07 Sep 2020 Reflections prompted by Elie Ayache’s ‘The Medium of Contingency’

Introduction

This article reflects on some of the ideas developed in Elie Ayache’s 2015 book, The Medium of Contingency. The book is stimulating and thought-provoking, but it makes, at times, for difficult reading. It is not my purpose here to attempt a faithful and accurate summary of the ideas it contains. This article would perhaps be better characterised as a discussion of some of the thoughts prompted by reading the book.

Risk, uncertainty and probabilities

Ayache takes it as axiomatic that the risks associated with changes in future financial market prices are fundamentally immeasurable. The book’s metaphysical discussions of this topic are dense, repetitive, and filled with rather obscure language (for example, there are a lot of sentences like this: “Folding back the infinite series in the static intensive strike of contingency is one way of regaining immanence.”[1]). But the essential idea, I think, is that future financial market prices are fundamentally uncertain in the Knightian sense of the word. That is, we cannot make reliable quantitative probabilistic statements about future price changes over any horizon, and this is true irrespective of how much historical data we have at our disposal. The ‘process’ driving changes in prices is too complex and non-stationary for us to reliably identify it; it is essentially unknowable.

Ayache emphasises the distinction between his philosophical perspective and that offered by Nassim Nicholas Taleb in his books on risk such as The Black Swan[2]. Like Ayache, Taleb rejects the use of ‘simple’ probability models (e.g. lognormal distributions) to describe the dynamics of financial market prices (and many other things). But Taleb’s solution is to find more complex probability models (in particular, fat-tailed distributions rather than Gaussian distributions) that might be better suited to describing the probability distributions of changes in asset prices. To Ayache, this approach is wrong-headed: the ‘true distribution’ is not merely complex, it is inherently unknowable, and there is no point in pretending otherwise. In practical terms, this difference is perhaps less severe than it seems: one of Taleb’s key points is that an enormous amount of data is required to accurately calibrate the parameters of fat-tailed distributions, and so the ‘true’ distributions of asset returns is extremely difficult to identify.

Whilst Ayache’s characterisation of financial market risk and uncertainty may strike some as controversial or outlandish, the presence of Knightian uncertainty in financial asset returns isn’t a particularly unusual assertion. Moreover, the more basic point that the stochastic dynamics of financial market prices are more complex than, say, geometric Brownian motion is, of course, quite widely accepted. After all, if, say, equity index volatility really was constant and of the form specified by geometric Brownian motion, we could accurately estimate the volatility’s permanent value once and for all by looking at a few hours of minute-by-minute index changes in any given day. No one suggests that such an approach would be an adequate form of risk model calibration.

The difficult question that follows is ‘so what?’. Once we accept that the empirical behaviour of asset prices is more complex than the typical characterisations used in standard economic theory, what can still be said about quantitative probabilities of future price changes? To Ayache, every event and every moment in time is unique. Future events cannot be grouped into populations of homogenous occurrences. Statistics has no place in his ontology – there is no possibility of inferring the future of the market from its past.

Prices, not probabilities

A philosophy of probability has developed over the last one hundred years that is intended to offer a definition of probability that is applicable to the kind of unique events described by Ayache. First developed (independently) by the Cambridge polymath Frank Ramsey and the Italian mathematician Bruno De Finetti in the 1920s and 1930s, subjective probability defines the probability of an event as the probability implied by the price of an asset whose pay-off is contingent on the event occurring. So, for example, if someone requires odds of 2-1 to induce them to bet that it will rain tomorrow, this implies that their subjective probability that it will rain tomorrow is 1/3.

This definition of probability is closer to Ayache’s metaphysical viewpoint than the ‘traditional’ philosophy of objective probability, which defines probabilities as frequencies within sets of infinite numbers of homogenous events. Ayache argues, however, that probability is fundamentally uninteresting and redundant if the price of the contingent event exists: “But who said the definition of probability should be our purpose? Forget about probability being the purpose and then the only thing that will matter will be the price offered, not the probability that subjectively went into the head [of the person who traded at that price].”[3]

In Ayache’s metaphysical system, all assets have observable and tradable prices (and, as we will discuss further below, the universe of assets includes all possible derivative contracts on all assets). He regards the way a market determines prices as beyond our understanding: ‘The market finds these prices and they are given to me without explanation’[4]. In other places, he describes the market price of assets arising “literally through a miracle”[5]. So, in Ayache’s system, market prices are inputs, not outputs. No asset ever needs to be valued because its price is already observable. This is just as well, as economic theory does not have a good account of how to value assets in the presence of the uncertainty that Ayache argues is omnipresent in markets.

