arXiv subjects

Blanka Horvath

Publications and source records attributed to Blanka Horvath.

31 records · Page 2Linked to original sources

arXiv2023

Optimal Stopping via Distribution Regression: a Higher Rank Signature Approach

Distribution Regression on path-space refers to the task of learning functions mapping the law of a stochastic process to a scalar target. The learning procedure based on the notion of path-signature, i.e. a classical transform from rough path theory, was widely used to approximate weakly continuous functionals, such as the pricing functionals of path--dependent options' payoffs. However, this approach fails for Optimal Stopping Problems arising from mathematical finance, such as the pricing of American options, because the corresponding value functions are in general discontinuous with respect to the weak topology. In this paper we develop a rigorous mathematical framework to resolve this issue by recasting an Optimal Stopping Problem as a higher order kernel mean embedding regression based on the notions of higher rank signatures of measure--valued paths and adapted topologies. The core computational component of our algorithm consists in solving a family of two--dimensional hyperbolic PDEs.

math.PR↗

arXiv2021

Clustering Market Regimes using the Wasserstein Distance

The problem of rapid and automated detection of distinct market regimes is a topic of great interest to financial mathematicians and practitioners alike. In this paper, we outline an unsupervised learning algorithm for clustering financial time-series into a suitable number of temporal segments (market regimes). As a special case of the above, we develop a robust algorithm that automates the process of classifying market regimes. The method is robust in the sense that it does not depend on modelling assumptions of the underlying time series as our experiments with real datasets show. This method -- dubbed the Wasserstein $k$-means algorithm -- frames such a problem as one on the space of probability measures with finite $p^\text{th}$ moment, in terms of the $p$-Wasserstein distance between (empirical) distributions. We compare our WK-means approach with a more traditional clustering algorithms by studying the so-called maximum mean discrepancy scores between, and within clusters. In both cases it is shown that the WK-means algorithm vastly outperforms all considered competitor approaches. We demonstrate the performance of all approaches both in a controlled environment on synthetic data, and on real data.

q-fin.CP↗

arXiv2021

Hedging under rough volatility

In this chapter we first briefly review the existing approaches to hedging in rough volatility models. Next, we present a simple but general result which shows that in a one-factor rough stochastic volatility model, any option may be perfectly hedged with a dynamic portfolio containing the underlying and one other asset such as a variance swap. In the final section we report the results of a back-test experiment using real data, where VIX options are hedged with a forward variance swap. In this experiment, using a rough volatility model allows to almost completely remove the bias and reduce the overall hedging error by a factor of 27% compared to traditional diffusion-based models.

q-fin.MF↗

arXiv2021

Deep Hedging under Rough Volatility

We investigate the performance of the Deep Hedging framework under training paths beyond the (finite dimensional) Markovian setup. In particular we analyse the hedging performance of the original architecture under rough volatility models with view to existing theoretical results for those. Furthermore, we suggest parsimonious but suitable network architectures capable of capturing the non-Markoviantity of time-series. Secondly, we analyse the hedging behaviour in these models in terms of P\&L distributions and draw comparisons to jump diffusion models if the the rebalancing frequency is realistically small.

q-fin.CP↗

arXiv2020

A Data-driven Market Simulator for Small Data Environments

Neural network based data-driven market simulation unveils a new and flexible way of modelling financial time series without imposing assumptions on the underlying stochastic dynamics. Though in this sense generative market simulation is model-free, the concrete modelling choices are nevertheless decisive for the features of the simulated paths. We give a brief overview of currently used generative modelling approaches and performance evaluation metrics for financial time series, and address some of the challenges to achieve good results in the latter. We also contrast some classical approaches of market simulation with simulation based on generative modelling and highlight some advantages and pitfalls of the new approach. While most generative models tend to rely on large amounts of training data, we present here a generative model that works reliably in environments where the amount of available training data is notoriously small. Furthermore, we show how a rough paths perspective combined with a parsimonious Variational Autoencoder framework provides a powerful way for encoding and evaluating financial time series in such environments where available training data is scarce. Finally, we also propose a suitable performance evaluation metric for financial time series and discuss some connections of our Market Generator to deep hedging.

q-fin.ST↗

arXiv2019

Deep Learning Volatility

We present a neural network based calibration method that performs the calibration task within a few milliseconds for the full implied volatility surface. The framework is consistently applicable throughout a range of volatility models -including the rough volatility family- and a range of derivative contracts. The aim of neural networks in this work is an off-line approximation of complex pricing functions, which are difficult to represent or time-consuming to evaluate by other means. We highlight how this perspective opens new horizons for quantitative modelling: The calibration bottleneck posed by a slow pricing of derivative contracts is lifted. This brings several numerical pricers and model families (such as rough volatility models) within the scope of applicability in industry practice. The form in which information from available data is extracted and stored influences network performance: This approach is inspired by representing the implied volatility and option prices as a collection of pixels. In a number of applications we demonstrate the prowess of this modelling approach regarding accuracy, speed, robustness and generality and also its potentials towards model recognition.

