Presentation for QWAFAFEW, June 8, 2011
The Impact of the Federal Reserve's Interest Rate Target
Announcement on Stock Prices: A Closer Look at How the
Market Impounds New Information
Justin Birru
and
Stephen Figlewski*
New York University Stern School of Business
The Classic Event Study Market Response Plot
Price
News release
Figlewski
QWAFAFEW Presentation June 2011
Time
2
The Classic Event Study Market Response Plot, with Information Leakage
Price
News release
Figlewski
QWAFAFEW Presentation June 2011
Time
3
Figure 1: Intraday Behavior of the Forward Value of the S&P Index on Dec. 11, 2007
Figlewski
QWAFAFEW Presentation June 2011
4
Table 1: Federal Reserve Interest Rate Target Announcements
Figlewski
Date
Target Rate
Change
S&P 500 Index
Change
Fed Funds
Futures
"Surprise"
5/3/2005
6/30/2005
8/9/2005
9/20/2005
11/1/2005
12/13/2005
1/31/2006
3/28/2006
5/10/2006
6/29/2006
8/8/2006
9/20/2006
10/25/2006
12/12/2006
1/31/2007
3/21/2007
5/9/2007
6/28/2007
8/7/2007
9/18/2007
10/31/2007
12/11/2007
1/30/2008
3/18/2008
4/30/2008
6/25/2008
8/5/2008
9/16/2008
10/29/2008
12/16/2008
3
3.25
3.5
3.75
4
4.25
4.5
4.75
5
5.25
5.25
5.25
5.25
5.25
5.25
5.25
5.25
5.25
5.25
4.75
4.5
4.25
3
2.25
2
2
2
2
1
0.125*
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0.25
0
0
0
0
0
0
0
0
0
-0.5
-0.25
-0.25
-0.5
-0.75
-0.25
0
0
0
-0.5
-0.875
-0.99
-8.52
8.25
-9.68
-4.25
7.00
-5.11
-8.38
-2.29
26.87
-4.29
7.54
4.84
-1.48
9.42
24.10
4.86
-0.63
9.04
43.13
18.36
-38.31
-6.49
54.14
-5.35
7.68
35.87
20.90
-10.42
44.61
0
0
0
0.014
0.225
0
0
0
-0.007
-0.015
-0.039
0
0
0
0
0
0
0
0.025
-0.138
-0.020
0.007
-0.095
0.155
-0.055
-0.025
-0.006
0.056
-0.060
-0.110
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5
Hypotheses and Questions about the Impact of the Fed Announcement
Is the market's response to the announcement unbiased?
Does the information in the announcement all enter the market at the moment of the
public announcement?
• Information leakage beforehand?
• Sluggish adjustment or overshooting afterward?
The mean of the Risk Neutral Density (RND) and the forward value of the spot S&P index
are tied together by arbitrage. Do they behave identically on announcement day?
• Is one a better prediction of the announcement than the other?
The RND reveals the market's risk neutral expectation for the future value of the index
and also the variance around that expectation, a direct measure of uncertainty.
• How much uncertainty does a Fed announcement resolve on average?
• Does it matter if the announcement is viewed as positive or negative by the
market?
Figlewski
QWAFAFEW Presentation June 2011
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Hypotheses and Questions about Informational Efficiency
Information flow within a time interval can be proxied by the standard deviation of price
change over the interval or by volatility within the interval. What is information flow like
over an announcement day?
•
How much information does the announcement itself convey?
•
How is the day's price change distributed within the day?
•
Is there a distinct period of re-equilibration after the announcement?
•
How does an announcement day compare to a regular day?
Information flow is assumed to be serially independent, so price changes in an efficient
market should have zero autocorrelation.
•
Does this hold for the forward index level?
•
Does it hold for the RND mean?
Figlewski
QWAFAFEW Presentation June 2011
7
Answers Using the Risk Neutral Density for the S&P 500
We address these questions by exploring the behavior of the risk neutral probability
density for the market portfolio on days when the Federal Reserve announces its
interest rate target.
