Volatility-Based Pairs Trading: Empirical Evidence from U.S. Options Markets
Cara M. Marshall, Ph.D.; Queens College of the City University of New York;
300K Powdermaker Hall, 65-30 Kissena Boulevard, Flushing, NY 11367; Cara.Marshall@qc.cuny.edu
This paper examines how two different measures of implied volatility for
the same stock are related. The examination is motivated by the rapid growth of
pairs trading and statistical arbitrage, two related hedge fund strategies
commonly applied to trading stocks but potentially applicable to trading
volatility as well. One of the measures is the classic implied volatility extracted
directly from equity options on a stock. The other measure is of my own design
and is based on the hypothesis that the ratio of a stock’s historical realized
volatility to the historical realized volatility of the market might be a good
indicator of the long-term equilibrium to which the ratio of the implied
volatilities will tend. I refer to this measure of implied volatility as the betaimplied volatility.
I run empirical tests to compare levels and volatilities of the two
measures of implied volatility. I test the correlation between the two measures of
implied volatility, and explore evidence of mean reversion in an effort to
determine if pairs trading could have been a profitable strategy during the time
frame studied. The minimum conditions necessary for pairs trades involving the
two measures of implied volatility were met about 8% of the time.
Submitted to Journal of Financial and Economic Practice, 5/15/2013
INTRODUCTION
This paper examines how two different measures of implied volatility for the same stock
are related to one another. The examination is motivated by the rapid growth of pairs trading and
statistical arbitrage, two related hedge fund strategies commonly applied to trading stocks but
potentially applicable to trading volatility as well. One of the measures is the classic implied
volatility extracted directly from equity options on a stock. I refer to this as the option-implied
volatility or OIV. The other measure is of my own design and is based on the hypothesis that the
ratio of a stock’s historical realized volatility to the historical realized volatility of the market (as
proxied by the S&P 500) might be a good indicator of the long-term equilibrium to which the
ratio of the implied volatilities will tend. If so, by applying the calculated ratio for a particular
stock to the current implied volatility of the index obtained from index options, we can estimate
an alternative measure of the implied volatility of the stock. This critical ratio can be shown to be
equivalent to the stock’s historical beta divided by the historical return correlation between the
stock and the index. 1 For this reason, I refer to this measure of implied volatility as the betaimplied volatility or BIV.
This paper is structured as follows. I begin with a more formal development of the BIV
measure. I then provide a description of pairs trading and statistical arbitrage in the context of
stock trading (the traditional setting for these strategies). This involves an examination of the key
elements necessary for successful application of these strategies. I then extend this logic to pairs
trading of a single stock’s alternative measures of implied volatility and then extend it further to
statistical arbitrage involving these alternative volatility measures. Next I examine four specific
dimensions (tests) of the two measures of a stock’s implied volatility. In Test 1, I examine the
comparable levels of the two measures of implied volatility. In Test 2, I examine the correlation
between the two measures of implied volatility. In Test 3, I compare the volatilities of the two
measures of implied volatility using three different approaches. In Test 4, I examine each of 413
stocks for which there is sufficient data for evidence of mean reversion. Tests 1 and 3 are not
critical to the viability of volatility-based pairs trading but they do help to round out the study.
Tests 2 and 4 are the critical components for establishing the necessary conditions for volatilitybased pairs trading of the type described in the next section. Detailed results of the critical tests
are provided and discussed in a separate section. Finally, conclusions with respect to the viability
of the trading strategy are suggested. Critical to drawing these conclusions is setting the “hurdle
correlation” necessary for acceptable risk management. I make an assumption as to a reasonable
hurdle correlation, but I am fully aware that this is an arbitrary “line in the sand” and that
different traders would set different hurdle correlations.
DEVELOPMENT OF BETA-IMPLIED VOLATILITY
The BIV measure of implied volatility rests on the hypothesis that the most recently
observed ratio of a stock’s realized volatility to the realized volatility of the market (as measured
by an index) will tend to persist. Thus, the same ratio can be expected to describe forwardlooking (i.e., implied) volatilities. Let σi and σm denote the realized volatilities of stock i and the
index, respectively. Let BIV and IOIV represent the forward looking volatilities of the stock and
the index, respectively. Then by the argument above, the expected value of the BIV/IOIV ratio
for a given stock i, should be equal to the most recently observed ratio of their realized values.
This is given by Equation (i).
 BIVi   i


 IOIV   m
(i)
and, because IOIV may be viewed as a constant across all stocks, Equation (i) implies Equation
(ii):
BIV i  
i
 IOIV
m
(ii)
If, for example, a stock’s realized vol was 37.5 while the index’s realized vol was 15,
then the ratio (based on past behavior) is 2.5. That is, over the recent past the stock has been 2.5
times as volatile as the index. Then, if the index volatility is currently (as measured by the IOIV)
24, we would expect the stock’s BIV to be 60.
Keeping in mind that volatilities are, in general, standard deviations, it is easily shown
that this is equivalent to Equation (iii).
BIVi  
i
 IOIV
 i ,m
(iii)
Proof:
Starting with the definition of beta described by Equation (iv)2
i 
 i ,m
 m2
(iv)
which is equivalent to Equation (v).
i 
Which reduced to:
i 
Therefore,
 i m  i ,m
 m2
(v)
 i  i ,m
m
(vi)
i

 i
i , m  m
(vii)
Substituting into Equation (ii), renders Equation (iii).
BIVi  
i
 IOIV
 i ,m
(iii)
PAIRS TRADING, STATISTICAL ARBITRAGE AND STRUCTURE OF THE STUDY
A now classic hedge fund strategy known as “pairs trading” involves trading one security
against another security. Most discussions of pairs trading involve two stocks and, for the
moment, I will address the pairs trading approach in that context. Pairs trading, which is a form
of expected convergence trading, would only make sense when four conditions are satisfied.3
The first of these conditions is that the two stocks tend to move together. The second condition is
that the price behaviors of the two stocks occasionally deviate due to market anomalies. The
third condition is that the relationship between the two stocks’ prices tends to re-establish itself
following a pricing deviation. The fourth condition is that there presently exists a sufficiently
large discrepancy between the two prices to render a profit from transacting. The first three of
these conditions are necessary for a viable pairs trading strategy. But sufficiency requires the
fourth condition as well. The fourth condition is a function of several difficult to estimate inputs
and one impossible to estimate input (i.e., the trader’s utility function4), in this test I limit myself
to testing for the necessary conditions but make no attempt to render judgment on sufficiency.5
From a statistical perspective, conditions 1 and 2 require that the stocks’ returns be
positively correlated, but not perfectly so. Condition 3 requires that the relationship between the
two stocks’ prices be mean reverting. When testing for mean reversion, pairs traders do not
typically look at the prices per se, but rather at the log of the ratio of the two prices. The DickeyFuller test of stationarity is then often applied to this ratio.6
We can borrow the logic of pairs trading with stocks and apply the same approach to
trading volatility.7 That is, we would expect the two measures of a stock’s implied volatility,
OIV and BIV, to be positively correlated since both are looking, in some sense, at the volatility of
the same stock. However, because they are obtained in different ways, we would not expect them
to be perfectly correlated. Additionally, there is substantial evidence that volatilities are
approximately lognormally distributed (Marshall 2008). The lognormality of volatility dictates
the structure of some of the tests that follow.
