This article is authored by Dr. Rick Nason, Principal of RSD Solutions and Associate Professor of Finance at Dalhousie University, Canada
Arbitrage models are the basis of pricing for many investment structures. Arbitrage pricing is a very powerful and useful methodology for determining the fair value of many different types of structure – both simple and complex. In fact, arbitrage models are so useful that there is a great temptation to apply them to “almost arbitrage” situations by using increasing levels of mathematical sophistication. They work quite well most of the time when the necessary assumptions about the “almost arbitrage” conditions are not too wild or inconsistent with actual market activity. This memo will claim however that this is when reliance on arbitrage models is most dangerous. All available and necessary mathematical tools should be applied in attempts to price financial instruments when true arbitrage conditions apply. However mathematical sophistication should be used very warily when “almost” arbitrage conditions exist.
Arbitrage in finance is akin to finding a tautology. In other words, arbitrage is discovering two different investment strategies that produce identical outcomes under ALL future economic scenarios. Note that the definition does not rely on any assumptions whatsoever. Mathematically we can state that we have one investment portfolio A, which at some time in the future will be economically identical to an investment portfolio B. If the current price of portfolio A is different from that of portfolio B then a risk-free arbitrage profit is to be made.
It is rare for a risk-free arbitrage profit to exist for an extended period of time. Rational investors will exploit the situation which in turn forces the relative prices of portfolio A and portfolio B back into an equilibrium whereby the risk-free arbitrage profit will no longer exist. The fact that rational traders will quickly drive risk-free arbitrage situations out of existence and restore equilibrium to the markets is the basis for many of the pricing models and pricing formulas in existence. For instance the models for pricing index futures and interest rate swaps and currency forwards are all based on the principle of no risk-free arbitrage.
Arbitrage models tend to work very well; particularly over extended periods of time. Markets may occasionally stray from equilibrium, but when the possibility of setting up a risk-free arbitrage exists, it is virtually certain that rational traders will quickly pounce on the situation, take their profits, and in the process drive prices back into line. Therefore arbitrage models have become the staple of the financial modeler or pricing analyst. The success of arbitrage pricing has led to a natural increasing of the sophistication of the field, with modelers building ever more complex arbitrage structures, with ever increasing levels of mathematical sophistication in order to price ever more complex financial instruments and structures.
As arbitrage pricing models become ever more complex, it is almost always necessary to stray from “perfect arbitrage” to “almost arbitrage” by introducing either market based assumptions (such as markets trade continuously), or statistical assumptions (market returns are normally distributed), or behavioural assumptions (traders remain profit seeking). Generally all three types of assumption are made. Furthermore these very benign looking assumptions are made with little concern, and indeed such assumptions often prove to be suitably valid. This gives the analysts increasing confidence to raise the level of sophistication of their models and of their subsequent mathematical apparatus.
It is when there is a straying from perfect arbitrage to “almost arbitrage” that great danger lurks. Furthermore this great danger lurks under a heavy veil of sophisticated mathematics – mathematics which few people other than the analyst are likely to understand with any degree of confidence.
Take for example what is probably one of the most famous and the most successful “almost arbitrage” models that exists – the Black-Scholes Option Pricing Model (BSOPM). The BSOPM is based on the arbitrage situation that an option on an asset can be “almost” perfectly replicated by dynamically trading the underlying asset according to a set of parameters known to traders as the “Greeks”. While the underlying mathematics (stochastic calculus) that went into the development of the model is beyond the understanding of the vast majority of financial traders, the dynamic replication concept itself is relatively well understood and easily illustrated. The Nobel Prize winning BSOPM is widely used and successfully so. It has been so widely accepted that few traders ever pause to question its validity and most financial analysts will use it to benchmark other models they have under development.
The assumptions of the BSOPM are relatively benign. The first major assumption is you can trade the underlying – utilizing both long and short trades – without affecting the actual market price. With 24 hour trading, and the huge amount of liquidity available for most securities this appears to be an innocent assumption. The second assumption is that market prices move smoothly – that is that market prices do not gap from one price level to another. This too appears to be an innocent assumption when one considers that most exchanges require market specialists to limit price moves and have circuit breakers in place to limit extraordinary large daily moves. There are a few more equally innocent assumptions to the BSOPM, but these two will suffice to illustrate the argument for now.
The problem with BSOPM is that it works very well almost all of the time, but very poorly when you need it to work the most – i.e. during a market crisis. This was a lesson learned by Fisher Black and Myron Scholes in the early days of development of the formula, and then relearned (?) by Myron Scholes and others during the meltdown of Long Term Capital Management. Upon discovering the formula, Black and Scholes decided that they would exploit their knowledge by trading options. Their strategy was simply to buy underpriced (according to the formula) options and sell overpriced options. According to an article by Fisher Black , their results were less than profitable. Later when the hedge fund Long Term Capital Management (LTCM) started trading they were initially very profitable by using the BSOPM (amongst other models one assumes). However these two the assumptions came back to haunt LTCM traders. The supposedly benign assumptions of being able to trade continuously, and that the markets do not gap, broke down simultaneously. Since they had high confidence in their models (and why shouldn’t they have confidence as LTCM had been massively profitable from start-up) they were highly leveraged, and carrying very large positions. When they tried to trade the underlying assets (to capture the arbitrage of the replicating portfolio) LTCM’s traders realized that their assumption of being able to trade continuously without affecting market prices was invalid. This was compounded by the fact that the markets started to gap with the absence of liquidity which only made their situation exponentially worse. The result of course is that LTCM was wiped out.
Similar mistakes are currently being made in the credit markets. Sophisticated models, based on benign assumptions, are being utilized to price complex structures such as CDO’s. As with all progress, the mathematical tools being used are increasingly sophisticated, and their use is justified by the reasonableness of the assumptions, and the fact that the models appear to be valid as confirmed by market results. The issue is that these are “almost arbitrage” models, not true arbitrage models. What exists is not a tautology, but instead a financial construction (generally based on modeled cash flows) that looks “almost” like a tautology, but only after imposing a few innocent assumptions, such as default rates and default correlation levels will remain in their traditional range of values. Thus we find investors realizing that their AAA tranches might now be worth less than 10% of face value relatively soon after issuance.
The point of this article is not to argue against the use of sophisticated mathematics. In fact mathematics is not the problem at all. The issue is the assumption of arbitrage when a true arbitrage - a tautology - does not in fact exist. Mathematics is a field of tautology. Assumptions are not.
When a true arbitrage situation exists, then financial modelers should apply all the mathematics that is necessary (but not any more than is necessary) to successfully uncover the valid price of an asset or structure. However when an “almost arbitrage” situation exists, caution should be the keyword, as sophisticated mathematics will only obscure the situation and mislead the trader when the underlying assumptions start to slip – as they inevitably do just when you need them to remain valid the most. The tough part is figuring out what is a true arbitrage versus an “almost arbitrage”.