Data sample:
• Option Price Reporting Authority (OPRA) National Best Bid and Offer data
provides a continuous synchronized record of price movements in the markets for
all equity and equity index options and their underlyings
•
S&P 500 index calls and puts, with maturities in March, June, September, and
December, 28 announcement days 2005 – 2008
•
Puts and calls are combined to cover the range of traded exercise prices: Out of
the money puts for low X; out of the money calls for high X; a weighted
combination of calls and puts for X in the middle range.
•
Options with bid prices less than $0.50 were discarded
Figlewski
QWAFAFEW Presentation June 2011
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The Risk Neutral Probability Distribution for the S&P 500 Index
The market price of a risky asset depends on the market's assessment of the probability
distribution for its future payoff together with the market's risk aversion. Neither of these
can be observed directly, but the two can be combined and expressed as the "Risk
Neutral Probability Density" (RND).
Breeden and Litzenberger (Journal of Business, 1978) showed how the risk neutral
probability distribution for ST, the value of the underlying asset on option expiration day
can be extracted from a set of market option prices.
Two major problems in constructing a complete risk neutral density from a set of market
option prices:
1. How to smooth and interpolate option prices to limit pricing noise and produce a
smooth density
2. How to extend the distribution to the tails beyond the range of traded strike prices.
But one terrific advantage is that unlike implied volatility, the risk neutral density is modelfree.
Figlewski
QWAFAFEW Presentation June 2011
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Extracting the Risk Neutral Density from Options Prices in Theory
The value of a call option is the expected value under the risk neutral distribution of its
payoff on the expiration date T, discounted back to the present.

C   e rT (ST  X ) f (ST )dST
X
Taking the partial derivative in (1) with respect to X,

C
 rT  

e
(
S

X
)
f
(
S
)
dS
T
T
T

X
X  X

C
 rT
  e  f ( ST )dST   e  rT 1  F ( X ) 
X
X
F(X )  e
Figlewski
rT
C
 1
X
QWAFAFEW Presentation June 2011
10
Extracting the Risk Neutral Density from Options Prices in Theory
We will use three options with sequential strike prices Xn-1, Xn, and Xn+1 in order to obtain
an approximation to F(X) centered on Xn.
F(Xn )  e
rT
 Cn 1  Cn 1 

  1
 X n 1  X n 1 
Taking the derivative with respect to X a second time yields the risk neutral density
function at X.
which is approximated by
2C
f (X )  e
X 2
rT
f ( X n )  e rT
Figlewski
Cn1  2 Cn  Cn1
(X )2
QWAFAFEW Presentation June 2011
11
Extracting the Risk Neutral Density, a more intuitive approach
Consider a call option that allows you to buy a share of some underlying stock for a price
of 101 one month from now. If the stock price in one month is above 101, you will
exercise the option. The market price for this option is 5.00 .
There is a second call option that allows you to buy 1 share of the same stock for a price
of 100 in one month. The market price for Option 2 is 5.70.
Stock price
in 1 month
90
95
100
101
105
110
Option 1
value
0
0
0
0
4
9
Option 2
value
0
0
0
1
5
10
For every stock price above 101, the second option pays 1 more than the first option.
The market values that extra 1 that option 2 pays if the stock price is above 101 as being
worth 5.70 – 5.00 = 0.70. So (roughly speaking) the market is saying the probability the
stock price will be above 101 is 70%.