Because OIV is derived entirely from current option prices for an individual stock, one
might expect it to react almost instantaneously to new volatility-influencing information pertinent
to that stock. BIV, on the other hand, because it is derived in part from a stock’s historical beta
coefficient, might be expected to react more slowly. Because this latter measure is also a function
of index volatility, it could perhaps be argued that it too can react instantaneously to new
information. But new stock-specific volatility-influencing information would be expected to only
have a very small effect on the volatility of the index. On the other hand, BIV will react to any
changes in IOIV whether they are stock specific or not and, therefore one might expect BIV to be
more volatile than OIV. The upshot is we cannot say a priori whether BIV or OIV will be more
volatile. More importantly, at least to a volatility trader, if either OIV or BIV tends to over-react
(relative to the other) to stock-specific volatility-influencing information, then there would be a
tendency for the over-reacting measure to gravitate back toward the other following the
“deviation.” This would tend to show up as mean reversion in the log of the OIV/BIV ratio and
could be a source of alpha opportunities for volatility traders.
In the study covered by this paper, I sought to determine if necessary conditions are met
for trading OIV against BIV for the same stock. Positions in the former can be taken via long or
short positions in equity options on individual stocks. Positions in the latter can be taken via
short or long positions in index options with beta serving as a critical element in sizing the latter
position correctly. Importantly, there is no reason why OIV and BIV must converge over the life
of the option positions. Therefore, these sorts of pairs trades are expected convergence trades at
best, and the strategy is not, in any sense, risk free.8
Should mean reversion exist in the log of the OIV/BIV ratio for a particular stock, an
alternative approach to exploit the relationship would be to write equity options on that stock and
buy index options when the log of OIV/BIV exceeds its long-term average by a sufficient
amount. Similarly, one would buy equity options on a stock and write index options when the log
of OIV/BIV falls below its long-term average by a sufficient amount. This approach may negate
the need to trade in index options entirely. Indeed, on a sufficiently large scale, pairs trading
takes on the character of what hedge funds call “statistical arbitrage.”9 That is, the pairs trading
approach transforms into statistical arbitrage. Consider a simple example of this in the context of
volatility pairs trading. Suppose that the log of the ratio of OIV/BIV for General Electric (GE)
exceeds the long-term average of the ratio for GE and that the pairs trader exploits this by writing
10 GE calls and buying 22 index calls. At the same time, the log of the ratio of OIV/BIV for
Yahoo (YHOO) is below the long-term average of the ratio for YHOO and that the pairs trader
would exploit this by buying 10 YHOO calls and writing 27 index calls. Notice that the long and
short positions in the index options largely cancel out. When done on a grand enough scale, the
overall position in index options could be expected to be nearly nill leaving only a diversified
portfolio of long positions in equity options and a diversified portfolio of short positions in equity
options. It is at this point that the volatility-based pairs trading has been transformed into
volatility-based statistical arbitrage.
For the reasons noted above, my final test of the efficient pricing of volatility is limited to
determining if necessary conditions for volatility-based pairs trading (and by implication
volatility-based statistical arbitrage) were satisfied during the two-year study period. For
purposes of having a common statistical level of significance, in all four tests, I limited the stocks
employed to those for which there were a minimum of 501 daily observations on the relevant
variables. This criterion reduced the number of stocks examined from the 586 in my data set to
413. In the first test, I compared the levels of OIV and BIV for each of the stocks. In the second
test, I examined the correlation between OIV and BIV for each stock. In the third, I examined the
comparative volatilities of OIV and BIV for each stock. In the fourth, I examined the log of the
ratio of OIV to BIV for each stock for evidence of mean reversion. My purpose in conducting this
study was to determine if necessary (but not necessarily sufficient) conditions were satisfied for
pairs trading OIV against BIV. This would require (1) a sufficiently high degree of correlation
between OIV and BIV, but less than perfect correlation; and (2) clear evidence of mean reversion
in the log of the ratio of OIV to BIV.
DATA SELECTION, ACQUISITION, AND MANIPULATION
This study utilizes information on the stocks that comprise the S&P 500 each day from
October 31, 2005 through November 1, 2007. Because stocks are periodically removed from the
index and replaced by other stocks, there were actually 586 stocks incorporated in the study,
though only 500 on any given day. The study also uses daily end-of-day option-implied
volatilities for 30-day ATM calls and puts for individual stocks and for the S&P 500 index.
These are the “option-implied volatilities” (OIVs) and “index-option-implied volatilities”
(IOIVs), respectively. This data served as the primary inputs for the various components of the
study.
I also obtained implied volatilities from synthesized 30-day10 ATM S&P 500 index
options (ticker symbol SPX) based on the midpoint of the closing bid-ask option premiums.
Bloomberg extracts implied volatilities using a Black-Scholes Model for European options, a
Roll-Geske Model for American-type options if there is one dividend, and a Trinomial Model if
there is more than one dividend. This was done for both the calls and the puts. This data was
downloaded from Bloomberg on a per-day basis for the two-year study period. These are the
“Index-Option-Implied Volatilities” referred to earlier.
I next obtained at-the-close implied volatilities for each of the 586 individual stocks
included, at one time or another, in the S&P 500 from synthesized 30-day ATM equity options
for each of the 505 trading days in the two-year study period. This data was downloaded from
Bloomberg. Bloomberg uses the same methodology to extract implied volatilities from
individual equity options that it uses to extract implied volatility from index options.
