In 1980, Carl Pomerance and J. L. Selfridge proved D. J. Newman’s coprime mapping conjecture: If n is a positive integer and I is a set of n consecutive integers, then there is a bijection f :{1, 2, . . . , n}→ I such that gcd(i, f(i)) = 1 for 1 ≤ i ≤ n. The function f described in their theorem is called a coprime mapping. Around the same time, Roger Entringer conjectured that all trees are pr...

Integrating large quantities of supply‐driven renewable electricity generation remains a political and operational challenge. One of the main obstacles in Europe to installing at least 200 GWs of power from variable renewable sources is how to deal with the insufficient network capacity and the congestion that will result from new flow patterns. We model the current methodology for contr...

Causal independence modelling is a well-known method both for reducing the size of probability tables and for explaining the underlying mechanisms in Bayesian networks. Many Bayesian network models incorporate causal independence assumptions; however, only the noisy OR and noisy AND, two examples of causal independence models, are used in practice. Their underlying assumption that either at lea...

A binomial lattice based framework for the analysis of finite investment options with finite operational phase is developed. Solutions for European and American type finite horizon investment options with optimal capital structure and a multi-stage investment setting with multiple debt issues are discussed. The analysis shows that optimal leverage ratios are not affected by option moneyness at ...

Locational marginal pricing (LMP) is a widely employed method for pricing electricity in the wholesale electricity market. Although it is well known that the LMP mechanism is vulnerable to market manipulation, there is little literature providing a systematic analysis of this phenomenon. In the first part of this paper, we investigate the economic dispatch outcomes of the LMP mechanism with str...

In this paper, we present a non-commutative version of some portions of finance theory, including theory of arbitrage, asset princing, and optional decomposition in financial markets based on finite dimensional quantum probability spaces. The binomial model (or, the CRR-model) is studied in the non-commutative setting, and in particular, we prove that a single-step model in non-commutative sett...

In this paper, we discuss statistical families P with the property that if the distribution of a random variable X is in P, then so is the distribution of ZwBi(X, p) for 0%p%1. (Here we take ZwBi(X, p) to mean that given XZx, Z is a draw from the binomial distribution Bi(x, p).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probabilit...

Here we model the price of a stock in discrete time by a Markov chain of the recursive form Sn+1 = SnYn+1, n ≥ 0, where the {Yi} are iid with distribution P (Y = u) = p, P (Y = d) = 1 − p. Here 0 < d < 1 + r < u are constants with r the risk-free interest rate ((1 + r)x is the payoff you would receive one unit of time later if you bought $x worth of the risk-free asset (a bond for example, or p...

[points] 1. Consider the Cheuk-Vorst algorithm for pricing floating strike lookback put options. The truncated binomial tree for the process is constructed for Y t = S max t S t , Y t ≥ 1. Let V n j denote the numerical approximation to V t = V t /S t at the (n, j) th node of the binomial tree for Y t , where Y t = u j , j ≥ 0. (i) Explain how to derive the binomial formula V n j = e −r∆t [p V ...

This paper explores the utility of a discrete singular convolution (DSC) algorithm for solving the Black–Scholes equation. Both European and American style options, which include all nontrivial plain option pricing problems, are considered to test the accuracy and to examine the efficiency of the present algorithm. Adaptive meshes are constructed to enhance the performance of the DSC algorithm....