^{ 1 }

^{ 2 }

^{ 3 }

^{ 1 }1Emeritus Professor, Sorbonne’s Business School, Head of Research, Riskinnov Ltd., London

^{ 2 }2Global Head of Capital Markets Research, Allianz Investment Management SE, Munich

^{ 3 }3Associate Professor of Economics, The Sorbonne-Paris V, Social Science Department, Paris

^{ * }Corresponding author: Bertrand Munier,

^{ * }Date of submission: 01/01/2017

^{ * }Date of publication: 21/03/2017

JEL Codes: G11, G13, G14, G15

Keywords: volatility – time series – markets efficiency – VIX

Volatility has been a long debated issue in Finance. Whether it really is a sufficient and coherent measure of risk on markets has been discussed at length and we shall not dwell on this topic here. This paper rather aims at raising the issue of the relation between the ‘historical’ and the ‘implied’ concepts of volatility. Are historical volatility and implied volatility completely separate concepts, or can some relation between them be inferred from observation? It is usually admitted that anticipating the future – the task usually assigned to implied volatility - rests on psychological attitudes, having an impact on the assessment of the market, whereas examining the past – the very idea of historical volatility - deals only with factual observations. It therefore seems difficult, at first sight, to assess any clear relation between them. On another hand, ‘implied volatility’ is often represented by the VIX index, which is then taken as a form of expectation of market’s volatility in the near future, essentially the next month(s). Is such a view empirically validated? If not, can we suggest an intermediate variable, which could bridge the gap between historical volatility and the VIX?

Such a debate encompasses wider issues than purely financial ones. Indeed, economists – who used to rely on adaptive expectations – have come to develop several rational expectation hypotheses which they regarded for a long time as “the revolution in macroeconomic theory of the frame of forward looking anticipations rather than expectations based on past data only” (Tirole, 2012) and presumably impossible to link to the former. More recently, however, behavioral economics as well as behavioral finance have challenged this belief and brought some authors to change their mind on the issue. This paper wants to argue that Allais’s insightful developments of “psychological time” allow bridging that gap. More specifically, we claim that a “

In a first section, we shall recall the Allaisian world of relativity, i.e. of “psychological time” and “physical time”, either of these frames being possibly taken as a reference, but never independently from the other. More specifically, we show that, although developed for specific purposes, Allais’ framework opens on an explanation of

The next natural question bears on the issue of forecasting volatility using a similar framework. We briefly discuss this last issue and conclude that it calls for some hard questions to be solved.

It is little known that Maurice Allais came out in the Sixties and Seventies with an impressive theory of the demand for money. From a monetary point of view, his contribution amounts to a subtle and sophisticated reformulation of the quantitative theory of money. This development is outside the focus of the present paper and will not be further mentioned here. However, part of Allais’s contribution offers the basis for a model of market expectation formation. When properly used, this model can be regarded as the core of an original time series analysis, although the foundations of it owe more to a psychology of finance and economics than to statistics. Several financial phenomena can therefore be scrutinized, using Allais’s contribution as a tool, when looking for an explanatory reconstruction within a given sample of observations.

Time is a relative notion. Two different types of time do coexist: the “_{0,} is a constant equal to the rate i_{0} at which the future is discounted per unit of psychological time (Allais, 1966 p.1129). Within this time reference, we thus have _{0}= χ_{0}, a constant parameter to be estimated. Allais expresses a postulate of ‘temporal psychological symmetry’: “

However, the physical time reference does not exhibit the same property of a constant rate of memory decay, as is the case with psychological time. Experience and historical accounts suggest, in our usual time reference, that events of same length in physical time are being felt as having various subjective time lengths according to the circumstances. For example, common sense suggests that time elapses differently for someone sentenced to death and for someone in charge of repetitive tasks in a boring activity without any end horizon. Similarly, from an economic point of view, memory decay of changes in prices is much quicker if prices are strongly increasing (for example during the German hyper-inflation) than when prices remain stable (Allais, 1965 pp.23-25). Clearly, the instantaneous rate of memory decay is not constant in the physical time reference. To express that this rate is all the higher in physical time, because experienced changes happen at a high speed, one can write, following Allais:

with

where

is a logistic function mapping that psychological feeling into a rate at which the past is being forgotten. Differentiating (2), we can write:

In his monetary theory, for example, Allais takes

Consistently with the idea that any change in affairs in the past will be exponentially forgotten about at some

:

An equation in which

and where the instantaneous rate of memory decay at time

, varies through time in the interval

. As Philip Cagan, commenting on Allais’s thought, put it:

._{ }

in equation (1) such that

.

