This PDF document can be downloaded from http://derivatives.garven.com/wp-content/uploads/2024/03/CRR-Explanation.pdf.
Cox-Ross-Rubinstein Binomial Model (One Page Summary)
CRR Explanation
The Trillion Dollar Equation Video
The video “The Trillion Dollar Equation” explores the profound influence of the Black-Scholes/Merton equation on the theory and practice of modern finance. This equation, which originated from physics, transformed the derivatives market into a multi-trillion-dollar industry and revolutionized how risk is assessed and managed. This video, lasting just 31 minutes, offers an exceptionally clear and comprehensive explanation of options, enriched with historical background, unlike any I have encountered before.
Initially, the video narrates the historical context where Sir Isaac Newton, despite his mathematical prowess, suffered financial losses due to the unpredictable nature of the stock market. Contrasting Newton’s experience, it introduces Jim Simons, a mathematician who achieved unparalleled success in the stock market by leveraging mathematical models.
The narrative then shifts to the origins of financial models with Louis Bachelier, who proposed a mathematical solution to pricing options, laying the groundwork for modern financial theory. It explains the concept of options and their benefits. It introduces the concept of random walks in stock prices, likened to particles undergoing Brownian motion—a phenomenon first described by Albert Einstein.
The video discusses how Ed Thorp applied mathematical strategies from blackjack to the stock market, significantly impacting hedge fund strategies and introducing the concept of dynamic hedging. This laid the foundation for the groundbreaking work by Fischer Black, Myron Scholes, and Robert Merton, who developed a formula for pricing options, transforming the financial industry accurately.
The narrative highlights the rapid industry adoption of the Black-Scholes/Merton model, leading to the explosive growth of derivative markets and new financial instruments like credit default swaps and securitized debts. It illustrates how these tools can both mitigate and amplify market risks, impacting global financial stability.
There’s much to look forward to during the upcoming second half of Finance 4366!
Shameless plug…
During class today, I referred to a journal article that I published early in my academic career that Professor Robert C. Merton cites in his Nobel Prize lecture (Merton shared the Nobel Prize in economics in 1997 with Myron Scholes “for a new method to determine the value of derivatives”).
Here’s the citation (and link) to Merton’s lecture:
Merton, Robert C., 1998, Applications of Option-Pricing Theory: Twenty-Five Years Later, The American Economic Review, Vol. 88, No. 3 (Jun. 1998), pp. 323-349.
See page 337, footnote 11 of Merton’s paper for the reference to Neil A. Doherty and James R. Garven (1986)… (Doherty and I “pioneered” the application of a somewhat modified version of the BSM model to insurance pricing; thus, Merton’s reference to our Journal of Finance paper in his Nobel Prize lecture).
Here is the APA-compliant listing for the above-referenced Doherty-Garven:
Doherty, Neil A. and James R. Garven. 1986. “Price Regulation in PropertyLiability Insurance: A Contingent Claims Approach.” Journal of Finance, 41 (5):1031-1050.
WSJ Page 1: The Small University Endowment That Is Beating the Ivy League
From WSJ: “The more than $200 million Paul and Alejandra Foster Pavilion at Baylor University opened in January, with proceeds from the endowment helping to fund its construction.
WSJ (2/14/2024) Page 1 story about Baylor’s 2 billion dollar endowment, and how it is managed…
On the importance of “arbitrage-free” pricing in finance
The notion of “arbitrage-free” pricing is important not only in Finance 4366 but also in your other finance studies. In Finance 4366, we take it as given that investors are risk averse. However, it turns out that we don’t need to invoke the assumption of risk aversion to price risky securities such as options, futures, and other derivatives; all we need to assume is that investors are “greedy” in the sense that they prefer more return to less return, other things equal. Through a variety of trading strategies, we can synthetically replicate any security we want and do so in such a way that we (in theory anyway) can take no risk, incur zero net cost of investment, and yet earn positive returns.
Without question, the notion of arbitrage-free pricing is THE key concept in Finance 4366. However, it is also important in corporate finance. For example, the famous Modigliani-Miller Capital Structure Theorem; i.e., that the value of a firm’s shares is unaffected by how that firm is financed, is based upon this principle. For your personal enjoyment and intellectual edification, I attach a copy of a short (3-page) teaching note that provides a “no-arbitrage” proof based on a simple numerical example:
The Modigliani-Miller Capital Structure Theorem – A “No-Arbitrage” Proof
A Preview of Future Topics in Finance 4366
This week in Finance 4366, we introduced and described the nature of financial derivatives and motivated 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 where they are traded, we also considered various properties of forward and option contracts, since virtually all financial derivatives feature isomorphic payoffs 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 counterparties 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 invests, so-called “real” options to expand or abandon the investment in the future are 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 to 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 its 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.
Kids Explain Futures Trading
For a non-technical introduction to forward and futures contracts, it’s hard to beat the following video tutorial on this topic (apologies if an ad appears on the front-end of this video :-():
Finance: Insights into its meanings, etymology, and more from the Oxford English Dictionary
Kudos to my colleague Dr. Seward for bringing to our attention the many fascinating historical perspectives on the word “Finance” which appear in the Oxford English Dictionary:
Finance in the Oxford English DictionaryTo download this PDF document, click here.
On the relationship between the S&P 500 and VIX
Besides reviewing the course syllabus during the first day of class on Tuesday, January 16, 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 the VIX (the x-axis variable) are strongly negatively correlated with each other. The blue dots are based on 8,574 contemporaneous observations of daily returns for both variables, spanning 34 years starting on January 2, 1990, and ending on January 12, 2024. When we fit a regression line through this scatter diagram, we obtain the following equation:
where 
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.
The 17 equations that changed the course of history (spoiler alert: we use 4 of these equations in Finance 4366!)
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!
From Ian Stewart’s book, these 17 math equations changed the course of human history.