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Register a new userA game theoretical approach to homothetic robust forward investment performance processes in stochastic factor models
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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.
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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.
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.
We prove a maximum principle for mild solutions to stochastic evolution equations with (locally) Lipschitz coefficients and Wiener noise on weighted $L^2$ spaces. As an application, we provide sufficient conditions for the positivity of forward rates in the Heath-Jarrow-Morton model, considering the associated Musiela SPDE on a homogeneous weighted Sobolev space.
In the classical model of stock prices which is assumed to be Geometric Brownian motion, the drift and the volatility of the prices are held constant. However, in reality, the volatility does vary. In quantitative finance, the Heston model has been successfully used where the volatility is expressed as a stochastic differential equation. In addition, we consider a regime switching model where the stock volatility dynamics depends on an underlying process which is possibly a non-Markov pure jump process. Under this model assumption, we find the locally risk minimizing pricing of European type vanilla options. The price function is shown to satisfy a Heston type PDE.
We study the continuous time portfolio optimization model on the market where the mean returns of individual securities or asset categories are linearly dependent on underlying economic factors. We introduce the functional $Q_gamma$ featuring the expected earnings yield of portfolio minus a penalty term proportional with a coefficient $gamma$ to the variance when we keep the value of the factor levels fixed. The coefficient $gamma$ plays the role of a risk-aversion parameter. We find the optimal trading positions that can be obtained as the solution to a maximization problem for $Q_gamma$ at any moment of time. The single-factor case is analyzed in more details. We present a simple asset allocation example featuring an interest rate which affects a stock index and also serves as a second investment opportunity. We consider two possibilities: the interest rate for the bank account is governed by Vasicek-type and Cox-Ingersoll-Ross dynamics, respectively. Then we compare our results with the theory of Bielecki and Pliska where the authors employ the methods of the risk-sensitive control theory thereby using an infinite horizon objective featuring the long run expected growth rate, the asymptotic variance, and a risk-aversion parameter similar to $gamma$.
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