Although Ayache believes that the probabilities associated with future price changes are unknowable, he stipulates that markets are arbitrage-free and informationally efficient. All price changes are therefore random. This applies to all prices – derivatives as well as underlying assets (the derivative price change will have some form of sensitivity to the underlying asset price change, as well as other independent sources of randomness). Immeasurable randomness implies a qualitative rather than quantitative property: prices will go up or down, but we cannot say by how much.

In summary, in Ayache’s system, future asset price changes are subject to Knightian uncertainty, so probabilities cannot be meaningfully ascribed to them; and the current prices of all assets (and derivative contracts written on them) are observable, informationally efficient and arbitrage-free. We have no means of understanding or explaining these prices, they simply exist. The question then follows…what can this model of financial markets be used to do?

It is immediately clear that there are a lot of things that it cannot be used to do. In this world, any notion of an expected present value calculation would seem to be illegitimate. If there are no probabilities, there are no expectations. Conventional actuarial science and much of what actuaries do does not naturally fit into this perspective. Ayache argues, however, that probability can be used in financial problems where the probabilities are known and the risk is diversifiable. He suggests that much actuarial work has these properties, but this is clearly a simplification (insurance companies and pension funds typically choose to retain many non-diversifiable risks and uncertainties on their balance sheets, and much actuarial work is concerned with measuring and managing them).

Moreover, economic concepts such as expected returns and risk premia become undefinable in a world where probability does not exist. Even ordinal ranking of the expected returns of different types of asset classes may not be possible in Ayache’s world. How can we say equities are expected to generate a return in excess of the risk-free rate when no probabilities can be attached to future equity prices? This is not to say that Ayache’s characterisation must be in in some sense empirically unreasonable, but simply to point out that we may lose a lot of analytical capability when we embrace this perspective. In Ayache’s setting, we are price-takers. This is the perspective of the market-maker. It is the perspective of a market participant who trades risk, and who has no interest in retaining it. It is a framework that may be of little use to investors, in theory or in practice.

The essential idea implicit in the book, though it is never directly and tangibly developed, is that the hedging of any risk does not require any use of probability. That is, optimal hedging in the Ayache world can be done without subscribing to any beliefs whatsoever about the probabilities or stochastic processes driving asset prices (beyond anything implied by the assumption of no-arbitrage prices). This idea is not presented in any explicit theorem, but it ultimately can be the only thing that the book is about. It is an interesting idea and many questions are immediately prompted by it:

• Is it possible to identify dynamic hedging strategies only by referring to current prices, and without using any probabilistic assumptions about the future behaviour of prices? Any imperfect dynamic hedging strategy that is not an arbitrage must involve scenarios where the strategy creates a mix of positive and negative pay-offs across different scenarios. If we cannot specify any probabilities for these scenarios, in what sense can we say that one strategy is better than the other?

• Does uncertainty mean we have to do without probability all together? Must concepts such as expected present values, expected returns and risk premia be dismissed as conceptually meaningless? Is there a way of having our cake and eating it: that is, recognising that financial market risk is inherently uncertain in the Knightian sense, whilst still finding some use for economic theory that assumes it is not?

We explore these questions further below.

Derivatives and markets

Ayache argues that derivative valuation theory implies derivative assets are redundant. Derivative valuation theory is based on the idea that the price of the derivative can be found by identifying the dynamic mix of underlying asset and risk-free asset that can perfectly replicate the derivative’s pay-off. The principle of no-arbitrage implies that the cost of such a portfolio must be equal to the derivative value. But, Ayache argues, if replication strategies exist that can perfectly replicate the derivative pay-off, the derivative contract serves no useful economic purpose.

Whilst there is an interesting insight here, we should also bear in mind that derivative pay-offs are found not only in traded derivative contracts, but are embedded in many forms of economic exposure (a shareholder’s option to default on a debt repayment; an oilfield owner’s option to extract the oil, etc.). In these cases, the exposures do not exist to be traded, they arise within other forms of economic arrangement, and their valuation can serve useful purposes. But Ayache surely has a point: one of the fundamental reasons for the existence of traded derivative contracts may well be that the pay-offs they offer are otherwise impossible to obtain with certainty through the dynamic trading of other assets. But if this is true, how can these derivative contracts be valued? According to Ayache, they cannot. As noted above, for Ayache we can only observe their price – its explanation or derivation is beyond us.