The computation of log returns was straightforward except for one complication – the
monthly returns are “rolling returns.” This requires some explanation. The data set covered five
years with approximately 252 trading days per year. There were actually 1261 trading days in the
entire five-year period. This data was then subdivided (as described in Step 2) into two nonoverlapping segments. The latter two years was the study period and covered 505 days (October
31, 2005 through November 1, 2007). The prior three years was the pre-study period data
necessary to compute rolling returns. This period contained price data for 756 trading days
(October 30, 2002 through October 28, 2005). If I denote the first day of the study period as Day
1 (October 31, 2005) then the last day of the pre-study period is Day 0 (October 28, 2005) and the
first day of the pre-study period is Day -755 (October 30, 2002). This is depicted in Figure 1
below.
Figure 1: Timeline
Pre-Study
Period
Pre -Study
Period
Day 1
Day -755
Day 0
Day 1
Day 505
Day 0
Study Period
Study Period
A monthly return for a stock is calculated by taking the natural log of the ratio of the
price on a given day to the price of the same stock 21 trading days earlier. For this reason, the
first day for which a monthly return can be computed is day -734 (November 29, 2002).
As Equation (iii) suggests, once a stock’s historical beta has been obtained, we can use
this historical beta, together with the historical return correlation and the current implied volatility
of the index, to obtain the beta-implied volatility of the stock. Further manipulation of the data
was necessary to calculate the historical betas.
To calculate historical betas, I needed to select the time frame for historical data and the
length of the differencing interval. Reilly and Brown (2000) argue that that there is a trade-off
between a desire to have many observations, in order to improve statistical confidence, and a
desire to employ only recent data, in order to avoid estimation errors associated with the
continuous evolution of companies. There have been several studies of the effect of the
differencing intervals (often called time intervals) used to compute beta.11 Throughout this study,
the rolling historical betas were computed using monthly data (i.e., 21 trading-day intervals) for
the prior three years.
The beta-implied volatilities were extracted for each stock each day for the two-year
study period. This was done separately for calls and for puts. The call BIVs were calculated
from the implied vols (IOIVs) of the S&P 500 calls and the put BIVs were calculated from the
implied vols (IOIVs) of the S&P 500 puts.
The rolling historical betas were calculated directly from the standard deviations of the
monthly returns of the individual stocks, the monthly return correlations of the individual stocks
and the index, and the standard deviations of the index monthly returns.12 Henceforth these
rolling historical betas will simply be called betas.
Beta was computed for each stock on each of the 505 dates in the two-year study period.
While it is not critical to this study, except as a check on the accuracy of the calculations, it is
possible that the beta of the index computed as a weighted average of the betas of the individual
stocks in the index may not precisely equal 1 on a given day. There are two reasons why the
index beta may deviate from unity. First, the index beta on a given date is a weighted average of
the individual stock betas on that same date. However, the weights employed in the S&P 500
change, however slightly, on a daily basis. Thus, the historical index returns (based on past
weightings) may not accurately reflect the current weightings. Second, because of missing data
(missing stock price data and missing option-implied volatility data), the weights had to be
adjusted. With respect to historical stock price data, on average, on any given day, data was
missing for 10 stocks. 13 With respect to missing implied volatility data for individual stocks,
there was, on average, on any given day, data missing for nineteen stocks.14 This too can cause a
slight discrepancy in the computation of the index beta from the individual stock component
betas. As a check on the amount by which the index beta, computed as a weighted average of the
individual component betas, deviated from 1, I computed the index beta for each of the 505 days
in the study period using all the available stocks in the data set for the relevant date (i.e., not
limited to the 413 employed in the four tests described below). The largest value obtained for
the index beta was 1.042 (which occurred on May 2, 2006), the lowest value obtained was 0.964
(which occurred on August 8, 2006), and the average value over the 505 days was 0.999.
At this point the data file for a given date contained (1) the betas for each individual stock
on that given date; (2) the return correlations for each stock with the index; and (3) the IOIV for
the index derived once from a call on the index and once from a put on the index. By applying
Equation (iii), I then derived the BIV for each stock, separately for a call and put, on that date.
This was repeated for each of the 505 days. The resultant BIVs were then saved in the individual
stock files which also contained the option-implied volatilities (OIVs) for both calls and puts for
the corresponding dates. There were 586 of these stock files. For example, I now had a GE stock
file that had BIVs for calls and puts and OIVs for calls and puts on each trade date from October
31, 2005 through November 1, 2007. The other 585 files each represented another stock.
MISSING DATA
The data was now complete for the various tests in the study except for the problem of
missing data. In a day-by-day comparison of the OIVs to the BIVs for a given stock, the problem
exists that one or the other might be missing. If options on an individual stock were not quoted at
the end of a day, Bloomberg reported an OIV of either 0 or N/A for the stock for that day.
Further, if there was insufficient data to compute the beta of a stock, then the BIV for that stock
could not be computed. This would typically involve a series of days in which the BIV for that
stock could not be computed. For example, if on the first day of the study period (October 31,
2005), price data for a given stock did not go back a full three years, then BIVs would be missing
until such time as the stock price data did exist for three full years.
Because the purpose of this study is to compare these two different types of implied
volatilities, it is pointless to look at one absent the other. Therefore, the day-by-day comparison
of the two volatility measures simply skips those dates for those stocks in which either or both the
OIV or BIV is missing.
For 385 of the 586 stocks incorporated in this study, the data sets were complete and a
full 505 data points were available for the analysis. Other stocks were missing a few to a few
hundred data points. And, for 67 stocks, there were literally no data points at all to observe
because at no time during the study period did those stocks have three full years of prior price
data from which to compute the inputs to the BIV. I decided to limit my analysis of the joint
behavior of OIV and BIV to those stocks for which there was a minimum 501 data points. Of the
586 stocks, 413 met this criterion. The reason for requiring a minimum of 501 data points is that
one data point will be lost in a necessary manipulation of the data for Test 4, leaving a minimum
of 500 usable data points. The critical value I later use to judge significance is based on a
minimum sample size of 500.
TEST 1: OIV VERSUS BIV – LEVELS COMPARISON
As indicated earlier, my first day-by-day comparison of OIV and BIV dealt with the
levels of OIV and BIV for each stock. The levels for OIV and BIV for each stock for each day
were already saved in the stock files. The difference between OIV and BIV varied considerably
for many of the 413 stocks over the two-year study period. For example, for the first five stocks
(alphabetically) in the data set, the maximum difference and the minimum difference are provided
in Table 1.
