Robust option pricing with Neural SDEs

1.导论：数学建模在金融行业中无处不在，并驱动着关键的决策过程。每个模型只提供了对现实的粗略近似，使用不充分的模型的风险通常很难检测和量化。因此，模型不确定性是数学建模的一个重要组成部分，在数学金融和经济学中尤其重要，因为在这些领域，人们不能将模型建立在完善的物理定律之上。然而，金融领域的标准建模实践很少涉及这种模型不确定性。一种这样的范式是稳健的金融方法，其中模型风险是其哲学的核心。目前，机器学习技术正在为稳健和数据驱动的模型选择机制的不同方式打开大门。然而，大多数机器学习模型仍然被认为是所谓的“黑匣子”，因为单个参数没有意义的解释。因此，我们计划在期权定价的背景下，将经典建模与深度学习技术结合起来。直到最近，除了其他原因外，模型的复杂性是不可取的，因为增加了执行所需的计算工作量，特别是校准，但也包括定价和风险计算。随着更多的机器学习方法和更强的计算能力，现在可以使用更复杂的模型。在我们的方法中，我们让数据决定模型，同时仍然在模型表单上保留强大的优先级。这是通过使用模型动力学的随机微分方程组(SDE)来实现的，但我们不是为模型SDE选择固定的参数，而是允许通过过参数的神经网络来给出漂移和扩散。我们将这些称为神经性SDE。这些不仅为模型选择提供了一个系统的框架，而且非常值得注意的是，它还能对衍生品价格产生稳健的估计。在这里，校准和模型选择同时进行。由于神经SDE模型是过度参数化的，因此存在大量可能的模型，并且训练算法选择一个模型。2.与EPSRC的研究领域保持一致：本项目属于EPSRC的统计和应用概率研究领域。提出的方法结合了来自随机和概率建模的经典概率技术和来自数据科学，更具体地说是深度神经网络的新的机器学习方法。我们将重点放在应用概率上，结合稳健统计和人工智能，这与拟议的研究领域是一致的。我们强调，这种方法的应用远远不止期权定价，更广泛地说，也适用于金融。它涵盖了任何场景，包括在不同时间点具有遗传随机性和已知值或测量值(或其函数)的建模过程。这类应用包括数据分析、医疗保健建模和医疗统计、人工智能和不确定性量化等方面的问题。合作：该项目是与艾伦·图灵研究所(ATI)和爱丁堡大学联合开展的，尤其是与大卫·西斯卡和ATI的金融和经济项目主任Lukasz Szpruch及其研究团队成员Marc Sabate Vidales和Patryk Gierjatowicz共同开展的。

1. Introduction: Mathematical modelling is ubiquitous in the financial industry and drives key decision processes. Every model provides only a crude approximation to reality and the risk of using an inadequate model is usually hard to detect and quantify. Model uncertainty is, hence, an essential part of mathematical modelling and is particularly important in mathematical finance and economics, where one cannot base models on well-established physical laws. Nevertheless, standard modelling practices in finance rarely address this model uncertainty. One such paradigm, where the model risk is central to its philosophy, is the robust finance approach. Currently, machine learning techniques are opening doors to different ways of robust and data-driven model selection mechanisms. However, most machine learning models are still considered to be so-called "black-boxes" as individual parameters do not have meaningful interpretation. We therefore plan to focus on combining classical modelling with deep learning techniques in the context of option pricing. Until recently, model complexity was undesirable, amongst other reasons, for increasing the computational effort required to perform, in particular calibration, but also pricing and risk calculations. With greater uptake of machine learning methods and greater computational power more complex models can now be used. In our approach, we let the data dictate the model, while still keeping a strong prior on the model form. This is achieved by using stochastic differential equations (SDEs) for the model dynamics, but instead of choosing a fixed parametrization for the model SDEs we allow the drift and diffusion to be given by over-parametrised neural networks. We refer to these as Neural SDEs. These are shown to not only provide a systematic framework for model selection, but also, quite remarkably, to produce robust estimates on the derivative prices. Here, the calibration and model selection are done simultaneously. Since the neural SDE model is overparametrised, there is a large pool of possible models and the training algorithm selects a model. 2. Alignment to EPSRC research areas: This project falls within the EPSRC 'Statistics and applied probability' research area. Presented methodology combines classical probabilistic techniques from stochastic and probabilistic modelling and novel machine learning approaches from data-science, more specifically -- deep neural networks. We kept our focus on applied probability in combination with robust statistics and artificial intelligence, which is in line with the proposed research area. We emphasise this approach has applications well beyond just option pricing and, more broadly, finance. It covers any scenario including modelling processes with inherit randomness and known values or measurements (or functions thereof) at different points in time. Such applications include problems in data analytics, healthcare modelling and medical statistics, artificial intelligence and uncertainty quantification etc.3. Collaboration: The project was carried out jointly with the Alan Turing Institute (ATI) and University of Edinburgh, more particularly with David Siska and the Programme Director for Finance and Economics at ATI, Lukasz Szpruch and members of his research team Marc Sabate Vidales and Patryk Gierjatowicz.