Moment explosions in the rough Heston model

Abstract

We show that the moment explosion time in the rough Heston model [El Euch, Rosenbaum 2016, arXiv:1609.02108] is finite if and only if it is finite for the classical Heston model. Upper and lower bounds for the explosion time are established, as well as an algorithm to compute the explosion time (under some restrictions). We show that the critical moments are finite for all maturities. For negative correlation, we apply our algorithm for the moment explosion time to compute the lower critical moment. arXiv:1801.09458v3 [q-fin.MF]

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 These cases can be analyzed by applying the familiar theory of quadratic equations to the polynomial Rpu, wq. Following the notation from [GGP18], we rewrite Rpu, wq as…”

“…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 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 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).…”