Portfolio Optimization in Fractional and Rough Heston Models

Abstract

We consider a fractional version of the Heston volatility model which is inspired by [16]. Within this model we treat portfolio optimization problems for power utility functions. Using a suitable representation of the fractional part, followed by a reasonable approximation we show that it is possible to cast the problem into the classical stochastic control framework. This approach is generic for fractional processes. We derive explicit solutions and obtain as a by-product the Laplace transform of the integrated volatility. In order to get rid of some undesirable features we introduce a new model for the rough path scenario which is based on the Marchaud fractional derivative. We provide a numerical study to underline our results.

Mean–Variance Portfolio Selection Under Volterra Heston Model

“…An alternative definition for Heston model with rough paths in Guennoun et al (2018) is known as fractional Heston model. The power utility maximization with the fractional Heston model has been investigated in Bäuerle and Desmettre (2018) for the case of zero correlation. Our consideration is much general because we incorporate the market leverage effect with a non-zero ρ in the Volterra Heston model.…”

“…The literature tends to focus more on the option pricing problems and portfolio optimization under rough volatility models is still at an early stage. However, some recent works do exist (Fouque and Hu, 2018a,b;Bäuerle and Desmettre, 2018;Glasserman and He, 2019;Han and Wong, 2019). The studies on (Fouque and Hu, 2018a,b) consider the expected power utility portfolio maximization with slow or fast varying stochastic factors driven by the fOU processes whereas the fractional Heston model is used in Bäuerle and Desmettre (2018) with the same objective function.…”

Integral representation of generalized grey Brownian motion

“…the standard Wiener process; compare the works of Carmona, Coutin, Montseny, and Muravlev [ 5 , 6 , 32 ] or also the monograph of [ 23 ]. Recently, further applications of this representation have for instance been investigated in [ 14 ] with a focus on finance and in [ 2 ] in the context of optimal portfolios.…”

Developing a two‐stage stochastic programming model in tolerance optimization using a guard band approach

“…From an applied standpoint, much of the recent research has focused on energy systems, such as hybrid renewable energy systems, which have stochastic reserve constraints as well as probabilistic energy sources based on the weather such as solar and wind power 39‐41 . Groundwater remediation and portfolio optimization are also topics that have seen a recent interest as well 42‐45 …”