FEAR GREED IN VOLATILITYMARKETS - Columbia Universitysmirnov/Derman.pdfThe skew represent the...

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QuantitativeStrategies

Fear&Greed.fm

FEAR & GREED

IN

VOLATILITY MARKETS~

Emanuel Derman

Quantitative Strategies GroupGoldman Sachs & Co.

Are There Patterns to Volatility Changes?

e persistently skewed.nge as markets move?

ted or unarticulatedr volatility change:flectsGreed;flectsModeration;flectsFear.

luing & hedging

ble. They focus ont, but they own

ata through the prism

“regimes”) in which

oscillate between the

t the short.
of 24

❏ Since 1987, global index options markets arHow do/should volatilities and the skew cha

❏ Every description of data involves an articulamodel. There are at least three “models” fo

❍ An apocryphalSticky-Strike Rule, that re❍ An apocryphalSticky-Delta Rule, that re❍ A theoreticalImplied Tree Model, that re

❏ Each rule leads to different predictions for vaoptions. Which works best? And why?

_______________________________❏ Traders’ daily reports are sometimes unrelia

liquid at-the-money volatility, a moving targedefinite strikes.

❏ Therefore, ignore everyone and look at the dof models.

________________❏ There appear to be several distinct periods (

different rules seem to hold.

❏ Often, S&P 500 implied volatilities seems toFear Rule and the Greed Rule.....

❏ Producing Moderation in the long run, but no

Contents

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1. INTRODUCTION: GLOBAL IMPLIED VOLATILITIES

2. GREED(STICKY STRIKE)

3. MODERATION (STICKY DELTA)

4. FEAR (STICKY IMPLIED TREE)

5. WHAT REALLY HAPPENS: MODEL REGIMES

INGIMV

P

ES

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TRODUCTION:LOBAL INDEXPLIED

OLATILITIES

ART I

PART I

INTRODUCTION:GLOBAL INDEX

IMPLIED VOLATILITI

INGIMV

P

A Persistent Negative Global Skew

tent with Black-

225

00

Sendg

100

0

0

g

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OLATILITIES

ART I

A persistent large skew,almost linear, and inconsisScholes.

Global Three-Month Volatility Skews

Nikkei

S&P 5

Hang

FTSE

DAX

CAC 4

MIB 3

SMI

AEX25D Put Atm 25D Call

20

25

30

35

40

45

50

Mar 99

Σ K( ) Σatm b K S0–( )–=

INGIMV

P

A Negative Correlation with the Index

nth implied

u own a fixed strike.

ons

INDEX

ATM

50005000500050000050100150200

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OLATILITIES

ART I

The S&P 500 index and itsat-the-money three-movolatility, Sep 1 1997 through Nov 2 1998.

Note - you don’t own at-the-money volatility, yo

Three-Month Implied Volatilities of SPX Opti

09

-01

-97

1

0-0

1-9

7

11

-03

-97

1

2-0

1-9

7

01

-02

-98

0

2-0

2-9

8

03

-02

-98

0

4-0

1-9

8

05

-01

-98

0

6-0

1-9

8

07

-01

-98

0

8-0

3-9

8

09

-01

-98

1

0-0

1-9

8

11

-02

-98

15

20

25

30

35

40

45

50

55

60

65

677889911111

INGIMV

P

Volatility Behavior By Strike Is Complex

INDEX

ATM

750"

800"

850"

900"

950"

1000"

1050"

1100"

1150"

1200"

11

-02-

98

65070075080085090095010001050110011501200

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OLATILITIES

ART I

What’s going on here?

Three Month Implied Volatilities of SPX Options

09-0

1-97

10

-01-

97

11-0

3-97

12

-01-

97

01-0

2-98

02

-02-

98

03-0

2-98

04

-01-

98

05-0

1-98

06

-01-

98

07-0

1-98

08

-03-

98

09-0

1-98

10

-01-

98

15

20

25

30

35

40

45

50

55

60

65

INGIMV

P

What’s The Future Skew?

) = Σ(S,S).

th SPX Options

9 98 97

?

?

?

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OLATILITIES

ART I

We know the current skewΣ(K) = Σatm – b(K - S0).

❏ What will happen when the index moves?

❏ What’s the S-dependence inΣ(S,K)?❏ Distinguish carefully betweenΣ(S,K) andΣatm(S

Hypothetical Implied Volatility of Three-Mon

Index 103 102 101 100 9

Strike

103 17

102 ? 18

101 19

100 ? 20

99 21

98 ? 22

97 23

G(S

P

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REEDTICKY STRIKE)

ART II

PART II

GREED(STICKY STRIKE)

Complacency or Greed: Sticky Strike “Model”

ging the impliedto change it at all.” the closest thing

-the-money volatility

s representing

.

latility falls, and youore cheaply, asex declines.

b K S0–( )–

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PART II

The simplest & most convenient model for chanvolatility of an option as the index moves is not This is the orcomplacency model, or“sticky strike,to Black-Scholes. It’s also thelazy-tradermodel.

