arXiv subjects

Zongxia Liang

Publications and source records attributed to Zongxia Liang.

At least 55 records · Page 3Linked to original sources

arXiv2024

Equilibria for Time-inconsistent Singular Control Problems

We study a time-inconsistent singular control problem originating from irreversible reinsurance decisions with non-exponential discount. A novel definition of equilibrium for time-inconsistent singular control problems is introduced. For the problem with non-exponential discount, both sufficient and necessary conditions are derived, providing a thorough mathematical characterization of the equilibrium. Specifically, the equilibrium can be characterized by an extended HJB system, which is a coupled system of non-local parabolic equations with free boundaries. Finally, by showing the existence of classical solutions to the extended HJB system, the existence of equilibria is established under some additional assumptions.

math.OC↗

arXiv2024

Optimal VPPI strategy under Omega ratio with stochastic benchmark

This paper studies a variable proportion portfolio insurance (VPPI) strategy. The objective is to determine the risk multiplier by maximizing the extended Omega ratio of the investor's cushion, using a binary stochastic benchmark. When the stock index declines, investors aim to maintain the minimum guarantee. Conversely, when the stock index rises, investors seek to track some excess returns. The optimization problem involves the combination of a non-concave objective function with a stochastic benchmark, which is effectively solved based on the stochastic version of concavification technique. We derive semi-analytical solutions for the optimal risk multiplier, and the value functions are categorized into three distinct cases. Intriguingly, the classification criteria are determined by the relationship between the optimal risky multiplier in Zieling et al. (2014 and the value of 1. Simulation results confirm the effectiveness of the VPPI strategy when applied to real market data calibrations.

econ.GN↗

arXiv2024

Monotone Mean-Variance Portfolio Selection in Semimartingale Markets: Martingale Method

We use the martingale method to discuss the relationship between mean-variance (MV) and monotone mean-variance (MMV) portfolio selections. We propose a unified framework to discuss the relationship in general financial markets without any specific setting or completeness requirement. We apply this framework to a semimartingale market and find that MV and MMV are consistent if and only if the variance-optimal signed martingale measure keeps non-negative. Further, we provide an example to show the application of our result.

math.OC↗

arXiv2024

Dynamic portfolio selection under generalized disappointment aversion

This paper addresses the continuous-time portfolio selection problem under generalized disappointment aversion (GDA). The implicit definition of the certainty equivalent within GDA preferences introduces time inconsistency to this problem. We provide the sufficient and necessary condition for a strategy to be an equilibrium by a fully nonlinear integral equation. Investigating the existence and uniqueness of the solution to the integral equation, we establish the existence and uniqueness of the equilibrium. Our findings indicate that under disappointment aversion preferences, non-participation in the stock market is the unique equilibrium. The semi-analytical equilibrium strategies obtained under the constant relative risk aversion utility functions reveal that, under GDA preferences, the investment proportion in the stock market consistently remains smaller than the investment proportion under classical expected utility theory. The numerical analysis shows that the equilibrium strategy's monotonicity concerning the two parameters of GDA preference aligns with the monotonicity of the degree of risk aversion.

q-fin.MF↗

arXiv2024

Retirement decision with addictive habit persistence in a jump diffusion market

This paper investigates the optimal retirement decision, investment, and consumption strategies in a market with jump diffusion, taking into account habit persistence and stock-wage correlation. Our analysis considers multiple stocks and a finite time framework, intending to determine the retirement boundary of the ``wealth-habit-wage" triplet $(x, h, w)$. To achieve this, we use the habit reduction method and a duality approach to obtain the retirement boundary of the primal variables and feedback forms of optimal strategies. { When dealing with the dual problem, we address technical challenges in the proof of integral equation characterization of optimal retirement boundary using a $C^1$ version of It$\hat{\rm o}$'s formula.} Our results show that when the so-called ``de facto wealth" exceeds a critical proportion of wage, an immediate retirement is the optimal choice for the agent. Additionally, we find that the introduction of jump risks allows for the possibility of discontinuous investment strategies within the working region, which is a novel and insightful finding. Our numerical results effectively illustrate these findings by varying the parameters.

q-fin.MF↗

arXiv2024

Stackelberg reinsurance and premium decisions with MV criterion and irreversibility

