Martino Grasselli, a doctor in Quantitative Finance and professor-researcher at ESILV School of Engineering, is the co-author of an article titled “Fast Hybrid Schemes for Fractional Riccati Equations (Rough Is Not so Tough)”, published in the international journal “Mathematics of Operations Research”.
Martino Grasselli is the head of the Finance Group at the De Vinci Research Center in Paris La Defense and a full professor at the Mathematics Department of the University of Padua, Italy. He co-wrote a paper with Giorgia Callegaro and Gilles Pagès on “Fast Hybrid Schemes for Fractional Riccati Equations (Rough Is Not so Tough)”, accepted for publication in “Mathematics of Operations Research”, a research journal edited by the Institute for Operations Research and the Management Sciences (INFORMS).
Fast Hybrid Schemes for Fractional Riccati Equations (Rough Is Not so Tough)
In this paper, Giorgia Callegaro, Martino Grasselli and Gilles Pagès have introduced a very efficient numerical scheme to solution fractional Riccati equations, which speeds up the computation remarkably. The main tool was a series development of the solution, where the authors were able to control the truncation error very precisely. This result opens the door to a new technology that was difficult to exploit for rough models due to numerical complexity. In fact, it is now possible not only to price but also to calibrate the rough Heston model to market data in a reasonable time.
Volatility follows a less regular path than charted by existing models: this stylized fact induced quants to turn away from the maths of classical Brownian motion used in other approaches. Instead, the rough volatility model uses a variant called fractional Brownian motion, introduced by French mathematician Benoit Mandelbrot in the late 1960s but scarcely used since.
Fractional Brownian motion uses a parameter, the Hurst parameter (H), that regulates the dependency of current observations on previous ones.
For H greater than 0.5, variations are positively correlated. As H approaches one, the process generates a path that gets smoother and smoother. At H equals 0.5 the variations are uncorrelated, as in normal Brownian motion with its random increments. And for H smaller than 0.5, variations are negatively correlated and the resulting path draws the characteristic rough pattern.
Starting from a pioneering paper of 2014, Jim Gatheral and Mathieu Rosenbaum published a remarkable number of contributions, to the point that they have recently awarded as 2020 Quants of the year by Risk.net.
Using the fractional Brownian motion as a driver for the stochastic volatility, they extended the Heston model to the rough specification and they showed that the procedure for pricing still works similarly as in a classic way, namely by integrating the Fourier transform of the asset price.
This involves the computation of a quadratic (Riccati) ordinary differential equation, that becomes a fractional Riccati equation in the rough Heston case, which is far to be trivial.