Mathematical Finance

I wrote this piece for the National Institute for Economic and Social Research's Rebuilding Macroeconomics project.

There are three types of mathematicians: those that can count and those that can’t. This aphorism challenges the public perception of mathematics as being concerned with calculation and is liked by mathematicians because it enables them to highlight what mathematics is concerned with, which is identifying and describing relationships between objects.

A more sophisticated misunderstanding relates to the way mathematics is conducted. The error originates in how mathematicians present their work, as starting with definitions and assumptions from which ever more complex theorems are deduced. This is the convention that Euclid established in his Elements of Geometry and led Kant to believe that synthetic a priori knowledge was possible. Euclid actually started with Pythagoras’ Theorem, and all the other geometric ‘rules’ that had emerged out of practice, and broke them into their constituent parts until he identified the elements of geometry. It was only having completed this analysis did he then reconstruct geometry in a systematic way in The Elements. Today the consensus within mathematics is that the discipline is analytic, from observations, not synthetic, outside of mathematics there persists a belief in the power of pure deductive, synthetic a priori reasoning.

Physical sciences are in tune with what mathematicians do. This is exemplified by Newton who gathered observations on the planets and invented calculus to interpret the data. On this basis he concluded that momentum was being conserved and deduced the gravitational law. The key idea originating with Newton is that momentum is an invariant in a dynamic system. This is understood most clearly when presented using calculus, the mathematics Newton invented. Since Newton, all significant advances in physics have been associated with the identification of an invariant (momentum, energy, increase in entropy, speed of light) and inventing clear and succinct ways of describing objects (mathematics) that re-presents nature based on an invariant.

Finance has developed a mathematical theory in the Fundamental Theorem of Asset Pricing that has the same status in mathematical finance as Newton’s Laws have in classical physics. The central principle, analogous to the conservation of momentum, is that of ‘no-arbitrage’. The Fundamental Theorem of Asset Pricing states that if an asset is priced on the principle of no-arbitrage then there is a reciprocal relationship in the exchange. There are at least two ways of understanding this principle. It is a version of Euclid’s ‘First Common Notion’: if A=B and C=B, then A=C. Money takes the role of “B” and arbitrates the value of A relative to C. Alternatively, it is a version of the scholastic argument that a riskless profit is a shameful gain (turpe lucrum).

The no-arbitrage principle is justified through Ramsey’s ‘Dutch Book Argument’ that requires markets are mediated by jobbers (market-makers or dealers in the US) rather than brokers. When a jobber quotes a price, they do not know whether the counter-party is looking to buy or sell at the price. The jobber will quote a price at which they will buy and a higher price at which they will sell. They signify confidence in their quote by having a narrow difference between the prices. If a jobber quotes a price that another trader believes is wrong, the trader will take the quote, immediately moving the market. These jobber-mediated markets are, therefore, essentially discursive. Jobbers are engaged in making assertions as to prices, which are challenged when others take the quote; this is ‘market making’. If the market agrees that a jobber has correctly priced the asset, no trading will take place - silence is consent - and the market dissolves.

Jobbers do not hold assets and prefer trading in financial contracts rather than hold physical assets, they have no commitment to the assets they trade and identify themselves as taking long and short positions rather than buying or selling. While they lack commitment to assets, jobbers must be sincere in their statements, they must believe the quote is right. This means, that in the face of radical uncertainty, a jobber’s price quote is reliable, it can be trusted.

The significance of reciprocity in markets rests on the no-arbitrage principle that can only be justified if exchange is being conducted by jobbers, who will buy and sell at the quoted prices. Markets in economics tend to be based on brokers who bring property owners, one a buyer, one a seller, together. The focus on broker-mediated markets rather than jobber-mediated markets means that the importance of reciprocity in exchange is obscured. The different emphasis is rooted in financial markets being concerned with uncertain futures whereas economic markets are concerned with immediate scarcities.

In modern business, if a manufacturer can sell a product at an enormous profit, creating an arbitrage, they are succeeding. Economic theory argues that in the presence of these excess profits, competitors can come in and the price of the product will fall. This appears to be no different to the situation in a jobber-mediated market: jobbers will bid (buy) at the cost of production and offer (sell) at that cost plus a risk premium, just as manufacturers will do in a competitive broker-mediated market. However, while the ultimate point might be the same for jobber and broker mediated markets, the routes to the point are different. For jobbers, no-arbitrage, and hence reciprocity, are iron laws that must not be breached, ever. In broker-mediated markets, arbitrages are transitory and the ideal is to capture them before they disappear; it is a virtue to break the principle of reciprocity. Prices at which exchange takes place in jobber-mediated markets are always disputed prices, but sincere; in broker-mediated markets, prices are always accepted, if not fair.

Consider some thought experiments. If a manufacturer, making arbitrage profits, was obliged to buy identical goods, manufactured by others, at the prices they themselves quoted, would they quote the same price? If a slum-landlord had to live in the accommodation they rented, would they rent inferior quality accommodation? Public services are often expensive because they are of a quality that the providers would like to receive. These examples highlight the ethical nature of dual-quoting, it imposes the categorical imperative: do unto others as you would have them do unto you.

Financial instability has long been blamed on jobbers, who trade ‘paper’ and lack commitment to material assets, they are 'disinterested'. However, bubbles are a consequence of property owners ‘ramping’ assets and selling them above their intrinsic value. The failure of Long Term Capital Management in 1997 was precipitated by an apparent arbitrage, in the ‘asset swap’ strategy involving rock-solid US government debt, being an illusion. The Credit Crisis was a result of investment banks believing they could construct mortgage backed securities (MBS), out of ‘real’ assets, for less than their worth, not realising the inherent risks because they believed in arbitrages. Investment banks have been fined for selling MBS above their internally recognised value; they were being profit maximisers but insincere. There is evidence that the ‘Bitcoin’ bubble of December 2017 was a consequence of it being easy to buy Bitcoin, but difficult to sell; something not possible in a jobber-mediated market. These are all situations where financial instability originates in a belief that the no-arbitrage principle could be ignored or that prices could be insincere, and so it was possible to earn risk-less profits.

Recognising that the no-arbitrage principle is analogous to Euclid’s First Common Notion means that arbitrageurs should be regarded in the same way as promoters’ perpetual-motion-machines are: mis-guided cranks. It also emphasises that exchange should be reciprocal, it should not involve profiting at another’s expense. Mathematics only works on the basis of Euclid’s First Common Notion; markets only work well on the basis of reciprocity.