Lord Copper gave us an enticing commentary on the book (‘When Genius Failed’) about the LTCM debacle, and then followed this up with a perceptive article on the maths of market pricing. As Lord Copper reflects, some aspects of trading, and indeed economics, went mathematical some decades ago. Later chaos theory invited some challenging angles to their maths and econometric modelling.
I think we can all agree that a ton of copper is a ton of copper. We can put it in a warehouse, and it will still be a ton of copper next month when placed on the scales. We can weigh it, measure it, assay it, ship it. Immanuel Kant (1724 - 1804), the German philosopher, would even agree that it is a ton of copper, except that he would say it is copper as we describe it (what he called the phenomenal world), as opposed to what it may be ‘in itself’, which we can never know (the noumenal world). So a ton of copper does have a philosophical foundation per Kant. Even quantum physicists will assign it an exact value, despite copper’s quantum particles whizzing around probabilistically within that ton.
Based on this underlying real world, maths works very well, starting with the digits of our fingers and moving on to Einstein’s somewhat more challenging calculations. There is an exactitude of predictions. The results are always precise and the same, whether delving down into billionths of billionths or moving up exponentially to numbers on a grand scale.
When it comes to economics and markets this precision is not the case. Consider the simple mechanism of price. The trader in the bazaar will say the price is what I can get someone to pay - start high to maximise psychological effect and close high. The economist will say the price is based on supply and demand, so arguably it has a mathematical basis that can be calculated. Both may be right in their way, though I defy anyone to map out oil prices from 1970 to 2000 based on the figures for oil supplied and used. Certainly, we cannot put a price in a warehouse and retrieve that price a month later unchanged. In that sense price does not belong to the same real world as a ton of copper. Does this have implications for the use of price as input for a mathematical model? If we go a step further, then the option price is yet a further step removed from the ton of copper than the price, being based on price changes of the underlying copper. Traditionally the option price may have been taken as a kind of estimate of market trends, but once mathematical models were introduced, price volatility was brought explicitly into the determination and quantified, changing the nature of the estimate made by those using the model.
Arithmetic has problems with non-digital stuff like curves, so we need strange numbers like pi to match a circle’s diameter to its circumference. On top of that, we have calculus which allows infinitesimal gradations along curves to be computed. We end up with a very sophisticated computational mechanism with fine tuned results. This works extraordinarily well in the world of physics, and I would agree that a sophisticated model can be effective in helping to evaluate outcomes in a like for like environment in the man-made financial world. The model can quantify and compare complex scenarios more precisely than gut feel and has a common language for its users. Where the difficulty arises is with the fundamental inputs that these mathematical models require for financial markets, be they to evaluate risk or establish option prices. This is where I come back to price and price volatility. Past prices and their volatilities will suffice as the input of the model in a normal environment with its fluctuations that any chartist or bell curve aficionado knows well. However, aberrations soon lead to catastrophic deviations that are well known to chaos theory. The more perfect, fine tuned model is likely to exacerbate the catastrophe, especially if it were believed by the user to be producing absolute, conclusive values, or to be capable of identifying market inconsistencies that must converge.
I am not suggesting that we avoid mathematical pricing models. They are useful in their way and with general usage effectively become part of the market’s dynamics. Pragmatically, we have to accept markets the way they are and live with what is served up. This includes the influences of fundamentals, sentiments, models, economic fads and, of course, politics and taxation, not to mention wars.
You can slavishly apply your model and win, just as you can have a winning streak in the casino or the foreign exchange markets - eventually your ‘luck’ turns and you lose all. History shows that scepticism of financial models’ deification has not been universal, but I still find it hard to believe that investor and bankers gave LTCM so many billions to squander on their narrow maths vision of the world of financial markets.