Table 1: Maximum and Minimum OIV-BIV Difference for A Sampling of Stocks
CALLS (OIV-BIV)
PUTS (OIV-BIV)
MAX
MIN
MAX
MIN
Ticker Symbol
A
13.4547
-70.9647
8.4255
-62.9035
AA
19.4282
-55.8553
19.5186
-50.2282
ABC
16.4063
-30.6926
16.0549
-29.5528
ABI
38.0488
-31.9695
32.9555
-29.7867
ABK
38.2235
-14.8323
73.7239
-19.0513
More importantly, from the OIV and BIV data saved in the stock files, the average OIV
and the average BIV for each stock over the two-year study period was computed and saved in a
new file that I designated the “Results File.” A summary of the average values of OIV and BIV
for each of the 413 stock is reported in the Results Tables (Tables 3 and 4) provided later. I also
saved the maximum and minimum differences between OIV and BIV over the 505 days of the
study period for each stock.
For OIVs and BIVs derived from calls, the average BIV exceeded the average OIV for
259 stocks. This was 63% of the total. On average for the 413 stocks, the call OIVs were 31.61
and the call BIVs were 33.30. For puts, the average BIV exceeded the average OIV for 222
stocks. This was 54% of the total. On average for the 413 stocks, the put OIVs were 31.85 and
the put BIVs were 32.39.
To determine whether or not the differences in the OIV and BIV levels were statistically
significant it was necessary to make a distributional assumption. It is generally assumed that
volatilities have lognormal distributions. Indeed this is the standard assumption in most modeling
of stochastic volatility. However, because each OIVavg and each BIVavg is an average of at least
501 observations, we can appeal to the central limit theorem to justify treating the OIVavg and the
BIVavg as being normally distributed.
Table 2: Test 1 Metrics
OIVavg
BIVavg
Calls
Sample Size
413
Mean
31.6118
33.2997
StDev
9.4507
11.4226
StError of the mean
0.4650
0.5621
Correlation
0.7732
Puts
Sample Size
413
Mean
31.8525
32.3948
StDev
9.5736
11.1184
StError of the mean
0.4711
0.5471
Correlation
0.7588
We now seek to test whether the mean value of OIVavg and the mean value of BIVavg are
the same. This is equivalent to the null hypothesis that d = OIVavg - lBIVavg = 0 versus the
alternative hypothesis that d  0. Given that OIVavg and BIVavg should be normally distributed we
can conduct a test of “the difference in means for paired observations.” This test, more fully
described in Mood, Graybill and Boes (1974, page 387), is applicable to bivariate normal
distributions. If X and Y are normally distributed with correlation ρ then the difference between
X and Y is normally distributed with mean X - Y and standard error of the mean σd given by

2
OIVa vg
2
  BIV
 2   OIVa vg , BIVa vg   OIVa vg   BIVa vg
a vg
N
.
For calls, the mean difference (d) between OIVavg and BIVavg was -1.6879. For puts, the
mean difference was -0.5423. The z score for the null hypothesis for the calls is therefore -4.718
and the z score for the puts was therefore -1.504.
Based on these z scores, we can reject the null hypothesis of no difference between the
mean of OIVavg and BIVavg for calls at a 5% and at a 1% level of significance in favor of the
alternative hypothesis of a difference in the means. We cannot, however, reject the null
hypothesis of no difference in the means of OIVavg and BIVavg for puts at either the 5% or 1%
levels of significance.
While the difference between the mean OIVavg and mean BIVavg for calls may be
statistically significant, it hardly appears economically important given transaction costs. I would
therefore conclude that there is no systematic economically important difference between the
mean OIVavg and the mean of BIVavg for either calls or puts.
TEST 2: CORRELATION AND COINTEGRATION OF OIV AND BIV
Because OIV and BIV are both measures of a single stock’s volatility, but derived in
different ways, we would expect them to be positively correlated. At the same time, because they
are derived in different ways, we would not expect them to be perfectly correlated. The
calculation of the correlation between the OIV for a stock and the BIV for that stock (“OIV,BIV
correlation”) was, again, straightforward. The process involved opening each stock file,
computing the OIV,BIV correlation, and saving the results in the same “Results File” described in
Test 1 above. For the 413 stocks for which there are at least 501 data points, the results are
reported in the Results Tables, Table 3 and 4 (shown later).15
Because correlation is a necessary condition for pairs trading, the results of the
correlation analysis will be discussed separately in the section titled Results of Critical Tests.
While it would seem reasonable to expect a trader to want to employ the most recent indicators of
implied volatility and therefore to focus on contemporaneous correlation between BIV and OIV, I
felt it would also be useful to examine the correlation between OIV and a lagged value of BIV.
For this reason, correlation between BIV and OIV for each stock was calculated twice, one on a
contemporaneous basis and once on a one-month (21-trading day) lagged basis. This was done
separately for calls and puts. Note that only the contemporaneous correlations are reported in the
Results Tables. The results of the lagged correlations are discussed in the Results of Critical
Tests section.
Because, as discussed later in the Critical Results section, the OIV and BIV for most of
the 413 stocks covered by this study exhibited positive correlation, I decided to run a two-step
Engle-Granger test on OIV and BIV for cointegration. This was done for each of the 413 stocks
for both calls and puts. The procedure is useful for eliminating the possibility of spurious
correlation. Specifically, if two series are individually non-stationary, but a linear combination of
the two series is stationary, cointegration exists. The test procedure is to first define two new
variables denoted ln(OIVt) and ln(BIVt). As the first step, we regress ln(OIVt) against ln(BIVt).
Denote the residuals from this regression et. As the second step, we define a new variable Δet = et
– et-1. Next, we regress Δet against et-1. From the t-statistic of this second regression, we can test
for cointegration. In my case, in which I have at least 500 data points for each regression, the
critical value for rejecting the null hypothesis of unit root (i.e., non-stationarity) at a 1% level of
significance is –3.90. That is, we would reject the null hypothesis of nonstationarity if the t-
statistic from the second regression is less than -3.90 (that is, more negative than -3.90). The two
step process involved a total of 1652 regressions. That is, for calls, the first step involved 413
regressions for step 1 and 413 regressions for step 2. The same number of regressions were
conducted for puts.
The results of the Engle-Granger test of cointegration are discussed in the Results of
Critical Tests section.
TEST 3: OIV VERSUS BIV COMPARATIVE VOLATILITIES
Based on the intuitive arguments made earlier, I can plausibly argue the case that OIV
would be more volatile than BIV. But I can also plausibly argue the opposite case. To be clear,
we are looking at the “volatilities of the volatilities.” There are, however, some issues in how to
best measure the comparative volatilities of OIV and BIV. For this reason, I chose to measure it
in three different ways. In each case, I used the 413 stocks for which there were a minimum of
501 data points.
