Spanning With Binary Options: A Vector Lattice Approach

The issue of portfolio insurance is one of the prime concerns of the investors who want to insure their asset at minimum or appropriate cost. Static hedging with binary options is a popular strategy that has been explored in various option models (see e.g. (2; 3; 4; 7)). In this thesis, we propose a static hedging algorithm for discrete time models. Our algorithm is based on a vector lattice technique. In chapter 1, we give the necessary background on the theory of vector lattices and the theory of options. In chapter 2, we reveal the connection of lattice-subspaces with the minimum-cost portfolio insurance strategy. In chapter3, we outline our algorithm and give applications to binomial and trinomial option models. In chapter 4, we perform simulations and analyze the hedging errors of our algorithm for European, Barrier, Geometric Asian, Arithmetic Asian, and Lookback options. The study has revealed that static hedging could be suitable strategy for the European, Barrier, and Geometric Asian options as these options have shown less inclination to the rollover effect.

Spanning With Binary Options: A Vector Lattice Approach

The issue of portfolio insurance is one of the prime concerns of the investors who want to insure their asset at minimum or appropriate cost. Static hedging with binary options is a popular strategy that has been explored in various option models (see e.g. (2; 3; 4; 7)). In this thesis, we propose a static hedging algorithm for discrete time models. Our algorithm is based on a vector lattice technique. In chapter 1, we give the necessary background on the theory of vector lattices and the theory of options. In chapter 2, we reveal the connection of lattice-subspaces with the minimum-cost portfolio insurance strategy. In chapter3, we outline our algorithm and give applications to binomial and trinomial option models. In chapter 4, we perform simulations and analyze the hedging errors of our algorithm for European, Barrier, Geometric Asian, Arithmetic Asian, and Lookback options. The study has revealed that static hedging could be suitable strategy for the European, Barrier, and Geometric Asian options as these options have shown less inclination to the rollover effect.

New Models for Solving Time-Varying LU Decomposition by Using ZNN Method and ZeaD Formulas

In this paper, by employing the Zhang neural network (ZNN) method, an effective continuous-time LU decomposition (CTLUD) model is firstly proposed, analyzed, and investigated for solving the time-varying LU decomposition problem. Then, for the convenience of digital hardware realization, this paper proposes three discrete-time models by using Euler, 4-instant Zhang et al. discretization (ZeaD), and 8-instant ZeaD formulas to discretize the proposed CTLUD model, respectively. Furthermore, the proposed models are used to perform the LU decomposition of three time-varying matrices with different dimensions. Results indicate that the proposed models are effective for solving the time-varying LU decomposition problem, and the 8-instant ZeaD LU decomposition model has the highest precision among the three discrete-time models.

Comparison of stabilizing mechanisms between discrete- and continuous-time consumer-resource models

There is rich literature on using continuous-time and discrete-time models for studying population dynamics of consumer-resource interactions. The continuous-time framework is generally used to model populations with overlapping generations and all year-round reproduction. In contrast, discrete-time models are more suited for populations with non-overlapping generations that reproduce in a discrete pulse determined by season. Inspired by the Nicholson-Bailey/Lotka-Volterra modeling formalisms in discrete-time/continuous-time, respectively, we consider host-parasitoid interactions with an arbitrary parasitoid attack rate that is a function of both the host and parasitoid population densities. We characterize and compare stability regimes in both modeling frameworks for analogous host reproduction and attack rates. Our analysis shows that a Type II functional response is stabilizing in both modeling frameworks only when combined with other mechanisms, such as mutual interference between parasitoids. This stability regime related to a Type II functional response is smaller in the discrete-time framework compared to continuous-time, and shrinks with increasing host reproduction. A Type III functional response is by itself stabilizing, but the extent of attack-rate acceleration needed is much higher in the discrete-time framework, and its stability regime expands with increasing host reproduction. Finally, our results show that while mutual parasitoid interference can stabilize population dynamics, cooperation between parasitoids to handle hosts is destabilizing in both frameworks. However, a combination of a Type III functional response together with parasitoid cooperation can create stability. In summary, our comparative analysis systematically characterizes diverse ecological processes driving stable population dynamics in discrete-time and continuous-time consumer-resource models.

Asymptotic synthesis of contingent claims with controlled risk in a sequence of discrete‐time markets

We examine the connection between discrete‐time models of financial markets and the celebrated Black–Scholes–Merton (BSM) continuous‐time model in which “markets are complete.” Suppose that (a) the probability law of a sequence of discrete‐time models converges to the law of the BSM model and (b) the largest possible one‐period step in the discrete‐time models converges to zero. We prove that, under these assumptions, every bounded and continuous contingent claim can be asymptotically synthesized, controlling for the risks taken in a manner that implies, for instance, that an expected‐utility‐maximizing consumer can asymptotically obtain as much utility in the (possibly incomplete) discrete‐time economies as she can at the continuous‐time limit. Hence, in economically significant ways, many discrete‐time models with frequent trading resemble the complete‐markets model of BSM.