Incomplete Markets

Experimental Analysis of Deep Hedging Using Artificial Market Simulations for Underlying Asset Simulators ArXiv ID: 2404.09462 “View on arXiv” Authors: Unknown Abstract Derivative hedging and pricing are important and continuously studied topics in financial markets. Recently, deep hedging has been proposed as a promising approach that uses deep learning to approximate the optimal hedging strategy and can handle incomplete markets. However, deep hedging usually requires underlying asset simulations, and it is challenging to select the best model for such simulations. This study proposes a new approach using artificial market simulations for underlying asset simulations in deep hedging. Artificial market simulations can replicate the stylized facts of financial markets, and they seem to be a promising approach for deep hedging. We investigate the effectiveness of the proposed approach by comparing its results with those of the traditional approach, which uses mathematical finance models such as Brownian motion and Heston models for underlying asset simulations. The results show that the proposed approach can achieve almost the same level of performance as the traditional approach without mathematical finance models. Finally, we also reveal that the proposed approach has some limitations in terms of performance under certain conditions. ...

Optimal portfolio under ratio-type periodic evaluation in incomplete markets with stochastic factors ArXiv ID: 2401.14672 “View on arXiv” Authors: Unknown Abstract This paper studies a type of periodic utility maximization for portfolio management in an incomplete market model, where the underlying price diffusion process depends on some external stochastic factors. The portfolio performance is periodically evaluated on the relative ratio of two adjacent wealth levels over an infinite horizon. For both power and logarithmic utilities, we formulate the auxiliary one-period optimization problems with modified utility functions, for which we develop the martingale duality approach to establish the existence of the optimal portfolio processes and the dual minimizers can be identified as the “least favorable” completion of the market. With the help of the duality results in the auxiliary problems and some fixed point arguments, we further derive and verify the optimal portfolio processes in a periodic manner for the original periodic evaluation problems over an infinite horizon. ...

Representation of forward performance criteria with random endowment via FBSDE and its application to forward optimized certainty equivalent ArXiv ID: 2401.00103 “View on arXiv” Authors: Unknown Abstract We extend the notion of forward performance criteria to settings with random endowment in incomplete markets. Building on these results, we introduce and develop the novel concept of \textit{“forward optimized certainty equivalent (forward OCE)”}, which offers a genuinely dynamic valuation mechanism that accommodates progressively adaptive market model updates, stochastic risk preferences, and incoming claims with arbitrary maturities. In parallel, we develop a new methodology to analyze the emerging stochastic optimization problems by directly studying the candidate optimal control processes for both the primal and dual problems. Specifically, we derive two new systems of forward-backward stochastic differential equations (FBSDEs) and establish necessary and sufficient conditions for optimality, and various equivalences between the two problems. This new approach is general and complements the existing one for forward performance criteria with random endowment based on backward stochastic partial differential equations (backward SPDEs) for the related value functions. We, also, consider representative examples for both forward performance criteria with random endowment and for forward OCE. Furthermore, for the case of exponential criteria, we investigate the connection between forward OCE and forward entropic risk measures. ...

Data-Driven Merton’s Strategies via Policy Randomization ArXiv ID: 2312.11797 “View on arXiv” Authors: Unknown Abstract We study Merton’s expected utility maximization problem in an incomplete market, characterized by a factor process in addition to the stock price process, where all the model primitives are unknown. The agent under consideration is a price taker who has access only to the stock and factor value processes and the instantaneous volatility. We propose an auxiliary problem in which the agent can invoke policy randomization according to a specific class of Gaussian distributions, and prove that the mean of its optimal Gaussian policy solves the original Merton problem. With randomized policies, we are in the realm of continuous-time reinforcement learning (RL) recently developed in Wang et al. (2020) and Jia and Zhou (2022a, 2022b, 2023), enabling us to solve the auxiliary problem in a data-driven way without having to estimate the model primitives. Specifically, we establish a policy improvement theorem based on which we design both online and offline actor-critic RL algorithms for learning Merton’s strategies. A key insight from this study is that RL in general and policy randomization in particular are useful beyond the purpose for exploration – they can be employed as a technical tool to solve a problem that cannot be otherwise solved by mere deterministic policies. At last, we carry out both simulation and empirical studies in a stochastic volatility environment to demonstrate the decisive outperformance of the devised RL algorithms in comparison to the conventional model-based, plug-in method. ...

On optimal tracking portfolio in incomplete markets: The reinforcement learning approach ArXiv ID: 2311.14318 “View on arXiv” Authors: Unknown Abstract This paper studies an infinite horizon optimal tracking portfolio problem using capital injection in incomplete market models. The benchmark process is modelled by a geometric Brownian motion with zero drift driven by some unhedgeable risk. The relaxed tracking formulation is adopted where the fund account compensated by the injected capital needs to outperform the benchmark process at any time, and the goal is to minimize the cost of the discounted total capital injection. When model parameters are known, we formulate the equivalent auxiliary control problem with reflected state dynamics, for which the classical solution of the HJB equation with Neumann boundary condition is obtained explicitly. When model parameters are unknown, we introduce the exploratory formulation for the auxiliary control problem with entropy regularization and develop the continuous-time q-learning algorithm in models of reflected diffusion processes. In some illustrative numerical example, we show the satisfactory performance of the q-learning algorithm. ...

Adversarial Deep Hedging: Learning to Hedge without Price Process Modeling ArXiv ID: 2307.13217 “View on arXiv” Authors: Unknown Abstract Deep hedging is a deep-learning-based framework for derivative hedging in incomplete markets. The advantage of deep hedging lies in its ability to handle various realistic market conditions, such as market frictions, which are challenging to address within the traditional mathematical finance framework. Since deep hedging relies on market simulation, the underlying asset price process model is crucial. However, existing literature on deep hedging often relies on traditional mathematical finance models, e.g., Brownian motion and stochastic volatility models, and discovering effective underlying asset models for deep hedging learning has been a challenge. In this study, we propose a new framework called adversarial deep hedging, inspired by adversarial learning. In this framework, a hedger and a generator, which respectively model the underlying asset process and the underlying asset process, are trained in an adversarial manner. The proposed method enables to learn a robust hedger without explicitly modeling the underlying asset process. Through numerical experiments, we demonstrate that our proposed method achieves competitive performance to models that assume explicit underlying asset processes across various real market data. ...