Figlewski
QWAFAFEW Presentation June 2011
12
Strike
price
500
550
S&P 500 Index
600
Options Prices
700
750
Jan. 5, 2005
800
825
S&P close
850
900
1183.74
925
950
Option expiration
975
995
3/18/05
1005
1025
1050
1075
1100
1125
1150
1170
1175
At the 1180
money 1190
1200
1205
1210
1215
1220
1225
1250
1275
1300
1325
1350
1400
1500
Figlewski
Best bid
134.50
111.10
88.60
67.50
48.20
34.80
31.50
28.70
23.30
18.60
16.60
14.50
12.90
11.10
9.90
4.80
1.80
0.75
0.10
0.15
0.00
0.00
Best
offer
Calls
Average
price
136.50
113.10
90.60
69.50
50.20
36.80
33.50
30.70
25.30
20.20
18.20
16.10
14.50
12.70
10.90
5.30
2.30
1.00
0.60
0.50
0.50
0.50
135.500
112.100
89.600
68.500
49.200
35.800
32.500
29.700
24.300
19.400
17.400
15.300
13.700
11.900
10.400
5.050
2.050
0.875
0.350
0.325
0.250
0.250
Implied
volatility
0.118
0.140
0.143
0.141
0.135
0.131
0.129
0.128
0.126
0.123
0.123
0.121
0.122
0.120
0.119
0.117
0.114
0.115
0.116
0.132
0.157
0.213
Best
bid
Best
offer
0.00
0.00
0.00
0.00
0.00
0.10
0.00
0.00
0.00
0.20
0.50
0.85
1.30
1.50
2.05
3.00
4.50
6.80
10.10
15.60
21.70
23.50
25.60
30.30
35.60
38.40
41.40
44.60
47.70
51.40
70.70
92.80
116.40
140.80
165.50
-
QWAFAFEW Presentation June 2011
0.05
0.05
0.05
0.10
0.15
0.20
0.25
0.50
0.50
0.70
1.00
1.35
1.80
2.00
2.75
3.50
5.30
7.80
11.50
17.20
23.70
25.50
27.60
32.30
37.60
40.40
43.40
46.60
49.70
53.40
72.70
94.80
118.40
142.80
167.50
-
Puts
Average
price
0.025
0.025
0.025
0.050
0.075
0.150
0.125
0.250
0.250
0.450
0.750
1.100
1.550
1.750
2.400
3.250
4.900
7.300
10.800
16.400
22.700
24.500
26.600
31.300
36.600
39.400
42.400
45.600
48.700
52.400
71.700
93.800
117.400
141.800
166.500
-
Implied
volatility
0.593
0.530
0.473
0.392
0.356
0.331
0.301
0.300
0.253
0.248
0.241
0.230
0.222
0.217
0.208
0.193
0.183
0.172
0.161
0.152
0.146
0.144
0.142
0.141
0.139
0.139
0.138
0.138
0.136
0.137
0.139
0.147
0.161
0.179
0.198
-
13
Extracting the Risk Neutral Density from Options Prices in Practice
Obtaining a well-behaved risk neutral density from market option prices is a nontrivial
exercise. Here are the main steps we follow.
1. Use bid and ask quotes, rather than transactions prices. Eliminate options too far in
or out of the money.
2. Construct a smooth curve in strike-implied volatility space
3. Interpolate the IVs using a 4th degree smoothing spline
4. Fit the spline to the bid-ask spread
5. Use out of the money calls, out of the money puts, and a blend of the two at the
money
6. Convert the interpolated IV curve back to option prices and extract the middle portion
of the risk neutral density
7. Append tails to the Risk Neutral Density from a Generalized Extreme Value
distribution
Figlewski
QWAFAFEW Presentation June 2011
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Strike
price
500
550
S&P 500 Index
600
Options Prices
700
750
Jan. 5, 2005
800
825
S&P close
850
900
1183.74
925
950
Option expiration
975
995
3/18/05
1005
1025
1050
1075
1100
1125
1150
1170
1175
At the 1180
money 1190
1200
1205
1210
1215
1220
1225
1250
1275
1300
1325
1350
1400
1500
Figlewski
Best bid
134.50
111.10
88.60
67.50
48.20
34.80
31.50
28.70
23.30
18.60
16.60
14.50
12.90
11.10
9.90
4.80
1.80
0.75
0.10
0.15
0.00
0.00
Best
offer
Calls
Average
price
136.50
113.10
90.60
69.50
50.20
36.80
33.50
30.70
25.30
20.20
18.20
16.10
14.50
12.70
10.90
5.30
2.30
1.00
0.60
0.50
0.50
0.50
135.500
112.100
89.600
68.500
49.200
35.800
32.500
29.700
24.300
19.400
17.400
15.300
13.700
11.900
10.400
5.050
2.050
0.875
0.350
0.325
0.250
0.250
Implied
volatility
0.118
0.140
0.143
0.141
0.135
0.131
0.129
0.128
0.126
