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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 More Information.

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".

FracOpts1.jpg (176907 bytes)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.


FracOpts2.jpg (160759 bytes)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 More Information and please indicate specifics of interest to you.




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Last modified: July 25, 2011