We study a reinsurance Stackelberg game in which both the insurer and the reinsurer adopt the mean-variance (abbr. MV) criterion in their decision-making and the reinsurance is irreversible. We apply a unified singular control framework where irreversible reinsurance contracts can be signed in both discrete and continuous times. The results theoretically illustrate that, rather than continuous-time contracts or a bunch of discrete-time contracts, a single once-for-all reinsurance contract is preferred. Moreover, the Stackelberg game turns out to be centering on the signing time of the single contract. The insurer signs the contract if the premium rate is lower than a time-dependent threshold and the reinsurer designs a premium that triggers the signing of the contract at his preferred time. Further, we find that reinsurance preference, discount and reversion have a decreasing dominance in the reinsurer's decision-making, which is not seen for the insurer.

q-fin.MF↗

arXiv2024

A Mean Field Game Approach to Relative Investment-Consumption Games with Habit Formation

This paper studies an optimal investment-consumption problem for competitive agents with exponential or power utilities and a common finite time horizon. Each agent regards the average of habit formation and wealth from all peers as benchmarks to evaluate the performance of her decision. We formulate the n-agent game problems and the corresponding mean field game problems under the two utilities. One mean field equilibrium is derived in a closed form in each problem. In each problem with n agents, an approximate Nash equilibrium is then constructed using the obtained mean field equilibrium when n is sufficiently large. The explicit convergence order in each problem can also be obtained. In addition, we provide some numerical illustrations of our results.

math.OC↗

arXiv2023

A Framework of State-dependent Utility Optimization with General Benchmarks

Benchmarks in the utility function have various interpretations, including performance guarantees and risk constraints in fund contracts and reference levels in cumulative prospect theory. In most literature, benchmarks are a deterministic constant or a fraction of the underlying wealth variable; thus, the utility is also a function of the wealth. In this paper, we propose a general framework of state-dependent utility optimization with stochastic benchmark variables, which includes stochastic reference levels as typical examples. We provide the optimal solution(s) and investigate the issues of well-definedness, feasibility, finiteness, and attainability. The major difficulties include: (i) various reasons for the non-existence of the Lagrange multiplier and corresponding results on the optimal solution; (ii) measurability issues of the concavification of a state-dependent utility and the selection of the optimal solutions. Finally, we show how to apply the framework to solve some constrained utility optimization problems with state-dependent performance and risk benchmarks as some nontrivial examples.

math.OC↗

arXiv2023

Dynamic portfolio selection for nonlinear law-dependent preferences

This paper addresses the portfolio selection problem for nonlinear law-dependent preferences in continuous time, which inherently exhibit time inconsistency. Employing the method of stochastic maximum principle, we establish verification theorems for equilibrium strategies, accommodating both random market coefficients and incomplete markets. We derive the first-order condition (FOC) for the equilibrium strategies, using a notion of functional derivatives with respect to probability distributions. Then, with the help of the FOC we obtain the equilibrium strategies in closed form for two classes of implicitly defined preferences: CRRA and CARA betweenness preferences, with deterministic market coefficients. Finally, to show applications of our theoretical results to problems with random market coefficients, we examine the weighted utility. We reveal that the equilibrium strategy can be described by a coupled system of Quadratic Backward Stochastic Differential Equations (QBSDEs). The well-posedness of this system is generally open but is established under the special structures of our problem.

q-fin.MF↗

arXiv2023

Comparison Between Mean-Variance and Monotone Mean-Variance Preferences Under Jump Diffusion and Stochastic Factor Model

This paper compares the optimal investment problems based on monotone mean-variance (MMV) and mean-variance (MV) preferences in the Lévy market with an untradable stochastic factor. It is an open question proposed by Trybuła and Zawisza. Using the dynamic programming and Lagrange multiplier methods, we get the HJBI and HJB equations corresponding to the two investment problems. The equations are transformed into a new-type parabolic equation, from which the optimal strategies under both preferences are derived. We prove that the two optimal strategies and value functions coincide if and only if an important market assumption holds. When the assumption violates, MMV investors act differently from MV investors. Thus, we conclude that the difference between continuous-time MMV and MV portfolio selections is due to the discontinuity of the market. In addition, we derive the efficient frontier and analyze the economic impact of the jump diffusion risky asset. We also provide empirical evidences to demonstrate the validity of the assumption in real financial market.