In Ayache’s world, there is an infinity of arbitrage-free asset prices, including all derivative contracts that can be written on all assets. Let’s conceptualise this as follows. Take a given asset, suppose for example it is a single equity stock. This asset has an observable price. There also exists call options and put options on this asset across the whole range of strikes and terms to maturity, again with observable prices. Note that this universe already encapsulates forwards and risk-free bonds, as these assets are easily obtained or constructed from portfolios of the stock and options of various strikes.

Now, suppose we use the Black-Scholes formula to calculate these option prices’ implied volatilities. This does not require us to subscribe to the ‘truth’ of the Black-Scholes model – the formula is merely being used as an instrument with which to express the price. The implied volatility is being used as a unit of measurement for option prices. Implied volatility is clearly observable, irrespective of whether or not we believe the Black-Scholes model is an accurate representation of reality (assuming the other inputs into the model are observable). The ‘actual’ volatility of the underlying asset, and the asset’s stochastic process more generally, are not necessarily observable.

Next, suppose there are options written on the implied volatilities of the options, and these options have observable prices. Again, these options and their prices exist for all terms and strikes. And then suppose there are options written on those options. And so on, ad infinitum.

Ayache’s key conjecture

Ayache argues that the presence of such an infinity of asset prices allows any of the assets to be perfectly replicated by some dynamic strategy involving some of the other assets. Moreover, this is true irrespective of the nature of the probabilistic process driving prices; and the replication strategy can be identified only by referring to current market prices, without making any assumptions about the probabilistic process driving prices (other than that price changes are random and all prices are arbitrage-free). The presence of a perfect replication strategy removes the considerable difficulty that arises in finding an ‘optimal’ imperfect replication strategy in a world where no probabilities can be attached to future states of the world.

This is an interesting conjecture. As noted above, it is not presented formally, but simply as a rhetorical, metaphysical argument. Let’s consider a simple example to see if this idea can make sense.

Let us suppose there is an equity stock and all options on this asset have the same implied volatility for all terms and strikes. Further suppose all the options on the implied volatilities of those options imply a volatility of implied volatility of zero (that is, all the out-the-money options on the implied volatilities have a price of zero), as do all the options on those options and so on. In this case, how would we construct a replicating strategy for, say, a call option on the underlying asset? By the assumption of no-arbitrage, the prices of the options on the implied volatilities necessitate that the future values of the implied volatility of the option on the underlying asset must be always be the same as today’s implied volatility (otherwise unlimited profits could be obtained by taking a position in the free options on the implied volatility). So, in this especially simple case, we have used the arbitrage-free prices assumption to make an inference about the actual probability distribution of future option-implied volatility: in the future, implied volatility will always be the same as today’s with probability 1.

Armed with this knowledge of permanently static implied volatility, we can then say that the only source of randomness in the price of the call option must be given by changes in the price of the underlying asset (assuming the other inputs into the Black-Scholes option pricing formula cannot change). Now, we want to avoid making any probabilistic assumptions about the behaviour of the underlying asset price change. But we know that the instantaneous price sensitivity of an option with a static Black-Scholes option-implied volatility is, by definition, given by the Black-Scholes delta (using the observed implied volatility). The dynamic replication strategy for the call option here must therefore simply involve dynamically delta hedging with the underlying asset. (And to be on the safe side, we could always load the portfolio up with unlimited amounts of free out-the-money options on the implied volatility.)

Note that we have not explicitly estimated the probability distribution of the underlying asset in this example. We have simply worked with the prices we can observe. But what if something very unpredictable happened to the underlying asset price? For example, what if the underlying asset price made a large discontinuous jump, resulting in errors in the hedge portfolio performance? How can the vanilla delta hedging strategy deliver a perfect hedge in such a circumstance? It cannot, but the essential point is that such a possibility is incompatible with this set of observed market prices. If the discontinuity could happen, today’s prices would not be what they are, as they would then be offering an arbitrage. The arbitrage-free prices assumption allows us to identify the scenarios that must be impossible according to the prices, and that therefore are not relevant to the hedging strategy.

So, in this setting, we seem to have been able to identify a dynamic hedging strategy that will perfectly replicate the call option pay-off in all possible states of the world that are compatible with the current prices being arbitrage-free. We have done so by only referring to current market prices and assuming they are arbitrage-free, and we have not made any assumptions about the probabilistic dynamics of any of the prices beyond those that can be inferred from the arbitrage-free property of current prices. This is what Ayache says will happen for any set of market prices.

