Quantitative
Strategies
FEAR & GREED
IN
VOLATILITY MARKETS
~
Emanuel Derman
Quantitative Strategies Group
Goldman Sachs & Co.
Page 1 of 24
Fear&Greed.fm
Are There Patterns to Volatility Changes?
Quantitative
Strategies
❏ Since 1987, global index options markets are persistently skewed.
How do/should volatilities and the skew change as markets move?
❏ Every description of data involves an articulated or unarticulated
model. There are at least three “models” for volatility change:
❍ An apocryphal Sticky-Strike Rule, that reflects Greed;
❍ An apocryphal Sticky-Delta Rule, that reflects Moderation;
❍ A theoretical Implied Tree Model, that reflects Fear.
❏ Each rule leads to different predictions for valuing & hedging
options. Which works best? And why?
_______________________________
❏ Traders’ daily reports are sometimes unreliable. They focus on
liquid at-the-money volatility, a moving target, but they own
definite strikes.
❏ Therefore, ignore everyone and look at the data through the prism
of models.
Page 2 of 24
________________
❏ There appear to be several distinct periods (“regimes”) in which
different rules seem to hold.
❏ Often, S&P 500 implied volatilities seems to oscillate between the
Fear Rule and the Greed Rule.....
❏ Producing Moderation in the long run, but not the short.
Contents
1. INTRODUCTION: GLOBAL IMPLIED VOLATILITIES
Quantitative
Strategies
2. GREED (STICKY STRIKE)
3. MODERATION (STICKY DELTA)
4. FEAR (STICKY IMPLIED TREE)
5. WHAT REALLY HAPPENS: MODEL REGIMES
Page 3 of 24
PART I
Quantitative
Strategies
PART I
INTRODUCTION:
GLOBAL INDEX
IMPLIED
VOLATILITIES
Page 4 of 24
INTRODUCTION:
GLOBAL INDEX
IMPLIED VOLATILITIES
A Persistent Negative Global Skew
Quantitative
Strategies
A persistent large skew, almost linear, and inconsistent with BlackScholes.
Global Three-Month Volatility Skews
Mar 99
PART I
50
Nikkei 225
INTRODUCTION:
GLOBAL INDEX
IMPLIED
VOLATILITIES
45
S&P 500
g
Hang Sendg
40
FTSE 100
35
DAX
30
25
MIB 30
20
SMI
25D Put
Page 5 of 24
CAC 40
Atm
25D Call
AEX
Σ ( K ) = Σ atm – b ( K – S 0 )
A Negative Correlation with the Index
Quantitative
Strategies
The S&P 500 index and its at-the-money three-month implied
volatility, Sep 1 1997 through Nov 2 1998.
Three-Month Implied Volatilities of SPX Options
PART I
65
60
INTRODUCTION:
GLOBAL INDEX
IMPLIED
VOLATILITIES
55
50
1200
1150
1100
1050
1000
950
900
850
800
750
700
650
45
40
35
30
25
20
Page 6 of 24
ATM
11-02-98
10-01-98
09-01-98
08-03-98
07-01-98
06-01-98
05-01-98
04-01-98
03-02-98
02-02-98
01-02-98
12-01-97
11-03-97
10-01-97
09-01-97
15
INDEX
Note - you don’t own at-the-money volatility, you own a fixed strike.
Volatility Behavior By Strike Is Complex
Quantitative
Strategies
Three Month Implied Volatilities of SPX Options
65
1200
1150
1100
1050
1000
950
900
850
800
750
700
650
60
55
50 I
PART
45
40
INTRODUCTION
:
35
GLOBAL
INDEX
30
IMPLIED
25
VOLATILITIES
20
What’s going on here?
Page 7 of 24
11-02-98
10-01-98
09-01-98
08-03-98
07-01-98
06-01-98
05-01-98
04-01-98
03-02-98
02-02-98
01-02-98
12-01-97
11-03-97
10-01-97
09-01-97
15
INDEX
ATM
750"
800"
850"
900"
950"
1000"
1050"
1100"
1150"
1200"
What’s The Future Skew?
