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Optimal Rebalance Strategies and

Market Conditions Dependent Risk Limits

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.


It is quite easy to demonstrate (with real world P&Ls) that market convention position keeping strategies are imperfect and, amongst other things, lead to predictably preventable losses (or conversely, lead to repeatable arbitrage opportunities).  There are many reasons contributing to these circumstances, such as poor model assumptions, poor position keeping practices that are inconsistent with mandates and valuation/risk methodologies, not to mention many "real world human factors" issues1.  One contributing factor is the manner in which risk limits are set.  It cannot be emphasised too strongly that the entire valuation/position keeping/risk limits/mandate cycle is a holistic process, and so one must not set any one component in isolation of the other components (see TG2RM1st for introductory guidelines, or Request More Information).


Having said so, consider just the "static nature" of the risk limits process.  Static risk limits are by far the most common practice2.   A deep analysis is required to fully illustrate the issues, but a few examples should be sufficient to demonstrate some key concerns.  One of the most notable and easy to demonstrate difficulties in this category is the almost exclusive use of "linear hedging" for "non-linear positions" such as in options trading, though these problems exist with almost every type of contract/book3


Black-Scholes-Merton risk-neutral valuation theory shows that (an infinite) sequence of linear hedges equates to the non-linear option profile.  This is perfectly true under the many luxurious assumptions of a Black-Scholes "universe" such as no Vega risk, no transactions costs, infinite liquidity, etc.  Of course, those assumptions do not hold in the real world.  As such, one critical problem is the "mismatch" (slippage) between the "hedge book/trade" and the "target book/trade" resulting from a finite number of (costly) rebalances (e.g. instead of an infinite number of free rebalances) transacted at market levels somehow tied to risk limits (e.g. must rebalance when Delta exceeds some level, etc). 


Much worse, however, is that the slippage results in "non-linear P&L effects" as well, and that the slippage depends strongly on not only rebalance strategy but also on market conditions.  For example, visualise the circumstance of Delta hedging a short vol position during calm vs. choppy markets.  "Chasing the Delta" during wobbly conditions tends to lock-in "non-linearly increasing" losses compared to calm conditions (e.g. see TG2RM1st - Chapters 9 and 10). 


Fortunately, there is no need to "guess", since as is customary at the ARBLab, PaR analysis can be used with various rebalancing strategies and many market conditions to obtain a precise and quantifiable measure of the risk-adjusted P&L effects due to risk limits.  Other examples of PaR analysis and the  Pr/rO software are provided in the ARBLab Samples section, such as ARBLab: P&L Optimal Options Rebalancing - 1, and all of TG2RM1st - Chapter 12 is dedicated to the introduction of PaR analysis.


The image to right (click to ENLARGE) shows 1738 net-P&L's resulting from 1738 trades, each being held and rebalanced as required by the structure/strategy under consideration.  The target portfolio is of a "generic" nature and with common characteristics/composition as may be found on almost any trading floor.  Here only two of the many "market condition dimensions"  (Factors X & Y) are examined4.  These factors are real world measurable quantities such as prices, vols, moving averages, etc.  The results have been annotated with coloured circles to help focus attention on a few key aspects of the position keeping/risk management performance.  The blue circle shows a collection of entire trade-P&Ls that appear to have a low risk profile (the P&L is relatively flat), and in this region the P&Ls are also relatively "constant" over a range of market conditions.  The fuchsia and red circles show that P&L's behaviour can be markedly different under different market conditions, and indeed the position keeping performance is rather poor under some market conditions with this particular rebalancing strategy/limit structure.  Much worse is that the P&L performance is not only non-linear, but also asymmetric (i.e. the slippage is mostly against you and so the overall risk-adjusted performance is, by and large, unacceptable).  Notice in particular both the "curvature" of the P&Ls  around the region of the fuchsia circle and how quickly the position keeping performance under these risk limits/strategy can fall-off for even small changes in market conditions.


The image to right (click to ENLARGE) shows an interpolated surface for the 1738 net-P&L's shown above.  This particular interpolation uses "multiple" surfaces to capture various important additional effects.  For the moment, though, it is sufficient that the surfaces may make easier visualisation of the "market conditions/risk limits" issues depicted by the "cloud/point diagram" above.


The image to right (click to ENLARGE) shows the 1738 entire trade net-P&L's resulting from above, but  under a different limit structure (though using the same strategy and range of market conditions).  This particular limit structure is rather "tighter" than that used above, and shows a lower variability in the P&Ls.  Unfortunately this is not quite as good as it sounds.  First, there is still quite considerable variation in position keeping/risk management performance over the range of market conditions shown.  Moreover the range of P&Ls though less variable than above, is even more asymmetric than above, and shows an even less acceptable "overall risk-adjusted performance".  This implies that, net-net, just moving to stricter limits may be worse.


The image to right (click to ENLARGE) shows an interpolated surface for these 1738 net-P&L's.  Comparison of the two interpolated surfaces further emphasises the notion of blind application of tighter/changed limits may not in itself be a good thing.


Moreover, the manner in which the "P&L behaviour" transitions as limits and limit structures are altered is also non-linear, and can be counter intuitive.  For example, a static limit set careful chosen between to the two limit sets above actually out performs both (though is still not as effective as a dynamic approach).


Just a few of the key observations, once again, are:

Limits must be chosen in concert with mandate, models, strategies, etc

Static limits may be suboptimal

Blind alteration of limit structures may make matters worse

Optimal strategies/limit structures may not be obvious without careful considerations, and may require a formalised dynamic review process.

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.


For detailed research results on this issue please Request More Information and please feel free to indicate specifics of interest to you.


1 For example, traders behave differently near bonus dates than at other times, and they behave differently again depending on the year-to-date P&L.

2 In many cases there is some flexibility applied in how the "static limits" are employed, but generally breaking limits is a "career limiting" event, and in practice the "official" limits are almost always independent of market dynamics with the exception of "really big" events (when they are reset haphazardly driven by fear, not reason).

3 Notice that it can be shown that blind application of non-linear (e.g. Gamma/Vega etc) hedging strategies may not eliminate such problems, and indeed in some cases may exacerbate the losses.

4 Actually, the images shows several market factors (e.g. the colouring of the points/surfaces is obtained by applying additional factors), and the graphical machinery has the power to display up to 7-dimensional effects, but that is beyond the current scope.


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Last modified: July 25, 2011