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07 March 2018

More Market Mathematics



Writing last week about LTCM and its mathematical modelling of financial markets started me thinking a bit more about whole topic. At first sight, it seems eminently reasonable to project the certainties of statistics onto the movement of markets. After all, like a coin or a dice, there are a fixed number of possibilities for each price change - it can go up, go down or remain the same. We don’t know in advance which will be the outcome on any given occasion, and from there it’s a small step to assume that with sufficient results we will see the bell curve of a normal distribution. Armed with that  assumption, we can then go on to build lots of models to predict market behaviour. Then, we can apply the models and go on to be hugely successful traders and money managers, because we know that, deep down, our implicit assumption of rational markets must be right, because the mathematics and the models tell us so.

Well, yes, up to a point. Want to price an option? Easy. Run it through the Black-Scholes model, and you’ll get your answer. And you know what? Everyone else who prices the same option will come to the same answer. That just reinforces our belief that it’s right. But let’s turn that around for a moment. Everyone uses the same model, and that’s why they all come to the same price. That doesn’t make it empirically right, though; it just means everyone is using the same model. For trading purposes, that generally works, of course, because it creates an agreed basis on which value can be predicated. ‘Generally’ is the important word there, though; but what’s more critical are the occurrences outside that ‘general’ area. One of the the conditions set for the Black-Scholes model is ‘ceteris paribus’, which means ‘other things remaining the same’. In other words, we can vary the inputs to the model, but for it to work, it assumes that the relationships between the inputs - in other words, the framework of the world in which it lives - remain the same.

I would suggest that that is unrealistic when we transfer from the theoretical to the practical. The time when we need the model to function best is precisely when the market moves violently and becomes illiquid. When markets are stable and liquidity is good, then the dynamic hedging required by the model is all pretty straightforward. But when things lose that serenity, then the option trader - or, let’s be clearer, the option seller - can suddenly be in a position where the required hedge sales or purchases are impossible to transact at the necessary price levels. Now, one could argue that that is just where the skill and experience of the trader becomes relevant; but that should frankly be irrelevant, because the purpose of the modelling is precisely to remove that human element - that emotive trader’s gut feeling - from the business. 

If liquidity (or its absence) was LTCM’s Scylla, then Charybdis was the excess of leverage. If we can accept my proposition that violently moving markets baffle the model, then to multiply one’s position many times using borrowed money or credit lines will lead into even further problems. Where a position supported by adequate underlying capital may be able to weather the storms (and bear in mind that the model ideally looks for better than just not going under as its expectation) one that’s multiplying the risk of that capital by twenty, thirty or more times will almost certainly not manage to survive. 

I’m not a mathematician like Miller, Merton, Black and Scholes; I can pick holes in the models, based on my experience of markets, but I don’t actually have anything to offer in replacement. What I can say is that we use the Black-Scholes model because we use it, not because it is intrinsically a demonstrably correct picture of market behaviour. That’s important to remember. What I can also say is that any model which purports to remove the human element from a market is going to have problems. The coin and the dice have no memory; but the market does (which is probably the basic fallacy of the position of Miller, Merton, Black, Scholes and the rest). Traders remember what happened last time and modify their behaviour accordingly. Thus, ceteris paribus does not apply. The framework, the architecture, of the market changes under the influence of human interaction and that, in a nutshell, is probably why we can’t apply mathematics to economics in the way we can to physics. Economics - of which finance is a part - sits like so much else in life in that half dark, half light between art and science.

Will the influence of AI change this? After all, the computers have no memory……or do they? That’s a subject for another article, but, just as a preview, if anything, I suspect AI will result in greater volatility and liquidity issues.    

  


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