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What Goes into Risk-Neutral
Volatility? Empirical Estimates
of Risk and Subjective
Risk Preferences
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volatility input. As an alternative to statistical
estimation, market makers and many others
rely on implied volatilities extracted from
option market prices.
Implied volatility (IV) makes a theoretical
model consistent with pricing in the market, but
one has to specify which model “the market” is
using. Black–Scholes IV is nearly always chosen,
even though an enormous amount of research
has shown that a constant volatility lognormal
diffusion for returns is greatly oversimplified.1
Constraining IV to be equal across strike prices,
as the BS model requires, fails immediately in
the face of the ubiquitous volatility “smile” (or
“smirk” or “skew,” depending on its shape).
Equity market volatility changes over time;
returns are not really lognormal, especially in
the tails; and there appear to be nondiffusive
jumps in both returns and volatility.
Time variation in the returns distribution should matter to investors, beyond
the obviously greater difficulty of hitting a
moving target, because stochastic volatility is
unspanned (meaning it can’t be fully hedged
by other securities). Once more complex
returns processes are allowed, numerous possibilities arise for unspanned stochastic factors
that are not in the BS model but that can
affect actual options and may be priced in
the market. The resulting risk premia may
themselves vary stochastically over time.
Implied volatility is a risk-neutral value,
in which all risk premia in the market have
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he Black–Scholes (BS) model
showed that option value depends
heavily on the volatility of the
underlying stock, which is assumed
to follow a constant-volatility logarithmic
diffusion. With this process, a single parameter σ captures “risk” in all its manifestations. These include (1) instantaneous
volatility, which determines delta hedging
performance; (2) the standard deviation of
the stock price around its mean on option
expiration day, which is what implied volatility extracted from pricing model measures;
(3) the stock’s probable trading range over the
option’s lifetime, which determines whether a
speculator can profit from a short-term trade
and how much financing risk an arbitrage
trade is exposed to; (4) the frequency and
size of tail events, which risk managers worry
about most; and more. Different types of
investors care about some of these risks more
than others. For example, a market maker
hedging his trading book worries about very
short-run f luctuations and price jumps that
are of little consequence to a buy-and-hold
investor. But under Black–Scholes, they all
care about the same volatility parameter.
Volatility in the BS model is a known
parameter, but real world volatility varies
randomly over time and is hard to predict
accurately. And no matter how volatility is
estimated, market prices invariably differ
from model values computed with that
A
is a professor at the
New York University
Stern School of Business
in New York, NY.
[email protected]
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address the question in the title of this article: “What
Goes Into Risk-Neutral Volatility?”
The market bases its volatility forecasts and
required risk premia on current and historical data. But
these estimates apply to the stock’s behavior over the
future lifetime of the options involved. Thus, we are
interested in both what information gleaned from historical data goes into the market’s RND and how that
RND relates to the volatility actually realized in the
future. We have divided the investigation into two parts.
This article focuses on what past and current information appears to enter today’s volatility estimates and risk
preferences. A second article will consider risk-neutral
volatility as a forecast, addressing issues like the following: How accurate is it? How does it compare with
other estimators, such as generalized autoregressive conditional heteroscedasticity (GARCH) and the CBOE
Volatility Index (VIX)? How does the market’s forecasting horizon compare to the option’s remaining life?
In this article, we address several broad questions.
Which return- and volatility-related factors are most
important to investors in forecasting the empirical probability density embedded in the RND? What factors influence the process of risk neutralization? Do the answers to
these questions differ for long maturity versus short maturity options, or in different market conditions? Since risk
premia are significant components of total return, there
is considerable value in understanding risk neutralization
just to be able to estimate how much extra return the
market requires for bearing different sorts of risks; or on
the flip side of the same question, how much would the
market pay to hedge those risks? For example, Bollerslev
and Todorov [2011] argued that much of the volatility risk
premium during the fall of 2008 was compensation for lefttail “crash risk,” whereas Kozhan et al. [2010] found that
about half of the implied volatility “skew” reflects a risk
premium for skewness (i.e., the third moment of returns).
We use the Breeden and Litzenberger [1978] technology to extract RNDs from daily S&P 500 Index
option prices. The primary objective is not to build a
formal model of option risk premia, but to establish a
set of empirical stylized facts that any such model will
need to be consistent with. Two significant problems,
which we have addressed in a technical appendix (available online), are how to smooth and interpolate market
option prices to limit the effect of pricing noise and how
to extend the distribution into the tails beyond the range
of traded strike prices. An enormous advantage of the
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been impounded into a modified probability distribution
for returns, known as the risk-neutral density (RND).
In an important paper, Harrison and Kreps [1979]
proved that in a world free of profitable arbitrage opportunities, there will always be an RND that combines
investors’ risk preferences with their objective forecast
of the probability density for the stock price at option
expiration. The RND is often called the “Q measure,”
while the expected true probability distribution is the
“P measure.” Implied volatility and other parameters
computed under the Q measure will rarely be equal to
their real world expected values under the P measure.
