Abstract. This paper studies dynamic mean-variance (MV) asset allocation problems in general incomplete markets. Besides the conventional MV objective on a portfolio’s terminal wealth, our framework can accommodate running MV objectives with general (nonexponential) discounting factors and, in general, any time-dependent preferences. We attempt the problem with a game-theoretic framework and decompose the equilibrium control policies into two parts: the first part is a myopic strategy characterized by a linear Volterra integral equation of the second kind, and the second part reveals the hedging demand governed by a system of nonlocal backward stochastic differential equations. We manage to establish the global well-posedness of the solutions to the two aforementioned equations in tailored Bananch spaces by the fixed-point theorem. It allows us to devise a numerical scheme for solving for the equilibrium control policy with a guarantee and to conclude that the dynamic (equilibrium) mean-variance policy in general settings is well-defined. Our probabilistic approach allows us to consider a broad range of stochastic factor models, such as the Chan–Karolyi–Longstaff–Sanders (CKLS) model. For this model, we verify all technical assumptions and provide a sound numerical scheme. Numerical examples are provided to illustrate our framework.
Abstract. We introduce predictable relative forward performance processes (PRFPP) as a new framework for studying portfolio management within a competitive and incomplete market environment. Each agent trades a distinct stock following a binomial distribution with probabilities for a positive return depending on the market regime characterized by a nontraded stochastic factor. For both the finite population and mean field games, we construct and analyse PRFPPs for initial data of the CARA class along with the associated equilibrium strategies. We find that relative performance concerns do not necessarily lead to more investment in the risky asset compared to when there are no such concerns. Under some parameter constellations, agents short a stock with positive expected excess return. The binomial market setting facilitates a straightforward adjustment of risky asset skewness, enabling an analysis of its impact on investment behavior—an aspect that continuous-time frameworks cannot capture.
Abstract. The lifted Heston model [E. Abi Jaber, Quantitative Finance, 19 (2019), pp. 1995–2013] is a stochastic volatility model emerging as a Markovian lift of the rough Heston model [O. El Euch and M. Rosenbaum, Mathematical Finance, 29 (2019), pp. 3–38] and the class of rough volatility processes. The model encodes the path dependency of volatility on a set of [Formula: see text] square-root latent factors driven by a common stochastic factor. While the system is Markovian, simulation schemes such as the Euler scheme exist but require a small-step, multidimensional simulation of the latent factors and are therefore numerically challenging. We propose a novel simulation scheme of the class of implicit integrated variance schemes. The method exploits the near-linear nature between the stochastic driver and the conditional integrated variance process, which allows for a consistent and efficient sampling of the integrated variance process using an inverse Gaussian distribution. Since we establish the linear relation using a linear projection in the [Formula: see text] space, the method is optimal in an [Formula: see text] sense and offers a significant efficiency gain over similar methods. We demonstrate that our scheme achieves near-exact accuracy even for coarse discretizations and allows for efficient pricing of volatility options with large time steps.
Abstract. We prove explicit stability bounds for CRRA portfolio functionals in one-dimensional stochastic-exponential additive return models with bounded no-short-sale portfolio fractions. The main estimate compares the Hamiltonians through [Formula: see text] perturbations of the drift and diffusion coefficients and through total variation of the finite weighted kernels [Formula: see text], where [Formula: see text]. This weight is tailored to the CRRA integrand: Compensated small jumps are quadratic, while large positive jumps contribute at order [Formula: see text]. The results yield uniform-in-control Hamiltonian convergence, a value bound, and an [Formula: see text] optimizer bound under strong concavity. For the associated CRRA problem, deterministic pointwise maximizers are optimal among all bounded predictable controls. A rare-large-jump example shows that UCP convergence and even a vanishing unweighted variation for finite jump kernels need not imply value stability; a complementary small-jump example shows that both exponents in [Formula: see text] are sharp.
We develop a model for financial contagion in collateralized networks in which two channels of fire sales interact. We consider a financial market with multiple assets that can be used for both investment purposes and to satisfy collateral requirements. In our model, a fire sale can be triggered both before default, when illiquid assets are sold to satisfy payment obligations, and after default, when collateral of defaulted institutions is sold. We investigate contagion that arises from the overlap in assets used for investment purposes and as collateral. In particular, we illustrate how fire sales triggered prior to default can reduce the effectiveness of collateralization after a default. Our results highlight the importance of using high-quality assets as collateral to improve financial stability.
