** Model Arbitrage: P&L Performance of
fractal-Adjusted Options Trading**

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 factors. Of course all of the analyses is based on real market
conditions and real world trading considerations. For access to the
"un-sanitised" results, and for analysis tailored to your needs please
submit an email via Request
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It is a generally accepted fact that
options valuation models are imperfect due to a variety of (simplifying)
assumptions that are made during their development. The
Black-Scholes formula includes a number of shortcomings including its
rebalancing assumptions (see for example P&L Optimal Options Rebalancing - 1,
and all of TG2RM1st
- Chapter 12 is dedicated to
the introduction of PaR analysis),
market risk assumptions and so forth.

One interesting assumption in the
Black-Scholes model (and virtually all options models, term-structure
models, etc) is the assumption about the behaviour of the underlying
forward market, and the time evolution of these prices/rates.
Notably these models assume that forward prices have a Gaussian
distribution and evolve with a variance proportional to the square root of
time, sometimes referred to as geometric Brownian Motion. A key
reason for this is assumption is that while it does produce a forward
price distribution that "appears OK", in reality changing this
assumption can make the mathematics intractable. In any case,
changing market convention formulations has certain difficulties, for
example market makers "must show the market convention price"
(or end up with one way flows), and have some serious explaining to do
when management and regulators review the P&L.

However, even a cursory comparison
of true market evolution illustrates that the market does not actually
evolve in this simplified manner, and in some markets such shortcomings
have been long recognised and traded (e.g. fat tailed FX options).
So is it possible to produce a formulation that in some consistent manner
out performs the marker convention, and still allows traders to carry out
their business in a sensible manner?

### An approach to model arbitrage strategies

Here we propose one approach for
model arbitrage strategies that has two components. First, the
strategy relies on synthetic replication, so that the trading is done at
market prices, but the rebalances are done via the "new
model". This obviates any question of tradable prices and
reporting/regulatory issues. If the "model" is indeed
better than the market convention, then the arbitrage profits will arise
over the holding period of the strategy (and also nobody is "bonused"
prior to the profits being booked).

Second, we examine a fractal
dimension adjusted options (FAO) valuation approach on the hypothesis that
this is a "better" model of the evolution of forward prices than
pure geometric Brownian Motion on its own via this a synthetic replication
approach. There are many other aspects to the model that may be
tested as well, such as jump diffusion etc, but for present purpose the
discussion is focused on FAO (also please see *ARTicles*
Chaos and Predictability
in Finance - Parts 1 through 3:
for a lucid introduction to chaos and fractals, though of course, the machinery
that produced the results here is very considerably more sophisticated
than the simple illustrative examples in those *ARTicles*).

Of course at some stage we will need
to "strap on" a few such trades and see the result, but first we
back test the model against different synthetic replication
strategies. The replication strategies can take on many forms from
simple delta, delta/gamma trading to other more sophisticated
methodologies.

The back testing is performed by
actually running the position with the strategies against real market
data. All options prices are taken from the market data, but the
sensitivities and rebalance calculations are made using the FAO. All
rebalances are done at market prices, and adjusted for bid/offer, funding,
transactions costs, etc. At the end of such a holding period
calculation, the net holding period P&L is calculated. This
process is repeated for many positions using a very large database of
market prices.

Figure 1 shows the results for one
particular series of such holding period calculations against a well known
and liquid contract using FAO. Each dot in the image in Figure 1 a)
is a the net holding period for an entire position simulation, and there
are 4,881 such holding period simulations shown. The vertical axis
shows the net P&L, while the x-axis, y-axis, and the
"colouring" show the affect of "trading factors".

Figure
1 a) <click the image to enlarge> on the right shows a
"cloud" of points. Each point is the net P&L for an
entire trading strategy, over an entire holding period, and these points
have been plotted against two "market conditions", and have
been coloured by a third "market condition". The min
plots on the right hand side of the image are the "edge-on"
views of the main 4-D plot.

Figure
1 b) <click the

These "trading factors"
have been sanitised for the purposes of this presentation, but we track
dozens of such factors and may include items such as underlying prices,
volatility, moving averages, position details (e.g. strikes, structure,
etc), and many possibly more sophisticated components that can be tracked
by the trader sitting in front of a screen. In this way, if a
trading opportunity is found, it is immediately expressed in "trading
terms". Often at least 7 or 8 such factors and additional
analysis are required to decide if indeed there is a cheap/dear or
arbitrage condition.

Performing an 8-dimensional analysis
is a bit involved, but the 4-dimensional image in Figure 1 a) illustrates
the basic idea. So what does it mean anyway?

With a bit additional machinery it
can be shown that the "cloud" of P&Ls in Figure 1 a) is not
merely a random "blob" of points, but rather it has "structure".
If the market convention was in fact "efficient", then this
cloud would be essentially random with a negative bias due to trading
costs. But it is not.

Moreover, fitting a surface to these
types of clouds results in images such as that in Figure 1 b). This particular
image is plotted against "trading factors" different from those
in Figure 1 a). It shows that there is a kind of valley running down
the length of the surface. Importantly the valley is always in the
"loss". This means that for the range of "trading
factors" covering the valley, the FAO trading strategy/position
tested here should have been "done the other way around" (i.e.
bough instead of sold, or sold instead of bought).

These results suggest that the FAO
strategy does appear to have some benefits over the market convention
valuation, and there measurable "trading factors" that at
certain levels imply an "opportunity".

So is it time to "strap it
on" ... may be. Though in our view such analysis requires
considerably more depth than shown here (e.g. translating back testing to
results to trading is imperfect and so requires attention, and there are
other issues to do with assessing the risk of the position and relating to
the required return on capital etc.). There are also many more
issues with trading decisions that are mandate specific that need to be
addressed.

If you are interested in obtaining research results on this issue please
Request
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interest to you.

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