The Reasonable Ineffectiveness of Mathematics in Trading
Mathematics affords virtually unlimited precision, but is limited in its scope.
—paraphrase of Eugene Wigner,
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
Is mathematics a quantum leap forward compared to other methods of thinking? Sure. Its precision beats every other possible “system” and human intuition is limited by experience. One cannot “see” curves without tangents, nor intuit an n- dimensional space. But everyone lives and thinks with intuition in ordinary life. Successful trading is about buying cheap and selling dear: mathematical thinking is an indispensible means to that end. But it can also obscure intuition that necessarily deals with the inexact definitions of everyday life. Further, a mindset wholly engrossed in the mathematical development of an axiom base can take one far from practical relevance.
Don’t Be a (Norbert) Wiener
Janos (Johnny) von Neumann reinvented himself multiple times before he died of cancer. He was a key contributor to the foundations of mathematics, the pioneer of ergodic theory (Birkhoff tried to cheat von Neumann here), did some slick investigation on the invariant subspace problem, the expositor of the underappreciated subject of continuous geometry, the one who systematically investigated game theory, the human calculator behind components of the first nuclear weapon, and considered the father of machine computing. This doesn’t even cover all his achievements, much less capture their depth. He knew how to party too, but that comes further down.
There was nothing superhuman about him: he just had a solid, thoroughly normal personality with a tremendous work ethic. If there was any genius in him, it was his instinctive drive to stay close to directly observed facts, and thus application. This is not to say that he didn’t theorize. In fact, his axiomatization of quantum theory is still one of the best reads on the subject because his axioms are stylized from observation, straightforward, and easy to relax and restrict depending on where you want to go.
This stands in stark contrast to Norbert Wiener’s approach to constructing a view of the world. He was the first to rigorously characterize Brownian motion, but he exclusively used a (then) challenging and formal measure-theoretic construction. Wiener avoided drawing connections to probabilistic methods because probability theory had a bad reputation among mathematical colleagues of the day. Probabilistic intuition and measure theory reinforce each other: many who could have benefitted most by the advance were left out of the discussion.
Perhaps these differences in approach are based on personality. Von Neumann was reasonably courteous, somewhat intolerant of pretension and illusions of superiority, generally friendly, and a fun-loving people person. Wiener was by accounts childish; overly-sensitive; a self-absorbed, difficult person to be around. The divide between their personalities was diametric.
The current divide between mathematics and hard sciences couldn’t be much wider, either. Mathematicians talk amongst themselves. Engineers talk amongst themselves. As the respective dialects grow mutually unintelligible, both tend to ignore natural problems that anyone can understand. Instead, both delve into their own franchised abstractions. For example, Lie group representations permit an abstraction of Fourier methods for solving differential equations, but the generalization is, with some exceptions, limited to equations amenable to separation of variables. Lie groups do provide a vantage point where one can survey interesting mathematical vistas, but for close-up details, one often has to fall back on solution methods engineers routinely use. These methods also fail to overcome current challenges. Engineers busy themselves in altering one assumption after another to gain slightly more general solution methods (or accuracies) that persistently fail to adhere to demands of some real-life problems.
Long gone are the days when Isaac Newton built the foundations of calculus by observing the effect of gravity on an apple, or a wave equation to measure the speed of sound. Laplace corrected Newton’s work, not based on symbolic manipulation, but because of the values his experiments yielded. Cauchy elaborated Newton’s theory of light dispersion using the same prism as a starting point. Well into this century, Born, Heisenberg, and Schrodinger used harmonic analysis merely as a tool to substantiate quantum observations.
Von Neumann stepped back and modeled experimental statements via a Hilbert space and deduced how the statements logically connect. Wiener refrained from deploying an expository method in common use and wide relevance because it was distasteful to colleagues of the day. Everybody loses when experts write mutual admiration notes to each other, coded such that they either bore others to death or exclude them from grasping the point.
This is just the tip of the iceberg. If one thinks the difference between mathematics and engineering is huge, then consider the difference between them and softer subjects like economics (no offense meant). Isolation leads to shallow thinking and research in applied settings, and mathematics needs relevant practical problems to renew itself.
When the Emperor announced the Japanese surrender to the Allies after World War II, his subjects couldn’t understand what he was saying. After centuries of distance from his subjects, the Imperial court no longer spoke Japanese. The recording preserves a jumble of lilting tones and strange cadences that left everyone but a handful scratching their heads.
Action Precedes Data
Johnny von Neumann was terrible at poker. This wasn’t because he got too drunk during the games: Stefan Banach once got Neumann so drunk on vodka that he had to run to the toilet to puke. After the purge, he reentered the Scottish Café and resumed the functional analysis discussion without missing a beat.
The point is that a human calculator with unrivalled skills of preparation like von Neumann could easily compute the highest probability opportunities of a poker hand, but if your hand sucks, it sucks. Sometimes all options are bad, and the lesser evil is still disaster. The only option: action. But sometimes nothing is a pretty cool hand.
Poker involves human interaction and strategy, which means above all mind games. Mind games, as their name implies, are designed precisely to throw reasonable thinking out of whack. Must there be head games? Well, basic reality is described by the observable arrangements of particles governed by replicable physical laws of motion (classical and quantum). These laws are indeed imperfect, conditioned on a limited knowledge of the world. But it is unconscious bits of brain matter that cause unobservable consciousness.
