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Tatiana Lozovaia and Helen Hizhniakova
HOW TO EXTEND MODERN PORTFOLIO THEORY TO MAKE
MONEY FROM TRADING EQUITY OPTIONS
How to read dispersion numbers, or market is the biggest portfolio
one could ever manage
Contents
INTRODUCTION
Every trader, market-maker or financial analyst knows what risk
is and methods to estimate it. There are various theories on
how to estimate risk in the world of finance. The most popular
are thus by definition the most widely used. We do not endeavor
in this forum to evaluate some of the more sophisticated
theories of risk estimation but a brief discussion of what is
probably the best known theory of portfolio risk, Value At Risk,
is warranted. Value At Risk has become so prevalent that it is
almost impossible to find professionals in the financial markets
that are not at least remotely familiar with the measure.
What is commonly referred to as VAR is the risk expressed in dollar
terms showing what amount of money your portfolio can lose
during a defined interval with a given probability. The terms
which are most commonly mentioned along with VAR and are
similar in meaning to VAR are dispersion, variance and volatility.
So, VAR is in essence the volatility of portfolio expressed in dollar
terms. Beginning with the basics of VAR analysis we can then demonstrate how this concept can be applied not only to
estimate the risks of a portfolio of assets, but as to how this theory can be applied to trading the market itself.
In general, we could consider the markets as one large portfolio of various assets with each having their respective weights
in the global market place. Furthermore, we can estimate the global market risk, draw comparisons with historical data and
perform many types of empirical studies etc. The only significant practical difficulties that one would encounter would be
the large quantity of historical information required and the substantial attendant technical resources to manage it.
We can more efficiently explore the same concepts however by opting to focus on a few subsets of the global markets. Let’s
consider a few major indices within the markets which were introduced specifically to provide an assessment of the general
market conditions, and let’s venture to apply dispersion analytics to these indices.
An index measures the price performance of a portfolio of selected stocks. It allows us to consider an index as a portfolio
of stock components. From that point of view, index risk can be evaluated as the risk of the portfolio of stock constituents.
It is well known that the portfolio risk is a weighted sum of covariation of all stocks in the portfolio. Thus, it can be calculated
by the following formula that hereinafter will be referred to as the main formula.
Here  is a weight of an asset in the portfolio or a component weight in the index, is dispersion of a stock component.
NOTE
The formula above can be used in several ways. First, remark that all the data in this equation are provided by
observations of the market. But if we were to input actual data on the left and right side, we would not find exact
equality. This leads to idea that substituting some parameters with actual data, we can determine theoretical values
of the remaining ones. We can substitute historical or implied volatilities in this formula in place of  , and
correlations between stock prices or between implied volatilities of stocks in place  of to calculate theoretical
risk of an index.
With this formula the portfolio or index risk can be calculated for different terms, since we can use volatilities and correlations
calculated on any desired historical time interval for price data and different forecast times for implied data. Also, by
using this formula we can calculate a value that expresses the correlation level between the implied volatilities of the
stocks and the index implied volatility observed on the market.
All charts introduced in the paper are provided by EGAR Dispersion and IVolatility.com database.
FUNDAMENTAL INDEX PARAMETERS
Let’s discuss the types of values that can be
employed in the dispersion strategy. These values
enable traders to determine whether current conditions
are suitable for a dispersion trade. We will distinguish
amongst these three kinds of values: realized,
implied, and theoretical.
Realized values can be calculated on the basis of
historical market data, e.g. prices observed on the
market in the past. For example, values of historical
volatility, correlations between stock prices are realized
values.
Implied values are values implied by the option
prices observed on the current day in the market.
For example, implied volatility of stock or index is
volatility implied by stock/index option prices,
implied index correlation is an internal correlation
implied by the market. More details on this value
type can be found below.
Theoretical values are values calculated on the basis of some theory, so they depend on the theory you choose to calculate
them. Index volatilities calculated on the basis of the portfolio risk formula are theoretical, and can differ from realized
or implied volatilities.
A comparison of theoretical values with realized ones allows traders to determine what market behavior are best applied to
the actual trading environment. By studying and analyzing the historical relationship between these two types of values
one can make an informed decision about related forecasts.
By comparing implied and historical volatilities, theoretical and realized, or theoretical and implied values of the index risk,
we can attempt to ascertain the best time to employ the dispersion strategy or to choose to continue to monitor the markets.
