Special Session 118: Recent advances in mathematical finance

Irreversible Capital Accumulation with Economic Impact

Hessah Al-Motairi Kuwait University Kuwait Co-Author(s):    Mihail Zervos

Abstract: \begin{abstract} We consider an irreversible capacity expansion model in which additional investment has a strictly negative effect on the value of an underlying stochastic economic indicator. The associated optimisation problem takes the form of a singular stochastic control problem that admits an explicit solution. A special characteristic of this stochastic control problem is that changes of the state process due to control action depend on the state process itself in a proportional way. \end{abstract}

Investments in Mining Farms under Uncertainty: Real Options Approach

ahmed alqubaisi khalifa university United Arab Emirates Co-Author(s):    Ahmed Alqubaisi, Yerkin Kitabpayev

Abstract: This paper tackles the problem of optimal stopping on building a Bitcoin mining farm based on the real options framework. The framework uses the LSM method to value the spread option between stochastic Bitcoin and electricity price. Bitcoin prices and electricity prices are simulated for a farm in Texas and the option valuation framework is studied over time for a given project lifetime. We present option values for different scenarios for Bitcoin and electricity while also back-testing the model.

Ergodic optimization

Alexandre V Antonov ADIA United Arab Emirates Co-Author(s):    Ahmed AlQubaisi

Abstract: We propose a special form of optimization with a stationary utility function and corresponding weights. We considerer ergodic asset returns process and derive an analytical expression for optimal weights for certain natural extensions of the Markowitz formulation. As immediate applications we apply the proposed technique to sovereign fund needs.

Optimal hedging of the interest rate swap book

Jorgen Blomall Linkoping university Sweden Co-Author(s):

Abstract: With an optimization model interest rate curves are measured with increased accuracy from Overnight Index Swaps. Principal Component Analysis identifies the significant risk factors in interest rate markets. With these a Stochastic Programming model is formulated to determine the optimal hedge of the Overnight Index Swap book, where significant improvements are found relative to traditional delta hedging.

Dynamic portfolio risk budgeting through reinforcement learning

Giorgio Consigli Khalifa University of Science and Technology United Arab Emirates Co-Author(s):    Sanabel Bisharat, Alvaro Gomez

Abstract: In an early paper we have studied the correspondence between second order interval stochastic dominance (ISD-2) and interval conditional value-at-risk (ICVaR), a tail risk measure carrying specific properties and generalizing the popular conditional value-at-risk. Relying on the ICVaR, in this paper we present a reinforcement learning approach to solve a trade-off problem based on one side on a risk parity paradigm and on the other on an ICVaR function enforcing second order dominance with respect to a benchmark strategy. The bi-criteria objective helps clarifying the risk-budgeting implications induced by a progressive switch from risk parity towards SD against the benchmark for portfolio construction in a dynamic model. We consider a 1-year investment problem with an asset universe, or decision space of the problem, based on exchange traded funds (ETF) and market benchmarks spanning different risk classes. An extended in- and out-of-sample validation is performed on market data with a discussion on the computational properties of the problem.

Deep prediction and XAI on Financial Market Sequence for Enhancing economic policies

Massimiliano Ferrara University Mediterranea of Reggio Calabria Italy Co-Author(s):    Massimiliano Ferrara

Abstract: Numerous sectors are greatly impacted by the quick advancement of image and video processing technologies. Investors can make informed investment decisions based on the analysis and projection of ffnancial market income, and the government can create accurate policies for various forms of economic control. This study uses an artiffcial rabbits optimization algorithm in image processing technology to examine and forecast the returns on ffnancial markets and various indexes using a deep learning LSTM network. To successfully record the regional correlation properties of ffnancial market data, this research uses the time series technique. Convolution pooling in LSTM is then used to gather signiffcant details hidden in the time series data, generate the data`s trend curve, and incorporate the features using technology for image processing to ultimately arrive at the prediction of the ffnancial sector`s moment series earnings index. A popular kind of artiffcial neural network used in time series analysis is the Long Short-Term Memory (LSTM) network. By processing data with numerous input and output timesteps, it can accurately forecast ffnancial market prices. The correctness of ffnancial market predictions can be increased by optimizing the hyperparameters of an LSTM model using metaheuristic algorithms like the Artiffcial Rabbits Optimization Algorithm (ARO). This research presents the development of an optimized deep LSTM network with the ARO model (LSTM-ARO) for stock price prediction. The research`s deep learning system for ffnancial market series prediction is efffcient and precise, according to the ffndings. Technologies for data analysis and image processing offer useful approaches and signiffcantly advance the study of ffnance.

