Computational Finance
We are interested in the numerical pricing of financial derivatives. The numerical approximation techniques used are adaptive finite difference methods and radial basis functions. We consider pricing of both European and American options as well as some exotic options such as Turbo warrants, bubbles and the term structure equation. The whole range of the solution process is considered, from the numerical discretization to the solution of the resulting linear systems using iterative methods and preconditioning.
We have also in 2013 started research on model calibration and parameter estimation for financial models. This is needed for model consistent risk management of financial derivatives (hedging, risk assessment like "conditional value at risk" computations), something which is emphasized by new stricter regulations. Our idea is to combine advanced numerical methods with statistics in order to propose new statistical tools that are very much needed, especially in light of the rapid increase of the volume of the financial derivative markets, leading to huge volumes of trading data and large numbers of transactions in need of accurate statistical information.
Since 2014 we are leading the large international collaboration project BENCHOP. The purpose and aim of this project is to provide sets of benchmark problems that can be used for comparison and evaluation of methods and to serve as a take off for future development of methods in option pricing. The project has so far resulted in two publications
- BENCHOP–SLV: The BENCHmarking project in Option Pricing – Stochastic and local volatility problems. In International Journal of Computer Mathematics, volume 96, pp 1910-1923, 2019. (DOI, Fulltext).
- BENCHOP—The BENCHmarking project in Option Pricing. In International Journal of Computer Mathematics, volume 92, pp 2361-2379, 2015. (DOI, fulltext:postprint).
Activities organized by the group
- The second BENCHOP publication has appeared:
- BENCHOP–SLV: The BENCHmarking project in Option Pricing – Stochastic and local volatility problems. In International Journal of Computer Mathematics, volume 96, pp 1910-1923, 2019. (DOI, Fulltext).
- August 22-23 2016, BENCHOP - complex models and beyond - a continuation of the original BENCHOP had a kick-off in Uppsala with 14 participants.
- Lina von Sydow was editor for a special issue Topics in Computational and Algorithmic Finance.
- The workshop at Mittag-Leffler (see below) resulted in a large collaborative project BENCHOP - The BENCHmarking project in Option Pricing lead by the research group. The outcome of the project was a set of benchmarking problems an MATLAB-codes available here, and a joint publication
- BENCHOP—The BENCHmarking project in Option Pricing. In International Journal of Computer Mathematics, volume 92, pp 2361-2379, 2015. (DOI, fulltext:postprint).
- A workshop on Mathematical and Numerical Modeling in Finance was organized by Josef Höök, Elisabeth Larsson and Lina von Sydow June 9-11 2014 at Institut Mittag-Leffler in Djursholm. It was well attended by people from academia and financial institutions. There were research presentations with a mix of applied mathematics and numerical methods in the morning. New problems were discussed in the afternoon sessions, which resulted in at least one new major collaboration project.
- May 8-9 2014, Slobodan Milovanovic and Victor Shcherbakov accompanied by Elisabeth Larsson took part in the organization of and attended Postgraduate Workshop on Approximation Theory (with emphasis on Financial Applications) at Birkbeck, University of London. Doctoral students and researchers from Birkbeck, Leicester University, and Uppsala University had a chance to share their research results and exchange ideas amongst a larger audience. The broader aim of the workshop was to explore potential collaborative projects in this field for the future.
- On October 8 2013 the group organized a Workshop in Computational and Mathematical Finance. The workshop was very successful with many interesting presentations in the field that attracted 50 participants from business and academia. We plan to make this a recurrent event.
Refereed publications
- BENCHOP–SLV: The BENCHmarking project in Option Pricing – Stochastic and local volatility problems. In International Journal of Computer Mathematics, volume 96, pp 1910-1923, 2019. (DOI, Fulltext).
- The Kolmogorov forward fractional partial differential equation for the CGMY-process with applications in option pricing. In Computers and Mathematics with Applications, volume 76, pp 2330-2344, 2018. (DOI).
- Radial basis function generated finite differences for option pricing problems. In Computers and Mathematics with Applications, volume 75, pp 1462-1481, 2018. (DOI).
- Forward deterministic pricing of options using Gaussian radial basis functions. In Journal of Computational Science, volume 24, pp 209-217, 2018. (DOI, fulltext:postprint).
- A least squares radial basis function partition of unity method for solving PDEs. In SIAM Journal on Scientific Computing, volume 39, pp A2538-A2563, 2017. (DOI).
