I especially like the fact that Ian Stewart includes the famous Black-Scholes equation (equation #17) on his list of the 17 equations that changed the course of history; Equations (2), (3), (7), and (17) play particularly important roles in Finance 4366!
During this past week’s statistics tutorials, we discussed (among other things) the concept of statistical independence, and focused attention on some important implications of statistical independence for probability distributions such as the binomial and normal distributions.
Here, I’d like to call everyone’s attention to an interesting (non-finance) probability problem related to statistical independence. Specifically, consider the so-called “Birthday Paradox”. The Birthday Paradox pertains to the probability that in a set of randomly chosen people, some pair of them will have the same birthday. Counter-intuitively, in a group of 23 randomly chosen people, there is slightly more than a 50% probability that some pair of them will both have been born on the same day.
To compute the probability that two people in a group of n people have the same birthday, we disregard variations in the distribution, such as leap years, twins, seasonal or weekday variations, and assume that the 365 possible birthdays are equally likely. Thus, we assume that birth dates are statistically independent events. Consequently, the probability of two randomly chosen people not sharing the same birthday is 364/365. According to the combinatorial equation, the number of unique pairs in a group of n people is n!/2!(n-2)! = n(n-1)/2. Assuming a uniform distribution (i.e., that all dates are equally probable), this means that the probability that no pair in a group of n people shares the same birthday is equal to p(n) = (364/365)^[n(n-1)/2]. The event of at least two of the n persons having the same birthday is complementary to all n birthdays being different. Therefore, its probability is p’(n) = 1 – (364/365)^[n(n-1)/2].
Given these assumptions, suppose that we are interested in determining how many randomly chosen people are needed in order for there to be a 50% probability that at least two persons share the same birthday. In other words, we are interested in finding the value of n which causes p(n) to equal 0.50. Therefore, 0.50 = (364/365)^[n(n-1)/2]; taking natural logs of both sides and rearranging, we obtain (ln 0.50)/(ln 364/365) = n(n-1)/2. Solving for n, we obtain 505.304 = n(n -1); therefore, n is approximately equal to 23.
The following graph illustrates how the probability that a pair of people share the same birthday varies as the number of people in the sample increases: It is worthwhile noting that real-life birthday distributions are not uniform since not all dates are equally likely. For example, in the northern hemisphere, many children are born in the summer, especially during the months of August and September. In the United States, many children are conceived around the holidays of Christmas and New Year’s Day. Also, because hospitals rarely schedule C-sections and induced labor on the weekend, more Americans are born on Mondays and Tuesdays than on weekends; where many of the people share a birth year (e.g., a class in a school), this creates a tendency toward particular dates. Both of these factors tend to increase the chance of identical birth dates since a denser subset has more possible pairs (in the extreme case when everyone was born on three days of the week, there would obviously be many identical birthdays!).
Note that since 26 students are enrolled in Finance 4366 this semester, this implies that the probability that two Spring 2021 Finance 4366 students share the same birthday is p’(26) = 1 – (364/365)^[26(5)/2] =59%, although given footnote 1’s caveats, it’s likely that there may be one or more shared birthday pairs.
Probability and statistics, along with the basic calculus principles covered last Thursday, are foundational for the theory of pricing and managing risk with financial derivatives, which is what this course is all about. During yesterday’s class meeting, we introduced discrete and continuous probability distributions, calculated parameters such as expected value, variance, standard deviation, covariance, and correlation, and applied these concepts to measure expected returns and risks for portfolios comprising risky assets. During tomorrow’s class meeting, we will take a deeper dive into discrete and continuous probability distributions, in which the binomial and normal distributions will be showcased.
On Tuesday, February 2, we will introduce and describe the nature of financial derivatives, and motivate their study with examples of forwards, futures, and options. Derivatives are so named because they derive their values from one or more underlying assets. Underlying assets typically involve traded financial assets such as stocks, bonds, currencies, or other derivatives, but derivatives can derive value from pretty much anything. For example, the Chicago Mercantile Exchange (CME) offers exchange-traded weather futures and options contracts (see “Market Futures: Introduction To Weather Derivatives“). There are also so-called “prediction” markets in which derivatives based upon the outcome of political events are actively traded (see “Prediction Market“).
Besides introducing financial derivatives and discussing various institutional aspects of markets in which they are traded, we’ll consider various properties of forward and option contracts, since virtually all financial derivatives feature payoffs that are isomorphic to either or both schemes. For example, a futures contract is simply an exchange-traded version of a forward contract. Similarly, since swaps involve exchanges between counter-parties of payment streams over time, these instruments essentially represent a series of forward contracts. In the option space, besides traded stock options, many corporate securities feature “embedded” options; e.g., a convertible bond represents a combination of a non-convertible bond plus a call option on company stock. Similarly, when a company makes an investment, so-called “real” options to expand or abandon the investment at some future is often present.
Perhaps the most important (pre-Midterm 1) idea that we’ll introduce is the concept of a so-called “arbitrage-free” price for a financial derivative. While details will follow, the basic idea is that one can replicate the payoffs on a forward or option by forming a portfolio comprising the underlying asset and a riskless bond. This portfolio is called the “replicating” portfolio, since, by design, it replicates the payoffs on the forward or option. Since the forward or option and it’s replicating portfolio produce the same payoffs, then they must also have the same value. However, suppose the replicating portfolio (forward or option) is more expensive than the forward or option (replicating portfolio). If this occurs, then one can earn a riskless arbitrage profit by simply selling the replicating portfolio (forward or option) and buying the forward or option (replicating portfolio). However, competition will ensure that opportunities for riskless arbitrage profits vanish quickly. Thus the forward or option will be priced such that one cannot earn arbitrage profit from playing this game.
“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.”
This blog entry provides a helpful follow-up for a couple of calculus-related topics that we covered during today’s Mathematics Tutorial.
- See page 12 of the above-referenced lecture note. There, the equation for a parabola () appears, and the claim that is corroborated by solving the following expression:
In the 11-minute Khan Academy video at https://youtu.be/HEH_oKNLgUU, Sal Kahn takes on the solution of this problem in a very succinct and easy-to-comprehend fashion.
- In his video lesson entitled “Visualizing Taylor polynomial approximations”, Sal Kahn replicates the tail end of today’s Finance 4366 class meeting in which we approximated y = ex with a Taylor polynomial centered at x=0 (as also shown in pp. 18-23 of the Mathematics Tutorial lecture note). 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.
Since many of the topics covered in Finance 4366 require a basic knowledge and comfort level with algebra, differential calculus, and probability & statistics, the second class meeting during the Spring 2021 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 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 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 Introduction to the Normal Distribution.
Finally, if your algebra skills are generally a bit on the rusty side, I would also recommend checking out the Khan Academy’s review of algebra.