Violations of the BS assumptions, such as fatter than
lognormal tails, cause BS implied volatilities to exhibit
the customary but theoretically anomalous smile shape.
Researchers on volatility risk, for example, Carr
and Wu [2008], have shown that the market places a
negative risk premium on volatility risk. The market dislikes volatility, so expected returns are lower and prices
are bid up for options and other securities that can hedge
volatility risk, because they rise in price with higher
volatility. They will be priced (under the Q measure)
as if volatility were expected to be higher than investors
really think.
IVs that differ across strikes are inconsistent
with BS model assumptions. But when a cross-section
of option prices is available, the full RND can be
extracted without specifying a pricing model. Breeden
and Litzenberger [1978] showed how this could be done
very simply, given prices for options with exercise prices
that span a broad range of future stock prices. Option
prices are now being used to extract not just BS implied
volatilities, but the entire risk-neutral probability density
for the price of the underlying on expiration day.2 The
RND ref lects the current state of supply and demand
for options with the same maturity and different exercise
prices. It will impound the market’s beliefs about the
true probabilities of all volatility-related properties of
the underlying stock’s returns, such as time-variation in
instantaneous volatility, fat tails, and jumps; and it will
also ref lect risk premia on those factors and any others
that investors care about.
Risk-neutral volatility is the standard deviation
under this density. We analyze these densities to explore
how the RND, and by extension option pricing in the
market, is affected by various volatility-related properties of the true returns distribution and by factors that
may ref lect risk preferences. In so doing, we begin to
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The next section explores the model’s robustness
by fitting it on data subsets, according to option maturity
and time period. The final section offers some concluding comments.
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The variable we wish to explain is the standard
deviation of the S&P 500 Index on option expiration
day under the risk-neutral probability density. The
RND aggregates the individual risk-neutralized subjective probability beliefs within the investor population.
The resulting density is not a simple transformation of
the true but unobservable distribution of realized returns
on the underlying asset, as in BS, nor should we expect
it to obey any particular probability law. Our model-free
fitting procedure allows maximum f lexibility to ref lect
option market pricing. To compare RNDs from options
with different maturities, the RND standard deviation
is converted into an annualized percentage value. This
is what we call risk-neutral volatility.
The basic strategy of nonparametric RND extraction is well known by now, so to save space, the specifics
of our implementation are sketched out in the Appendix,
which provides more details about this and other technical aspects of the empirical work.5
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RND is that it does not require us to know the market’s
option pricing model. It is model-free.
The VIX, which measures implied volatility for
the overall stock market portfolio, is roughly based on
the same model-free methodology that we employ. 3
However, the VIX focuses on a single 30-day maturity, whereas maturities in our sample range from 14 to
199 days. The VIX calculation also ignores several issues
of data handling that we try to deal with more carefully,
including extending the tails of the density beyond the
range of available strike prices.4 Still, it is worth noting
that over our 15-year sample period, the VIX averaged 22.1% while realized volatility was only 20.6%
on average. This implies a volatility risk premium on
index options equivalent to about 1.5 volatility points.
Before moving on, we must consider how Ross’s
[2015] recent theoretical breakthrough in this area
applies to this research. It has long been thought that
objective probability estimates and risk premia cannot
be separately deduced from option prices without further
information about either the market’s risk preferences or
the true return-generating process. Ross proves a deep
theoretical result showing this is not strictly true. But
the assumptions required are strong enough that the
immediate problem we are addressing remains, which
is how to understand better how objective measures of
different aspects of price risk and behavioral proxies for
market “sentiment” get into real world option prices.
The next section defines the variables we conjecture could inf luence investors’ demands for options. A
practical issue rarely considered is how far back in time
investors look in calculating volatility and other statistics
from past returns. In this research, we have investigated
different lag definitions as well as different cutoffs to
define tail events and assume investors pay most attention to those that contain the most information about
future volatility. Due to space limitations, this material is
relegated to the online appendix.
The following section first looks at univariate relationships in the form of simple correlations between the
volatility measures and the explanatory variables and
then presents a grand regression with all of the variables together. Among the interesting results from this
exercise is strong evidence that several distinct aspects
of volatility seem to be independently important and
variables that were hypothesized to be correlated with
investor risk attitudes do in fact appear to have significant explanatory power for RND volatility.
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RND and Realized Volatility
Option prices. We take the midpoints of the closing
bid and ask quotes for all traded S&P 500 Index options,
downloaded from OptionMetrics that satisfied the following criteria:
• Observation dates and option expirations between
January 4, 1996, and April 29, 2011.
• Option bid prices ≥ 0.50. The average index level
in the sample is 1151.
• Maturities 14 < T < 199 calendar days, allowing
about three RND maturities per day.