We know the current skew Σ(K) = Σatm – b(K - S0).
Quantitative
Strategies
Hypothetical Implied Volatility of Three-Month SPX Options
Index
103
102
101
100
99
98
97
Strike
PART I
INTRODUCTION:
GLOBAL INDEX
IMPLIED
VOLATILITIES
103
102
17
?
101
100
97
Page 8 of 24
?
19
?
99
98
18
20
?
21
?
22
?
23
❏ What will happen when the index moves?
❏ What’s the S-dependence in Σ(S,K)?
❏ Distinguish carefully between Σ(S,K) and Σatm(S) = Σ(S,S).
PART II
Quantitative
Strategies
PART II
GREED
(STICKY STRIKE)
Page 9 of 24
GREED
(STICKY STRIKE)
Complacency or Greed: Sticky Strike “Model”
Quantitative
Strategies
The simplest & most convenient model for changing the implied
volatility of an option as the index moves is not to change it at all.
This is the or complacency model, or “sticky strike,” the closest thing
to Black-Scholes. It’s also the lazy-trader model.
“STICKY STRIKE”
PART II
GREED
(STICKY STRIKE)
Σ ( S, K ) ≡ Σ ( K ) = Σ atm – b ( K – S 0 )
Characteristics
❏ Fixed-strike volatility is independent of S.
❏ Therefore, because of the negative skew, at-the-money volatility
falls with rising S.
❏ ∆ = ∆BS.
____________
In a rising market, you can think of this model as representing
Irrational Exuberance or Greed:
At-the-money options are the most liquid.
When the market rises, at-the-money volatility falls, and you
are selling the most liquid options more and more cheaply, as
though you need never worry about future index declines.
Page 10 of 24
How Options Trees Evolve
In The Sticky Strike Model
Quantitative
Strategies
Index
90
Strike
100
110
Known
90
90
PART II
GREED
(STICKY STRIKE)
100
100
110
110
❏ Fixed-strike volatility is independent of S.
❏ Therefore, because of the negative skew, at-the-money volatility
falls with rising S.
❏ ∆ = ∆BS.
Page 11 of 24
PART III
Quantitative
Strategies
PART III
MODERATION
(STICKY DELTA)
Page 12 of 24
MODERATION
(STICKY DELTA)
Rational Moderation
At-the-money volatility is the rational estimate for the future cost of
replicating liquid options issued now. On average, over the long run,
at-the-money volatility should be independent of index level.
Quantitative
Strategies
If you have no special expectations about the future, you should keep
at-the-money volatility unchanged.
Given the negative skew, as the index rises, you need to raise every
strike’s volatility to keep at-the-money volatility unchanged.
Traders refer to this as the Sticky Moneyness or Sticky Delta Model.
PART III
MODERATION
(STICKY DELTA)
“STICKY DELTA”: Σ = Σ ( K ⁄ S ) = Σ
Characteristics
❏ Atm vol is independent of S.
❏ Fixed-strike vol increases with S.
❏ ∆ > ∆BS.
____________
Page 13 of 24
atm – b ( K – S )
How Options Trees Evolve
In The Sticky Delta Model
.
Quantitative
Strategies
Index
90
Strike
PART III
100
Known
90
MODERATION
(STICKY DELTA)
90
100
100
110
Page 14 of 24
110
❏ Atm vol is independent of S.
❏ Fixed-strike vol increases with S.
❏ ∆ > ∆BS.
110
PART IV
Quantitative
Strategies
PART IV
FEAR
(STICKY IMPLIED
TREE)
Page 15 of 24
FEAR
(STICKY IMPLIED TREE)
Why The Skew? Fear of Index Declines!
The skew represent the premium for the fear of a downward market
move and an increase in realized and implied volatility.
Quantitative
Strategies
Relation between the current skew and the expected future volatility.
