Model Arbitrage: Fat Tailed FX Options

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It is widely known that most markets do not follow a perfect Normal distribution, and that FX markets in particular often exhibit pronounced leptokurtic (fat tailed) phenomenon.

An approach to model arbitrage verification

In another ARBLab analysis, Model Arbitrage: Risk/P&L for Holding Period Strategies with fractal-adjusted Options Methods (and all of TG2RM1st - Chapter 12 is dedicated to the introduction of PaR analysis),  it was shown that there may be various fundamental reasons for, and strategies with which, to examine and exploit (model) arbitrage.  Notably, the analysis relies on using traded options prices and traded underlying prices (thus eliminating model pricing assumptions), but using different models for the rebalancing calculations (with a variety of rebalancing strategies/structures) to analyse the net-P&L left over after some holding period.  The analysis is repeated many times for many markets, over many market periods, etc, with both forward-testing and back-testing methodologies.

Consider for example the net-P&L's from a particular options structure that is being rebalanced with a specific strategy.  The figure to the right (just click on to ENLARGE it) shows 3214 points.  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".  Instead, notice that there is a very definite structure to these P&L's, and importantly the pattern indicates that increasing time to expiration is "out of whack" by something that looks similar to a "Root-2" discrepancy, while the discrepancy increases with increasing implied volatility.  Both of these factors support the contention the contention of fat tails, and now it can be seen in terms of P&L.

The pattern in the "dots" can be more easily seen with a surface fitting approach as seen to the right.  This result implies that for the given market conditions, the person trading on the model prices as and rebalance formulation will be making money consistently if they are "this way around on this structure", and so implies that arbitrage exists (or excess profits exist).

By comparison, consider the P&L results with a special model that permits adjustment to the structure of the distribution for the identical conditions from the experiment above.  Now the net-P&L's are more evenly distributed in three-space, without any obvious bias one way or the other, and so no apparent arbitrage opportunities exist, implying that this modelling methodology is "efficient".

Again, a surface fit may be used to better illustrate the trends.  In this case there are two surfaces showing the "upper-" and "lower-envelope" that defines the space filled by the net-P&L points.  Here again, it is clear that there is essentially no bias.

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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