Sense, nonsense and the S&P500
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Abstract
The theory of financial markets is well developed, but before any of it can be applied there are statistical questions to be answered: Are the hypotheses of proposed models reasonably consistent with what data show? If so, how should we infer parameter values from data? How do we quantify the error in our conclusions? This paper examines these questions in the context of the two main areas of quantitative finance, portfolio selection and derivative pricing. By looking at these two contexts, we get a very clear understanding of the viability of the two main statistical paradigms, classical (frequentist) statistics and Bayesian statistics.
Keywords
Bayesian statistics Frequentist statistics Derivative pricing HedgingJEL Classification
C11 C18 C581 Introduction
So Sect. 2 gives us some kind of framework for answering the question, ‘How should we invest in this market?’ In Sect. 3, we look at the question we should have asked first, namely ‘Are the modelling assumptions reasonable?’—in other words, can we suppose that returns are IID multivariate Gaussian? Not surprisingly, the answer is ‘No’. However, as we shall see, this is not as bad as it seems, because simple transformations change the data into something that is reasonably like IID Gaussian, and the theory developed in Sect. 2 may actually be fairly relevant. We then take a look in Sect. 4 at investing in the S&P500 and see what some of these ideas give us.
The next section of the paper looks at derivative pricing, and once again the inconsistencies of classical statistics surface in a big way almost immediately. Once again, Bayesian statistics offers an escape.
2 Portfolio selection
2.1 What does classical statistics say?
So just by thinking briefly about classical statistics in the context of the portfolio selection problem (3), we see that it just cannot work!
2.2 What does Bayesian statistics say?
More has been said about the choice of the prior than we could ever summarize, but my view is that this is a relatively innocent subjective choice. In practice, one would run the analysis for a number of widely different priors as a diagnostic; if the answers are broadly similar, then the choice of prior was not particularly critical, and if the answers vary a lot, then we learn that there was not so much information in the data, again useful to know. A far more important subjective choice, already mentioned, is the choice of the family of models allowed.
As we shall see, taking a Bayesian view deals completely with all the theoretical aspects of statistical inference, but the price we end up paying is that the computational aspects become a lot more onerous.
3 Are S&P500 returns IID Gaussian?
We continue to illustrate the themes of this paper by simplifying the model (1) we began with to one asset, the S&P500 index. The model assumption is that the daily returns are IID Gaussian, but are they?
3.1 Are returns identically distributed?
If the returns were IID, we should expect to see a plot that goes up roughly as a straight line, and this is obviously not the case. However, we can transform the returns into something much closer to IID by the simple trick of vol rescaling, which goes like this.
These plots show that the rescaled returns look quite time homogeneous.
3.2 Are returns Gaussian?
3.3 Are returns independent?
nonconstant vol does not matter, nonconstant mean returns do.
4 How well does Bayesian model averaging work?
In Fig. 9, we see the P&L generated when we take just two models, the first of which thinks that \(\mu _t(1) = 0.15\) for all t and the second of which thinks that \(\mu _t(2) = 0.15\) for all t—in other words, the index is either growing at 15% per annum, or shrinking at 15% per annum. Taking the transactions costs to be 3 bp, we find that the Sharpe ratio of the strategy is 41.00%, substantially higher than the two constantdollar strategies. The P&L shown in Fig. 9 displays relatively little drawdown. The positions shown in Fig. 10 fluctuate between \(\,0.15\) and 0.15, the extremes we would expect if the posterior probabilities were at their extreme values. It is interesting to see that periods when the position is strongly negative, such as 1973–1974, 2001–2003, 2008–2010, correspond to periods when the global economy was under significant stress.
This looks like (and is) an impressive demonstration of the power of Bayesian modelling techniques. But it is worth underlining that some cherrypicking has been going on here; if we include a third model into the comparison which says \(\tilde{X}_t \vert {\mathcal {F}}_{t1} \sim N(0,1)\), then the same analysis leads to a Sharpe ratio of 30.18%. Changing the various parameters of the model can make a big difference to the conclusion, and we need to be aware of this; searching around for a ‘sweet spot’ is a form of data snooping. We would be outraged if someone proposed a trading strategy that needed to know all future returns, but if we search for ‘good’ parameter values in some parametric model, we are in effect making use of information about the entire future evolution of returns, even if the individual model selected at the end does not. I have seen this done in practice; a model gets adopted and then fails to deliver the returns that historical analysis gave. It is good practice to leave several years of data locked up until the model has been chosen, and then see what happens once those data are unlocked—outofsample testing. Even so, the future may not cooperate.
5 Derivative pricing
Some derivatives are very liquid, so their prices are taken to be the market prices—any model should match those prices very closely, if not perfectly. More exotic derivatives on the other hand are made to order, and there is no market price, so the price has to come from some (parametric) model, as a function of observable state variables \(X_t\) and unobserved parameters \(\theta \). The parameter \(\theta \) of the model will not be known, so we have to carry out some statistical procedure to identify it, and as with portfolio selection, there are the two main paradigms to consider.nonconstant mean returns do not matter, nonconstant vol does.
5.1 What does classical statistics say?
 1.
The model prices \(\varphi ^a(X_t,\theta )\) may not exactly match market prices \(Y^a_t\).
 2.
Tomorrow we recalibrate and arrive at a value \(\theta ^*_{t+1}\)—so how do we marktomarket and hedge a derivative that we sold on day t? Using \(\theta = \theta ^*_t\)? Using \(\theta ^*_{t+1}\)? Using some other \(\theta \) value?
 3.
Would some other model be ‘better’?
 4.
\(\theta ^*_t\) is an estimate—what account do we take of estimation error?
So overall the conventional calibration approach is inconsistent and cannot account for estimation error.
5.2 What does Bayesian statistics say?

