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Optimal Rebalance Strategies and

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Market Conditions Dependent Risk Limits

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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" issues^{1}. 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 practice^{2}. 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/book^{3}.

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 examined^{4}. 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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