Markets perceive the future in very distorted ways

Discounting paradigms should adapt to be more realistic, says Jean-Philippe Bouchaud

Key points Summary Q&A

• Forward rate curves may be likened to a structure of elastic strings, with different maturities constrained like beads on a polymer.
• Perceived time appears to follow a logarithmic function of real time – compatible with hyperbolic discounting. The perception of the width of a future time interval tends to shrink as the future becomes more distant. One week in 10 years’ time is perceived by bond markets as just one week.
• A two‑parameter elastic‑manifold model, first proposed by Belal Baaquie and Bouchaud in 2004 and revisited by Victor Le Coz, reproduces forward rate correlations from 1994–2023 with a relative error below 2%.

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

Transferring ideas and methods from physics to finance has a long history. From the Bachelier-Einstein theory of Brownian motion for financial prices, through the Feynman-Kac path integral and Mandelbrot’s multifractals to random matrix theory – the examples of successful grafts are many.

My focus for this column is on so-called elastic manifolds – theoretical models in statistical physics – and their somewhat unexpected application to the dynamics of the forward rate curve and the volatility surface.

In its original construct, an elastic manifold is a lattice of beads tethered by springs. A one-dimensional elastic manifold is just a string of such beads – think of a polymer chain, such as polyethylene. A two-dimensional manifold is like a sheet, a famous example being graphene – a single layer of carbon atoms arranged in a honeycomb planar nanostructure.

Scientists have been studying the dynamics of such structures for many years. In the case of polymers, each bead is subject to Brownian noise, but is also tied to its nearest neighbours – such that the whole string maintains its integrity over time – and the possible conformations of the string are heavily constrained.

So, what does this have to do with finance? 

Our most striking result is the confirmation that perceived time in interest rate markets follows a logarithmic function of real time

Well, nearly 30 years ago, while watching an animation of the daily variations of the US forward rate curve, my colleagues and I were struck by the similarity of their movement to the random motion of a polymer. The analogy made immediate sense: forward rates move constantly, buffeted by random supply and demand shocks. Yet the forward rate curve has a clear one-dimensional structure – the rate for a three-month loan in three years and six months and the corresponding rate in three years and nine months simply cannot diverge.

The curve maintains its integrity, like a string. This constraint imposes a special kind of correlation between different maturities that is not random noise – rather, it is structured noise.

This idea was formalised by different groups at the time: Pedro Santa-Clara and Didier Sornette; Rama Cont; and Raphael Douady. Note: the fact that log time provides a better parameterisation of the forward rate curve is mentioned in passing in Douady’s paper.

But this line of research culminated in a model that Belal Baaquie and I proposed in 2004, which accounted remarkably well for the full correlation matrix of daily forward rate curve returns during 1994–96. One interesting twist, however, was that assuming that time flows uniformly led to a very poor goodness-of-fit. In order to achieve good calibration, we had to postulate that market participants in fact perceive future time in a very distorted way. Perhaps unsurprisingly, the parameterisation amounted to saying that one month, 10 years from now, therefore appears much shorter than one month, one year from now. The model then fell into almost complete oblivion.

Fortunately, Victor Le Coz, then a PhD candidate at the Econophysics Lab, decided to revisit the model with new ideas and, more importantly, new data. With only two parameters, the model accurately reproduces the entire correlation structure of the forward rate curve over a 30-year period from 1994 to 2023, with a relative error below 2%. 

The parameters are remarkably stable over time, with one notable exception: during the quantitative easing period of 2009–2014, they shift – suggesting that central bank intervention left a detectable imprint on how markets process information along the yield curve. The model also faithfully reproduces how correlations depend on time resolution – the so-called Epps effect – from which one can estimate a cross-tenor information propagation time of around 10 minutes.

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But perhaps our most striking result is the confirmation that perceived time in interest rate markets follows a logarithmic function of real time, as surmised in our 2004 paper. More specifically, our results are fully compatible with hyperbolic (rather than exponential) discounting, in line with the recent behavioural literature. Interest rate markets may in fact offer the best possible laboratory to test such ideas, given the depth and liquidity of the data. 

From our calibration, the crossover between normal time flow and logarithmic time flow occurs around two months into the future. Beyond this point, the distortion becomes extreme: one year, 10 years from now, is perceived by bond markets as roughly one week of real time. This reflects a striking myopia about the distant future – one the data seems to insist upon.

Similar ideas extend naturally to volatility surfaces, as anticipated by Cont and José Da Fonseca. The temptation was strong to apply the elastic manifold framework here, too, with two dimensions – maturity and moneyness. The anisotropic two-dimensional generalisation of our model fits the correlation structure of implied volatility moves very well – and, again, perceived maturity grows as the logarithm of real time.¹

Abandoning classical exponential discounting in favour of hyperbolic discounting is no small matter, both for financial economics and for macroeconomic models. What is perhaps most surprising is not the result itself – behavioural economists have long argued for hyperbolic discounting in individual decision-making – but how strongly it appears to be compelled by empirical data from the most liquid financial market in the world.

Somehow, the aggregate behaviour of bond markets encodes a collective distortion of time that no individual participant may be consciously aware of.

The elastic string was always there in plain sight. What took longer to realise is that its beads are not spaced in real time, but in its logarithm. 

“For in a minute there are many days.” As Shakespeare’s teenage heroine Juliet was keenly aware.

Note

1. As will be seen in a paper I am preparing with V Le Coz, P Ioselevich and M Smerlak.

Jean-Philippe Bouchaud is the chairman and head of research at Capital Fund Management

Editing by Louise Marshall