Search

The main contribution of this thesis is to examine binary options within the British payoff mechanism introduced by Peskir and Samee. This includes British cash-or-nothing put, British asset-or-nothing put, British binary call and American barrier binary options. We assume the geometric Brownian motion model and reduce the optimal stopping problems to free-boundary problems under the Markovian nature of the underlying process. With the help of the local time-space formula on curves, we derive a closed form expression for the arbitrage-free price in terms of the rational exercise boundary and show that the rational exercise boundary itself can be characterised as the unique solution to a non-linear integral equation. We begin by investigating the binary options of American-type which are also called `one-touch' binary options. Then we move on to examine the British binary options. Chapter~2 reviews the existing work on all different types of the binary options and sets the background for the British binary options. We price and analyse the American-type (one-touch) binary options using the risk-neutral probability method. In Chapters~3 ~4 and ~5, we present the British binary options where the holder enjoys the early exercise feature of American binary options whereupon his payoff is the `best prediction' of the European binary options payoff under the hypothesis that the true drift equals a contract drift. Based on the observed price movements, if the option holder finds that the true drift of the stock price is unfavourable then he can substitute it with the contract drift and minimise his losses. The key to the British binary option is the protection feature as not only can the option holder exercise at unfavourable stock price to a substantial reimbursement of the original option price (covering the ability to sell in a liquid option market completely endogenously) but also when the stock price movements are favourable he will generally receive high returns. Chapters~3 and~4 focus on the British binary put options and Chapter~5 on call options. We also analyse the financial meaning of the British binary options and show that with the contract drift properly selected the British binary options become very attractive alternatives to the classic European/American options. Chapter~6 extends the binary options into barrier binary options and discusses the application of the optimal structure without a smooth-fit condition in the option pricing. We first review the existing work for the knock-in options and present the main results from the literature. Then we examine the method in \cite{dai2004knock} in the application to the knock-in binary options. For the American knock-out binary options, the smooth-fit property does not hold when we apply the local time-space formula on curves. We transfer the expectation of the local time term into a computational form under the basic properties of Brownian motion. Using standard arguments based on Markov processes, we analyse the properties of the value function.

This thesis consists of three self-contained parts, each with its own abstract, body, references and page numbering. Part I, 'Potential theory, path integrals and the Laplacian of the indicator', finds the transition density of absorbed or reflected Brownian motion in a d-dimensional domain as a Feynman-Kac functional involving the Laplacian of the indicator, thereby relating the hitherto unrelated fields of classical potential theory and path integrals. Part II, 'The problem of alternatives', considers parallel investment in alternative technologies or drugs developed over time, where there can be only one winner. Parallel investment accelerates the search for the winner, and increases the winner's expected performance, but is also costly. To determine which candidates show sufficient performance and/or promise, we find an integral equation for the boundary of the optimal continuation region. Part III, 'Optimal support for renewable deployment', considers the role of government subsidies for renewable technologies. Rapidly diminishing subsidies are cheaper for taxpayers, but could prematurely kill otherwise successful technologies. By contrast, high subsidies are not only expensive but can also prop up uneconomical technologies. To analyse this trade-off we present a new model for technology learning that makes capacity expansion endogenous. There are two reasons for this standalone structure. First, the target readership is divergent. Part I concerns mathematical physics, Part II operations research, and Part III policy. Readers interested in specific parts can thus read these in isolation. Those interested in the thesis as a whole may prefer to read the three introductions first. Second, the separate parts are only partially interconnected. Each uses some theory from the preceding part, but not all of it; e.g. Part II uses only a subset of the theory from Part I. The quickest route to Part III is therefore not through the entirety of the preceding parts. Furthermore, those instances where results from previous parts are used are clearly indicated.

In this dissertation, we study the motion of a relativistic barotropic fluid with a free boundary. Relativistic barotropic fluids are fluids whose density and pressure are directly related. Such fluids are used as models in a wide range of scientific applications from meteorology to astrophysics. We prove that such a system of equations is locally well-posed, and the unique solution has the same regularity as the initial data. We use two methods to achieve the result. The first method is to construct approximate solutions to the equation by mollifying the nonlinear equation directly in the Lagrangian coordinate, and then passing to the limit so that the approximate solutions converge to the exact solution. The second method is to first consider the equation on a linear level, solve the linear equation using Galerkin approximation, and then solve the nonlinear equation by iterating on the solutions to these linear equations and passing to the limit.