Interest rates and dividend yield. Riskless interest rates,
interpolated to match the option maturity, and S&P 500
dividend yields were obtained from OptionMetrics.
Realized volatility to expiration. An option’s price
should embed the market’s best forecast of the volatility
that will be realized from the present through option
expiration, but this is not the volatility under the RND
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(3)
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Like the absolute return for the day, this variable
was included in our initial exploration, but the results
seemed odd. In regressions with both the absolute return
and the trading range, trading range comes in with a
highly significant positive coefficient, but the coefficient on absolute return was consistently about the same
size but strongly negative. Subsequently, market makers
explained that managing their risk on a day when the
market simply moves strongly up or down is much easier
than on a day when there is high uncertainty and prices
f luctuate over a broad range without finding a new
equilibrium much different from the initial level. They
suggested using the following instead.
Date t trading range minus absolute return. The range
minus the absolute change produced a single variable
with a strongly positive and significant coefficient.
Date t left- and right-tail returns. One of the most
apparent differences between the empirical return
distribution and the lognormal is that large absolute
returns—tail events—are more common than the lognormal allows for, especially on the downside, so the
occurrence and size of a tail event might be especially
important to investors. But what constitutes a tail event?
Rather than making ad hoc assumptions, as is usually
done, we treated it as an empirical question, which we
explored by comparing the explanatory power of alternative tail definitions, both in univariate regressions of
RND volatility on a tail variable and also along with
our other explanatory variables in multivariate specifications. Based on this investigation, as detailed in the
online technical appendix, we classify the return on
date t as a left- (right-) tail event if it falls in the 2% left(right-) tail (relative to a normal density). The cutoffs
for tail returns are set at ±2.054 times the previous day’s
one-day GARCH volatility forecast. If the return is not
in the tail, the value of the tail variable is set to zero.
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This equation assumes 252 trading days in a year and
adopts the standard approach of treating the daily mean
return as equal to zero.6 Realized volatility is expressed
in annualized percent, for example, 20% is 20.0.
We consider explanatory variables in three sets.
The first are variables related to volatility and extreme
price movements that do not require analyzing historical
data. In the second set are volatility-related variables calculated from historical data. The third set are variables
chosen to ref lect behavioral factors that we feel might
affect the risk-neutralization process.
⎛
S − St ,LOW ⎞
Range(t ) = 100 × log 1 + t ,HI
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Date t trading range. The intraday trading range
contains a lot of volatility information.8 Also the cost
for market makers to hedge their “Greek letter risks”
depends on the range over which the stock price travels
in a day. For consistency with the other variable definitions, the date t range is converted into a logarithmic
percentage in the following way:
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when there is a volatility risk premium. We explore
the real world differences by regressing RND volatility
and future realized volatility on explanatory variables
that we hypothesize are important to investors. Realized
volatility for date T maturity is defined as
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Date t Volatility Variables
σ t zt , zt ~ N (0,1)
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Date t return. The date t return is defined as 100 ×
log(St /St−1). Volatility, both empirical and implied, has
a strong tendency to go up following a negative return.
Models with separate stochastic factors for returns
and volatility typically find correlation of about −0.7
between them.
Absolute value of date t return. A large return shock
of either sign is consistent with high current volatility,
but this variable was replaced with a different but related
one, as described in the following.
GARCH model forecast. We adopt the Glosten,
Jagannathan, and Runkle [GJR, 1993] specification:
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where rt and σ t2 are the date t log return and conditional
variance, and zt is a standard normal i.i.d. random variable. The return equation contains no constant term,
consistent with Equation 1. The variance σ t2 is a function of the date t − 1 conditional variance and squared
return shock, plus a term that increases σ t2 when rt−1 is
negative. GJR-GARCH model forecasts were provided
by the NYU Volatility Institute’s VLAB.7
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Risk Attitude Variables
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The variables discussed so far are all tied to statistical characteristics of returns. We now turn to the
risk-neutralization process. We have selected a few factors that are potentially promising, but there are many
others that it might make sense to explore.
University of Michigan Index of Consumer Sentiment.
This is a classic survey of consumers’ confidence level.
It is not specifically tied to finance, but it is widely
known and followed by investors. The daily value is set
to the monthly sentiment index value for every day of
the month.
Baker–Wurgler Index of Investor Sentiment. Baker and
Wurgler [2006] devised a measure of investor confidence based on factors such as the discount on closedend mutual funds, the first day returns on initial public
offering (IPOs), etc.9 With all monthly series, the value
is repeated for every day of the month.
The price/earnings for the S&P 500 Index portfolio.
The ratios of price to earnings (P/E) will be higher
when investors anticipate favorable conditions and strong
earnings growth in the future. We hypothesize that optimistic investors also accept lower volatility risk premia.
The monthly P/E for the S&P Index was downloaded
from Robert Shiller’s “Irrational Exuberance” website.10
The Baa–Aaa corporate bond yield spread. This daily
credit spread series is the average yield spread between
corporate bonds rated Baa by Moody’s and AAA bonds.