The dispersion strategy typically consists of short selling
options on a stock index while simultaneously buying
options on the component stocks, the reverse dispersion
strategy consist of buying options on a stock index and
selling options on a the component stocks.
Over the last year the index options were priced quite
high (as shown on the charts below), while historical
volatility was lower. As a result, it was generally profitable
to sell rich index options. However, there also were
short periods, when the reverse scenario occurred, thus
buying index options in those periods would have generally
resulted in profits.
Chart 1: IV Index and HV for OEX |
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Chart 2: IV Index and HV for DJX |
As you can see on the second chart, historical volatility of DJX was greater than implied volatility at times and reached a
local maximum in the last week of September 2001. The explanation for this lies in the tragic events of September 11. A
similar jump in volatility which was driven by very different factors can be observed in the September 2002. This particular
time was quite good for buying cheap index options, and thus to engage in the reverse dispersion strategy.
2.1 Realized Values
Historical Volatility %
As mentioned above, historical volatility is calculated on the basis of stock price
changes observed over a given time period. Prices are observed at fixed intervals
of time (named terms): every day, every week, every month etc.
Historical Volatility is calculated as the standard deviation of a stock’s returns for
the last N days. Return is defined as the natural logarithm of close-to-close price
observations.
2.2 Implied Values
IV Index %
Implied Volatility Index, or IVIndex, is the main parameter used for implied data
in dispersion strategy analysis.
Implied Volatility is Volatility which is implicit in the option prices observed
within the markets. Implied Volatility can be used to monitor the market’s opinion
about the Volatility of a particular stock or index. Also implied volatility values
may be used to estimate the price of one option from the price of another option.
As implied volatilities are different for different options, it is useful to have a composite Volatility for a stock/index by taking
suitable weighted individual volatilities. Such a composite volatility, calculated on the basis of 16 Vega weighted ATM
options and normalized to a fixed maturity, is called the Implied Volatility Index. Hereinafter when we refer to implied
volatility of a stock or index we mean the Implied Volatility Index.
The chart below shows the time history of Implied Volatility Index for OEX, DJX, SPX, SOX, NDX, and OSX indexes. As
one can observe on the chart:
- In whole implied volatilities of indices move synchronously.
- The Implied Volatility Index of global equity indices such as SPX, OEX, and DJX are almost coincident, since such indexes
reflect the state of economics in general, and so their performances are affected by the similar factors.
- The implied volatilities of global equity indexes are lower than those of sector indexes, e.g. see SOX and OSX. It can be
explained by that changes in one sector considerable affect indexes from this sector, but may slightly enough influence
on the global market indexes. So risk for index that consists of equities from within the same industry is higher.
- As you can see NASDAQ-100 (violet line on the chart) has higher implied volatility that other major market indexes. It
can be explain by the fact that this index includes companies listed on the NASDAQ Stock Market only, while DJX, OEX,
SPX cover broader group of stocks listed on different exchanges.
As was mentioned above, major market indices have in whole lower implied volatility in comparison with sector indices. But
it should not be considered as some indexes are better, and other is worse for the dispersion strategy. The important role
in the strategy plays not absolute value of implied volatility, but historical relationship between implied and realized index
volatility.
Chart 3: Time history of Implied Volatility for OEX, DJX, SPX, SOX, NDX, OSX. |
EXAMPLE
Chart 4: 30 day IVIndex and HV of DJX index. |
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You can see the Implied Volatility Index and Historical Volatility of the
DJX index on the chart.
The most recent implied volatility of index (31%) is not far from the
recently registered maximum value (41%), and it reaches its local maxim
and higher than realized (historical) volatility. It suggests that it is a
good time for selling options on the DJX Index in the dispersion strategy,
because it means selling relatively rich options.
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Note: If the implied volatility of the index was relatively low and lower
than historical volatility it would mean a good time for buying options
on the index, i.e. to engage in the reverse dispersion strategy. |
Implied Index Correlation %
Implied Index Correlation defines correlation level between the actual implied volatility of the index and the implied
volatility of its stock components. In other words, it is a component-averaged correlation between implied volatilities calculated
from the formula of the portfolio risk, where and are actual index and stock components implied volatilities  ,
is a weight of a component in the index.