Let us now add one more layer of complexity to our market prices. Now let’s suppose the option-implied volatilities of the options on the underlying asset vary by term and strike. Moreover, options on the implied volatility are now priced with some time value, i.e. out-the-money options on implied volatility have positive prices. Options on those options, however, have no time value, and the same applies to all ‘higher-level’ options. Future changes in the call option price will now vary as a function of random changes in the price of the underlying asset and random changes in the implied volatility of the call option. But the zero prices for the higher-level options imply that these are the only two sources of randomness that will impact on the call option price. In this case, the dynamic replicating strategy for the call option on the underlying asset will consist not only of the underlying asset and risk-free asset, but also a position in the option on the implied volatility. It seems intuitive that the dynamic hedging strategy can be identified from the given set of prices, though I have not attempted to prove it.

As we increase the layers of option prices with non-zero time value, the number of sources of random variation in the call option price will increase, and the number of assets required in its replicating portfolio will increase. But, according to Ayache’s conjecture, it is always possible to perfectly replicate the call option price with a dynamic strategy that can be determined by reference to current market prices. It is beyond my technical skills to attempt to prove or dis-prove this conjecture, but it would certainly be interesting to see an attempt.

State prices

You might have detected a contradiction in the above discussion – on the one hand, Ayache criticises conventional derivatives valuation theory for assuming perfect replication is possible and argues that this makes derivative assets redundant; but then his own theory is also based on the theoretical possibility of perfectly replicating any derivative pay-off. Ayache’s market ‘completeness’ property arises, however, not by specifying stochastic processes for market prices that can be shown will facilitate perfect replication of the option pay-off; rather, Ayache’s completeness is obtained without specifying any stochastic processes and instead relying on an observable infinity of arbitrage-free market prices for assets of all levels of complexity.

It may be interesting to compare and contrast Ayache’s infinite universe of observable prices with the neoclassical economic theory of state prices (sometimes called Arrow-Debreu prices). In an economy with continuous states, there is an infinity of state prices. State prices depend on risk-free interest rate, risk premia and probabilities. Probabilities cannot be directly observed from state prices, as they also depend on risk aversion, which is not directly observable either[6].

So, both the Arrow-Debreu and Ayache perspectives postulate the existence of an infinite set of asset prices, and that future price probabilities cannot be observed directly from them. Are these prices in some sense doing the same thing?

The economic theory that derives a particular structural form for the state price equation relies on explicit assumptions about the nature of the probability distributions for asset prices. Ayache rejects the specification of any such probabilistic assumptions, but the infinite set of option prices observed in Ayache’s system are only compatible (i.e. arbitrage-free) with some set of probabilistic properties implied by the observed prices. In both cases, the price of any contingent claim can be expressed as some knowable sum of the prices of the observable assets (state prices or option prices), and this sum is obtained through the principle of no-arbitrage and replication. The two sets of infinite prices seem to be doing the same job. The difference seems to boil down to whether fundamental asset prices’ stochastic processes are assumed to be exogenous and known, or whether the fundamental asset’s arbitrage-free derivative prices are assumed to be exogenous and known. But in both cases perfect dynamic replication of derivative pay-offs is possible and derivative prices / values are consistent with the costs of their replicating strategies.

Uncertainty and Theory

In my own view, I find the characterisation of financial market risk as uncertainty in the Knightian sense quite compelling. Earlier we asked if there was a way of recognising that financial market risk is inherently uncertain in the Knightian sense, whilst still finding some use for economic theory that assumes it is not. I believe this is certainly the case. As I have explored in more depth elsewhere[7], derivative valuation theory, and economic theory more generally, is useful not as positive science but as an interpretative study. That is, the theory is deliberately and unavoidably counter-factual, and its usefulness cannot be measured by the accuracy of its empirical predictions. Nonetheless, using the theory to understand the dynamics of this simplified counterfactual world is insightful and interesting, even if there exists an unbridgeable epistemic gap between the theory’s counterfactual ideal and our empirical reality. Saying something is infinitely complex does not mean we can understand absolutely nothing about it.

[1] The Medium of Contingency, p. 127.

[2] Both Ayache and Taleb are Lebanese retired options traders.

[3] Ibid, p. 164.

[4] Ibid, p. 88.

[5] Ibid, p. 156.

[6] Subjective probability (i.e. probability derived from observed prices) was developed independently of (and prior to) Arrow-Debreu’s state prices. It was understood from its inception that subjective probability has a limitation: prices, in general, depend on risk-aversion, and this complicates the derivation of the probabilities implied by prices.

[7] See Chapter 4 of https://craigturnbullfia.com/some-notes-on-the-methodology-of-actuarial-science/