Karel In't Hout professor

Research Karel In't Hout

Research team

Expertise

Analysis and development of numerical methods for time-dependent differential equations with applications to financial mathematics.

Numerical analysis and simulation of exotic energy derivatives. 01/01/2023 - 31/12/2026

Abstract

We consider numerical methods for the efficient and stable solution of advanced, multidimensional partial integro-differential equations (PIDEs) and partial integro-differential complementarity problems (PIDCPs) arising in financial energy option valuation. Here the underlying uncertain factors, e.g. the electricity price, are modelled by exponential Lévy processes to account for the jumps that are often observed in the markets. These jumps give rise to the integral term in the PIDEs and PIDCPs and is nonlocal. For the effective numerical solution, we investigate in this project operator splitting methods. This broad class of methods has already been successfully applied and analysed in the special case of partial differential equations (PDEs). Many questions about their adaptation to PIDEs and PIDCPs are, however, still largely open. In this project we consider two main research topics: (1) infinite activity jumps and (2) swing options. The first topic concerns exponential Lévy processes with an infinite number of jumps in every time interval. The second topic deals with a popular type of exotic energy option that has multiple exercise times. We develop novel, second-order operator splitting methods of the implicit-explicit (IMEX) and alternating direction implicit (ADI) kind. We analyse their fundamental properties of stability, consistency, monotonicity and convergence. The acquired theoretical results are validated by ample numerical experiments.

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  • Research Project

Modelling and simulation with applications in finance, insurance and economics 01/01/2021 - 31/12/2025

Abstract

This FWO scientific research network will focus on interdisciplinary research (mathematics – physics) in the area of stochastic modelling based on the interaction between theory, numerical computations and applications in financial markets. Hereto the network will make use of the complementary expertise present in the participating research groups.

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Support maintenance scientific equipment (Computational Mathematics). 01/01/2007 - 31/12/2024

Abstract

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  • Research Project

Convergence analysis and application of ADI schemes for partial differential equations from financial mathematics. 01/10/2016 - 31/07/2017

Abstract

In this research project our aim is to investigate the convergence of ADI schemes in the numerical solution of multi-dimensional time-dependent PDEs arising in financial mathematics. As mentioned above, these PDEs possess essentially different features from those in other application areas. In particular, mixed spatial derivative terms are pervasive in finance and ADI schemes were not originally developed for PDEs with such terms. Recently, however, various natural adaptations, of increasing sophistication, have been defined: the Douglas (Do) scheme, the Craig-Sneyd (CS) scheme, the Modified Craig-Sneyd (MCS) scheme and the Hundsdorfer-Verwer (HV) scheme, see [3,11,12,16]. ADI schemes now constitute a main class of numerical methods in academic and industrial finance, and their range of financial applications continues to extend.

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Developing and calibrating tractable cutting-edge multivariate financial models. 01/03/2016 - 31/12/2019

Abstract

The increased trading in multi-name financial products has required state-of-the-art multivariate models that are, at the same time, computationally tractable and flexible enough to explain the stylized facts of asset returns and of their dependence structure. The project is aimed at developing and calibrating multivariate models that can replicate financial market data whatever the level of investor's fear in the market. To this end, we will use advanced stochastic processes, such as Lévy processes, Sato processes and continuous time Markov chains. We will also develop fast and accurate calibration algorithms based on series expansions and on the matching of market implied moments and co-moments extracted from current market quotes. Particular attention will be given to the models' ability to explain the asset dependence structure, which plays a crucial role in the assessment of correlation risk. A correct management of this new kind of financial risk, which is inherent to any multi-name financial product, has indeed appeared to be vital during recent systemic crashes, such as the global financial crisis of 2007-2008.

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Convergence analysis and application of ADI schemes for partial differential equations from financial mathematics. 01/10/2014 - 30/09/2016

Abstract

In this research project our aim is to investigate the convergence of ADI schemes in the numerical solution of multi-dimensional time-dependent PDEs arising in financial mathematics. As mentioned above, these PDEs possess essentially different features from those in other application areas. In particular, mixed spatial derivative terms are pervasive in finance and ADI schemes were not originally developed for PDEs with such terms. Recently, however, various natural adaptations, of increasing sophistication, have been defined: the Douglas (Do) scheme, the Craig-Sneyd (CS) scheme, the Modified Craig-Sneyd (MCS) scheme and the Hundsdorfer-Verwer (HV) scheme, see [3,11,12,16]. ADI schemes now constitute a main class of numerical methods in academic and industrial finance, and their range of financial applications continues to extend.