Strike Implied Volatility (%)
PART IV
FEAR
(STICKY IMPLIED
TREE)
100
20%
99
21%
98
22%
97
23%
You can deduce the local volatility at different market levels by
treating the implied volatility as an average over local (future at-themoney) volatilities.
Index Level Local volatility (%)
Page 16 of 24
100
20%
99
22%
98
24%
97
26%
These local volatilities are the future at-the-money volatilities feared
to occur in a decline. Note that local volatilities increase twice as fast
with index changes as implieds increase with strike.
Sticky Implied Tree Extracts Local Volatilities
Quantitative
Strategies
There is one market-consistent tree - the implied tree - whose
expectations of future volatilities match all current options prices and
the skew. In this view, the skew is attributable to an expectation of
higher volatility as the market moves (jumps?) down.
You can use this tree to price all options consistently off future
implied local volatilities. This is similar to pricing all off-the-run
bonds off current forwards.
PART IV
stock
price
variable local
volatility σ(S,t)
in the future
FEAR
(STICKY IMPLIED
TREE)
time
several different constant volatility
trees are equivalent to one implied tree
When the index moves, to find the new skew, you roll along the local
vols. This is similar to rolling along the forward curve to get future
yields as time passes.
STICKY IMPLIED TREE:
Page 17 of 24
Σ ( K , S ) = Σ atm – b ( K + S )
How Options Trees Evolve
In The Sticky Implied Tree Model
Quantitative
Strategies
Index
90
Strike
100
110
1 Current Tree
90
110
110
100
90
90
PART IV
FEAR
(STICKY IMPLIED
TREE)
100
110
100
90
90
110
110
100
90
Page 18 of 24
110
90
❏ Fixed-strike volatility decreases as K or S increases.
❏ Atm vol falls twice as rapidly as skew.
❏ ∆ < ∆BS.
110
PART V
Quantitative
Strategies
PART V
MODEL SUMMARY
Page 19 of 24
MODEL SUMMARY
The Properties of the Models
Quantitative
Strategies
Behavior of
Stickiness
Model
PART V
Strike
Equation for Σ ( S, K )
Σ atm ( t ) – b ( t ) ( K – S 0 ) independent of
MODEL SUMMARY
Delta
Σ
Implied tree Σ
Page 20 of 24
Fixed-strike
Option Volatility
index level
At-the-money
Option Volatility
Delta
decreases as
= ∆BS
index level increases
atm ( t ) – b ( t ) ( K – S )
increases as
independent of index > ∆BS
index level increases level
atm ( t ) – b ( t ) ( K + S )
decreases as
decreases twice as
< ∆BS
index level increases rapidly as index level
increases
PART VI
Quantitative
Strategies
PART VI
WHAT REALLY
HAPPENS: MODEL
REGIMES
Page 21 of 24
WHAT REALLY HAPPENS:
MODEL REGIMES
Which Model Reigns in Which Regime?
Quantitative
Strategies
Three-Month S&P 500 Implied Volatilities
Fear
sticky
strike
or
sticky
implied
tree
60
Greed
Correction
index trends;
should be
sticky delta,
seems to be
sticky strike
jumpy index:
sticky
implied tree
PART VI
65
Greed
CORRECTION
vols rise to
sticky delta
level
Fear
index trends;
should be
sticky delta,
seems to be
sticky strike
stable
Greed
Correction
index trends; CORRECTION
should be
vols rise to
sticky delta, sitcky delta level
seems to be
sticky strike
jumpy index:
sticky implied tree
sticky implied tree
1400
1300
1200
ATM
1100
45
1000
?
