility of volatility”.

n

n To estimate we analyze historical SPX and VIX data.

( )1

2

dx dt v dZ

dv v v dZβ

µ

α η

= +

= +

1 2,dZ dZ dtρ=

x

( )vα ρη

12

3 32 2

gives the Heston model

=1 gives Wiggins' lognormal model

gives Lewis's model

β

β

β

=

=

v

β

Page 14: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

SPX from 1/1/1990 to 4/25/2003

Correlation of vol changes with log returns

y = -1.0953x + 0.0003R2 = 0.6177

-15.00%

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

-8.00% -6.00% -4.00% -2.00% 0.00% 2.00% 4.00% 6.00% 8.00%

Log Return

Vo

lati

lity

Ch

ang

e

Page 15: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

n We regress vs for different values of .n Fit quality is plotted below:

Regressing VIX vs SPX

-1 -0.5 0.5 1 1.5 2a

0.52

0.54

0.56

0.58

0.62

R2

VIX VIXα ∆ ln(SPX)∆ α

2R 0α =

0

11

ααα

== −=

is lognormal

is 3/2 model

is Heston

0α =

n We note that is maximized at roughly which corresponds to alognormal process for volatility.

n At , so . Then, from the slope of theregression, with .

2 0.62R ≈ 0.79ρ ≈ −v v t Zη∆ ≈ ∆ 2.8η ≈

Page 16: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Historical SPX implied volatility

VIX Index

0

10

20

30

40

50

60

Jan-90 Jan-91 Jan-92 Jan-93 Jan-94 Jan-95 Jan-96 Dec-96 Dec-97 Dec-98 Dec-99 Dec-00

Page 17: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

n We regress the 20-day moving average std of vs fordifferent values of .

n Fit quality is plotted below:

Empirical dynamics of VIX

VIX∆ VIXα

2R

1

20

ααα

===

is lognormal

is 3/2 model

is Heston

n We note that is maximized at roughly . This would put thevolatility dynamics between lognormal and 3/2 with

n Imposing lognormality, the slope of the regression should give usvolatility of volatility directly.

α

0.5 1 1.5 2 2.5a

0.35

0.4

0.45

0.5

0.55

0.6

R2

1.4α =( 1) / 2 1.2β α= + ≈

Page 18: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Regression of VIX volatility vs VIX level

y = 0.0635xR2 = 0.5629

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

0.0% 5.0% 10.0% 15.0% 20.0% 25.0% 30.0% 35.0% 40.0% 45.0% 50.0%

20 day moving average

20 d

ay s

tand

ard

devi

atio

n

n The regression gives . Then,12 0.0635tσ η σ σ∆ ≈ ∆ ≈ 0.0635 16 2 2.0η ≈ × × ≈

Page 19: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Unconditional distribution of VIX vs lognormal

VIX from 1/1/1990 to 4/25/2003

0

50

100

150

200

250

-3 -2.5 -2 -1.5 -1 -0.5 0

ln(VIX)

Freq

uenc

y

n Consistent with lognormal volatility dynamics!

Page 20: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Dynamics of the volatility skew

n Our empirical investigations so far indicate that the appropriatespecification of stochastic volatility is lognormal with volatility ofvolatility somewhere between 2.0 and 3.0.

n This choice of specification has implications for the dynamics of theimplied volatility skew. We can check to see whether or not thevolatility skew behaves consistently with this specification.

η

Page 21: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Vol term structure and skew under stochastic volatility

n All stochastic volatility models generate volatility surfaces withapproximately the same shape

n The Heston model has an impliedvolatility term structure that looks to leading order like

It’s easy to see that this shape should not depend very much on theparticular choice of model.

n Also, Gatheral (2002) shows that the term structure of the at-the-moneyvolatility skew has the following approximate behavior for all stochasticvolatility models of the form

n So we can estimate by regressing volatility skew against volatilitylevel.

( )1 / 2

2

0

1/2

(1 ), 1

with /2

T

BSx

v ex T

x T T

v

β λ

β

ρησ

λ λ

λ λ ρη

′− −

=

∂ −≈ − ′ ′∂

′ = −

( )dv v v dt v dZβλ η= − − +

( )dv v v dt v dZλ η= − − +

( ) ( )2 (1 ),

T

BSe

x T v v vT

λ

σλ

−−≈ + −

β

Page 22: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

SPX 3-month ATM volatility skew vs ATM 3m volatility

0.25 0.3 0.35ATM vol

-0.13

-0.12

-0.11

-0.09

-0.08

ATM vol skew

Page 23: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Interpreting the regression of skew vs volatility

n Recall that if the variance satisfies the SDE

at-the-money variance skew should satisfy

and at-the-money volatility skew should satisfy

n The graph shows volatility skew to be roughly independent of volatilitylevel so again consistent with lognormal volatility dynamics.

