1. ### A Robust Derivation of the Black-Scholes Partial Differential Equation System without the Self-Financing Hypothesis

Jeong-Hoon Kim1, Graeme Wake2

1. Department of Mathematics, Yonsei University, Seoul, Republic of Korea.
2. Institute of Natural and Mathematical Sciences, Massey University, Auckland, New Zealand.

Abstract: Delta hedging is the core of the derivation of the well-known Black-Scholes formula for the price of European options. When Ito calculus is used faithfully without the self-financing hypothesis, the dependence of the delta on both time and the underlying asset price variables is induced and subsequently a consistent version of the Black-Scholes partial differential equation system is derived for the option price and the delta. This paper proposes the system as a possible starting point of a more robust study of mathematical option pricing.

Keywords: Black-Scholes partial differential equation, Ito calculus, Self-financing, No arbitrage, Option pricing.

2. ### Identifying Hard-to-Survey Provinces in Burkina Faso

Baguinébié Bazongo

Department of statistical coordination and research, Institut national de la statistique et de la démographie, Ouagadougou, Burkina Faso

Abstract:

Probabilistic sampling is theoretical the best method to select representative sample from target population. However, this result does not always hold in practice because some selected sampling units are missed during survey. The aim of this paper is to categorize the 45 provinces of Burkina Faso in terms of household surveys complexity in order to adapt specific methodology to each group of similar provinces.

We applied hierarchical clustering based on principal components analysis on provinces data and found five clusters. According to these results, two provinces (Kouritenga and Loroum) were the most hard-to-survey. Therefore, adapted methodology should be applied to each group of provinces during survey implementation to maximize data quality.

Keywords: hard-to-survey, hierarchical clustering, FactoMineR.

3. ### Direct Estimation of Confidence Intervals for Proportion by Means of Continuity Simulation of Binomial Distribution

Niuman M Comas

Departamento de Matemática, Universidad de Holguín, Ave XX Aniversario, 82100, Holguín, Cuba.

Abstract:

This paperexplains a methodology to obtain confidence intervals for proportion without any normal approximation. It is based on the use ofbinomial distribution in order to calculate the confidence bounds using polynomial interpolation techniques to estimate “artificial” probabilities associated to fractional distribution arguments.

The methodology includes the estimation of cumulative probabilities with continuity simulation and the respective inverse root. Hypothesis test is also incorporated. Abrief review of other common methods is previously shown.

The implementation of the “Direct” method wascarried outunder open source by means of JavaScript programming and it is available in the on-line statistical laboratory DynStats.com.

The results were compared with other procedures like the Wald, Clopper-Pearson and Mid-P methods. Most similarities were found for the Wald and Mid-p methods according to the proportion’s region.

Keywords: Confidence interval, binomial proportion, binomial distribution, polynomial interpolation.