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Mathematics, Finance & Ethics

Tim Johnson

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 really 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; 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.

Physical sciences are in tune with mathematics. This is exemplified by Newton who gathered observations on the planets and invented calculus to interpret the data. He concluded that momentum was being conserved and deduced the gravitational law. The key idea originating from Newton is that momentum is an invariant in a dynamic system.

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.

No arbitrage as an invariant principle

Finance has developed a mathematical theory in the Fundamental Theorem of Asset Pricing that has the same status in finance as Newton’s Laws have in classical physics. The central invariant 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, which we come to, is a version of the scholastic argument that a riskless profit is a shameful gain (turpe lucrum).


he no-arbitrage principle is justified through Ramsey’s ‘Dutch Book Argument’ that requires markets are mediated by jobbers (market-makers in the UK, 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 that price. The jobber will quote a price at which they will buy and a slightly 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 trade on the basis of 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 takes place – silence is consent.

Jobbers 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.

Brokered markets and no-excess profits

Markets in economics tend to be based on brokers who bring property owners (one a buyer and one a seller) together. The focus on broker-mediated markets rather than jobber-mediated markets means that 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 business, if a manufacturer can sell a product at an enormous profit they succeed in making excess profits. Theory holds that in the presence of excess profits, competitors enter the market and the price of the product falls. This appears to be no different to the situation in a jobber mediated market. Jobbers buy (bid) at the cost of production and sell (offer) at that cost plus a risk premium, just as manufacturers will do in a competitive broker-mediated market.

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 or invariant 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 sincere prices. In broker mediated markets, prices are accepted, if not sincere.

Consider some thought experiments. First, if a manufacturer, making excess profits, was obliged to buy identical goods manufactured by others at the prices they themselves quoted, would they quote the same price? Second, if a slum-landlord had to live in the accommodation they rented, would they rent inferior quality accommodation? Third, 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.

Insincere prices and financial instability

Financial instability has long been blamed on those who trade ‘paper’ or lack commitment to material assets. 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. The ‘asset swap’ strategy involving rock-solid US government debt was an illusion.

The trigger for the Global Financial Crisis was investment banks constructing mortgage backed securities (MBS) from claims on ‘real’ assets for more than their worth. Because they believed in arbitrages, they had not realised the inherent risks. Investment banks have been fined for selling MBS above their internally recognised value. They were being profit maximizers but insincere.

Is the ‘Bitcoin bubble’ a consequence of being easy to buy Bitcoin but difficult to sell? This is 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 and that prices could be insincere and it is possible to earn risk-less profits. Recognising that the no-arbitrage principle is analogous to Euclid’s First Common Notion 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.

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