q-fin.MF↗

arXiv2019

On deep calibration of (rough) stochastic volatility models

Techniques from deep learning play a more and more important role for the important task of calibration of financial models. The pioneering paper by Hernandez [Risk, 2017] was a catalyst for resurfacing interest in research in this area. In this paper we advocate an alternative (two-step) approach using deep learning techniques solely to learn the pricing map -- from model parameters to prices or implied volatilities -- rather than directly the calibrated model parameters as a function of observed market data. Having a fast and accurate neural-network-based approximating pricing map (first step), we can then (second step) use traditional model calibration algorithms. In this work we showcase a direct comparison of different potential approaches to the learning stage and present algorithms that provide a suffcient accuracy for practical use. We provide a first neural network-based calibration method for rough volatility models for which calibration can be done on the y. We demonstrate the method via a hands-on calibration engine on the rough Bergomi model, for which classical calibration techniques are diffcult to apply due to the high cost of all known numerical pricing methods. Furthermore, we display and compare different types of sampling and training methods and elaborate on their advantages under different objectives. As a further application we use the fast pricing method for a Bayesian analysis of the calibrated model.

q-fin.MF↗

arXiv2019

Volatility options in rough volatility models

We discuss the pricing and hedging of volatility options in some rough volatility models. First, we develop efficient Monte Carlo methods and asymptotic approximations for computing option prices and hedge ratios in models where log-volatility follows a Gaussian Volterra process. While providing a good fit for European options, these models are unable to reproduce the VIX option smile observed in the market, and are thus not suitable for VIX products. To accommodate these, we introduce the class of modulated Volterra processes, and show that they successfully capture the VIX smile.

q-fin.PR↗

arXiv2018

Short-time near-the-money skew in rough fractional volatility models

We consider rough stochastic volatility models where the driving noise of volatility has fractional scaling, in the "rough" regime of Hurst parameter $H < 1/2$. This regime recently attracted a lot of attention both from the statistical and option pricing point of view. With focus on the latter, we sharpen the large deviation results of Forde-Zhang (2017) in a way that allows us to zoom-in around the money while maintaining full analytical tractability. More precisely, this amounts to proving higher order moderate deviation estimates, only recently introduced in the option pricing context. This in turn allows us to push the applicability range of known at-the-money skew approximation formulae from CLT type log-moneyness deviations of order $t^{1/2}$ (recent works of Alòs, León & Vives and Fukasawa) to the wider moderate deviations regime.

q-fin.PR↗

arXiv2018

Dirichlet Forms and Finite Element Methods for the SABR Model

We propose a deterministic numerical method for pricing vanilla options under the SABR stochastic volatility model, based on a finite element discretization of the Kolmogorov pricing equations via non-symmetric Dirichlet forms. Our pricing method is valid under mild assumptions on parameter configurations of the process both in moderate interest rate environments and in near-zero interest rate regimes such as the currently prevalent ones. The parabolic Kolmogorov pricing equations for the SABR model are degenerate at the origin, yielding non-standard partial differential equations, for which conventional pricing methods ---designed for non-degenerate parabolic equations--- potentially break down. We derive here the appropriate analytic setup to handle the degeneracy of the model at the origin. That is, we construct an evolution triple of suitably chosen Sobolev spaces with singular weights, consisting of the domain of the SABR-Dirichlet form, its dual space, and the pivotal Hilbert space. In particular, we show well-posedness of the variational formulation of the SABR-pricing equations for vanilla and barrier options on this triple. Furthermore, we present a finite element discretization scheme based on a (weighted) multiresolution wavelet approximation in space and a $θ$-scheme in time and provide an error analysis for this discretization.

q-fin.MF↗

arXiv2017

Functional Analytic (Ir-)Regularity Properties of SABR-type Processes

The SABR model is a benchmark stochastic volatility model in interest rate markets, which has received much attention in the past decade. Its popularity arose from a tractable asymptotic expansion for implied volatility, derived by heat kernel methods. As markets moved to historically low rates, this expansion appeared to yield inconsistent prices. Since the model is deeply embedded in market practice, alternative pricing methods for SABR have been addressed in numerous approaches in recent years. All standard option pricing methods make certain regularity assumptions on the underlying model, but for SABR these are rarely satisfied. We examine here regularity properties of the model from this perspective with view to a number of (asymptotic and numerical) option pricing methods. In particular, we highlight delicate degeneracies of the SABR model (and related processes) at the origin, which deem the currently used popular heat kernel methods and all related methods from (sub-) Riemannian geometry ill-suited for SABR-type processes, when interest rates are near zero. We describe a more general semigroup framework, which permits to derive a suitable geometry for SABR-type processes (in certain parameter regimes) via symmetric Dirichlet forms. Furthermore, we derive regularity properties (Feller- properties and strong continuity properties) necessary for the applicability of popular numerical schemes to SABR-semigroups, and identify suitable Banach- and Hilbert spaces for these. Finally, we comment on the short time and large time asymptotic behaviour of SABR-type processes beyond the heat-kernel framework.

q-fin.MF↗

arXiv2016

Mass at zero in the uncorrelated SABR model and implied volatility asymptotics

We study the mass at the origin in the uncorrelated SABR stochastic volatility model, and derive several tractable expressions, in particular when time becomes small or large. As an application--in fact the original motivation for this paper--we derive small-strike expansions for the implied volatility when the maturity becomes short or large. These formulae, by definition arbitrage free, allow us to quantify the impact of the mass at zero on existing implied volatility approximations, and in particular how correct/erroneous these approximations become.

q-fin.PR↗

arXiv2016

On the probability of hitting the boundary for Brownian motions on the SABR plane

Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models--related to the SABR model in mathematical finance--which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change methods.

math.PR↗