In my first approach, I simply calculated the standard deviation of the raw OIV for a
single stock from the 500+ daily observations on that stock’s OIV stored in the stock files. I then
did the same for that stock’s raw BIV. These results were saved in the Results File. The process
was repeated for each of the 413 stocks.
The results clearly indicate that, on average, a stock’s raw BIV is considerably more
volatile than its raw OIV. For the calls, the standard deviation of a stock’s raw BIV exceeded the
standard deviation of the same stock’s raw OIV in 388 cases which is 94% of the time.
Additionally, on average for the calls, the standard deviation of the BIV was 1.83 times as high as
the standard deviation of the OIV. For the puts, the standard deviation of a stock’s raw BIV
exceeded the standard deviation of the same stock’s raw OIV in 390 cases which is also 94% of
the time. Additionally, on average for the puts, the standard deviation of the BIV was 1.86 times
as high as the standard deviation of the OIV.
Because these results struck me as counterintuitive, I considered what might have
distorted the measures. One obvious possibility was the difference in the levels of OIV and BIV
that was shown in Test 1. That is to say, two variables can have very different standard
deviations simply due to their levels. Given that the average levels of BIV were slightly higher
than the average levels of OIV for both the calls and the puts (statistically significantly so in the
case of the calls, but not in the case of the puts), I decided to neutralize the effect of difference in
levels. For this reason, in my second method for measuring comparative volatilities of OIV and
BIV, I scaled each observation on a stock’s OIV by its average value over the two-year study
period. I did the same for the stock’s BIV. For example, consider GE. I took the first
observation on GE’s OIV and divided by the average OIV for GE. I did the same for GE’s BIV.
I then computed the standard deviations of the scaled OIV and BIV measures.
For both the calls and the puts, the scaled BIV’s were far more volatile than the scaled
OIVs. For the calls, the standard deviation of the scaled BIV’s exceeded the standard deviation
of the scaled OIVs in 399 cases or 97% of the time. Additionally, on average, the standard
deviation of the scaled BIV was 1.77 times that of the standard deviation of the scaled OIV. For
puts, the standard deviation of the scaled BIVs exceeded the standard deviation of the scaled
OIVs in 402 cases or 97% of the time. Additionally, on average, the standard deviation of the
scaled BIV was 1.87 times that of the standard deviation of the scaled OIV.
In my third method for measuring the comparative volatilities of OIV and BIV, I
computed the standard deviations of the daily percentage changes in the two volatility measures,
measured as the log of the ratio of two consecutive days OIVs and BIVs. For example, for GE, I
calculated a new series defined as ln(OIVt/OIVt-1). I then calculated the standard deviation of this
series. I then did the same for BIV for GE.
Once again, the BIVs were considerably more volatile than the OIVs. For the calls, the
standard deviation of the percentage change in the BIV exceeded the standard deviation of the
percentage change in the OIV in 392 cases or 95% of the time. Additionally, on average the
standard deviation of the percentage change in the BIV was 1.43 times that of the standard
deviation of the percentage change in the OIV. For puts, the standard deviation of the percentage
change in the BIV exceeded the standard deviation of the percentage change in the OIV in 401
cases or 97% of the time. Additionally, on average the standard deviation of the percentage
change in the BIV was 1.53 times that of the standard deviation of the percentage change in the
OIV.
By any of the three measures of the volatilities of OIV and BIV, the BIVs were
considerably more volatile and most of this difference could not be explained by differences in
the levels of the two measures.
The results for each of the three approaches for measuring the comparative volatilities of
the OIV and BIV for each of the 413 stocks were stored in the Results File. The detailed results
for the third method only are reported in the Results Tables (Tables 3 and 4).
TEST 4: TEST OF MEAN REVERSION
As discussed in the beginning of this chapter, pairs traders trading stocks often rely on the
Dickey-Fuller test to determine if the log of the ratio of the two stock prices exhibit mean
reversion. I borrowed this logic and applied it to determine if the log of the ratio of OIV to BIV
exhibits mean reversion. This required that I first engage in one last manipulation of the data
stored in the stock files.
The Dickey-Fuller test for non-stationarity (i.e., unit root), as applied to the log of the
OIV/BIV ratio, required me to define a new variable Yt such that:
Yt  ln(OIV t /BIVt )
(viii)
 ln(OIV t ) - ln(BIV t )
(ix)
Next, I defined a new series Yt which represents the difference between two successive
observations on Y . That is:
Yt  Yt - Yt -1
(x)
This required sequentially opening each stock file and generating four new time series for each
stock. These were the log of OIV, the log of BIV, Y and Y .16 I saved the resultant four series
in the respective stock files. Because of the lagged structure of the regressions for which two of
the series will be used (see below), it was necessary to offset one series by one data point to
reflect the lag. Notice that this results in the loss of one data point. Thus, the maximum sample
sizes for the regressions described below are 504. And, as described in the section above on
missing data, the minimum sample sizes for the regressions described below are 500.17
Dickey and Fuller (1981) offered three different test models. The first model does not
include an intercept term (i.e., a constant term), the second model does include an intercept, and
the third model adds a drift factor. There is also an Augmented Dickey Fuller Model that adjusts
for serial correlation in the error terms. My review of the trade literature suggests that most pairs
traders who use the Dickey-Fuller methodology to test for mean reversion use Model 2. I chose
to do the same. In Model 2 of the Dickey- Fuller test, we regress the change in Y against the
lagged value of Y with inclusion of an intercept:
Yt    Yt 1   t
(11)
In an effort to determine if the log of the ratio of OIV to BIV is mean reverting, we seek to reject
the null hypothesis   0 in favor of the alternative hypothesis   0 . If   0 , we have a unit
root and non-stationarity.18 If   0 , we have stationarity which implies mean reversion. The
more negative the t-statistic associated with  , the more certain we can be of mean reversion.
However, t-statistics obtained from Dickey-Fuller regressions do not have t distributions and,
hence, t-tables cannot be used to correctly judge statistical significance. Instead, we employ
critical values from Dickey-Fuller tables. These critical values are dependent upon the size of the
sample. I uniformly employed levels of significance of 10%, 5%, 2.5%, and 1%. For example,
for a 1% level of significance, with a sample size greater than 500, and the inclusion of an
intercept term (i.e., Dickey-Fuller Model 2), the critical value to reject the null hypothesis of nonstationarity and accept the alternative of mean reversion is -3.43. That is, if the t-statistic
associated with the regression coefficient γ is less than (i.e., is more negative than) -3.43, we can
reject the null hypothesis and conclude that the log of the OIV/BIV ratio is indeed mean
reverting.
