Yuh-Dauh Lyuu

On the diameter vulnerability of Kautz digraphs

Abstract We show that in the Kautz digraph K ( d , t ) with d t + d t −1 vertices each having out degree d , there exist d vertex-disjoint paths between any pair of distinct vertices, one of length at most t , d − 2 of length at most t + 1, and one of length at most t + 2.

Spreading messages

We model a network in which messages spread by a simple directed graph G=(V,E) and a function @a:V->N mapping each v@?V to a positive integer less than or equal to the indegree of v. The graph G represents the individuals in the network and the communication channels between them. An individual v@?V will be convinced of a message when at least @a(v) of its in-neighbors are convinced. Suppose we are to convince a message to the individuals by first convincing a subset of individuals, called the seeds, and then let the message spread. We study the minimum number min-seed (G,@a) of seeds needed to convince all individuals at the end. In particular, we prove a lower bound on min-seed (G,@a) and the NP-completeness of computing min-seed (G,@a). We also analyze the special case, called the strict-majority scenario, where each individual is convinced of a message when more than half of its in-neighbors are convinced. For the strict-majority scenario, we prove three results. First, we show that with high probability over the Erdos-Renyi random graphs G(n,p), @W(min{n,1/p}) seeds are needed to convince all individuals at the end. Second, if G=(V,E) is undirected, then a set of s uniformly random samples from V convinces no more than an expected s(2|E|+2|V|)|V| individuals at the end. Third, in a digraph G=(V,E) with a positive minimum indegree, one can find in polynomial (in |V|) time a set of at most (23/27)|V| seeds convincing all individuals.

The Bino-Trinomial Tree: A Simple Model for Efficient and Accurate Option Pricing

A model with a closed-form solution is the Holy Grail of derivatives valuation, because as computers have become increasingly powerful, exact answers to even very complicated formulas can typically be obtained almost instantaneously. Unfortunately, as derivative instruments have become increasingly complicated, closed-form valuation has become increasingly rare. Researchers are now trying to develop more efficient and accurate approximation techniques. Lattice models, such as the classic binomial model of Cox, Ross, and Rubinstein, are among the workhorses of this effort. Lattice models converge to accurate values as the number of node calculations increases, but convergence is often erratic and slow. The biggest problem is that the option payoff between two price nodes can be highly nonlinear, for example when the critical price barrier for a knock-out option falls between two layers of nodes, and a large jump in the computed option value occurs when a small change in the number of time steps causes the critical price to hop from one side of a node to the other. A variety of tricks have been proposed to deal with this problem by adapting the geometry of the lattice to make the nodes land directly on top of the critical prices. Generally this has required the additional flexibility afforded by a trinomial structure rather than the binomial, but that is costly in terms of efficiency. In this article, Dai and Lyuu introduce a new approach that achieves remarkable improvement in efficiency by combining binomial and trinomial structures. Here the trick is to construct a binomial lattice with nodes that land on top of the key prices, but to use a very small amount of trinomial lattice to connect the initial price—the model option value—to the binomial structure. This allows both the best placement of the tree relative to the critical areas and also great efficiency gains because binomial lattice probabilities for the terminal nodes can be computed directly using combinatorial results, skipping over calculations for all of the intermediate time steps.

Optimal Buy-and-Hold Strategies for Financial Markets with Bounded Daily Returns

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.

Line digraph iterations and connectivity analysis of de Bruijn and Kautz graphs

A graph has spread (m, k, l) if for any m+1 distinct nodes x, y/sub 1/, . . ., y/sub m/ and m positive integers r/sub 1/, . . ., r/sub m/, such that Sigma /sub i/r/sub i/=k, there exist k node-disjoint paths of length at most 1 from x to the y/sub i/, where r/sub i/ of them end at y/sub i/. This concept contains, and is related to many important concepts used in communications and graph theory. The authors prove an optimal general theorem about the spreads of digraphs generated by line digraph iterations. Useful graphs, like the de Bruijn and Kautz digraphs, can be thus generated. The theorem is applied to the de Bruijn and Kautz digraphs to derive optimal bounds on their spreads, which implies previous results and resolves open questions on their connectivity, diameter, k-diameter, vulnerability, and some other measures related to length-bound disjoint paths. >

Very Fast Algorithms for Barrier Option Pricing and the Ballot Problem

Combinatorial methods prove extremely useful towards designing blazingly fast yet simple algorithms for pricing European-style barrier options. Closed-form formulae to standard European-style barrier options can then be easily derived. Combinatorial formulae under the trinomial model are also presented. The common practice in the literature compares algorithms based on their respective numbers of time steps towards convergence. We illustrate the pitfalls of this custom by evaluating the performance of our binomial model-based algorithm and the trinomial tree algorithm, whose superiority over the binomial model is widely accepted. Contrary to common beliefs, however, our algorithm emerges as a clear winner. In fact, the performance gap is two orders of magnitude. Also shattered is the myth that the binomial model converges extremely slowly when the current stock price is very close to the barrier.

Accurate pricing formulas for Asian options

Asian options have payoffs that depend on the average price of the underlying asset such as stocks, commodities, or financial indices. As exact closed-form formulas do not exist for these popular options, how to price them numerically in an efficient and accurate manner has been extensively investigated. There are two types of Asian options, fixed-strike and floating-strike Asian options. Excellent lower-bound formulas for both types of options have been derived by Rogers and Shi. These formulas are extremely easy to calculate, but they restrict the option’s maturity to exactly 1 year. This paper extends the Rogers–Shi formulas to general maturities. Numerical experiments are performed to compare the formulas with many other numerical methods in the literature and under a wide variety of situations. They confirm the extreme accuracy of the formulas.

The transition to perfect generalization in perceptrons

Several recent papers (Gardner and Derrida 1989; Gyrgyi 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 = 2.0821, that if at least n 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 wtx 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 as low as 1.5.

Bounding the number of tolerable faults in majority-based systems

Consider the following coloring process in a simple directed graph G(V,E) with positive indegrees. Initially, a set S of vertices are white. Thereafter, a black vertex is colored white whenever the majority of its in-neighbors are white. The coloring process ends when no additional vertices can be colored white. If all vertices end up white, we call S an irreversible dynamic monopoly (or dynamo for short). We derive upper bounds of 0.7732|V| and 0.727|V| on the minimum sizes of irreversible dynamos depending on whether the majority is strict or simple. When G is an undirected connected graph without isolated vertices, upper bounds of ⌈|V|/2 ⌉ and

Fast-fault-tolerant parallel communication and on-line maintenance using information dispersal

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