Price Formation Models with Common Noise: A Variational Approach

Diogo Gomes KAUST Saudi Arabia Co-Author(s):    Majid Almarhoumi, Julian Gutierrez, Ricardo Ribeiro

Abstract: In this work, we investigate price formation models under the influence of common noise, where agents continuously trade a commodity in a stochastic environment. The model incorporates stochastic supply dynamics and agent preferences, aiming to determine a market-clearing price that balances supply and demand. By employing a variational formulation, we derive a system of stochastic partial differential equations (SPDEs) that govern the price evolution, agent behavior, and asset distribution. We highlight key results, including the decoupling of variance dynamics in the quadratic case and the implications for market equilibrium.

Finite Element Method for HJB in Option Pricing with Stock Borrowing Fees

Rakhymzhan Kazbek Astana IT University Kazakhstan Co-Author(s):    Aidana Abdukarimova

Abstract: In mathematical finance, many derivatives from markets with frictions can be formulated as optimal control problems in the HJB framework. Analytical optimal control can result in highly nonlinear PDEs, which might yield unstable numerical results. Accurate and convergent numerical schemes are essential to leverage the benefits of the hedging process. In this study, we apply a finite element approach with a non-uniform mesh for the task of option pricing with stock borrowing fees, leading to an HJB equation that bypasses analytical optimal control in favor of direct PDE discretization. The time integration employs the theta-scheme, with initial modifications following Rannacher`s procedure. A Newton-type algorithm is applied to address the penalty-like term at each time step. Numerical experiments are conducted, demonstrating consistency with a benchmark problem and showing a strong match. The CPU time needed to reach the desired results favors P2-FEM over FDM and linear P1-FEM, with P2-FEM displaying superior convergence. This paper presents an efficient alternative framework for the HJB problem and contributes to the literature by introducing a finite element method (FEM)-based solution for HJB applications in mathematical finance.

A Coupled Optimal Stopping Approach to Pairs Trading over a Finite Horizon

Yerkin Kitapbayev Khalifa University United Arab Emirates Co-Author(s):

Abstract: We study the problem of trading a mean-reverting price spread over a finite horizon with transaction costs and an unbounded number of trades. Modeling the price spread by the Ornstein-Uhlenbeck process, we formulate a coupled optimal stopping problems to determine the optimal timing to switch positions. We analyze the corresponding free-boundary system for the value functions. Our solution approach involves deriving a system of Volterra-type integral equations that uniquely characterize the boundaries associated with the optimal timing decisions. These integral equations are used to numerically compute the optimal boundaries. Numerical examples are provided to illustrate the optimal trading boundaries and examine their sensitivity with respect to model parameters.

An exploration of different machine learning algorithms for financial forecasting in crypto markets

Davide La Torre SKEMA Business School, Cote d`Azur University France Co-Author(s):    Filippo Dal Lago, Charles Webb, Jan Broekaert, Faizal Hafiz

Abstract: In this talk, we will explore the challenge of forecasting Bitcoin price movements over various horizons - 1, 7, 14, and 30 days - from two perspectives: computer science and trading. We evaluated three distinct models: Support Vector Machines (SVM), Random Forest (RF), and Multi-Layer Perceptron (MLP). We begin with an overview of the current state of financial forecasting using machine learning, highlighting key findings from previous studies and the limitations they faced. The computer science segment will detail our rigorous approach to the problem, starting with the dataset construction and the features included. We will outline a data preparation pipeline for training the models, followed by a forecasting algorithm designed to train, evaluate, and optimize hyperparameters for the three models. Our findings will reveal that SVM exhibited superior predictive capabilities. In the trading section, we will discuss how we leveraged the SVM forecasts to create a long-only trading strategy. This part will demonstrate that while the SVM performs well in theory, it can also be applied to develop a potentially profitable trading strategy in practice. Join us to discover how these insights contribute to the intersection of machine learning and trading in the cryptocurrency market.

An optimal stopping problem for variable annuities

Alessandro Milazzo University of Turin Italy Co-Author(s):    T. De Angelis and G. Stabile.

Abstract: Variable annuities are life-insurance contracts designed to meet long-term investment goals. Such contracts provide several financial guarantees to the policyholder. A minimum rate is guaranteed by the insurer in order to protect the policyholder`s capital against market downturns. Moreover, the policyholder has the right to early terminate the contract (early surrender) and to receive the account value. In general, a penalty, which decreases in time, is applied by the insurer in case of early surrender. We provide a theoretical analysis of variable annuities with a focus on the holder`s right to an early termination of the contract. We obtain a rigorous pricing formula and the optimal exercise boundary for the surrender option. We also illustrate our theoretical results with extensive numerical experiments. The pricing problem is formulated as an optimal stopping problem with a time-dependent payoff, which is discontinuous at the maturity of the contract. This structure leads to non-monotone optimal stopping boundaries, which we prove nevertheless to be continuous. Because of this lack of monotonicity, we cannot use classical methods from optimal stopping theory and, thus, we contribute a new methodology for non-monotone stopping boundaries.

