Breaking Mathematical Sense

Introduction

Mathematicians, as far as I can see, are not terribly interested in the philosophy of mathematics. They often have philosophical views, but they are usually not very keen on challenging or developing them—they don’t usually consider this as worthy of too much effort. They’re also very suspicious of philosophers. Indeed, mathematicians know better than anyone else what it is that they’re doing. The idea of having a philosopher lecture them about it feels kind of silly, or even intrusive. (Roi 2017, 3)

So we turn to people who have something to do with mathematics in their professional or daily lives, but are not focused on mathematics. Such people often have some sort of vague, sometimes naïve, conceptions of mathematics. One of the most striking manifestations of these folk views is the following: If I say something philosophical that people don’t understand, the default assumption is that I use big pretentious words to cover small ideas. If I say something mathematical that people don’t understand, the default assumption is that I’m saying something so smart and deep that they just can’t get it. (Roi 2017, 3-4)

There’s an overwhelming respect for mathematics in academia and wider circles. So much so that bad, trivial, and pointless forms of mathematization are often mistaken for important achievements in the social sciences, and sometimes in the humanities as well. It is often assumed that all ambiguities in our vague verbal communication disappear once we switch to mathematics, which is supposed to be purely univocal and absolutely true. But a mirror image of this approach is also common. According to this view, mathematics is a purely mechanical, inhuman, and irrelevantly abstract form of knowledge. (Roi 2017, 4)

I believe that the philosophy of mathematics should try to confront such naïve views. To do that, one doesn’t need to reconstruct a rational scheme underlying the way we speak of mathematics, but rather paint a richer picture of mathematics, which tries to affirm, rather than dispel, its ambiguities, humanity, and historicity. (Roi 2017, 4)

(….) The uncritical idolizing of mathematics as the best model of knowledge, just like the opposite trend of disparaging mathematics as mindless drudgery, are both detrimental to the organization and evaluation of contemporary academic knowledge. Instead, mathematics should be appreciated and judged as one among many practices of shaping knowledge. (Roi 2017, 4-5)

A Vignette: Option Pricing and the Black-Sholes Formula

The point of the following vignette is to give a concrete example of how mathematics relates to its wider scientific and practical context. It will show that mathematics has force, and that its force applies even when actual mathematical claims to not quite work as descriptions of reality…. The context of this vignette is option pricing. An “option” is the right (but not the obligation) to make a certain transaction at a certain cost at a certain time. For example, I could own the option to buy 100 British pounds for 150 US dollars three months from today. If I own the option, and three months from today 100 are worth more than 150 dollars, I will most probably simply discard it. Such options could be used as insurance. The preceding option, for example, would insure me against a drop in the dollar-pound exchange rate, if I needed such insurance. It could also serve as a simple bet for financial gamblers. But what price should one put on this kind of insurance or bet? There are two narratives to answer this question. The first says that until 1973, no one really knew how to price such options, and prices were determined by supply, demand, and guesswork. More precisely, there existed some reasoned means to price options, but they all involved putting a price on the risk one was willing to take, which is a rather subjective issue. (Roi 2017, 6)

In two papers published in 1973, Fischer Black and Myron Sholes, followed by Robert Merton, came up with a reasoned formula for pricing options that did not require putting a price on risk. This feat was deemed so important that in 1977 Scholes and Merton were awarded the Nobel Prize in economics [see The Nobel Factor] for their formula (Black had died two years earlier). Indeed, “Black, Merton and Scholes thus laid the foundation for the rapid growth of markets for derivatives in the last ten years”—at least according to the Royal Swedish Academy press release (1977). (Roi 2017, 6-7)

But there’s another way to tell the story. This other way claims that options go back as far as antiquity, and option pricing has been studied as early as the seventeenth century. Option pricing formulas were established well before Black and Scholes, and so were various means to factor out putting a price on risk (based on something called put-call parity rather than the Nobel-winning method of dynamic hedging, but we can’t go into details here). Moreover, according to this narrative, the Black-Sholes formula simply doesn’t work and isn’t used (Derman and Taleb 2005; Haug and Taleb 2011).

If we wanted to strike a compromise between the two narratives, we could say that the Black-Scholes model was a new and original addition to existing models and that it works under suitable ideal conditions, which are not always approximated by reality. But let’s try to be more specific. (Roi 2017, 7)

The idea behind the Black-Scholes model is to reconstruct the option by a dynamic process of buying and selling the underlying assets (in our preceding example, pounds and dollars). It provides an initial cost and a recipe that tells you how to continuously buy and sell these dollars and pounds as their exchange rate fluctuates over time in order to guarantee that by the time of the transactions, that money one has accumulated together with the 150 dollars dictated by the option would be enough to buy 100 pounds. This recipe depends on some clever, deep, and elegant mathematics. (Roi 2017, 7)

This recipe is also risk free and will necessarily work, provided some conditions hold. These conditions include, among others, the capacity to always instantaneously buy and sell as many pounds/dollars as I want and a specific probabilistic model for the behavior of the exchange rate (Brownian motion with a fixed and known future volatility, where volatility is a measure of the fluctuations of the exchange rate). (Roi 2017, 7)

