arXiv subjects

Jan Pospíšil

Publications and source records attributed to Jan Pospíšil.

11 recordsLinked to original sources

arXiv2025

Challenges in automatic differentiation and numerical integration in physics-informed neural networks modelling

In this paper, we numerically examine the precision challenges that emerge in automatic differentiation and numerical integration in various tasks now tackled by physics-informed neural networks (PINNs). Specifically, we illustrate how ill-posed problems or inaccurately computed functions can cause serious precision issues in differentiation and integration. A major difficulty lies in detecting these problems. A simple large-scale view of the function or good-looking loss functions or convergence results may not reveal any potential errors, and the resulting outcomes are often mistakenly considered correct. To address this, it is critical to determine whether standard double-precision arithmetic suffices or if higher precision is necessary. Three problematic use-cases for solving differential equations using PINNs are analysed in detail. For the case requiring numerical integration, we also evaluate several numerical quadrature methods and suggest particular numerical analysis steps to choose the most suitable method.

math.NA↗

arXiv2025

Computation of Greeks under rough Volterra stochastic volatility models using the Malliavin calculus approach

Using Malliavin calculus techniques, we obtain formulas for computing Greeks under different rough Volterra stochastic volatility models. Due to the fact that underlying prices are not always square integrable, we extend the classical integration by parts formula to integrable but not necessarily square integrable functionals. First of all, we obtain formulas for general stochastic volatility (SV) models, concretely the Greeks Delta, Gamma, Rho, Vega and we introduce the Greek with respect to the roughness parameter. Then, the particular case of rough Volterra SV models is analyzed. Finally, three examples are treated in detail: the family of alpha-RFSV models, that includes rough versions of SABR and Bergomi models, a mixed alpha-RFSV model with two different Hurst parameters representing short (roughness) and long memory, and the rough Stein-Stein model. For different models and Greeks we show a numerical convergence of our formulas in Monte Carlo simulations and depict for example a dependence of the Greeks on the roughness parameter.

q-fin.MF↗

arXiv2023

Monotone iteration scheme for nonlinear PDEs in risk models

In this paper we study nonlinear partial differential equations (PDEs) that are used to model different value adjustments denoted generally as xVA. These adjustments are nowadays commonly added to the risk-free financial derivative values and the PDE approach allows their easy incorporation. The aim of this paper is to apply the method of monotone iterations with sub- and supersolutions to the nonlinear Black-Scholes-type equation that occurs especially in the counterparty risk models. We introduce a monotone iteration scheme with semi-explicit solution formulas for each iteration step. Moreover, we show that the problem greatly simplifies for contracts with non-negative payoffs. To show the viability of the approach we apply our method to the call option, the forward, and the gap option.

math.AP↗

arXiv2023

Robustness and sensitivity analyses for rough Volterra stochastic volatility models

In this paper, we analyze the robustness and sensitivity of various continuous-time rough Volterra stochastic volatility models in relation to the process of market calibration. Model robustness is examined from two perspectives: the sensitivity of option price estimates and the sensitivity of parameter estimates to changes in the option data structure. The following sensitivity analysis consists of statistical tests to determine whether a given studied model is sensitive to changes in the option data structure based on the distribution of parameter estimates. Empirical study is performed on a data set consisting of Apple Inc. equity options traded on four different days in April and May 2015. In particular, the results for RFSV, rBergomi and $α$RFSV models are provided and compared to the results for Heston, Bates, and AFSVJD models.

q-fin.PR↗

arXiv2022

On simulation of rough Volterra stochastic volatility models

Rough Volterra volatility models are a progressive and promising field of research in derivative pricing. Although rough fractional stochastic volatility models already proved to be superior in real market data fitting, techniques used in simulation of these models are still inefficient in terms of speed and accuracy. This paper aims to present accurate and efficient tools and techniques for Monte-Carlo simulations for a wide range of rough volatility models. In particular, we compare three commonly used simulation methods: the Cholesky method, the Hybrid scheme, and the rDonsker scheme. We also comment on the implementation of variance reduction techniques. In particular, we show the obstacles of the so-called turbocharging technique whose performance is sometimes counter-productive. To overcome these obstacles, we suggest several modifications.