Multilayer heat equations and their solutions via oscillating integral transforms

Dmitry Muravey ADIA United Arab Emirates Co-Author(s):    Andrey Itkin, Alexander Lipton

Abstract: By using series expansion of the Dirac delta function into eigenfunctions of the corresponding Sturm-Liouville problem we construct some new (oscillating) integral transforms. These transforms are then used for solving various problems in finance, physics and mathematics which could be characterized by existence of a multilayer spatial structure and moving (time-dependent) boundaries (internal interfaces) between the layers. Thus constructed solutions are semi-analytical and extend previous work of the authors.

The Boltzmann Equation in Finance

Giulio Occhionero Al Ramz PSJC United Arab Emirates Co-Author(s):    Michele Bogliardi, Zoubida Charif Khalifi, Yerkin Kitapbayev, Miquel Noguer Alonso, Giulio Occhionero, and Jorge P Zubelli

Abstract: This article seeks to bypass the reliance on the Kolmogorov Partial Differential Equations (PDEs) typically rooted in Markov stochastic processes by proposing a more flexible formula to represent different functions of random variables. This novel framework will result in the formulation of integro-differential and integro-difference equations, also referred to as Boltzmann equations, in the space of probability distributions. This approach offers greater flexibility as it governs several kinds of stochastic processes, including cases of both diffusion and concentration. Additionally, this framework allows for the derivation of the probability distribution of the price of a European contingent claim at maturity.

Algorithmic Differentiation - Artificial Intelligence

Adil Reghai ADIA United Arab Emirates Co-Author(s):    Adil Reghai

Abstract: Algorithmic Differentiation is a key component in the computation and performance quality that underpins recent advances in financial mathematics, making a real impact on the business. It is also the pillar for machine learning algorithms. Furthermore, by integrating certain mathematical results, we open up new areas of application for solving complex optimal control problems, particularly those involving American options and CVA.

Optimal trading with regime switching: Numerical and analytic techniques applied to valuing storage in an electricity balancing market

David Zoltan Szabo Corvinus University of Budapest Hungary Co-Author(s):    Paul Johnson, Peter Duck

Abstract: Accurately valuing storage in the electricity market recognizes its role in enhancing grid flexibility, integrating renewable energy, managing peak loads, providing ancillary services and improving market efficiency. In this paper we outline an optimal trading problem for an Energy Storage Device trading on the electricity balancing (or regulating) market. To capture the features of the balancing (or regulating) market price we combine stochastic differential equations with Markov regime switching to create a novel model, and outline how this can be calibrated to real market data available from NordPool. By modelling a battery that can be filled or emptied instantaneously, this simplifying assumption allows us to generate numerical and quasi analytic solutions. We implement a case study to investigate the behaviour of the optimal strategy, how it is affected by price and underlying model parameters. Using numerical (finite-difference) techniques to solve the dynamic programming problem, we can estimate the value of operating an Energy Storage Device in the market given fixed costs to charge or discharge. Finally we use properties of the numerical solution to propose a simple quasi-analytic approximation to the problem. We find that analytic techniques can be used to give a benchmark value for the storage price when price variations during the day are relatively small.

No-arbitrage perturbations of implied volatility

Michael Tehranchi University of Cambridge England Co-Author(s):

Abstract: Possible shapes of the implied volatility smile are constrained by the absence of static arbitrage. For the sake of generating stress-testing scenarios, it is useful to consider prices after a perturbation of an observed implied volatility smile. However, some perturbations (for example, parallel shifts and scalings) do not respect the no-arbitrage constraints. A family of admissible perturbations is proposed.

Reinforcement learning for optimal constant proportion portfolio management

Jorge P Zubelli Khalifa University United Arab Emirates Co-Author(s):    Giorgio Consigli

Abstract: A reinforcement learning (RL) approach is presented to address a multi-period optimization problem whereby the portfolio manager requires an optimal constant proportion portfolio strategy by minimizing a tail risk measure consistent with second order stochastic dominance (SSD) principles. As a risk measure, we consider the particular case of the Interval Conditional Value-at-Risk (ICVaR). By including the ICVaR in the reward function of an RL method we show that an optimal fixed-mix policy can be derived as solution of short- to medium-term allocation problems through an accurate specification of the learning parameters under general statistical assumptions. The methodology is tested in- and out-of-sample on market data showing good performance relative to the SP500, adopted as benchmark policy. The talk is based on joint work with Giorgio Consigli and Alvaro Gomez which appeared in the journal {\em Engineering Applications of Artificial Intelligence}.