The greater Implied Index Correlation, the stronger correlation between the index implied volatility and that of its constituent
stocks, and therefore the more suitable the market conditions for deploying a dispersion strategy. However, this is
not an absolute measure and should therefore be examined in the light of its historical (realized) performance.
Realized Index Correlation can be calculated from the same formula by using the historical volatilities of stocks and the
index instead of implied volatilities.
Empirically sector indices usually have exhibited higher implied and realized correlations than major market indices. As you
can see in the table below, the 30 day implied index correlation of sector indexes SOX and OSX are 75.75% and 93.90% correspondingly,
while the implied correlation of broader market indexes such as the NDX, OEX, and DJX are lower (60.87%,
66.40% and 68.50% correspondingly). This phenomena has a reasonable explanation in that sector indexes consist of equities
from within the same industry, thus they are much more co-dependant on the same conditions rather than equities
from different industries.
Looking at the time history of Implied Correlation for global market indexes (DJX, OEX,SPX etc) one can observe, that the
implied index correlation for them has a tendency to increase (certainly, there are some jump and drops, but in whole it
rises), while average implied correlation of sector indexes (OSX , SOX etc) remain on the same high level (see charts 5
and 6).
Chart 5: Implied Index Correlation of OSX |
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Chart 6: Implied Index Correlation of OEX |
NOTE:
The implied index correlation is calculated on the basis of the implied volatilities of the stock components and the
index options which are implicit in the option prices observed in the market. So, the implied index correlation can
be greater than 1, or 100%, since the individual stock options and index options markets are in reality separate
markets. Implied Index Correlation greater than one means that actual implied volatility of the index substantially
exceeds the theoretical volatility, i.e. theoretically index options are too overpriced (see formula below). But as
a rule, for American indexes Implied Index Correlation is less than 100%.
In this formula  is actual implied volatility of index calculated from the market conditions, and  is theoretical
implied volatility of index calculated from the formula of the portfolio risk.
EXAMPLE
Chart 7: Implied Index Correlation for DJX |
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As mentioned above, the greater the implied index correlation (IC), the
greater the components and index volatilities are moving in the same
direction. Therefore, the higher the average correlation between index
volatility and volatilities of constituent stocks, the better the timing of
a dispersion strategy. Let’s look at the chart 7 which shows the 30 day
Implied Index Correlation for the DJX index. Last year correlation was
positive, and current value of 70% is close to the maximum. It suggests
an opportune time for a dispersion strategy. |
2.3 Theoretical Volatility Values
There are several ways to calculate the risk of the index. Theoretical Index
Volatility can be calculated from the formula of portfolio dispersion, or as a
weighted sum of components’ volatilities. The calculation of theoretical volatilities
is not difficult, but it is a very laborious task, since it requires processing of a
substantial quantity of historical volatility data.
WtdCompIV %
The simplest method to calculate an index volatility is to consider it as a weighted
sum of volatilities of its components. This sum will be called weighted components implied volatility, or weighted
volatility of index.
The weighted volatility of index calculated this way expresses overall implied volatility of the index components, but
ignores correlation between component stocks. The ratio of the components implied volatility (WtdCompIV %) to actual
implied index volatility hereinafter will be referred to as a first volatility level coefficient.
Let’s clear up connection between the weighted component implied volatility and implied index correlation described
above. As it follows from the formula of implied index correlation
, where  and are actual values of the index and stock components implied volatilities, IC is implied index correlation.
It is not difficult to re-arrange this expression to the following form
So if implied index correlation is 1 the actual implied volatility of index is exactly the weighted sum of the individual stock
constituent implied volatilities (see the formula above). As was mentioned above, generally, IC is less than one, so, as a rule,
the first volatility level coefficient is greater than 100%, i.e. theoretical volatility of index calculated without taking correlations
between index components is greater than actual implied volatility (see chart 8). But since implied index correlation
can be greater then 100%, e.g. this happens for European indexes, the first volatility level coefficient can be less
than one.
