An important issue in finance is model calibration. The calibration problem is the inverse of the option pricing problem. Calibration is performed on a set of option prices generated from a given exponential L´evy model. By numerical examples, it is shown that the usual formulation of the inverse problem via Non-linear Least Squares is an ill-posed problem. To achieve well-posedness of the problem, some regularization is needed. Therefore a regularization method based on relative entropy is applied.

The purpose of this paper is threefold. First, the paper builds on the work done previously done in the area of option implied probability distribution functions (PDFs) by extending the methods described by Breeden and Litzenberger (1978) to individual equity options. Second, it describes a closed-form, onto mapping from a two-dimensional volatility surface to the risk-neutral PDF. Lastly the paper performs an event study on the implied risk-neutral PDFs of companies which are the target of corporate takeover. While there was not sufficient data to determine any statistical relationship, there is observational evidence that option market implied PDFs may be predictive of future takeovers.

The Fast Fourier Transform (FFT) has been used in many scientific and engineering applications. The use of FFT for financial derivatives has been gaining momentum in the recent past. In this thesis, i) we have improved a recently proposed model of FFT for pricing financial derivatives to help design an efficient parallel algorithm. The improved mathematical model put forth in our research bridges a gap between quantitative approaches for the option pricing problem and practical implementation of such approaches on modern computer architectures. The thesis goes further by proving that the improved model of fast Fourier transform for option pricing produces accurate option values. ii) We have developed a parallel algorithm for the FFT using the classical Cooley-Tukey algorithm and improved this algorithm by introducing a data swapping technique that brings data closer to the respective processors and hence reduces the communication overhead to a large extent leading to better performance of the parallel algorithm. We have tested the new algorithm on a 20 node SunFire 6800 high performance computing system and compared the new algorithm with the traditional Cooley-Tukey algorithm. Option values are calculated for various strike prices with a proper selection of strike-price spacing to ensure fine-grid integration for FFT computation as well as to maximize the number of strikes lying in the desired region of the stock price. Compared to the traditional Cooley-Tukey algorithm, the current algorithm with data swapping performs better by more than 15% for large data sizes. In the rapidly changing market place, these improvements could mean a lot for an investor or financial institution because obtaining faster results offers a competitive advantages. / October 2004

The evolution of the price of two financial assets may be modeled by correlated geometric Brownian motion with additional, independent, finite activity jumps. Similarly, the evolution of the price of one financial asset may be modeled by a stochastic volatility process and finite activity jumps. The value of a contingent claim, written on assets where the underlying evolves by either of these two-factor processes, is given by the solution of a linear, two-dimensional, parabolic, partial integro-differential equation (PIDE). The focus of this thesis is the development of new, efficient numerical solution approaches for these PIDE's for both linear and non-linear cases. A localization scheme approximates the initial-value problem on an infinite spatial domain by an initial-boundary value problem on a finite spatial domain. Convergence of the localization method is proved using a Green's function approach. An implicit, finite difference method discretizes the PIDE. The theoretical conditions for the stability of the discrete approximation are examined under both maximum and von Neumann analysis. Three linearly convergent, monotone variants of the approach are reviewed for the constant coefficient, two-asset case and reformulated for the non-constant coefficient, stochastic volatility case. Each monotone scheme satisfies the conditions which imply convergence to the viscosity solution of the localized PIDE. A fixed point iteration solves the discrete, algebraic equations at each time step. This iteration avoids solving a dense linear system through the use of a lagged integral evaluation. Dense matrix-vector multiplication is avoided by using an FFT method. By using Green's function analysis, von Neumann analysis and maximum analysis, the fixed point iteration is shown to be rapidly convergent under typical market parameters. Combined with a penalty iteration, the value of options with an American early exercise feature may be computed. The rapid convergence of the iteration is verified in numerical tests using European and American options with vanilla payoffs, and digital, one-touch option payoffs. These tests indicate that the localization method for the PIDE's is effective. Adaptations are developed for degenerate or extreme parameter sets. The three monotone approaches are compared by computational cost and resulting error. For the stochastic volatility case, grid rotation is found to be the preferred approach. Finally, a new algorithm is developed for the solution of option values in the non-linear case of a two-factor option where the jump parameters are known only to within a deterministic range. This case results in a Hamilton-Jacobi-Bellman style PIDE. A monotone discretization is used and a new fixed point, policy iteration developed for time step solution. Analysis proves that the new iteration is globally convergent under a mild time step restriction. Numerical tests demonstrate the overall convergence of the method and investigate the financial implications of uncertain parameters on the option value.