It measures the default-risk premium on high-grade
corporate debt. The series was downloaded from the
St. Louis Fed’s FRED (Federal Reserve Economic
Data) database.11
GARCH model error over a recent period equal to the
option’s remaining lifespan. A positive (negative) value
EP
252 t − 65 2
∑ rτ .
65 t −1
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σ Hist =
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Past realized volatility and similar factors are computed from historical returns, but it is not known a priori
what sampling horizon the market focuses on. As with
tail definitions, we used the behavior of the RND as a
window onto this issue. Most of the variables showed
similar performance over a fairly broad range of sample
lengths. In the end, we selected estimation windows that
gave reasonable performance in the various exploratory
regressions and that “made sense” in terms of the behavior
one might expect from an intelligent investor. Details are
given in the Appendix. The final choices are as follows.
Left and right-tail events. The empirical distribution’s fat tails can be due to more frequent tail returns
than under the normal and/or unexpectedly large tail outcomes conditional on falling in the tail. As mentioned
previously, 2% and 98% cutoffs define a tail event. A
historical window of two years (500 trading days) was
chosen to gauge frequency for both tails.
Return and absolute return from date t − n to t − 1.
Recent index return is defined as 100 × log(St−1/St−n),
but the “right” choice of n was not clear cut. We selected
the return over just the last five trading days because it
had good explanatory power and less chance of proxying
for some other factor.
Historical volatility. Historical volatility is computed
from the squared log returns over the last 65 trading
days, treating the expected drift as zero. Specifically,
volatility uncertainty may require a larger risk premium
when their forecasting model has been performing badly,
so we incorporate two variables measuring historical
accuracy of the GARCH model. A positive coefficient
on the average GARCH error from date t − n to t − 1
means that RND volatility is higher when the GARCH
model has been underpredicting realized variance,
whereas a positive GARCH RMSE coefficient increases
risk-neutral volatility following large model errors on
either side. The historical sample length over which
investors are assumed to gauge the GARCH model’s
accuracy is set equal to the option’s remaining lifetime.
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Historical Volatility Variables
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Average daily range t − n to t − 1. The daily range
is a proxy for potential hedging costs. Each day’s range
is computed as in Equation 3. In the online technical
appendix, we find that averaging the daily range over a
window the same size as the option’s remaining lifetime
performs distinctly better than any fixed horizon.
Range over t − n to t − 1. If the underlying stock
trades over a wide range during an option’s lifetime,
an investor may make a short-term profit on an option
early even if it ultimately expires out of the money. The
range the index has traversed in the most recent 25 days
is computed as a logarithmic percentage relative to the
average price in the interval.
GARCH error and GARCH RMSE (root mean
squared error). Investors who dislike volatility and also
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EXHIBIT 1
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Sample Correlations
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corresponds to average under- (over-) prediction of
the squared returns. GARCH model errors in previous
periods are more likely to affect current risk-neutral
volatility through risk aversion than through the objective conditional forecast of future volatility.
GARCH model RMSE. This variable is described
earlier and in the online technical appendix.
RND volatility premium on date t − 1. If “market
sentiment” is persistent, perhaps because of excluded
exogenous variables, that dependence can be captured
by including the previous day’s volatility risk premium,
which we measure as date t − 1 RND volatility minus
the GARCH forecast of volatility to expiration. But
simply “explaining” risk-neutral volatility by its own
lagged value is not very revealing, so this variable is not
included in the “all variables” specification. It is useful,
however, to see if yesterday’s volatility risk premium contains information for predicting realized volatility and
also to see how the coefficients on the other variables
are affected if this variable is included in the regression.
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RESULTS
Notes: The table shows the simple correlation coefficients among the
variables considered in the study. The sample includes data from 10,152
date-option maturity pairs from January 4, 1996, through April 29,
2011, with bad data points excluded as described in the text.
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We first look at univariate correlations among the
volatility-related dependent variables and the explanatory variables. This will show the typical explanatory
power of each factor when considered in isolation. We
then fit a grand regression on all variables together to
gauge each one’s marginal contribution when included
with the others. Because RND volatility impounds a
forecast of volatility over an option’s remaining lifetime, there is potential correlation in residuals from any
two options whose lifetimes overlap. This does not bias
coefficient point estimates, but ordinary least squares
(OLS) standard errors are inconsistent. The standard
errors are corrected using an adaptation of the Newey
and West [1987] model, as detailed in the online technical appendix.
Simple Correlations
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Exhibit 1 shows simple correlations between the
dependent variables and each of the explanatory variables.
