Multi-Asset Option Pricing with Levy Process

Ph.DDOCTOR OF PHILOSOPH

B-Spline Based Methods: From Monotone Multigrid Schemes for American Options to Uncertain Volatility Models

In the first part of this thesis, we consider B-spline based methods for pricing American options in the Black-Scholes and Heston model. The difference between these two models is the assumption on the volatility of the underlying asset. While in the Black-Scholes model the volatility is assumed to be constant, the Heston model includes a stochastic volatility variable. The underlying problems are formulated as parabolic variational inequalities. Recall that, in finance, to determine optimal risk strategies, one is not only interested in the solution of the variational inequality, i.e., the option price, but also in its partial derivatives up to order two, the so-called Greeks. A special feature for these option price problems is that initial conditions are typically given as piecewise linear continuous functions. Consequently, we have derived a spatial discretization based on cubic B-splines with coinciding knots at the points where the initial condition is not differentiable. Together with an implicit time stepping scheme, this enables us to achieve an accurate pointwise approximation of the partial derivatives up to order two. For the efficient numerical solution of the discrete variational inequality, we propose a monotone multigrid method for (tensor product) B-splines with possible internal coinciding knots. Corresponding numerical results show that the monotone multigrid method is robust with respect to the refinement level and mesh size. In the second part of this thesis, we consider the pricing of a European option in the uncertain volatility model. In this model the volatility of the underlying asset is a priori unknown and is assumed to lie within a range of extreme values. Mathematically, this problem can be formulated as a one dimensional parabolic Hamilton-Jacobi-Bellman equation and is also called Black-Scholes-Barenblatt equation. In the resulting non-linear equation, the diffusion coefficient is given by a volatility function which depends pointwise on the second derivative. This kind of non-linear partial differential equation does not admit a weak H^1-formulation. This is due to the fact that the non-linearity depends pointwise on the second derivative of the solution and, thus, no integration by parts is possible to pass the partial derivative onto a test function. But in the discrete setting this pointwise second derivative can be approximated in H^1 by L^1-normalized B-splines. It turns out that the approximation of the volatility function leads to discontinuities in the partial derivatives. In order to improve the approximation of the solution and its partial derivatives for cubic B-splines, we develop a Newton like algorithm within a knot insertion step. Corresponding numerical results show that the convergence of the solution and its partial derivatives are nearly optimal in the L^2-norm, when the location of volatility change is approximated with desired accuracy

Mathematical analysis and numerical methods for pricing some pension plans and mortgages

