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We derive high-order compact finite difference schemes for option pricing in stochastic volatility models on non-uniform grids. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical study we obtain high-order numerical convergence also for non-zero correlation and non-smooth payoffs which are typical in option pricing. In all numerical experiments a comparative standard second-order discretisation is significantly outperformed. We conduct a numerical stability study which indicates unconditional stability of the scheme.
We present high-order compact schemes for a linear second-order parabolic partial differential equation (PDE) with mixed second-order derivative terms in two spatial dimensions. The schemes are applied to option pricing PDE for a family of stochastic volatility models. We use a non-uniform grid with more grid-points around the strike price. The schemes are fourth-order accurate in space and second-order accurate in time for vanishing correlation. In our numerical convergence study we achieve fourth-order accuracy also for non-zero correlation. A combination of Crank-Nicolson and BDF-4 discretisation is applied in time. Numerical examples confirm that a standard, second-order finite difference scheme is significantly outperformed.
This book discusses the state-of-the-art and open problems in computational finance. It presents a collection of research outcomes and reviews of the work from the STRIKE project, an FP7 Marie Curie Initial Training Network (ITN) project in which academic partners trained early-stage researchers in close cooperation with a broader range of associated partners, including from the private sector. The aim of the project was to arrive at a deeper understanding of complex (mostly nonlinear) financial models and to develop effective and robust numerical schemes for solving linear and nonlinear problems arising from the mathematical theory of pricing financial derivatives and related financial products. This was accomplished by means of financial modelling, mathematical analysis and numerical simulations, optimal control techniques and validation of models. In recent years the computational complexity of mathematical models employed in financial mathematics has witnessed tremendous growth. Advanced numerical techniques are now essential to the majority of present-day applications in the financial industry. Special attention is devoted to a uniform methodology for both testing the latest achievements and simultaneously educating young PhD students. Most of the mathematical codes are linked into a novel computational finance toolbox, which is provided in MATLAB and PYTHON with an open access license. The book offers a valuable guide for researchers in computational finance and related areas, e.g. energy markets, with an interest in industrial mathematics.
We extend the scheme developed in B. Düring, A. Pitkin, ”High-order compact finite difference scheme for option pricing in stochastic volatility jump models”, 2017, to the so-called stochastic volatility with contemporaneous jumps (SVCJ) model, derived by Duffie, Pan and Singleton. The performance of the scheme is assessed through a number of numerical experiments, using comparisons against a standard second-order central difference scheme. We observe that the new high-order compact scheme achieves third order convergence alongside improvements in efficiency and computation time.
We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility jump models, e.g. in Bates model. In such models the option price is determined as the solution of a partial integro-differential equation. The scheme is fourth order accurate in space and second order accurate in time. Numerical experiments for the European option pricing problem are presented. We validate the stability of the scheme numerically and compare its efficiency and hedging performance to standard finite difference methods. The new scheme outperforms a standard discretisation based on a second-order central finite difference approximation in all our experiments. At the same time, it is very efficient, requiring only one initial LU-factorisation of a sparse matrix to perform the option price valuation. It can also be useful to upgrade existing implementations based on standard finite differences in a straightforward manner to obtain a highly efficient option pricing code.
This book presents a collection of papers emphasizing applications of mathematical models and methods to real-world problems of relevance for industry, life science, environment, finance and so on. The biannual Conference of ECMI (the European Consortium of Mathematics in Industry) held in 2014 focused on various aspects of industrial and applied mathematics. The five main topics addressed at the conference were mathematical models in life science, material science and semiconductors, mathematical methods in the environment, design automation and industrial applications, and computational finance. Several other topics have been treated, such as, among others, optimization and inverse problems, education, numerical methods for stiff pdes, model reduction, imaging processing, multi physics simulation, mathematical models in textile industry. The conference, which brought together applied mathematicians and experts from industry, provided a unique opportunity to exchange ideas, problems and methodologies, bridging the gap between mathematics and industry and contributing to the advancement of science and technology. The conference has included a presentation of EU-Maths-In (European Network of Mathematics for Industry and Innovation), a recent joint initiative of ECMI and EMS. The proceedings from this conference represent a snapshot of the current activity in industrial mathematics in Europe, and are highly relevant to anybody interested in the latest applications of mathematics to industrial problems.
The Heston stochastic volatility model is one extension of the Black-Scholes model which describes the money markets more accurately so that more realistic prices for derivative products are obtained. From the stochastic differential equation of the underlying financial product a partial differential equation (p.d.e.) for the value function of an option can be derived. This p.d.e. can be solved with the finite difference method (f.d.m.). The stability and consistency of the method is examined. Furthermore a boundary condition is proposed to reduce the numerical error. Finally a non uniform structured grid is derived which is fairly optimal for the numerical result in the most interesting point.
The book contains a selection of high quality papers, chosen among the best presentations during the International Conference on Spectral and High-Order Methods (2012), and provides an overview of the depth and breath of the activities within this important research area. The carefully reviewed selection of the papers will provide the reader with a snapshot of state-of-the-art and help initiate new research directions through the extensive bibliography. ​
ECMI, the European Consortium for Mathematics in Industry, is the European brand associated with applied mathematics for industry and organizes highly successful biannual conferences. In this series, the ECMI 2010, the 16th European Conference on Mathematics for Industry, was held in the historic city hall of Wuppertal in Germany. It covered the mathematics of a wide range of applications and methods, from circuit and electromagnetic device simulation to model order reduction for chip design, uncertainties and stochastics, production, fluids, life and environmental sciences, and dedicated and versatile methods. These proceedings of ECMI 2010 emphasize mathematics as an innovation enabler for industry and business, and as an absolutely essential pre-requiste for Europe on its way to becoming the leading knowledge-based economy in the world.
We propose a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise from stochastic volatility models in option pricing. Our approach combines different high-order spatial discretisations with Hundsdorfer and Verwer's ADI time-stepping method, to obtain an efficient method which is fourth-order accurate in space and second-order accurate in time. Numerical experiments for the European put option pricing problem using Heston's stochastic volatility model confirm the high-order convergence.