~dv v dZβ

1/2

0k

vv

kβ −

=

∂∝

2 2

0

BSBS

kkβσ

σ −

=

∂∝

1β ≈

Page 24: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Implied vs empirical stochastic volatility parameters

n It is often claimed that stochastic volatility parameters obtained by fittingstochastic volatility models to option prices are inconsistent withhistorical parameters. Specifically, the implied volatility of volatility isoften thought to be extreme.

n The typical volatility of volatility returned from a fit of the Heston modelto option prices is .

n The relationship between lognormal vol of vol and Heston vol of volshould be . With an implied volatility level of around 30%,would give which should be compared with fromour empirical analysis.

1Hη ≈

H BSη η σ≈ × 1Hη ≈3.3η ≈ (2.0,3.0)η ∈

Page 25: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Do stochastic volatility models fit option prices?

n Once again, we note that the shape of the implied volatility surfacegenerated by a stochastic volatility model does not strongly depend onthe particular choice of model.

n Given this observation, do stochastic volatility models fit the impliedvolatility surface? The answer is “more or less”. Moreover, fittedparameters are reasonably stable over time.

n For very short expirations however, stochastic volatility models certainlydon’t fit as the next slide will demonstrate.

Page 26: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Short Expirations

n Here’s a graph of the SPX volatility skew on 17-Sep-02 (just beforeexpiration) and various possible fits of the volatility skew formula:

n We see that the form of the fitting function is too rigid to fit the observedskews.

n Jumps could explain the short-dated skew!

0.5 1 1.5 2t

-1

-0.8

-0.6

-0.4

-0.2

ATM skew

Page 27: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

More reasons to add jumps

n The statistical (historical) distribution of stock returns and the optionimplied distribution have quite different shapes.

n The size of the volatility of volatility parameter estimated from fits ofstochastic volatility models to option prices is too high to be consistentwith empirical observations

Page 28: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

J

instantaneous variancelong-term mean variancerate of mean reversion of variancevolatility of volatilitycorrelation between stock and volatility changesrate of arrival of jumps

mean log-jump

vvbarληρλ

α sizestandard deviation of log-jump sizeδ

Fitting STOXX50 volatilities using Heston and SVJ

n Heston and SVJ models were fitted to SX5E implied volatility data from18-Oct-2002.

n Heston and SVJ parameter definitions are as follows:

Page 29: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Heston FitsThe orange line is a parametric fit to the implied volatility data; the dashed blue line is the Heston fit.

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.5

0.6

0.7

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.45

0.5

0.55

0.6

0.65

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.35

0.4

0.45

0.5

0.55

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.35

0.4

0.45

0.5

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.35

0.4

0.45

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.25

0.35

0.4

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.25

0.275

0.325

0.35

0.375

0.4

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.26

0.28

0.32

0.34

0.36

0.38

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.26

0.28

0.32

0.34

0.36

l ® 3.9, r ® -.7, h ® 1.15, v® 0.542, vbar ® .312

Page 30: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

SVJ FitsThe orange line is a parametric fit to the implied volatility data; the dashed blue line is the SVJ fit.

l ® 3.5, r ® -.75, h ® 1.15, v® 0.552, vbar ® .252, lJ ® 1, a ® -.1, d ® .1

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.5

0.6

0.7

-0.6 -0.4 -0.2 0.2 0.4 0.60.35

0.45

0.5

0.55

0.6

0.65

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.35

0.4

0.45

0.5

0.55

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.35

0.4

0.45

0.5

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.25

0.35

0.4

0.45

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.25

0.35

0.4

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.25

0.275

0.325

0.35

0.375

0.4

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.24

0.26

0.28

0.32

0.34

0.36

-0.6 -0.4 -0.2 0.2 0.4 0.6

0.26

0.28

0.32

0.34

Page 31: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Observations on the fits

n We note that the Heston model cannot fit volatility levels at allexpirations simultaneously nor can it fit the skew at the front.

n On the other hand, the SVJ model allows us to fit the short term skewand also more or less all of the at-the-money volatility levels.

n Both fits return volatility-of-volatility parameters that are larger thanseem reasonable.