0.123
0.123
0.121
0.122
0.120
0.119
0.117
0.114
0.115
0.116
0.132
0.157
0.213
Best
bid
Best
offer
0.00
0.00
0.00
0.00
0.00
0.10
0.00
0.00
0.00
0.20
0.50
0.85
1.30
1.50
2.05
3.00
4.50
6.80
10.10
15.60
21.70
23.50
25.60
30.30
35.60
38.40
41.40
44.60
47.70
51.40
70.70
92.80
116.40
140.80
165.50
-
QWAFAFEW Presentation June 2011
0.05
0.05
0.05
0.10
0.15
0.20
0.25
0.50
0.50
0.70
1.00
1.35
1.80
2.00
2.75
3.50
5.30
7.80
11.50
17.20
23.70
25.50
27.60
32.30
37.60
40.40
43.40
46.60
49.70
53.40
72.70
94.80
118.40
142.80
167.50
-
Puts
Average
price
0.025
0.025
0.025
0.050
0.075
0.150
0.125
0.250
0.250
0.450
0.750
1.100
1.550
1.750
2.400
3.250
4.900
7.300
10.800
16.400
22.700
24.500
26.600
31.300
36.600
39.400
42.400
45.600
48.700
52.400
71.700
93.800
117.400
141.800
166.500
-
Implied
volatility
0.593
0.530
0.473
0.392
0.356
0.331
0.301
0.300
0.253
0.248
0.241
0.230
0.222
0.217
0.208
0.193
0.183
0.172
0.161
0.152
0.146
0.144
0.142
0.141
0.139
0.139
0.138
0.138
0.136
0.137
0.139
0.147
0.161
0.179
0.198
-
Not used
Blended
15
Figure 2: Risk Neutral Density from Raw Options Prices
0.025
Probability
0.02
0.015
0.01
0.005
0
-0.005
800
900
1000
1100
1200
1300
1400
S&P 500 Index
Density from put prices
Figlewski
Density from call prices
QWAFAFEW Presentation June 2011
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Figure 3: Market Option Prices with Cubic Spline Interpolation
180
160
140
Option price
120
100
80
60
40
20
0
500
600
700
800
900
1000
1100
1200
1300
1400
1500
S&P 500 Index
Spline interpolated call price
Figlewski
Spline interpolated put price
QWAFAFEW Presentation June 2011
Market call prices
Market put prices
17
Figure 4: Densities from Option Prices with Cubic Spline Interpolation
0.05
0.04
0.03
Density
0.02
0.01
0.00
-0.01
-0.02
-0.03
-0.04
800
900
1000
1100
1200
1300
1400
S&P 500 Index
Density from interpolated put prices
Figlewski
Density from interpolated call prices
QWAFAFEW Presentation June 2011
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Figure 5: Implied Volatilities with Spline and
4th degree Polynomial Interpolation
0.70
Implied volatility
0.60
0.50
0.40
0.30
0.20
0.10
0.00
500
600
700
800
900
1000
1100
1200
1300
1400
1500
S&P 500 Index
4th degree polynomial call IV
Figlewski
4th degree polynomial put IV
Call IVs
QWAFAFEW Presentation June 2011
Put IVs
Call spline IVs
Put spline IVs
19
Figure 6: Densities from Interpolated Implied Volatilities
0.012
0.010
0.008
Density
0.006
0.004
0.002
0.000
-0.002
-0.004
800
900
1000
1100
1200
1300
1400
S&P 500 Index
Calls w. 4th deg poly
Figlewski
Puts w. 4th deg poly
Calls w. spline IVs
QWAFAFEW Presentation June 2011
Puts w. spline IVs
20
4th Degree Smoothing Spline with Bid-Ask Spread Adjustment
Figlewski
QWAFAFEW Presentation June 2011
21
Empirical Risk Neutral Density January 5, 2005
with IV Interpolation using 4th Degree Polynomial
0.012
0.010
Density
0.008
0.006
0.004
0.002
0.000
800
900
1000
1100
1200
1300
1400
S&P 500 Index
Figlewski
QWAFAFEW Presentation June 2011
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The Generalized Extreme Value (GEV) Distribution
We complete the risk neutral density by adding tails from a GEV density.
The GEV distribution has three parameters:
μ = location
σ = scale
ξ = tail shape
FGEV (x ; , , )


 
 x   
exp  1   







1/  




If ξ > 0, the GEV is a Fréchet distribution, that has a heavier tail than the normal
distribution;
ξ = 0, the GEV is a Gumbel distribution with tails like the normal;
ξ < 0, the density is a Weibull, with a finite tail that does not extend to infinity.