At this point, my data sets were complete and I could proceed to the application of the
Dickey-Fuller test, as described above. This involved opening the first stock file, running the
regression described above (once for calls and once for puts), extracting the key summary
statistics and saving them in the “Results File.” I then moved to the second stock file, and so on,
until all the regressions had been performed and all the results saved. A total of 826 regressions
were performed (i.e., 413 for calls and 413 for puts). Note that in the summary sections at the
ends of Tables 3 and 4, I report the number of stocks for which I can reject the null hypothesis of
non-stationarity at levels of significance of 10%, 5%, 2.5%, and 1%.
Because mean reversion is a critical condition for pairs trading, the results are discussed
below in the Results of Critical Tests section.
RESULTS OF CRITICAL TESTS
A summary of the results of all four tests, which collectively constitute the study, are
provided in the Results Tables (Table 3 for calls and Table 4 for puts). Full detailed results can
be found in the appendix for each of the 413 stocks for which a minimum of 501 daily
observations were available.
The results of the non-critical tests (at least with respect to pairs traders) have already
been discussed. Here we focus on the two tests that constitute the necessary conditions for
successful pairs trading, and which therefore constitute critical tests for the presence of alpha
opportunities. These are (1) a relatively high degree of correlation between OIV and BIV (Test
2), and (2) a strong tendency toward mean reversion in the log of OIV/BIV (Test 4).
Table 3: Summary of Results for Calls
Table 4: Summary of Results for Puts
CRITICAL TEST: CORRELATION AND COINTEGRATION
For calls, the average contemporaneous correlation between stocks’ OIVs and BIVs was
0.4897. But this ranged from a high of 0.9350 (AMBAC Financial Group, Inc.; ticker symbol
ABK) to a low of -0.2721 (Dow Jones & Co., Inc.; ticker symbol DJ). For puts, the average
contemporaneous correlation between stocks’ OIVs and BIVs was 0.4424. But this ranged from
a high of 0.9169 (Merrill Lynch; ticker symbol MER) to a low of -0.2238 (Dow Jones & Co.;
ticker symbol DJ). When a one month (21-day) lag structure was employed, that is, when the
correlation was between OIVt and BIVt-21, the correlations declined in most cases. Specifically,
the lagged correlation was lower for 403 of the 413 stocks when using calls (98% of the cases),
and lower for 385 of the 413 stocks when using puts (93% of the cases). On average, for calls,
the lagged correlation was 0.3360 compared to the average contemporaneous correlation of
0.4897. On average, for puts, the lagged correlation was 0.3056 compared to the average
contemporaneous correlation of 0.4424.
It is instructive to visually examine the relationship between OIV and BIV for at least a
handful of the 413 stocks. I have depicted below, in Figure 2, the day-by-day behavior of OIV
and BIV for the two calls having the highest OIV,BIV correlations and the two puts having the
highest OIV,BIV correlations. Figure 3 does the same, but depicts the two calls and the two puts
with the middling correlations. Figure 4 also does the same but depicts the two calls and the two
puts with the lowest correlations.
Correlation: 0.923
OIV
BIV
Correlation: 0.914
OIV
180
160
140
120
100
80
60
40
20
0
BIV
10/31/2007
AMBAC Financial Group Inc. (ABK)
10/31/2007
Correlation: 0.917
8/31/2007
BIV
8/31/2007
Merrill Lynch & Co. (MER)
6/30/2007
AMBAC Financial Group Inc. (ABK)
6/30/2007
Puts
4/30/2007
2/28/2007
12/31/2006
10/31/2006
8/31/2006
6/30/2006
4/30/2006
2/28/2006
12/31/2005
Calls
4/30/2007
OIV
2/28/2007
12/31/2006
10/31/2006
BIV
8/31/2006
6/30/2006
4/30/2006
Merrill Lynch & Co. (MER)
2/28/2006
90
80
70
60
50
40
30
20
10
0
12/31/2005
Correlation: 0.935
10/31/2005
10/31/2007
8/31/2007
6/30/2007
4/30/2007
2/28/2007
12/31/2006
10/31/2006
8/31/2006
6/30/2006
4/30/2006
2/28/2006
12/31/2005
10/31/2005
160
140
120
100
80
60
40
20
0
10/31/2005
10/31/2007
8/31/2007
6/30/2007
4/30/2007
OIV
2/28/2007
12/31/2006
10/31/2006
8/31/2006
6/30/2006
4/30/2006
2/28/2006
12/31/2005
10/31/2005
Figure 2: Cases with the Highest OIV,BIV Correlations
80
70
60
50
40
30
20
10
0
Correlation: 0.526
OIV
0
BIV
OIV
Correlation: 0.489
BIV
Correlation: 0.489
King Pharmaceuticals Inc. (KG)
160
140
120
100
80
60
40
20
0
10/31/2007
BIV
10/31/2007
The Charles Schwab Corp. (SCHW)
8/31/2007
Pactiv Corp. (PTV)
8/31/2007
Puts
6/30/2007
4/30/2007
2/28/2007
12/31/2006
10/31/2006
8/31/2006
6/30/2006
4/30/2006
2/28/2006
12/31/2005
Calls
6/30/2007
4/30/2007
OIV
2/28/2007
10
12/31/2006
BIV
10/31/2006
20
8/31/2006
30
6/30/2006
Consolidated Edison Inc. (ED)
4/30/2006
40
2/28/2006
50
12/31/2005
Correlation: 0.529
10/31/2005
10/31/2007
8/31/2007
6/30/2007
4/30/2007
2/28/2007
12/31/2006
10/31/2006
8/31/2006
6/30/2006
4/30/2006
2/28/2006
12/31/2005
10/31/2005
140
120
100
80
60
40
20
0
10/31/2005
10/31/2007
8/31/2007
6/30/2007
4/30/2007
OIV
2/28/2007
12/31/2006
10/31/2006
8/31/2006
6/30/2006
4/30/2006
2/28/2006
12/31/2005
10/31/2005
Figure 3: Cases with the Middling OIV,BIV Correlations
90
80
70
60
50
40
30
20
10
0
Correlation: -0.272
OIV
BIV
Correlation: -0.22
OIV
OIV
120
100
80
60
40
20
0
BIV
10/31/2007
Dow Jones & Co. Inc. (DJ)
10/31/2007
Correlation: -0.192
8/31/2007
BIV
8/31/2007
EOG Resources Inc. (EOG)
6/30/2007
Harrah's Entertainment Inc. (HET)
6/30/2007
Puts
4/30/2007
2/28/2007
12/31/2006
10/31/2006
8/31/2006
6/30/2006
4/30/2006
2/28/2006
12/31/2005
Calls
4/30/2007
2/28/2007
12/31/2006
BIV
10/31/2006
8/31/2006
6/30/2006
Dow Jones & Co. Inc. (DJ)
4/30/2006
2/28/2006
140
120
100
80
60
40
20
0
12/31/2005
Correlation: -0.205
10/31/2005
10/31/2007
8/31/2007
6/30/2007
4/30/2007
2/28/2007
12/31/2006
10/31/2006
8/31/2006
6/30/2006
4/30/2006
2/28/2006
12/31/2005
10/31/2005
90
80
70
60
50
40
30
20
10
0
10/31/2005
10/31/2007
8/31/2007
6/30/2007
4/30/2007
OIV
2/28/2007
12/31/2006
10/31/2006
8/31/2006
6/30/2006
4/30/2006
2/28/2006
12/31/2005
10/31/2005
Figure 4: Cases with the Lowest OIV,BIV Correlations
120
100
80
60
40
