127 research outputs found
A Fourier-Based Valuation Method for Bermudan and Barrier Options under Heston’s Model
No full textWe develop an efficient Fourier-based numerical method for pricing Bermudan and discretely monitored barrier options under the Heston stochastic volatility model. The two-dimensional pricing problem is dealt with by a combination of a Fourier cosine series expansion, as in [F. Fang and C.W. Oosterlee, SIAM J. Sci. Comput., 31 (2008), pp. 826–848, F. Fang and C. W. Oosterlee, Numer. Math., 114 (2009), pp. 27–62], and high-order quadrature rules in the other dimension. Error analysis and experiments confirm a fast error convergence.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc
Two-Dimensional Fourier Cosine Series Expansion Method for Pricing Financial Options
No full textThe COS method for pricing European and Bermudan options with one underlying asset was developed in [F. Fang and C. W. Oosterlee, SIAM J. Sci. Comput., 31 (2008), pp. 826--848] and [F. Fang and C. W. Oosterlee, Numer. Math., 114 (2009), pp. 27--62]. In this paper, we extend the method to higher dimensions, with a multidimensional asset price process. The algorithm can be applied to, for example, pricing two-color rainbow options but also to pricing under the popular Heston stochastic volatility model. For smooth density functions, the resulting method converges exponentially in the number of terms in the Fourier cosine series summations; otherwise we achieve algebraic convergence. The use of an FFT algorithm, for asset prices modeled by Lévy processes, makes the algorithm highly efficient. We perform extensive numerical experiments.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc
Pricing Early-Exercise and Discrete Barrier Options by Fourier-Cosine Series Expansions
No full textWe present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Levy asset price models. The error convergence is exponential for processes characterized by very smooth transitional probability density functions. The computational complexity is with a (small) number of terms from the series expansion, and , the number of early-exercise/monitoring dates.
Pricing early-exercise and discrete barrier options by fourier-cosine series expansions
No full textWe present a pricing method based on Fourier-cosine expansions for early-exercise and discretely-monitored barrier options. The method works well for exponential Lévy asset price models. The error convergence is exponential for processes characterized by very smooth () transitional probability density functions. The computational complexity is O((M ? 1)N log N) with N a (small) number of terms from the series expansion, and M, the number of early-exercise/monitoring dates. This paper is the follow-up of (Fang and Oosterlee in SIAM J Sci Comput 31(2):826–848, 2008) in which we presented the impressive performance of the Fourier-cosine series method for European options.Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc
Accurate and Robust Numerical Methods for the Dynamic Portfolio Management Problem
No full textThis paper enhances a well-known dynamic portfolio management algorithm, the BGSS algorithm, proposed by Brandt et al. (Review of Financial Studies, 18(3):831–873, 2005). We equip this algorithm with the components from a recently developed method, the Stochastic Grid Bundling Method (SGBM), for calculating conditional expectations. When solving the first-order conditions for a portfolio optimum, we implement a Taylor series expansion based on a nonlinear decomposition to approximate the utility functions. In the numerical tests, we show that our algorithm is accurate and robust in approximating the optimal investment strategies, which are generated by a new benchmark approach based on the COS method (Fang and Oosterlee, in SIAM Journal of Scientific Computing, 31(2):826–848, 2008).Delft Institute of Applied MathematicsElectrical Engineering, Mathematics and Computer Scienc
Calculation of exposure profiles and sensitivities of options under the Heston and the Heston Hull-White models
No full textCredit Valuation Adjustment (CVA) has become an important field as its calculation is required in Basel III, issued in 2010, in the wake of the credit crisis. Exposure, which is defined as the potential future loss on a financial contract due to a default event, is one of the key elements for calculating CVA. This paper provides a backward dynamics framework for assessing exposure profiles of European, Bermudan and barrier options under the Heston and Heston Hull-White asset dynamics. We discuss the potential of the Stochastic Grid Bundling Method (SGBM), which is based on the techniques of simulation, regression and bundling (Jain and Oosterlee, Applied Mathematics and Computation, 269:412–431, 2015). By SGBM we can relatively easily compute the Potential Future Exposure (PFE) and sensitivities over the whole time horizon. Assuming independence between the default event and exposure profiles, we give here examples of calculating exposure, CVA and sensitivities for Bermudan and barrier options
Pricing Bermudan options under local Lévy models with default
No full textWe consider a defaultable asset whose risk-neutral pricing dynamics are described by an exponential Lévy-type martingale. This class of models allows for a local volatility, local default intensity and a locally dependent Lévy measure. We present a pricing method for Bermudan options based on an analytical approximation of the characteristic function combined with the COS method. Due to a special form of the obtained characteristic function the price can be computed using a fast Fourier transform-based algorithm resulting in a fast and accurate calculation. The Greeks can be computed at almost no additional computational cost. Error bounds for the approximation of the characteristic function as well as for the total option price are given
The evolution of electron overdensities in magnetic fields
No full textWhen a neutral gas impinges on a stationary magnetized plasma an enhancement in the ionization rate occurs when the neutrals exceed a threshold velocity. This is commonly known as the critical ionization velocity effect. This process has two distinct timescales: an ion–neutral collision time and electron acceleration time. We investigate the energization of an ensemble of electrons by their self-electric field in an applied magnetic field. The evolution of the electrons is simulated under different magnetic field and density conditions. It is found that electrons can be accelerated to speeds capable of electron impact ionization for certain conditions. In the magnetically dominated case the energy distribution of the excited electrons shows that typically 1% of the electron population can exceed the initial electrostatic potential associated with the unbalanced ensemble of electrons
On cross-currency models with stochastic volatility and correlated interest rates
No full textWe construct multi-currency models with stochastic volatility and correlated stochastic interest rates with a full matrix of correlations. We frst deal with a foreign exchange (FX) model of Heston-type, in which the domestic and foreign interest rates are generated by the short-rate process of Hull-White [HW96]. We then extend the framework by modeling the interest rate by a stochastic volatility displaced-diffusion Libor Market Model [AA02], which can model an interest rate smile. We provide semi-closed form approximations which lead to effcient calibration of the multi-currency models. Finally, we add a correlated stock to the framework and discuss the construction, model calibration and pricing of equity- FX-interest rate hybrid payoffs.Foreign-exchange (FX); stochastic volatility; Heston model; stochastic interest rates; interest rate smile; forward characteristic function; hybrids; affne diffusion; effcient calibration.