Page 32: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Volatility skew from the characteristic function

n We can compute the volatility skew directly if we know thecharacteristic function .

n The volatility skew is given by the formula (Gatheral (2002)):

n The Heston characteristic function is given by

where and are the familiar Heston coefficients withparameters .

n Adding a jump in the stock price (SVJ) gives the characteristic function

with

( ) E[ ]Ti u xT u eφ =

2 / 820

0

2 1 Im[ ( / 2)]1 / 4

BS TBS T

k

u u ie du

k uTσσ φ

π

∞−

=

∂ −= −

∂ +∫

{ }( ) exp ( , ) ( , )SVT u C u T v D u T vφ = +

( , )C u T ( , )D u T, ,λ η ρ

( ) ( ) ( )SVJ SV JT T Tu u uφ φ φ=

( ) ( ){ }2 2 2/ 2 / 2( ) exp 1 1J i u uT J Ju iu T e T eα δ α δφ λ λ+ −= − − + −

Page 33: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

SVJJ

n Adding a simultaneous jump in the volatility gives the characteristicfunction (Matytsin (2000)):

with

where

with (usual Heston notation)

( ) ( ) ( ) ( )VSVJ SV J JT T T Tu u u uφ φ φ φ=

( ){ }2 2 / 2( ) exp ( ) 1VJ i u uT Ju v T e I uα δφ λ −= −

( )( )( , )( , )

0 0

21( )

1 / 1 /V

V

zT D u TD u T V e dzI u d t e

T p p z p z p

γγ γ −−

+ − + −

= = −+ +∫ ∫

{ }2Vp b iu d

γρη

η± = − ±

Page 34: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

2 4 6 8 10

-0.15

-0.125

-0.075

-0.05

-0.025

Comparing ATM skews from different models

n With parameters

we get the following plots of ATM variance skew vs time to expirationl ® 2.03, r ® -0.57, h ® 0.38, v® 0.1, vè ® 0.04, lJ ® 0.59, a ® -0.05, d ® 0.07, gv® 0.1

SVJJ

SVJ

SV

Page 35: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

0.05 0.1 0.15 0.2 0.25

-0.15

-0.125

-0.1

-0.075

-0.05

-0.025

Short expiration detail

n SV and SVJ skews essentially differ only for very short expirations

SVJJ

SVJSV

Page 36: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Other problems for diffusion models

n If the underlying stochastic process for the stock is a diffusion, weshould be able to get from the statistical measure to the risk neutralmeasure using Girsanov’s Theorem

• This change of measure preserves volatility of volatility.

n However, historical volatility of volatility is significantly lower than theSV fitted parameter.

n Finally, in the few days prior to SPX expirations, out-of-the moneyoption prices are completely inconsistent with the diffusion assumption

• For example a 5 cent bid for a contract 10 standard deviations out-of-the-money.

Page 37: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Simple jump diffusion models don’t work either

n Although jumps may be necessary to explain very short dated volatilityskews, introducing jumps introduces more parameters and this is notnecessarily a good thing. For example, Bakshi, Cao, and Chen (1997and 2000) find that adding jumps to the Heston model has little effect onpricing or hedging longer-dated options and actually worsens hedgingperformance for short expirations (probably through overfitting).

n Different authors estimate wildly different jump parameters for simplejump diffusion models.

Page 38: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

SPX large moves from 1/1/1990 to 4/25/2003

Log returns over 4%

-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

8.00%

-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%

Volatility Change

Log

Ret

urn

Page 39: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

SPX large moves from 1/1/1990 to 4/25/2003

Vol changes over 6%

-8.00%

-6.00%

-4.00%

-2.00%

0.00%

2.00%

4.00%

6.00%

-15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00%

Volatility Change

Log

Ret

urn

Page 40: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Empirical jump observations

n A large move in the SPX index is invariably accompanied by a largemove in volatility

• Volatility changes and log returns have opposite sign

n This is consistent with clustering• If there is a large move, more large moves follow i.e. volatility must jump

n We conclude that any jumps must be double jumps!

Page 41: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

What about pure jump models?n Dilip Madan and co-authors have written extensively on pure jump

models with stable increments• The latest versions of these models involve subordinating a pure jump

process to the integral of a CIR process – trading time again.

n Pure jump models are more aesthetically pleasing than SVJJ• The split between jumps and and diffusion is somewhat ad hoc in SVJJ

n However, large jumps in the stock price don’t force an increase inimplied volatilities.

Page 42: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

The instantaneous volatility impact of option trades

n Recall our simple market price impact model

where is the number of days’ volume represented by the trade.n We can always either buy an option or replicate it by delta hedging until

expiration: in equilibrium, implied volatilities should reflect thisn If we delta hedge, each rebalancing trade will move the price against us

by

n In the spirit of Leland (1985), we obtain the shifted expectedinstantaneous volatility

n What does this mean for the implied volatility?

x σ ξ∆ =

ξ

n SS

ADV ADVδ

σ σ σ ξ∆ Γ ∆

= = Γ ∆

2ˆ 1 2S

tσ ξ

σ σπ δ

Γ ≈ ±

Page 43: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

The implied volatility impact of an option trade

n To get implied volatility from instantaneous volatilities, integrate localvariance along the most probable path from the current stock price todayto the strike price at expiration (see Gatheral (2002))

with .