[Resumen] El objetivo principal de esta tesis se centra en el análisis matemático y la solución numérica de algunos modelos de ecuaciones en derivadas parciales (EDPs) para valorar planes de pensiones con beneficios definidos e hipotecas a tipo de interés fijo. En los planes de pensiones, los beneficios por jubilación dependen del salario medio. Cuando no se permite jubilación anticipada, se plantea un problema de Cauchy asociado con un operador de Kolmogorov degenerado, en otro caso aparece un problema de complementariedad. Bajo ambas alternativas, obtenemos la existencia y unicidad de solución en los espacios funcionales adecuados. Además, si incorporamos saltos en el salario, entonces los problemas de Cauchy y de complementariedad están asociados a un operador integro-diferencial. En las hipotecas a tipo de interés fijo, consideramos las opciones de amortización anticipada e impago. Los factores estocásticos subyacentes son el tipo de interés y el valor del colateral. La valoración del contrato requiere resolver una serie de problemas de frontera libre enlazados. Los valores del seguro y de la fracción de pérdida potencial no cubierta por dicho seguro son la solución de una serie de problemas de Cauchy enlazados. Para obtener una solución numérica, se propone un método de Lagrange-Galerkin para la discretización de las EDPs, que se combina con un método iterativo de conjunto activo basado en una formulación de tipo lagrangiana aumentada para las restricciones de desigualdad. El término integral en el operador integro-diferencial se trata con técnicas de integración numérica apropiadas.[Resumo] O obxectivo principal desta tese céntrase na análise matemática e a solución numérica dalgúns modelos de ecuacións en derivadas parciais (EDPs) para valorar plans de pensións con beneficios definidos e hipotecas a tipo de xuro fixo. Nos plans de pensións, os beneficios por xubilación dependen do salario medio. Cando non se permite xubilación anticipada, formúlase un problema de Cauchy asociado cun operador de Kolmogorov dexenerado, noutro caso aparece un problema de complementariedade. Baixo as dúas alternativas, obtemos a existencia e unicidade de solución nos espazos funcionais axeitados. Ademais, se incorporamos saltos no salario, entón os problemas de Cauchy e de complementariedade están asociados cun operador integro-diferencial. Nas hipotecas a tipo de xuro fixo, consideramos as opcións de amortización antecipada e incumprimento. Os factores estocásticos subxacentes son o tipo de xuro e o valor do colateral. A valoración do contrato require resolver unha serie de problemas de fronteira libre enlazados. Os valores do seguro e da fracción de perda potencial non cuberta polo devandito seguro son a solución dunha serie de problemas de Cauchy enlazados. Para obter unha solución numérica, proponse un método de Lagrange-Galerkin para a discretización das EDPs, que se combina cun método iterativo de conxunto activo baseado nunha formulación de tipo lagranxiana aumentada para as restricións de des-igualdade. O termo integral no operador integro-diferencial trátase con técnicas de integración numérica axeitadas.[Abstract] The main objective of this thesis concerns to the mathematical analysis and numerical solution of some partial differential equation (PDE) models for pricing defined benefit pension plans and fixed-rate mortgages. In pension plans, the benefits at retirement depend on the average salary. When early retirement is not allowed a Cauchy problem associated with a degenerated Kolmogorov operator is posed, otherwise a complementarity problem arises. Under both alternatives, we obtain the existence and uniqueness of solution in appropriate functional spaces. If we incorporate jumps in the salary, then the Cauchy and complementarity problems are associated with a partial integro-differential operator (PIDE). In fixed-rate mortgages, we consider the possibility of prepayment and default. The underlying stochastic factors are the interest rate and the value of the collateral. The mortgage valuation requires solving a sequence of linked free boundary problems. The values of the insurance and coinsurance are the solution of respective sequences of linked Cauchy problems. For the numerical solution a Lagrange-Galerkin method for PDEs discretization is proposed, which is also combined with an active set iterative method based on an augmented Lagrangian formulation for inequality constraints. The integral term in the PIDE operator is treated with appropriate numerical integration techniques

Adaptive Finite Element Methods for Variational Inequalities: Theory and Applications in Finance

We consider variational inequalities (VIs) in a bounded open domain Omega subset IR^d with a piecewise smooth obstacle constraint. To solve VIs, we formulate a fully-discrete adaptive algorithm by using the backward Euler method for time discretization and the continuous piecewise linear finite element method for space discretization. The outline of this thesis is the following. Firstly, we introduce the elliptic and parabolic variational inequalities in Hilbert spaces and briefly review general existence and uniqueness results. Then we focus on a simple but important example of VI, namely the obstacle problem. One interesting application of the obstacle problem is the American-type option pricing problem in finance. We review the classical model as well as some recent advances in option pricing. These models result in VIs with integro-differential operators. Secondly, we introduce two classical numerical methods in scientific computing: the finite element method for elliptic partial differential equations (PDEs) and the Euler method for ordinary different equations (ODEs). Then we combine these two methods to formulate a fully-discrete numerical scheme for VIs. With mild regularity assumptions, we prove optimal a priori convergence rate with respect to regularity of the solution for the proposed numerical method. Thirdly, we derive an a posteriori error estimator and show its reliability and efficiency. The error estimator is localized in the sense that the size of the elliptic residual is only relevant in the approximate noncontact region, and the approximability of the obstacle is only relevant in the approximate contact region. Based on this new a posteriori error estimator, we design a time-space adaptive algorithm and multigrid solvers for the resulting discrete problems. In the end, numerical results for $d=1,2$ show that the error estimator decays with the same rate as the actual error when the space meshsize and the time step tend to zero. Also, the error indicators capture the correct local behavior of the errors in both the contact and noncontact regions

