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 :-():

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:

{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 period (specifically, 48.87%) can be statistically “explained” by changes in volatility, and the correlation between {R_{SP500}} and {R_{VIX}} during this period is -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^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.