Optimal portfolio under fractional stochastic environment

Fouque, Jean‐Pierre; Hu, Ruimeng

doi:10.1111/mafi.12195

Cited by 28 publications

(41 citation statements)

References 41 publications

(92 reference statements)

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“…. Plugging this ansatz in the HJB equation (6) and canceling terms of zero order in ε gives that the function Ψ (0) satisfies (7) and is given by (9). Cancelling the terms of order one in ε, we deduce that the function Ψ (1) must solve:…”

Section: Asymptotics Around the Fully Correlated Casementioning

confidence: 99%

“…The goal is now to construct sub-and super-solution and to obtain bounds for the value function. Consider Ψ (0) and Ψ (1) given as a solutions to (7) and (12). Using those, define…”

Section: Building Sub-and Super-solutionsmentioning

confidence: 99%

“…where M > 0 is a constant to be determined later. Since we want to use the fact that Ψ (i) , i = 0, 1 are solutions to (7) and (12), we first need to show that these exists and are unique. We will additionally show that they have smooth and bounded derivatives, as this will be useful in future calculations.…”

Section: Building Sub-and Super-solutionsmentioning

confidence: 99%

“…Specifically, the authors heuristically derived the asymptotic approximation to the value function and the optimal strategy for general utility functions. This analysis is complemented in [6] and in [7] in a non-Markovian context. The multiscale feature is essential to be able to consider multiple factors, because each factor requires a unique time scale.…”

Section: Introductionmentioning

confidence: 99%

“…The multiscale feature is essential to be able to consider multiple factors, because each factor requires a unique time scale. The analysis simplifies considerably in the case of a single factor and power utilities thanks to a distortion transformation which linearizes the problem (see [18], [9], [7]).…”

Section: Introductionmentioning

confidence: 99%

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Optimal Investment with Correlated Stochastic Volatility Factors

Bichuch

Fouque

2019

SSRN Journal

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The problem of portfolio allocation in the context of stocks evolving in random environments, that is with volatility and returns depending on random factors, has attracted a lot of attention. The problem of maximizing a power utility at a terminal time with only one random factor can be linearized thanks to a classical distortion transformation. In the present paper, we address the problem with several factors using a perturbation technique around the case where these factors are perfectly correlated reducing the problem to the case with a single factor. We illustrate our result with a particular model for which we have explicit formulas. A rigorous accuracy result is also derived using sub-and super-solutions of the HJB equation involved. In order to keep the notations as explicit as possible, we treat the case with one stock and two factors and we describe an extension to the case with two stocks and two factors.

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Section: Asymptotics Around the Fully Correlated Casementioning

confidence: 99%

“…The goal is now to construct sub-and super-solution and to obtain bounds for the value function. Consider Ψ (0) and Ψ (1) given as a solutions to (7) and (12). Using those, define…”

Section: Building Sub-and Super-solutionsmentioning

confidence: 99%

Section: Building Sub-and Super-solutionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optimal Investment with Correlated Stochastic Volatility Factors

Bichuch

Fouque

2019

SSRN Journal

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Transition density function expansion methods for portfolio optimization

Lu,

Zhou,

et al. 2024

Optim Control Appl Methods

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In this study, we introduce transition density function expansion methods inspired from Yang et al. (J Econom. 2019;209(2):256–288.) to stochastic control issues related to utility maximization, without imposing limitations on the variety of asset price models and utility functions. Utilizing Bellman's dynamic programming principle, we initially recast the conditional expectation via the transition density function pertinent to the diffusion process. Subsequently, we employ the Itô‐Taylor expansion and Delta expansion techniques to the transition density function associated with the multivariate diffusion process, facilitated by a quasi‐Lamperti transformation, aiming to derive explicit recursive expressions for expansion coefficient functions. Our main contributions are that we articulate detailed algorithms, stemming from the backward recursive formulations of the value function and optimal strategies, achieved through discretization methodologies with rigorous proof of expansion convergence in portfolio optimization. Both theoretical and practical demonstrations are presented to validate the convergence of these approximate techniques in addressing stochastic control challenges. To underscore the efficiency and precision of our proposed methods, we apply them to portfolio selection problems within several benchmark models, and highlight the reduced complexity in comparison to the current methodologies.

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Mean–Variance Portfolio Selection Under Volterra Heston Model

Han

Wong

2020

Appl Math Optim

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This paper investigates Merton's portfolio problem in a rough stochastic environment described by Volterra Heston model. The model has a non-Markovian and nonsemimartingale structure. By considering an auxiliary random process, we solve the portfolio optimization problem with the martingale optimality principle. The optimal strategy is derived in a semi-closed form that depends on the solution of a Riccati-Volterra equation. Numerical studies suggest that investment demand decreases with the roughness of the market. 1 specific example in Abi Jaber et al. (2017). In addition, the rough Heston model becomes a special case of Volterra Heston model under the so-called fractional kernel, K(t) = t H−1/2 Γ(H+1/2) . The structure of characteristic functions in El Euch and Rosenbaum (2019) can be extended to affine Volterra processes using Riccati-Volterra equations as shown in Abi Jaber et al. (2017). Therefore, this paper focuses on the financial market with the Volterra Heston model.We are interested in a question: how does the roughness of the market volatility affect investment demands? We address this by investigating the optimal investment demand with the Merton problem as it is probably the most classic financial economic approach to do so. 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. Due to some market insights, it is suggested in Glasserman and He (2019) to use roughness as a trading signal. To the best of our knowledge, portfolio selection with Volterra Heston model is firstly studied in Han and Wong (2019) in the context of mean-variance objective.In this paper, we investigate the Merton portfolio problem with an unbounded risk premium which is in contrast to the studies in (Fouque and Hu, 2018a,b) of assuming an essentially bounded risk premium. Therefore, their results are not directly applicable to our problem. 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...

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Optimal portfolio under fractional stochastic environment

Cited by 28 publications

References 41 publications

Optimal Investment with Correlated Stochastic Volatility Factors

Optimal Investment with Correlated Stochastic Volatility Factors

Transition density function expansion methods for portfolio optimization

Mean–Variance Portfolio Selection Under Volterra Heston Model

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