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Bean. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature

Numerical and optimal control methods for partial differential equations arising in computational finance

The chosen title for my PhD thesis is "Numerical and optimal control methods for partial differential equations arising in computational finance". The body of my research is divided into two parts. The first part is devoted to the application of an alternating direction implicit numerical method for solving stochastic volatility option pricing models. The second part focuses on a partial-integro differential equation constrained optimal control approach to parameter estimation for the forward jump-diffusion option pricing model. The body of the thesis is preceded by an extensive introduction, which seeks to contextualize my work with respect to the field of computational finance, this is followed by a brief conclusion. Finally, the thesis is completed by a list of refer ences. The first project proposes a new high-order alternating direction implicit (ADI) finite difference scheme for the solution of initial-boundary value problems of convection-diffusion type with mixed derivatives and non-constant coefficients, as they arise from stochastic volatility models in option pricing. The approach combines different high-order spatial discretisations with Hundsdorfer and Verwer's ADI time-stepping method, to obtain an efficient method which is fourth-order accurate in space and second-order accurate in time. Numerical experiments for the European put option pricing problem using Heston's stochastic volatility model confirm the high-order convergence. The second project proposes to solve a parameter calibration problem for the forward jump-diffusion option pricing model proposed by Andersen and Andreasen. A distributed optimal control approach is employed, with a partial-integro differential equation as our state equation. By approaching the problem from a functional analysis perspective, I investigate the necessary regularity conditions for our parameters of interest. Following this, the existence of optimal solutions is proven under certain analytical conditions. Furthermore, the first-order necessary conditions for optimality are also established. Finally, a projected-gradient optimization method is applied numerically to empirical market data and results are given

Numerical singular perturbation approaches based on spline approximation methods for solving problems in computational finance

Philosophiae Doctor - PhDOptions are a special type of derivative securities because their values are derived from the value of some underlying security. Most options can be grouped into either of the two categories: European options which can be exercised only on the expiration date, and American options which can be exercised on or before the expiration date. American options are much harder to deal with than European ones. The reason being the optimal exercise policy of these options which led to free boundary problems. Ever since the seminal work of Black and Scholes [J. Pol. Econ. 81(3) (1973), 637-659], the differential equation approach in pricing options has attracted many researchers. Recently, numerical singular perturbation techniques have been used extensively for solving many differential equation models of sciences and engineering. In this thesis, we explore some of those methods which are based on spline approximations to solve the option pricing problems. We show a systematic construction and analysis of these methods to solve some European option problems and then extend the approach to solve problems of pricing American options as well as some exotic options. Proposed methods are analyzed for stability and convergence. Thorough numerical results are presented and compared with those seen in the literature.South Afric

Mesh Free Methods for Differential Models In Financial Mathematics

Philosophiae Doctor - PhDMany problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston's volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided

Mesh free methods for differential models in financial mathematics

Philosophiae Doctor - PhDMany problems in financial world are being modeled by means of differential equation. These problems are time dependent, highly nonlinear, stochastic and heavily depend on the previous history of time. A variety of financial products exists in the market, such as forwards, futures, swaps and options. Our main focus in this thesis is to use the numerical analysis tools to solve some option pricing problems. Depending upon the inter-relationship of the financial derivatives, the dimension of the associated problem increases drastically and hence conventional methods (for example, the finite difference methods or finite element methods) for solving them do not provide satisfactory results. To resolve this issue, we use a special class of numerical methods, namely, the mesh free methods. These methods are often better suited to cope with changes in the geometry of the domain of interest than classical discretization techniques. In this thesis, we apply these methods to solve problems that price standard and non-standard options. We then extend the proposed approach to solve Heston' volatility model. The methods in each of these cases are analyzed for stability and thorough comparative numerical results are provided.South Afric