With more than 20 of them, multicollinearity is a concern, but only a few correlations are above 0.7, and these
tend to be where one would expect them. For example,
variables like historical volatility and the daily trading
range are highly correlated with the GARCH forecast,
34
as one might expect. Yet, these correlations are not so
high that they preclude precise coefficient estimates.12
Recall that in a one-variable regression, the
correlation and regression coefficient have the same sign,
and the R 2 is the square of the correlation coefficient. We
first note that the correlation between RND volatility
and realized volatility to expiration is only 0.613. The
low value ref lects the fact that risk neutralization modifies the market’s prediction of the empirical density
and, moreover, future volatility is just hard to forecast
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The next four variables relate to the frequency and
size of past extreme events. Seeing many returns in the
tails should increase investors’ perceptions of volatility
risk, and the exhibit shows positive correlations with the
number of both left and right 2% tail events. The average
size of a tail return is another dimension of tail risk.
The average left-tail return is negative, so negative correlation means volatility increases when there is a large
negative left-tail return. The right-tail return shows the
expected positive correlation. Realized volatility is also
related to these tail variables but with substantially lower
correlations than RND volatility shows, in each case.
The last section of Exhibit 1 contains variables
related to the risk-neutralization process. Both RND
and realized volatility are negatively correlated with the
University of Michigan consumer sentiment survey and
the P/E for the S&P index. The Baker–Wurgler measure, however, is less correlated with RND volatility
and has the “wrong” sign for realized volatility, although
the size is quite small. To the extent that these variables
ref lect investor confidence, negative correlation with
RND volatility is expected. The Baa–Aaa corporate
bond yield spread measures market-perceived default
risk. The variable is strongly positively correlated with
the two volatility variables, which suggests that wider
credit spreads in the bond market correspond to higher
volatility in equity options.
The next two variables capture how accurate the
GARCH model forecast has been in the recent past.
Investors perceive more volatility risk, and there seems
to be more actual volatility, following a period in which
volatility models exhibited large errors, especially when
volatility was underpredicted during the sample period
(i.e., mean GARCH error was positive). Both RND
and future realized volatility are strongly correlated with
GARCH RMSE.
The final variable in Exhibit 1 is the previous day’s
RND volatility premium. There is almost no correlation
between yesterday’s volatility risk premium and today’s
RND volatility level (although the premium itself is
highly autocorrelated), and the correlation with realized volatility is actually negative. Thus, the difference
between risk-neutral volatility and a good forecast of
empirical volatility does not appear to contain information about future realized volatility or even about
tomorrow’s risk-neutral volatility.
The correlations in Exhibit 1 show that the factors expected to inf luence RND volatility and realized
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accurately—that is, the GARCH forecast and historical
volatility show similar correlation levels with realized
volatility.
The next section contains the date t variables. Date
t return is negatively correlated with the two dependent variables, but only weakly, probably because the
exhibit reports correlations among levels, not changes.
The absolute value of date t return is distinctly more
important than the signed return, especially for RND
volatility. The GARCH forecast of volatility through
expiration is an extremely important variable. Its correlation with both RND and realized volatility is close to
the largest of any of the exogenous variables.13 Extreme
negative and positive “tail” returns show the expected
signs—a large negative left-tail return leads to higher
volatility and so does a positive return in the 2% right
tail—but correlations are low.
Next are the volatility-related variables computed
from historical prices and returns. With negative correlations on last week’s return and positive correlations
on the absolute return, when the market went up (down)
over the past week, volatility fell (increased), but volatility was higher after a week with a large return of either
sign. Historical volatility should be similar to GARCH,
although less accurate as a forecast because it uses the same
past returns, but in a less sophisticated way. Still, investors
may well incorporate it in RND volatility because it is
more accessible than a full-f ledged GARCH model to
many market participants. As expected, the correlations
between historical volatility over the last 65 days and the
two dependent variables are very high; but for realized
volatility, they are a little lower than the GARCH forecast.
As we look backward, two different trading range
concepts could be important. If the index moves over a
broad range intraday, it signals high volatility and also
greater hedging risk for market makers. The daily range
averaged over a longer time period measures the latter
effect in recent returns, and the correlations with the
two volatility variables are strongly positive, more so
for risk-neutral volatility.14 The second trading range
concept is of more interest to a longer-term investor.
Trading over a wide range during the option’s lifetime
can be desirable if it provides an opportunity for the
investor to gain a short-run profit, even when the option
in question will eventually expire out of the money.
Although RND volatility is very highly correlated with
both range variables, they have less explanatory power
for the volatility that is actually realized.
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columns feature key results and some interesting differences between RND volatility and realized volatility.
Both show a significant leverage effect, with negative date
t returns leading to higher volatility, and highly significant coefficients on the GARCH forecast. However, the
coefficient on GARCH is more than twice as large for
realized volatility as for RND volatility and both are
well below 1.0.
Trading on date t over a much wider range than
the change from yesterday’s close is associated with
higher risk-neutral volatility and even more strongly
with subsequent realized volatility. But left- or righttail returns had no significant effect on subsequent
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Exhibit 2 presents the results from four regressions run with the full set of explanatory variables, with
t-statistics corrected for cross-correlation. The first four
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Regressions with All Variables
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volatility appear to do so with the anticipated signs.