20
0
It is quite clear that the stocks with the highest OIV,BIV correlations would be the ones
of most interest to a pairs trader looking to trade OIV against BIV. One must then ask, how high
does the correlation need to be before a volatility trader would consider a pairs trade. This can be
thought of as a “hurdle correlation.”19 The lower the hurdle correlation, the more risk the
volatility trader takes on. The higher the hurdle correlation, the less risk the volatility trader takes
on. The choice of a hurdle correlation is a bit arbitrary and a function of the degree of risk
aversion of the volatility trader. At the very minimum, classic hedging theory would require a
minimum correlation of 0.50 before a risk-neutral volatility trader would consider a pairs trade.20
Because pairs traders are not risk neutral, one would expect a hurdle correlation above 0.50.
Pairs traders trading stocks (i.e., the classic pairs trades discussed earlier) often set a hurdle
correlation of about 0.80.21 I will consider both levels of hurdle correlation.
For the calls, of the 413 stocks examined, 224 had an OIV,BIV correlation of 0.50 or
higher. At the 0.80 hurdle correlation level, the number of stocks drops to 38. For the puts, of
the 413 stocks examined, 192 had an OIV,BIV correlation of 0.50 or higher. At the 0.80 hurdle
correlation level, the number of stocks drops to 29. Using the more restrictive and more realistic
level of 0.80, we can conclude that less than 10 percent of stocks meet the hurdle correlation
criterion. These are the stocks that would be further considered as candidates for volatility-based
pairs trading of the OIV vs BIV variety.
With respect to the Engle-Granger cointegration test, I found that I could reject the null
hypothesis of unit root and therefore non-stationarity at a 1% level of significance in almost every
case. Specifically, for calls, of the 413 stocks, tested for cointegration of OIV and BIV, fully 412
exhibited cointegration. For puts, the number was only slightly lower at 406. These results
clearly allow us to reject any possibility of spurious correlation between OIV and BIV.
CRITICAL TEST: MEAN REVERSION
The second critical test representing a necessary condition for viable pairs trading is the
test for mean reversion. As described in Test 4 above, the critical t-statistic values for rejecting
the null hypothesis are taken from Dickey-Fuller tables as applicable with at least 501
observations. For the calls, the null hypothesis can be rejected at a 10% level in 412 of the 413
cases. For the puts, the null hypothesis can be rejected at a 10% level of significance in all 413
cases. For calls, the null hypothesis can be rejected at a 1% level of significance in 409 of the
413 cases. For puts, the null hypothesis can be rejected at a 1% level of significance in 405 of the
413 cases.
These results make it clear that the log of the OIV/BIV ratio is mean reverting in almost
every case and we can reject the null hypothesis of non-stationarity in favor of the alternative
hypothesis of mean reversion at a 1% level of significance in almost all cases.
COMBINED CRITICAL TESTS: CORRELATION AND MEAN REVERSION
The first critical test considered OIV,BIV correlation of at least 0.80. The second critical
test considered demonstrated mean reversion at a 1% level of significance. However, for a pairs
trade of the type contemplated in this study to be viable, both conditions must be satisfied
simultaneously. Given that almost all of the stocks exhibited mean reversion, it is not surprising
that 38 stocks met both criteria for the calls, and 29 stocks met both criteria for the puts.
CONCLUSIONS
We may conclude that the minimum conditions necessary for convergence trades
involving the two measures of implied volatility were met about 8% of the time over the two year
study period based on end of day quoted prices. We cannot conclude, however, from these
results that profitable trading opportunities at an acceptable level of risk are available—only that
we cannot rule them out. We would also need to consider (1) the size of transaction costs, (2) the
risk parameters of the trades as they present themselves (since these are not arbitrage trades –
only expected convergence trades), and (3) the expected profit from convergence. Because (1)
the risk factors can be extremely complex when options are involved and have to be assessed in
the context of real-time data, and (2) different traders would have different risk-reward
tolerances, we cannot conclude that hedge funds would trade OIV against BIV. Nevertheless, the
methodology of volatility-based pairs trading, as described at the beginning of this chapter, would
seem to offer some promise of generating the sorts of alphas that hedge funds and proprietary
traders seek.
It would make an interesting follow-up study to take a small sample of promising stocks
and to attempt to acquire all the data necessary to factor in all of the risks, the transaction costs,
and the trader’s risk tolerances to see if profitable trading opportunities of the type described do
in fact exist.
With respect to the non-critical tests, BIV is modestly higher than OIV. This is not
surprising. BIV is based on the premise that the ratio of a single stock’s volatility to the market’s
volatility tends to persist. BIV is obtained by multiplying this ratio by the IOIV. In Marshall
(2009) it was shown that IOIV is higher than MIV at a statistically significant level. Since MIV
was derived from the individual stocks’ OIVs, and BIV was derived in part from IOIV, one
would expect BIV to exceed OIV to a small degree (i.e., it has an amplifying effect).
Much more difficult to explain is the difference between the volatility of OIV and the
volatility of BIV. All three methods of comparing volatility that I used revealed BIV to be
considerably more volatile than OIV. The volatility of BIV will be influenced by any emerging
information that impacts broad market volatility, as measured by IOIV, and this effect on BIV
seems to dominate the idiosyncratic volatility drivers where the effect is largely limited to OIV.
