رويكرد يادگيري ماشين براي قيمت گذاري سبد مالي و مديريت ريسك براي مسائل با ابعاد بالا

Machine Learning Approach to Portfolio Pricing an‎d Risk Management for High-Dimensional Problems

رياضي كاربردي و علوم كامپيوتر

In this thesis, we study the problem of risk management an‎d pricing of financial portfolios within a discrete-time framework. Although the problem may appear straightforward at first glance, it becomes computationally challenging in practice, particularly in high-dimensional settings. The central idea of this work is to describe the portfolio value function in a manner that is simple, stable, an‎d learnable from data. To achieve this goal, we introduce a general framework based on the concept of a replicating martingale, which enables the representation of the portfolio value process through its terminal information. Within this framework, instead of directly computing conditional expectations at each time step, we first approximate the terminal value function of the portfolio. Relying on the martingale structure of the problem, the intermediate values are then consistently recovered. This approximation is learned in a supervised learning setting using simulated data. The main challenge arises as the dimensionality of the problem increases, since Monte Carlo–based methods typically suffer from severe computational costs in high dimensions. In this thesis, we show that by learning effective features, this difficulty can be largely mitigated, leading to accurate an‎d stable approximations even in high-dimensional settings. In particular, data-driven dimension reduction proves to be especially beneficial when polynomial basis functions are employed. In addition, neural networks provide an alternative learning approach an‎d play a different role in enhancing the flexibility an‎d accuracy of the model for approximating the terminal value of the financial portfolio. In the numerical section, the performance of the proposed method is investigated in Gaussian models an‎d compared with stan‎dard techniques such as nested Monte Carlo an‎d regress-now Least-Squares Monte Carlo. These comparisons demonstrate that the proposed framework, while conceptually simple, delivers superior accuracy an‎d competitive computational efficiency, an‎d can serve as a practical tool for risk analysis an‎d solvency capital calculations in realistic applications.

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