Financial Mathematics Text

Wednesday, January 14, 2015

Financial Mathematics: Statistics - Properties of Distributions

Probability distributions have a number of properties which help us summarize and characterize them. We'll look at some of these properties, how they are calculated and what they are used for.


Contents:


I) Mean
II) Median
III) Variance
IV) Skewness
V) Kurtosis

Thursday, January 1, 2015

The Obsolescence of Time

Every year we perform a collection of rituals which include, but are not limited to:
  • Staying up late.
  • Drinking, lots of drinking.1
  • Watching a big colorful ball drop. 
  • Writing "resolutions", a list of things we'll fail to do in the upcoming year (or week.)
We dub these rituals "New Year's Celebrations" for the New Year's holiday.

Wednesday, December 17, 2014

Some Ambiguities Regarding the Cost of Capital

So I'd like to suggest there's an ambiguity in measuring the cost of capital. As far as I can tell, there isn't always a straight forward consensus on how this should be handled. There are a number of arguments in favor of particular positions.

Today I'm going to very briefly present the issue.

Friday, November 21, 2014

ROE, ROA and Leverage (Update)

Sometimes I don't see alternative ways of writing expressions until after the fact. So this will just be a brief modification of a previous post: Relating ROE with ROA and Leverage.

Monday, November 17, 2014

Option Returns - Empirical Results

Previously, I looked at what we'd expect call and put options would be if we assumed that stock returns follow a normal distribution (see the Expected Return of a Call Option and Put Option).

My findings indicated that the underlying assumptions of the Black-Scholes pricing model are inconsistent with the mean-variance view of risk. This was not an empirical result, mind you. Empirically, I've yet to find a single set of financial data that was normally distributed. It was a theoretical result; the theory is inconsistent with a mean-variance view of risk and return.

Today I'll be looking at some odd empirical results. I wanted to see what actual returns actually looked like. As it turns out, they're even worse than what the theory predicts.

Friday, November 14, 2014

On Counting (Exploring Operational Definitions Part III)

For the first two parts in this series see:

Exploring Operational Definitions: Part I
Exploring Operational Definitions: Part II - Distance

Perhaps the "simplest" procedure that most folks have learned is the technique(s) of counting. What I would like to explore is that there are a variety of techniques that we call counting. In some cases they build on one another. In other cases, they are techniques which give "approximate" solutions.

Of course not all societies count things (see here). Nonetheless, I suspect that many of our "intuitions" about mathematics ultimately stem from our earlier experience with counting. Our attachment to such intuitions will somewhat determine how willing we are able to accept alternative definitions and techniques for counting. Today I'll explore a few of these definitions.

Monday, November 10, 2014

Financial Mathematics: Statistics - Moments

In statistics, there are a variety of calculations referred to as moments. We'll be discussing three types of moments: Raw Moments, Central Moments and Standardized Moments.

Monday, November 3, 2014

Some Comments on the Quantity Theory of Money

As I mentioned in my previous post - Inflation - Why the official numbers are wrong! - I pointed out that the general theory for which inflation is based upon implies that the "price level" is a vector but measures of inflation represent this as a scalar. Today I want to explore how that complicates the picture for the Quantity Theory of Money.