This relativity of time perception in the precise sense defined by equations (1) to (3) above is what allowed Allais applying a same

At time _{t}

which forms the best possible opinion we may have on the phenomenon studied, i.e. further down, the market volatility – what we could best expect regarding that volatility. For this reason, we shall use further below the term of “expected variability” for

Let us now assume that, at time (

Clearly, the ideal would be that our surprises along time were all equal to zero. In the standard case, alas, such is not the case. Could we then reformulate our notion of

How do we correct our expectation, once we have taken notice of the _{t}

More specifically, starting from equation (3), keeping (1) in mind and developing (2) as a Taylor’s expansion, we can write, using notations _{t}

_{ }in place of

for simplicity’s sake:

Setting then

, we can write:

or

Which can be rewritten as:

And finally, keeping equation (6) in mind:

if we set:

The discussion of this last equation could lead to quite interesting issues, but it is out of the scope of the present paper. The interested reader is referred to Barthalon (2014) for a more complete exposition as well as to Allais’s papers to be found in the references at the end of this paper.

Finally, past absolute psychological changes reach at time _{t}

We have hastily sketched, in section I above, the Allaisian dual framework of time reference. One could wonder where to these equations will lead us. We claim that

For our present purpose, we examine the time series of

This is indeed the from under which it is most of the time computed in trading rooms, where the time series of the P_{n}
_{t}

To make our argument explicit, we consider now a sample of our monthly data stretching over twenty-six years, namely from January1990 to December 2016. We assume to know all values taken by the S&P500 index over that entire period and compute, month by month, the historical volatility series. On that basis we use the equations mentioned in the preceding section to compute the expected variability

To fulfill these computations, we have to choose a parameter _{0} to initialize the process. As already mentioned above, we choose _{0} as dimensionless, the rule of choice chosen here being to minimize the absolute sum of surprises (7) over the whole time span under scrutiny. Note that this rule of setting _{0} is not logically compelling – we can pick up another rule, leading to slightly different results in the very first periods under scrutiny, but converging quickly to practically equivalent results. In this framework, we claim that several propositions can be established, out of which we select here five. We now take on them and illustrate our results successively.

Figure 1 shows the results of our computation. We confront ^{2} further below) between the actual volatility observed ^{2} in this context of high autocorrelation, the Root of the Mean Squared Error (RMSE further below) is at a 4.5%, a low figure when considering financial volatility on such a sample.

Why do we lag ^{2} deteriorates and the RMSE increases to 5.03%. Intuitively equation (5) above shows that the weighting of past observations is of an exponential type and it is quite clear that the last observation, say _{t}

A frequently encountered rule of thumb among traders has it that a “normal” implied volatility is equal to the average of recent months historical volatility. There is, of course, no scientific ground whatsoever to such a rule, which is frequently invalidated by experience itself. On the other hand, in line of most of the literature, we find that implied volatility, far from being a reliable forecast of future volatility, appears to rise once an accident has happened, in a somehow similar way as many insurance premia do. This is but a kind of new illustration of the saying used as a subtitle to this paper. And still, this saying is a generous judgment, for, as seen from Proposition 4 above, the index has followed a trend inversely related to the risk-premium for some twenty years now.

What this paper has shown is that the “missing link” between historical and implied volatility lies in the Allaisian analysis of time series derived from that great author’s monetary theory. This missing link yields what we have termed above the “expected variability”, computed from Allais’s subtle and sophisticated algorithm using historical prices as input and based on the theory of psychological time.

More generally, our results support the idea that expectations are not really grounded on prospective insights, but more so on past representation of data. Between today and tomorrow, the link is a subtle representation of the past. This was Maurice Allais’ firm conviction. Our data support it, even on a market driven by sophisticated investors (not everybody trades on derivatives) and even considering a relatively short horizon, as is the case with the VIX. The magic of Allais’ psychological time theory bridges the gap on this market, as it does with even greater force in other domains. Between today and tomorrow, the link is a subtle representation of the past.

Further research is called for regarding our results on over- and under-evaluation of the VIX, which, contrary to what the rule of thumb evoked at the beginning of this paragraph suggests, can only be judged with respect to the near future’s figures of volatility. The situation can therefore only be back-tested, as has been done in this paper. But the factors of over- or under-valuation are still to be determined. We could only offer hypotheses in this paper.

An entirely different question is the one of volatility forecasting. It is quite clear that the Allaisian algorithm briefly developed in section I of this paper cannot be any tool of forecasting, as has been noted above: Allais’s algorithm assumes implicitly that all data of the time span under consideration are known with certainty, which cannot obviously be the case if one is to consider forecasting future unknown data. Clearly, analytical complements of different types have to be added to the initial algorithm, if one wants to base a prediction tool on similar ideas. Riskinnov has fine-tuned such a computation procedure, which performs better than the best-known sophisticated models of market volatility, including GJR-Garch. But that is in itself the topic of another paper.