40
900
35
800
800
850
900
950
1000
1050
1100
1150
1200
1250
1300
30
1350
700
25
Page 22 of 24
4/29/99
4/7/99
4/19/99
3/26/99
3/16/99
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2/9/99
2/19/99
1/28/99
1/6/99
1/18/99
12/24/98
12/2/98
12/14/98
11/20/98
11/10/98
10/29/98
10/7/98
10/19/98
9/25/98
9/2/98
9/14/98
8/21/98
8/11/98
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4/21/98
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12/3/97
12/15/97
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10/29/97
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500
10/17/97
15
9/23/97
600
9/1/97
20
9/11/97
Volatility
WHAT REALLY
55
HAPPENS: MODEL
50
REGIMES
S&P 500 Level
70
INDEX
Conclusions
Quantitative
Strategies
PART VI
WHAT REALLY
HAPPENS: MODEL
REGIMES
Sticky strike (complacency)
Sticky delta (moderation)
Sticky implied tree (fear)
are intuitively useful ways of thinking about variations in implied
volatility that sometimes correspond to modes of market behavior.
❏ When times are good, and the index keeps rising, the options
market keeps every strike’s volatility roughly fixed, and so the
pendulum of at-the-money volatility drops.
❏ When times get bad, and the index jumps down a few percentage
points, the market has to compensate for having let at-the-money
volatility drop too far. The pendulum reverses, and moves at-themoney volatility up at twice the rate as the index collapses.
❏ On average, over the long haul, the pendulum oscillations between
sticky-strike Greed and sticky-implied-tree Fear average out to
sticky-delta Moderation.
Will these conjectured regimes extend through time and across
markets?
Is there a model of stochastic volatility that encompasses this?
Page 23 of 24
Recent Update: July-August ‘99
Quantitative
Strategies
PART VI
WHAT REALLY
HAPPENS: MODEL
REGIMES
Page 24 of 24
8/25/99
8/24/99
8/23/99
8/20/99
8/19/99
8/18/99
8/17/99
8/16/99
8/13/99
1450
1400
1350
1300
1250
1200
1150
Index
19.85
22.08
22.13
19.14
22.78
22.79
20
25.08
24.99
19.32
24.25
24.37
1500
1550
1600
Rising index, Atm vol falls
19.5 again
19.26
19.13
19.23
22.36
22.49
19.64
25.03
24.7
18.84
21.81
S&P rises 21.27
80 pts
18.86
23.22
22.57
18.3
22.68
23.16
18.22
22.59
23.16
18.91
23.07
23.59
19.11
24.73
25.43
18.95
22.65
22.45
18.89
vols
by strike22.91
remain 23.81
18.17 unchanged
21.41 again
21.82
roughly
16.94
20
21.1
17.35
17.23
16.8
17.97
16.94
17.31
16.86
16.92
16.63
16.54
16.79
16.38
ATM vol again drops
17.36
15.98
17.96
about 3 pts as index
16
17.29
20.87
rises
16.86
17.65
18.22
17.57
17.13
18.03
17.23
16.9
16.91
16.5
15.85
16.32
16.06
14.88
16.1
15.76
14.93
15.54
16.1
14.37
15.91
8/12/99
8/11/99
8/9/99
20.45
20.56
20.45
20.04
20.05
19.65
19.78
20.2
20.4
20.21
20.03
19.1
18.59
18.73
18.9
17.94
17.77
18.33
17.45
18.11
18.43
18.43
18.07
17.38
17.58
17.56
8/10/99
