Utility Maximization

Asymptotic methods for transaction costs ArXiv ID: 2407.07100 “View on arXiv” Authors: Unknown Abstract We propose a general approximation method for determining optimal trading strategies in markets with proportional transaction costs, with a polynomial approximation of the residual value function. The method is exemplified by several problems from optimally tracking benchmarks, hedging the Log contract, to maximizing utility from terminal wealth. Strategies are also approximated by practically executable, discrete trades. We identify the necessary trade-off between trading frequency and trade sizes to have satisfactory agreement with the theoretically optimal, continuous strategies of infinite activity. ...

Leveraging IS and TC: Optimal order execution subject to reference strategies ArXiv ID: 2401.03305 “View on arXiv” Authors: Unknown Abstract The paper addresses the problem of meta order execution from a broker-dealer’s point of view in Almgren-Chriss model under execution risk. A broker-dealer agency is authorized to execute an order of trading on some client’s behalf. The strategies that the agent is allowed to deploy is subject to a benchmark, referred to as the reference strategy, regulated by the client. We formulate the broker’s problem as a utility maximization problem in which the broker seeks to maximize his utility of excess profit-and-loss at the execution horizon, of which optimal feedback strategies are obtained in closed form. In the absence of execution risk, the optimal strategies subject to reference strategies are deterministic. We establish an affine structure among the trading trajectories under optimal strategies subject to general reference strategies using implementation shortfall (IS) and target close (TC) orders as basis. Furthermore, an approximation theorem is proposed to show that with small error, general reference strategies can be approximated by piece-wise constant ones, of which the optimal strategy is piece-wise linear combination between IS and TC orders. We conclude the paper with numerical experiments illustrating the trading trajectories as well as histograms of terminal wealth and utility at investment horizon under optimal strategies versus those under TWAP strategies. ...

Optimal Portfolio with Ratio Type Periodic Evaluation under Short-Selling Prohibition ArXiv ID: 2311.12517 “View on arXiv” Authors: Unknown Abstract This paper studies some unconventional utility maximization problems when the ratio type relative portfolio performance is periodically evaluated over an infinite horizon. Meanwhile, the agent is prohibited from short-selling stocks. Our goal is to understand the impact of the periodic reward structure on the long-run constrained portfolio strategy. For power and logarithmic utilities, we can reformulate the original problem into an auxiliary one-period optimization problem. To cope with the auxiliary problem with no short-selling, the dual control problem is introduced and studied, which gives the characterization of the candidate optimal portfolio within one period. With the help of the results from the auxiliary problem, the value function and the optimal constrained portfolio for the original problem with periodic evaluation can be derived and verified, allowing us to discuss some financial implications under the new performance paradigm. ...

Optimal fees in hedge funds with first-loss compensation ArXiv ID: 2310.19023 “View on arXiv” Authors: Unknown Abstract Hedge fund managers with the first-loss scheme charge a management fee, a performance fee and guarantee to cover a certain amount of investors’ potential losses. We study how parties can choose a mutually preferred first-loss scheme in a hedge fund with the manager’s first-loss deposit and investors’ assets segregated. For that, we solve the manager’s non-concave utility maximization problem, calculate Pareto optimal first-loss schemes and maximize a decision criterion on this set. The traditional 2% management and 20% performance fees are found to be not Pareto optimal, neither are common first-loss fee arrangements. The preferred first-loss coverage guarantee is increasing as the investor’s risk-aversion or the interest rate increases. It decreases as the manager’s risk-aversion or the market price of risk increases. The more risk averse the investor or the higher the interest rate, the larger is the preferred performance fee. The preferred fee schemes significantly decrease the fund’s volatility. ...

Mind the Cap! – Constrained Portfolio Optimisation in Heston’s Stochastic Volatility Model ArXiv ID: 2306.11158 “View on arXiv” Authors: Unknown Abstract We consider a portfolio optimisation problem for a utility-maximising investor who faces convex constraints on his portfolio allocation in Heston’s stochastic volatility model. We apply the duality methods developed in previous work to obtain a closed-form expression for the optimal portfolio allocation. In doing so, we observe that allocation constraints impact the optimal constrained portfolio allocation in a fundamentally different way in Heston’s stochastic volatility model than in the Black Scholes model. In particular, the optimal constrained portfolio may be different from the naive capped portfolio, which caps off the optimal unconstrained portfolio at the boundaries of the constraints. Despite this difference, we illustrate by way of a numerical analysis that in most realistic scenarios the capped portfolio leads to slim annual wealth equivalent losses compared to the optimal constrained portfolio. During a financial crisis, however, a capped solution might lead to compelling annual wealth equivalent losses. ...

Exponential Utility Maximization in a Discrete Time Gaussian Framework ArXiv ID: 2305.18136 “View on arXiv” Authors: Unknown Abstract The aim of this short note is to present a solution to the discrete time exponential utility maximization problem in a case where the underlying asset has a multivariate normal distribution. In addition to the usual setting considered in Mathematical Finance, we also consider an investor who is informed about the risky asset’s price changes with a delay. Our method of solution is based on the theory developed in [“4”] and guessing the optimal portfolio. ...

Machine Learning for Trading ArXiv ID: ssrn-3015609 “View on arXiv” Authors: Unknown Abstract In multi-period trading with realistic market impact, determining the dynamic trading strategy that optimizes expected utility of final wealth is a hard problem Keywords: Market Impact, Optimal Execution, Dynamic Trading, Utility Maximization, Algorithmic Trading, Equities / Quantitative Trading Complexity vs Empirical Score Math Complexity: 8.5/10 Empirical Rigor: 3.0/10 Quadrant: Lab Rats Why: The paper uses advanced multi-period optimal control theory, utility theory, and Hamilton-Jacobi-Bellman equations, indicating high mathematical complexity, but focuses on theoretical proof-of-concept in a simulated market with no real-world data, backtests, or implementation details, resulting in low empirical rigor. flowchart TD Start(["Research Goal"]) --> Method["Dynamic Trading Strategy<br/>Optimization with Market Impact"] Start --> Input["Realistic Market Data<br/>& Historical Prices"] Method --> Process["Computational Process:<br/>Multi-Period Optimization<br/>Maximizing Expected Utility"] Input --> Process Process --> Outcome1["Novel Optimal<br/>Execution Algorithms"] Process --> Outcome2["Quantified Market<br/>Impact Costs"] Process --> Outcome3["Dynamic Strategy<br/>Constraints Analysis"] Outcome1 --> End(["Key Findings"]) Outcome2 --> End Outcome3 --> End