Moment Explosions in the Rough Heston Model

A comparison principle between rough and non-rough Heston models—with applications to the volatility surface

“…In the power-law case κ " κ α this has already been shown in [GGP18]. However, our goal is not just to characterize the domains where ψ κ exists globally, but rather to give a more refined comparison principle between ψ κ and the non-rough Heston solution ψ 1 .…”

“…The properties of the solution ψ κ , in particular its maximal life-time, crucially depend on the nature of Rpu, wq. As in [GGP18], we distinguish between the following cases, illustrated in Figure 1 (A) Rpu, 0q ą 0 and B w Rpu, 0q ě 0, ∆puqq. In the case ρ ă 0 (which is typical in applications) this leads to the following classification, which is illustrated in Figure 3 below: Lemma 3.2.…”

“…4.1). Our new bound is usually tighter than the upper bound of [GGP18] and, more importantly, given by a transformation of the classic Heston explosion time, thus allowing for direct comparison between rough and non-rough Heston models. The result rests on a comparison principle for non-linear Volterra integral equations and leads to a number of further comparison results: For the moment generating functions of rough and non-rough Heston model (Thm.…”

“…Starting with the results of [Lee04] (see also [BFL09]) it has become well-understood that wing asymptotics of implied volatility are intimately connected to moment explosions in the underlying stochastic model (see also [FKR10]). Consequently, a first study of moment explosions in the rough Heston model has been undertaken by Gerhold, Gerstenecker and Pinter [GGP18]. The authors derive a lower 1 and upper bound for moment explosion times and a method for their numerical approximation (valid in a certain parameter range).…”

“…The authors derive a lower 1 and upper bound for moment explosion times and a method for their numerical approximation (valid in a certain parameter range). We build on the results of [GGP18] and derive a new upper bound for the moment explosion time in the rough Heston model (Thm. 4.1).…”