Figlewski
QWAFAFEW Presentation June 2011
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Figure 9: Risk Neutral Density and Fitted GEV Tail Functions
0.012
0.010
Density
0.008
0.006
0.004
95%
92%
0.002
0.000
800
2%
900
5%
95%
98%
1000
1100
1200
1300
1400
S&P 500 Index
Empirical RND
Figlewski
Left tail GEV function
Right tail GEV function
QWAFAFEW Presentation June 2011
Attachment points
24
Figure 10: Full Estimated Risk Neutral Density Function for Jan. 5, 2005
0.012
0.010
Density
0.008
0.006
0.004
95%
0.002
98%
0.000
800
900
1000
1100
1200
1300
1400
S&P 500 Index
Empirical RND
Figlewski
Left tail GEV function
QWAFAFEW Presentation June 2011
Right tail GEV function
25
The Risk Neutral Density in a Black-Scholes World
An Aside: The RND under Black-Scholes Assumptions
Risk neutral valuation was first developed in the context of the Black-Scholes model. The
returns process is modeled as:
dS
  dt   dz
S
The empirical distribution of the date T asset price is lognormal.
Risk neutralization simply replaces the empirical drift μ with the riskless rate r:
dS
 r dt   dz
S
The risk neutral distribution is still lognormal with the same volatility. It is simply shifted to
the left, and only the mean changes.
Is the curve on the previous slide lognormal? No! It can't be, because it is skewed
to the left while the lognormal is skewed to the right.
Figlewski
QWAFAFEW Presentation June 2011
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Figure 2: Initial Impact of the Fed Announcement on the Risk Neutral Density, Dec. 11, 2007
Figlewski
QWAFAFEW Presentation June 2011
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Figure 3: Re-equilibration of the Stock Market after the Fed Announcement, Dec. 11, 2007
Figlewski
QWAFAFEW Presentation June 2011
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Information Flow and the Resolution of Uncertainty
New information entering the market will produce a change in the market price. The size
of the price move measures the flow of new information during a given time interval.
We estimate the rate of information flow over different periods of the day by the standard
deviation of interval price changes across announcement dates.
The RND mean is the market's expectation for the level of the S&P index on option
expiration day, and the RND variance is the market's uncertainty about S(T), given
S(t). The risk neutral variance measures how much new information the market
anticipates will arrive over the time remaining to expiration.
If there are T days to expiration, on average 1/T of the uncertainty should be resolved
each day and the RND variance is should fall by the fraction 1/T per day. Research
shows that this relationship holds very closely in the data.
We look at the change in RND variance to measure how much the market's uncertainty is
resolved by the information contained in the Fed's announcement
Figlewski
QWAFAFEW Presentation June 2011
29
Excerpt from Table 2
Levels and Changes of Key Variables on Fed Announcement Days
mean
std dev
5.70
17.93
Overnight:
Close date t-1
to 10:00 AM
date t
2.12
8.46
mean
std dev
4.02
20.07
0.52
9.31
2.70
4.95
-0.66
7.47
1.45
12.23
mean
std dev
-4.97
9.10
-3.51
5.50
-1.16
2.18
-0.43
3.04
0.13
4.65
18.37
18.36
-7.50
3.94
4.13
-4.00
3.36
3.12
-1.29
1.99
1.93
-1.39
9.07
9.17
-0.82
Market Down after Announcement
change in S&P forward
-6.97
change in RND mean
-10.32
change in RND variance
-2.45
0.30
-3.08
-3.03
2.21
2.28
-1.02
-3.25
-3.25
0.52
-6.23
-6.27
1.08
Full day:
Close date t-1
to Close date t
Change in
Forward S&P
Change in
RND mean
RND variance
Relative change
Market Up after Announcement
change in S&P forward
change in RND mean
change in RND variance
Figlewski
PreAnnouncement
announcement:
impact:
10:00 AM to 2:14 2:14 PM to 2:24
PM
PM
2.79
-0.63
4.96
7.61
QWAFAFEW Presentation June 2011
Reequilibration:
2:24 PM to
Close date t
1.42
12.03
30
Table 6: Volatility of the Forward S&P Index and the RND Mean during the Re-Equilibration Period, by Sub-Intervals
Impact
From
To
Full ReEquil
Re-Equilibration
2:14 P.M.