q-fin.CP↗

arXiv2021

A note on a PDE approach to option pricing under xVA

In this paper we study partial differential equations (PDEs) that can be used to model value adjustments. Different value adjustments denoted generally as xVA are nowadays added to the risk-free financial derivative values and the PDE approach allows their easy incorporation. The aim of this paper is to show how to solve the PDE analytically in the Black-Scholes setting to get new semi-closed formulas that we compare to the widely used Monte-Carlo simulations and to the numerical solutions of the PDE. Particular example of collateral taken as the values from the past will be of interest.

q-fin.RM↗

arXiv2020

Solution of option pricing equations using orthogonal polynomial expansion

In this paper we study both analytic and numerical solutions of option pricing equations using systems of orthogonal polynomials. Using a Galerkin-based method, we solve the parabolic partial diferential equation for the Black-Scholes model using Hermite polynomials and for the Heston model using Hermite and Laguerre polynomials. We compare obtained solutions to existing semi-closed pricing formulas. Special attention is paid to the solution of Heston model at the boundary with vanishing volatility.

q-fin.PR↗

arXiv2019

Robustness and sensitivity analyses for stochastic volatility models under uncertain data structure

In this paper we perform robustness and sensitivity analysis of several continuous-time stochastic volatility (SV) models with respect to the process of market calibration. The analyses should validate the hypothesis on importance of the jump part in the underlying model dynamics. Also an impact of the long memory parameter is measured for the approximative fractional SV model. For the first time, the robustness of calibrated models is measured using bootstrapping methods on market data and Monte-Carlo filtering techniques. In contrast to several other sensitivity analysis approaches for SV models, the newly proposed methodology does not require independence of calibrated parameters - an assumption that is typically not satisfied in practice. Empirical study is performed on data sets of Apple Inc. equity options traded in April and May 2015.

q-fin.PR↗

arXiv2019

Isogeometric analysis in option pricing

Isogeometric analysis is a recently developed computational approach that integrates finite element analysis directly into design described by non-uniform rational B-splines (NURBS). In this paper we show that price surfaces that occur in option pricing can be easily described by NURBS surfaces. For a class of stochastic volatility models, we develop a methodology for solving corresponding pricing partial integro-differential equations numerically by isogeometric analysis tools and show that a very small number of space discretization steps can be used to obtain sufficiently accurate results. Presented solution by finite element method is especially useful for practitioners dealing with derivatives where closed-form solution is not available.

q-fin.CP↗

arXiv2019

Decomposition formula for rough Volterra stochastic volatility models

The research presented in this article provides an alternative option pricing approach for a class of rough fractional stochastic volatility models. These models are increasingly popular between academics and practitioners due to their surprising consistency with financial markets. However, they bring several challenges alongside. Most noticeably, even simple non-linear financial derivatives as vanilla European options are typically priced by means of Monte-Carlo (MC) simulations which are more computationally demanding than similar MC schemes for standard stochastic volatility models. In this paper, we provide a proof of the prediction law for general Gaussian Volterra processes. The prediction law is then utilized to obtain an adapted projection of the future squared volatility -- a cornerstone of the proposed pricing approximation. Firstly, a decomposition formula for European option prices under general Volterra volatility models is introduced. Then we focus on particular models with rough fractional volatility and we derive an explicit semi-closed approximation formula. Numerical properties of the approximation for a popular model -- the rBergomi model -- are studied and we propose a hybrid calibration scheme which combines the approximation formula alongside MC simulations. This scheme can significantly speed up the calibration to financial markets as illustrated on a set of AAPL options.

q-fin.PR↗

arXiv2019

Decomposition formula for jump diffusion models

In this paper we derive a generic decomposition of the option pricing formula for models with finite activity jumps in the underlying asset price process (SVJ models). This is an extension of the well-known result by Alos (2012) for Heston (1993) SV model. Moreover, explicit approximation formulas for option prices are introduced for a popular class of SVJ models - models utilizing a variance process postulated by Heston (1993). In particular, we inspect in detail the approximation formula for the Bates (1996) model with log-normal jump sizes and we provide a numerical comparison with the industry standard - Fourier transform pricing methodology. For this model, we also reformulate the approximation formula in terms of implied volatilities. The main advantages of the introduced pricing approximations are twofold. Firstly, we are able to significantly improve computation efficiency (while preserving reasonable approximation errors) and secondly, the formula can provide an intuition on the volatility smile behaviour under a specific SVJ model.

q-fin.PR↗