But correlation among the explanatory variables is high,
so it is not clear how much any individual factor contributes beyond the fact that all of them are correlated
with “volatility” broadly defined. To gauge the marginal
inf luence of each variable, we now combine them all
into a single grand regression.
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EXHIBIT 2
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Full Sample Regressions
Notes: Results from OLS regressions of RND volatility and realized volatility from observation date through option expiration, regressed on the full
set of explanatory variables. t-Statistics are corrected for cross-correlation caused by overlapping option lives.
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we do not favor this specification for the RND volatility regression because this difference is a large part of
what we are trying to explain with exogenous factors.
We include it here simply to show that it has little effect
on the other coefficients. There were changes in coefficient sizes and significance levels, but just a few variables
changed sign and only the date t right-tail return went
from significantly positive to significantly negative.
The lagged RND volatility premium in the realized
volatility regression is nearly significant but negative, and
it contributes almost nothing to raising the regression
R 2. This variable ref lects the risk-neutralization process, which is not a prediction of future volatility at all.
A negative coefficient suggests that holding the other
predictive variables constant, future realized volatility
tends to be lower (higher) when the volatility risk
premium is high (small).
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SUBSAMPLE RESULTS
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This section explores the robustness of the relationships uncovered in the last section in subsamples
broken down by maturity and in different economic
environments.
Exhibit 3 runs the regression of RND volatility
on the full set of explanatory variables separately for
contracts with less than 75 days to expiration and for
longer maturity contracts. The results for short-term
and longer-term options are remarkably similar. Only
the average size of left-tail events changes sign and is
insignificant in both subsamples. The average right-tail
return in the last two years becomes significant in the
longer maturity regression, while the coefficient on
GARCH RMSE becomes insignificant (with the wrong
sign in both regressions).
Exhibit 4 compares regressions for both RND
volatility and realized volatility on all variables over three
different time periods. The first is January 1996–June
2003, which includes the Russian debt crisis and the
Internet bubble, followed by the bear market of 2001–
2003. It was a period of medium volatility with a sharp
run-up in stock prices and a sharp drop in the second
half. The second subsample, July 2003–December 2007,
was a time of persistently rising stock prices and extraordinarily low volatility. The third subperiod, January
2008–April 2011, includes the market crash in 2008,
when volatility soared to extreme heights, followed by
the aftermath.
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realized volatility. The t-statistics in the RND volatility
regression are much higher, but the anomalous positive
coefficient on a negative tail return implies that a shock
to the downside reduced RND volatility.
For RND volatility, all nine historical return
variables except the average left-tail return were significant with the expected signs, although only three
had significant explanatory power for realized volatility.
A behavioral explanation is that different volatilityrelated aspects of the returns process may be independently relevant to, and valued by, certain classes of
investors beyond their connection to future realized volatility. Market makers may focus especially on the recent
daily range, whereas speculators may favor options on
stocks that have traded over a wide range in the past
month, and the most risk-averse investors might be particularly worried about tail events.
Including historical volatility and GARCH in the
same specification allows for the expected result that
GARCH, as a superior forecaster, should dominate
historical volatility in the realized volatility equation.
Indeed, historical volatility gets a negative coefficient in
the realized volatility equation, even though the simple
correlation between the two is 0.589. Yet, historical
volatility could be important to investors who are less
comfortable with sophisticated econometrics. And in
fact, both GARCH and historical volatility do get significantly positive coefficients in the RND equation,
albeit with the GARCH coefficient being more than
three times as large.
Among the risk-neutralization variables, strength
in the Michigan survey, the Baker–Wurgler investor
confidence measure, and the market P/E were associated with lower RND volatility, although only the
Michigan survey was significant. The Michigan sentiment measure was also significant for realized volatility,
but Baker–Wurgler investor sentiment came out strongly
significant with the wrong sign. The Baa–Aaa bond
yield spread was not significant for RND volatility, but
it had the expected positive sign and was quite large and
nearly significant for realized volatility. In contrast to
Exhibit 1, the two variables related to accuracy of the
GARCH model were insignificant, and three of four
had the wrong signs.
The four rightmost columns present results from
the same regressions with yesterday’s volatility premium
added as an explanatory variable. Although yesterday’s
volatility premium is obviously highly significant,
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EXHIBIT 3
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Regressions of RND Volatility on All Variables for Short and Long Maturities Contracts
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Note: The table reports results from the same RND volatility regressions shown in Exhibit 2, broken down according to option maturity.
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Not surprisingly, there is considerable dispersion,
but the important full sample results largely hold across
subperiods for both RND and realized volatility. The
differences tend to be more in statistical significance
levels than in the fitted coefficients. The GARCH
forecast remains highly significant, except for realized
volatility in the second subperiod. Historical volatility is
positive and significant in two of three RND volatility
regressions, but insignificant in two subperiods for realized volatility, and significant but with a negative sign
in the third.
