The Truncated Multivariate Normal Distribution in Finance and Econometrics
Published on 24. October 2014
Book
XVI, 94 pages
978-3-86624-612-6 (ISBN)
Description
This dissertation presents three papers and an introduction chapter on the truncated multivariate normal distribution, which has been applied to recent problems in derivatives pricing and spatial econometrics.
This collection of articles seeks to provide a survey on the wide range of applications for this distribution; it is accompanied by powerful implementations in R, one of today's leading statistical software packages. It introduces a new valuation methodology for autocallables and analyses empirical pricing and investment performance, thereby generating many new insights into the functioning of these complex derivatives. Another major contribution to spatial econometrics is the improved implementation of the Bayesian estimation of spatial probit models according to LeSage and Pace (2009) in R, allowing for the estimation of large spatial models.
This work should be of interest to scholars working with the truncated multivariate normal distribution, particularly those interested in sampling issues and sparse matrices. It should also be of interest to practitioners of many disciplines, wishing to apply the proposed algorithms in R. Chapters 1 and 3 are original, unpublished, independent work by the author, Stefan Wilhelm. Chapters 2 and 4 are collaborative articles with Manjunath B. G. from the University of Siegen and Miguel Godinho de Matos from Carnegie Mellon University, Pittsburgh, respectively.
Both chapters have been published in The R Journal. The corresponding R packages tmvtnorm and spatialprobit have been submitted to the "Comprehensive R Archive Network" (CRAN). I started my work on truncation in 2008 when I was first challenged with the valuation of autocallables, which are also known as express certificates in Germany. I modeled the dependence structure of asset prices, their constraints, and autocallables payoffs at different points in time in the framework of the truncated multivariate normal distribution. This work forms the basis for Chapter 3. At that time, my favorite statistical software R did not provide any packages that covered this statistical distribution, so I was left to implement some methods for sampling or marginal distributions myself. Because I found that it might be of general interest for other R users, I decided to publish these methods as an R package. Shortly after I submitted a first version of the R package tmvtnorm in 2009, I received some feedback and questions regarding the sampling from Manjunath B. G., a PhD student at the University of Siegen. We agreed to collaborate on Gibbs sampling and moments computation in order to extend the package. After doing so, we wrote a manuscript that introduces the package to all R users. Our manuscript was published in The R Journal in 2010.
In late 2011, I was contacted by Miguel Godinho de Matos, a PhD student in social engineering at Carnegie Mellon University, who wanted to use the tmvtnorm package in a Bayesian estimation of spatial probit models in social networks. He contacted me because his implementation of this iterative procedure was extremely slow and could not estimate models for the larger sample sizes that are found in today's data sets. With my previous knowledge, I quickly realized that his truncated normal Gibbs sampling step as the innermost loop inside the outer Bayesian iteration was responsible for the poor performance. We optimized the estimation procedure with the help of the precision matrix approach and the use of sparse matrices by magnitudes. I proposed publishing all functions as a new R package spatialprobit and writing an article on how to estimate spatial probit models in R. This was eventually published in The R Journal in 2013. In both joint works I was the project lead, responsible for all major areas of concept formation, code implementation, and R package maintenance, as well as manuscript composition and journal correspondence. Manjunath B. G. was involved throughout the project in concept formation and manuscript composition. Miguel Godinho de Matos was involved throughout the project in concept formation and manuscript edits. He contributed a first R implementation of the LeSage and Pace (2009) spatial probit estimation algorithm, which I rebuilt and revised in many respects, especially in order to improve its performance and make it compatible with common R standards.
This collection of articles seeks to provide a survey on the wide range of applications for this distribution; it is accompanied by powerful implementations in R, one of today's leading statistical software packages. It introduces a new valuation methodology for autocallables and analyses empirical pricing and investment performance, thereby generating many new insights into the functioning of these complex derivatives. Another major contribution to spatial econometrics is the improved implementation of the Bayesian estimation of spatial probit models according to LeSage and Pace (2009) in R, allowing for the estimation of large spatial models.
This work should be of interest to scholars working with the truncated multivariate normal distribution, particularly those interested in sampling issues and sparse matrices. It should also be of interest to practitioners of many disciplines, wishing to apply the proposed algorithms in R. Chapters 1 and 3 are original, unpublished, independent work by the author, Stefan Wilhelm. Chapters 2 and 4 are collaborative articles with Manjunath B. G. from the University of Siegen and Miguel Godinho de Matos from Carnegie Mellon University, Pittsburgh, respectively.
Both chapters have been published in The R Journal. The corresponding R packages tmvtnorm and spatialprobit have been submitted to the "Comprehensive R Archive Network" (CRAN). I started my work on truncation in 2008 when I was first challenged with the valuation of autocallables, which are also known as express certificates in Germany. I modeled the dependence structure of asset prices, their constraints, and autocallables payoffs at different points in time in the framework of the truncated multivariate normal distribution. This work forms the basis for Chapter 3. At that time, my favorite statistical software R did not provide any packages that covered this statistical distribution, so I was left to implement some methods for sampling or marginal distributions myself. Because I found that it might be of general interest for other R users, I decided to publish these methods as an R package. Shortly after I submitted a first version of the R package tmvtnorm in 2009, I received some feedback and questions regarding the sampling from Manjunath B. G., a PhD student at the University of Siegen. We agreed to collaborate on Gibbs sampling and moments computation in order to extend the package. After doing so, we wrote a manuscript that introduces the package to all R users. Our manuscript was published in The R Journal in 2010.
In late 2011, I was contacted by Miguel Godinho de Matos, a PhD student in social engineering at Carnegie Mellon University, who wanted to use the tmvtnorm package in a Bayesian estimation of spatial probit models in social networks. He contacted me because his implementation of this iterative procedure was extremely slow and could not estimate models for the larger sample sizes that are found in today's data sets. With my previous knowledge, I quickly realized that his truncated normal Gibbs sampling step as the innermost loop inside the outer Bayesian iteration was responsible for the poor performance. We optimized the estimation procedure with the help of the precision matrix approach and the use of sparse matrices by magnitudes. I proposed publishing all functions as a new R package spatialprobit and writing an article on how to estimate spatial probit models in R. This was eventually published in The R Journal in 2013. In both joint works I was the project lead, responsible for all major areas of concept formation, code implementation, and R package maintenance, as well as manuscript composition and journal correspondence. Manjunath B. G. was involved throughout the project in concept formation and manuscript composition. Miguel Godinho de Matos was involved throughout the project in concept formation and manuscript edits. He contributed a first R implementation of the LeSage and Pace (2009) spatial probit estimation algorithm, which I rebuilt and revised in many respects, especially in order to improve its performance and make it compatible with common R standards.