Efficient tree methods for option pricing

The aim of this dissertation is the study of efficient algorithms based on lattice procedures for dealing with two relevant issues arising in the recent literature on option pricing: the pricing of complex barrier-type options and the pricing of options when the equity model takes into account a stochastic interest rate. This research is developed with a twofold perspective: first, we propose a good solution from a numerical point of view through the introduction of efficient lattice procedures and secondly, we study the theoretical aspects related to the tackled problems (such as the convergence and the rate of convergence of the scheme proposed)

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

Meshless Methods for Option Pricing and Risks Computation

In this thesis we price several financial derivatives by means of radial basis functions. Our main contribution consists in extending the usage of said numerical methods to the pricing of more complex derivatives - such as American and basket options with barriers - and in computing the associated risks. First, we derive the mathematical expressions for the prices and the Greeks of given options; next, we implement the corresponding numerical algorithm in MATLAB and calculate the results. We compare our results to the most common techniques applied in practice such as Finite Differences and Monte Carlo methods. We mostly use real data as input for our examples. We conclude radial basis functions offer a valid alternative to current pricing methods, especially because of the efficiency deriving from the free, direct calculation of risks during the pricing process. Eventually, we provide suggestions for future research by applying radial basis function for an implied volatility surface reconstruction

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

Geobase Information System Impacts on Space Image Formats

As Geobase Information Systems increase in number, size and complexity, the format compatability of satellite remote sensing data becomes increasingly more important. Because of the vast and continually increasing quantity of data available from remote sensing systems the utility of these data is increasingly dependent on the degree to which their formats facilitate, or hinder, their incorporation into Geobase Information Systems. To merge satellite data into a geobase system requires that they both have a compatible geographic referencing system. Greater acceptance of satellite data by the user community will be facilitated if the data are in a form which most readily corresponds to existing geobase data structures. The conference addressed a number of specific topics and made recommendations

Essays on Numerical Evaluation of Derivatives

In general, this thesis contemplates the potential of series expansion methods in evaluating financial derivatives. Recently, instead of relying on closed-form solutions, the usage of numerical methods became increasingly popular. The contribution of the thesis at hand is three-folded: First, we present and analyze a new method to price European options that are written on a single underlying asset by introducing Gabor series methods into option pricing. The resulting procedure shows to be a very robust pricing tool with a special strength in calculating short-term contracts. Second, we dedicate our attention to multi-asset derivatives. Multi-asset contracts are notoriously hard to deal with in case the user demands both, advanced stochastic processes and fast evaluation. Compared to single underlying derivatives, these multi-asset exotic options are studied to a much lower degree. We focus on European multi-asset options as well as on discrete barrier multi-asset options. To the best of our knowledge, the valuation of multi-asset barrier options in terms of multi-dimensional Fourier series methods has not been addressed in literature before. Third, as recent events in the market for credit risk have shown, there is a need for further research in methods to evaluate credit derivatives. Therefore, we focused on extending the standard Gaussian factor model to price synthetic collateralized debt obligations. The new models are able to cope with a wide range of market conditions. However, given a crisis as severe as the financial crisis of 2008, questions, such as market liquidity, that are outside the scope of pure modeling overlay the approximation quality. Nevertheless, the models presented are flexible instruments to price synthetic collateralized debt obligations while still staying in the intuitive framework of a factor model

Multilevel Monte Carlo for jump processes

This thesis consists of two parts. The first part (Chapters 2-4) considers multilevel Monte Carlo for option pricing in finite activity jump-diffusion models. We use a jump-adapted Milstein discretisation for constant rate cases and with the thinning method for bounded state-dependent rate cases. Multilevel Monte Carlo estimators are constructed for Asian, look-back, barrier and digital options. The computational efficiency is numerically demonstrated and analytically justified.\ud \ud The second part (Chapter 5) deals with option pricing problems in exponential Levy models where the increments of the underlying process can be directly simulated. We discuss several examples: Variance Gamma, Normal Inverse Gaussian and α-stable processes and present numerical experiments of multilevel Monte Carlo for Asian, lookback, barrier options, where the running maximum of the Levy process involved in lookback and barrier payoffs is approximated using discretely monitored maximum. To analytically verify the computational complexity of multilevel method, we also prove some upper bounds on $L^p$ convergence rate of discretely monitored error for a broad class of Levy processes