math.OC↗

arXiv2023

A Unified Formula of the Optimal Portfolio for Piecewise Hyperbolic Absolute Risk Aversion Utilities

We propose a general family of piecewise hyperbolic absolute risk aversion (PHARA) utilities, including many classic and non-standard utilities as examples. A typical application is the composition of a HARA preference and a piecewise linear payoff in asset allocation. We derive a unified closed-form formula of the optimal portfolio, which is a four-term division. The formula has clear economic meanings, reflecting the behavior of risk aversion, risk seeking, loss aversion and first-order risk aversion. We conduct a general asymptotic analysis to the optimal portfolio, which directly serves as an analytical tool for financial analysis. We compare this PHARA portfolio with those of other utility families both analytically and numerically. One main finding is that risk-taking behaviors are greatly increased by non-concavity and reduced by non-differentiability of the PHARA utility. Finally, we use financial data to test the performance of the PHARA portfolio in the market.

q-fin.MF↗

arXiv2023

Weak equilibria for time-inconsistent control: with applications to investment-withdrawal decisions

This paper considers time-inconsistent problems when control and stopping strategies are required to be made simultaneously (called stopping control problems by us). We first formulate the timeinconsistent stopping control problems under general multi-dimensional controlled diffusion model and propose a formal definition of their equilibria. We show that an admissible pair $(\hat{u},C)$ of controlstopping policy is equilibrium if and only if the auxiliary function associated with it solves the extended HJB system, providing a methodology to verify or exclude equilibrium solutions. We provide several examples to illustrate applications to mathematical finance and control theory. For a problem whose reward function endogenously depends on the current wealth, the equilibrium is explicitly obtained. For another model with a non-exponential discount, we prove that any constant proportion strategy can not be equilibrium. We further show that general non-constant equilibrium exists and is described by singular boundary value problems. This example shows that considering our combined problems is essentially different from investigating them separately. In the end, we also provide a two-dimensional example with a hyperbolic discount.

math.OC↗

arXiv2023

Modern Tontine with Transaction Costs

In this paper, we propose a new type of reversible modern tontine with transaction costs. The wealth of the retiree is divided into a bequest account and a tontine account. And consumption can only be withdrawn from the bequest account. Each transaction between the two accounts incurs fixed and proportional transaction costs depending on the transaction volume. The retiree dynamically controls the allocation policy between the two accounts and the consumption policy to maximize the consumption and bequest utilities. We formulate the optimization problem as a combined stochastic and impulse control problem with infinite time horizon, and characterize the value function as the unique viscosity solution of a HJBQVI (Hamilton-Jacobi-Bellmen quasi-variational inequality). The numerical results exhibit the V-shaped transaction region which consists of two stages. In the former stage, since longevity credits increase gradually, the retiree decreases the proportion of wealth in the tontine account to smooth the wealth and reduce the volatility. However, in the latter stage, the retiree increases the proportion of wealth in the tontine account to gamble for the considerable longevity credits.

q-fin.MF↗

arXiv2023

Optimal Mix Among PAYGO, EET and Individual Savings

In order to deal with the aging problem, pension system is actively transformed into the funded scheme. However, the funded scheme does not completely replace PAYGO (Pay as You Go) scheme and there exist heterogeneous mixes among PAYGO, EET (Exempt, Exempt, Taxed) and individual savings in different countries. In this paper, we establish the optimal mix by solving a Nash equilibrium between the pension participants and the government. Given the obligatory PAYGO and EET contribution rates, the participants choose the optimal asset allocation of the individual savings and the consumption policies to achieve the objective. The results extend the ``Samuelson-Aaron" criterion to age-dependent preference orderings. And we identify three critical ages to distinguish the multiple outcomes of preference orderings based on heterogeneous characteristic parameters. The government is fully aware of the optimal feedback of the participants. It chooses the optimal PAYGO and EET contribution rates to maximize the overall utility of the participants weighted by each cohort's population. As such, the negative population growth rate leads to the decline of the PAYGO attractiveness as well as the increase of the older cohorts' weight in the government decision-making. The optimal mix is the comprehensive result of the two effects.

q-fin.MF↗

arXiv2023

Optimal management of DB pension fund under both underfunded and overfunded cases