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  • Research Project

Convergence analysis and application of ADI schemes for partial differential equations from financial mathematics. 01/10/2013 - 30/09/2014

Abstract

In this research project our aim is to investigate the convergence of ADI schemes in the numerical solution of multi-dimensional time-dependent PDEs arising in financial mathematics. As mentioned above, these PDEs possess essentially different features from those in other application areas. In particular, mixed spatial derivative terms are pervasive in finance and ADI schemes were not originally developed for PDEs with such terms. Recently, however, various natural adaptations, of increasing sophistication, have been defined: the Douglas (Do) scheme, the Craig-Sneyd (CS) scheme, the Modified Craig-Sneyd (MCS) scheme and the Hundsdorfer-Verwer (HV) scheme, see [3,11,12,16]. ADI schemes now constitute a main class of numerical methods in academic and industrial finance, and their range of financial applications continues to extend.

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  • Research Project

Novel methods in computational finance (STRIKE). 01/01/2013 - 31/12/2016

Abstract

In recent years the computational complexity of mathematical models employed in financial mathematics has witnessed a tremendous growth. Advanced numerical techniques are imperative for the most present-day applications in financial industry. The motivation for this training network is the need for a network of highly educated European scientists in the field of financial mathematics and computational science, so as to ex-change and discuss current insights and ideas, and to lay groundwork for future collaborations.

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Stability of finite difference methods on non-uniform grids for partial differential equations from financial mathematics. 01/10/2011 - 31/01/2013

Abstract

In this research project we derive theoretical stability results for finite difference methods on general non-uniform grids. We consider several applications from financial mathematics. First of all we start with the well-known 1-dimensional Black-Scholes equation. After that we move on to higher dimensional partial differential equations, for example the Heston-Hull-White model. Our theoretical results are supported by numerical experiments.

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Stochastic modeling with applications in financial markets. 01/01/2011 - 31/12/2020

Abstract

This FWO scientific research network will focus on interdisciplinary research (mathematics – physics) in the area of stochastic modelling based on the interaction between theory, numerical computations and applications in financial markets. Hereto the network will make use of the complementary expertise present in the participating research groups.

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  • Research Project

Stability of finite difference methods on non-uniform grids for partial differential equations from financial mathematics. 01/10/2009 - 30/09/2011

Abstract

In this research project we derive theoretical stability results for finite difference methods on general non-uniform grids. We consider several applications from financial mathematics. First of all we start with the well-known 1-dimensional Black-Scholes equation. After that we move on to higher dimensional partial differential equations, for example the Heston-Hull-White model. Our theoretical results are supported by numerical experiments.

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  • Research Project

Design of new models and techniques for high performance financial applications. 01/01/2008 - 31/12/2011

Abstract

In the past decennia the international financial markets are witnessing a huge increase in the trading of more and more complex products, such as exotic options and interest products, and this growth is only amplifying. For the exchanges and banks it is of crucial importance to be able to price these products accurately, and as fast as possible. The simulation of the current, sophisticated pricing models is, however, very time consuming with classical techniques such as Monte Carlo methods or binomial trees, and practical pricing formulas are often not at hand. This project is concerned with new models and techniques for robustly and efficiently pricing modern financial products. We investigate two complementary approaches: the first is based on partial differential equations and the second on quantum mechanical path integrals. In the first approach, we will consider operator splitting methods and meshfree methods for the effective numerical solution of these, often multi-dimensional, equations. In the second approach, path integral formulas for financial products will be studied by using the present theory concerning physical multi-particle systems and the comonotonicity coefficient. The obtained models and computational techniques will continually be mutually validated.

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Stability analysis of numerical processes for time-dependent partial differential equations. 01/07/2007 - 31/12/2011

Abstract

The goal of this project is to analyze the stability of numerical processes for time-dependent partial differential equations. We investigate important open stability questions concerning both space- and time-discretization methods. In answering these questions, we employ among others the recent numerical stability theory based on resolvent conditions.

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Numerical solution of multi-dimensional convection-diffusion-reaction equations. 01/07/2007 - 30/06/2010

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This research project concerns the design and analysis of numerical methods for multi-dimensional, time-dependent convection-diffusion-reaction equations with applications to financial mathematics. We study operator splitting methods, in particular ADI schemes, which are highly promising for the numerical solution of these, large-scale problems. Our analysis deals with fundamental properties such as stability and convergence.

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