During today’s class meeting, we will discuss (among other things) the famous St. Petersburg Paradox. The source for this is Daniel Bernoulli’s famous article entitled “Exposition of a New Theory on the Measurement of Risk“. As was the standard practice in academia at the time, Bernoulli’s article was originally published in Latin in 1738. It was subsequently translated into English in 1954 and published a second time that same year in Econometrica (Volume 22, No. 1): pp. 22–36. Considering that this article was published 282 years ago in an obscure (presumably peer-reviewed) academic journal, it is fairly succinct and surprisingly easy to read.
Also, the Wikipedia article about Bernoulli’s article is worth reading. It provides the mathematics for determining the price at which the apostle Paul would have been indifferent about taking the apostle Peter up on this bet. The original numerical example proposed by Bernoulli focuses attention on Paul’s gamble per se and does not explicitly consider the effect of Paul’s initial wealth on his willingness to pay. However, the quote on page 31 of the article (“… that any reasonable man would sell his chance … for twenty ducats”) implies that Bernoulli may have assumed Paul to be a millionaire, since (as shown in the Wikipedia article) the certainty-equivalent value of this bet to a millionaire who has logarithmic utility comes out to 20.88 ducats.
During class last week, I mentioned that the September 1 assigned reading entitled “The New Religion of Risk Management” (by Peter Bernstein, March-April 1996 issue of Harvard Business Review) is a synopsis of the same author’s 1996 book entitled “Against the Gods: The Remarkable Story of Risk”. Here’s a fascinating quote from page 33 which explains the ancient origin of the word “algorithm”:
“The earliest known work in Arabic arithmetic was written by alKhowarizmi, a mathematician who lived around 825, some four hundred years before Fibonacci. Although few beneficiaries of his work are likely to have heard of him, most of us know of him indirectly. Try saying “alKhowarizmi” fast. That’s where we get the word “algorithm,” which means rules for computing.”
Note: The book cover shown above is a copy of a 1633 oil-on-canvas painting by the Dutch Golden Age painter Rembrandt van Rijn.
In his video lesson entitled “Visualizing Taylor polynomial approximations”, Sal Kahn replicates the tail end of yesterday’s Finance 4335 class meeting (beginning at the 1:06:13 mark in the Section 1 Zoom recording) in which we approximate y = ex with a Taylor polynomial centered at x=0. Sal approximates y = ex with a Taylor polynomial centered at x=3 instead of x=0, but the same insight obtains in both cases, which is that the accuracy of Taylor polynomial approximations increases as the order of the polynomial increases (as also shown in pp. 18-23 of the Mathematics Tutorial lecture note).
There is a section in the assigned “Optimization” reading due Thursday, August 27 on pp. 74-76 entitled “Lagrangian Multipliers” which (as noted in footnote 9 of that reading) may be skipped without loss of continuity. The primary purpose of this chapter is to re-acquaint students with basic calculus and how to use calculus to solve so-called optimization problems. Since the course only requires solving unconstrained optimization problems, there’s no need for Lagrangian multipliers.
Since many of the topics covered in Finance 4335 require a basic knowledge and comfort level with algebra, differential calculus, and probability & statistics, the second class meeting during the Spring 2020 semester will include a mathematics tutorial, and the third and fourth class meetings will cover probability & statistics. I know of no better online resource for brushing up on (or learning for the first time) these topics than the Khan Academy.
So here are my suggestions for Khan Academy videos which cover these topics (unless otherwise noted, all sections included in the links which follow are recommended):