- Pricing of basket options using dimension reduction and adaptive finite differences in space, and discontinuous Galerkin in time. In Numerical Mathematics and Advanced Applications: ENUMATH 2015, volume 112 of Lecture Notes in Computational Science and Engineering, pp 607-615, Springer, 2016. (DOI).
- Radial basis function partition of unity operator splitting method for pricing multi-asset American options. In BIT Numerical Mathematics, volume 56, pp 1401-1423, 2016. (DOI, fulltext:postprint).
- Radial basis function partition of unity methods for pricing vanilla basket options. In Computers and Mathematics with Applications, volume 71, pp 185-200, 2016. (DOI, fulltext:postprint).
- Preconditioning for radial basis function partition of unity methods. In Journal of Scientific Computing, volume 67, pp 1089-1109, 2016. (DOI, fulltext:postprint).
- Adaptive finite differences and IMEX time-stepping to price options under Bates model. In International Journal of Computer Mathematics, volume 92, pp 2515-2529, 2015. (DOI, fulltext:postprint).
- BENCHOP—The BENCHmarking project in Option Pricing. In International Journal of Computer Mathematics, volume 92, pp 2361-2379, 2015. (DOI, fulltext:postprint).
- Numerical option pricing without oscillations using flux limiters. In Computers and Mathematics with Applications, volume 70, pp 1-10, 2015. (DOI, fulltext:postprint).
- A radial basis function partition of unity collocation method for convection–diffusion equations arising in financial applications. In Journal of Scientific Computing, volume 64, pp 341-367, 2015. (DOI, fulltext:postprint).
- An IMEX-scheme for pricing options under stochastic volatility models with jumps. In SIAM Journal on Scientific Computing, volume 36, pp B817-B834, 2014. (DOI).
- Composition schemes for the stochastic differential equation describing collisional pitch-angle diffusion. In Computer Physics Communications, volume 185, pp 590-594, 2014. (DOI).
- On discontinuous Galerkin for time integration in option pricing problems with adaptive finite differences in space. In Numerical Analysis and Applied Mathematics: ICNAAM 2013, volume 1558 of AIP Conference Proceedings, pp 2373-2376, American Institute of Physics (AIP), Melville, NY, 2013. (DOI).
- Radial basis function methods in computational finance. In Proc. 13th International Conference on Computational and Mathematical Methods in Science and Engineering: Volume III, pp 895-906, Universidad de Almería, Spain, 2013.
- Iterative methods for pricing American options under the Bates model. In Procedia Computer Science, volume 18, pp 1136-1144, 2013. (DOI).
- A multigrid preconditioner for an adaptive Black–Scholes solver. In BIT Numerical Mathematics, volume 51, pp 217-233, 2011. (DOI).
- Numerical option pricing in the presence of bubbles. In Quantitative finance (Print), volume 11, pp 1125-1128, 2011. (DOI).
- Pricing American options using a space-time adaptive finite difference method. In Mathematics and Computers in Simulation, volume 80, pp 1922-1935, 2010. (DOI).
- Boundary values and finite difference methods for the single factor term structure equation. In Applied Mathematical Finance, volume 16, pp 253-259, 2009. (DOI).
- A highly accurate adaptive finite difference solver for the Black–Scholes equation. In International Journal of Computer Mathematics, volume 86, pp 2104-2121, 2009. (DOI).
- Multi-dimensional option pricing using radial basis functions and the generalized Fourier transform. In Journal of Computational and Applied Mathematics, volume 222, pp 175-192, 2008. (DOI).
- Improved radial basis function methods for multi-dimensional option pricing. In Journal of Computational and Applied Mathematics, volume 222, pp 82-93, 2008. (DOI).
- Pricing European multi-asset options using a space-time adaptive FD-method. In Computing and Visualization in Science, volume 10, pp 173-183, 2007. (DOI).
- Space-time adaptive finite difference method for European multi-asset options. In Computers and Mathematics with Applications, volume 53, pp 1159-1180, 2007. (DOI).
- Option pricing using radial basis functions. In Proc. ECCOMAS Thematic Conference on Meshless Methods, pp C24.1-6, Departamento de Matemática, Instituto Superior Técnico, Lisboa, Portugal, 2005.
PhD Theses
- Radial Basis Function generated Finite Difference Methods for Pricing of Financial Derivatives. Ph.D. thesis, Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology nr 1702, Acta Universitatis Upsaliensis, Uppsala, 2018. (fulltext, preview image).
