Efficient Broadcast in All-Port Wormhole-Routed Hypercubes

This chapter presents the binomial tree approach to the option pricing problem. We first illustrate the basic ideas of option pricing by considering the one-period binomial tree model and then extend to a multi-period binomial tree model. We then show that, by taking limits in an appropriate way, the binomial expression for the option price converges to the Black–Scholes option price and pricing equation. Alternatively, the continuous time model can be discretised in a way that yields the same expressions as obtained by the binomial tree approach.

Carbon option pricing based on uncertain fractional differential equation: A binomial tree approach

In this paper, a network-partitioning approach for one-to-all broadcasting on wormhole-routed networks is proposed. To broadcast a message, the scheme works in three phases. First, a number of data-distributing networks (DDNs), which can work independently, are constructed. Then the message is evenly divided into submessages, each being sent to a representative node in one DDN. Second, the submessages are broadcast on the DDNs concurrently. Finally, a number of data-collecting networks (DCNs), which can work independently too, are constructed. Then, concurrently on each DCN, the submessages are collected and combined into the original message. Our approach, especially designed for wormhole-routed networks, is conceptually similar but fundamentally very different from the traditional approach of using multiple edge-disjoint spanning trees in parallel for broadcasting in store-and-forward networks. One interesting issue is on the definition of independent DDNs and DCNs, in the sense of wormhole routing. We show how to apply this approach to tori, meshes, and hypercubes. Thorough analyses and comparisons based on different system parameters and configurations are conducted. The results do confirm the advantage of our scheme, under various system parameters and conditions, over other existing broadcasting algorithms.

With the increasing popularity of wireless mesh networks (WMNs), broadcasting traffic (e.g. IP-TV) will contribute a large portion of network load. In this paper, we consider a multi-channel multi-interface WMN with real time broadcast call arrivals. Aiming at maximizing the call acceptance rate of the network, an efficient broadcast tree construction algorithm, called Schedule-based Greedy Expansion (S-Expand), is designed. Unlike the existing time fraction approach, which focuses on assigning time fractions to tree links to guarantee the existence of a feasible schedule, we follow the approach of joint routing and scheduling. The proposed S-Expand algorithm packs non-interfering transmissions to use the same time slots; this would allow more flexibility in accepting future calls. Simulation results show that S-Expand achieves higher call acceptance rate than the traditional time fraction approach.

The recombining binomial tree approach, which has been initiated by Cox et al. (J Financ Econ 7: 229–263, 1979) and extended to arbitrary diffusion models by Nelson and Ramaswamy (Rev Financ Stud 3(3): 393–430, 1990) and Hull and White (J Financ Quant Anal 25: 87–100, 1990a), is applied to the simultaneous evaluation of price and Greeks for the amortized fixed and variable rate mortgage prepayment option. We consider the simplified binomial tree approximation to arbitrary diffusion processes by Costabile and Massabo (J Deriv 17(3): 65–85, 2010) and analyze its numerical applicability to the mortgage valuation problem for some Vasicek and CIR-like interest rate models. For fixed rates and binomial trees with about thousand steps, we obtain very good results. For the Vasicek model, we also compare the closed-form analytical approximation of the callable fixed rate mortgage price by Xie (IAENG Int J Appl Math 39(1): 9, 2009) with its binomial tree counterpart. With respect to the binomial tree values one observes a systematic underestimation (overestimation) of the callable mortgage price (prepayment option price) analytical approximation. This numerical discrepancy increases at longer maturities and becomes impractical for a valuable estimation of the prepayment option price.

We study here deterministic broadcasting in geometric radio networks (GRN) whose nodes have complete knowledge of the network. Nodes of a GRN are deployed in the Euclidean plane (R 2) and each of them can transmit within some range r assigned to it. We adopt model in which ranges of nodes are non-uniform and they are drawn from the predefined interval 0 ≤ r min ≤ r max . All our results are in the conflict-embodied model where a receiving node must be in the range of exactly one transmitting node in order to receive the message.We derive several lower and upper bounds on the time of deterministic broadcasting in GRNs in terms of the number of nodes n, a distribution of nodes ranges, and the eccentricity D of the source node (i.e., the maximum length of a shortest directed path from the source node to another node in the network). In particular:(1) We show that D + Ω(log(n − D)) rounds are required to accomplish broadcasting in some GRN where each node has the transmission range set either to 1 or to 0. We also prove that the bound D + Ω(log(n − D)) is almost tight providing a broadcasting procedure that works in this type of GRN in time D + O(logn).(2) In GRNs with a wider choice of positive node ranges from r min , ...,r max , we show that broadcasting requires \(D+\Omega (\min\{\log {\frac {r_{max}}{r_{min}}},\log (n-D)\})\) rounds and that it can be accomplished in \(O (D\log^2 \frac {r_{max}} {r_{min}})\) rounds subsuming the best currently known upper bound \(O (D (\frac {r_{max}} {r_{min}})^4)\) provided in [15].(3) We also study the problem of simulation of minimum energy broadcasting in arbitrary GRNs. We show that energy optimal broadcasting that can be completed in h rounds in a conflict-free model may require up to h/2 additional rounds in the conflict-embodied model. This lower bound should be seen as a separation result between conflict-free and conflict-embodied geometric radio networks. Finally, we also prove that any h-hop broadcasting algorithm with the energy consumption \({\cal E}\) in a GRN can be simulated within O(hlogψ) rounds in the conflict-embodied model using energy \(O({\cal E})\), where ψ is the ratio between the largest and the shortest Euclidean distance between a pair of nodes in the network.

