We develop a method for incorporating relevant non-Euclidean geometric information into a broad range of classical filtering and statistical or machine learning algorithms. We apply these techniques to approximate the solution of the non-Euclidean filtering problem to arbitrary precision. We then extend the particle filtering algorithm to compute our asymptotic solution to arbitrary precision. Moreover, we find explicit error bounds measuring the discrepancy between our locally triangulated filter and the true theoretical non-Euclidean filter. Our methods are motivated by certain fundamental problems in mathematical finance. In particular we apply these filtering techniques to incorporate the non-Euclidean geometry present in stochastic volatility models and optimal Markowitz portfolios. We also extend Euclidean statistical or machine learning algorithms to non-Euclidean problems by using the local triangulation technique, which we show improves the accuracy of the original algorithm. We apply the local triangulation method to obtain improvements of the (sparse) principal component analysis and the principal geodesic analysis algorithms and show how these improved algorithms can be used to parsimoniously estimate the evolution of the shape of forward-rate curves. While focused on financial applications, the non-Euclidean geometric techniques presented in this paper can be employed to provide improvements to a range of other statistical or machine learning algorithms and may be useful in other areas of application.
We present an artificial neural network (ANN) approach to value financial derivatives. Atypically to standard ANN applications, practitioners equally use option pricing models to validate market prices and to infer unobserved prices. Importantly, models need to generate realistic arbitrage-free prices, meaning that no option portfolio can lead to risk-free profits. The absence of arbitrage opportunities is guaranteed by penalizing the loss using soft constraints on an extended grid of input values. ANNs can be pre-trained by first calibrating a standard option pricing model, and then training an ANN to a larger synthetic dataset generated from the calibrated model. The parameters transfer as well as the non-arbitrage constraints appear to be particularly useful when only sparse or erroneous data are available. We also explore how deeper ANNs improve over shallower ones, as well as other properties of the network architecture. We benchmark our method against standard option pricing models, such as Heston with and without jumps. We validate our method both on training sets, and testing sets, namely, highlighting both their capacity to reproduce observed prices and predict new ones.
Data analytics using machine learning (ML) has become ubiquitous in science, business intelligence, journalism and many other domains. While a lot of work focuses on reducing the training cost, inference runtime and storage cost of ML models, little work studies how to reduce the cost of data acquisition, which potentially leads to a loss of sellers' revenue and buyers' affordability and efficiency. In this paper, we propose a model-based pricing (MBP) framework, which instead of pricing the data, directly prices ML model instances. We first formally describe the desired properties of the MBP framework, with a focus on avoiding arbitrage. Next, we show a concrete realization of the MBP framework via a noise injection approach, which provably satisfies the desired formal properties. Based on the proposed framework, we then provide algorithmic solutions on how the seller can assign prices to models under different market scenarios (such as to maximize revenue). Finally, we conduct extensive experiments, which validate that the MBP framework can provide high revenue to the seller, high affordability to the buyer, and also operate on low runtime cost.
It began, late last year, with Silver boasting about the success of his election models and Taleb shooting back that Silver doesn't "know how math works." Silver said Taleb was "consumed by anger" and hadn't had any new ideas since 2001. The argument has gotten personal, with Silver calling Taleb an "intellectual-yet-idiot" (an insult taken from Taleb's own book) and Taleb calling Silver "klueless" and "butthurt." Here is a recap of what they're fighting about so you can know who's right (Silver, mostly) and who's wrong (Taleb). The origin of Taleb's ire can be found in Silver's success since 2008--and his some-time failures.
We construct realistic equity option market simulators based on generative adversarial networks (GANs). We consider recurrent and temporal convolutional architectures, and assess the impact of state compression. Option market simulators are highly relevant because they allow us to extend the limited real-world data sets available for the training and evaluation of option trading strategies. We show that network-based generators outperform classical methods on a range of benchmark metrics, and adversarial training achieves the best performance. Our work demonstrates for the first time that GANs can be successfully applied to the task of generating multivariate financial time series.