- Localised Radial Basis Function Methods for Partial Differential Equations. Ph.D. thesis, Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology nr 1600, Acta Universitatis Upsaliensis, Uppsala, 2018. (fulltext, preview image).
- Accurate Finite Difference Methods for Option Pricing. Ph.D. thesis, Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology nr 206, Acta Universitatis Upsaliensis, Uppsala, 2006. (fulltext).
Lic Thesis
- Radial basis function methods for pricing multi-asset options. Licentiate thesis, IT licentiate theses / Uppsala University, Department of Information Technology nr 2016-001, Uppsala University, 2016. (fulltext).
Reports (nonoverlapping with refereed publications)
- A high order method for pricing of financial derivatives using radial basis function generated finite differences. In Mathematics and Computers in Simulation, volume 174, pp 205-217, 2020. (DOI).
- Pricing financial derivatives using radial basis function generated finite differences with polyharmonic splines on smoothly varying node layouts. In Computing Research Repository, number 1808.02365, 2018. (External link). Publication status: Submitted
- Pricing derivatives under multiple stochastic factors by localized radial basis function methods. In Computing Research Repository, number 1711.09852, 2017. (External link). Publication status: Submitted
- A radial basis function partition of unity collocation method for convection-diffusion equations. Technical report / Department of Information Technology, Uppsala University nr 2013-023, 2013. (External link).
- Pricing turbo warrants. Technical report / Department of Information Technology, Uppsala University nr 2006-015, 2006. (External link).
Supervised Master thesis projects
- Stephane Dumanois, Least Squares Radial Basis Function generated Finite Differences for Option Pricing, U.U.D.M. project report, 2016:51, Department of Mathematics, Uppsala University, 2016. (Advisor S. Milovanovic, L. von Sydow)
- Gustav Ludvigsson, Numerical methods for option pricing under the CGMY process, UPTEC Report F 15 014, School of Engineering, Uppsala University, 2015.
- Chi Zhang, Fast Fourier Transforms in IMEX-schemes to price options under Bates model, U.U.D.M. project report, 2014:37, Department of Mathematics, Uppsala University, 2014. (Advisor L. von Sydow)
- Xi Wang, Radial basis function generated finite differences for option pricing, U.U.D.M. project report; 2014:36, Department of Mathematics, Uppsala University, 2014. (Advisor: L. von Sydow)
- Shiraz Farouq, Analysis of Empirical Investor Networks and Information Events in Stock Market, IT Report 13 088, Department of Information Technology, Uppsala University, 2013. (Advisor: J. Waldén)
- Alexander Sjöberg, Adaptive finite differences to price European options under the Bates model, IT Report 13 063, Department of Information Technology, Uppsala University, 2013. (Advisor: L. von Sydow)
- Paria Ghafari, Dimension Reduction and Adaptivity to Price Basket Options, U.U.D.M. project report; 2013:3, Department of Mathematics, Uppsala University, 2013. (Advisor: L. von Sydow).
- Cong Wang, Evaluation of a least-squares radial basis function approximation method for solving the Black-Scholes equation for option pricing, IT nr 12 051, Department of Information Technology, Uppsala University, 2012. (Advisor: E. Larsson).
- Wei Hu, Stable Treatment of Discontinuities in the Numerical Pricing of Options with Dividends, IT Report 11 071, Department of Information Technology, Uppsala University, 2011. (Advisor: P. Lötstedt).
