Optimal Portfolio under Fast Mean-Reverting Fractional Stochastic Environment

Abstract

Empirical studies indicate the existence of long range dependence in the volatility of the underlying asset. This feature can be captured by modeling its return and volatility using functions of a stationary fractional Ornstein-Uhlenbeck (fOU) process with Hurst index H ∈ ( 1 2 , 1). In this paper, we analyze the nonlinear optimal portfolio allocation problem under this model and in the regime where the fOU process is fast mean-reverting. We first consider the case of power utility, and rigorously give first order approximations of the value and the optimal strategy by a martingale distortion transformation. We also establish the asymptotic optimality in all admissible controls of a zeroth order trading strategy. Then, we consider the case with general utility functions using the epsilon-martingale decomposition technique, and we obtain similar asymptotic optimality results within a specific family of admissible strategies..Here, ǫ is a small parameter to make the process Y ǫ,H t fast-varying and its natural time scale to be of order ǫ (that is, its mean-reversion time scale proportional to ǫ), and W (H) t is a fractional Brownian motion

Mean–Variance Portfolio Selection Under Volterra Heston Model

“…Compared with Bäuerle and Desmettre (2018), we allow the correlation between stock and volatility to be non-zero, reflecting the well-known market leverage effect. Although the power utility maximization with the classic Heston model has been studied in Kraft (2005), the non-Markovian and non-semimartingale characteristic in Volterra Heston model prevents the use of the Hamilton-Jacobi-Bellman (HJB) framework for the classical model in Kraft (2005).To overcome the aforementioned difficulty, we apply the martingale optimality principle and construct the Ansatz, which is inspired by the martingale distortion transformation (Zariphopoulou, 2001;Fouque and Hu, 2018a) and the exponential-affine representations Abi Jaber et al (2017). The key finding is the auxiliary process M t in (3.5) and the properties of it presented in Theorem 3.1 below.…”

Optimal portfolio under fractional stochastic environment

“…The case of fast varying fractional stochastic environment with is the topic of the paper Fouque and Hu ().…”

“…Finally, we extend our analysis to the case of general utilities where we can derive the first-order asymptotic optimality within a specific subclass of strategies , which is of the form̃0 +̃1, with 0 and̃1 being of feedback forms and > 0. The case of fast varying fractional stochastic environment with ∈ ( 1 2 , 1) is the topic of the paper Fouque and Hu (2018). (i) The slowly varying fractional factor , defined in (29) is a stationary Gaussian process with zero mean and variance [ (…”

“…() for further discussion of the time scales involved), we only consider one slowly varying fractional stochastic factor denoted by for . The case with fast mean‐reverting fractional environment is treated in Fouque and Hu (), while multiscale models are studied in the paper in preparation Hu (). As in Garnier and Sølna (), we model by a fractional Ornstein–Uhlenbeck (fOU) process, which satisfies the following stochastic differential equation (SDE) where δ is a small parameter and is an fBm with Hurst index H .…”

“…As in Fouque and Hu (2017), and in particular because of the relevance for longterm investments (see Fouque et al (2015) for further discussion of the time scales involved), we only consider one slowly varying fractional stochastic factor denoted by , for 0 < < 1. The case with fast mean-reverting fractional environment is treated in Fouque and Hu (2018), while multiscale models are studied in the paper in preparation Hu (2018). As in Garnier and Sølna (2017), we model , by a fractional Ornstein-Uhlenbeck (fOU) process, which satisfies the following stochastic differential equation (SDE)…”