Efficient Multigrid Methods based on Improved Coarse Grid Correction Techniques
No full textMultigrid efficiency often suffers from inadequate coarse grid correction in different prototypic situations. We select a few problems, where coarse grid correction issues arise because of anisotropic coefficients, non-equidistant or non-uniform grid stretching, or inherent indefiniteness in the partial differential equation. Most of the work in this thesis can be classified as an attempt to increase multigrid efficiency by analysing and developing novel grid coarsening techniques that ensure sufficient coarse grid correction for the multigrid algorithm. Anisotropy in discrete systems can stem from various continuous and discrete features of the problem and has to have its negative effects countered before a successful multigrid solution can be brought about. We select multidimensional stationary diffusion equation as the first important problem to be treated in this context. The work for dimensions higher than three, is aimed at developing grid coarsening strategies for discretization on rectangular hyper-grids that differ greatly in their dimensions, and thus induce the so-called grid-aligned anisotropies in the system. Coarse grids formed through standard coarsening fail to provide sufficient coarse grid correction, and alternative block relaxation techniques are expensive in high dimensions. We also investigate and test coarsening strategies with the aim that their use would allow point based relaxation to stay effective in this non-equidistant multigrid scenario. Through local Fourier analysis we also analyze w-RB Jacobi, and implement a computer program through which we compute the optimal relaxation parameters. There are three important inferences in this regard. (1) Partial (and grid dependent) coarsening strategies allow the successful use of point relaxation methods for this problem. (2) Quadrupling along a few dimensions is a very attractive partial coarsening choice. (3) Optimal relaxation parameters have a significant enhancement effect on multigrid convergence in high dimensions. The efficient solution of time-dependent multidimensional equations (discretized with implicit time-integration schemes) is also a challenge. We first use the sparse grid technique to reduce the exponential complexity of the discrete problem, and then use the d-dimensional multigrid techniques to solve the sparse grid sub problems. In this situation, i.e., with a multitude of different non equidistant grids, evaluating and using optimal relaxation parameters on the fly is not an option any more. As a multigrid solver in high dimensions depends on optimal attributes to quite a large extent, we employ the method as a preconditioner, instead of a solver. This results in a very robust and efficient multigrid preconditioned Bi-CGSTAB solver. Another coarsening strategy that we develop in this thesis is aimed at two-dimensional grids that are non-uniformly stretched. We investigated different experimental coarsening strategies. A strategy based on improving individual mesh aspect ratios of grid cells proves successful both theoretically as well as experimentally. It is based on adaptive coarsening so that on each successive coarsening step the grid cells become more square. We can also successfully use point relaxation methods with the proposed technique and get nice multigrid convergence in this case. This bit of work also has consequence for locally refined grids. Efficient multigrid techniques for the indefinite Helmholtz equation form a separate research theme included in the thesis. We employ the complex shifted operator preconditioning technique for model problems that stem from quantum mechanics applications. These model problems have strongly varying wave-numbers which perturbs the solution. The mesh size requirement in the region of this perturbation is quite demanding. This requirement can be eased by saturating the grid in that area. We find that standard coarsening in this situation works well. This is in contrast to the existing strategies where coarsening is only done in the region of refinement, until the grid is regularized. We discretize the model problems both on equidistant and also on locally refined grids, and the efficiency of the multigrid preconditioner is tested in both these situations. Experiments point to the fact that existing techniques for the indefinite Helmholtz are still not satisfactory and must be enhanced. Some conclusions and outlook mark the end of the thesis.Applied mathematicsElectrical Engineering, Mathematics and Computer Scienc
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