- Andreas Hall, Pricing financial derivatives using radial basis functions and the generalized Fourier transform, UPTEC Report IT 05 036, School of Engineering, Uppsala University, 2005. (Advisors: E. Larsson and K. Åhlander)
- Gunilla Linde, High-order adaptive space-discretizations for the Black-Scholes equation, UPTEC Report F 05 019, School of Engineering, Uppsala University, 2005. (Advisors: L. von Sydow and J. Persson)
- Gunnar Marcusson, Option pricing using radial basis functions, UPTEC Report F 04 078, School of Engineering, Uppsala University, 2004. (Advisors: E. Larsson and L. von Sydow)
Supervised Bachelor thesis projects
- Teodor Fredriksson, Fokker Planck for the Cox-Ingersoll-Ross Model, U.U.D.M. project report, 2017:36, Department of Mathematics, Uppsala University, 2017. (Advisor: Elisabeth Larsson)
- Håkan Öhrn and Adam Lindell, Discretization of the Dirac delta function for application in option pricing, TVE Report 16 067, Department of Engineering Sciences, Uppsala University, 2016. (Advisors: L. von Sydow and E. Larsson)
- Robin Eriksson, Stencil Study for RBF-FD in Option Pricing, TVE Report 16 051, Department of Engineering Sciences, Uppsala University, 2016. (Advisors: S. Milovanovic and L. von Sydow)
- Andreas Abrahamsson and Rasmus Pettersson, Smoothing of initial conditions for high order approximations in option pricing, TVE Report 16 029, Department of Engineering Sciences, Uppsala University, 2016. (Advisors: L. von Sydow and S. Milovanovic)
- Denniz Falk Soylu, Dimension Reduction Methods for Predicting Financial Data, U.U.D.M. project report, 2015:17, Department of Mathematics, Uppsala University, 2015. (Advisor: J. Höök, T. Rydén, P. Lötstedt, L. von Sydow and E. Larsson)
- William Gustafsson, Evaluating the Longstaff-Schwartz method for pricing of American options, U.U.D.M. project report, 2015:13, Department of Mathematics, Uppsala University, 2015. (Advisor: J. Höök)
- Tomas Sundvall and David Trång, Examination of Impact from Different Boundary Conditions on the 2D Black-Scholes Model, TVE Report 14 045, Department of Engineering Sciences, Uppsala University, 2014. (Advisors: L. von Sydow and S. Milovanovic)
- Johan Ekegren Gunnarsson and Martin Pettersson, Discontinuous Galerkin and BDF-2 for time-integration and adaptive finite differences to price options, TVE Report 13 034, Department of Engineering Sciences, Uppsala University, 2013. (Advisor: L. von Sydow)
- Victor Nilsson, Discontinuous Galerkin methods for initial value problems and option pricing, TVE Report 12 036, Department of Engineering Sciences, Uppsala University, 2012. (Advisors: L. von Sydow and A. Målqvist)
Supervised projects in project course in Scientific Computing
- Carl Jonéus and Andreas Abrahamsson, Evaluating the efficiency of radial basis function methods for option pricing, Department of Information Technology, Uppsala University, 2017. (Advisors E. Larsson and V. Shcherbakov)
- Basel Aiesh, Ludvig Backlund, Perry Hansler, Forward and backward option pricing, Department of Information Technology, Uppsala University, 2016. (Advisors L. von Sydow, E. Larsson, J. Höök, S. Milovanovic, and V. Shcherbakov)
- Arvid Westlund, An IMEX-method for pricing options under Bates model using adaptive finite differences, Department of Information Technology, Uppsala University, 2014. (Advisor L. von Sydow)
- David Henriksson, Gustav Ludvigsson, and Andreas Yacob, Finite difference and Monte-Carlo methods for pricing of options in markets with jumps, Department of Information Technology, Uppsala University, 2014. (Advisors: L. von Sydow and J. Höök)
- Emil Larsson, Option pricing using the discontinuous Galerkin method for time integration, Department of Information Technology, Uppsala University, 2013. (Advisor: L. von Sydow)
- Robert Skogberg and Jakob Markhed, Pricing European Options using Stochastic Volatility and Integral Jump Term, Department of Information Technology, Uppsala University, 2009. (Advisors: L. von Sydow and M. Neytcheva)
- Erik Ekedahl, Erik Hansander and Erik Lehto, Dimension Reduction for the Black-Scholes Equation - Alleviating the Curse of Dimensionality, Department of Information Technology, Uppsala University, 2007. (Advisor: L. von Sydow)
- Per Landin, Snabb prissättning av optioner, Department of Information Technology, Uppsala University, 2006. (Advisor: L. von Sydow)
- Gunilla Linde and Martin Åberg, Pricing European Options Using a Pseudospectral Method, Department of Information Technology, 2004. (Advisors: J. Persson and C. Peterson)
- Per Lindberg, Gunnar Marcusson and Gustav Nordman, Numerical Analysis of the Optimal Exercise Boundary of an American Put Option, Department of Information Technology, Uppsala University, 2002. (Advisors: J. Tysk and J. Persson)
- Linnéa Klar and Jonas Jacobson, Pricing of European Call Options, Department of Information Technology, Uppsala University, 2001. (Advisors: J. Tysk, L. von Sydow and J. Persson)