fear greed in volatilitymarkets - columbia universitysmirnov/derman.pdfthe skew represent the...
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Page 1 of 24
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.
Page 2 of 24QuantitativeStrategies
❏ 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
Page 3 of 24
QuantitativeStrategies
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
Page 4 of 24
QuantitativeStrategies
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
Page 5 of 24
QuantitativeStrategies
TRODUCTION:LOBAL INDEXPLIED
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
Page 6 of 24
QuantitativeStrategies
TRODUCTION:LOBAL INDEXPLIED
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
Page 7 of 24
QuantitativeStrategies
TRODUCTION:LOBAL INDEXPLIED
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
?
?
?
Page 8 of 24
QuantitativeStrategies
TRODUCTION:LOBAL INDEXPLIED
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
Page 9 of 24
QuantitativeStrategies
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–( )–
Page 10 of 24
QuantitativeStrategies
GREED(STICKY STRIKE)
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
Page 11 of 24
QuantitativeStrategies
GREED(STICKY STRIKE)
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
Page 12 of 24
QuantitativeStrategies
MODERATION(STICKY DELTA)
PART III
MODERATION(STICKY DELTA)
Rational Moderation
the future cost ofer the long run,dex level.
re, you should keep
to raise everychanged.
Delta Model.
S– )
Page 13 of 24
QuantitativeStrategies
MODERATION(STICKY DELTA)
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
Page 14 of 24
QuantitativeStrategies
MODERATION(STICKY DELTA)
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)
Page 15 of 24
QuantitativeStrategies
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.
Page 16 of 24
QuantitativeStrategies
FEAR(STICKY IMPLIEDTREE)
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+( )
Page 17 of 24
QuantitativeStrategies
FEAR(STICKY IMPLIEDTREE)
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
Page 18 of 24
QuantitativeStrategies
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
Page 19 of 24
QuantitativeStrategies
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
Page 20 of 24
QuantitativeStrategies
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:
Page 21 of 24
QuantitativeStrategies
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
Page 22 of 24
QuantitativeStrategies
HAT REALLYAPPENS: MODELEGIMES
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
index trends;should be sticky delta, seems to be sticky strike
index trends;should be sticky delta, seems to be sticky strike
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?
Page 23 of 24
QuantitativeStrategies
HAT REALLYAPPENS: MODELEGIMES
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
Page 24 of 24
QuantitativeStrategies
HAT REALLYAPPENS: MODELEGIMES
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
/99
8/25
/99
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%