**Historical Volatility vs.
Implied Volatility Strategies - Part 1**

Please note, as *ART Consulting/Research* is a fee based
service, in the following the results have been "sanitised" to
disguise the specific markets, trading factors, strategy parameters and many
other essentials. Of course, all of the analyses is based on real market
conditions and real world trading considerations (trans cost, funding, etc). For access to the
"un-sanitised" results, and for analysis tailored to your needs please
submit an email via Request
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Also see, **
ARTicles**:
ARB101 - An Abridged Introduction to Arbitrage Trading.

One particularly compelling aspect
of options trading is the question of the relationship between historical
volatility (v_{H}) and implied volatility (v_{I}), if any.
Many traders have a sense that there might be some connection between
these two measures that could be helpful in forecasting or
trading. This is a complex matter, and the first step (and so
this "the Part 1") is to discover if there are possibilities for
improving the P&L based on such considerations (i.e. "*is there
any chance at all of making money with this approach*?").

The
image to right (click to ENLARGE)
illustrates histories of both historical volatility (v_{H}) and implied volatility
(v_{I}) for a CME traded contract. In this particular
history of data, there appears to be consistent drift in "volatility
spread" (v_{H} - v_{I}).

Can this be modelled?

Can this be (predictably) exploited for profit?

Can this be done consistently, and with sufficiently high risk-adjusted
returns to warrant assignment of capital, limits, salaries, etc.

This area is particularly tricky in
part due to the complications both of understanding and modelling
volatility, and that of the coupled complications of assessing the correct
or best trading strategy to exploit any potential benefit from such
relationships.

1)
__Modelling Considerations__: notice that options models are
required for both valuation and rebalancing calculations for any trading
strategy aiming to profit from a v_{H} vs. v_{I}.
There are many such issues to consider. For example, there is no
guarantee that an empirical measure such as v_{H} is consistent
with the measure v_{I} assumed in a theoretical model such
Black-Scholes (see TG2RM1st
Chapters 8-12 for the
"mountain range" metaphor for theoretical uncertainty vs.
reality and P&L impact). Another consideration is that empirical
measures necessarily measure the past, while implied measures
(partially) reflect expectations about the future. For these
and many other reasons, just arriving at a suitable and practical level
of "relationship assessment" is a very complex process.

2)
__Trading Strategy Considerations__: The modelling considerations may
or may not be performed independently from trading strategy
assumptions. Assume for a moment that those relationships exist (a
very big assumption). Now, there is still the matter of assessing
the actual trading strategy for the (risk-adjusted) P&L
performance. In options trading, the strategies to exploit such
relationships almost always rely on dynamic/synthetic replication.
This means that the holding period rebalancing sequence of trades (i.e.
the strategy) will vary depending on how the volatility relationships
above are expressed. Adding to the complications is the
observation that options position cannot isolate volatility, and so
there will be other effects to contend with (e.g. at least the usual
Greeks etc.). This means that the strategy consideration and the
resultant risk-adjusted P&L's will require very careful
interpretation. For example, a high frequency rebalance Delta neutral
vanilla call option strategy's P&L may or may not be directly
comparable to a Delta/Vega neutral straddle strategy given certain
specific issues in the nature of any volatility relationships determined
in step 1) above.

The first step is to determine if
there is any hope of a P&L advantage at all. If such exists,
then more resources might be committed for further analysis or
trading. One answer to this question follows from a PaR analysis of
various (v_{H} vs. v_{I}) relationships combined with
various trading strategies.

As always, PaR analysis with
the Pr/rO ®
software employs very realistic
holding period forward and backward simulations of market scenarios
and trading strategies (including all of the usual *nitty gritties*
faced by traders doing real trading, such as transactions costs, funding,
liquidity constraints, credit, etc ... ). Other examples of PaR
analysis are provided in the *ARBLab*
Samples section, such as *ARBLab*:
P&L Optimal Options Rebalancing - 1,
while all of TG2RM1st
- Chapter 12 is
dedicated to the introduction of PaR analysis.

####

#### A first analysis

Assume
that some (v_{H} vs. v_{I}) relationships combined
with various trading strategies have been determined. The PaR
holding period net-P&L's from simulating
4,676 holding period trading strategies is shown in the figure to the right
(click to ENLARGE). Each
point is a net-P&L for the stated conditions. If the market
pricing convention was truly arbitrage free, then the points in this graph
should be distributed evenly in "three space". The
"trading factors" *Factor X* and *Factor Y* are real world trading
parameters as might be used by any trader but have been disguised for the
purposes of this discussion. A third "trading
factor" has been used for colouring the points (so, in effect this is
4-dimensional plot).

Two of the more important observations are:

1)
Notice that the lower-right image shows that increasing level of *Factor
Y* lead to consistently negative P&L's. Meaning that
"doing this trade the other way around" would lead to consistently
increasing P&L (though there is some asymmetry due to operating
costs etc).

2)
Notice that the "*Colouring Factor*" also shows a P&L
bias. In particular, the points at the green-end of the spectrum
are consistently in the negative (and so again implies that the inverse
trade would be consistently profitable).

The pattern in the "dots" can be more
easily seen with a surface fitting approach as in the image to the right (click to ENLARGE).
This plot illustrates more clearly that increasing levels of *Factor Y*
lead to consistent P&L bias (and so a consistently tradable condition).
Using an Upper and Lower surface (in addition to other data verification
methods, as is provided here), provides statistical verification of the
"credibility" of the results.

The implication of this "Step 1" analysis is that there is
indeed a possibility to trade "some"
(v_{H} vs. v_{I}) relationships profitable (or at
least justify further analysis).

As usual, caution is required.
The analysis here, though including thousands of trades, and incorporating
many real world factors cannot be taken as any perfect predictor of the
future, and additional specific analysis may be required for your due
diligence.

If you are interested in obtaining research results on this issue please
Request
More Information and please feel free to indicate a few specifics of
interest to you.

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