2:24 P.M.
2:24 P.M.
2:36 P.M.
2:36 P.M.
2:48 P.M.
2:48 P.M.
3:00 P.M.
3:00 P.M.
3:12 P.M.
3:12 P.M.
3:24 P.M.
3:24 P.M.
3:36 P.M.
3:36 P.M.
3:48 P.M.
3:48 P.M.
4:00 P.M.
2:24 P.M.
4:00 P.M.
Announcement Days
Std dev of change over
full interval
S&P forward
RND mean
7.61
7.47
3.72
3.83
5.69
5.88
4.89
5.21
2.84
2.89
5.51
5.57
3.01
3.04
3.94
4.22
8.61
8.44
12.03
12.23
Interval std dev relative
to full trading day
S&P forward
RND mean
0.47
0.46
0.23
0.24
0.35
0.36
0.30
0.32
0.18
0.18
0.34
0.34
0.19
0.19
0.25
0.26
0.54
0.52
0.75
0.76
Std dev of 1-minute
changes in interval
S&P forward
RND mean
1.84
2.01
1.07
1.18
1.10
1.31
0.88
1.07
0.77
1.06
0.77
0.96
0.72
1.01
0.72
0.99
0.75
1.03
0.91
1.14
Interval 1-minute std
dev relative to full day
S&P forward
RND mean
2.76
2.11
1.61
1.24
1.65
1.38
1.32
1.13
1.15
1.12
1.15
1.01
1.09
1.07
1.07
1.04
1.12
1.09
1.37
1.20
Autocorrelation of 1minute changes
S&P forward
RND mean
0.17
0.10
0.14
0.08
0.03
-0.01
-0.02
-0.07
0.03
-0.03
0.05
-0.06
-0.02
-0.13
0.12
-0.05
0.12
-0.01
0.06
-0.06
Non-Announcement Days
Std dev of change over
full interval
S&P forward
RND mean
2.25
2.31
2.54
2.42
1.86
2.14
1.96
2.12
2.39
2.62
1.22
1.07
3.69
4.53
2.55
2.29
2.35
2.96
7.21
7.13
Interval std dev relative
to full trading day
S&P forward
RND mean
0.20
0.21
0.23
0.22
0.17
0.19
0.18
0.19
0.21
0.23
0.11
0.10
0.33
0.40
0.23
0.20
0.21
0.26
0.64
0.64
Std dev of 1-minute
changes in interval
S&P forward
RND mean
0.55
1.13
0.58
1.24
0.52
0.90
0.48
0.91
0.55
1.29
0.54
1.05
0.57
1.30
0.59
1.03
0.52
1.06
0.56
1.15
Interval 1-minute std
dev relative to full day
S&P forward
RND mean
1.07
0.97
1.13
1.07
1.00
0.78
0.94
0.79
1.06
1.12
1.05
0.91
1.10
1.12
1.14
0.89
1.00
0.91
1.08
1.00
Autocorrelation of 1minute changes
S&P forward
RND mean
0.08
-0.22
0.00
-0.18
0.06
-0.12
0.03
-0.19
0.12
-0.26
0.03
-0.25
0.06
-0.16
-0.05
-0.14
0.20
-0.06
0.04
-0.24
Figlewski
QWAFAFEW Presentation June 2011
31
Variance Diminishes Gradually During Re-equilibration
To explore the time decay of volatility further, we regress the log of the absolute price
change in each minute relative to the volatility within the impact period, as a function of
the number of minutes since the end of the impact period.
 is a minute within date t, with 0 representing 2:24 P.M.