This paper investigates the optimal management of an aggregated defined benefit pension plan in a stochastic environment. The interest rate follows the Ornstein-Uhlenbeck model, the benefits follow the geometric Brownian motion while the contribution rate is determined by the spread method of fund amortization. The pension manager invests in the financial market with three assets: cash, bond and stock. Regardless of the initial status of the plan, we suppose that the pension fund may become underfunded or overfunded in the planning horizon. The optimization goal of the manager is to maximize the expected utility in the overfunded region minus the weighted solvency risk in the underfunded region. By introducing an auxiliary process and related equivalent optimization problems and using the martingale method, the optimal wealth process, optimal portfolio and efficient frontier are obtained under four cases (high tolerance towards solvency risk, low tolerance towards solvency risk, a specific lower bound, and high lower bound). Moreover, we also obtain the probabilities that the optimal terminal wealth falls in the overfunded and underfunded regions. At last, we present numerical analyses to illustrate the manager's economic behaviors.

q-fin.PM↗

arXiv2023

The continuous-time pre-commitment KMM problem in incomplete markets

This paper studies the continuous-time pre-commitment KMM problem proposed by Klibanoff, Marinacci and Mukerji (2005) in incomplete financial markets, which concerns with the portfolio selection under smooth ambiguity. The decision maker (DM) is uncertain about the dominated priors of the financial market, which are characterized by a second-order distribution (SOD). The KMM model separates risk attitudes and ambiguity attitudes apart and the aim of the DM is to maximize the two-fold utility of terminal wealth, which does not belong to the classical subjective utility maximization problem. By constructing the efficient frontier, the original KMM problem is first simplified as an one-fold expected utility problem on the second-order space. In order to solve the equivalent simplified problem, this paper imposes an assumption and introduces a new distorted Legendre transformation to establish the bipolar relation and the distorted duality theorem. Then, under a further assumption that the asymptotic elasticity of the ambiguous attitude is less than 1, the uniqueness and existence of the solution to the KMM problem are shown and we obtain the semi-explicit forms of the optimal terminal wealth and the optimal strategy. Explicit forms of optimal strategies are presented for CRRA, CARA and HARA utilities in the case of Gaussian SOD in a Black-Scholes financial market, which show that DM with higher ambiguity aversion tends to be more concerned about extreme market conditions with larger bias. In the end of this work, numerical comparisons with the DMs ignoring ambiguity are revealed to illustrate the effects of ambiguity on the optimal strategies and value functions.

q-fin.MF↗

arXiv2023

Equilibrium Portfolio Selection for Smooth Ambiguity Preferences

This paper investigates the equilibrium portfolio selection for smooth ambiguity preferences in a continuous-time market. The investor is uncertain about the risky asset's drift term and updates the subjective belief according to the Bayesian rule. Two versions of the verification theorem are established and an equilibrium strategy can be decomposed into a myopic demand and two hedging demands. When the prior is Gaussian, the closed-form equilibrium solution is obtained. A puzzle in the numerical results is interpreted via an alternative representation of the smooth ambiguity preferences.

math.OC↗

arXiv2022

A Stackelberg reinsurance-investment game under $α$-maxmin mean-variance criterion and stochastic volatility

This paper investigates a Stackelberg game between an insurer and a reinsurer under the $α$-maxmin mean-variance criterion. The insurer can purchase per-loss reinsurance from the reinsurer. With the insurer's feedback reinsurance strategy, the reinsurer optimizes the reinsurance premium in the Stackelberg game. The financial market consists of cash and stock with Heston's stochastic volatility. Both the insurer and reinsurer maximize their respective $α$-maxmin mean-variance preferences in the market. The criterion is time-inconsistent and we derive the equilibrium strategies by the extended Hamilton-Jacobi-Bellman equations. Similar to the non-robust case in Li and Young (2022), excess-of-loss reinsurance is the optimal form of reinsurance strategy for the insurer. The equilibrium investment strategy is determined by a system of Riccati differential equations. Besides, the equations determining the equilibrium reinsurance strategy and reinsurance premium rate are given semi-explicitly, which is simplified to an algebraic equation in a specific example. Numerical examples illustrate that the game between the insurer and reinsurer makes the insurance more radical when the agents become more ambiguity aversion or risk aversion. Furthermore, the level of ambiguity, ambiguity attitude, and risk attitude of the insurer (reinsurer) have similar effects on the equilibrium reinsurance strategy, reinsurance premium, and investment strategy.

q-fin.PM↗