Degenerate diffusion equations and their interface dynamics have received a lot of attention in the past couple of decades. In particular, surfaces moving with curvature dependent velocities and discontinuous diffusion intensities naturally appear in physical and biological models. In this dissertation, we study global well-posedness and geometry of two equations: mean curvature flows with forcing and degenerate nonlinear parabolic equations with discontinuous diffusion coefficients. Both problems have gradient flow structures in the space of sets and the Wasserstein space, respectively, which are useful to study the global-time behavior.In Chapter 1, we develop a parabolic version of the Aleksandrov and Serrin's moving plane methods for mean curvature flow with forcing. With the class of forcing which bounds the volume of evolving sets away from zero and infinity, we show that a strong version of star-shapedness is preserved over time. Based on this geometric property, we prove that volume preserving mean curvature flow starting from a star-shaped set converges to a ball.Chapter 2 is devoted to the study of degenerate parabolic equations with discontinuous diffusion intensities. We show the existence and uniqueness of the solutions in the sense of distributions. Our notion of solutions allows us to give a fine characterization of the emerging critical regions, observed previously in numerical experiments.

We consider the continuity equation for a population density subject to (i) a density upper-bound that depends on space and time and (ii) a velocity that minimizes the kinetic energy. A solution is constructed via the Wasserstein minimizing movement scheme for a corresponding time-dependent energy. Motion of the solution is driven by a decreasing density constraint.With a few assumptions, we prove this solution moves according to a free boundary problem of modified Hele-Shaw type that depends on the density constraint. In order to do this, we utilize a modified porous medium equation as an approximation to the original problem. Viscosity solution arguments are used to prove that given a decreasing density constraint, the porous medium equation solutions converge to the Hele-Shaw free boundary problem solution. By analyzing the Wasserstein gradient flow structure of the time-dependent energies involved, we next show that the porous medium equation solutions also converge to a solution of the original problem, thus identifying it with the Hele-Shaw description.In addition, we consider complications of the analysis without each assumption, perform numerical simulations supporting the results, and explore some limiting situations of the dynamics.

Ductal carcinoma in situ (DCIS) is the earliest form of breast cancer. Three mathematical models in the one dimensional case arising from DCIS are proposed. The first two models are in the form of parabolic equation with initial and known moving boundaries. Direct and inverse problems are considered in model 1, existence and uniqueness are proved by using tool from heat potential theory and Volterra integral equations. Also, we discuss the direct problem and nonlocal problem of model 2, existence and uniqueness are proved. And approximation solution of these problems are implemented by Ritz-Galerkin method, which is the first attempt to deal with such problems. Based on the finding of the previous two models, the more general free boundary problem model - nonlinear parabolic partial differential equation with initial, boundary and free boundary condition is presented. Well-posedness theorems are proved by applying knowledge of semigroup solution operators. Illustrative examples are included to demonstrate the validity and applicability of the technique for all three models.

We study nonlinear partial differential equations describing Hele-Shaw flows for which the source of the fluid, which classically is at a single point, instead expands as the moving boundary uncovers new sources. This problem is reformulated as a type of quasi- variational inequality, for which we show there exists a unique weak solution. Our procedure utilizes a Baiocchi integral transform. However, unlike in traditional applications, the inequality solved by the transformed variable continues to depend on the free boundary, and thus standard theory of variational inequalities cannot be immediately applied. Instead, we develop convergent time-stepping scheme. As a further application, we generalize and study a related problem with a growing density function obeying a hard constraint on the maximum density.

We analyze the geometry and regularity of the largest subsolution of a Free Boundary Problem. We showed that the largest subsolution is a viscosity solution of (1) with Lipschitz and Non-Degenerate properties under a very general free boundary condition. In addition to this, we provide density bounds for the positivity set and its complement near the free boundary.