“STICKY STRIKE”

Characteristics❏ Fixed-strike volatility is independent of S.

❏ Therefore, because of the negative skew, atfalls with rising S.

❏ ∆ = ∆BS.

____________

In a rising market, you can think of this model aIrrational Exuberance or Greed:

At-the-money options are the most liquid

When the market rises, at-the-money voare selling the most liquid options more and mthough you need never worry about future ind

Σ S K,( ) Σ K( )≡ Σatm=

How Options Trees Evolveel

-the-money volatility

110

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PART II

In The Sticky Strike Mod

❏ Fixed-strike volatility is independent of S.

❏ Therefore, because of the negative skew, atfalls with rising S.

❏ ∆ = ∆BS.

Index 90 100

Strike Known

90

100

110

90

100

110

PART III

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MODERATION(STICKY DELTA)

PART III


Rational Moderation

the future cost ofer the long run,dex level.

re, you should keep

to raise everychanged.

Delta Model.

S– )

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PART III

At-the-money volatility is the rational estimate for replicating liquid options issued now.On average, ovat-the-money volatility should be independent of in

If you have no special expectations about the futuat-the-money volatility unchanged.

Given the negative skew, as the index rises, you need strike’s volatility to keep at-the-money volatility un

Traders refer to this as theSticky Moneyness or Sticky

“STICKY DELTA”:

Characteristics❏ Atm vol is independent of S.

❏ Fixed-strike vol increases with S.

❏ ∆ > ∆BS.

____________

Σ Σ K S⁄( ) Σatm b K(–= =

How Options Trees Evolve

10

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PART III

In The Sticky Delta Model

.

❏ Atm vol is independent of S.

❏ Fixed-strike vol increases with S.

❏ ∆ > ∆BS.

Index 90 100 1

Strike Known

90

100

110

90

100

110

PART IV

E)

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FEAR(STICKY IMPLIEDTREE)

PART IV

FEAR(STICKY IMPLIED TRE

Why The Skew? Fear of Index Declines!

downward markettility

ted future volatility.

t levels bycal (future at-the-

volatilities fearedse twice as faste.

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PART IV

The skew represent the premium for the fear of a move and an increase in realized and implied vola.

Relation between the current skew and the expec

You can deduce the local volatility at different marketreating the implied volatility as an average over lomoney) volatilities.

These local volatilities are the future at-the-moneyto occur in a decline. Note thatlocal volatilities increawith index changes as implieds increase with strik

Strike Implied Volatility (%)

100 20%

99 21%

98 22%

97 23%

Index Level Local volatility (%)

100 20%

99 22%

98 24%

97 26%

Sticky Implied Tree Extracts Local Volatilities

- whose options prices andn expectation ofn.

tly off futurell off-the-run

roll along the localve to get future

time

variable localvolatility σ(S,t)

t volatility

in the future

ne implied tree

K S+( )

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PART IV

There is onemarket-consistent tree - the implied treeexpectations of future volatilities match all currentthe skew. In this view, the skew is attributable to ahigher volatility as the market moves (jumps?) dow

You can use this tree to price all options consistenimplied local volatilities. This is similar to pricing abonds off current forwards.

When the index moves, to find the new skew, youvols. This is similar to rolling along the forward curyields as time passes.

STICKY IMPLIED TREE:

stockprice

several different constan trees are equivalent to o

Σ K S,( ) Σatm b–=

F(ST

P

How Options Trees Evolveodel

creases.

110

110

110

110

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EARTICKY IMPLIED

REE)

ART IV

In The Sticky Implied Tree M

❏ Fixed-strike volatility decreases as K or S in

❏ Atm vol falls twice as rapidly as skew.

❏ ∆ < ∆BS.

Index 90 100

Strike 1 Current Tree

90

100

110

90100

110

90

90100

110

90

90100

110

90

M

P

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ODEL SUMMARY

ART V

PART V

MODEL SUMMARY

M

P

The Properties of the Models

oneylatility

Delta

S asincreases

= ∆BS

D t of index> ∆BS

I twice asndex level

< ∆BS

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ODEL SUMMARY

ART V

Behavior of

StickinessModel

Equation for Fixed-strikeOption Volatility

At-the-mOption Vo

trike independent ofindex level

decreases index level

elta increases asindex level increases

independenlevel

mplied tree decreases asindex level increases

decreases rapidly as iincreases

Σ S K,( )

Σatm t( ) b t( ) K S0–( )–

Σatm t( ) b t( ) K S–( )–

Σatm t( ) b t( ) K S+( )–

WHR

PNS:

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HAT REALLYAPPENS: MODELEGIMES

ART VI

PART VI

WHAT REALLY HAPPEMODEL REGIMES

WHR

P

Which Model Reigns in Which Regime?

