985 research outputs found
Efficient Numerical Methods for Pricing American Options under Lévy Models
Two new numerical methods for the valuation of American and Bermudan options are proposed, which admit a large class of asset price models for the underlying. In particular, the methods can be applied with Lévy models that admit jumps in the asset price. These models provide a more realistic description of market prices and lead to better calibration results than the well-known Black-Scholes model. The proposed methods are not based on the indirect approach via partial differential equations, but directly compute option prices as risk-neutral expectation values. The expectation values are approximated by numerical quadrature methods. While this approach is initially limited to European options, the proposed combination with interpolation methods also allows for pricing of Bermudan and American options. Two different interpolation methods are used. These are cubic splines on the one hand and a mesh-free interpolation by radial basis functions on the other hand. The resulting valuation methods allow for an adaptive space discretization and error control. Their numerical properties are analyzed and, finally, the methods are validated and tested against various single-asset and multi-asset options under different market models
Efficient Multigrid Preconditioners for Atmospheric Flow Simulations at High Aspect Ratio
Many problems in fluid modelling require the efficient solution of highly
anisotropic elliptic partial differential equations (PDEs) in "flat" domains.
For example, in numerical weather- and climate-prediction an elliptic PDE for
the pressure correction has to be solved at every time step in a thin spherical
shell representing the global atmosphere. This elliptic solve can be one of the
computationally most demanding components in semi-implicit semi-Lagrangian time
stepping methods which are very popular as they allow for larger model time
steps and better overall performance. With increasing model resolution,
algorithmically efficient and scalable algorithms are essential to run the code
under tight operational time constraints. We discuss the theory and practical
application of bespoke geometric multigrid preconditioners for equations of
this type. The algorithms deal with the strong anisotropy in the vertical
direction by using the tensor-product approach originally analysed by B\"{o}rm
and Hiptmair [Numer. Algorithms, 26/3 (2001), pp. 219-234]. We extend the
analysis to three dimensions under slightly weakened assumptions, and
numerically demonstrate its efficiency for the solution of the elliptic PDE for
the global pressure correction in atmospheric forecast models. For this we
compare the performance of different multigrid preconditioners on a
tensor-product grid with a semi-structured and quasi-uniform horizontal mesh
and a one dimensional vertical grid. The code is implemented in the Distributed
and Unified Numerics Environment (DUNE), which provides an easy-to-use and
scalable environment for algorithms operating on tensor-product grids. Parallel
scalability of our solvers on up to 20,480 cores is demonstrated on the HECToR
supercomputer.Comment: 22 pages, 6 Figures, 2 Table
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The mathematical modelling and numerical solution of options pricing problems
Accurate and efficient numerical solutions have been described for a selection of financial options pricing problems. The methods are based on finite difference discretisation coupled with optimal solvers of the resulting discrete systems. Regular Cartesian meshes have been combined with orthogonal co-ordinate transformations chosen for numerical accuracy rather than reduction of the differential operator to constant coefficient form. They allow detailed resolution in the regions of interest where accuracy is most desired, and grid coarsening where there is least interest. These transformations are shown to be effective in producing accurate solutions on modest computational grids. The spatial discretisation strategy is chosen to meet accuracy requirements as sell as to produce coefficient matrices with favourable sparsity and stability properties.
In the case of single factor European options, a modified Crank-Nicolson, second order accurate finite difference scheme is presented, which uses adaptive upwind differences when the mesh Peclet conditions are violated. The resulting tridiagonal system of equations is solved using a direct solver. A careful study of grid refinement displays convergence towards the true solution and demonstrates a high level of accuracy can be obtained with this approach. Laplace inversion methods are also implemented as an alternative solution approach for the one-factor European option. Results are compared to those produced by the direct solver algorithm and are shown to be favourable.
