Parallel Solution of Diffusion Equations using Laplace Transform Methods with Particular Reference to Black-Scholes Models of Financial Options

Diffusion equations arise in areas such as fluid mechanics, cellular biology, weather forecasting, electronics, mechanical engineering, atomic physics, environmental science, medicine, etc. This dissertation considers equations of this type that arise in mathematical finance. For over 40 years traders in financial markets around the world have used Black-Scholes equations for valuing financial options. These equations need to be solved quickly and accurately so that the traders can make prompt and accurate investment decisions. One way to do this is to use parallel numerical algorithms. This dissertation develops and evaluates algorithms of this kind that are based on the Laplace transform, numerical inversion algorithms and finite difference methods. Laplace transform-based algorithms have faced a legitimate criticism that they are ill-posed i.e. prone to instability. We demonstrate with reference to the Black-Scholes equation, contrary to the received wisdom, that the use of the Laplace transform may be used to produce reasonably accurate solutions (i.e. to two decimal places), in a fast and reliable manner when used in conjunction with standard PDE techniques. To set the scene for the investigations that follow, the reader is introduced to financial options, option pricing and the one-dimensional and two-dimensional linear and nonlinear Black-Scholes equations. This is followed by a description of the Laplace transform method and in particular, four widely used numerical algorithms that can be used for finding inverse Laplace transform values. Chapter 4 describes methodology used in the investigations completed i.e. the programming environment used, the measures used to evaluate the performance of the numerical algorithms, the method of data collection used, issues in the design of parallel programs and the parameter values used. To demonstrate the potential of the Laplace transform based approach, Chapter 5 uses existing procedures of this kind to solve the one-dimensional, linear Black-Scholes equation. Chapters 6, 7, 8, and 9 then develop and evaluate new Laplace transform-finite difference algorithms for solving one-dimensional and two-dimensional, linear and nonlinear Black-Scholes equations. They also determine the optimal parameter values to use in each case i.e. the parameter values that produce the fastest and most accurate solutions. Chapters 7 and 9 also develop new, iterative Monte Carlo algorithms for calculating the reference solutions needed to determine the accuracy of the LTFD solutions. Chapter 10 identifies the general patterns of behaviour observed within the LTFD solutions and explains them. The dissertation then concludes by explaining how this programme of work can be extended. The investigations completed make significant contributions to knowledge. These are summarised at the end of the chapters in which they occur. Perhaps the most important of these is the development of fast and accurate numerical algorithms that can be used for solving diffusion equations in a variety of application areas

Variance-optimal hedging for processes with stationary independent increments

We determine the variance-optimal hedge when the logarithm of the underlying price follows a process with stationary independent increments in discrete or continuous time. Although the general solution to this problem is known as backward recursion or backward stochastic differential equation, we show that for this class of processes the optimal endowment and strategy can be expressed more explicitly. The corresponding formulas involve the moment, respectively, cumulant generating function of the underlying process and a Laplace- or Fourier-type representation of the contingent claim. An example illustrates that our formulas are fast and easy to evaluate numerically.Comment: Published at http://dx.doi.org/10.1214/105051606000000178 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Local time and the pricing of time-dependent barrier options

A time-dependent double-barrier option is a derivative security that delivers the terminal value $\phi(S_T)$ at expiry $T$ if neither of the continuous time-dependent barriers b_\pm:[0,T]\to \RR_+ have been hit during the time interval $[0,T]$. Using a probabilistic approach we obtain a decomposition of the barrier option price into the corresponding European option price minus the barrier premium for a wide class of payoff functions $\phi$, barrier functions $b_\pm$ and linear diffusions $(S_t)_{t\in[0,T]}$. We show that the barrier premium can be expressed as a sum of integrals along the barriers $b_\pm$ of the option's deltas \Delta_\pm:[0,T]\to\RR at the barriers and that the pair of functions $(\Delta_+,\Delta_-)$ solves a system of Volterra integral equations of the first kind. We find a semi-analytic solution for this system in the case of constant double barriers and briefly discus a numerical algorithm for the time-dependent case.Comment: 32 pages, to appear in Finance and Stochastic