Empirical Tests of Canonical Nonparametric American Option Pricing Methods

This paper proposes a new efficient algorithm for the computation of Greeks for options using the binomial tree. We also show that Greeks for European options introduced in this article are asymptotically equivalent to the discrete version of Malliavin Greeks. This fact enables us to show that our Greeks converge to Malliavin Greeks in the continuous time model. The computation algorithm of Greeks for American options using the binomial tree is also given in this article. There are three advantageous points to use binomial tree approach for the computation of Greeks. First, mathematics is much simpler than using the continuous time Malliavin calculus approach. Second, we can construct a simple algorithm to obtain the Greeks for American options. Third, this algorithm is very efficient because one can compute the price and Greeks (delta, gamma, vega, and rho) at once. In spite of its importance, only a few previous studies on the computation of Greeks for American options exist, because performing sensitivity analysis for the optimal stopping problem is difficult. We believe that our method will become one of the popular ways to compute Greeks for options.

Strategic capital investment decisions are being made every day in an increasingly uncertain world. While the traditional NPV approach does a reasonable job of valuing simple, passively managed projects, it does not capture the many ways in which a highly uncertain project might evolve, and the ways in which active managers will influence this evolution. In cases where managerial flexibility is a major source of strategic value, companies will want to use real options valuation methods.This article serves as a managerial tutorial on this newer, less understood approach. It uses simple examples to illustrate the essence of four basic categories of real options—timing, growth, production, and abandonment. The examples begin by taking a “binomial” approach to option valuation, in which the value of an investment initiative is allowed to take on two possible future values. Besides being used to illustrate the distinctive features of a real option, the binomial approach also serves to help the reader understand the alternative Black‐Scholes valuation approach (though without requiring the reader to master the complex mathematics underlying Black‐Scholes). Basic instructions for implementing both approaches are provided, along with a discussion of how to set appropriate discount rates and the important role of volatility assessment in the valuation process.

Studies the problem of parallel implementation of divide-and-conquer algorithms on binary de Bruijn network using a temporal binomial tree (rather than the usual binary tree) computation structure. Two cases of message volumes are considered: (i) uniform, and (ii) logarithmically decreasing (increasing) weights. A single mapping is proposed for both cases. It has average extra dilation 1 and is communication link contention-free. A lower bound for the total extra dilation of any mapping from uniform-weighted binomial tree to an arbitrary degree-4 network is also developed to show that the mapping is asymptotically optimal with respective to the average extra dilation. The implementation is well suited to a binary de Bruijn network with a wormhole or circuit switching communication scheme. >

Learning Options and Binomial Trees

The option features embedded in many bonds and fixed-income securities have made the binomial interest rate tree approach to bond valuation a valuable model for pricing debt securities. Fundamental to the application of the binomial model to bond valuation is the assumption about the underlying stochastic process. There are two general approaches to modelling stochastic interest rate movements using a binomial model – the equilibrium model and the calibration model. Both models assume that the interest rate's logarithmic return is normally distributed. However, a number of empirical studies have provided evidence that the return distributions of a number of securities exhibit persistent skewness. In modelling interest rate patterns, the existence of skewness in a binomial process impacts not only the values of the up and down parameters, but also the probabilities of the underlying rate increasing or decreasing each period. The purpose of this study is to show how a binomial model of interest rates can be...

Decision support systems that help physicians are becoming a very important part of medical decision making. They are based on different models and the best of them are providing an explanation together with an accurate, reliable, and quick response. One of the most viable among models are decision trees, already successfully used for many medical decision-making purposes. Although effective and reliable, the traditional decision tree construction approach still contains several deficiencies. Therefore we decided to develop and compare several decision support models using four different approaches. We took statistical analysis, a MtDeciT, in our laboratory developed tool for building decision trees with a classical method, the well-known C5.0 tool and a self-adapting evolutionary decision support model that uses evolutionary principles for the induction of decision trees. Several solutions were evolved for the classification of metabolic and respiratory acidosis (MRA). A comparison between developed models and obtained results has shown that our approach can be considered as a good choice for different kinds of real-world medical decision making.

With the wide application of mobile smart devices carried by humans, content dissemination by using Device-to-Device wireless communication technologies in Mobile Opportunistic Networks (MONs) attracts numerous attentions. However, traditional content dissemination approaches have not considered the efficiency of broadcasting interest packets to the providers in MONs with unreliable links. To increase the efficiency of broadcasting interest packets, we propose two data-request-packets broadcasting approaches with the goal of decreasing energy and communication consumption, in which the policy of self-adapting broadcasting time interval is applied. The extensive simulation results show that our proposed approaches greatly prolong the network lifetime, while maintaining higher delivery ratios compared with traditional approach.

Efficient Multicast in All-Port Wormhole-Routed Hypercubes