The law of \(X_{t+h}\) conditional on \({\mathcal {F}}_t\) has density \(\sum _j \pi _j(t)\, p_j(X_t, \cdot )\);
 If model j gives the price of an exotic to be \(\xi _j\), then take the overall price to bethe posterior mean;$$\begin{aligned} \bar{\xi }\equiv \sum _j \pi _j(t) \, \xi _j, \end{aligned}$$(16)

What is the error in \(\bar{\xi }\)? It is the mean of a discrete distribution over the values \(\xi _j\) with weights \(p_j(t)\), so we know the variance and all other moments;

If model j gives delta hedge^{2} \(H_j\), then to first order we have a delta hedge given by \(\sum _j \pi _j(t) H_j\).
 1.
The model prices \(\varphi ^a(X_t,\theta )\) may not exactly match market prices \(Y^a_t\). The Bayesian approach does not say that the prices must be any particular value—it says that any price is a random variable whose distribution we know completely.
 2.
Tomorrow we recalibrate and arrive at a value \(\theta ^*_{t+1}\)—so how do we marktomarket and hedge a derivative that we sold on day t? Using \(\theta = \theta ^*_t\)? Using \(\theta ^*_{t+1}\)? Using some other \(\theta \) value? At all times, the price from the Bayesian approach is the posterior mean of the price—there is no inconsistency;
 3.
Would some other model be ‘better’? Other models can be compared simply by adding them to the universe of models in the Bayesian comparison;
 4.
\(\theta ^*_t\) is an estimate—what account do we take of estimation error? Nothing is estimated in the Bayesian approach.
6 Summary
This survey has taken a look at how statistical methodology helps in the analysis of financial asset returns, whether for portfolio selection or for derivative pricing. The conclusion is that statistics helps up to a point, but falls far short of what we would like to be able to do. The classical paradigm is an unworkable conceptual framework for studying data; its shortcomings may be hidden when we look at experimental data from the physical sciences, where the signaltonoise ratio is much smaller than in financial data, but once we try to use it in finance and economics, it simply fails. Nevertheless, the methods of classical statistics provide very useful exploratory tools; if we were given ten years of daily returns on 2000 assets, we would almost certainly begin by calculating sample mean returns, and the sample covariance matrix, then we might try to pull out some principal components. Such calculations would very quickly tell us stylized facts of the data and direct our attention to questions of interest.
Hopefully, this article has well made the point that if we want a statistical methodology that is consistent, then it has to be Bayesian. Sadly, when it comes to trying to use Bayesian statistics in practice, the computational challenges quickly become overwhelming. Nevertheless, with patience and computational resources, we can make progress. As always, choosing very simple models pays off, and it is here that some judicious use of classical methodology to discover stylized facts and then using a more thorough Bayesian analysis of a simple model expressing those facts can be successful. Though the tools of statistics have changed little over time, there is no uniform recipe for using them; in the end, applying experience and an openminded approach to a new data context is the best we can do.
Footnotes
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