1/1

8/9

9

1/2

8/9

9

2/9

/99

2/1

9/9

9

3/4

/99

3/1

6/9

9

3/2

6/9

9

4/7

/99

4/1

9/9

9

4/2

9/9

9

500

600

700

800

900

1000

1100

1200

1300

1400

ATM

800

850

900

950

1000

1050

1100

1150

1200

1250

1300

1350

INDEX

N vel

sticky implied tree

S&

P 5

00 L

evel

Vol

atili

ty

tion

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ART VI

Three-Month S&P 500 Implied Volatilities

15

20

25

30

35

40

45

50

55

60

65

70

9/1

/97

9/1

1/9

7

9/2

3/9

7

10

/6/9

7

10

/17

/97

10

/29

/97

11

/10

/97

11

/20

/97

12

/3/9

7

12

/15

/97

12

/25

/97

1/7

/98

1/2

0/9

8

2/2

/98

2/1

2/9

8

2/2

4/9

8

3/6

/98

3/1

8/9

8

3/3

0/9

8

4/9

/98

4/2

1/9

8

5/1

/98

5/1

3/9

8

5/2

5/9

8

6/4

/98

6/1

6/9

8

6/2

6/9

8

7/8

/98

7/2

0/9

8

7/3

0/9

8

8/1

1/9

8

8/2

1/9

8

9/2

/98

9/1

4/9

8

9/2

5/9

8

10

/7/9

8

10

/19

/98

10

/29

/98

11

/10

/98

11

/20

/98

12

/2/9

8

12

/14

/98

12

/24

/98

1/6

/99

?

stickystrike

or sticky

implied tree

jumpy index:sticky

implied tree

index trends;should be sticky delta, seems to be sticky strike

CORRECTIOvols rise to

sitcky delta le

stable

jumpy index:sticky implied tree

CORRECTIONvols rise tosticky delta

level



Fear Greed Correction Greed Fear Greed Correc

WHR

P

Conclusions

ations in impliedf market behavior.ising, the options fixed, and so the

wn a few percentageaving let at-the-moneys, and moves at-the-

ndex collapses.

oscillations betweenear average out to

time and across

ompasses this?

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ART VI

Sticky strike (complacency)Sticky delta (moderation)Sticky implied tree (fear)

are intuitively useful ways of thinking about varivolatility that sometimes correspond to modes o❏ When times are good, and the index keeps r

market keeps every strike’s volatility roughlypendulum of at-the-money volatility drops.

❏ When times get bad, and the index jumps dopoints, the market has to compensate for hvolatility drop too far. The pendulum reversemoney volatility up at twice the rate as the i

❏ On average, over the long haul, the pendulumsticky-strike Greed and sticky-implied-tree Fsticky-delta Moderation.

Will these conjectured regimes extend through markets?

Is there a model of stochastic volatility that enc

WHR

P

Recent Update: July-August ‘99

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ART VI

52.37 43.52 40.86 38.47 35.84 33.59 30.88 28.36 26.14 23.97 22.09 20.29 19.85 22.08 22.1352.3 42.44 40.15 37.88 35.61 33.26 30.83 28.32 26.11 24.09 22.25 20.7 19.14 22.78 22.79

45.55 40.44 38.52 36.61 34.73 32.68 30.41 28.1 25.93 24.01 22.46 21.13 20 25.08 24.9946.01 40.48 38.45 36.44 34.39 32.24 30 27.63 25.51 23.55 21.7 20.27 19.32 24.25 24.37

900 950 1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 1550 1600

50.83 42.54 39.99 37.48 34.88 32.64 30.53 28.14 25.88 23.84 22.17 20.45 19.5 19.26 19.1349.57 41 38.74 36.59 34.23 32.12 30.05 27.84 25.74 23.79 22.05 20.56 19.23 22.36 22.4946.67 40.31 38.29 36.13 33.99 31.96 30.04 27.78 25.54 23.65 21.82 20.45 19.64 25.03 24.752.57 41.4 39.11 36.69 34.56 32.43 30.05 27.79 25.72 23.69 21.58 20.04 18.84 21.27 21.8148.21 40.6 38.37 36.11 33.5 31.62 29.7 27.42 25.18 23.33 21.51 20.05 18.86 23.22 22.5749.78 40.16 38.01 35.83 33.44 31.34 29.27 27.19 25.01 23.06 21.42 19.65 18.3 22.68 23.1650.38 40.21 37.98 35.75 33.43 31.41 29.28 27.02 24.95 23.07 21.49 19.78 18.22 22.59 23.1645.48 39.06 37.18 35.25 33.37 31.33 29.49 27.42 25.21 23.36 21.83 20.2 18.91 23.07 23.5945.37 39.26 37.34 35.42 33.33 31.36 29.44 27.22 25.15 23.3 21.85 20.4 19.11 24.73 25.4352.22 41.23 39.02 36.7 34.31 32.1 29.62 27.49 25.41 23.43 21.7 20.21 18.95 22.65 22.4551.29 41.06 38.65 35.84 34.02 31.8 29.61 27.48 25.4 23.45 21.9 20.03 18.89 22.91 23.81

