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Register a new userConstruction of Forward Performance Processes in Stochastic Factor Models and an Extension of Widders Theorem
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Added by
Mykhaylo Shkolnikov
Publication date
2018
fields
Financial
and research's language is
English
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We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. Given multiple traded assets, the prices of which depend on multiple observable stochastic factors, we construct a large class of forward performance processes with power-utility initial data, as well as the corresponding optimal portfolios. This is done by solving the associated non-linear parabolic partial differential equations (PDEs) posed in the wrong time direction, for stock-factor correlation matrices with eigenvalue equality (EVE) structure, which we introduce here. Along the way we establish on domains an explicit form of the generalized Widders theorem of Nadtochiy and Tehranchi [NT15, Theorem 3.12] and rely hereby on the Laplace inversion in time of the solutions to suitable linear parabolic PDEs posed in the right time direction.
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In an incomplete market, with incompleteness stemming from stochastic factors imperfectly correlated with the underlying stocks, we derive representations of homothetic (power, exponential and logarithmic) forward performance processes in factor-form using ergodic BSDE. We also develop a connection between the forward processes and infinite horizon BSDE, and, moreover, with risk-sensitive optimization. In addition, we develop a connection, for large time horizons, with a family of classical homothetic value function processes with random endowments.
This paper studies an optimal forward investment problem in an incomplete market with model uncertainty, in which the underlying stocks depend on the correlated stochastic factors. The uncertainty stems from the probability measure chosen by an investor to evaluate the performance. We obtain directly the representation of the homothetic robust forward performance processes in factor-form by combining the zero-sum stochastic differential game and ergodic BSDE approach. We also establish the connections with the risk-sensitive zero-sum stochastic differential games over an infinite horizon with ergodic payoff criteria, as well as with the classical robust expected utilities for long time horizons. Finally, we give an example to illustrate that our approach can be applied to address a type of robust forward investment performance processes with negative realization processes.
We consider the problem of optimal portfolio selection under forward investment performance criteria in an incomplete market. The dynamics of the prices of the traded assets depend on a pair of stochastic factors, namely, a slow factor (e.g. a macroeconomic indicator) and a fast factor (e.g. stochastic volatility). We analyze the associated forward performance SPDE and provide explicit formulae for the leading order and first order correction terms for the forward investment process and the optimal feedback portfolios. They both depend on the investors initial preferences and the dynamically changing investment opportunities. The leading order terms resemble their time-monotone counterparts, but with the appropriate stochastic time changes resulting from averaging phenomena. The first-order terms compile the reaction of the investor to both the changes in the market input and his recent performance. Our analysis is based on an expansion of the underlying ill-posed HJB equation, and it is justified by means of an appropriate remainder estimate.
This paper studies a robust portfolio optimization problem under the multi-factor volatility model introduced by Christoffersen et al. (2009). The optimal strategy is derived analytically under the worst-case scenario with or without derivative trading. To illustrate the effects of ambiguity, we compare our optimal robust strategy with some strategies that ignore the information of uncertainty, and provide the corresponding welfare analysis. The effects of derivative trading to the optimal portfolio selection are also discussed by considering alternative strategies. Our study is further extended to the cases with jump risks in asset price and correlated volatility factors, respectively. Numerical experiments are provided to demonstrate the behavior of the optimal portfolio and utility loss.
The problem of portfolio optimization when stochastic factors drive returns and volatilities has been studied in previous works by the authors. In particular, they proposed asymptotic approximations for value functions and optimal strategies in the regime where these factors are running on both slow and fast timescales. However, the rigorous justification of the accuracy of these approximations has been limited to power utilities and a single factor. In this paper, we provide an accuracy analysis for cases with general utility functions and two timescale factors by constructing sub- and super-solutions to the fully nonlinear problem such that their difference is at the desired level of accuracy. This approach will be valuable in various related stochastic control problems.
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