The preceding two conditions do not hold in reality. First, buying and selling is never really unlimited and instantaneous. Second, exchange rates do not adhere precisely to the specific probabilistic model. But if we can buy and sell fast enough, and the Brownian model is a good enough approximation, the pricing formula should work well enough. Unfortunately, prices sometimes follow other probabilistic models (with some infinite moments), where the Black and Scholes formula may fail to be even approximately true. The latter flaw is sometimes cited as an explanation for some of the recent market crashes—but this is a highly debated interpretation. (Roi 2017, 7-8)

Another problem is that the future volatility (a measure of cost fluctuations from now until the option expires) of whatever the option buys and sells has to be known for the model to work. One could rely on past volatility, but when comparing actual option prices and the Black-Sholes formula, this doesn’t quite work. The volatility rate that is required to fit the Black-Sholes formula to actual market option pricing is not simply past volatility. (Roi 2017, 8)

In fact, if one compares actual option prices to the Black-Sholes formula, and tries to calculate the volatility that would make them fit, it turns out that there’s no single volatility for a given commodity at a given time. The cost of wilder options (for selling or buying at a price far removed from the present price) reflects higher volatility than the more tame options. So something is clearly empirically wrong with the Black-Sholes model, which assumes a fixed (rather than a stochastic) future volatility for whatever the option deals with, regardless of the terms of the option. (Roi 2017, 8)

So the Black-Sholes formula is nice in theory, but needn’t work in practice. Haug and Taleb (2011) even argue that practitioners simply don’t use it, and have simpler practical alternatives. They go as far as to say that the Black-Sholes formula is like “scientists lecturing birds on how to fly, and taking credit for their subsequent performance—except that here it would be lecturing them the wrong way” (101, n. 13). So why did the formula deserve a Nobel prize? (Roi 2017, 8)

Looking at some informal exchanges between practitioners, one can find some interesting answers. The discussion I quote from the online forum Quora was headed by the question “Is the Black-Sholes Formula Just Plain Wrong?” (2014). All practitioners agree that the formula is not used as such. Many of them don’t quite see it as an approximation either. But this does not mean they think it is useless. One practitioner (John Hwang) writes:

Where Black-Sholes really shines, however, is as a common language between options traders. It’s the oldest, simplest, and the most intuitive option pricing model around. Every option trader understands it, and it is easy to calculate, so it makes sense to communicate implied volatility [the volatility that would make the formula fit the actual price] in terms of Black-Sholes…. As proof, the exchanges disseminate [Black-Sholes] implied volatility in addition to data.

Another practitioner (Rohit Gupta) adds that this “is done because traders have better intuition in terms of volatilities instead of quoting various prices.” In the same vein, yet another practitioner (Joseph Wang) added:

One other way of looking at this is that Black-Sholes provides something of a baseline that lets you compare the real world to a nonexistent ideal world…. Since we don’t live in an ideal world, the numbers are different, but the Black-Sholes framework tells us *how different* the real world is from the idealized world.

So the model earned its renown by providing a common language that practitioners understand well, and allowing them to understand actual contingent circumstances in relation to a sturdy ideal. (Roi 2017, 9)

Now recall that practitioners extrapolate the implied volatility by comparing the Black-Sholes formula to actual prices, rather than plug a given volatility into the formula to get a price. This may sound like data fitting. Indeed, one practitioner (Ron Ginn) states that “if the common denominator of the crowd’s opinion is more or less Black-Sholes … smells like a self fulfilling prophecy could materialize,” or, put in a more elaborate manner (Luca Parlamento):

I just want to add that CBOE [Chicago Board Options Exchange] in early ’70 was looking to market a new product: something called “options.” Their issue was that how you can market something that no one evaluate? You can’t! You need a model that helps people exchange stuff, turn[s] out that the BS formula … did the job. You have a way to make people easily agree on prices, create a liquid market and … “why not” generate commissions.

The tone here is more sinister: the formula is useful because it’s there, because it’s a reference point that allows a market to grow around it. (Roi 2017, 9)

But why did this specific formula attract the market, and become a common reference point, possibly even a self-fulfilling prophecy? Why not any of the other older or contemporary pricing practices, which are no worse? Why was this specific pricing model deemed Nobel worthy? (Roi 29017, 10)

The answer, I believe, lies in the mathematics. The formula depends on a sound and elegant argument. The mathematics it uses is sophisticated, and enjoys a record of good service in physics, which imparts a halo of scientific prestige. Moreover, it is expressed in the language of an expressive mathematical domain that makes sense to practitioners (and, of course, it also came at the right time).

This is the force of mathematics. It’s a language that the practitioners of the relevant niches understand and value. It feels well founded and at least ideally true. If it is sophisticated and comes with a good track record in other scientific contexts, it is assumed to be deep and somehow true. All this helps build rich practical networks around mathematical ideas, even when these ideas do not reflect empirical reality very well. (Roi 29017, 10)

(….) [I]f we want to understand the surprising force of mathematics demonstrated in this vignette, we need to engage in a more careful analysis of mathematical practice. (Roi 29017, 10)

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