22.17
22.05
21.82
21.58
21.51
21.42
21.49
21.83
21.85
21.7
21.9
20.92
20.19
20.26
20.33
19.56
19.39
19.79
18.95
19.79
20.01
19.95
19.65
18.93
19.33
19.17
8/6/99
20.29
20.7
21.13
20.27
1450
8/5/99
22.09
22.25
22.46
21.7
1400
8/4/99
8/3/99
8/2/99
7/30/99
7/29/99
7/28/99
7/27/99
7/26/99
7/23/99
7/22/99
7/21/99
7/20/99
7/19/99
7/16/99
7/15/99
7/14/99
7/13/99
7/9/99
7/12/99
7/8/99
7/7/99
7/6/99
7/5/99
7/2/99
7/1/99
6/30/99
6/29/99
6/28/99
40.86
38.47
35.84
33.59
30.88
28.36
26.14
23.97
40.15
37.88 Three-Month
35.61 S&P
33.26
30.83
28.32
26.11
24.09
500 Implied
Volatilities
38.52
36.61
34.73
32.68
30.41
28.1
25.93
24.01
38.45
36.44
34.39
32.24
30
27.63
25.51
23.55
1000
1050
1100
1150
1200
1250
1300
1350
Rising Index, Atm vol falls to 19%
Falling index, Atm vol rises to 25%
39.99
37.48
34.88
32.64
30.53
28.14
25.88
23.84
38.74
36.59
34.23
32.12
30.05
27.84
25.74
23.79
38.29
36.13
33.99
31.96
30.04
27.78
25.54
23.65
39.11
36.69
34.56
32.43
30.05
27.79
25.72
23.69
S&P
declines
140 pts
100 pt rise in S&P
38.37
36.11
33.5
31.62
29.7
27.42
25.18
23.33
38.01
35.83
33.44
31.34
29.27
27.19
25.01
23.06
37.98
35.75
33.43
31.41
29.28
27.02
24.95
23.07
37.18
35.25
33.37
31.33
29.49
27.42
25.21
23.36
37.34
35.42
33.33
31.36
29.44
27.22
25.15
23.3
39.02
36.7
34.31
32.1
29.62
27.49
25.41
23.43
38.65
35.84
34.02
31.8
29.61
27.48
25.4
23.45
vols by strike
rise about
3 pts
36.43
34.64
33.26remain31.15
28.23
26.08
24.19
22.29
vols by strike
36.06
34.07
31.93
29.8
27.68
25.63
23.93
22.03
unchanged
37.28
35.11
32.76
30.51
28.39
26.34
24.21
22.14
36.2
34.24
32.31
30.28
28.07
25.98
23.98
22.06
37.05
34.33
32.3
30.04
27.84
25.66 Atm 23.28
21.3as
vol rises twice
36.79
34
32.04
29.8
27.5
25.29 much,23.2
21.22
about 6 pts
35.54 ATM vol
33.6drops about
32.1 3 pts29.99
27.76
25.68
23.52
21.66
36.08 as index
33.73rises 31.48
29.3
27.05
24.9
22.81
20.9
36.38
34.28
31.79
29.72
27.6
25.57
23.44
21.59
34.32
33.35
31.55
29.62
27.85
25.86
23.7
21.94
37.34
33.24
31.4
29.67
27.57
25.54
23.5
21.58
35.29
33.7
31.61
29.61
27.51
25.43
23.39
21.42
Date
36.88
32.32
30.76
28.88
26.56
24.54
22.69
20.79
39.05
36.11
33.52
30.82 135028.39 1400 25.941450 23.71
21.52
1150
1200
1250
1300
1500
INDEX
35.3
33.35
30.76
29.32
27.26
25.19
22.88
21.29
6/25/99
6/16/99
6/15/99
6/14/99
6/11/99
6/9/99
6/10/99
6/8/99
6/7/99
6/4/99
6/3/99
6/2/99
15
6/24/99
20
6/23/99
25
42.54
41
40.31
41.4
40.6
40.16
40.21
39.06
39.26
41.23
41.06
36.56
37.84
39.5
37.72
39.85
39.63
37.32
38.27
38.34
39.46
39.8
39.7
39.71
42.47
1100
41.61
6/22/99
30
6/1/99
Volatility
35
50.83
49.57
46.67
52.57
48.21
49.78
50.38
45.48
45.37
52.22
51.29
42.2
51.9
42.4
42.45
41.98
41.89
42.4
40.45
38.84
42.22
42.28
42.72
42.88
46ATM
44.2
6/21/99
40
6/18/99
skew is about 4 vol pts
per 100 S&P pts points
43.52
42.44
40.44
40.48
950
6/17/99
45
52.37
52.3
45.55
46.01
900