2 2

0

1( , ) ( , )

T

BS tK T S t dtT

σ σ≈ ∫ %

/

00

t T

t

KS S

S

%

Page 44: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

20

70

120

170

3m6m

9m 12m 15

m 18m 21

m 24m 27

m 30m 33

m 36m 39

m 42m 45

m 48m 51

m 54m 57

m 60m

25.00%

26.00%

27.00%

28.00%

29.00%

30.00%

31.00%

32.00%

33.00%

32.00%-33.00%

31.00%-32.00%

30.00%-31.00%

29.00%-30.00%

28.00%-29.00%

27.00%-28.00%

26.00%-27.00%

25.00%-26.00%

The volatility impact of a 5 year 120 strike call

5 year 120 call, 10 days volume

(original surface flat 40% volatility)

Page 45: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

20

70

120

170

3m6m

9m 12m 15

m 18m 21

m 24m 27

m 30m 33

m 36m 39

m 42m 45

m 48m 51

m 54m 57

m 60m

35.00%

36.00%

37.00%

38.00%

39.00%

40.00%

41.00%

42.00%

41.00%-42.00%

40.00%-41.00%

39.00%-40.00%

38.00%-39.00%

37.00%-38.00%

36.00%-37.00%

35.00%-36.00%

The volatility impact of a 5 year 100/120 collar trade

5 years, 100/120 collar, 3 1/2 days volume

(original surface flat 40% volatility)

Page 46: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Liquidity and the volatility surface

n We see that the shape of the implied volatility surface should reflect thestructure of open delta-hedged option positions.

n In particular, if delta hedgers are structurally short puts and long calls,the skew will increase relative to a hypothetical market with no frictions.

• Part of what we interpret as volatility of volatility when we fit stochasticvolatility models to the market can be ascribed to liquidity effects.

Page 47: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

How option prices reflect the behavior of stock pricesn Short-dated implied vol. more

volatile than long-dated implied vol.n Significant at-the-money skew

n Skew depends on volatility level

n Extreme short-dated impliedvolatility skews

n High implied volatility of volatility

n Clustering - mean reversion ofvolatility

n Anti-correlation of volatility movesand log returns

n Volatility of volatility increases withvolatility level

n Jumps

n Stock volatility depends on thestrikes and expirations of open delta-hedged options positions.

Page 48: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Conclusions

n Far from being ad hoc, stochastic volatility models are naturalcontinuous time extensions of simple but realistic discrete-time modelsof stock trading.

n Stylized features of log returns can be related to empirically observedfeatures of implied volatility surfaces.

n By carefully examining the various stylized features of option prices andlog returns, we are led to reject all models except stochastic volatilitywith simultaneous index and volatility jumps.

n Investor risk preferences and liquidity effects also affect the observedvolatility skew so SV-type models may be misspecified and fittedparameters unreasonable.

Page 49: Modeling the Implied Volatility Surface - Baruch Collegefaculty.baruch.cuny.edu/jgatheral/barcelona2003.pdfModeling the Implied Volatility Surface ... n Empirical dynamics of SPX and

Jim Gatheral, Merrill Lynch, May-2003

Referencesn Clark, Peter K., 1973, A subordinated stochastic process model with finite

variance for speculative prices, Econometrica 41, 135-156.n Duffie D., Pan J., and Singleton K. (2000). Transform analysis and asset pricing

for affine jump-diffusions. Econometrica, 68, 1343-1376.n Geman, Hélyette, and Thierry Ané, 2000, Order flow, transaction clock, and

normality of asset returns, The Journal of Finance 55, 2259-2284.n Gatheral, J. (2002). Case studies in financial modeling lecture notes.

http://www.math.nyu.edu/fellows_fin_math/gatheral/case_studies.htmln Heston, Steven (1993). A closed-form solution for options with stochastic

volatility with applications to bond and currency options. The Review of FinancialStudies 6, 327-343.

n Leland, Hayne (1985). Option Pricing and Replication with Transactions Costs.Journal of Finance 40, 1283-1301.

n Lewis, Alan R. (2000), Option Valuation under Stochastic Volatility : withMathematica Code, Finance Press.

n Matytsin, Andrew (2000). Modeling volatility and volatility derivatives, CiranoFinance Day, Montreal.http://www.cirano.qc.ca/groupefinance/activites/fichiersfinance/matytsin.pdf

n Wiggins J.B. (1987). Option values under stochastic volatility: theory andempirical estimates. Journal of Financial Economics 19, 351-372.