Multivariate Quadratic Hawkes Processes
Volatility is now well known to be path dependent, i.e. dependent on the path taken by recent price changes (see e.g. Volatility is (mostly) path-dependent: Quantitative Finance: Vol 23, No 9). This is at odds with the popular, but deeply flawed local volatility models for which volatility only depends on the current value of the asset, but not on the path that brought it where it is now.
What matters is, on top of dependence on past realized volatility, what the past trend was. Interestingly the dependence is quadratic. For small enough trends, the correlation is negative for the S&P (i.e. vol goes up when price goes down, the so-called “leverage” effect). But for large trends, vol goes up independently of their sign — this is the Zumbach effect. This is captured by Quadratic Hawkes Processes, that we proposed in a univariate setting with Pierre Blanc and Jonathan Donier. An interesting question is to know whether cross-leverage and cross-Zumbach effects exist in a multivariate setting.
This is what our recent paper arxiv.org/pdf/2509.21244 with Cecilia Aubrun, PhD and Michael Benzaquen investigates. Our main findings are: (a) While cross-Hawkes feedback effects have been empirically studied previously, cross-Zumbach effects are clearly identified for the first time. The effect of recent trends of the E-mini futures contract onto the volatility of other futures contracts is especially strong; (b) We have identified a new type of feedback that couples past realized covariance between two assets and future volatility of these two assets, with the pair E-mini/Tbond as a case in point; (c) A cross-leverage effect, whereby the sign of the return of one asset impacts the volatility of another asset, is also clearly identified.
The cross-leverage effect between the E-mini and the residual volatility of single stocks is notable, and surprisingly universal across the universe of stocks that we considered, see Fig below where the hardly visible grey region shows the dispersion (over stocks) of the cross-leverage kernel. In other words: when the index goes down, the idiosyncratic part of the vol also goes up.
Jean-Philippe Bouchaud
Jean-Philippe Bouchaud is a renowned physicist whose career bridges groundbreaking research in statistical physics, complex systems, and quantitative finance. Widely regarded as one of the leading figures in the application of physics-inspired methods to economics and markets.
He is adjunct professor at École Normale Supérieure and co-director of the X-CFM chair of Econophysics & Complex Systems (Ecole Polytechnique). He is a member of the French Academy of Sciences, and held the Bettencourt Innovation Chair at Collège de France in 2020. Jean-Philippe graduated from Ecole Normale Supérieure in Paris, where he also obtained his PhD in physics.
Bouchaud is the Chairman and Chief Scientist of Capital Fund Management (CFM), a Paris-based quantitative investment firm he co-founded in 1994, where he has shaped research at the intersection of physics, mathematics, and financial modeling