Consciousness is subjective; conscious interaction with others shapes society. Society in increasing complexity brings with it money, property, government, and standards of right and wrong. But the foundation remains utterly subjective: they exist because enough people believe they exist in a reasonably concrete way. The changing quantum of “enough people” determines the basic building block of value, what is desired. Value presents itself when enough people who see it accept it. Price presents itself as attractive when an individual accepts it. The price is the price.
People can anchor their perceptions of value based on the prices of things now. Valuation becomes a part of the Gestalt, something given, not something for which to search. Subjective consciousness and basic reality dissolve into a unity, because an unchangeable world-view is assumed. Those who fail to “get it” simply do not understand. There is little place for rational disagreement because it is arguing against the self-evident. While the financial motivation for trading remains sure—the commitment to survive and thrive—there are opportunities within established parameters for anything and nothing. All action can be channeled into modes of “not missing out” and alternatively “not rushing in where fools tread”. For a time, just about everyone thought equity tranche MBS was good enough and super-senior was bullet-proof. Now only braver souls sniff non-agency securities.
Action precedes data. What drives price action is trading. Without trading, or in an illiquid market, no one knows how an asset is valued. Trading in a liquid market provides approximate knowledge of value up to an interval in which price fluctuates around the bid-ask spread needed to fund market-making. Valuation is always dynamic. Data that informs models comes after.
By virtue of their beauty and elegance, mathematical methods can make even things with thoroughly unobservable foundations seem natural, realistic, and inevitable. People manage voluminous information and complexity by sticking to a view of the world no matter how shaky that outlook is. That illusion easily breaks down.
Predictive mathematical models provide in a rigorous justification for behavior based on hindsight. They reinforce belief that the world cannot materially change from the present (stationarity) and in so doing obscure the distinction between the subjective and objective. Mathematics does have unparalleled precision, but unquantifiable events fall outside of its scope. There is a place for thinking outside of data at hand and exploring if never-seen-before scenarios could make it fail.
Using mathematics to model what is essentially subjective and intractable is like playing poker: it can identify high-probability strategies to play, but one has to work the table. This is why sell-side and forecasters exist. The outcome of a bad hand is determined by how you play it as well as what’s in it to some cases.
Poker or Bingo
The current antidote to this endless cycle of falling blindly in love—not with an asset, but with a dubious axiomatization that shapes how the world is perceived—and the inevitable bitter disillusioning break-up is called risk management.
Oversimplification alert: the focus here is hedge fund risks that make money by taking residual risk, not on making money through the flow business. Risk management is very different for the two. Dealers need risk models that calibrate to daily markets, because tomorrow the position will be unloaded to another customer. Risk management for those taking residual risk is managing liquidity (cash/near-cash provisioning), reducing return volatility (diversification), and hedging non-stationarity (cheap exposures to the other side of crowded trades).
But at the end of the day what good does a reasonably self-funding but imperfect instrument like VIX call spreads really do in the face of a melt-down? Risk management depends on the same world-view that leads to melt-downs in the first place. Take VaR, for example. VaR works like it should only when a book is liquid. One is once more forced to concede that liquidity simply means you know what your book is worth. Illiquidity means you can’t price it.
In addition, one has to take into account the size of the players at the table and the types of trades they make. Small investors are price takers and not price makers—they are able to enter the market, trade in the volume they choose, and leave, without disturbing the market price (however, any trade can shift the price to some degree under a microscope). By contrast large investors are price makers because the size of trades they need to make inevitably shift market prices (thus they lack the anonymity of the small trader, and this can be seriously damaging, especially when a trader is forced to trade through visible weakness. Considering that just about everything is a short-end borrow, the effect of illiquidity on everyone can be as extreme as a surfer swallowed by a tsunami. It doesn’t matter how the hand is played. You take what is given in this case.
Illiquidity is non-stationarity. Are there trades such that non-stationarity can be exploited? These types of trades carry inherent risks still, but a floating component moves with non-stationary moves. Meaningful diversification is a mix of receive float/receive fixed positions.
For fixed receiver example, consider CUSIP 38145X533, a 13 month reverse convertible note paying 6.5%. You lose money only if the index drops ~80%. Synthetics allow you to customize exposure with no float caps: take a bond and buy protection on it, and then swap the cash-flow for a floating rate. A simple concrete example: buy a 10Y JGB with protection, pay fixed on a 10Y IR swap. Pocket the bond coupon and any floating rate (less funding, protection costs, and fixed swap payment).
Beware of the risks as well as classic paralysis by analysis. Dangers grow as one broods on them. In the concrete example, there is an implicit assumption that the CDS contract has a near-perfect inverse relationship to the underlying bond. Even if the trade is well within one’s risk tolerance, serious leverage has to be deployed to make money. This leverage increases mark-to-market risk via funding pressure. There can be upfront issues with a dealer. There is counterparty risk on the float proceeds and the protection. Relatively inexpensive counterparty risk hedges like exposure to rising LIBOR-OIS spreads replicate the risk inefficiently. CDS on the protection seller is costly (overkill?). Even then, there is counterparty risk on that protection too. There are currency risks: cash serves as a reference point, but it certainly isn’t a risk-free one.
Only trivial axiomatic worldviews are consistent: there will always be problems that cannot be resolved in a closed-form way. The closer one looks, the more hedge needs come into view. It can become like plugging holes in a Cantor set. There are contingencies where hedging is futile. There are times when precision is needed, and there are times when precision gets in the way.