A meshfree approximation scheme based on the radial basis function methods is presented for the numerical solution of the options pricing model. This thesis deals with the valuation of the European, Barrier, Asian, American options of a single asset and American options of multi assets. The option prices are modeled by the Black-Scholes equation. The θ-method is used to discretize the equation with respect to time. By the next step, the option price is approximated in space with radial basis functions (RBF) with unknown parameters, in particular, we con- sider multiquadric radial basis functions (MQ-RBF). In case of Ameri- can options a penalty method is used, i.e. removing the free boundary is achieved by adding a small and continuous penalty term to the Black- Scholes equation. Finally, a comparison of analytical and finite difference solutions and numerical results from the literature is included.

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Radial basis function (RBF) approximation, is a new extremely powerful tool that is promising for high-dimensional problems, such as those arising from pricing of basket options using the Black-Scholes partial differential equation. The main problem for RBF methods have been ill-conditioning as the RBF shape parameter becomes small, corresponding to flat RBFs. This thesis employs a recently developed method called the RBF-QR method to reduce computational cost by improving the conditioning, thereby allowing for the use of a wider range of shape parameter values. Numerical experiments for the one-dimensional case are presented  and a MATLAB implementation is provided. In our thesis, the RBF-QR method performs better  than the RBF-Direct method for small shape parameters. Using Chebyshev points, instead of a standard uniform distribution, can increase the accuracy through clustering of the nodes towards the boundary. The least squares formulation for RBF methods is preferable to the collocation approach because it can result in smaller errors  for the same number of basis functions.

Spending on investments in integrative information systems (IIS) has considerably risen during the last few years due to a high need for linking various information systems. The demand for integrating the systems stems from developments like mergers and acquisitions and is typically satisfied in practice using Enterprise Application Integration solutions, Enterprise Resource Planning systems, Portals, or Data Warehouses. For the valuation of such an investment previous literature recommends the use of a real options analysis (ROA) since traditional capital budgeting methods such as the Net Present Value underestimate its value. Contrary, the ROA is able to conveniently account for managerial flexibility, represented by the possibility to implement follow-on opportunities, generated by the IIS. However, in practice ROA suffers from a lack of appliance mainly because of its complexity. This thesis precisely closes this gap and develops a simplified process model for a ROA by exactly tailoring the broad real options concept to the requirements of an investment valuation of IIS. For that, it reviews option pricing models from the financial world as well as previous research in the area of ROA and creates the desired model by conducting a ROA for four case studies in detail. The study reveals new findings concerning the question of how a decision-maker can apply the real options method and at the same time, when he/she is able to abandon a detailed ROA or a ROA at all. (author's abstract)

This thesis explores numerical methods for solving nonlinear partial differential equations (PDEs) that arise in option pricing problems. The goal is to develop or identify robust and efficient techniques that converge to the financially relevant solution for both one and two factor problems. To illustrate the underlying concepts, two nonlinear models are examined in detail: uncertain volatility and passport options. For any nonlinear model, implicit timestepping techniques lead to a set of discrete nonlinear equations which must be solved at each timestep. Several iterative methods for solving these equations are tested. In the cases of uncertain volatility and passport options, it is shown that the frozen coefficient method outperforms two different Newton-type methods. Further, it is proven that the frozen coefficient method is guaranteed to converge for a wide class of one factor problems. A major issue when solving nonlinear PDEs is the possibility of multiple solutions. In a financial context, convergence to the viscosity solution is desired. Conditions under which the one factor uncertain volatility equations are guaranteed to converge to the viscosity solution are derived. Unfortunately, the techniques used do not apply to passport options, primarily because a positive coefficient discretization is shown to not always be achievable. For both uncertain volatility and passport options, much work has already been done for one factor problems. In this thesis, extensions are made for two factor problems. The importance of treating derivative estimates consistently between the discretization and an optimization procedure is discussed. For option pricing problems in general, non-smooth data can cause convergence difficulties for classical timestepping techniques. In particular, quadratic convergence may not be achieved. Techniques for restoring quadratic convergence for linear problems are examined. Via numerical examples, these techniques are also shown to improve the stability of the nonlinear uncertain volatility and passport option problems. Finally, two applications are briefly explored. The first application involves static hedging to reduce the bid-ask spread implied by uncertain volatility pricing. While static hedging has been carried out previously for one factor models, examples for two factor models are provided. The second application uses passport option theory to examine trader compensation strategies. By changing the payoff, it is shown how the expected distribution of trading account balances can be modified to reflect trader or bank preferences.