Portfolio Optimization in Fractional and Rough Heston Models

We consider a fractional version of the Heston volatility model which is inspired by [H. Guennoun et al., SIAM J. Financial Math,, 9 (2018), pp. 1017-1045]. 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 in 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.