Chart 8: Theoretical and implied volatilities of DJX |
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The main factor affected the first volatility level coefficient is changing
correlation between stock and index volatility. Let’s look at the chart 8
which shows components implied volatility and actual implied volatility
for DJX. As you can see on the chart, in whole components implied
volatility and actual implied volatility move synchronously. But lately
the weighted components implied volatility verges towards the actual
implied volatility (see chart 9). It can be explained by growing implied
index correlation (see chart 10). |
Chart 9 displays the first volatility level coefficient for the DJX index. The coefficient falls over the last two years. The current
value of the coefficient is 1.2. It is close to its lowest value in two years. Thus, it is convenient time for dispersion
strategy, because it means that correlation level between implied volatilities of stocks and index is high.
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| Chart 9: First volatility level coefficient for DJX |
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Chart 10: Implied Index Correlation for DJX |
CorrWtdComponent IV%
Let’s consider the index as a portfolio of component stocks with the corresponding weights. Thus, by using the main formula
we can calculate risk of index as risk of portfolio.
We can calculate the theoretical value of index volatility on the basis of implied volatility index (IVIndex) of each component,
and correlations between components’ IVIndex values. This volatility further will be referred to as theoretical correlated
implied volatility of index.
Here  is a component weight in the index,  is implied volatility index of a component stock,
 - correlation between implied volatility indexes of two stocks.
The ratio of theoretical correlated implied volatility of the index to the actual implied volatility is calculated to estimate
the difference between theoretical and real prices of index options. Furthermore, this relationship will be called a second
volatility level coefficient.
If the second volatility level coefficient is less than one, and relatively low, it means that, theoretically, index options are
too overpriced. Thus it is profitable to sell index options. If the second volatility level coefficient is greater than one, and
relatively high, it means that, theoretically, index options are conservative priced and represent a value. Thus it is profitable
to buy them.
In practical terms, since the second volatility level coefficient is based on correlations of implied volatilities, it is a better
measure for a “vega” trade, i.e. trade in which you are trying to capture the relative value based solely on vega. Longer term
trades have lower gamma but higher vega, and so predominant risk factor is volatility. The second volatility level coefficient
tends to perform better for longer term trades where vega is the prevalent risk factor while gamma and theta are minor
risk factors. Comparing second volatility level coefficients for different terms allows to select the best term for dispersion
trades.
EXAMPLE
Chart 11: Second volatility level coefficient for SOX |
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The following chart shows second volatility level coefficient of SOX for a
30 day term. The current value is 0.57. So historically theoretical correlated
implied volatility is lower than actual implied volatility, and this
was observed for all indexes over last few years. Nevertheless, drops and
jumps over histories can show times where options theoretically were
more or less overpriced relatively. So selling index options is profitable
in these periods. |
ADDITIONALLY
It is simply to prove that the theoretical correlated implied volatility of an index, which is calculated on a correlation
adjusted basis, cannot be greater than weighted implied volatility, which ignores correlations.
Thus second volatility level coefficient is always less or equal to first coefficient. As you can see on the chart below, taking
into account the correlation between stocks essentially reduces overall theoretical index volatility.
Chart 12: Time history of WtdCompIV and CorrWtdCompIV for DJX. |
CorrWtdCompHV, %
The theoretical historical volatility of index can be calculated from the main formula on the basis of historical volatilities
of each component and correlations between stock prices.
In this formula  is historical volatility of constituent stocks calculated on the basis of recent 10, 20, 30, 60, 90, 120,
150, 180 days,  - correlation between stock prices for the corresponding term.
The ratio of theoretical historical volatility of index to actual volatility is shown on the chart 13. This ratio that henceforth
will be referred to as third volatility level coefficient indicates how much theoretical historical volatility differs from
actual volatility.
If the third coefficient is less than 1 and low, it means that theoretical performance of the index is less then actual volatility
and trader can gain profit selling index options. As a rule, for American indices the third volatility level coefficient is
less than 1. The higher the value of third volatility level coefficient, the better time for buying index option.
The shorter term options have much less vega risk but more gamma risk, i.e. risk to the price moves of the underlying. On
a practical level since the third volatility level coefficient is based on stock prices and correlations between them, it is a
better measure for a "gamma" trade, i.e. trade in which you are trying to capture the relative value based solely on gamma.
Thus this measure tends to be better suited for short term portfolios where gamma is the dominant factor.
EXAMPLE
Let’s look at the chart 13, which shows third volatility level coefficient for DJX. Current value is relatively high and less than
1, thus it is not a good time for selling short term index options, since the theoretical historical volatility is not far from
actual one. But because the current value of second volatility coefficient is relatively low and less than 1, selling long term
index options can be profitable, because theoretically they are substantially overpriced.
