Most derivatives do not have simple valuation formulas and must be priced by numerical methods such as tree models. Although the option prices computed by a tree model converge to the theoretical value as the number of time steps increases, the distribution error and the nonlinearity error may make the prices converge slowly or even oscillate significantly. This paper introduces a novel tree model, the bino-trinomial tree (BTT), that can price a wide range of derivatives efficiently and accurately. The BTT reduces the nonlinearity error sharply by adapting its structure to suit the derivative's specification; consequently, the pricing results converge smoothly and quickly. Moreover, the pricing of some European-style options on the BTT can be made extremely efficient by combinatorial tools, which are not available to most other tree models.Therefore, the BTT can efficiently reduce the distribution error by picking a large number of time steps. This paper uses a variety of options to demonstrate the effectiveness 1 of the BTT. Extensive numerical experiments show the superiority of the BTT to many other popular numerical models.
In the context of investment analysis, we formulate an abstract online computing problem called a planning game and develop general tools for solving such a game. We then use the tools to investigate a practical buy-and-hold trading problem faced by long-term investors in stocks. We obtain the unique optimal static online algorithm for the problem and determine its exact competitive ratio. We also compare this algorithm with the popular dollar averaging strategy using actual market data.
In 1989, Michael Rabin proposed a fundamentally new approach to the problems of fault-tolerant routing and memory management in parallel computation, based on the idea of information dispersal. Yuh-Dauh Lyuu developed this idea in a number of new and exciting ways in his PhD thesis. Further work has led to extensions of these methods to other applications such as shared memory emulations. This volume presents an extended and updated printing of Lyuu's thesis. It gives a detailed treatment of the information dispersal approach to the problems of fault-tolerance and distributed representations of information which have resisted rigorous analysis by previous methods.
Several recent papers (Gardner and Derrida 1989; Györgyi 1990; Sompolinsky et al. 1990) have found, using methods of statistical physics, that a transition to perfect generalization occurs in training a simple perceptron whose weights can only take values ±1. We give a rigorous proof of such a phenomena. That is, we show, for α = 2.0821, that if at least αn examples are drawn from the uniform distribution on {+1, −1}n and classified according to a target perceptron wt ∈ {+1, −1}n as positive or negative according to whether wt·x is nonnegative or negative, then the probability is 2−(√n) that there is any other such perceptron consistent with the examples. Numerical results indicate further that perfect generalization holds for α as low as 1.5.
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