RoughHeston

Section 3.1 of Perfect hedging in rough Heston models: Generalized rough Heston models as limit of nearly unstable HawkesProcesses

In this section, the authors explore the connection between generalized rough Heston models and nearly unstable Hawkes processes. They explain that a microscopic price model based on two-dimensional Hawkes processes converges to a rough Heston log-price with constant mean-reversion after suitable rescaling. However, this method has limitations when it comes to computing prices and hedging portfolios using classical Fourier inversion methods.

To address this issue, the authors propose an alternative approach similar to the one presented in another study. They consider a sequence of one-dimensional Hawkes processes with intensity given by:

λTt = µT + ∫(from 0 to t) aT * ϕ(t - s) * dNsT

Here, µT and aT are positive constants with aT < 1, and ϕ is an integrable function. The authors show that, under certain conditions on the parameters, the intensity process λTt asymptotically behaves as the variance process of a rough Heston model with constant mean-reversion and an initial variance equal to zero.

To obtain a time-dependent mean-reversion level and a non-zero starting value in the limit, the authors draw inspiration from another study, which suggests that a time-dependent µT is a way to modify some parameters in the limit. This approach helps to better understand the relationship between generalized rough Heston models and nearly unstable Hawkes processes, and ultimately aids in managing financial risks more effectively.