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| Chart 13: Third volatility level coefficient for DJX |
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Chart 14: Second volatility level coefficient for DJX. |
Let’s look at the chart 15 which shows theoretical implied volatility (CorrWtdCompIV) and theoretical historical volatility
(CorrWtdCompHV) for DJX. As a rule CorrWtdCompHV (calculated on the basis of price correlations and historical volatility)
is higher that CorrWtdCompIV (calculated on the basis of implied volatilities and correlations between them). It arises from
higher level of correlations between stocks’ prices in comparison with correlations of stocks’ implied volatilities.
Chart 15: CorrWtdCopmIV and CorrWtdCompHV for DJX |
HistCorrWtdCompIV %
Theoretical correlated implied volatility of an index is calculated on the basis of implied volatilities of its constituent and
correlations between them, the theoretical historical volatility of index is calculated from the main formula on the basis of
stock historical volatilities and correlations between stock prices.
But we can use mixed data, historical and implied, to calculate theoretical volatility of index. Such volatility is calculated
by using the main formula, where  is implied volatility index of a component, and  is correlation between stock
prices, not between implied volatilities.
The chart 16 shows the ratio of such a theoretical volatility to actual implied volatility of index. As a rule, this relation is
less than one.
Chart 16: HistCorrWtdCompIV/IV Index for DJX |
The only difference between CorrWtdCompIV and HistCorrWtdCompIV consists in different correlations used in calculations.
One can observe that as a rule HistCorrWtdCompIV is higher than CorrWtdCompIV since stock prices correlations are higher
than correlations between implied volatilities.
Chart 17: 30 day HistCorrWtdCompIV and CorrWtdCompIV for DJX. |
3. DISPERSION STRATEGY
Volatility Dispersion Strategy is considered to be one of the best
working strategies in sophisticated analytics. It can be explained
by the fact, that historically index volatility has traded rich, while
individual stock volatility has been fairly priced. Thus the dispersion
strategy allows traders to profit from price differences using
index options and offsetting options on individual stocks.
The dispersion strategy typically consists of short selling options
on a stock index while simultaneously buying options on the component
stocks, i.e. leaves short correlation and long dispersion.
The reverse dispersion strategy consists of buying options on a
stock index and selling options on the component stocks.
The success of the Volatility Dispersion Strategy lies in determining
whether the time is right to do a dispersion trade at all, and
selecting the best possible stocks for the offsetting dispersion
basket.
THE BEST TIME FOR THE DISPERSION STRATEGY
When selecting the best time to engage in the dispersion strategy, you should to pay attention to the following
parameters:
IV Index of index
If the relation of actual implied volatility of index to historical volatility is greater than 1 and relatively high, it is
good time for selling index options, since it means selling expensive options on the stock index.
If the relation of actual implied volatility of index to historical volatility is less than 1 and relatively low, it is good
time for buying index options, since it means buying relatively cheap options.
Implied Index Correlation
Implied Index Correlation should not be too far from the maximum registered value, since the dispersion strategy
works better if the implied volatility of the index is highly correlated with the implied volatilities of its stock components.
WtdCompIV/IV Index – first volatility level coefficient
The low value corresponds to high Implied Index Correlation. So the lower the value of the first volatility level coefficient,
the better the time to engage in a dispersion strategy.
CorrWtdCompIV/IV Index – second volatility level coefficient is a better measure for for longer term trade.
If the second volatility level coefficient is less than one and relatively low, it means that theoretical performance of
the index is less then implied by market and a trader can gain profit selling index options. Otherwise, if the second
coefficient is greater than one and relatively high, it is a better time for buying index options.
CorrWtdCompHV/HV - third volatility level coefficient is a better measure for shorter term trade.
If the third volatility level coefficient is less than one and relatively low, it means that theoretical performance of the
index is less then implied by market and a trader can gain profit selling index options. Otherwise, if the third coefficient
is greater than one and relatively high, it is a better time for buying index options.
COMPONENT STOCKS SELECTION FOR THE DISPERSION STRATEGY
Assuming that the timing is propitious for a dispersion trade, the next step is to select the best component stocks to sell
(or buy in the case of reverse dispersion strategy). This step will be discussed in the next article.
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