Ricardo J. Rodriguez

First Significant Digit Patterns From Mixtures of Uniform Distributions

Traditional tests searching for human influence in data assume that, barring such influence, first significant digits (FSD) are uniformly distributed. More recent tests rely on Benfords law, postulating that lower digits are more likely than higher ones. I show that both patterns belong to a family arising from mixtures of uniforms, and characterize the FSD patterns for a one-parameter subset of the family. I also show that all family members exhibit decreasing FSD probabilities. The empirical analysis suggests that although the uniform FSD pattern and Benfords law are reasonable models for some data, alternative family members better fit other data.

Reducing False Alarms in the Detection of Human Influence on Data

Benfords law states that the frequency of first significant digits (FSD) in a random sample decreases as those digits increase. While this curious law is increasingly used to test for human influence on data, including corporate fraud and psychological barriers in financial markets, it often produces frustratingly many false positives. I advance toward the goal of understanding and reducing the false alarm rate by showing that Benfords law is inadequate when data are drawn from various common distributions, including the ubiquitous normal distribution. I also prove that Benfords law is obeyed when the untainted data are lognormally distributed with a high variance parameter. In addition, I explain why data sets expressible as the product of two other sets often conform better to Benfords law than either multiplicand data set. The empirical analysis strongly supports these findings.

Option Pricing Bounds: Synthesis and Extension

The main option pricing bounds in the literature were originally obtained through various disparate methods. I show that those bounds can be derived from a single analytical framework. The key to this synthesis lies in the use of a general expression for the price of a call option depending on the corresponding put options discount factor. Although the puts discount factor is unknown, it can be bounded from below. I use this lower bound on the puts discount factor to derive traditional lower bounds for call prices. In addition, I extend the literature by finding a new tighter lower bound.

Lognormal option pricing for arbitrary underlying assets: a synthesis

All European option pricing formulas sharing the assumption of a lognormally distributed terminal price for the underlying asset are formally similar. It is thus natural to seek a single explicit general formula for this class of options. This paper provides such a synthesis. The key insight is recognizing that all option pricing equations depend explicitly on the expected terminal price of the arbitrary underlying asset, which is often obtained through basic financial reasoning. To illustrate the power and pedagogical value of this framework, I obtain several classical option pricing formulas as special cases of the general equation.

Finite sequences of St. Petersburg games : inferences from a simulation study

The St. Petersburg game consists of performing Bernoulli trials until a success occurs, with each additional trial doubling the gambler’s payoff. Despite the centuries-long history of this game, its analysis remains surprisingly incomplete. For example, the statistical properties of the average payoff distribution and the distribution of the maximum number of trials needed for success (the maximum, for short) over a finite sequence of games are unknown. Even more surprising is the absence of an extensive simulation study of finite St. Petersburg sequences from the literature. Such a study may be indispensable in the case of the average payoff because a theoretical analysis of this random variable may not be feasible. This article relies on extensive simulations to infer several new properties of finite St. Petersburg sequences. For instance, a statistical analysis of the simulations reveals that for large sequences, the suitably shifted average payoff distribution is stationary. The simulations also reveal that the ratio of the mode of the average payoff to the mode of the maximum is increasingly likely to be near 1 as the number of games increases. These inferred properties jointly imply that for large sequences, the fair entrance fee per game equals the mode of the maximum. A general formula for this mode is derived.

The After-Tax Yield to Maturity of a Premium Bond: Bias from a Simple Approximation

When finding the after-tax yield to maturity of a bond, it is customary to use the approximate relationship: after-tax yield = (1 - tax rate) × (before-tax yield). This paper provides a theoretical justification for this simplified formula by showing that, for premium bonds, it can be obtained from the complex implicit function defining the exact after-tax yield by using a first-order Taylor series approximation. Such simple approximations normally result in potentially large errors, and this is indeed what our simulation results show. Nevertheless, the approximation formulas error possesses the desirable properties that it is consistent in sign and bounded. Hence, the approximate formula, although easy to apply, should be used with caution.