Ft,impact is the standard deviation of 1-minute changes in the index forward during the
impact period on date t.
t-statistics are shown in parentheses. NOBS = 2649
(15)
Figlewski
log( | Ft,τ | / Ft,impact)
=
-0.562
(-4.84)
+
-0.276 log(  - 0)
(-8.82)
QWAFAFEW Presentation June 2011
R2 = 0.027
32
Variance Diminishes Gradually During Re-equilibration
Running this regression with the absolute change in the RND mean gives
log( | RNDmeant, | / RNDt,impact)
=
-0.749 + -0.183 log(  - 0)
(-7.99)
(-7.23)
R2 = 0.019
The RND variance also shrinks consistently during the re-equilibration period
log( RNDvart, / RNDvart,impact)
Figlewski
=
-0.012 + -0.0061 log(  - 0)
(-2.92)
(-5.39)
QWAFAFEW Presentation June 2011
R2 = 0.010
33
The Evolution of the RND on December 11, 2007
Movie Time
On December 11, 2007 the Federal Reserve announced that it was lowering its
interest rate target by 25 basis points. Normally a cut in the Fed funds rate
causes the stock market to rise, but this time the market was clearly
disappointed that the cut was not larger. Although the market had drifted
higher during the day before the announcement, the S&P index fell 27 points
(-1.74%) in the next 10 minutes, and a further 18 points by the end of the day.
This video shows how the Risk Neutral probability density behaved minute by
minute during the course of that day. The density is for the level of the stock
market on option expiration day, March 21, 2008. The vertical green line
shows the current forward level of the S&P index in the market at the same
time.
Figlewski
QWAFAFEW Presentation June 2011
34
Table 8: Autocorrelation in Intraday Index Option Mid-Quote Changes on Non-Announcement Days
4:00 PM
PreAnnouncement
10:00 AM
2:14 PM
2:14 PM
2:24 PM
2:24 PM
2:36 PM
2:36 PM
2:48 PM
2:48 PM
3:00 PM
3:00 PM
3:12 PM
3:12 PM
3:24 PM
3:24 PM
3:36 PM
3:36 PM
3:48 PM
3:48 PM
4:00 PM
Full Reequilibration
2:24 PM
4:00 PM
-0.347
-0.419
-0.354
-0.299
-0.280
-0.296
-0.271
-0.257
-0.331
-0.431
-0.544
-0.675
-0.362
-0.436
-0.375
-0.309
-0.292
-0.312
-0.279
-0.270
-0.349
-0.445
-0.548
-0.664
-0.386
-0.443
-0.372
-0.357
-0.331
-0.343
-0.327
-0.314
-0.390
-0.485
-0.597
-0.731
-0.323
-0.353
-0.295
-0.293
-0.279
-0.305
-0.298
-0.292
-0.382
-0.495
-0.612
-0.735
-0.394
-0.465
-0.380
-0.367
-0.314
-0.339
-0.311
-0.309
-0.421
-0.495
-0.580
-0.767
-0.298
-0.365
-0.286
-0.267
-0.225
-0.248
-0.232
-0.211
-0.285
-0.408
-0.613
-0.807
-0.313
-0.386
-0.333
-0.259
-0.225
-0.242
-0.242
-0.243
-0.275
-0.417
-0.591
-0.762
-0.248
-0.290
-0.214
-0.227
-0.221
-0.227
-0.198
-0.198
-0.209
-0.365
-0.570
-0.773
-0.316
-0.372
-0.311
-0.290
-0.257
-0.282
-0.254
-0.251
-0.304
-0.408
-0.596
-0.714
-0.300
-0.408
-0.303
-0.232
-0.212
-0.219
-0.199
-0.204
-0.263
-0.428
-0.536
-0.768
-0.314
-0.370
-0.305
-0.261
-0.258
-0.252
-0.281
-0.281
-0.373
-0.519
-0.570
-0.802
-0.307
-0.372
-0.297
-0.268
-0.242
-0.260
-0.247
-0.244
-0.303
-0.419
-0.544
-0.717
-0.287
-0.519
-0.428
-0.335
-0.279
-0.286
-0.264
-0.229
-0.236
-0.248
-0.251
-0.267
-0.302
-0.525
-0.447
-0.361
-0.292
-0.303
-0.278
-0.246
-0.246
-0.259
-0.268
-0.278
-0.330
-0.570
-0.391
-0.332
-0.308
-0.340
-0.335
-0.286
-0.303
-0.335
-0.306
-0.325
-0.286
-0.530
-0.420
-0.290
-0.289
-0.288
-0.274
-0.243
-0.246
-0.261
-0.238
-0.281
-0.318
-0.466
-0.421
-0.354
