Beyond epicycles: models must describe markets, not just fit them

Modelling needs to embrace complexity in volatility patterns, says Jean-Philippe Bouchaud

Key points Summary Q&A

• Volatility is largely endogenous, being rough and path dependent, driven by leverage effects and Zumbach trend effects.
• Cross‑asset effects are strong: trends in the E‑mini futures strongly influence other futures and raise the residual volatility of single stocks.
• The authors also find a new effect linking past realised covariance between two assets to their future volatility, illustrated by the E‑mini/T‑bond pair.
• Quants should build models that do not merely agree with markets today, but that still make sense when markets change tomorrow. 

These summaries are generated by an LLM, but edited by humans.

When Bruno Dupire proposed his celebrated local volatility model in 1994, it spread like wildfire among practitioners. 

No wonder. It preserves all the seductions of Black–Scholes’ near-magical delta hedge and, with limited numerical effort, can be calibrated to any surface of traded European option prices – simply by letting volatility depend on price level and time. When such a congenial quick fix becomes available, it is hard not to use it without restraint – in particular, to price exotics in a way that is supposedly perfectly consistent with vanilla markets.

Unfortunately, local volatility is at heart a sophisticated curve-fitting exercise, with little grounding in market reality. 

It brings to mind Ptolemy’s epicycles – ingenious geometric contraptions designed to reconcile planetary observations with a geocentric worldview. Each planet was made to move in a small circle, whose centre itself moved along a larger circle around Earth. The construction was remarkably effective: it reproduced complicated apparent paths and yielded accurate tables for planetary positions – accuracy that Newtonian dynamics initially struggled to beat.

But there are no epicycles. There are only planets tracing ellipses around the sun.

Volatility is the outcome of self-referential feedback loops

Likewise, it won’t do to pretend that markets behave as if volatility is merely a function of time and level – or as if economic agents are rational. A good theory should be, first and foremost, a faithful mental picture of what is going on in the world. 

As Doyne Farmer argues in his book Making Sense of Chaos, models should follow a principle of verisimilitude. “Models should fit the facts and their assumptions should be plausible,” he writes. “Assumptions that seem wrong from the outset are more likely to lead to false conclusions than plausible assumptions. We need to replace ‘as-if’ reasoning with ‘as-is’ reasoning.” 

Fast forward 25 years, and we know much more about what drives volatility. We also know that the much-ballyhooed risk-neutral probability distribution that option markets are said to reveal is, in practice, far closer to the true distribution of future paths than financial mathematics textbooks would have us believe. In technical lingo, Q is not so different from P.

So, what does volatility depend on? Beyond genuinely unpredictable, exogenous news that buffets markets, volatility is to a large extent endogenous. It is shaped by past volatility itself – but also by the path the price took to reach its current level. The so-called leverage effect is well known – when a stock goes down, its future volatility tends to rise. This effect is even stronger for equity indexes, because correlations between stocks increase at the same time, compounding the move.

Another, more recent effect – first unearthed by Gilles Zumbach – is that trends of either sign also generate higher future volatility. Put differently, a sequence of positively autocorrelated returns is followed by higher volatility than a sequence with the same distribution of returns but with a sideways, zigzagging structure.

Related

Pricing with a smile 

Path-dependent volatility 

In short, volatility is rough and path dependent, not merely level dependent as local volatility models assume. This matters in concrete settings – pricing Vix options, for example, or options with complicated path-dependent pay-offs – where the path dependence of volatility is (almost tautologically) of crucial importance.

But the story does not end there. The volatility of asset A can also depend on the path of asset B. In a recent paper with Cecilia Aubrun and Michael Benzaquen, we show that cross-leverage and cross-Zumbach effects exist in a multivariate setting. The influence of recent trends in the E-mini futures contract on the volatility of other futures contracts is especially strong. The cross-leverage effect between the E-mini and the residual volatility of single stocks is notable – and surprisingly universal across the stock universe. In other words, when the index falls, the idiosyncratic component of volatility rises.

Finally, we have uncovered a new effect coupling past realised covariance between two assets and future volatility of each asset – illustrated particularly clearly by the E-mini/T-bond pair.

These findings strengthen the view that volatility is the outcome of self-referential feedback loops – arising not only from each asset’s own dynamics, but from the subtle collective dance of multiple, correlated markets. They also illustrate how research moves forward – not by becoming transfixed by elegant, convenient solutions, but through careful, stubborn attention to empirical facts – facts that, as they accumulate, reveal deeper layers of structure and complexity.

Benoit Mandelbrot famously noted that in economics there can never be a “theory of everything, but that each attempt comes closer to a proper understanding of how markets behave”. Indeed, the right ambition is not a single final formula. It is to keep replacing epicycles with mechanisms – to build models that do not merely agree with markets today, but that still make sense when markets change tomorrow. 

The reward is not aesthetic elegance. It is understanding. 

Is the topic of next month’s column – generative AI ‘non-models’ – a glaring counter example?

Editing by Mauro Cesa