Matthieu Garcin and Martino Grasselli, professors at ESILV engineering school, have had their research into financial modelling and rough volatility published as an article in a highly-regarded web of science applied mathematics journal.
Matthieu Garcin and Martino Grasselli are professors in Financial Engineering at ESILV and members of the Finance Group at De Vinci Research Center. Their research, published in “Decisions in economics and finance” – a three-star journal – brings new insights on Fractional Brownian Motion volatility models.
“Long versus short time scales: the rough dilemma and beyond”
In this paper, the two ESILV professors outline new paths to improve sophisticated modelling tools such as Fractional Brownian Motion processes. The price of financial assets follows random movements whose average amplitude, called volatility, itself changes randomly.
Modelling this stochastic volatility is at the heart of the options market and of the hedging of their risk by banks. Recently, new empirical observations exploiting high-frequency quotes have revealed that volatility follows a fractional Brownian motion (fBm) which has the particularity of evolving very irregularly: rough volatility models were born.
The fBm, introduced by Benoît Mandelbrot and John Van Ness in 1968 to describe fractal properties, finds there a new application that has generated substantial literature in a short time. Fractals are self-similar objects: the details of the object, seen at a certain scale, are a simple transformation of the object itself.
Romanesco broccoli is a very concrete illustration of such an object. Thus, going back to finance, there is a very simple transformation linking daily changes in volatility to annual changes in volatility. It is the analysis of the magnitude of these variations at various time scales that allows us to find the fractal properties of volatility.
Capturing market behaviour needs more than fBm models
However, by exploiting a very large exchange rate dataset, Matthieu Garcin and Martino Grasselli were able to show that reality was more complex than an fBm. Indeed, by pushing the analysis of the amplitude of the variations of volatility to timescales larger than what is commonly done, one can observe a new phenomenon: the transformation from one scale to another is linear for an fBm, whereas empirical observations show a convex transformation beyond a certain scale.
Another difficulty has been addressed in this work. Studying the fractal properties of volatility is more complicated than for a Romanesco broccoli: you can touch and even eat a vegetable, but the volatility of a financial asset is an object that cannot be observed directly, we simply have an appreciation of it, disrupted by measurement noise. And it turns out that this noise biases the analysis of the object’s fractal properties. It was therefore necessary to build a statistical method capable of processing this noise in order to reach reliable conclusions.
Martino Grasselli has also been invited for a plenary talk on this subject to the conference RIO (Research In Options) 2021. It opens the door to more realistic models for the financial markets and finally to a more precise assessment and hedging of risks, in particular on the options market.