TAU METHOD FOR PRICING AMERICAN OPTIONS UNDER COMPLEX MODELS

The European option can be exercised only at the expiration date while an American option can be exercised on or at any time before the expiration date.In this paper, we will study the numerical solutions of a class of complex partial differential equations (PDE) systems with free boundary conditions. This kind of problems arise naturally in pricing (finite-maturity) American options, which is applies to a wide variety of asset price models including the constant elasticity of variance (CEV), hyper-exponential jump-diffusion (HEJD) and the finite moment log stable (FMLS) models. Developing efficient numerical schemes will have significant applications in finance computation. These equations have already been solve by the Hybrid Laplace transformfinite difference methods and the Laplace transform method(LTM). In this paper we will introduce a method to solve these equations by Tau method. Also, we will show that using this method will end up to a faster convergence. Numerical examples demonstrate the accuracy and velocity of the method in CEV models

Three Problems in Stochastic Control and Applications

This thesis mainly concludes three different projects that I am devoted to: Recombining Tree Approximations for Optimal Stopping for Diffusions(Chapter 2), Continuity of Utility Maximization under Weak Convergence(Chapter 3) and Disorder Detection with Costly Observations(Chapter 4). The first two projects are related work. The third one is based on [16]. In Chapter 2,we develop two numerical methods for optimal stopping in the framework of one dimensional diffusion. Both of the methods use the Skorohod embedding in order to construct recombining tree approximations for diffusions with general coefficients. This technique allows us to determine convergence rates and construct nearly optimal stopping times which are optimal at the same rate. Finally, we demonstrate the efficiency of our schemes on several models. In Chapter 3, we find sufficient conditions for the continuity of the utility maximization problem from terminal wealth under convergence in distribution of the underlying processes. We provide several examples which illustrate that without these conditions, we cannot generally expect continuity to hold. Finally, we apply our continuity results to numerical computations of the shortfall risk in the Heston model. In Chapter 4, we study the Wiener disorder detection problem where each observation is associated with a positive cost. In this setting, a strategy is a pair consisting of a sequence of observation times and a stopping time corresponding to the declaration of disorder. We characterize the minimal cost of the disorder problem with costly observations as the unique fixed point of a certain jump operator, and we determine the optimal strategy.PHDApplied and Interdisciplinary MathematicsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/166133/1/guojia_1.pd

Managing the Paradox of Growth in Brand Communities Through Social Media

The commercial benefits of online brand communities are an important focus for marketers seeking deeper engagement with increasingly elusive consumers. Managing participation in these socially bound brand conversations challenges practitioners to balance authenticity towards the community against corporate goals. This is important as social media proliferation affords communities the capacity to reach a scale well beyond their offline equivalents and to operate independently of brands. While research has identified the important elements of engagement in brand communities, less is known about how strategies required to maximise relationships in these circumstances must change with growth. Using a case study approach, we examine how a rapidly growing firm and its community have managed the challenges of a maturing relationship. We find that, in time, the community becomes self-sustaining, and a new set of marketing management strategies is required to move engagement to the next level

Managing the Paradox of Growth in Brand Communities Through Social Media

The commercial benefits of online brand communities are an important focus for marketers seeking deeper engagement with increasingly elusive consumers. Managing participation in these socially bound brand conversations challenges practitioners to balance authenticity towards the community against corporate goals. This is important as social media proliferation affords communities the capacity to reach a scale well beyond their offline equivalents and to operate independently of brands. While research has identified the important elements of engagement in brand communities, less is known about how strategies required to maximise relationships in these circumstances must change with growth. Using a case study approach, we examine how a rapidly growing firm and its community have managed the challenges of a maturing relationship. We find that, in time, the community becomes self-sustaining, and a new set of marketing management strategies is required to move engagement to the next level