The date t and “last week” return variable coefficients were negative in nearly all cases and significant for most of them. These variables were consistently
more signif icant in the RND volatility than the
38
realized volatility equations, suggesting that returns
may affect risk attitudes in addition to objective volatility expectations.
The date t trading range minus the absolute return
was positive in all six regressions and highly significant
in five of them. A large average daily range in the past
increased RND volatility significantly in every subperiod but increased realized volatility only in the earliest
one. The range covered by the index over the previous
25 days did not perform consistently. The coefficients in
the RND volatility regressions were positive and significant overall in Exhibit 2 but only significant in the final
subperiod; whereas for realized volatility, this variable
was significantly positive in the earliest subperiod and
significantly negative in the last.
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EXHIBIT 4
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Regressions of RND Volatility on All Variables over Three Subperiods
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Note: The table reports results from the same RND volatility regressions shown in Exhibit 2, broken down into subsamples by time period.
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The left- and right-tail variables did not perform
well. A date t left-tail return received a significantly
positive coefficient in all three RND volatility regressions, implying that an unusually large negative return
was associated with a decrease in risk-neutral volatility.
Frequent left-tail events in the past increased both riskneutral and realized volatility overall, but results varied
widely across subperiods. Similarly, the average size
of left-tail events showed inconsistent coefficient estimates and more than half had the wrong (i.e., positive)
sign. Right-tail events showed a similar lack of consistency between RND volatility and realized volatility
equations and over time.
Among the risk-neutralization variables, a higher
value for the Michigan survey reduced both RND volatility and realized volatility significantly, with a couple
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of exceptions, while the Baker–Wurgler measure of
investor confidence was more ambiguous—with both
volatility measures increasing when this confidence
measure was higher in the middle subperiod. By contrast, the S&P 500 P/E had the right (negative) signs in
Exhibit 2 but was not significant. Here, the coefficients
are negative in all regressions, and half are significant.
Overall, the coefficient on the bond yield spread was
insignificant for RND volatility but positive and significant for realized volatility; whereas in Exhibit 4, it
is significantly positive for realized volatility only in the
final subperiod, consistent with default risk becoming a
more important factor after 2008.
Finally, the GARCH model errors and the
GARCH RMSE continue to be largely insignificant and
anomalous in the subsamples, as they were in Exhibit 2.
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beyond expected future volatility, such as the recent
trading range, and the risk premia they require appear
to be related to measures of confidence and sentiment.
These non-BS factors are important determinants of
how options are priced in the real world.
A GARCH model forecast was found to be highly
significant for both realized and RND volatility. As a
less accurate predictor, historical volatility did not contribute significantly in forecasting realized volatility, but
it did in the RND volatility regressions. Investors appear
to pay attention to both estimators in pricing options.
The well-known negative relationship of volatility with
return was strongly manifested by risk-neutral volatility.
Both the date t return and also the return over the previous week were estimated with highly significant negative coefficients overall and, with a single exception, in
every subperiod regression. The larger and more significant coefficients in the RND volatility regressions
suggest that empirical (P measure) volatility is correlated
with returns, but transforming the P measure into the
Q measure increases the effect.
Trading over a wide range that produced little
change in the closing index level was highly significant
in increasing both RND and realized volatility. But a
large average daily range in the recent past or a large
total range also increased RND volatility substantially,
while having no significant effect on realized volatility.
These variables seem to be more strongly connected to
investors’ and market makers’ risk aversion than to their
objective forecasts of future volatility.
Six variables were specifically chosen for their possible correlation with risk preferences. RND volatility
was negatively correlated with the Michigan survey of
consumer sentiment and the Baker–Wurgler index of
investor sentiment, with the former showing strong performance for both RND volatility and realized volatility. It is not surprising to find high market volatility in
times when consumers and investors are less confident.
The effect appears distinctly stronger for RND
volatility than for realized volatility, supporting the
hypothesis that investor sentiment impacts risk neutralization. The P/E of the S&P 500 and the credit
spread in the bond market are direct measures of the
market’s willingness to bear risk. The consistently negative coefficient on the P/E suggests that it may also
be a useful ref lection of market sentiment, although it
was only significant in a few cases. However, the bond
yield spread results were odd overall and inconsistent
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A negative coefficient on GARCH RMSE implies
that RND volatility is lower when the GARCH
model has been less accurate in the recent past; and
a negative coefficient on the GARCH error means
that RND volatility was lower still when GARCH
underpredicted volatility.