It is shown how Semi-Lagrange time-integration can solve the path-dependent Asian pricing problem, by integrating out the average price term and simplifying the finite difference equations into a parameterised Black-Scholes form. The implicit equations that result are unconditionally stable, second order accurate and can be solved using standard tridiagonal solvers. The Semi-Lagrange method is shown to be easily used in conjunction with co-ordinate transformations applied in both spatial directions. A variable time-stepping scheme is implemented in the algorithm. Early exercise is also easily incorporated, the resulting linear complementarity problem can be solved using a projection or penalty method (the penalty method is shown to be slightly more efficient). Second order accuracy has been confirmed for Asian options that must be held to maturity. A comparison with published results for continuous-average-rate put and call options, with and without early exercise, shows that the method achieves basis point accuracy and that Richardson extrapolation can also be applied
VI Workshop on Computational Data Analysis and Numerical Methods: Book of Abstracts
The VI Workshop on Computational Data Analysis and Numerical Methods (WCDANM) is going to be held on June 27-29, 2019, in the Department of Mathematics of the University of Beira Interior (UBI), Covilhã, Portugal and it is a unique opportunity to disseminate scientific research related to the areas of Mathematics in general, with particular relevance to the areas of Computational Data Analysis and Numerical Methods in theoretical and/or practical field, using new techniques, giving especial emphasis to applications in Medicine, Biology, Biotechnology, Engineering, Industry, Environmental Sciences, Finance, Insurance, Management and Administration. The meeting will provide a forum for discussion and debate of ideas with interest to the scientific community in general. With this meeting new scientific collaborations among colleagues, namely new collaborations in Masters and PhD projects are expected. The event is open to the entire scientific community (with or without communication/poster)
Numerical Methods for Real Options in Telecommunications
This thesis applies modern financial option valuation methods to the problem of telecommunication network capacity investment decision timing. In particular, given a cluster of base stations (wireless network with a certain traffic capacity per base station), the objective of this thesis is to determine when it is optimal to increase capacity to each of the base stations of the cluster. Based on several time series taken from the wireless and bandwidth industry, it is argued that capacity usage is the major uncertain component in telecommunications. It is found that price has low volatility when compared to capacity usage. A real options approach is then applied to derive a two dimensional partial integro-differential equation (PIDE) to value investments in telecommunication infrastructure when capacity usage is uncertain and has temporary sudden large variations. This real options PIDE presents several numerical challenges. First, the integral term must be solved accurately and quickly enough such that the general PIDE solution is reasonably accurate. To deal with the integral term, an implicit method is suggested. Proofs of timestepping stability and convergence of a fixed point iteration scheme are presented. The correlation integral is computed using a fast Fourier transform (FFT) method. Techniques are developed to avoid wrap-around effects. This method is tested on option pricing problems where the underlying asset follows a jump diffusion process. Second, the absence of diffusion in one direction of the two dimensional PIDE creates numerical challenges regarding accuracy and timestep selection. A semi-Lagrangian method is presented to alleviate these issues. At each timestep, a set of one dimensional PIDEs is solved and the solution of each PIDE is updated using semi-Lagrangian timestepping. Crank-Nicolson and second order backward differencing timestepping schemes are studied. Monotonicity and stability results are derived. This method is tested on continuously observed Asian options. Finally, a five factor algorithm that captures many of the constraints of the wireless network capacity investment decision timing problem is developed. The upgrade decision for different upgrade decision intervals (e. g. monthly, quarterly, etc. ) is studied, and the effect of a safety level (i. e. the maximum allowed capacity used in practice on a daily basis—which differs from the theoretical maximum) is investigated
International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book
The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions.
This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more
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Stochastic models with random parameters for financial markets
This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University London.The aim of this thesis is a development of a new class of financial models with
random parameters, which are computationally efficient and have the same
level of performance as existing ones. In particular, this research is threefold.
I have studied the evolution of storable commodity and commodity futures
prices in time using a new random parameter model coupled with a Kalman
filter. Such a combination allows one to forecast arbitrage-free futures prices
and commodity spot prices one step ahead.
Another direction of my research is a new volatility model, where the
volatility is a random variable. The main advantage of this model is high
calibration speed compared to the existing stochastic volatility models such as
the Bates model or the Heston model. However, the performance of the new
model is comparable to the latter. Comprehensive numerical studies demonstrate
that the new model is a very competitive alternative to the Heston or
the Bates model in terms of accuracy of matching option prices or computing
hedging parameters.
Finally, a new futures pricing model for electricity futures prices was developed.
The new model has a random volatility parameter in its underlying
process. The new model has less parameters, as compared to two-factor models
for electricity commodity pricing with and without jumps. Numerical experiments
with real data illustrate that it is quite competitive with the existing
two-factor models in terms of pricing one step ahead futures prices, while being
far simpler to calibrate. Further, a new heuristic for calibrating two-factor
models was proposed. The new calibration procedure has two stages, offline
and online. The offline stage calibrates parameters under a physical measure,
while the online stage is used to calibrate the risk-neutrality parameters on
each iteration of the particle filter. A particle filter was used to estimate the
values of the underlying stochastic processes and to forecast futures prices one
step ahead.
The contributory material from two chapters of this thesis have been submitted
to peer reviewed journals in terms of two papers:
• Chapter 4: “A fast calibrating volatility model” has been submitted to
the European Journal of Operational Research.
• Chapter 5: “Electricity futures price models : calibration and forecasting”
has been submitted to the European Journal of Operational Research
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