42.2 36.56 36.43 34.64 33.26 31.15 28.23 26.08 24.19 22.29 20.92 19.1 18.17 21.41 21.8251.9 37.84 36.06 34.07 31.93 29.8 27.68 25.63 23.93 22.03 20.19 18.59 16.94 20 21.142.4 39.5 37.28 35.11 32.76 30.51 28.39 26.34 24.21 22.14 20.26 18.73 17.35 17.23 16.8

42.45 37.72 36.2 34.24 32.31 30.28 28.07 25.98 23.98 22.06 20.33 18.9 17.97 16.94 17.3141.98 39.85 37.05 34.33 32.3 30.04 27.84 25.66 23.28 21.3 19.56 17.94 16.86 16.92 16.6341.89 39.63 36.79 34 32.04 29.8 27.5 25.29 23.2 21.22 19.39 17.77 16.54 16.79 16.38

42.4 37.32 35.54 33.6 32.1 29.99 27.76 25.68 23.52 21.66 19.79 18.33 17.36 15.98 17.9640.45 38.27 36.08 33.73 31.48 29.3 27.05 24.9 22.81 20.9 18.95 17.45 16 17.29 20.8738.84 38.34 36.38 34.28 31.79 29.72 27.6 25.57 23.44 21.59 19.79 18.11 16.86 17.65 18.2242.22 39.46 34.32 33.35 31.55 29.62 27.85 25.86 23.7 21.94 20.01 18.43 17.57 17.13 18.0342.28 39.8 37.34 33.24 31.4 29.67 27.57 25.54 23.5 21.58 19.95 18.43 17.23 16.9 16.9142.72 39.7 35.29 33.7 31.61 29.61 27.51 25.43 23.39 21.42 19.65 18.07 16.5 15.85 16.3242.88 39.71 36.88 32.32 30.76 28.88 26.56 24.54 22.69 20.79 18.93 17.38 16.06 14.88 16.1

46 42.47 39.05 36.11 33.52 30.82 28.39 25.94 23.71 21.52 19.33 17.58 15.76 14.93 15.5444.2 41.61 35.3 33.35 30.76 29.32 27.26 25.19 22.88 21.29 19.17 17.56 16.1 14.37 15.91

Three-Month S&P 500 Implied Volatilities

15

20

25

30

35

40

45

6/1/

99

6/2/

99

6/3/

99

6/4/

99

6/7/

99

6/8/

99

6/9/

99

6/10

/99

6/11

/99

6/14

/99

6/15

/99

6/16

/99

6/17

/99

6/18

/99

6/21

/99

6/22

/99

6/23

/99

6/24

/99

6/25

/99

6/28

/99

6/29

/99

6/30

/99

7/1/

99

7/2/

99

7/5/

99

7/6/

99

7/7/

99

7/8/

99

7/9/

99

7/12

/99

7/13

/99

7/14

/99

7/15

/99

7/16

/99

7/19

/99

7/20

/99

7/21

/99

7/22

/99

7/23

/99

7/26

/99

7/27

/99

7/28

/99

7/29

/99

7/30

/99

8/2/

99

8/3/

99

8/4/

99

8/5/

99

8/6/

99

8/9/

99

8/10

/99

8/11

/99

8/12

/99

8/13

/99

8/16

/99

8/17

/99

8/18

/99

8/19

/99

8/20

/99

8/23

/99

8/24

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8/25

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Date

Vol

atili

ty

1150

1200

1250

1300

1350

1400

1450

Inde

x

ATM 1100 1150 1200 1250 1300 1350 1400 1450 1500 INDEX

100 pt rise in S&P

vols by strike remain unchanged

ATM vol drops about 3 pts as index rises

S&P declines 140 pts

vols by strike rise about 3 pts

Atm vol rises twice as much, about 6 pts

skew is about 4 vol pts per 100 S&P pts points

S&P rises 80 pts

vols by strike remain roughly unchanged again

ATM vol again drops about 3 pts as index rises

Rising Index, Atm vol falls to 19% Rising index, Atm vol falls again

Falling index, Atm vol rises to 25%

FEAR GREED IN VOLATILITYMARKETS - Columbia Universitysmirnov/Derman.pdfThe skew represent the...

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