-0.300
-0.318
-0.290
-0.248
-0.287
-0.312
-0.288
-0.314
-0.231
-0.553
-0.332
-0.273
-0.230
-0.218
-0.214
-0.158
-0.176
-0.213
-0.190
-0.217
-0.246
-0.594
-0.420
-0.257
-0.232
-0.227
-0.211
-0.208
-0.213
-0.223
-0.220
-0.216
-0.198
-0.473
-0.297
-0.226
-0.203
-0.231
-0.190
-0.127
-0.149
-0.146
-0.112
-0.180
-0.274
-0.535
-0.404
-0.267
-0.253
-0.276
-0.257
-0.228
-0.250
-0.251
-0.252
-0.268
-0.211
-0.568
-0.352
-0.226
-0.174
-0.199
-0.196
-0.162
-0.164
-0.200
-0.183
-0.192
-0.266
-0.593
-0.405
-0.302
-0.239
-0.249
-0.266
-0.221
-0.236
-0.232
-0.240
-0.239
-0.249
-0.545
-0.379
-0.272
-0.234
-0.246
-0.230
-0.192
-0.209
-0.224
-0.209
-0.232
Trading
Day
10:00 AM
Impact
Re-equilibration subintervals
Calls
All
0-5%
5-15%
15-25%
25-35%
35-45%
45-55%
55-65%
65-75%
75-85%
85-95%
95-100%
Puts
All
0-5%
5-15%
15-25%
25-35%
35-45%
45-55%
55-65%
65-75%
75-85%
85-95%
95-100%
Figlewski
QWAFAFEW Presentation June 2011
35
Simulation of Option Quotes when Stock Price Follows a Diffusion and Tick Sizes are Different
Figlewski
QWAFAFEW Presentation June 2011
36
Conclusions about the Impact of the Fed Announcement
Is the market's response to the announcement unbiased? YES, IT APPEARS TO BE
Does the information in the announcement all enter the market at the moment of the public
announcement?
• Information leakage beforehand? MAYBE SOME LEAKAGE
• Sluggish adjustment or overshooting afterward?
THERE IS A LOT OF FURTHER ADJUSTMENT DURING RE-EQUILIBRATION
AFTER THE ANNOUNCEMENT, BUT NO APPARENT BIAS
The mean of the Risk Neutral Density (RND) and the forward value of the spot S&P index
are tied together by arbitrage. Do they behave identically on announcement day? Is one a
better prediction of the announcement than the other?
NOT QUITE IDENTICAL BUT NEITHER SHOWS PREDICTIVE ABILITY
The RND reveals the market's risk neutral expectation for the future value of the index and
also the variance around that expectation, a direct measure of uncertainty.
• How much uncertainty does a Fed announcement resolve on average?
ABOUT THE SAME AS ON 5 ORDINARY DAYS
•
Does it matter if the announcement is viewed as positive or negative by the market?
YES. MUCH LESS UNCERTAINTY IS RESOLVED WHEN THE MARKET FALLS
AFTER THE ANNOUNCEMENT
Figlewski
QWAFAFEW Presentation June 2011
37
Conclusions about the Impact of the Fed Announcement, p.2
Information flow within a time interval can be proxied by the standard deviation of price
change over the interval or by volatility within the interval. What is information flow like
over an announcement day?
• How much information does the announcement itself convey?
STANDARD DEVIATION OVER 10 MINUTES ~ 30-40% OF FULL DAY PRICE CHANGE
•
How much of the day's price change occurs before the market opens?
SURPRISINGLY, ABOUT HALF
•
How much during trading hours before the announcement?
IN TOTAL ABOUT 1/3 LESS THAN IN THE 10 MINUTES OF IMPACT
•
How much in the re-equilibration period following the initial impact of the
announcement?
A LOT! 60 – 70% OF FULL DAY MOVE
•
Is there a time pattern of diminishing volatility during re-equilibration?
YES, AND VOLATILITY DROPS MORE FOR THE FORWARD THAN THE RND
•
How does an announcement day compare to a regular day?
THE TIME PATTERN IS MUCH DIFFERENT
Figlewski
QWAFAFEW Presentation June 2011
38