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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 More Information.

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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