Optimal Portfolio under Fractional Stochastic Environment

Fouque, Jean‐Pierre; Hu, Ruimeng

doi:10.48550/arxiv.1703.06969

Cited by 4 publications

(23 citation 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%

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Optimal Investment with Correlated Stochastic Volatility Factors

Bichuch

Fouque

2019

SSRN Journal

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Portfolio Optimization under Fast Mean-Reverting and Rough Fractional Stochastic Environment

Fouque

2018

Applied Mathematical Finance

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Fractional stochastic volatility models have been widely used to capture the non-Markovian structure revealed from financial time series of realized volatility. On the other hand, empirical studies have identified scales in stock price volatility: both fast-time scale on the order of days and slow-scale on the order of months. So, it is natural to study the portfolio optimization problem under the effects of dependence behavior which we will model by fractional Brownian motions with Hurst index H, and in the fast or slow regimes characterized by small parameters ǫ or δ. For the slowly varying volatility with H ∈ (0, 1), it was shown that the first order correction to the problem value contains two terms of order δ H , one random component and one deterministic function of state processes, while for the fast varying case with H > 1 2 , the same form holds at order ǫ 1−H . This paper is dedicated to the remaining case of a fast-varying rough environment (H < 1 2 ) which exhibits a different behavior. We show that, in the expansion, only one deterministic term of order √ ǫ appears in the first order correction.

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Optimal Portfolio under Fast Mean-Reverting Fractional Stochastic Environment

Fouque¹,

Hu²

2018

SIAM J. Finan. Math.

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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

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Optimal Portfolio under Fractional Stochastic Environment

Cited by 4 publications

References 0 publications

Optimal Investment with Correlated Stochastic Volatility Factors

Optimal Investment with Correlated Stochastic Volatility Factors

Portfolio Optimization under Fast Mean-Reverting and Rough Fractional Stochastic Environment

Optimal Portfolio under Fast Mean-Reverting Fractional Stochastic Environment

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