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.

In this dissertation we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conservative form and present a convergence analysis for the two-dimensional Black-Scholes equation arising in the Hull-White model for pricing European options with stochastic volatility. We formulate a non-conforming Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems defined on element edges. We show that the bilinear form of the finite element method is coercive and continuous and establish an upper bound of order O(h) on the discretization error of method, where h denotes the mesh parameter of the discretization. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presentd.

Option is an asymmetric contract between two parties with future payoff derived from the price of underlying asset. Methods of pricing di erent types of options under more or less general assumptions have been extensively studied since the Nobel price winning works of Black and Scholes [1] and Merton [12] were published in 1973. A new way of pricing options with the use of Mellin transforms have been recently introduced by Panini and Srivastav [15] in 2004. This thesis offers a brief introduction to option pricing with Mellin transforms and a revision of some of the recent research in this field.

We present two option pricing models, both different from the classical Black-Scholes-Merton model. The first model, suggested by Heston, considers the case where the asset price volatility is stochastic. For this model we study the asset price process and give in detail the derivation of the European call option price process. The second model, suggested by Brody-Hughston-Macrina, describes the observation of certain information about the claim perturbed by a noise represented by a Brownian bridge. Here we also study in detail the properties of this noisy information process and give the derivations of both asset price dynamics and the European call option price process.

Traditional financial modeling is based on semimartingale processes with stationary and independent increments. However, empirical investigations on financial data does not always support these assumptions. This contradiction showed that there is a need for new stochastic models. Fractional Brownian motion (fBm) was proposed as one of these models by Benoit Mandelbrot. FBm is the only continuous Gaussian process with dependent increments. Correlation between increments of a fBm changes according to its self-similarity parameter H. This property of fBm helps to capture the correlation dynamics of the data and consequently obtain better forecast results. But for values of H different than 1/2, fBm is not a semimartingale and classical Ito formula does not exist in that case. This gives rise to need for using the white noise theory to construct integrals with respect to fBm and obtain fractional Ito formulas. In this thesis, the representation of fBm and its fundamental properties are examined. Construction of Wick-Ito-Skorohod (WIS) and fractional WIS integrals are investigated. An Ito type formula and Girsanov type theorems are stated. The financial applications of fBm are mentioned and the Black&amp / Scholes price of a European call option on an asset which is assumed to follow a geometric fBm is derived. The statistical aspects of fBm are investigated. Estimators for the self-similarity parameter H and simulation methods of fBm are summarized. Using the R/S methodology of Hurst, the estimations of the parameter H are obtained and these values are used to evaluate the fractional Black&amp / Scholes prices of a European call option with different maturities. Afterwards, these values are compared to Black&amp / Scholes price of the same option to demonstrate the effect of long-range dependence on the option prices. Also, estimations of H at different time scales are obtained to investigate the multiscaling in financial data. An outlook of the future work is given.

This article examines the hedging positions derived from the Black-Scholes(B-S) model and the GARCH(1,1) models, respectively, when the log returns of underlying asset exhibits GARCH(1,1) process. The result shows that Black-Scholes and GARCH options deltas, one of the hedging parameters, are similar for near-the-money options, and Black-Scholes options delta is higher then GARCH delta in absolute terms when the options are deep out-of-money, and Black-Scholes options delta is lower then GARCH delta in absolute terms when the options are deep in-the-money. Simulation study of hedging procedure of GARCH(1,1) and B-S models are performed, which also support the above findings.

We deal with locally ill-posed nonlinear operator equations F(x) = y in L^2(0,1), where the Fréchet derivatives A = F'(x_0) of the nonlinear forward operator F are compact linear integral operators A = M ◦ J with a multiplication operator M with integrable multiplier function m and with the simple integration operator J. In particular, we give examples of nonlinear inverse problems in natural sciences and stochastic finance that can be written in such a form with linearizations that contain multiplication operators. Moreover, we consider the corresponding ill-posed linear operator equations Ax = y and their degree of ill-posedness. In particular, we discuss the fact that the noncompact multiplication operator M has only a restricted influence on this degree of ill-posedness even if m has essential zeros of various order.

This thesis develops a generic framework based on the Fourier transform for pricing and hedging of various options in equity, commodity, currency, and insurance markets. The pricing problem can be reduced to solving a partial integro-differential equation (PIDE). The Fourier Space Time-stepping (FST) framework developed in this thesis circumvents the problems associated with the existing finite difference methods by utilizing the Fourier transform to solve the PIDE. The FST framework-based methods are generic, highly efficient and rapidly convergent. The Fourier transform can be applied to the pricing PIDE to obtain a linear system of ordinary differential equations that can be solved explicitly. Solving the PIDE in Fourier space allows for the integral term to be handled efficiently and avoids the asymmetrical treatment of diffusion and integral terms, common in the finite difference schemes found in the literature. For path-independent options, prices can be obtained for a range of stock prices in one iteration of the algorithm. For exotic, path-dependent options, a time-stepping methodology is developed to handle barriers, free boundaries, and exercise policies. The thesis includes applications of the FST framework-based methods to a wide range of option pricing problems. Pricing of single- and multi-asset, European and path-dependent options under independent-increment exponential Levy stock price models, common in equity and insurance markets, can be done efficiently via the cornerstone FST method. Mean-reverting Levy spot price models, common in commodity markets, are handled by introducing a frequency transformation, which can be readily computed via scaling of the option value function. Generating stochastic volatility, to match the long-term equity options market data, and stochastic skew, observed in currency markets, is addressed by introducing a non-stationary extension of multi-dimensional Levy processes using regime-switching. Finally, codependent jumps in multi-asset models are introduced through copulas. The FST methods are computationally efficient, running in O(MN^d log_2 N) time with M time steps and N space points in each dimension on a d-dimensional grid. The methods achieve second-order convergence in space; for American options, a penalty method is used to attain second-order convergence in time. Furthermore, graphics processing units are utilized to further reduce the computational time of FST methods.

Although the Black-Scholes (BS) model and its alternatives have been widely applied in finance, their flaws have drawn the attention of many investors and risk managers. The Black-Scholes (BS) model fails to explain the volatility smile. Its alternatives, such as the BS model with a Poisson jump process, fail to explain the volatility clustering. Based on the literature, a novel dynamic regime-switching option-pricing model is developed in this thesis, to overcome the flaws of the traditional option pricing models. Five macroeconomic indicators are identified as the drivers of economic states over time. Two regimes are selected among all likely numbers of regimes under the Bayes Information Criterion (BIC). Both in-sample and out-of-sample tests are constructed to examine the prediction of the model. Empirical results show that the two-state regime-switching option-pricing model exhibits significant prediction power.

Recently, interest in the Graphics Processing Unit (GPU) for general purpose parallel applications development and research has grown. Much of the current research on the GPU focuses on the acceleration of regular problems, as irregular problems typically do not provide the same level of performance on the hardware. We explore the potential of the GPU by investigating four problems on the GPU with regular and/or irregular properties: lookback option pricing (regular), single-source shortest path (irregular), maximum flow (irregular), and the task matching problem using multi-swarm particle swarm optimization (regular with elements of irregularity). We investigate the design, implementation, optimization, and performance of these algorithms on the GPU, and compare the results. Our results show that the regular problem achieves greater performance and requires less development effort than the irregular problems. However, we find the GPU to still be capable of providing high levels of acceleration for irregular problems.