To summarize the results of these subsample
regressions, we note that there was not much difference
between short- and long-maturity contracts in their sensitivities to the explanatory variables. More diverse results
showed up when the sample was broken into subsamples
by time period. High predicted volatilities from both the
GARCH and historical volatility models were associated
with significantly higher risk-neutral and future realized
volatilities in most cases, although the relationship was
weaker for historical volatility in the realized volatility
regressions. The return variables showed the expected
strongly negative correlation between return and volatility, whereas absolute return was more ambiguous.
The variables relating to the trading range intraday and
over a longer historical period were for the most part
strongly positive: trading over a wider range increased
RND volatility, and realized volatility too, in most
cases. The tail event frequency and size showed inconsistent results, with some coefficients significant, but
with possibly different signs in different subperiods and
quite a few anomalous values. The results on proxies for
the market’s risk tolerance were promising, though not
overpoweringly strong. In most cases, positive sentiment
seems to be associated with lower volatility. However,
the Baa–Aaa bond yield spread and the two variables
measuring the accuracy of the GARCH model did not
add much explanatory power.
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CONCLUDING COMMENTS
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We set out to develop stylized facts about what
factors determine risk-neutral volatility. Let us
summarize what we have uncovered.
The first question was what return- and volatilityrelated factors are most important to investors. Under
BS, this amounts to getting the most accurate estimate
of the instantaneous volatility. We found that RND
volatility is strongly, but not perfectly, correlated with
realized volatility, with correlation around 0.6. Forecasts
of realized volatility from a GARCH model are highly
significant in explaining RND volatility. But investors
seem to care about other features of the returns process
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ENDNOTES
13
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Although the model is f itted on historical data,
GARCH is a “date t” variable because ambiguity over how
far back the historical data sample should go is not a major
issue with a parametric model like GARCH.
14
Based on the analysis in the online technical appendix,
the option’s remaining lifetime is selected as the horizon.
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in the breakdown across subperiods. Lastly, the two
variables designed to measure investors’ confidence in
the GARCH model’s accuracy did not add explanatory
power. The coefficients were mostly insignificant with
anomalous signs.
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Baker, M., and J. Wurgler. “Investor Sentiment and the
Cross-Section of Stock Returns.” Journal of Finance, 61 (2006),
pp. 1645-1680.
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Birru, J., and S. Figlewski. “Anatomy of a Meltdown: The
Risk Neutral Density for the S&P 500 in the Fall of 2008.”
Journal of Financial Markets, 15 (2012), pp. 151-180.
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Bliss, R., and N. Panigirtzoglou. “Testing the Stability of
Implied Density Functions.” Journal of Banking and Finance,
26 (2002), pp. 381-422.
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——. “Option Implied Risk Aversion Estimates.” Journal of
Finance, 59 (2004), pp. 407-446.
Bollerslev, T., and V. Todorov. “Tails, Fears and Risk Premia.”
Journal of Finance, 66 (2011), pp. 2165-2211.
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I thank the Q-Group and the NASDAQ OMX Derivatives Research Project at NYU for research support for this
project. Thanks to Rob Capellini for preparing the GARCH
forecasts. Valuable comments from Joanne Hill, Larry Harris,
and participants at the Q-Group 2012 Spring Seminar, the
University of Michigan finance seminar, and the ITAM
Finance Conference are greatly appreciated.
1
See Poon and Granger [2003] or Jackwerth [2004]
for reviews.
2
See, for example, Bliss and Panigirtzoglou [2002,
2004], Kozhan et al. [2010], Figlewski [2010], Christoffersen
et al. [2011], or Giamouridis and Skiadopoulos [2012].
3
See Chicago Board Options Exchange [2015].
4
Jiang and Tian [2007] demonstrate that ignoring these
features in the VIX methodology introduces substantial and
unnecessary inaccuracy.
5
See also the appendix to Birru and Figlewski [2012]
for full details on the RND extraction methodology.
6
The average daily return is only a few basis points,
which is much smaller than the average sampling error on
the mean, so it is generally felt that the small bias introduced
by treating the mean as zero is preferable to the much larger
errors that would result from using the sample mean in the
calculation.
7
A GJR-GARCH model was f itted for every day
during the sample and out-of-sample variance forecasts
were generated for 1 to 252 trading days ahead. These were
used to construct predictions of average variance from each
observation date through option expiration. The resulting
variances were converted to annualized percent volatilities.
See vlab.stern.nyu.edu/.
8
Parkinson [1980] showed that a volatility estimator
based on the intraday high and low can achieve the same
accuracy as one using only close-to-close returns with five
times as many observations.
9
The data were downloaded from Wurgler’s website,
“Investor sentiment data (annual and monthly), 1934–2010,”
people.stern.nyu.edu/jwurgler/.
10
www.econ.yale.edu/~shiller/data.htm.
11
research.stlouisfed.org/fred2/.
12
The full correlation matrix for all dependent and
exogenous variables is shown in the Appendix.
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To order reprints of this article, please contact Dewey Palmieri
at dpalmieri@ iijournals.com or 212-224-3675.
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