While other academic studies have examined less quantitatively sophisticated hedge fund
strategies, it is only within the last few years that the academy has considered the viability of
volatility and correlation-based strategies. This study contributes to that emerging body of
literature.
FOOTNOTES
Importantly, the concept of beta is ordinarily based on total returns which include dividends. But, for the
reasons discussed in Marshall (2009), option prices and volatilities are driven by price return only. Thus,
the various metrics derived and employed in this chapter use price returns unadjusted for dividends. These
include realized volatilities, realized correlations, and historical betas.
2
I remind the reader that all metrics derived from observed returns employed in this study are derived from
log price returns.
3
Pairs trading is a subset of a broader class of strategies known as long/short strategies. Pairs trading is
sometimes structured to be “cash neutral” and sometimes structured to be “market neutral.” In cash neutral
pairs trading, the dollar value of the long position taken is equal to the dollar value of the short position
taken at the time the trade is put on, but the betas of the two stocks do not have to match, thereby leaving
the trader with residual market risk. In a market neutral pairs trade, the trader matches the systematic risk
of the long position to the systematic risk of the short position. This is done by matching the “dollar beta”
of the long position to the “dollar beta” of the short position. Dollar beta is the dollar value of a position
times the beta of the position.
4
For example, see Kargin (2004).
5
The precise conditions I tested for are described later and differ slightly from what is described here for
stock pairs trading. I am adapting, with appropriate adjustments, the criteria for pairs trading with stocks to
fit the circumstances of pairs trading with volatilities.
6
For a more detailed discussion of the application of these measures in the context of pairs trading using
stocks, see Herlemont (2004).
7
The Lehman Brothers (2002) study of the volatility behavior of currency pairs is one such application.
8
The expected convergence rests on the assumption that the BIV i/IOIV ratio for a stock will return to the
long-term observed realized volatility ratio σi/σm. Any significant change in the product offerings of a
corporation, any changes in the employment of operating or financial leverage, any change in external
revenue/cost determining factors, etc., can temporarily or even permanently alter the volatility ratio. For
example, the 2007-2008 spike in oil prices and the credit crisis that began in 2007 would alter the ratio for
the transportation sector and financial services sector, respectively, if not on a permanent basis, then surely
on a somewhat protracted temporary basis. Further, even if convergence occurs, it may not take place prior
to the options expiration.
9
Statistical arbitrage involves simultaneously holding a diversified portfolio of long positions and a
diversified portfolio of short positions where each portfolio has been carefully selected so that the key
characteristics of the two portfolios are off-setting. Key characteristics would include such things as sector
weightings, capitalization weightings, betas, etc. When the key characteristics of the long and short
portfolios are fully matched, the risks largely off-set one another leaving only the alpha opportunities.
Thus, while each individual position is “risky” the overall combination of the two portfolios can be nearly
riskless.
10
To be consistent with the method of calculation on the VIX, 30-day options were used.
1
Reilly and Wright (1988) found that using monthly as opposed to weekly data is a cause for differences in
betas, but the effect is diminished as the size of the firm increases. Note that the present study is limited to
large cap stocks.
12
The standard deviations for the stocks and the index, and the correlations between the stocks and the
index are all computed as historical realized rolling values.
3
There are several reasons why stock price data might be missing. The stock was associated with a nonpublic company for a portion of the pre-study or study period. The company might have resulted from a
merger or acquisition and did not exist as a stand-alone company for a portion of the pre-study or study
periods. In still other cases, the stock might not have been trading due to an exchange-imposed trading
halt.
4
Again, there are several possible explanations for the missing volatility data. Bloomberg extracts the bidask midpoint volatility from four quoted option prices to synthesize 30-day ATM equity options. However,
if no market maker happens to be quoting the necessary option series from which Bloomberg synthesizes
the key call and put option, then the data is simply not available. Further, if there is an exchange-imposed
trading halt in the stock, market makers immediately cease quoting options on the stock. If a halt is in
effect at the close, there will be no closing option prices from which to extract the implied volatilities.
5
It is important not to confuse the correlation between OIV and BIV for a single stock, described here, with
the correlation between OIVavg and BIVavg across all stocks employed in Test 1.
6
Two of these four series had already been generated and saved as part of Test 3.
7
As it happened, no stock had exactly 500 data points in its sample, so the actual sample sizes ranged from
501 to 504.
8
Technically, the original regressions from which Dickey and Fuller derived their critical values were of
Yt    Yt -1   t . If the regression coefficient   1 , we have a unit root. If you deduct the
value Yt 1 from both sides of this equation, you get the equation Yt    Yt 1   t where   0 if
  1 . That is     1 . See Davidson and MacKinnon (2004, pages 613-620).
the form
The term “hurdle correlation” is often used by hedge funds to mean the minimum value the correlation
must reach before a trade is acceptable for further consideration. It is analogous to the term “hurdle rate”
as that term is used in capital budgeting decisions.
20
In classic hedging theory, which originated with commodity futures, a properly structured hedge done in
the futures markets would decrease risk if the correlation between the cash market price and the futures
price is greater than 0.50 and increase risk if the correlation between the cash market price and the futures
price is less than 0.50.
2
See Preston (2005) for example. For an excellent review of pairs trading in the academic literature, see
Gatev et al. (2006).
9
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Gatev, Evan, William N. Goetzmann, and K. Geert Rouwenhorst. “Pairs Trading:
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Herlemont, Daniel “Pairs Trading, Convergence Trading, Cointegration.” (2004)
<http://www.yats.com/doc/cointegration-en.pdf>.
Kargin, Vladislav. “Optimal Convergence Trading.” (2004)
<http://129.3.20.41/eps/fin/papers/0401/0401003.pdf>.
Lehman Brothers “On Mean Reversion in Implied Volatility Time Series,” Global
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Marshall, Cara M. (2008). “Volatility Trading: Hedge Funds and the Search for Alpha
(New Challenges to the Efficient Markets Hypothesis).” Fordham University Doctoral
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Marshall, Cara M. (2009). Dispersion trading: Empirical evidence from U.S. options
markets, Global Finance Journal, 20(3), 289-301.
Mood, Alexander, Franklin Graybill, and Duane Boes. Introduction to the Theory of
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APPENDIX
Detailed Results for Calls (Individual Stock Results that make up the summary in Table 3)
Detailed Results for Puts (Individual Stock Results that make up the summary in Table 4)