Math and Statistics – Risk Management Blog

August 30, 2023August 24, 2023

Visualizing Taylor polynomial approximations

On pp. 18-23 of the Mathematics Tutorial, I show how y = e^x can be approximated with a Taylor polynomial centered at x=0 for $\delta x$ values ranging from -2 to +2. In his video lesson entitled “Visualizing Taylor polynomial approximations”, Sal Kahn essentially replicates my work; the only difference between Sal’s numerical example and mine is that Sal approximates y = e^x with a Taylor polynomial centered at x=3 instead of x=0. The important insight provided in both cases is that the accuracy of Taylor polynomial approximations increases as the order of the polynomial increases.

August 25, 2023August 24, 2023

On the ancient origin of the word “algorithm”

The August 29th assigned reading entitled “The New Religion of Risk Management” (by Peter Bernstein, March-April 1996 issue of Harvard Business Review) offers a concise overview of the same author’s 1996 book entitled “Against the Gods: The Remarkable Story of Risk“. An intriguing excerpt from page 33 of “Against the Gods” elucidates the historical roots of the term “algorithm.” An intriguing excerpt from page 33 of “Against the Gods” elucidates the historical roots 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.

August 23, 2023August 23, 2023

The 17 equations that changed the course of history (spoiler alert: we use 3 of these equations in Finance 4335!)

Equations (2), (3), and (7) from this list play particularly important roles in Finance 4335!

The 17 equations that changed the course of history

From Ian Stewart’s book, these 17 math equations changed the course of human history.

August 23, 2023August 23, 2023

Rules for calculating (math) derivatives

Here’s a particularly useful list of rules for calculating (math) derivatives:

August 22, 2023August 23, 2023

On the relationship between the S&P 500 and the CBOE Volatility Index (VIX)

Besides going over the course syllabus during the first day of class on Tuesday, August 22, we will also discuss a particularly important “real world” example of financial risk. Specifically, we will study the relationship between realized daily stock market returns (as measured by daily percentage changes in the SP500 stock market index) and changes in forward-looking investor expectations of stock market volatility (as indicated by daily percentage changes in the CBOE Volatility Index (VIX)):
As indicated by this graph (which also appears in the lecture note for the first day of class), daily percentage changes in closing prices for the SP500 (the y-axis variable) and for the VIX (the x-axis variable) are strongly negatively correlated with each other. The blue dots are based on 8,470 contemporaneous observations of daily returns for both variables, spanning the 33-2/3-year period of time starting on January 2, 1990, and ending on August 15, 2023. When we fit a regression line through this scatter diagram, we obtain the following equation:

${R_{SP500}} = .00062 - .1147{R_{VIX}}$ ,

where ${R_{SP500}}$ corresponds to the daily return on the SP500 index and ${R_{VIX}}$ corresponds to the daily return on the VIX index. The slope of this line (-0.1147) indicates that on average, daily closing SP500 returns are inversely related to daily closing VIX returns. Furthermore, nearly half of the variation in the stock market return during this time period (specifically, 48.87%) can be statistically “explained” by changes in volatility, and the correlation between ${R_{SP500}}$ and ${R_{VIX}}$ came out to -0.70. While a correlation of -0.70 does not imply that daily closing values for ${R_{SP500}}$ and ${R_{VIX}}$ always move in opposite directions, it does suggest that this will be the case more often than not. Indeed, closing daily values recorded for ${R_{SP500}}$ and ${R_{VIX}}$ during this period moved inversely 78% of the time.

You can also see how the relationship between the SP500 and VIX evolves prospectively by entering http://finance.yahoo.com/quotes/^GSPC,^VIX into your web browser’s address field.

August 21, 2023August 23, 2023

Calculus and Probability & Statistics recommendations…

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 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 that cover these topics (unless otherwise noted, all sections included in the links which follow are recommended):

Algebra: Intro to the Binomial Theorem, Pascal’s Triangle and Binomial Expansion
Calculus: Taking derivatives, Optimization (profit maximization) with calculus, Visualizing Taylor Series for e^x
Probability and statistics: Basic probability, Compound, independent events, Permutations, Combinations, probability using combinatorics, Random variables and probability distributions, Binomial distribution, Law of Large Numbers, and Normal Distribution.

Finally, if your algebra skills are a bit on the rusty side, I would also recommend checking out the Khan Academy’s review of algebra.